Generated by All in One SEO v4.9.7.2, this is an llms.txt file, used by LLMs to index the site. # hyperfinecourse Course on hyperfine ## Sitemaps - [XML Sitemap](https://www.hyperfinecourse.org/sitemap.xml): Contains all public & indexable URLs for this website. ## Pages - [Home](https://www.hyperfinecourse.org/) - hyperfinecourse In condensed-matter physics, materials chemistry and biology, there is a family of experimental methods that relies on particular properties of the nucleus in order to extract information from atoms, molecules or solids. If you want to apply such methods yourself, or if you want to understand better research papers in your favourite field that - [Registration](https://www.hyperfinecourse.org/registration/) - [user_registration_form id="4837"] - [My Account](https://www.hyperfinecourse.org/my-account-2/) - [user_registration_my_account] - [My account](https://www.hyperfinecourse.org/my-account/) - Username or Email Address Password Remember Me Log In Lost your password? | Register Proceed to course! - [All Courses](https://www.hyperfinecourse.org/all-courses/) - Hyperfine course module 1 5 Lessons 4 Videos 2:00h Duration The Nucleus If you take this course in sync with the spring edition Ghent University, then please observe the due date for the activities of this week. We cannot talk about hyperfine interactions without discussing some properties of nuclei that go beyond the point charge. 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Monthly $49 /Month Course Learning Checks Course ## Forums - [toy model](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/electric-monopole-shift/top-model/) - [a nucleus with a general shape ?](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/a-nucleus-with-a-general-shape/) - [double ring](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/framework/double-ring/) - [expression and physics](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/electric-monopole-shift/expression-and-physics/) - [multipole moments](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/the-nucleus/multipole-moments/) - [nuclear properties](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/the-nucleus/nuclear-properties/) - [The nucleus](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/the-nucleus/) - [blablabla train use of 'post first' forum](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/blablabla-train-use-of-post-first-forum/) - [blablabla train use of normal forum](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/try-out-normal-forum/) - [The g-factor](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/refreshers/the-g-factor/) - [amplitude, probability, intensity](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/refreshers/amplitude-probability-intensity/) - [Refreshers](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/refreshers/) - [perturbed angular correlation spectroscopy](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/pac/perturbed-angular-correlation-spectroscopy/) - [PAC](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/pac/) - [NMR/ON](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/ltno-nmr-on/nmr-on/) - [LTNO & NMR/ON](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/ltno-nmr-on/) - [EPR on molecules and crystals (2)](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/erp/epr-on-molecules-and-crystals-2/) - [EPR on molecules and crystals (1)](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/erp/epr-on-molecules-and-crystals-1/) - [EPR on free atoms (2)](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/erp/epr-on-free-atoms-2/) - [EPR on free atoms (1)](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/erp/epr-on-free-atoms-1/) - [ERP](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/erp/) - [orientation: temperature and radiation](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/nmr-nqr/orientation-temperature-and-radiation/) - [NMR & NQR](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/nmr-nqr/) - [(optional) synchrotron Mössbauer spectroscopy at extreme conditions](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/synchrotron-methods/optional-synchrotron-mossbauer-spectroscopy-at-extreme-conditions/) - [nuclear resonant scattering](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/synchrotron-methods/nuclear-resonant-scattering/) - [synchrotron methods](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/synchrotron-methods/) - [realizing nuclear resonant scattering](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/mossbauer-spectroscopy/realizing-nuclear-resonant-scattering/) - [Mössbauer spectroscopy](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/mossbauer-spectroscopy/mossbauer-spectroscopy/) - [Mössbauer spectroscopy](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/mossbauer-spectroscopy/) - [laser spectroscopy](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/laser-spectroscopy/) - [miscellaneous topics](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/electric-quadrupole-interaction/miscellaneous-topics/) - [case studies / symmetry - task 1](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/electric-quadrupole-interaction/case-studies-symmetry-task-1/) - [case studies / symmetry - task 1](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/electric-quadrupole-interaction/case-studies-symmetry/) - [quadrupole operator](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/electric-quadrupole-interaction/quadrupole-operator/) - [from toy model to quantum](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/electric-quadrupole-interaction/from-toy-model-to-quantum/) - [electric quadrupole interaction](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/electric-quadrupole-interaction/) - [overlap contribution](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/magnetic-hyperfine-interaction/overlap-contribution/) - [in solids - task 2](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/magnetic-hyperfine-interaction/in-solids-task-2/) - [in solids - task 1](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/magnetic-hyperfine-interaction/in-solids/) - [in free atoms - task 1](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/magnetic-hyperfine-interaction/in-free-atoms-task-1/) - [in free atoms - task 2](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/magnetic-hyperfine-interaction/in-free-atoms/) - [reading task magnetic hyperfine interaction](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/magnetic-hyperfine-interaction/reading-task-magnetic-hyperfine-interaction/) - [magnetic hyperfine interaction](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/magnetic-hyperfine-interaction/) - [electric monopole shift](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/electric-monopole-shift/) - [framework](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/framework/) - [Exit (week 1)](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/the-nucleus/exit-week-1/) - [Quick start](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/quick-start/) - [multipole radiation](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/the-nucleus/multipole-radiation/) - [why are odd electric moments zero ?](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/the-nucleus/why-are-odd-electric-moments-zero/) - [nuclear moment tabulation](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/the-nucleus/nuclear-moment-tabulation/) - [About you](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/quick-start/about-you/) - [Hyperfine course](https://www.hyperfinecourse.org/forums/forum/hyperfine-course/) ## Topics - [EPR experiment](https://www.hyperfinecourse.org/forums/topic/epr-experiment-3/) - You have a sample with an anisotropic g-factor, and you want to measure the 3 principal components of the g-factor. So you put this sample in an EPR setup, characterised by the 2 electromagnets. Then, you can measure the g-factor in 2 ways: You use a fixed magnetic field, and scan the EM spectrum till - [Short-short EPR](https://www.hyperfinecourse.org/forums/topic/short-short-epr/) - On page 11, the technique of field modulation is explained: "Because the spectrometer is tuned to only detect signals that change amplitude as the field changes, the resultant signal appears as a first derivative." I don't understand why the spectrometer is tuned in this way. They say the magnetic field is made to change by - [EPR introduction document](https://www.hyperfinecourse.org/forums/topic/epr-introduction-document/) - In paragraph 1.5 "Line Schape", there is stated: It has to do with one of the selection rules in EPR, namely that only the magnetic moments from the sample in the direction of the external field (to be more precise: perpendicular to the direction of the magnetic field created by the microwaves) are detected. I - [EPR operation](https://www.hyperfinecourse.org/forums/topic/epr-operation/) - First, one places a single-crystal sample on a rotating stage within the spectrometer's microwave cavity. Then, one fixes the microwave frequency and sweeps the external magnetic field to determine the resonance positions, where microwave absorption occurs. Next, we calculate the effective g-value for each observed peak using the resonance condition. Since the defects are anisotropic, - [EPR on free atoms](https://www.hyperfinecourse.org/forums/topic/epr-on-free-atoms-2/) - First task: When a free La atom in an externally applied magnetic field absorbs a microwave photon, it undergoes a transition between quantized energy levels. This will cause the orientation of the magnetic moment of the electron cloud to change relative to the external field. Second task: Without hyperfine interaction, the energy of a magnetic - [EPR short-short introduction](https://www.hyperfinecourse.org/forums/topic/epr-short-short-introduction-2/) - On page 8 they say EPR has the selection rule only the magnetic moments from the sample in the direction of the external field (to be more precise: perpendicular to the direction of the magnetic field created by the microwaves) are detected. I understand that you need a perpendicular magnetic field to create a perturbation. - [EPR spectrum](https://www.hyperfinecourse.org/forums/topic/epr-spectrum/) - You place your one-crystal in the cavity (one-crystal because for powder you don't need to rotate as all orientations are present at once) and apply an external field. Simultaneously microwaves with a constant frequency are reflecting on the cavity. You scan over the external magnetic field to find the peaks. After this scan is finished - [Second answer](https://www.hyperfinecourse.org/forums/topic/second-answer-2/) - With no hyperfine interaction: There is only a shift because of the nuclear spin so not important for transitions. So that means we use the Zeemann splitting of the electron cloud which are equidistant levels with a distance of g_J \mu_B B_0. Using g_J = 0.8, B_0 = 2 T and \mu_B = 57.88 \mu - [First answer](https://www.hyperfinecourse.org/forums/topic/first-answer-3/) - When an external magnetic field is applied this splits all levels of the system. So when the correct frequency photon hits a La atom, that is not yet in the highest of thse split states, it can transition to the next level. This corresponds with a gradual change in orientation of the magnetic moment of - [EPR on free atoms](https://www.hyperfinecourse.org/forums/topic/epr-on-free-atoms/) - 1. Describe what happens to a free La atom, subject to an external magnetic field, when it absorbs a microwave photon that satisfies the EPR resonance condition. Consider the case of a point-like nucleus (i.e. no hyperfine interactions). When EM radiation is absorbed, the orientation of the intrinsic spin of the nucleus changes. The case - [Orientation: temperature and radiation](https://www.hyperfinecourse.org/forums/topic/orientation-temperature-and-radiation-3/) - 1) Inferred information: From the diagrams, we can infer that the nucleus has a spin quantum number of I = 1. Furthermore, the splitting of energy levels shown is characteristic of a magnetic dipole interaction. 2) Orientation at low temperature: As the temperature decreases, the nuclei lose thermal energy and preferentially occupy the lowest available - [Orientation dependence on T](https://www.hyperfinecourse.org/forums/topic/orientation-dependence-on-t/) - 1. Which information about the nucleus ánd which information about the hyperfine interaction can you infer, when given the picture above? The amount of hyperfine levels tells something about the spin of the nucleus. When the levels are further apart, the magnetic field is larger. We can also know which orientation has the lowest energy, - [nucleus information and low T orientation](https://www.hyperfinecourse.org/forums/topic/nucleus-information-and-low-t-orientation/) - 1) You can infer that it is a nucleus with spin I = 1. The splitting in 3 levels can be explained by the magnetic dipole interaction which immediately splits them in all scenario's, but the electric quadrupole operator can also split all levels of an integer-spin nucleus if it is deformed. Or it can - [NRS](https://www.hyperfinecourse.org/forums/topic/nrs-8/) - Nuclear resonant scattering (NRS) is an advanced spectroscopic technique that is used in condensed matter research to study electronic and magnetic properties of materials on atomic scales. The technique utilizes the high-brilliance X-rays produced by synchrotron radiation facilities to excite resonant states of atomic nuclei. NRS is considered the time-domain analog of traditional Mössbauer spectroscopy. - [NRS with synchrotron radiation](https://www.hyperfinecourse.org/forums/topic/nrs-with-synchrotron-radiation/) - Nuclear resonant scattering using synchrotron radiation is an experimental technique based on nuclear resonant scattering (NRS). NRS is the process in which gamma ray radiation emitted by a decaying nucleus is resonantly re-absorbed by another nucleus of the same isotope. This article is about the detection of NRS using . When NRS is observed using - [NRS](https://www.hyperfinecourse.org/forums/topic/nrs-7/) - Nuclear resonance scattering (NRS) is a spectroscopic technique based on the Mössbauer effect. It enables the investigation of hyperfine interactions in solids, including the isomer shift (arising from differences in electron density at the nucleus), quadrupole splitting (caused by electric field gradients, EFGs), and magnetic hyperfine splitting (due to internal magnetic fields). Basic principle NRS - [Temperature influence on magnetic hyperfine field](https://www.hyperfinecourse.org/forums/topic/temperature-influence-on-magnetic-hyperfine-field/) - The magnetic hyperfine field is smaller at higher temperatures. This is clear from the pictures shown in the video: The lower the T, the more influence you see from the magnetic field. The magnetic moments tend to align themselves for a minimal energy configuration. In this minimal energy configuration, all moments add up and the - [Fast train](https://www.hyperfinecourse.org/forums/topic/fast-train/) - The emitter and absorber need a high relative velocity. It is not useful to put them both on a fast track, because then they have both the same velocity. However, it is possible to make 2 tracks in opposite directions. Then each track needs only half the speed. I would realize this speed by conveyor - [Mössbauer spectrum for a magnetic hyperfine interaction](https://www.hyperfinecourse.org/forums/topic/mossbauer-spectrum-for-a-magnetic-hyperfine-interaction/) - The hyperfine field at 50 K is equal in magnitude to the field at 18 K because the overall width of the velocity splitting is identical in both spectra. As the sample cools from 50 K to 18 K, the internal magnetic moments transition from a fluctuating state to a static alignment. At 50 K, - [Experiment with train](https://www.hyperfinecourse.org/forums/topic/experiment-with-train/) - Maintaining a precise 319 m/s is incredibly difficult because even tiny speed fluctuations would miss the extremely narrow natural linewidth required for resonance. A linear "train" setup would need detailed synchronization and alignment between the moving source and the stationary scatterer. It could be made practically feasible by mounting the source on the tip of - [Mössbauer spectrum](https://www.hyperfinecourse.org/forums/topic/mossbauer-spectrum-3/) - The dips have the same position (all be it less intense) for T=50K and T=18K except for the middle. This tells us the following: 1. The hyperfine field has the same strength because the levels are split exactly the same 2. At higher temperatures relaxation effects lead to reduced intensity for some transitions but averaging - [train experiment](https://www.hyperfinecourse.org/forums/topic/train-experiment-2/) - The biggest challenge to me seems that you would need a lot of space to allow the source to travel that speed towards the scatterer without slamming into the scatterer and destroying it. However if you use a circular motion at that speed you don't have this issue. Now there will be a certain time - [PAS and Symmetry in Fe_4N](https://www.hyperfinecourse.org/forums/topic/pas-and-symmetry-in-fe_4n/) - For Fe-I (corner site), we find four 3-fold axes (the cubic body diagonals). According to theorem 2, because there are at least two axes of order greater than 3, the EFG vanishes. For Fe-II (face-centered site), there is a 4-fold rotation axis along the N-Fe-N bond. The z-axis is aligned with the 4-fold rotation axis - [Quantum - Classical](https://www.hyperfinecourse.org/forums/topic/quantum-classical-2/) - To evolve the quantum picture (I=1) into the classical model, you must increase the spin I toward infinity. As I increases, the number of discrete states (2I+1) becomes so large that the energy gaps between them vanish. This transforms the quantized steps into a continuous spectrum, matching the behavior of the classical toy model. - [PAS for Fe](https://www.hyperfinecourse.org/forums/topic/pas-for-fe/) - We start by discussing the FeI position. This ion has no rotation axes. There is no PAS for this ion at this position. Next, we look to the FeII positions. The FeIIa ion has only one 2-fold rotation axis (going from FeIIa through N to the FeIIa ion on the other side). This axis is - [Classical case](https://www.hyperfinecourse.org/forums/topic/classical-case/) - On one of the slides, a picture shows the energy levels for increasing spin I. For half-integer spin, the energy levels remain degenerate. For integer spin, they split for \eta going to 1 (no axial symmetry). This means that the classical picture (all energy values are allowed) will be reached when considering high integer spin - [Presence of axial symmetry](https://www.hyperfinecourse.org/forums/topic/presence-of-axial-symmetry-4/) - From the formula, one can see that only 2 free parameters (V_zz and \eta) remain. Q was determined by experiment. When one would not work in a PAS, 5 free parameters would pop up (spherical tensor of rank 2). From the figure, we see that the system is identical when mirroring about the x-axis or - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-35/) - Axial symmetry is represented in the matrix because the diagonal elements V_xx and V_yy are equal, resulting in an asymmetry parameter eta = 0. Visually, the toy model is axially symmetric because the charge distribution is invariant under any rotation around the z-axis. - [Misleading vertical axis](https://www.hyperfinecourse.org/forums/topic/misleading-vertical-axis/) - The misleading aspect is that while the three graphs look visually similar, the vertical axis scale significantly decreases as l/d decreases. Consequently, as the nucleus becomes smaller, the absolute value of the quadrupole term decreases and becomes a negligible fraction of the total energy. - [Misleading fact](https://www.hyperfinecourse.org/forums/topic/misleading-fact/) - For a smaller l/d value (thus a small nucleus), the influence of the quadrupole term becomes smaller as well. By the chosen values for the vertical axis, it looks like the quadrupole term is greatest for high l/d values. - [Fe4N symmetries](https://www.hyperfinecourse.org/forums/topic/fe4n-symmetries/) - For the corner site we find 3 4-fold rotation axis along the edges of the crystal so the EFG tensor is zero (undefined PAS here and also eta because V_zz=0). It also has 2-fold rotations on the diagonals of the faces of the cube and 3-fold rotations on the body-diagonals. For the face-centered site the - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-27/) - To go from the discrete levels in quantum mechanics to the continuous levels of the classical system we must lift the limited amount of orientations that are possible in quantum mechanics and allow all orientations. Which would be true if I, the nuclear spin, is infinite. - [axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-34/) - from the equation) If the perturbing hamiltonian only has diagonal elements that means eta, the assymmetry parameter, will be zero and that there thus is axial symmetry (V_xx = V_yy). from visual) the problem remains the same when rotated around the z-axis -> axial symmetry - [Quadrupole term](https://www.hyperfinecourse.org/forums/topic/quadrupole-term-13/) - The smaller the nucleus, the smaller the quadrupole term and deviation from the monopole term. This makes sense as the convergence of the multipole expansion also makes only the monopole term a more and more appropriate approximation. In the limit of a point-nucleus we expect only monopole terms. - [Energy correction](https://www.hyperfinecourse.org/forums/topic/energy-correction-9/) - No, there won't be an energy correction. Both the Fermi contact & Bohr-Weisskopf corrections are only present if electrons penetrate inside the nucleus. Maybe there will be higher-order corrections. - [Carousel](https://www.hyperfinecourse.org/forums/topic/carousel-23/) - We have a happy child on a turning carousel. The child represents an electron, so s/he holds a bar magnet (representing the spin of the electron, dipole contribution) and an electrically charged ball (representing the charge of the electron, orbital contribution). The magnetometer is on the axis of the carousel and represents the nucleus, which - [Exercise on g-factor and magnetic moment](https://www.hyperfinecourse.org/forums/topic/exercise-on-g-factor-and-magnetic-moment/) - We can use the formula µ = gIµ_N, where µ is the magnetic moment, g is the g-factor and I is the nuclear spin. µ_N = (e \hbar) / (2m_p) is a nuclear magneton, a natural unit to express the magnetic moments. Its numerical value is 5.05 10^(-27) J/T a) For the ground state, we - [Experimentally determine µ_2](https://www.hyperfinecourse.org/forums/topic/experimentally-determine-µ_2/) - For the first isotope, we know µ_1 and I_1. We can measure the "a" value. This gives us info about the B_J / J value for this isotope. This is a property of the electron cloud, so it will be the same for the 2nd isotope. The spin I_2 is again given. The B_J / - [Orientation of L & S](https://www.hyperfinecourse.org/forums/topic/orientation-of-l-s/) - The value of J can be seen as a vector sum of L and S. - If L & S are parallel, the resulting J-vector will be maximal. - If L & S are anti-parallel (parallel but pointing in opposite directions), the resulting J will be minimal. - If L & S are perpendicular, an - [Energy corrections due to the finite size](https://www.hyperfinecourse.org/forums/topic/energy-corrections-due-to-the-finite-size/) - There would be no energy correction if all electron charges stay outside the nuclear volume. This is because the first-order correction for an 'extended nucleus' vanishes if there is no overlap between the electron density and the nuclear charge. - [Carousel illustration](https://www.hyperfinecourse.org/forums/topic/carousel-illustration-2/) - Place the magnetometer at the carousel's center to represent the nucleus sensing its magnetic environment. - Spinning the carousel with a child holding an electrically charged ball mimics the orbital field created by moving electrons. - A child holding a bar magnet illustrates the spin-dipolar field common in p- or d- orbitals. - Moving that - [Task 1: g-factors and magnetic moments](https://www.hyperfinecourse.org/forums/topic/task-1-g-factors-and-magnetic-moments/) - In order to find the g-factors, we use the formula: g = mu / I (where mu is expressed in nuclear magnetons). For the ground state of Cd-111, this results in -0.59485 / (1/2) = -1.1897. For the 245 keV level of Cd-111, we find -0.766 / (5/2) = -0.306. For the free electron, we - [Task 2: experimentally determine mu2](https://www.hyperfinecourse.org/forums/topic/task-2-experimentally-determine-mu2/) - First, measure the hyperfine coupling constant a1 and a2 for both isotopes using the experimental setup. Since both are the same element, the electronic configuration and total angular momentum J are identical, and the magnetic field B_J produced by the electrons remains constant. However, the nuclear spin I and magnetic moment mu vary because they - [Task 1: relative orientation](https://www.hyperfinecourse.org/forums/topic/task-1-relative-orientation/) - For J=2, L and S are parallel and pointing in the same direction, representing the maximum possible vector sum of the two angular momenta. J=1 corresponds to the orientation where L and S are perpendicular to each other. J=0 occurs when L and S are parallel and pointing in opposite directions, canceling each other out - [Energy correction](https://www.hyperfinecourse.org/forums/topic/energy-correction-8/) - A perfectly spherical nucleus undermines any higher than monopole orders for the electric multipoles so no corrections there. There is also no overlap however meaning no Fermi-contact contribution and if all electrons stay outside of the nucleus the hyperfine magnetic field is quasi-constant. Or from the electrons perspective it needs to enter the nucleus to - [Carousel](https://www.hyperfinecourse.org/forums/topic/carousel-22/) - Place the magnetometer at the center of the carousel, this can be considered as the position of the nucleus, the origin. You can let the child hold both the bar magnet and the electrically charged ball. This simulates them as the electron with a spin and electric charge. If the carousel wouldn't move you would - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-41/) - from the two different formulas for energy one can find that the g-factor equals \frac{\mu}{I \mu_N} a) for the ground state g = -1.1897 (I = 1/2 and \mu = -0.59485 \mu_N) for the excited state g = -0.3064 (I = 5/2 and \mu = -0.766 \mu_N) b) we find g = 2 (I = - [Determine magnetic moment](https://www.hyperfinecourse.org/forums/topic/determine-magnetic-moment/) - We could use a spectroscopic experiment where we measure absorption of light. We will find a resonance when the energy of the incident light corresponds with the energy difference between two levels. However because these levels are very close in energy you must work at very low temperatures for this kind of measurement because the - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-40/) - J=2 means L and S are oriented in the same direction as each other J = 0 means L and S are oriented anti-parallel to one another (opposite direction) and J = 1 means they're pointing perpendicular to one another - [Loophole](https://www.hyperfinecourse.org/forums/topic/loophole-12/) - In tm0, the total charge of the nucleus is +2e and the total charge of the electrons is -2e. In tmA, the total charge of the nucleus stays the same, but the total charge of the electrons changes to -2e - ε. This introduces extra negative charge into the problem, lowering the energy of the - [Time-averaged position of 1s electron](https://www.hyperfinecourse.org/forums/topic/time-averaged-position-of-1s-electron/) - Hi, where do you think a 1s electron will be in an atom? "On a circle around the nucleus, of course!" (or "Euh... I don't know") No, quantum mechanics is more fun! First of all, the nucleus and the electron cloud are the same in every direction of space. Not only to the left and - [(Apparent) contradiction](https://www.hyperfinecourse.org/forums/topic/apparent-contradiction-2/) - "Monopole shift is always up" means: if you only spread the same nuclear charge over a finite volume (no extra charge added), the energy moves up. Toy model A is different because it adds an extra negative charge inside the nucleus (without compensating elsewhere), which just makes the atom more bound, so the energy goes - [Observer inside hydrogen nucleus](https://www.hyperfinecourse.org/forums/topic/observer-inside-hydrogen-nucleus-2/) - If you were sitting inside the hydrogen nucleus, you would see that the 1s electron does not just circle around you, but actually spends some of its time right there at the center with you. It's time-averaged position looks like a 'charge cloud' that overlaps with the space the nucleus occupies. - [Li^6 state higher energy](https://www.hyperfinecourse.org/forums/topic/li6-state-higher-energy/) - In the video of expression and physics it seems like Li with 6 nucleons has a higher energy than Li with 7 nucleons. Intuitively I would think with more nucleons that would be larger and thus a larger positive correction. Is this caused by how we defined the nuclear state with charge distribution (which - [Contradiction](https://www.hyperfinecourse.org/forums/topic/contradiction-13/) - By adding electric charge in the center of the nucleus we do expect a positive shift caused by the monopole shift. But this addition has also already shifted the monopole field in the zeroth order. If we take a look at the zeroth order we see that it is a negative contribution that scales with - [intuitive explanation 1s electron behavior](https://www.hyperfinecourse.org/forums/topic/intuitive-explanation-1s-electron-behavior/) - The 1s electron sits way closer to the nucleus compared to others. On top of that it rotates way slower around the nucleus as well. If we average all the movement out over time we can determine what the chance is of finding the electron in a certain position. We will find that there is - [Complications from not using multipole expansion](https://www.hyperfinecourse.org/forums/topic/complications-from-not-using-multipole-expansion/) - The interaction energy is determined by a denominator that connects every point in the nucleus to every point in the electron cloud. For a nucleus of general shape, this expression cannot be simplified into separate terms. This prevents you from isolating the nuclear properties from the electronic environment, making it impossible to set up a - [complications if no multipole expansion](https://www.hyperfinecourse.org/forums/topic/complications-if-no-multipole-expansion/) - The goal of the multipole expansion is to decompose a complex charge distribution into simpler scalar/vectorial quantities. You make it easy to work with the charge distribution. If you use perturbation theory before making the multipole expansion, you will run into the problem that you don't have these simple expressions for the charge distribution. You - [Lowest-energy orientation](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-14/) - In all cases, the energy does not depend on the azimuthal orientation phi. In the case of alpha>0, the energy will be lowest if theta=0°, while in the case of alpha - [lowest-energy orientation of the dumb-bell](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-of-the-dumb-bell-5/) - alpha > 0: this is the case for which the plot of E_quad(theta) was made. The lowest E configuration is when the dumb-bell lies parallel to the z-axis (theta = 0° or theta = 180°). alpha = 0: there is no energy correction, the quadrupole energy is zero. Every orientation has the same energy. alpha - [Perturbation theory without multipole expansion](https://www.hyperfinecourse.org/forums/topic/perturbation-theory-without-multipole-expansion-3/) - When using perturbation theory the perturbation hamiltonian should be a relatively small correction to your problem. For example when solving atomic energy levels that are in eV the perturbation should only add a small contribution to this, like a couple of meV. In the case of a general shape of a nucleus, you even include - [minimal quadrupole energy](https://www.hyperfinecourse.org/forums/topic/minimal-quadrupole-energy-2/) - This is checked the simplest by looking at extreme values. If the rings coincide or h=0 and thus alfa - [Multipole moments](https://www.hyperfinecourse.org/forums/topic/multipole-moments-11/) - Multipole moments are used to decompose the CMB's temperature fluctuations across the sky into specific angular scales. By analyzing the power at each multipole, cosmologists can measure the universe's geometry and composition from the resulting peaks. - [Nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-44/) - Nuclear properties: - Number of protons - Number of neutrons - Mass - Spin - Magnetic moment - Size - [Multipole moments](https://www.hyperfinecourse.org/forums/topic/multipole-moments-10/) - In this video, the multipole moments of electric charge distributions are mentioned for the nucleus. They also exist for mass distributions at a very large scale, such as for fast-rotating neutron stars or even (Kerr) black holes. - [Nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-43/) - - number of nucleons (determines mass and size) - number of protons (electric charge) - number of neutrons - nuclear spin - Isospin - parity - stability (including magic numbers for nuclei which are extremely stable) - [example multipole moment](https://www.hyperfinecourse.org/forums/topic/example-multipole-moment-4/) - Quadrupole or even higher poles magnets are used to focus a particle beam in a particle accelerator - [properties of the nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-the-nucleus-16/) - - the number of nucleons, which can be once more seperated by number of protons and neutrons - the shape of the nucleus, is it spherical or deformed - is it a stable isotope or is it unstable - the nucleus is held together by the strong nuclear force which is mediated by gluons - - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-26/) - Hi! My name is Daan and I'm taking this course as part of my Master in Physics and Astronomy. I'm primarly focused on the underlying principles of experimental techniques, with a strong interest in interdisciplinary connections. I expect this course to broaden my foundation in applied physics, supporting my long-term goal of mastering interdisciplinary scientific - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-25/) - Hi, I'm Jeroen, and I study Physics and Astronomy at Ghent University (Belgium). I'm now in my first master's year, but I still don't know what I want to do with physics in the coming years/decades. I live in Harelbeke, and you should really view the Sint-Rita church. There is no other building in the - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-24/) - Hi my name is Sander, and I'm taking this course as part of my physics master at Gent University in Belgium. Gent is the largest city in East Flanders with a lot of cultural and economic history that impacted the entire country. It's also super cyclist friendly which is why I love cycling around in - [Properties of a nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-a-nucleus-7/) - - Magnetic quadrupole moment - Electric dipole moment - Atomic number - Mass number - Binding energy - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-23/) - My name is Juan Francisco Grillo I am Cuban, and I am taking this course from Havana You should visit my country because you will find the best beaches I just finished my Bachelor's Degree in Nuclear Physics I expect from this course to provide a solid foundation in hyperfine structure that will allow me - [Orientation, temperature and radiation](https://www.hyperfinecourse.org/forums/topic/orientation-temperature-and-radiation-2/) - 1) We can infer a pear shaped nucleus ensemble, with 3 orientations (along z, horizontal and along -z). We should have only magnetic contributions to hyperfine interaction, without any quadrupolar - electrical field gradient interaction. 2) At low T, we should expect the lowest level to be the most populated, with above levels almost empty - [Nuclear resonant scattering with synchrotron radiation](https://www.hyperfinecourse.org/forums/topic/nuclear-resonant-scattering-with-synchrotron-radiation/) - The idea is the same as the Mössbauer spectroscopy, which is to excite the smallest nuclear (hyperfine) levels of a sample and to observe its emission spectrum. The difference is mainly due to the nature of the excitation. Instead of a simple source giving only one energy, we have a very coherent source giving off - [From 50 to 18 K](https://www.hyperfinecourse.org/forums/topic/from-50-to-18-k/) - My guess is that the field is roughly the same, as the peaks are not displaced when temperature is lowered. At 50K, the higher levels are maybe populated by thermal fluctuations? Which means less transitions are possible, leading to the observed spectrum with reduced peaks. - [Train of thoughts](https://www.hyperfinecourse.org/forums/topic/train-of-thoughts/) - We would need a humongous experiment to make that work, with the initial distance between the source and the scatterer necessarily in the 100 to 1000 km minimum. Secondly, we would need all the space traveled more or less "clean", to avoid losing our initial signal with the huge distance it has to fly. Finally, - [PAS through rotation axis in Fe4N](https://www.hyperfinecourse.org/forums/topic/pas-through-rotation-axis-in-fe4n/) - If we consider only the chemical structure, there is 3 four-fold rotational symmetry axis through the FeI-N-FeI axis, which means it's the PAS for the electrical field gradient. If we now take into account the magnetic structure (ferro, along z), then we only have one 4-fold rotational axis, along z. We can then say it's - [Quantum transition to classical](https://www.hyperfinecourse.org/forums/topic/quantum-transition-to-classical/) - If we have a quantum system that has a nearly infinite nuclear spin, then the levels once splitted become almost like a continuum? Allowing for all possible energies to be explored by the sytem, giving us something similar to the classical picture? Not sure I understood the question, sorry. - [Presence of axial symmetry](https://www.hyperfinecourse.org/forums/topic/presence-of-axial-symmetry-3/) - If we take a look at the expression for the perturbing Hamiltonian, we clearly see that the contribution from EFG tensor is present through Vzz and eta. Knowing that eta ~ Vxx-Vyy and that our toy model (its quantum allowed states at least) is equivalent along x and y axis, we should end up with - [quadrupole moment change with nuclear dimension](https://www.hyperfinecourse.org/forums/topic/quadrupole-moment-change-with-nuclear-dimension/) - If we look closely at the energy levels, we see that when we reduce the ratio l/d (which means reducing the nucleus size), the energy splitting is reduced as well, despite all charges being constant. This is probably a shortcoming of the simple toy model used. - [Corrections](https://www.hyperfinecourse.org/forums/topic/corrections-4/) - The hyperfine field created by electrons can vary inside the nucleus. Depending on the way its charges and magnetic moments are distributed, they may experience a different field, leading to a different energy than if it was a point nucleus. The correction is expected to be very small though. - [Carousel as the hyperfine field](https://www.hyperfinecourse.org/forums/topic/carousel-as-the-hyperfine-field/) - We must first place the magnetometer at the center of the carousel, to symbolize the nucleus experiencing the hyperfine field. Then the child will hold both the ball and the magnet in his hand, and will wander around on the carousel. Most of the time, he is on a horse, meaning the magnet and the - [g-factors and µ](https://www.hyperfinecourse.org/forums/topic/g-factors-and-µ/) - We know that µ=gI in (µN/hbar unit), so if we know both the spin I and the magnetic moment µ, we can obtain g=µ/I. a) for the ground state, µ = -0.59485, I = 1/2 so g = -1.1897 for the first excited state, µ = -0.766, I = 5/2 so g = -0.3064 b) - [finding mewtwo](https://www.hyperfinecourse.org/forums/topic/finding-mewtwo/) - If we know µ1 and I1, then with the experiment we determine a1=µ1Bj/I1J1, and we can identify Bj/J1. Then when we perform the same experiment, assuming J2~J1 (ok, same isotope + free atoms experiment), we have a similar Bj/J2 and the only thing unknown will be µ2 = a2I2J2/Bj. - [mutual orientation of L and S](https://www.hyperfinecourse.org/forums/topic/mutual-orientation-of-l-and-s/) - When J=2, we have J=L+S which means L and S are parallel and pointing in the same direction. For J=1, we have L and S perpendicular to each other, only one contributes to J. For J=0, we have L and S parallel and pointing in opposite directions, as there contributions sum up to 0. - [loophole](https://www.hyperfinecourse.org/forums/topic/loophole-11/) - The toy model A adds a charge from nowhere compared to toy model 0. My guess is that by rearranging the negative charges in toy model 0 to have a small negative charge in the center (but a total negative charge of -2e), the energy wouldn't change so much. - [Position of 1s electron](https://www.hyperfinecourse.org/forums/topic/position-of-1s-electron/) - The 1s electron in hydrogen atom is over time exploring all of space. When I look at it from the hydrogen nucleus, I see it most of the time in the short distance zone, from the nucleus to the hydrogen atom radius. If we wait long enough, it will explore the entire sphere with center - [Paper](https://www.hyperfinecourse.org/forums/topic/paper-8/) - I got lost at page 8-9 where they introduced Powder method and rhombic class - [Epr](https://www.hyperfinecourse.org/forums/topic/epr-6/) - To obtain the G6 and G7 defect in an EPR experiment with spatial anisotropy, I would need to prepare a sample with anisotropic g-tensors, acquire its EPR spectrum at different orientations, analyze the spectra to obtain the g-tensor components , I have to repeat the experiment with different direction of applied field - [First and second answer](https://www.hyperfinecourse.org/forums/topic/first-and-second-answer/) - First one :- if the energy is right amount it will change it's orientation Second one :- 1) Without hyperfine interaction △Eo= gμBBo=1×(5.78838 × 10-5)x2= 1.15767636 × 10-4 eV. 1.1577x 10-4 eV 2) With hyperfine term written as Hnf = A,mrm] The other a BUT when a transition has △m=+1 and a fixed nuclear projection - [Nucleus info and temprature](https://www.hyperfinecourse.org/forums/topic/nucleus-info-and-temprature/) - 1) we can say that value of I=1 and based on 2I+1 it split into three levels 2) At low temperature, the nuclei will mostly occupy the lowest energy substate, causing the ensemble to become polarized along the field direction. Causing pears like shape - [NRS](https://www.hyperfinecourse.org/forums/topic/nrs-6/) - Nuclear Resonant Scattering (NRS) is a synchrotron X-ray technique that uses very sharp, resonant X-ray energies to excite specific atomic nuclei (like iron-57). When these nuclei re-emit the X-rays, the signal reveals hyperfine interactions—tiny magnetic and electric effects at the atomic scale. It’s powerful for studying magnetism, vibrations, and local environments in materials and biological - [Spectrum](https://www.hyperfinecourse.org/forums/topic/spectrum-3/) - At 50 k there is monopole shift the hyperfine quaderpole effect is not there but at 18k it's much visible and create a sexet Because at 50k there is thermal energy which overplay magnetic effect but as we lower the temperature the thermal energy becomes much lower - [Problem](https://www.hyperfinecourse.org/forums/topic/problem-2/) - The practical difficulty would be to make source fast as speed of sound I have to put so much energy it's really costly and require special type of equipment, to make it practically feasible we can use particle accelerator synchrotron etc but still here it's also energy consuming but it's much feasible. - [Task 2](https://www.hyperfinecourse.org/forums/topic/task-2-22/) - Fe-I have 4 fold rotation Fe-II a have C2 and C4 Fe - II b have only C2 - [Quantum to clasical](https://www.hyperfinecourse.org/forums/topic/quantum-to-clasical/) - We need to increase the value of L that will create continuous band - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-33/) - If an object or system looks the same when rotated around a central axis (usually the z-axis), then it exhibits axial symmetry. - [Quaderpole contribution](https://www.hyperfinecourse.org/forums/topic/quaderpole-contribution/) - As the distance between the nucleus decrease the contribution of quaderpole term become more and more exact and really small distance we can ignore quaderpole term and treat as a point charge - [Correction](https://www.hyperfinecourse.org/forums/topic/correction-5/) - According to me the answer is no The Bohr–Weisskopf effect requires the electron wavefunction to be nonzero inside the nucleus. It does not depend on nuclear size alone — it depends on the interaction between the extended nuclear magnetic moment and the electron’s spin density inside the nucleus. - [Carousel](https://www.hyperfinecourse.org/forums/topic/carousel-21/) - Here The carousel that is a rotating platform which simulates orbital motion child represents a spin or magnetic moment A bar magnet which is real magnetic moment represents external magnetic field An electrically charged ball that can generates electric field shows spin–orbit coupling A magnetometer that to detect fields, like a nucleus "feeling" the hyperfine - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-39/) - 1) g factor ground state for ¹¹¹Cd is -1.19 g factor for 245 kev for ¹¹¹Cd is -0.31 2) for free electron g factor is 2 3) for a nucleus with g factor of -3.826 the magnetic moment will be -1.9 - [Task 2](https://www.hyperfinecourse.org/forums/topic/task-2-21/) - Let’s say: Isotope 1: I₁ = 3/2, μ₁ = +2.0 μ_N, a₁ = 1000 MHz Isotope 2: I₂ = 1, a₂ = 800 MHz, μ₂ = ? since μ2 = μ1.(I2/I1).(a2/a1) We can determine μ2 Which will be μ2 = 1.07 μ_N - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-38/) - Let’s say: Isotope 1: I₁ = 3/2, μ₁ = +2.0 μ_N, a₁ = 1000 MHz Isotope 2: I₂ = 1, a₂ = 800 MHz, μ₂ = ? since μ2 = μ1.(I2/I1).(a2/a1) We can determine μ2 Which will be μ2 = 1.07 - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-37/) - Let’s say: Isotope 1: I₁ = 3/2, μ₁ = +2.0 μ_N, a₁ = 1000 MHz Isotope 2: I₂ = 1, a₂ = 800 MHz, μ₂ = ? since μ2 = μ1.(I2/I1).(a2/a1) We can determine - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-36/) - For the J=2, L and S are parallel and pointing in the same direction; For the J=1, L and S are perpendicular to each other; For the J =0, L and S are parallel and pointing in opposite directions. - [Toy model](https://www.hyperfinecourse.org/forums/topic/toy-model-9/) - We say the monopole shift gives a positive contribution because it increases binding, making the system more stable. In toy model A, the added negative charge inside the nucleus lowers the total energy, which may seem contradictory. However, this energy decrease actually reflects stronger attraction, not a negative shift in the physical sense. The “positive” - [For 15 year old](https://www.hyperfinecourse.org/forums/topic/for-15-year-old/) - this electron isn't like a little planet going in a circle—it's more like a fuzzy, invisible cloud that can pop up anywhere around you. Sometimes it's very close, sometimes it's farther away, and sometimes—very rarely—it even shows up right where you are, inside the nucleus Over time, as you keep watching, you notice a pattern. - [Complications](https://www.hyperfinecourse.org/forums/topic/complications-11/) - 1)Perturbation theory requires you to identify a small correction (perturbation) to a well-defined, solvable system. But A nucleus of general shape doesn't have a simple, analytical potential. Without multipole expansion, the perturbing potential due to the deformed charge distribution of the nucleus remains intractable 2) (1)The truncated expansion is valid when the electron is far - [Lowest energy](https://www.hyperfinecourse.org/forums/topic/lowest-energy/) - For alpha = 0 there will be no preferred orientations. For alpha > 0 xy plane. For alpha < 0 along the z axis - [destructive measurement](https://www.hyperfinecourse.org/forums/topic/destructive-measurement/) - When we add a sensor measuring which slit the electron passed through, the interference fringes disappear. The electron then behaves like a classical particle — we only see the sum of two diffraction patterns. This is because knowing the path, we cannot sum the amplitudes, we only sum the probabilities I(x)=|A_1|^2+|A_2|^2, interference only occurs when - [Mossbauer temperature](https://www.hyperfinecourse.org/forums/topic/mossbauer-temperature/) - The hyperfine magnetic field is on average smaller at T=50K than at T=18K. Due to thermal vibrations of the structure the hyperfine magnetic field decreases with increasing temperature, the magnetic field for a selected nucleus does not disappear but averaging over all nuclei gives a smaller value at a location and on average points in - [resonant velocity](https://www.hyperfinecourse.org/forums/topic/resonant-velocity/) - If I wanted to conduct a resonance scattering experiment in this way, I would have difficulty firstly with arranging the space for a suitably long track, and secondly with stopping the source. To overcome these problems, a synchrotron-type track could be made, in which the source would move with a constant velocity, would not have - [Bohr-Weisskopf effect](https://www.hyperfinecourse.org/forums/topic/bohr-weisskopf-effect/) - Is Bohr-Weisskopf effect really an "overlap" contribution, if no charge inside the nucleus is required for this effect to occur? Does the effect become stronger/weaker if there is charge penetrating into the nucleus? - [B11-05 - perturbed angular correlation spectroscopy](https://www.hyperfinecourse.org/forums/topic/b11-05-perturbed-angular-correlation-spectroscopy/) - 1-a) Why do you observe an exponential decay ? This is always the case for radioactive decay, this comes from the differential equation for the decay. Which depends on the amount of excited states that are left. 1-b) Why is the number of events/counts for detector combination 1-2 smaller than for 1-3 ? As we - [B11-03: NMR/ON](https://www.hyperfinecourse.org/forums/topic/b11-03-nmr-on/) - The experimental observation shows that the resonance frequency ν_L​ decreases with increasing external magnetic field B_{ext}. This means that the hyperfine field is negative and when we increase the strength of the external field we counteract this field more. For this the hyperfine field has to be negative. - [Paper “short-short introduction to EPR spectroscopy"](https://www.hyperfinecourse.org/forums/topic/paper-short-short-introduction-to-epr-spectroscopy-2/) - On page 16, I do not understand how they get the figure from the nuclear spins. - [EPR](https://www.hyperfinecourse.org/forums/topic/epr-5/) - 1) Prepare a crystalline sample containing paramagnetic defects 2) Mount the sample in an EPR spectrometer on a goniometer that allows precise rotation around specific crystallographic axes. 3) Align the crystal initially with its crystallographic axis (like [001]) parallel to the static magnetic field of the EPR spectrometer. 4) Set the microwave frequency to a - [Second answer](https://www.hyperfinecourse.org/forums/topic/second-answer/) - Without Hyperfine Interaction For a system with J=3/2, g=1, and B_0=2 T, the energy levels are: E(m_J)=−gμ_B B_0 m_J For transitions ΔmJ= 1; ΔE=g_μ_B B_0 = 1×5.788×10−5 eV/T×2 T = 1.16×10−4 eV This applies to both mJ=−3/2→−1/2 and mJ​=−1/2→+1/2 transitions. With Hyperfine Interaction Adding hyperfine field B_{hf}=10 T introduces the term A*m_I*m_J where: A = \frac{g\mu_N B_{hf}}{J} = - [First answer](https://www.hyperfinecourse.org/forums/topic/first-answer-2/) - When the resonance condition is satisfied there is the right amount of energy to let it change to a different level and its orientation would change. - [second task](https://www.hyperfinecourse.org/forums/topic/second-task-6/) - For the first case, without hyperfine interaction we use the EPR resonance condition E = -gJ*μB*B0*mJ, with gJ = 1, B0 = 2 T, and the Bohr magneton μB = 5.7883x10^-5 eV/T, we calculat the energy for levels with mJ = -3/2, -1/2, and +1/2. This gives us a transition energy of 0.116 meV between - [paper](https://www.hyperfinecourse.org/forums/topic/paper-7/) - For me it would be "As a result of this insensitivity, clear so-called turning-point features appear in the powder spectrum that closely correspond with the position of the g-values. Therefore the g-values can be read from the absorption-type spectra." Because at this point I realied I didnt actually understand what the poweder spectrum. Also what - [Answer](https://www.hyperfinecourse.org/forums/topic/answer-31/) - First, you need to determine the local g factor for that orientation; then, we change the orientation of the applied field and determine the g factor again. And get the g factor for horizontal and vertical orientation. - [Second task](https://www.hyperfinecourse.org/forums/topic/second-task-5/) - 1)Both situations, photon should be 4miuB. 2)If I=+1/2, photon should be 24miuB. If I=-1/2, photon should be -16miuB. - [First task](https://www.hyperfinecourse.org/forums/topic/first-task-5/) - There would be a transition between the hyperfine levels. - [Epr](https://www.hyperfinecourse.org/forums/topic/epr-4/) - To get a graph like this, put the irradiated silicon sample in an EPR spectrometer and select a constant microwave frequency. Align the external magnetic field to an initial direction and start sweeping the field strength to find resonance peaks. From these, calculate the g-factors. Then, rotate the crystal slightly, repeat the measurement, and continue - [First answer](https://www.hyperfinecourse.org/forums/topic/first-answer/) - When a free La atom is placed in an external magnetic field, its unpaired electron’s magnetic moment aligns with or against the field, whichs splits the energy levels. If it absorbs a microwave photon that matches the energy difference between these levels, the electron changes its spin, transitioning to the higher energy state. This absorption - [Nucleus info and temperature](https://www.hyperfinecourse.org/forums/topic/nucleus-info-and-temperature/) - (1) The nucleus has spin I=1, giving three magnetic substates (m_I = -1, 0, +1). The energy splitting between them shows the presence of a hyperfine interaction with an external field. (2) At low temperature, the nuclei will mostly occupy the lowest energy substate, causing the ensemble to become polarized along the field direction. - [Nucleus ensemble at low temperature](https://www.hyperfinecourse.org/forums/topic/nucleus-ensemble-at-low-temperature/) - 1) the picture indicates that the spin of the nucleus is one and the hyperfine splitting is present 2) due to lack of thermal excitation, the lowest energy state would be preferential, and the orientation would be anisotropic. - [task 2](https://www.hyperfinecourse.org/forums/topic/task-2-20/) - we have that delta_E = g mu_B B_0 = 1.6 mu_B T for the -3/2 --> -1/2 we have that delta_E1 = deltaE + A/2 delta_E2 = deltaE - A/2 for the -1/2 --> 1/2 delta_E1 = deltaE + A/2 delta_E2 = deltaE - A/2 - [information and temperature](https://www.hyperfinecourse.org/forums/topic/information-and-temperature/) - - (1) Which information about the nucleus and which information about the hyperfine interaction can you infer, when given the picture above? We can see that it has spin 1 and that hyperfine splitting has occurred. - (2) Describe the orientation of a nuclear ensemble in the conditions given in the picture above, at low - [answer](https://www.hyperfinecourse.org/forums/topic/answer-30/) - 1.First I can know that the orientation of each corresponding energy level. 2.In the low temperature, most nuclei would be in the lowest energy state, so I would say it’s polarized ensemble. - [paper](https://www.hyperfinecourse.org/forums/topic/paper-6/) - on page 11 they talk about field modulation. I get the concept and why it measures the first derivate, but I do not really understand why they do this. - [anwer](https://www.hyperfinecourse.org/forums/topic/anwer/) - to achive this picture you have to preform an EPR experiment. You start by placing you sample into the EPR cavity and by taking the the EPR pattern. It is important that you place sample in the cavity with the external magnetic field of the EPR machine is according to for example the OO1 direction - [photon absorbtion](https://www.hyperfinecourse.org/forums/topic/photon-absorbtion/) - When the atom absorbs the photon, the orientation between the electron spin and the orbital momentum changes. How much it changes depends on the value of J. - [paper](https://www.hyperfinecourse.org/forums/topic/paper-5/) - I understand everything up until right after the powder spectrum. In a powder, all orientations are in the sample simultaneously so you have many different g-tensors so you detect the sum of many different absorption lines. I start having trouble at page 9: "Fig. 8 shows the absorption and first-derivative spectra for three different classes - [answer](https://www.hyperfinecourse.org/forums/topic/answer-29/) - Assume you have a sample of a solid. First you apply a magnetic field in a certain direction wrt to the sample (eg. along the 100 direction). You irradiate the sample with microwaves and find the magnetic field strength for which you have an EPR resonance (absorption). From this initial spectrum, you can calculate the - [Second task answer](https://www.hyperfinecourse.org/forums/topic/second-task-answer/) - WITHOUT hyperfine interaction: We consider a transition between m_J = -3/2 and m_J = -1/2. E(m_J) = -g*mu_B*B*m_J so E(-3/2) = 3*mu_B and E(-1/2) = mu_B. The difference between them is 2*mu_B which is about 0.116 meV. The transition energy between m_J = -1/2 and +1/2 is the same. WITH hyperfine interaction (I took A - [First task answer](https://www.hyperfinecourse.org/forums/topic/first-task-answer/) - The m_J value changes if the system absorbs a photon, but the total angular moment J stays the same. This means the orientation of J changes along the z-axis, but only in discrete increments - [answers](https://www.hyperfinecourse.org/forums/topic/answers-5/) - 1. Which information about the nucleus ánd which information about the hyperfine interaction can you infer, when given the picture above? The nucleus has spin 1 (also on the picture) because the energy level splits in 3 distinct energy levels due to the magnetic hyperfine splitting. From the difference in energy, you can find the - [NRS](https://www.hyperfinecourse.org/forums/topic/nrs-5/) - Introduction Nuclear Resonant Scattering (NRS) is a spectroscopic technique that utilizes the Mössbauer effect to investigate hyperfine interactions. This is done by irradiating high-brilliance synchrotron X-ray pulses on a sample material containing Mossbauer-active nuclei. The X-rays are tuned to specific energies corresponding to the various nuclear transitions of the nuclei. Experimental setup An NRS experiment - [NRS](https://www.hyperfinecourse.org/forums/topic/nrs-4/) - Nuclear Resonant Scattering is a spectroscopic technique which exploits synchrotron radiation and the resonant absorption and re-emission of X-rays by atomic nuclei. It is an extension of Mössbauer spectroscopy to the time domain and uses the extremely narrow nuclear transitions of certain isotopes to study their hyperfine interactions, dynamics, and environments in materials with a - [Related to Nuclear and parity operator](https://www.hyperfinecourse.org/forums/topic/related-to-nuclear-and-parity-operator/) - I learn why Nucleus cannot have classical dipole moment, learn how and what a parity operator is and how it's applied - [Momentum of Cd-111](https://www.hyperfinecourse.org/forums/topic/momentum-of-cd-111/) - Quadrupole at 245 kev level of Cd-111 is +0.64(3)b. The magnetic dipole moment of the 245 kev level of Cd-111 is -0.766(3)nm. - [Daily life example](https://www.hyperfinecourse.org/forums/topic/daily-life-example-2/) - Remote control, compass, microwave, Mobile etc - [answers](https://www.hyperfinecourse.org/forums/topic/answers-4/) - 1) you can deduce that the nucleus had spin 1 and that the dominant contribution to the hyperfine splitting is the magnetic dipole interaction. the electric field gradient is equal to zero in this example because there is no splitting de to the electric quadrupole interaction 2) at low temperature, everything will be in the - [Nuclear Resonant Scattering](https://www.hyperfinecourse.org/forums/topic/nuclear-resonant-scattering-2/) - Nuclear Resonant Scattering Nuclear resonant scattering (NRS) is a synchrotron-based spectroscopic technique that investigates the interaction of X-rays with atomic nuclei, primarily utilizing the Mössbauer effect in the time domain. This method enables high-precision studies of hyperfine interactions in condensed matter systems, particularly in materials containing Mössbauer isotopes such as ^57Fe, ^121Sb, or ^125Te. Overview - [paper](https://www.hyperfinecourse.org/forums/topic/paper-4/) - I don't quite understand the sixth figure on page 7. Also the sentence before: in low viscosity the anisotropy is averaged? i don't get this - [answer1](https://www.hyperfinecourse.org/forums/topic/answer1/) - 1) Using electromagnets that generate a tunable magnetic field we form absorption spectra. 2) These spectra determine the peaks of the EPR sensitive defects 3) With this orientation we determine a g-factor 4) Now we change the orientation of the magnetic field ( strength at absorption peak i think) and remeasure the new g factor. - [answer](https://www.hyperfinecourse.org/forums/topic/answer-28/) - 1) I would say the La will change in energy and thus the orientation (spin) would change. 2) With zeeman splitting we have (due to I=1/2) E = m_J *g*mu_N * B because we only look at delta_m_J = 1--> delta_E = g*mu_N*B = 1*3.152 *10e-8 *eV/T *2T = 6.304 10e-8 eV With extra hyperfine - [answer](https://www.hyperfinecourse.org/forums/topic/answer-27/) - from the image you see that nucleus has spin one with 3 possible orientations. (So hyperfine splitting occurred). I would expect for low temperature an ensemble for the nuclei to all reside in the lowest energy level so all have the same direction, the higher temperature, more different directions - [Nuclear Resonant Scattering (NRS)](https://www.hyperfinecourse.org/forums/topic/nuclear-resonant-scattering-nrs/) - Nuclear Resonant Scattering (NRS) is a specialized technique used to study materials by examining nuclear energy levels and interactions. The technique utilizes hyperfine interactions that split nuclear energy levels. This method is simply put Mossbauer spectroscopy with an extra time dependent component. First the system is excited by synchrotron radiation pulses at the same time, - [different splittings](https://www.hyperfinecourse.org/forums/topic/different-splittings/) - Myabe the thermal motion of the atoms decreases, allowing the magnetic moments to align better. This alignment causes the splitting to increase. So, with less motion at lower temperatures, the system can achieve a more ordered magnetic structure, which results in larger hyperfine splittings and a clearer, more precise magnetic field at 18 K. - [Thermal effect on hyperfine field](https://www.hyperfinecourse.org/forums/topic/thermal-effect-on-hyperfine-field/) - Judging from the positions of the peaks, the magnitude of the splitting is the same at both temperature, but the effects of hyperfine splitting become more prominent for lower temperatures. My guess is that it could be due to the fact that at high temperatures the thermal broadening is larger than the hyperfine splitting, hence - [Experimental issues](https://www.hyperfinecourse.org/forums/topic/experimental-issues/) - Moving a gamma ray source at high speed is difficult, because the input of the energy required to accelerate the system has to happen in a way that doesn't disturb the emission process. A possible way to produce gamma rays of appropriate frequency could be synchrotron radiation. - [Mössbauer spectrum](https://www.hyperfinecourse.org/forums/topic/mossbauer-spectrum-2/) - The peaks lay at the same position, so the effect is the same. The strength is different, which could be because at lower temperatures there is less thermal noise. - [Train problem](https://www.hyperfinecourse.org/forums/topic/train-problem-3/) - A problem lays in the fact that you need them to go 319 m/s relative to each other. This means that we are rapidly approaching are detector or going away from it. Shooting something very fast at a sensitive detector would most certainly break something and to speed up something away from the detector is - [NRS](https://www.hyperfinecourse.org/forums/topic/nrs-3/) - NRS (nuclear resonant scattering) is a technique used in solid state physics to determine different properties of atoms and nuclei and their interaction. In NRS an atom and nucleus get excited by synchrotron radiation, because of the broad spectrum multiple hyperfine levels and the nucleus get excited. After this the system will decay and will - [Train problem](https://www.hyperfinecourse.org/forums/topic/train-problem-2/) - It would be difficult to make the source move at those speeds so maybe a workaround would be to make the source oscillate really fast - [answer](https://www.hyperfinecourse.org/forums/topic/answer-26/) - The hyperfine field has the same strength for both the 50 K and 18 K because the positions of the valleys remain the same. What does change with temperature is the intensity of the valleys which means more nuclei will contribute to the resonant scattering at low temperature. Maybe the thermal energy causes a fluctuation - [problem](https://www.hyperfinecourse.org/forums/topic/problem/) - An experiment like that would take up a lot of space and it would be difficult to make a container filled with gas have that speed. Maybe we could use ions and use a smaller version of a CERN-like setup with target nuclei at the sides of the ring. - [answer](https://www.hyperfinecourse.org/forums/topic/answer-25/) - I think it would stay the same. Between 50K en 18K the mossbauer effect becomes bigger so the lines become deeper. - [train](https://www.hyperfinecourse.org/forums/topic/train-2/) - It will be very difficult to accelerate a gass to 314m/s for a long time. A solution could be to shake it very rapidly. Only part of the emitted photons will have a high enough shift but your sample would not move to much. - [mossbauer](https://www.hyperfinecourse.org/forums/topic/mossbauer/) - I would say the hyperfine field stays the same, because the positions of the peaks in the spectra don't change ( no change in hyperfine splitting ). The peaks probably get higher because the temperature drop results in more alligned nuclei in reference to the hyperfine field - [train problem](https://www.hyperfinecourse.org/forums/topic/train-problem/) - I think in reality the 'train experiment' is difficult due to a number of reasons: First, eventually the train will go past the nuclei and not have the same relative speed anymore, it's negative. Also if the speed is exactly relative to the nuclei, both nuclei could collide? Furthermore, if the speed is to close - [task 2](https://www.hyperfinecourse.org/forums/topic/task-2-19/) - Looking at Fe_4N from the ppt and we only look at 1 primitive cell, you find for the Fe-I site a 4-fold rotational axis (through N and Fe-I) and then 2 1-fold rotational axis' (on the plane with the 4 Fe-II). Then for the Fe-II site you find a 3-fold rotational axis (through N and - [task 1](https://www.hyperfinecourse.org/forums/topic/task-1-35/) - The V_zz would have to be small for the quadrupole lift to run almost smoothly with the exact solution I think. - [axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-32/) - The properties of working with axial symmetry is that we can make a field symmetric matrix with trace=0. In the picture you can also intuitively recognize it if you look at the xy plane from above, you'll notice that there is no change in energy if you rotate the dumbbell around the z-axis. - [Quadrupole term](https://www.hyperfinecourse.org/forums/topic/quadrupole-term-12/) - As the radius of the nucleus becomes smaller and smaller, the quadrupole term approaches the exact solution. As long as the radius is finite the energy will not flatten out like in the monopole term. - [task2](https://www.hyperfinecourse.org/forums/topic/task2-3/) - Fe-I (Center Position)：C₄ ([001], [010], [100]), C₃ ([111])；High symmetry, cubic environment, 4-fold rotation. Fe-II (Face-Centered Position)：C₂ ([110], [101], [011])；Lower symmetry, lacks C₄ rotation, only 2-fold rotation remains. - [task 1](https://www.hyperfinecourse.org/forums/topic/task-1-34/) - We can add the number of I. - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-31/) - 1. By analyzing physical quantities. 2. By examining the geometric symmetry of the system. - [axis 2](https://www.hyperfinecourse.org/forums/topic/axis-2/) - assuming that we need to take N as the centre of symmetry: Fe-II has 2 two-fold axis and 1 4-fold rotation rotation axis which is the z axis of the system with eta= 0 Fe-I has 3 4-fold rotation axis thus the EFG tensor is 0. if the site itself is the symmetry point then - [1](https://www.hyperfinecourse.org/forums/topic/1-4/) - to change the quantum picture to be more like the classical one you would need to take instead of m+-1 and m0 all values between m+-1 and m0. You can maybe realise this by working with superpositions between m0 and m+-1 - [Rotation axes in Fe4N](https://www.hyperfinecourse.org/forums/topic/rotation-axes-in-fe4n/) - Fe-I: three four-fold axes, hence EFG tensor is zero Fe-II: a four-fold and two two-fold axes, so the PAS is defined by them and η=0. - [From quantum situation to classical](https://www.hyperfinecourse.org/forums/topic/from-quantum-situation-to-classical/) - What qualitatively distinguishes the quantum situation from the classical one, is that the energy correction can only take discrete values, instead of a continuum of values. For a very large value of nuclear spin, there would be so many discrete energy levels, that the difference between them would become negligibly small, and the spectrum would - [1](https://www.hyperfinecourse.org/forums/topic/1-3/) - we can tell the tensor is axially symmetric because the potential is identical for the xx direction and yy direction. we can also see that in the image because the electron cloud is 1 dimensionally distributed and the dumbell is a 1 dimentional object. leaving 2 degrees of freedom unrelated to eachother giving them equal - [1](https://www.hyperfinecourse.org/forums/topic/1-2/) - as the nucleus shrinks in comparison to the electron cloud, the quadrupole term also shrinks with respect to the monopole term - [rotation axes and PAS](https://www.hyperfinecourse.org/forums/topic/rotation-axes-and-pas-3/) - The Fe-I sites are at the cube corners and have cubic symmetry with three 4-fold rotation axes along the x, y, and z directions, which makes the asymmetry parameter eta = 0. The Fe-II sites, which are located at the face centers, have lower symmetry, with one 4-fold axis along the z-direction and two 2-fold - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-30/) - One way to determine the presence of the axial symmetry, is to note that V_xx = V_yy, hence the symmetry parameter η=0, which indicates axial symmetry. The axial symmetry also follows from the symmetry of the charge distribution, which is axially symmetric. - [The quadrupole term](https://www.hyperfinecourse.org/forums/topic/the-quadrupole-term/) - It seems that the magnitude of the quadrupole term decreases rapidly for smaller l/d ratios. - [quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-26/) - To make the quantum picture more similar to the classical result we would need to increase the quantum number I. In the image, I = 1 gives only three discrete energy levels while a classical system has a continuous range of orientations and energies. If we increase I to a much bigger value, the amount - [Rotation axis and PAS](https://www.hyperfinecourse.org/forums/topic/rotation-axis-and-pas/) - The Fe-I atoms have at least 3 four-fold axes, making the asymmetry parameter eta 0 and giving us three axes for our PAS. The Fe-II atoms have 1 four-fold axis and 2 two-fold axes, these could be our PAS. - [Quantum picture to classical](https://www.hyperfinecourse.org/forums/topic/quantum-picture-to-classical/) - To go from a quantized system to a classical one, such that it would be practically indistinguishable you would need to have a very high spin and a small asymmetry parameter. This would make the energy shifts practically continous. - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-29/) - From the equation it is clear that you can switch the x-axis witht the y-axis and it would be invariant, since Vxx=Vyy and Vxy=Vyx. From the picture it is clear that there is axial symmetry around the z-axis, since the electrons are situated on a line going through the centre. - [rotation axis](https://www.hyperfinecourse.org/forums/topic/rotation-axis-4/) - FeI : 4 fold rotation axis in the direction of the yellow arrow : z axis of PAS. In other directions : x and y there is also a 4 fold rotation axis FeII : 4 fold rotation axis in the diraction of the yellow arrow:z axis of PAS. In the perpendicular direction there are - [Rotational axis](https://www.hyperfinecourse.org/forums/topic/rotational-axis-2/) - Fe-1: it has 3 four-fold axes, thus resulting in a zero EFG-tensor. Fe-2: If we treat 2a and 2b differently, it has 3 two-fold axes and thus a non-zero EFG-tensor. If we treat 2a and 2b the same, we have 2 two-fold axes and 1 four-fold axes and thus a zero EFG-tensor - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-25/) - In order to go from a quantum (discrete) to a classical (continuous) case, one would need to create a system where I can go to infinity and thus create a continuum - [quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-24/) - As in slide 10, we need integer spin and eta non zero to lift the degeneracies in higher orders of m. When lifting the degeneracy we come close to a classical system. - [axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-28/) - On the previous slide, the first term is already axial symmetric. The second term is only axial symmetric if eta = 0, this is when Vxx == Vyy (which is the case when looking at the matrix). When looking at the toy model we see that there is rotation symmetry around at least one axis, - [Quadrupole term](https://www.hyperfinecourse.org/forums/topic/quadrupole-term-11/) - With a smaller nuclear size, the quadrupole interaction plays a lesser role because the charge distribution can be more accurately approximated as a point charge. However, in the three graphs, the quadrupole term appears equally distant from E0, which is a result of the vertical axis scaling. By comparing the vertical axes, you can see - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-27/) - 1. When Vxx, Vyy, etc. are zero, it is easy to conclude axial symmetry. 2. The elextrons are on the z-axis, and the center of the whole contraption is at the origin of the axis system. - [Quadrupole term](https://www.hyperfinecourse.org/forums/topic/quadrupole-term-10/) - When the nucleus becomes smaller, the quadrupole term also becomes smaller - [Fe4N](https://www.hyperfinecourse.org/forums/topic/fe4n-2/) - FeI: 3 times 4-fold, so EFG-tensor is zero according to theorem 2 FeII: 3 times 2-fold, so the z-axis of the PAS can be chosen along one of these 2-folds. Eta is non-zero. - [quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-23/) - We want a continuous spectrum instead of a quantized spectrum, so we want as many levels, as close together as possible. We can take: - high I, so there are more levels - integer spin and non-zero eta, so the degeneracy is lifted - Q and Vzz small so the energy levels are smaller and - [Quadrupole term](https://www.hyperfinecourse.org/forums/topic/quadrupole-term-9/) - The size of the quadrupole term itself becomes smaller if the nucleus becomes smaller. Looking at the three pictures might give you the misleading view that only the relative difference between the exact and E0+E2 approximation grows smaller, but the energy scale itself also changes drastically. - [from toy model to quantum](https://www.hyperfinecourse.org/forums/topic/from-toy-model-to-quantum-9/) - If the truncation after the quadrupole term is shown to give better results when the nucleus is smaller, this could be misleading. The reason is that the quadrupole term itself becomes smaller, not because truncating higher-order terms improves the approximation, but because the underlying charge distribution is becoming more symmetric as the nucleus gets smaller. - [axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-26/) - 1) There is axial symmetry when Vxx = Vyy and Vxy = 0. So eta = 0 2) The electrons are on the z-axis, which implies axial symmetry. - [Rotational Axis](https://www.hyperfinecourse.org/forums/topic/rotational-axis/) - FeI: In this case we see that FeI has 3 four-fold rotational axes. We can choose any one of them to be the z-axis in PAS, which will result in η=0, and thus axial symmetry. Furthermore, EFG will be zero, because we have more than 2 four-fold symmetry. FeII: In this case, we have 1 - [Quantum to Classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-22/) - To go from a quantum picture to a classical picture, we need to make our spectrum continous again, instead of quantised. To do this, we need an infinite number of levels, and thus an infinite spin as we have 2I+1 levels for each spin. - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-25/) - 1. On slide 4 we saw that η = 0 if V_xx = V_yy, which means we have axial symmetry. 2. On the figure on slide 8, we can see that the electrons are on the z-axis, which means we have axial symmetry around the z axis. - [Quadrupole term](https://www.hyperfinecourse.org/forums/topic/quadrupole-term-8/) - We can see that the scale on the vertical axis gets smaller if the nucleus becomes smaller compared to the electron cloud. This means that the quadrupole contribution gets smaller if l/d becomes smaller. - [Rotatiotal axis](https://www.hyperfinecourse.org/forums/topic/rotatiotal-axis/) - Fe-I has fourfold rotational symmetry around the axis through the orange atoms or through 2 red ones. For the Fe-IIa atoms there is the same fourfold rotational symmetry as the Fe-I. For Fe-IIb we only do not have a fourfold symmetry. This only has the same symmetry if we lay the rotation axis through both. - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-21/) - We go to a classical picture by changing from a quantized to a continuous spectrum of energy levels. For this we need l to go to infinity. To have an l with an infinite amount of possible values, we could also take l itself continuous. This would mean that the spin could point in any - [nuclear quadrupole interaction](https://www.hyperfinecourse.org/forums/topic/nuclear-quadrupole-interaction-3/) - If the nucleus becomes smaller, the size of the quadrupole term also becomes smaller. - [Rotation axis](https://www.hyperfinecourse.org/forums/topic/rotation-axis-3/) - Fe-I has 3 4-fold rotation axis => eta=0 if these axis are chosen as a basis Fe-IIa has a 4-fold rotation axis (z axis of the PAS) and 2 2-fold rotation axis => eta=0 Fe-IIb has a infinite-fold rotation axis (z axis of the PAS) and 2 2-fold rotation axis => eta=0 - [Quantum to Classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-20/) - The number of levels depends on the spin. Thus, to find an infinite number of level, the spin would need to be infinite. - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-24/) - 1) V_{xx}=V_{yy}=/=V_{zz} for a proper axis system 2) If it exists a rotation axis on which every rotations leave the equations invariant. - [Nuclear Quadrupole interaction](https://www.hyperfinecourse.org/forums/topic/nuclear-quadrupole-interaction-2/) - The smaller the nucleus with respect to the electron cloud, the smaller the electric quadrupole contribution. - [rotation axes and PAS for Fe](https://www.hyperfinecourse.org/forums/topic/rotation-axes-and-pas-for-fe/) - For Fe-I we have 3 4-fold rotation axes. Each rotation axis is a line through the Fe-I you focus on and through the closest Fe-I neighbour (for example like the yellow arrow). Any of these rotation axes can be taken as the z-axis of your PAS and in this case eta will be 0. For - [Fe4N](https://www.hyperfinecourse.org/forums/topic/fe4n/) - Fe-I: there are 3 four-fold rotation axes and so eta = 0, and the EFG = 0 for all three axes being used as z-axes of the system. Fe-II: 1 four-fold and 2 two-fold rotation axes, so eta = 0 for the four-fold axis as rotation axis. - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-19/) - The number of levels to which the quantum system splits is proportional to the spin of the nucleus (I). If we take then I -> inf there will be a very large number of energy levels, all infinitesimally close to each other, and so we get a continuum of energy values as in the classical - [Axial Symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-23/) - 1. From the equation we have axial symmetry if V_xx = V_yy, i.e. if eta = 0. 2. From the picture we can recognise the axial symmetry by the fact that the configuration is symmetric around the z-axis, i.e. a rotation in the x-y plane does not change the configuration - [Nuclear Quadrupole interaction](https://www.hyperfinecourse.org/forums/topic/nuclear-quadrupole-interaction/) - The size of the quadrupole moment is proportional to the ratio l/d, and so for a small nucleus l/d becomes small and so also the quadrupole moment will be small (relatively). - [Symmetry](https://www.hyperfinecourse.org/forums/topic/symmetry-5/) - 1. We can tell by the fact that the Vxx and Vyy components are equal, so the assymetry parameter is 0 2. By the fact that the charges are located along the Z axis - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-18/) - In the quantum case for I = 1, the nucleus can only have 3 possible orientations while the classical system (dumb-bell) can have all possible orientations between parallel and anti-parallel. In general we have 2I + 1 possible orientations so if we take I very large we can have many possible orientations that are very - [Axial symmetry of EFG tensor](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-of-efg-tensor-2/) - 1. If eta = 0, the tensor has axial symmetry. This happens when V_xx = V_yy. 2. The electrons causing the EFG are situated on the z-axis so rotating around the z-axis does not change anything. - [Quadrupole contribution](https://www.hyperfinecourse.org/forums/topic/quadrupole-contribution-3/) - The size of the quadrupole term decreases as the size of the nucleus decreases - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-22/) - This can easily be seen from the fact that the tensor has the same values for V_{xx} and V_{yy} but has a different value for V_{zz}. The -e charges are symmetric around the Z-axis, the dumbbell does not look symmetric around this axis but will be depending on the orientation. - [Quadrupole contribution](https://www.hyperfinecourse.org/forums/topic/quadrupole-contribution-2/) - When the nucleus becomes smaller, the quadrupole becomes smaller and the energy does not go as far away from the E0 term. - [question about PAS of MDI](https://www.hyperfinecourse.org/forums/topic/question-about-pas-of-mdi/) - In the last quiz, I was not sure aboit how to see what the PAS is for the MDI. - [rotation axis](https://www.hyperfinecourse.org/forums/topic/rotation-axis-2/) - FeI: its has a 4-fold rotation axis along the yellow arrow, so the z-axis of the pAS is along the yellow arrow FeII: it has 4-fold rotation axis perpendicular to the yellow arrow, out of the figure, so the z-axis of the PAS is also perpendicular to the yellow arrow and the normal on the - [task 1](https://www.hyperfinecourse.org/forums/topic/task-1-33/) - If you want to go from the QM case to the classical case you have to let the spin go to infinity. In this case the amount of levels for the quantumsystem becomes very large and can it take any value between its minimum and its maximum. - [Quadrupole contribution](https://www.hyperfinecourse.org/forums/topic/quadrupole-contribution/) - Whats misleading about it is that by how the scales are shown, it might give a wrong view of what the actual magnitude of the quadrupole contribution term is. So, while the truncation is better as the nucleus gets smaller, the quadrupole contribution also gets smaller. - [Corrections](https://www.hyperfinecourse.org/forums/topic/corrections-3/) - Yes, there will be corrections, due to the orbital and the spin dipolar contributions to the hyperfine field operator. - [Rotation axes and PAS](https://www.hyperfinecourse.org/forums/topic/rotation-axes-and-pas-2/) - Fe-I: We try to imagine the unit cell copy and pasted around the Fe-I atom, and we determine that there are three 4-fold symmetry axis along the edges of the unit cell. There are also three 2-fold symmetry axes along the diagonals passing through the Fe-I atom in each of the sides of the unit - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-17/) - The key difference between the quantum model and the classical toy model is that there are only 2I+1 possible values for m, leading to the discrete splitting of energy. We could increase the value for I, so that there are lots of values m can take. In this way, the spacing between the possible orientations - [Carousel](https://www.hyperfinecourse.org/forums/topic/carousel-20/) - Give the charged ball and the bar magnet to the child, put him on the carousel and let him run around while the carousel is spinning to simulate 1s orbital, and put the magnetometer in the centre. - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-32/) - 1a) g(gr. st.) = -1.188 g(ex. st.) = -0.3064 1b) g=2 1c)-1.913 \mu_N - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-21/) - We have axial symmetry in 2 situations: 1) when eta = 0 we have V_xx = V_yy, leading to axial symmetry. 2) In this simple toy model we will always have axial symmetry on the z-axis since the electrons are located on the z-axis. - [From toy model to quantum](https://www.hyperfinecourse.org/forums/topic/from-toy-model-to-quantum-8/) - Inspecting the vertical axes on the three graphs, we notice that if l becomes smaller, the quadrupole term itself also becomes smaller. So a smaller nucleus will have a smaller quadrupole term. - [Task 2](https://www.hyperfinecourse.org/forums/topic/task-2-18/) - Having the experimental setup to find out what the constant 'a' is and knowing the nuclear spin I_2, we could find what \mu_2 is, as J doesn't change due to an extra (or 1 less) neutron in the nucleus. - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-31/) - J=L+S,...,|L-S|. So J=2 means L and S are parallel and pointing in the same direction for them to sum up; J=1 is when L and S are perpendicular, which you can see as only S or L contributing to J and thus J=1; J=0 means L and S are antiparallel (parallel pointing in opposite directions), - [axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-20/) - 1) if V_xx = Vyy or eta = 0 2) if it is invariant under rotation around the z axis - [task2](https://www.hyperfinecourse.org/forums/topic/task2-2/) - By comparing the hyperfine structure of two isotopes of the same element, we can determine the unknown nuclear magnetic moment μ2​ using a simple ratio formula. This method is widely used in atomic physics and nuclear spectroscopy to extract nuclear magnetic moments of isotopes with high precision. - [task1](https://www.hyperfinecourse.org/forums/topic/task1/) - For J=2, L and S are parallel and pointing in the same direction; For J=1, L and S are perpendicular to each other; For J =0, L and S are parallel and pointing in opposite directions. - [task 1](https://www.hyperfinecourse.org/forums/topic/task-1-30/) - For the first task, the g-factor is found using 𝑔=𝜇/𝐼. For the ground state of 111𝐶𝑑 , with 𝜇=−0.5940𝜇_𝑁 and 𝐼=1/2, the g-factor is 𝑔=−1.188g=−1.188. For the 245 keV level, where 𝜇=−0.766μ_N and 𝐼=5/2 , the g-factor is 𝑔=−306. For the second task, the g-factor of a free electron is found using 𝑔=𝜇ℎ/𝜇𝐵 𝐼. Since - [corrections](https://www.hyperfinecourse.org/forums/topic/corrections-2/) - The fermi contact contribution vanishes (because no electrons inside the nucleus) But the Bohr-Weisskopf contribution does not vanish. Because it only depends on size, not on electrons penetrating the nucleus. - [carousel](https://www.hyperfinecourse.org/forums/topic/carousel-19/) - carousel = movement of electron around nucleus child in centre of carousel = represents the nucleus place the electrically charged ball next to the carousel = represents electron circling around nucleus -> orbital contribution place bar magnet next to carousel = electron as bar magnet creates B-field at nucleus -> spin dipolar contribution - [task 1](https://www.hyperfinecourse.org/forums/topic/task-1-29/) - (a) g = hbar*mu/(muN*I) so g_gorund = -1.19 hbar/muN g_245 = -0.306 hbar/muN (b) g = hbar*muB / (muN/2) = 2 (c) mu = g*muN*I/hbar = -1.913 muN/hbar - [carousel](https://www.hyperfinecourse.org/forums/topic/carousel-18/) - The carousel represents the electron cloud, with the child as the nucleus. Attaching a bar magnet to the child be like the nuclear magnetic moment, generating a dipole field as the carousel spins, like the spin dipolar contribution. Holding an electrically charged ball simulates the Fermi contact interaction, where electron density overlaps with the nucleus. - [Overlap](https://www.hyperfinecourse.org/forums/topic/overlap/) - In this case there would be no Fermi contact contribution because the electron would never reach the nuclear. There would, however, still be a correction for the nucleus' finite size i.e Bohr-Weisskopf. - [corrections](https://www.hyperfinecourse.org/forums/topic/corrections/) - the Bohr-Weisskoph effect would still be present since that correction is a consequence of the spacial distribution of the magnetic moment of the nucleus. the fermi correction however would vanish - [carousel](https://www.hyperfinecourse.org/forums/topic/carousel-17/) - suppose you have a carousel with a magnetometer inside the centre of the carousel. now imagine a child holding a bar magnet riding the carousel on a specific horse. the magnetic field that is measured in the centre is the dipolar hyperfine field. Now if the child was holding a charged ball, the magnetometer would - [value of μ2](https://www.hyperfinecourse.org/forums/topic/value-of-μ2/) - We have B1/J1 = a1*I1/mu1 B1/J1 is the magnetic field from the electron cloud. For an isotope we have different amount of neutrons, but equal amount of protons, so also equal amount of electrons. The electron cloud is therefore the same, so B1/J1 = B2/J2 So a1*I1/mu1 = a2*I2/mu2 So we find mu2 = a2*I2*mu1/(a1*I1) - [orientation](https://www.hyperfinecourse.org/forums/topic/orientation/) - parallel and pointing in opposite directions: J=0 perpendicular to each other: J=1 parallel and pointing in the same direction: J=2 - [Energy correction](https://www.hyperfinecourse.org/forums/topic/energy-correction-7/) - Bohr-Weisskopf correction would still be present, but the Fermi correction would vanish, since no electrons are present in the nuleus. - [g factors](https://www.hyperfinecourse.org/forums/topic/g-factors-17/) - 1.a) µ = g*µ_N*I/hbar µ = -0.5940 µ_N I = hbar* 1/2 g = -0.5940*2 = -1.188 g = -0.766 * 2/5 = -0.3064 1.b) µ = g*µ_B*S*hbar S = 1/2 * hbar µ = 1 µ_B g*1/2 = 1 g = 2 1.c) µ = g*µ_n*S/hbar S = hbar * 1/2 g = - [contributions to the hyperfine field](https://www.hyperfinecourse.org/forums/topic/contributions-to-the-hyperfine-field-4/) - A magnetometer placed in the center of the carousel would represent the nucleus. A child, equipped with a bar magnet and a charged ball, would ride the carousel. The magnetic field from the bar magnet and the magnetic field from the moving charge would be detected by the magnetometer. They represent dipolar and orbital contributions, - [g-factors](https://www.hyperfinecourse.org/forums/topic/g-factors-16/) - 1) The g-factor can be found from the following expression: g = mu/I. For the ground state, I = 1/2 and mu = -0.5940, hence g = −1.188. Analogously, for the 245 keV level we find g = -0.766/(5/2)= -0.3064. 2) g_e = mu/S_e = 2 (mu=1 in units of mu_B) 3) mu = g*S - [Energy corrections](https://www.hyperfinecourse.org/forums/topic/energy-corrections-9/) - There will still be a correction. The Bohr-Weisskopf effect is dependent on the magnetic field created by the electrons and how this interacts with the nuclear magnetic moment. The electrons do not have to be located inside the core for their magnetic field to penetrate the core and interact with the nuclear magnetic moments. - [Contributions to the hyperfine field](https://www.hyperfinecourse.org/forums/topic/contributions-to-the-hyperfine-field-3/) - Have the kid measure the magnetic field from the bar magnet attached to the a pole of the carousel with the magnetometer. This is the dipole term. Have the kid measure the field at the center of the carousel with the charged ball fastened to a pole while spinning. This is from the orbital motion. - [Experimental procedure](https://www.hyperfinecourse.org/forums/topic/experimental-procedure/) - The magnetic moment and the hyperfine coupling constant are related by the following formula: (μB_J)/(ℏ²IJ)=a. Assuming B_J can be determined experimentally, the value of J can be calculated for the first isotope. For another isotope, B_J and J remain the same, because the electron cloud is unaffected by adding a neutron to the nucleus. Hence, - [g-factors](https://www.hyperfinecourse.org/forums/topic/g-factors-15/) - a) mû = g*mu_N/h_bar*Î => g.s. g = -0.5940Nm*2/1Nm = -1.188; e.s. g = 2/5*(-0.766) = 0.306 b) mu e- = g*mu_B/h_bar*S_hat = 1mu_B => g = 1*1/2 = 2 c) mu = -3.826*I/h_bar*mu_N = -3.826mu_N/2 = -1.913mu_N - [µ_2 determination](https://www.hyperfinecourse.org/forums/topic/µ_2-determination/) - we know the spin I_1, magnetic moment µ_1. We use this to determine the coupling constant a = µB_J/(IJ) If we consider an isotope of the same element, J and J^2 remain invariant. µ_1 = a*J/(B_J*hbar)*I_1 µ_2 = a*J/(B_J*hbar)*I_2 µ_2/µ_1 = I_2/I_1 µ_2 = µ_1*I_2/I_1 - [L and S](https://www.hyperfinecourse.org/forums/topic/l-and-s-5/) - J=0 correspronds to S and L anti-aligning because J=L+S so L=-S and thus m_s = -m_l = -1, 0, 1 J=1 corresponds to S and L being perpendicular making the projection of one of the other on the z axis 0 leaving the remaining factor J=2 corresponds to S and L aligning for the same - [1](https://www.hyperfinecourse.org/forums/topic/1/) - If the nucleus becomes smaller, the quadrupole term becomes larger and larger. - [Value of mu_2](https://www.hyperfinecourse.org/forums/topic/value-of-mu_2/) - You start of by calculating the J value for the first isotope, which you can do since you can measure the 'a' and have the mu and I values. The J value will be the same for different isotopes, since only the nucleus changes, not the electron cloud. If you now measure the 'a2', and - [L-S orientation](https://www.hyperfinecourse.org/forums/topic/l-s-orientation-2/) - When L and S are parallel and pointing in the same direction, they add together to J = 2. When L and S are parallel and pointing in the opposite direction, they are subtracted and J = 0. When L and S are perpendicular they sum to J = 1. The perturbing term depends on - [determine mu_2](https://www.hyperfinecourse.org/forums/topic/determine-mu_2/) - For isotope 1 we have a_1 = (mu_1 / I_1) * (B_J / J). a_1 can be experimentally measured and mu_1, I_1 is known which allows us to determine the ratio of (B_J / J). For isotope 2 we have a_2 = (mu_2 / I_2) * (B_J / J). The ratio of (B_J / J) - [L and S orientations](https://www.hyperfinecourse.org/forums/topic/l-and-s-orientations-3/) - L and S are parallel and pointing in opposite directions when J = 0. They are parallel and pointing in the same direction when J = 2. When J = 1, L and S are perpendicular to each other. - [High-fequency noise](https://www.hyperfinecourse.org/forums/topic/high-fequency-noise/) - In almost every video there are sporadic high frequency noises, making it difficult to keep my concentration on the lesson. These noises have a frequency of around 15kHz - [Energy corrections](https://www.hyperfinecourse.org/forums/topic/energy-corrections-8/) - The contact correction will vanish, because there are no charges inside the nucleus. The Bohr-Weisskopf correction does not vanish as this depends on the finite size of the nucleus and not on the electrons penetrating inside the nucleus. - [Carousel](https://www.hyperfinecourse.org/forums/topic/carousel-16/) - We want to put the magnetometer in the center of the carousel (in the atom we want to know the hyperfine contribution in the nucleus). Then we can let the child on the carousel and hold the electric ball and bar magnet. The electric ball mimics the charge of the electron, the bar magnet mimics - [g factors](https://www.hyperfinecourse.org/forums/topic/g-factors-14/) - a) Cd-111: g = -0.595*2 = -1.19 Cd-111 (245 keV): g = -0.766 * 2/5 = -0.3064 b) g = 2 c) mu = mu_N * g * I / hbar = -3.826 * 1/2 * mu_N = -1.913 mu_N - [Carousel](https://www.hyperfinecourse.org/forums/topic/carousel-15/) - Put the magnetometer at the center of the carousel. Place the child on the carousel (near the edge) and let the child hold the ball and the bar magnet. This serves as a model for the orbital and dipolar contribution to the hyperfine field. To model the Fermi contribution, you would have to place the - [answers to exercises](https://www.hyperfinecourse.org/forums/topic/answers-to-exercises/) - 1. The g factor = (mu / mu_N) * (h_bar / I). For the ground state of 111Cd we have mu = -0.5940 nm and I = 1/2 * h_bar. Filling this in the formula above gives a g factor of -1.188 (dimensionless). For the 245 keV level, mu = -0.766 nm and I = - [energy corrections](https://www.hyperfinecourse.org/forums/topic/energy-corrections-7/) - Fermi contact contribution will be 0 because the same number of spin up and spin down electrons are in the nucleus, 0. The Bohr-Weisskopf effect will still be present even if there are no electrons in the nucleus. - [Correction](https://www.hyperfinecourse.org/forums/topic/correction-4/) - Yes. The key point is that while the Fermi contact interaction is zero (since no electrons are inside the nucleus), the Bohr-Weisskopf effect still leads to corrections. This happens because the nuclear magnetization is distributed over a finite volume, and the electron’s magnetic field still penetrates the nucleus, leading to non-uniform hyperfine interactions across the - [Carousel](https://www.hyperfinecourse.org/forums/topic/carousel-14/) - Let the child hold the ball on the carrousel. Use the ball near the magnetometer while the carousel spins. This rotating charge simulates an electric field gradient, which can induce a hyperfine quadrupole interaction in an atom with nuclear spin - [g factor](https://www.hyperfinecourse.org/forums/topic/g-factor-7/) - For Cd111 ground state I= 1/2 hbar and magnetic momentis Mu= -0.5940 * mu_N . With formula Mu = g * mu_N *I/hbar we find that g= mu/mu_N *hbar/I = -0.5940 * 2=-1.188. For 245keV we have g= -0.766 * 2/5 =-0.3064 . Free electron magnetic moment = 1 mu_B mu_B= e*hbar/2m_e. Bcs electron is - [Energy corrections?](https://www.hyperfinecourse.org/forums/topic/energy-corrections-6/) - If we would have a perfectly spherical nucleus, there would be no energy corrections due to the contact correlation, but there would be corrections because of the Borh-Weisskopf effect, sicne this only goes to 0 if the radius itself goes to 0. - [Carousel methaphor](https://www.hyperfinecourse.org/forums/topic/carousel-methaphor/) - One way to mimic contributions to the hyperfine fields can be to put the magnetometer in the middle of the carousel, as we would want to know the magnetic field that acts on the nucleus. We can put the child on the carousel while letting it hold the electric ball first. This moving electric charge - [Determine g-factors and magnetic moment](https://www.hyperfinecourse.org/forums/topic/determine-g-factors-and-magnetic-moment/) - 1.a) We know that µ = g * µ_N * I / hbar, so g = hbar * µ / µ_N * I. For the ground state of 111Cd we have µ = -0.594 µ_N and I = 1/2 hbar, so g = -1.188. For the state with energy 245 keV we have µ = - [Determine µ_2](https://www.hyperfinecourse.org/forums/topic/determine-µ_2/) - We know that a = µ * B_J / I*J, and assuming J_1 = J_2 and B_J1 = B_J2, we can write: B_J1 / J_1 = B_J2 / J_2 a_1 * I_1 / µ_1 = a_2 * I_ 2 / µ_2, which can be rewritten to find µ_2 = µ_1 * a_2 * I_2 - [relative orientations of L and S](https://www.hyperfinecourse.org/forums/topic/relative-orientations-of-l-and-s/) - J = 0: L and S are antiparallel J = 1: L and S are orthogonal J = 2: L and S are parallel - [energy correction](https://www.hyperfinecourse.org/forums/topic/energy-correction-6/) - There will be no energy correction coming from the fermi-contact effect;µ, because the electron density at the origin is always zero. The Bohr-Weiskopf effect will give rise to an energy correction, because the p and n in the nucleus will freely move around and give rise to a distribution in space and will thus inetract - [Energy corrections](https://www.hyperfinecourse.org/forums/topic/energy-corrections-5/) - I think there will still be an energy correction due to the Bohr-Weisskopf effect, because of the magnetic field generated by the electrons. I don't think there will be any hyperfine corrections though, as the nucleus needs to deformed for those to show up. - [carousel](https://www.hyperfinecourse.org/forums/topic/carousel-13/) - You would put the magnetometer in the centre of the carousel, as the we want to know the magnetic field at the centre of the atom i.e. at the nucleus. If you have a bar magnet that a child is holding on the carousel, then wether or not it is moving, the magnetometer will measure - [task 1](https://www.hyperfinecourse.org/forums/topic/task-1-28/) - 1) GS: g[µ_N/hbar] = -1.188, 245keV: g[µ_N/hbar] = -1.532 2) g = µ_b/µ_b * 2 = 2 3) µ = 2*-3.826*µ_N = -7.652 µ_N - [Correction](https://www.hyperfinecourse.org/forums/topic/correction-3/) - The fermi contact correction will vanish (psi(0)=0) but not the Bohr-Weisskopf (only depend on the nucleus) one. - [Carousel](https://www.hyperfinecourse.org/forums/topic/carousel-12/) - I would put the child in the carousel with the charged ball (the electron) and put the magnet in the center of the carousel while leaving it free to rotate. Then, I would use the magnetometer to measure the direction of the magnet. - [g factors and magnetic moments](https://www.hyperfinecourse.org/forums/topic/g-factors-and-magnetic-moments-3/) - a) \mu=-0.594 nm => g=\frac{\hbar^µ}{Iµ_N}=1.188/µ_N b) Using the same formula, g=2mu_B/µ_N c) Using the same formula, µ=gµ_N/2=-1.913µ_N - [Carousel](https://www.hyperfinecourse.org/forums/topic/carousel-11/) - We put the magnetometer at the centre of the carousel and perform 3 experiments. One wher the child has the charged ball, one where it has the bar magnet, and one where it has both. The magnetometer represents the nucleus and the child the electron. In the first experiment, we give the child the charged - [Determination of µ_2](https://www.hyperfinecourse.org/forums/topic/determination-of-µ_2/) - If B_j_1=B_j_2, we have µ_2=(µ_1I_1a_2)/(a_1I_2) In this, the only unknown is a_2 which can be measured. - [L and S orientation](https://www.hyperfinecourse.org/forums/topic/l-and-s-orientation-2/) - L and S are vectors. So if the sum of them is 0 (J=0) it means they are antiparallel. Using the same reasoning, we would find L and S are perpendicular for J=1 and parallel for J=2. - [g factors and magnetic moment](https://www.hyperfinecourse.org/forums/topic/g-factors-and-magnetic-moment/) - 1a) g = µ*h_bar / µ_N*I, with µ = -0.59403µ_N, I = 1/2 h_bar to give g = -0.59403 / (1/2) = -1.18806 for the ground state. For the 245 keV state, µ = -0.766µ_N and I = 5/3 h_bar to give g = -0.766 / (5/2) = -0.3064. 1b) µ = g*µ_B*I, with µ - [Energy corrections](https://www.hyperfinecourse.org/forums/topic/energy-corrections-4/) - There is an extended nucleus which normally causes corrections. As there is no overlap in this example, we will not have any corrections. I also think there will be no hyperfine interactions because we have a perfect spherical system. (We need deformation for that.) - [Discription of hyperfine field](https://www.hyperfinecourse.org/forums/topic/discription-of-hyperfine-field/) - For the Fermi interaction, we simply put the bar magnet in the middle and have the child move around with the magnetometer. For the dipool interaction we let the child ride the carousel holding the bar magnet. For the Orbital interaction we also let the child ride the carousel but now holding the charged ball. - [Determination of µ2](https://www.hyperfinecourse.org/forums/topic/determination-of-µ2/) - The coupling constant a = µ*B_J / I*J. We know a1, a2; I1, I2, µ1, and we assume that J and B_J are the same for both experiments. Then a1/a2 = µ1*I2 / µ2*I1 which we can rewrite to get µ2 = µ1*I2*a2 / I1*a1. - [g factor and moment](https://www.hyperfinecourse.org/forums/topic/g-factor-and-moment/) - $g = \frac{\mu}{I \mu_N} $ ground state g-factor: $g = \frac{-0.5940 \mu_N}{1/2 \mu_N} = -1.188 $ 245 keV level g-factor: $g = \frac{-0.766 \mu_N}{5/2 \mu_N} = -0.3064 $ For the free electron, we have $g = \frac{1 \mu_B}{1/2 \mu_B} = 2 $ For the neutron we have: $\mu = g I \mu_N = -3.826 * - [experimental determination](https://www.hyperfinecourse.org/forums/topic/experimental-determination-2/) - From the slides we get: $a = \frac{ \mu B_j}{I J}$. Here a is measurable and $B_j$ is a property of the element, not the isotope. As we also know I, we can easily find $\mu$. - [L-S coupling](https://www.hyperfinecourse.org/forums/topic/l-s-coupling/) - J = 2 means that they are parallel and pointing in the same direction, we take the sum of them. J = 1 means that they are perpendicular to each other, the S does not contribute to J as the vector is perpendicular. J = 0 means that they are parallel but pointing in opposite - [J-value orientation](https://www.hyperfinecourse.org/forums/topic/j-value-orientation/) - J=0: L, S parallel and opposite direction J=1: L, S perpendicular J=2: L,S parallel and same direction - [Spherical nucleus energy correction](https://www.hyperfinecourse.org/forums/topic/spherical-nucleus-energy-correction/) - The Bohr-Weisskopf effect will still give a non-zero correction, since this only goes to zero as the radius of the nucleus goes to zero. - [Carousel](https://www.hyperfinecourse.org/forums/topic/carousel-10/) - Put the child on the carousel, and give them the charged ball or bar magnet, or both. This is then a model for the electron orbiting the nucleus, where if the child is free to move they will also simulate the random movement of the electron in orbit i.e. it does not have a determinate - [g-factors](https://www.hyperfinecourse.org/forums/topic/g-factors-13/) - a) gs: gamma = -1.188, (254keV): gamma = -0.3064 b) g = -2 since mu_N = mu_B for the electron c) −1.913 mu_N - [Experimental determination](https://www.hyperfinecourse.org/forums/topic/experimental-determination/) - We have that a = c * mu/I, where c is some constant which does not depend on the isotope. We can then take a_1/a_2 to find that mu_2 = mu_1 * (a_2*I_2) / (a_1*I_1) - [J - value](https://www.hyperfinecourse.org/forums/topic/j-value/) - J = L+S, and so if L and S are parallel, they can be aligned to get J = 2 from l=s= +/- 1 or anti-aligned to get J = 0 from l = -s = +/- 1. If they are perpendicular we have that one of them is either +/- 1 and the other - [Energy level orderings](https://www.hyperfinecourse.org/forums/topic/energy-level-orderings/) - J^2 =(L+S)^2, so the largest value of J corresponds to the case where L and S are aligned, J=1 corresponds to perpendicular L and S and J=0 to antiparallel L and S - [Experimental determination of mu2](https://www.hyperfinecourse.org/forums/topic/experimental-determination-of-mu2/) - We can measure the hyperfine coupling constant of the isotope. From this value and the known nuclear spin I2, we can calculate the the magnetic moment as followed: mu_2 = (aJh_bar^2I_2)/(B_J) - [L, S orientations](https://www.hyperfinecourse.org/forums/topic/l-s-orientations/) - I assume that for J=0, the spins are alligned and pointing in opposite direction. For J=1, I think the spins are perpendicular to each other. For J=2, I think the spins are alligned and pointing in the same direction. - [experiment](https://www.hyperfinecourse.org/forums/topic/experiment-2/) - To determine the \mu2 of tge isotope, we use the fact that the hyperfine interaction constant a is proportional to the nuclear magnetic moment and the magnetic field at the nucleus, which we assume is very similar for isotopes of the same element. After measuring a_2 for the unknown isotope and comparing it to a - [Loophole](https://www.hyperfinecourse.org/forums/topic/loophole-10/) - The monopole shift is found by taking into account some properties of the material already there, like the size of the nucleus or the distribution of the electron cloud. The difference between toy model 0 and toy model A is not just the difference in property, it is a something more intrinsic, namely the monopole - [Task 1 electron monopole shift](https://www.hyperfinecourse.org/forums/topic/task-1-electron-monopole-shift/) - An electron in the 1s orbital in the hydrogen atom is the ground state of the electron, and because of some math the probability to find the electron around the nucleus forms a spherical cloud. So if you go and measure the position of the electron in space taking the nucleus as your reference point - [Experiment](https://www.hyperfinecourse.org/forums/topic/experiment/) - Coupling constant id related by magnetic moment (mu) divided by nuclear spin (I). So that we can use both coupling constants and I2, I1, mu1 to find mu2 with : mu2= a2 * I2 * mu1 /( a1 * I1) - [L, S orientation](https://www.hyperfinecourse.org/forums/topic/l-s-orientation/) - If L and S parallel -> J=2 , if perpendicular J= 1, if opposing J=0 - [mu_2](https://www.hyperfinecourse.org/forums/topic/mu_2/) - you first measure a_1. You then measure a_2. In the formula for the termsplitting, only the I and yhe mu are different in the 2 cases. So you can divide a_2 by a_1 which gives mu_2 very easily. - [L S](https://www.hyperfinecourse.org/forums/topic/l-s/) - J = 2: pointing in the same direction, which follows intuitively from the classical model. J = 0: pointing in opposite direction, same reason. J = 1: perpendicular - [L and S orientations](https://www.hyperfinecourse.org/forums/topic/l-and-s-orientations-2/) - Having both L and S pointing to the same direction would lead to the least energy and therefore the maximum value for J, so J = 2. Similarly, in the case where L and S have opposite orientations then the energy would be the highest and it would correspond to J = 0. Lastly the - [Contradiction](https://www.hyperfinecourse.org/forums/topic/contradiction-12/) - Adding this extra charge epsilon means we change the total charge of the system. In the original system the total charge is zero, but in model A it is -epsilon. We could solve this by adding epsilon/2 to each electron. This would mean that the constant C in our E_0 would change (it will grow: - [1s electron to a 15yo](https://www.hyperfinecourse.org/forums/topic/1s-electron-to-a-15yo/) - There's an electron and it is found somewhere around the nucleus, at a certain distance r. Looking at this distance from the nucleus or you as observer and noting down the distance everytime you check the electron, a pattern will emerge/a nice graph will be produced. Sometimes this electron will be inside of the observer, - [loophole](https://www.hyperfinecourse.org/forums/topic/loophole-9/) - if we assume the electron is not a point particle but has a size, it being inside the nucleus would take up space making the radius larger. Since the nucleon-nucleon interaction is stronger than the coulomb interaction gives this increase in radius a larger positive contribution than the negative one from the coulomb - [Toy model problem](https://www.hyperfinecourse.org/forums/topic/toy-model-problem/) - We add a charge in the nucleus. This charge will lower the nucleus charge instead of really adding a charge to the surrounding area around the nucleus. - [Contradiction](https://www.hyperfinecourse.org/forums/topic/contradiction-11/) - The contradiction arises because the monopole shift usually refers to modifications of a purely positive nucleus, however in the toy model, an additional explicitly negative charge is introduced which is not just a “finite-size correction” but rather a change in charge distribution, which can lower the energy of the system. - [Time average position of an electron](https://www.hyperfinecourse.org/forums/topic/time-average-position-of-an-electron/) - If you’d watch how the electron moves around you when you’re at the nucleus it would look like most of its time is spent nearby, less and less time is spent further and further away. It’s never on the same spot, and it doednt follow an orbit. - [where is the electron of hydrogen?](https://www.hyperfinecourse.org/forums/topic/where-is-the-electron-of-hydrogen/) - Suppose you are inside the proton of a hydrogen atom. this proton is pretty large for you and has a certain volume although you're not sure how much because you cannot see its edges. You know there is an electron near you that is attracted to the proton. this means that the electon will try - [Expression and physics](https://www.hyperfinecourse.org/forums/topic/expression-and-physics-3/) - If we sit in the nucleus, we can see the time-averaged position most near the nucleus and as we see farther from the nucleus we will see the electron less. - [Expression and physics](https://www.hyperfinecourse.org/forums/topic/expression-and-physics-2/) - If we sit inside the nucleus, we will not see the electron sit statistic in one position. We would see the electron can appear inside and outside the nucleus. It can appear anywhere around us if we are only points that do not take any possession of space. If we avarage the positions over time, - [Contradiction](https://www.hyperfinecourse.org/forums/topic/contradiction-10/) - The contradiction may come from ignoring how the extra charge interacts with the electrons outside the nucleus. We only looked at how it attracts the nucleus, which lowers the energy, but in a real atom, it would also repel nearby electrons and that would shift the energy back up. - [Contradiction](https://www.hyperfinecourse.org/forums/topic/contradiction-9/) - In the toy model A we add an electric charge. In the real world we do not add charge but change the charge distribution. The addition of an extra charge gives an increased repulsion, which lowers the energy. If we were to take the charge from the dumbbell, so the overall charge stays the same, - [1s electron](https://www.hyperfinecourse.org/forums/topic/1s-electron-6/) - If you were sitting inside the hydrogen nucleus watching the 1s electron, you wouldnt actually see it as a tiny particle orbiting around you. Instead, it would look like a blurry cloud surrounding you in all directions, sometimes coming very close, sometimes a bit farther, but never staying in one spot. Over time, if you - [What the electron looks like](https://www.hyperfinecourse.org/forums/topic/what-the-electron-looks-like/) - If you could stand on the nucleus and look at the electron without interacting with it. It would look like a huge cloud all around you. They start at a certain distance, get thicker and then decrease in density as far as you can see. - [Loophole to resolve contradiction](https://www.hyperfinecourse.org/forums/topic/loophole-to-resolve-contradiction/) - In toy model A we add an extra negative charge compared to toy model 0. The total energy of the system is shifted negatively compared to tm0 and the monopole shift is -2εC/2l. This could be because we need to take the interactions of the outer negative charges with the negative charge inside the dumbell - [Time-average electron position](https://www.hyperfinecourse.org/forums/topic/time-average-electron-position-2/) - The electron moves with time and does not stay in one specific position, or on 1 specific circular orbit. Instead, it's path is more of a cloud around and sometimes even inside the nucleus. Technically, the electron could go everywhere in space, but there is a cloud, called the orbital of the electron, where it - [1s electron](https://www.hyperfinecourse.org/forums/topic/1s-electron-5/) - If you were sitting inside the hydrogen nucleus watching the 1s electron, you wouldnt actually see it as a tiny particle orbiting around you. Instead, it would appear sort of as a blurry cloud surrounding you in all directions, sometimes coming very close, sometimes a bit farther, but never staying in one fixed spot. Over - [Contradiction](https://www.hyperfinecourse.org/forums/topic/contradiction-8/) - When we introduced a negatively charged distribution in the center, we didn't consider its interaction with the outer negatively charged distribution. This could lead to a negative contribution. - [Time-average electron position](https://www.hyperfinecourse.org/forums/topic/time-average-electron-position/) - The electron does not stay still in one specific position. It moves around. There is a fairly spherical cloud, called the orbital, surrounding the nucleus where the electron is more likely to be. The orbital can overlap with the nucleus. The time-averaged postion of the electron would be the entire orbital in space. - [1s electron](https://www.hyperfinecourse.org/forums/topic/1s-electron-4/) - The time average position of a 1s electron is most of the time out of the nucleus because even though the maximum of the wavefunction is at 0, =0 for r=0. But as the nucleus isn't point like, there is still a probability to find the electron inside the nucleus. =/=0 for r close to - [Toy model](https://www.hyperfinecourse.org/forums/topic/toy-model-8/) - The interaction energy of the electrons if you will (the outer negative charges) depends only on the charge present at the center, but not on how it is distributed (Gauss's law) This means that adding the negative charge in model A will lower the center charge and so lowers the interaction energy. - [Electron position](https://www.hyperfinecourse.org/forums/topic/electron-position/) - The electron would not orbit the nucleus like the earth does the sun. Instead, if you noted where you have seen the electron appear it would form a spherical cloud around you (the nucleus). The most probable position would be the Bohr radius, but in reality the electron can be anywhere in this cloud. - [complications of perturbation theory on non multipole expanded nucleus](https://www.hyperfinecourse.org/forums/topic/complications-of-perturbation-theory-on-non-multipole-expanded-nucleus/) - perturbation theory is, like in the name, an approximation of an unknown system by the assumption that it is a little different from a known system. In our case, is multipole expansion the known system to approximate the charge distribution. iirc, we dont have a "correct" model of the nucleus which means we cannot use - [contradiction](https://www.hyperfinecourse.org/forums/topic/contradiction-7/) - When considering toy model A, we only considered the interaction between the positive charges and the small negative charge at the center, and not between the small negative charge and the ones that represent the electrons, which could have a repelling effect, maybe leading to a positive contribution. - [Observer inside hydrogen nucleus](https://www.hyperfinecourse.org/forums/topic/observer-inside-hydrogen-nucleus/) - If you would imagine yourself being inside of the nucleus of a hydrogen atom, then the 1s electron would not look like it orbits the nucleus (as pictured mostly in high-school text books), but rather a fuzzy cloud-like region where the electron could be at any time. This cloud would be most dense at your - [contradiction](https://www.hyperfinecourse.org/forums/topic/contradiction-6/) - I think the energy lowers because we add a small negative charge in between the positive charges, but for the monopole shift of a nucleus we did not add an extra charge, just considered the overlap between the charge distributions. - [Atoms orientation](https://www.hyperfinecourse.org/forums/topic/atoms-orientation/) - FeI atoms lie at the corners of the cube, so they have three axes 4 fold axis lying along the edges of the unit cell and one 3 fold axis on the diagonal of the cube, this means that Vzz for this nucleus will be 0. In the case of FeII, iron atoms, let's consider - [Quantum toy](https://www.hyperfinecourse.org/forums/topic/quantum-toy/) - In the quantum result for spin I=1 we have a splitting into two levels when Vzz is different from zero these are the levels corresponding to m=+-1 and m=0. Such a case corresponds to the splitting of energy levels for the situation in which the nucleus (two positive charges) in toy model 0 is positioned - [Apparent contradiction](https://www.hyperfinecourse.org/forums/topic/apparent-contradiction/) - If I understand correctly, the orange curve represents the monopole contribution of the model zero, hence it does not take into account the extra charge at the origin, which is present in the model A. The monopole shift would still give a positive correction with respect to the correct monopole contribution, where all the charges - [Time-averaged position of the electron](https://www.hyperfinecourse.org/forums/topic/time-averaged-position-of-the-electron-3/) - If you would sit at the nucleus of the hydrogen atom and observe a 1s electron, you wouldn't see it moving in a continuous trajectory, like a planet around a sun. Instead, imagine you can only locate the electron by taking a picture of your surroundings, and determining from it where the electron was at - [behavior of 1s electron](https://www.hyperfinecourse.org/forums/topic/behavior-of-1s-electron/) - Imagine you close your eyes and each time you open your eyes, a small dot appears where the electron is. After a while, you would see a cloud of dots in a spherical shape. Each dot then represents the position of the electron when you opened your eyes. Most of the dots you see would - [axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-19/) - (1) from the equation on the previous slide only if \eta is equal 0 (2) because the system stays invariant by rotation them along z axis - this is precisely axial symmetry. - [A nuclear with a general shape](https://www.hyperfinecourse.org/forums/topic/a-nuclear-with-a-general-shape/) - 1.Many quantum mechanical problems are unsolvable if there’s no simplifying assumptions. 2.It is difficult to express the full nuclear potential. - [double ring](https://www.hyperfinecourse.org/forums/topic/double-ring/) - 1. For alpha>0，the dumb-bell aligns along the orientation along the principal axis. 2. For alpha=0, the energy is same in free direction. 3. For alpha0. - [Lowest energy configurations](https://www.hyperfinecourse.org/forums/topic/lowest-energy-configurations/) - I would say: for alpha < 0: Dumb-bell along z-axis for alpha = 0: any orientation for alpha > 0: Dumb-bell in the xy-plane - [loophole](https://www.hyperfinecourse.org/forums/topic/loophole-8/) - we only looked at the interaction between the positive and the negative charges. When calculating the energy of the system you also have to take into account that the negative charges will get "closer" or more like they will repell more. This will lead to a less bound system. - [hydrogen nucleus simple](https://www.hyperfinecourse.org/forums/topic/hydrogen-nucleus-simple/) - I would start from the more classical case as the concept of orbitals is rather difficult. You could look at the situation as you sitting on a big ball and there is a small ball moving around the big ball. Sometimes the small ball is really far and sometimes the small ball is inside the - [Complication](https://www.hyperfinecourse.org/forums/topic/complication-2/) - I don't know how you would perform perturbation without expanding in multipoles, as it is with these higher order multipoles that you are making the small (but significant enough) corrections. - [Lowest energy orientation](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-13/) - \alpha > 0 --> dumbbell aligned with z-axis \alpha = 0 --> no quadrupole contribution, maybe have to look in higher orders \alpha < 0 --> dumbbell parallel with xy-plane - [Task, toy to quanta](https://www.hyperfinecourse.org/forums/topic/task-toy-to-quanta/) - For different values ​​of l/d (0.6, 0.2, 0.05) we have different ranges of possible energies (-1.22, -0.6), (-0.832, -0.795), (-0.808, -0.796) respectively. We see that with decreasing size of the nucleus/(distance between central charges in the core) ranges of possible energies shrink, for the extreme case in which l=0 we would have exactly the energy - [Complications](https://www.hyperfinecourse.org/forums/topic/complications-10/) - If we didnt do first a multipole expansion, the problem would become too complex to handle. The nucleus-electron interaction wouldnt have a simple form, which would make it difficult to calculate small corrections. It would be really difficult because monopole expansion allows us to break the interaction into monopole, quadrupole, and without it we wouldnt - [Lowest energy orientations](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientations/) - For alpha = 0 there will be no preferred orientations. For alpha > 0 the biggest pull would come from above and below so the dumbbell would stabilise in the xy plane. For alpha < 0 there would be a strong pull in the sides as well so the dumbbell would stabilize along the z - [complications](https://www.hyperfinecourse.org/forums/topic/complications-9/) - Calculating the energy of the system with an unexpanded hamiltonian will be practically impossible. Performing pertubation theory without a multipole expansion makes it kind of impossible to have a perturbing part to begin with, since this perturbing part is originating from the multipole expansion. - [Lowest-energy orientation](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-12/) - For alpha > 0: theta = 0° or the dumbell is in the xy-plane For alpha = 0: all theta values will give the lowest energy For alpha < 0: theta = 90° or the dumbell is aligned with the z-axis - [Complications](https://www.hyperfinecourse.org/forums/topic/complications-8/) - If we'd consider a general shape, then the possibility of the vectors corresponding to r_< and r_> could swap at any point, meaning an expansion would be very complicated. Also, the interactions between the nucleus and the electron cloud would be very hard to compute analytically (and/or numerically) since we aren't able to split up - [Apparent loophole](https://www.hyperfinecourse.org/forums/topic/apparent-loophole/) - When we introduce the extra negative charge, this charge can create a negative contribution to the energy. However, the overal energy might increase, because this extra negative charge screens the other charges for each other. For example the interaction between the 2 positive charges in the middle will be less because there is a negative - [Lowest energy configurations for different alpha](https://www.hyperfinecourse.org/forums/topic/lowest-energy-configurations-for-different-alpha/) - If alpha > 0: theta = 90° If alpha = 0: independent of theta (all values give the same quadrupole energy contribution, which is 0 If alpha < 0: theta = 0° - [Average position of the electron](https://www.hyperfinecourse.org/forums/topic/average-position-of-the-electron/) - When looking out you see the electron as a cloud. It does not matter in which direction you look, up, down, right, left, etc. everywhere you see the electron. However, you never see it as a point or a ball, just a fuzzy image. It is not directly in front of your face, but also - [complications multipole expansion](https://www.hyperfinecourse.org/forums/topic/complications-multipole-expansion/) - -It would be nearly impossible to solve it without multipole expansion due to the complexity of the potential. - A multipole expansion simplifies complex nuclear shapes by breaking them into smaller, manageable terms, but it only works well if higher-order terms shrink quickly, ensuring the expansion converges. Perturbation theory applies when the unperturbed system's solutions - [Perturbation theory without multipole expansion](https://www.hyperfinecourse.org/forums/topic/perturbation-theory-without-multipole-expansion-2/) - The original (unexpanded) Hamiltonian is quite complex. Perturbing this Hamiltonian is non-trivial, first the corrections might be large. Second, we could perturbe this Hamiltonian in a lot of different ways, thus we need to order these different kinds of perturbations in order of magnitude to correctly use perturbation theory. - [Complications](https://www.hyperfinecourse.org/forums/topic/complications-7/) - Without using the multipole expansion, the complications would be in the definition of an operator able to describle the electric/magnetic field for a random shape of nucleus. - [Lowest energy orientation](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-11/) - For alpha0: the lowest state is when the dumbell is parallel. - [complications without expansion](https://www.hyperfinecourse.org/forums/topic/complications-without-expansion/) - The effects might be too large to do a good perturbative approach. This is also a very complicated expression that makes something already very complex a lot harder. - [energy orientations](https://www.hyperfinecourse.org/forums/topic/energy-orientations/) - alpha>0: lowest energy if theta is 0 degrees alpha=0: There is no quadrupole moment. alpha - [loophole](https://www.hyperfinecourse.org/forums/topic/loophole-7/) - Adding a negative charge in the nucleus makes the attraction between nucleus and electrons smaller, so this can lead to a negative contribution. Maybe the higher order multipole terms can 'lift' this apparent contradiction. - [Lowest energy orientation](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-10/) - For alpha > 0: the lowest energy orientation will be that the axis of the dumbbell is perpendicular to the planes of the rings For alpha > 0: we want the axis of the dumbbell to be parallel to the planes of the rings For alpha = 0: all orientations are good since the quadrupole - [Complications](https://www.hyperfinecourse.org/forums/topic/complications-6/) - Simply just applying perturbation theory will be very hard since the form of the Hamiltonian is very complex. It would also be possible for perturbations to be not small enough for perturbation theory to work. - [Lowes energy orientation](https://www.hyperfinecourse.org/forums/topic/lowes-energy-orientation/) - For alpha > 0, the lowest energy configuration is perpendicular to the rings, for alpha < 0 it is parallel to the planes and for alpha = 0 all orientations carry the same energy since the quadrupole contribution vanishes. - [Perturbation theory without multipole expansion](https://www.hyperfinecourse.org/forums/topic/perturbation-theory-without-multipole-expansion/) - If we take QxV as a perturbation Hamiltonian, it is not necessarily going to be a small perturbation, since it still contains the monopole term. Furthermore, choosing how many terms to retain in multipole expansion allows us to control the amount of detail we want to resolve and reduce the computational cost by discarding higher - [Lowest energy orientation](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-9/) - alpha>0: dumbbell along z-axis; alpha - [time-averaged position](https://www.hyperfinecourse.org/forums/topic/time-averaged-position-4/) - You would not see a tiny ball floating around, but a sort of misty cloud that represents the probability of the electron being found there. It is a spherical cloud. The electron can by mostly found close to the nucleus, but can also appear further away. - [lowest-energy orientation](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-8/) - For alpha > 0, the lowest energy is found when the dumbbell is parallel to the z-axis (previous slide). When alpha < 0, the situation on the previous slide is reversed so now the lowest energy is found when the dumbbell is in the xy-plane. When alpha = 0, the particular configuration of the dumbbell - [Complications without multipole expansion](https://www.hyperfinecourse.org/forums/topic/complications-without-multipole-expansion-4/) - If we don't do a multipole expansion first, the nucleus-electron interaction would be much more complicated to calculate, and possibly even impossible. Multipole expansion allows us to see the different energy level seperations so you can focus on which part you are interested in. Without multipole expansion, this is not possible, and you would only - [Lowest energy orientation](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-7/) - For α = 0, the orientation doesn't matter, as the Quadrupole energy will be zero regardless of orientation. For α < 0, the lowest energy orientation would be a 90° angle with respect to the z-axis, so lying in the xy-plane. For α > 0, the lowest energy orientation is parallel to the z-axis at - [multipole moments](https://www.hyperfinecourse.org/forums/topic/multipole-moments-9/) - An example of multipole moments in science is the fact that gravitational waves are only produced by quadrupole sources, so a perfectly symmetric pulsating star will not give rise to gravitational waves. - [nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-42/) - mass, spin, amount of n and p, structure, charge, lifetime - [minimal quadrupole energy](https://www.hyperfinecourse.org/forums/topic/minimal-quadrupole-energy/) - if alpha =0, it doesn't matter if alpha >0: lowest energy contribution at angle =0 or angle = 180 --> along the z-axis if alpha in the xy plane - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-22/) - hey, my name is Senne. I am a first year master student at the university of Ghent in Belgium. I am really interested in solid state physics and the different methods to characterize materials. So I expect that this course will help me with the research I want to do later and eventually will land - [Task](https://www.hyperfinecourse.org/forums/topic/task-3/) - for the lowest level in the 2D multiplet of the free La atom j=3/2 for the highest level in the 2D multiplet of the free La atom j=5/2 so |l+s|=5/2 and |l-s|=3/2 by adding both we have 2l=4 so l=2 and s=1/2 if we substitute into the equation gJ = 1 + (j(j + 1) - [multiple moments](https://www.hyperfinecourse.org/forums/topic/multiple-moments/) - Water Molecule. The oxygen atom is more electronegative than the hydrogen atoms. - [nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-41/) - charge mass spin lifetime - [Multipole moment in daily life or science](https://www.hyperfinecourse.org/forums/topic/multipole-moment-in-daily-life-or-science/) - The magnetic field of the earth can be seen (first order approximation of the true Earth's magnetic field) as a dipole. - [Role of dipole moments in climate change](https://www.hyperfinecourse.org/forums/topic/role-of-dipole-moments-in-climate-change/) - A very interesting example of the role of a dipole moment I saw recently in another class I'm taking was in the greenhouse gases that trap the Earth's outgoing radiation and emit it back in. The dipole moment plays its role in the absorption of infrared radiation by greenhouse gases because for a molecule to - [Real life example](https://www.hyperfinecourse.org/forums/topic/real-life-example/) - An electrostatic air filter uses statis electric dipoles for catching dust particles. - [origin of multipole expansion calculation](https://www.hyperfinecourse.org/forums/topic/origin-of-multipole-expansion-calculation/) - Why does it matter where you put your origin if you calculate your multipole expansion. Take for example the dipole expansion for the first part of the exercises. If we take the origin at the centre we get Qx= Qy = 0. if we take the origin at the bottom left charge, our Qx becomes - [Monopole radiation](https://www.hyperfinecourse.org/forums/topic/monopole-radiation/) - It was said that there is no monopole radiation for the electro-magnetic field. However, maybe as a pure monopole will not occur, it can still be observed by viewing at the top view of an antenna right? Looking at the top part of the antenna we see an onscillating point charge (if the antenna is - [About Cd-111](https://www.hyperfinecourse.org/forums/topic/about-cd-111/) - The isotope Cd-111 comes in a variety of excited energies. We are interested in 145 kEv. For this specific energy of this isotope, the nuclear magnetic moment is -0.766 nm. The (recommended) electric quadrupole moment is given by +0.64 b. The site is not too difficult to navigate. - [Example of multipole moments](https://www.hyperfinecourse.org/forums/topic/example-of-multipole-moments-8/) - Galaxy mass distributions can be approximated by evaluating a first few terms in the multipole expansion. - [Some things about the nucleus](https://www.hyperfinecourse.org/forums/topic/some-things-about-the-nucleus/) - Very small (femtometer range) Protons, neutrons and gluons Stong and weak force Decay of nucleotides Positive charge Quarks W and Z bosons - [Properties of a nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-a-nucleus-6/) - Number of protons and neutrons, charge, form factor, mass. - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-21/) - Ward Moortgat Leuven, Belgium University of Leuven is a must visit I'm a master student I expect a general understanding of the mechanics of hyperfine interactions My master's diploma - [Nucleus properties](https://www.hyperfinecourse.org/forums/topic/nucleus-properties-4/) - The properties I could think of related to the Nucleus are amount of protons and neutrons, mass of the nucleus, energy, binding energy, electrostatic repulsion. - [moments of 111Cd at 245keV level](https://www.hyperfinecourse.org/forums/topic/moments-of-111cd-at-245kev-level/) - magnetic dipole moment = -0.766(3) nuclear magneton electric quadrupole moment = +0.64(3) barn - [Multipole moment for continous charge distribution](https://www.hyperfinecourse.org/forums/topic/multipole-moment-for-continous-charge-distribution/) - Is it possible to get an example for a calculation of the multipoles for a continous charge distribution? I am not sure how to get the charge distribution for a random continous charge, so an example to see how and how to calculate the integrals would be appreciated. - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-20/) - Hi there, my name is Diego Martínez and I'm taking this course as an exchange student from Colombia in Ghent University so I'm currently based in, well, Ghent. So far I haven't been able to explore much of the city because of the extreme cold but from what I've been able to see so far - [Cd-111](https://www.hyperfinecourse.org/forums/topic/cd-111/) - magnetic: µ=-0.766(3) nm electric: Q=+0.64(3) barn - [Multipole moment example in science](https://www.hyperfinecourse.org/forums/topic/multipole-moment-example-in-science/) - The only example that comes right into my head is the way that gecko's can stick on to walls, which is thanks to (if I remember correctly) the electric dipole moment they create between their legs and the walls. - [Nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-40/) - Life-time, parity, spin, global or individual nucleon behavior, mass, N and Z number, deformation,... - [Nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-39/) - Spin, parity, excitation level, deformation, mass, Z and N numbers, does it show individual nucleon behavior (shell model) or a global behavior (rotation or vibration), life-time,... - [Example of Multipole Moments](https://www.hyperfinecourse.org/forums/topic/example-of-multipole-moments-7/) - Gravitational radiation. Since multipoles can described by spherical harmonics, and gravitational radiation uses spherical harmonics, multipole expansion is a way to describe gravitational radiation. - [First forum](https://www.hyperfinecourse.org/forums/topic/first-forum-13/) - first forum - [First forum](https://www.hyperfinecourse.org/forums/topic/first-forum-12/) - Just trying out my first forum - [First forum](https://www.hyperfinecourse.org/forums/topic/first-forum-11/) - trying out first forum - [moments of 245 keV level of 111Cd](https://www.hyperfinecourse.org/forums/topic/moments-of-245-kev-level-of-111cd/) - magnetic dipole moment of the 245 keV state of 111Cd: µ = -0.7656(25) µ_N nuclear quadrupole moment of the 245 keV state of 111Cd: Q = +0.77(12) barn - [complications](https://www.hyperfinecourse.org/forums/topic/complications-5/) - - lack of small parameter that is needed for perturbation theory - r_e > r_n might be possible such that 1/|r_e-r_n| is too big - complex potentials may not be written as a small perturbation of a simple, solvable system - [Example multipole moment](https://www.hyperfinecourse.org/forums/topic/example-multipole-moment-3/) - The use of a dipole in fluid dynamics to model the flow of a uniform flowing field around a circle/cilinder. - [properties of a nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-a-nucleus-5/) - -Number of nucleons (protons and neutrons) -Spin -Magnetic dipole moment -Electric quadrupole moment -Deformation - [Example of multipole moments](https://www.hyperfinecourse.org/forums/topic/example-of-multipole-moments-6/) - We can consider the earth's magnetic field as a dipole. Locally it could be described by octupole and even higher order terms in the multipole expansion. - [Properties of nuclei](https://www.hyperfinecourse.org/forums/topic/properties-of-nuclei-6/) - number of protons Z number of neutrons N mass number A electric charge eZ constant charge and mass density mass m small radius spin I parity P deformation (beta and gamma from Lund description) K-bands and their energies electric and magnetic moments - [Properties of nuclei](https://www.hyperfinecourse.org/forums/topic/properties-of-nuclei-5/) - number of protons Z number of neutrons N mass number A electric charge eZ constant charge and mass density mass m small radius spin I parity P deformation (beta and gamma from Lund description) K-bands and their energies electric and magnetic moments - [forum test](https://www.hyperfinecourse.org/forums/topic/forum-test/) - test test !!!! - [lowest-energy orientation](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-6/) - alpha > 0 : dumbbell axis along z-axis alpha = 0 : no preference alpha > 0 : dumbbell axis in xy-plane - [Moments of 245 keV 111Cd](https://www.hyperfinecourse.org/forums/topic/moments-of-245-kev-111cd-2/) - Magnetic dipole moment = -0.766 nm (nuclear magneton) and the quadrupole moment = 0.64 b (barn) - [magnetic and dipole moment Cd111](https://www.hyperfinecourse.org/forums/topic/magnetic-and-dipole-moment-cd111/) - magnetic dipole moment = -0.766(3)nm, electric quadrupole moment = (+)0.74(8)b - [Nuclear Properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-38/) - I wrote down: Spin, parity, isospin, angular momentum, multipole moment, magnetic moment, charge, mass, binding energy, #neutrons, #protons, #nucleons - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-19/) - Hi, I'm Stef I'm following this course in my first master year in Leuven but I am not familiar with the city itself even though I live nearby. I hope to obtain more insight on the working of the atom and nucleon-nucleon interactions and I'm curious how the content of this course will connect to - [First forum](https://www.hyperfinecourse.org/forums/topic/first-forum-10/) - biblibli - [First forum](https://www.hyperfinecourse.org/forums/topic/first-forum-9/) - blablabla - [Properties of the Nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-the-nucleus-15/) - The nucleus contains almost all the mass of the atom It consists of protons and neutrons It is held together by the Strong Interaction It can have different shapes (spherical, squeezed, elongated) It has a definite spin and parity defined by the numbers of nucleons (J^P) It has a magnetic moment The stability of the - [Properties of the Nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-the-nucleus-14/) - The nucleus contains almost all the mass of the atom It consists of protons and neutrons It is held together by the Strong Interaction It can have different shapes (spherical, squeezed, elongated) It has a definite spin and parity due to the number of nucleons (J^P) It has a magnetic moment The stability of the - [Properties of the Nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-the-nucleus-13/) - It contains pretty much all the mass of the atom Is held together by the Strong Interaction It consists of protons and neutrons The nucleus can have different shapes The stability of the nucleus depends on the amount of nucleons (odd, even, N=Z, ...) The nucleus has a spin and parity (J^P) defined by the - [Properties of the Nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-the-nucleus-12/) - It contains pretty much all the mass of the atom Is held together by the Strong Interaction It consists of protons and neutrons The nucleus can have different shapes The stability of the nucleus depends on the amount of nucleons (odd, even, N=Z, ...) The nucleus has a spin and parity (J^P) defined by the - [nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-37/) - Charge, mass, number of protons and neutrons, spin - [moments of 111Cd](https://www.hyperfinecourse.org/forums/topic/moments-of-111cd-2/) - μ = -0.766(3) Q = +0.64(3) - [111Cd](https://www.hyperfinecourse.org/forums/topic/111cd/) - nuclear quadrupole moment: 0.74b magnetic dipole moment: -0.766nm - [Application](https://www.hyperfinecourse.org/forums/topic/application-2/) - MRI: magnetic dipole moments in protons in hydrogen - [nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-36/) - protons neutrons mass number atomic number nuclear forces - [Do these work?](https://www.hyperfinecourse.org/forums/topic/do-these-work/) - I do not see the rest of the topics although I already sent in an answer - [test](https://www.hyperfinecourse.org/forums/topic/test-2/) - I forgot to test this first - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-18/) - - I'm Guillaume Smet - I live in Belgium, and I'm taking this course from Ghent University. - Ghent is a very nice city with lots of history and organizes many fun things to visit. The 'Gentse Feesten' are a perfect time for a visit. - I'm a master student from Ghent University. I also - [properties of the nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-the-nucleus-11/) - Amount of protons and neutrons Total mass and charge stability, binding energy electric en magnetic moments - [properties of the nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-the-nucleus-10/) - Amount of neutrons/protons shape and moments stability - [daily life example](https://www.hyperfinecourse.org/forums/topic/daily-life-example/) - Nuclear quadrupole resonance. The quadrupole moment will interact with the electric field. These deformities can be used as a medical imaging technique. This technique is similar to the MRI technique, but there the spins of hydrogen atoms are used. - [moment of Cd-111](https://www.hyperfinecourse.org/forums/topic/moment-of-cd-111/) - Q is +0.64 b The magnetic dipole moment is -0.766 nm - [Properties of nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-nucleus-7/) - Heavy (contains most of the atom's mass), contains neutrons and protons, very densely packed. Moves slow compaired to electrons. - [multipole moments](https://www.hyperfinecourse.org/forums/topic/multipole-moments-8/) - The electric dipole moment helps to understands some molecules like H2O. - [Nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-35/) - -A -Z -mass -spin -parity -Shape (deformation) -life-time -Energy level - [perfectly spherical nucleus](https://www.hyperfinecourse.org/forums/topic/perfectly-spherical-nucleus/) - Yes, because the field generated by electrons is not uniform in the space occupied by the nucleus (for example, look at the electric field generated by a uniformly charged ring). At the center of the nucleus r=0 the field can have a different sign and value than at the surface of the nucleus and at - [This is me](https://www.hyperfinecourse.org/forums/topic/this-is-me/) - Hi, I'm Victor Navarro de la Torre and this is my little "about me" discussion in the forum. I study physics in Leuven and that's also where I am taking this course from. I live in Brussels, but as I have family here in Leuven and I am most of the time here I consider - [First post test](https://www.hyperfinecourse.org/forums/topic/first-post-test/) - bladibladibla test - [First forum test](https://www.hyperfinecourse.org/forums/topic/first-forum-test/) - bladibladibla - [carousel](https://www.hyperfinecourse.org/forums/topic/carousel-9/) - I ask the child to sit on the carousel with a magnet in their hand, I place the magnetometer at the carousel's axis of rotation, as close as possible, then I ask the child to look at the magnetometer reading - in this way I illustrate the contribution of the B dipolar hyperfine field. In - [Task One](https://www.hyperfinecourse.org/forums/topic/task-one/) - (Ia) \hat{\mu}=\frac{g \mu_N}{\{hbar}}\hat{I} expect value of \mu for eigenvalues of I in z dierction we have \mu=\frac{g \mu_N}{\{hbar}}I, then g=\frac{\mu}{\mu_N I}. For Cd111 in ground state I=1/2, \mu=-0.5940 n.m., so g=-0.5940/0.5=-1.188 For Cd111 in 245keV level I=5/2, \mu=-0.766 n.m., so g=-0.766/2.5=-1.188=-0.3064 (Ib) Magnetic moment \mu of an electron is related to its spin and the - [Multipole moments](https://www.hyperfinecourse.org/forums/topic/multipole-moments-7/) - In complex charge distributions, where we then use the multipole expansion to model the field. - [Nucleus properties](https://www.hyperfinecourse.org/forums/topic/nucleus-properties-3/) - Number of neutrons and protons, spin, magnetic and electric moments, excited or not (energy) - [multipole moments](https://www.hyperfinecourse.org/forums/topic/multipole-moments-6/) - 1. A daily life example is that of our earth's gravitational field. It is not perfectly symmetric due to the spinning of the earth. These imperfection can be described using the deformation parameter beta, using the quadrupole moment. - [nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-34/) - Mass, charge, volume, spin, ... - [nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-33/) - Nuclei have a mass, a charge, a volume, an angular momentum... - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-17/) - Hi! My name is Ester Devlieghere, my username is EsterD. I live in Antwerp and study at the KU Leuven. Coming to Antwerp can be interesting for anyone, whether you prefer to be in a museum or at the bar, Antwerp has it all. I did my bachelors of physics in Antwerp, specializing in Condensed - [first forum](https://www.hyperfinecourse.org/forums/topic/first-forum-8/) - test 2 - [first forum](https://www.hyperfinecourse.org/forums/topic/first-forum-7/) - test - [Application](https://www.hyperfinecourse.org/forums/topic/application/) - Magnetic dipole moments in MRI - [nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-32/) - protons neutrons nuclear forces atomic number mass - [hyperfine experiment](https://www.hyperfinecourse.org/forums/topic/hyperfine-experiment/) - 1. Using experimental setup, measure the hyperfine coupling constant a_1 for the isotope witch known I_1 and \mu_1 2. Again measure the hyperfine coupling constant but now a_2 for the isotope witch known I_2 but unknown \mu_2 3. Assume that J and B(0) is the same for both isotope I_1 and I_2, then by dividing - [energy levels](https://www.hyperfinecourse.org/forums/topic/energy-levels-2/) - for J=0 L and S are parallel and pointing in opposite direction for J=1 L and S are perpendicular to each other for J=2 L and S are parallel and pointing in the same direction - [puzzle](https://www.hyperfinecourse.org/forums/topic/puzzle/) - This is because we have a different total charge value, in toy model 0 the total charge was 0, in model A we attracted the -epsilon charge from infinity which required additional work (energy), therefore the binding energy will be higher which results in a shift of the interaction energy towards negative values. - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-27/) - I am in the very center of the nucleus of an atom that has only one proton and electron - hydrogen. Now I will tell you where in such an atom there is an electron. I open and close my eyes as fast as I can, each time I see an electron in a different - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-26/) - Listen up, juvenile. I am in the very center of the nucleus of an atom that has only one proton and electron - hydrogen. I will now tell you where the electron is in such an atom. I open and close my eyes as fast as I can, each time I see an electron in - [properties of nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-nucleus-6/) - mass number of protons Z nuber of nucleons N parity spin quadrupole moment mean squere radius life time - [complications without multipole expansion](https://www.hyperfinecourse.org/forums/topic/complications-without-multipole-expansion-3/) - Without multipole expansion the Hamiltonian (charge-charge) takes the following form \frac{1}{|\Vec{r}_e-\Vec{r}_n|}. In such a case, I would not know what to adopt as the first correction to the Hamiltonian \hat{H}_0, term |\Vec{r}_e-\Vec{r}_n| is the distance of a given nucleon from a given electron and does not give us collective information about the system. - [lowest energy of special configurations](https://www.hyperfinecourse.org/forums/topic/lowest-energy-of-special-configurations/) - alpha>0 the dumbel lies along the z axis alpha=0 the same energy for all orientations alpha - [Properties of the nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-the-nucleus-9/) - * amount of protons * amount of neutrons * mass * spin or angular momentum * lifetime * charge * volume * density - [configuration 3 homework](https://www.hyperfinecourse.org/forums/topic/configuration-3-homework/) - Do you know any trick to calculate the quadrupole moment for distribution 3 (continuous ellipsoidal charge distribution) without counting complicated integrals in spherical coordinates? - [No multipole expansion complications](https://www.hyperfinecourse.org/forums/topic/no-multipole-expansion-complications/) - Without the multipole expansion, you would not know which part is negligible and which part is relevant. On top of that, if you don't know the shape of your nucleus you won't be able to calculate anything. - [lowest-energy orientation](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-5/) - For alpha>0, we saw it in the video : Theta=0° or dumbbell along z axis. For alpha=0, the quadrupole energy is 0 so no correction : all theta configurations have the same energy. For alpha - [Properties of nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-nucleus-5/) - Spin, charge,mass etc - [Properties of nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-nucleus-4/) - Spin , number of protons and Neutron etc - [about me](https://www.hyperfinecourse.org/forums/topic/about-me-16/) - i wish to study hyperfine interaction to understand spectroscopy in more details - [post first forum](https://www.hyperfinecourse.org/forums/topic/post-first-forum-3/) - really excited - [first forum](https://www.hyperfinecourse.org/forums/topic/first-forum-6/) - really excited - [Paper](https://www.hyperfinecourse.org/forums/topic/paper-3/) - 1 - 8 There is only one step left to understand the shape of spectra, measured with a frozen enzyme solution. It has to do with one of the selection rules in EPR, namely that only the magnetic moments from the sample in the direction of the external field (to be more precise: perpendicular to - [Recording data](https://www.hyperfinecourse.org/forums/topic/recording-data/) - To record such a spectrum the sample has to be systematically analyzed. Meaning in small angular steps the sample has to be rotated, after each measurement. For each measurement by reduction of intensity, the g factors can be determined, - [First task](https://www.hyperfinecourse.org/forums/topic/first-task-4/) - 1-a: I believe since nuclear decay is modeled by an exponential decay relating material remaining to the starting quantity, and half-life is the time it takes to reach half of the original material, it makes sense to me that lifetime would be related to this and that the range of possible decay lifetimes would also - [answer](https://www.hyperfinecourse.org/forums/topic/answer-24/) - I don't know the answer to this question. - [NMR/ON Hyperfine Field Sign](https://www.hyperfinecourse.org/forums/topic/nmr-on-hyperfine-field-sign/) - I notice it takes less energy to make the transition in stronger externally applied fields. This doesn't seem to have anything to do with the sign though... I'm looking back at my NMR notes and see deltaE = g*B*mu_N I also have written that the g factor is the relation between the spin vector and - [answer](https://www.hyperfinecourse.org/forums/topic/answer-23/) - 1. If an initial population in an excited state is not replenished , and the probability of emission from the excited state is proportional to the amount of nuclei in this state, then we expect the number of emission events per unit to decrease exponentially. 2.Supposedly 1-3 is in line with the nuclear spin and - [Answers](https://www.hyperfinecourse.org/forums/topic/answers-3/) - From the image, it can be deducted that: The nucleus has spin 1, that splits into three levels +1,0,-1 . The hyperfine field is unequeal to zero as a splitting is occuring. It is not possible for me to deduct if the nucleus is inside a material. - [First task](https://www.hyperfinecourse.org/forums/topic/first-task-3/) - 1) The lifetime of the excited states follow an exponential decay 2) Detector combination 1-3 are located parallel to the direction of the nuclear spin, causing to observe more detections than the perpendicular direction of combination 1-2. - [NMR/ON](https://www.hyperfinecourse.org/forums/topic/nmr-on-7/) - In the figure, we see that the resonance frequency decreases, as the external magnetic field increases. This indicates that the sign of the hyperfine field of Sr in iron is negative. - [exponential decay question](https://www.hyperfinecourse.org/forums/topic/exponential-decay-question/) - 1a) Because the nucleus experiences a nuclear decay from state 1 to state two. After one halftime has passed, there is a 50 percent chance that the nucleus has decayed. After one more halftime, there is a 75 percent change that it has decayed and so on. This means that right after the first decay, - [NMR/ON](https://www.hyperfinecourse.org/forums/topic/nmr-on-6/) - I believe that the shift of the resonance frequency for an increasing external magnetic field has something to do with this, but I can't tell what the sign would be for a frequency shift to the left. - [Mößbauer Train Problem (MTB](https://www.hyperfinecourse.org/forums/topic/mosbauer-train-problem-mtb/) - The relative velocity has to be 341m/s. Therefore, it is not possible to keep for several time this velocity constant, due to space issues. One possible solution would be to take two absorbers/scatterers and but them in series. So the photons have to travel through both of them. Now you could shake them contradicting to - [PAC](https://www.hyperfinecourse.org/forums/topic/pac/) - 1) Because the lifetime of the excited state follows an exponential curve behavior. 2) The probability of gamma ray emission is higher along the direction of the nuclear spin, which coincides with detectors 1 and 3 alignment, therefore the number of events detected will be higher than for detectors 1 and 2. - [axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-18/) - First, two entries of the quadrupole tensor are the same. Second, the electrons lye in the same z plane with mirror symmetry - [NMR/ON](https://www.hyperfinecourse.org/forums/topic/nmr-on-5/) - The picture shows how the frequency of the resonance peak increases as the intensity of the external magnetic field decreases, this inversely proportionality indicates that the hyperfine field of Sr is negative with respecto to the direction of the external field. - [EPR on molecules and crystals](https://www.hyperfinecourse.org/forums/topic/epr-on-molecules-and-crystals-2/) - To replicate spectra similar to those observed, scientists employ a systematic procedure. First, the sample is loaded into the EPR spectrometer mounted on a rotating support. Then, a magnetic field is applied, and the sample is irradiated with microwaves, with data collection commencing. Subsequently, the sample is incrementally rotated by a set number of degrees, - [paper](https://www.hyperfinecourse.org/forums/topic/paper-2/) - Page 7 : in a low viscosity solution, all of this anisotropy is averaged out. I don't understand immediatly without trying to make it very explicit, the relation between the viscosity of the solution and the 'averaging out'. How do i relate the viscosity (that I only remember by the ability to slide parallel plates - [La free atom](https://www.hyperfinecourse.org/forums/topic/la-free-atom/) - When a free lanthanum atom, containing unpaired electrons, absorbs a microwave photon matching the electron paramagnetic resonance (EPR) condition, the unpaired electrons transition between their spin states. This transition, dictated by the external magnetic field strength and the atom's g-factor, leads to the detection of an EPR signal. - [photon_energy](https://www.hyperfinecourse.org/forums/topic/photon_energy/) - Assuming a nuclear g-factor g=1, a nuclear spin I=1/2 and applied field B0=2 T, determine the photon energy for photons that will be absorbed in the case without hyperfine interaction : delta_E = 0.15 meV --- A = 0.21\mueV Transitions between the former mJ=-1/2 and -1/2 levels, delta_E(m_i=+1/2,+1/2) = A*1/2*/1/2 = 0.05 \mueV Transitions between - [lanthanium EPR](https://www.hyperfinecourse.org/forums/topic/lanthanium-epr/) - Due to the B-field the electronic energy levels will split up, zeeman effect. An atom in a certain energy eigenstate (with corresponding quantum numbers) will on absorption (not interaction, because then we could also have stimulated emission) transition to a higher energy eigenstate. - [Short-Short introduction.](https://www.hyperfinecourse.org/forums/topic/short-short-introduction/) - I became a little confused by the end of page 8. It states that the spectrum in figure 7 is the sum of all the orientations in the frozen sample, and that seems logical. But I'm not sure as to why the x and y is preferred in this example. My reasoning is that it - [EPR experiment](https://www.hyperfinecourse.org/forums/topic/epr-experiment-2/) - EPR works by letting the sample undergo different orientations of the external magnetic field. The sample will thus be measured multiple times, each time with a different orientation of the magnetic field (by using a rotating support). Then it is each time radiated by an appropriated microwave frequency. From the peaks of these spectra, the - [EPR introduction](https://www.hyperfinecourse.org/forums/topic/epr-introduction/) - I don't entirely understand figure 7. - [EPR](https://www.hyperfinecourse.org/forums/topic/epr-3/) - The sample needs to be measured multiple times, while changing the orientation of the external magnetic field. It needs to be exposed to the appropriate microwave photons. This will result in an EPR spectrum, from which the g-factors can be determined. - [Second task: Photon energy](https://www.hyperfinecourse.org/forums/topic/second-task-photon-energy/) - First we calculate the energy without hyperfine interaction: for the transition energy, we calculate the separate energy levels E(mJ=-1/2) and E(mJ=-3/2) E(m=-1/2) = -g * mu_N * B * mJ (with g = 1, mu_N = 3.15e-8 eV/T, B = 2T, mJ=1/2) E(mJ=-1/2) = 3.15e-8 eV Same for E(mJ=-3/2) gives 9.45e-8 eV, so we got - [EPR on free La atom.](https://www.hyperfinecourse.org/forums/topic/epr-on-free-la-atom/) - If the La atom absorbs a photon, the spin orientation of the electrons shifts, thus moving the state to a higher energy. - [transition energies](https://www.hyperfinecourse.org/forums/topic/transition-energies-2/) - For the non-hyperfine interaction part: E(J) = -g*mu_N*B0*m_J => delta E = E(-3/2) - E(-1/2) = -g*mu_N*B0(-3/2+1/2) = 1*1*5.788*10^-4 eV/T * 2T = 0.12 meV. For the hyperfine interaction part: E(J) = -g*mu_N*B0*m_J + A*mI*mJ = -g*mu_N*B0*m_J + (g*mu_N*B_hf)/J * mI*mJ for mJ=-3/2 and -1/2: delta E = 0.12 meV + mI*mJ*g*mu_N*B_hf(-2/3 + 6/3) = - [my answer](https://www.hyperfinecourse.org/forums/topic/my-answer-11/) - I believe that the electron cloud will change its orientation compared to the nucleus. - [EPR Short-short introduction](https://www.hyperfinecourse.org/forums/topic/epr-short-short-introduction/) - I start to get a little confused with the discussion of the powder spectrum, but I think I've justified all the statements to myself. Then the orientations start to get confusing on page 8. It reminds me of our discussion of the orientations of nuclear ensembles, but seems to be different. Smaller things that confused - [EPR multiple orientations](https://www.hyperfinecourse.org/forums/topic/epr-multiple-orientations/) - I imagine the scientists follow a procedure roughly as follows: 1. Load sample into EPR spectrometer on a rotating support. 2. Apply magnetic field, irradiate with microwaves and collect data. 3. Increment the rotation of the sample by some number of degrees and repeat step 2. 4. Collect data over 360 degrees, maybe again rotating - [Answer](https://www.hyperfinecourse.org/forums/topic/answer-22/) - 1-a) Exponential decay is observed due to the characteristic lifetime of excited states, commonly seen in processes like radioactive decay. 1-b) The fewer events/counts in detector combination 1-2 compared to 1-3 stem from the orientations relative to the nuclei. Detector 1, parallel to the nuclei, has high detection probability. Detector 2, perpendicular to both nuclei - [Answer](https://www.hyperfinecourse.org/forums/topic/answer-21/) - The absorption frequency decreases with increasing magnetic field strength in NMRON: This would indicate a negative sign of the hyperfine field for the oriented nuclei. It suggests that the local magnetic field experienced by the nuclei is oriented opposite to the direction of the external magnetic field. - [Transition Energies](https://www.hyperfinecourse.org/forums/topic/transition-energies/) - Without Hyperfine Interaction, the difference in energy between two levels =-g_J*mu_B*B_0 So the difference is -2*mu_B, -2 times the Bohr magneton With Hyperfine Interaction, there is a correction to the energy of each level = A*m_I*m_J where A =(g * mu_N * B_hf)/J so for the transition from -3/2 to -1/2, assuming m_I = 1/2, - [EPR on La](https://www.hyperfinecourse.org/forums/topic/epr-on-la/) - I had written "I think electrons are moving into different suborbitals without transitioning between shells". I figured that the different m_J levels represented different arrangements of electrons in orbitals and suborbitals and their spins. Since the spins can't change, and a change in orbital would mean a larger energy transition, I figured the only thing - [answer](https://www.hyperfinecourse.org/forums/topic/answer-20/) - Since the energy is lowest for a nucleus pointing along the positive z-axis (m=+1) the B-field is applied along the positive z-axis. If you take a sample consisting of many nuclei, at 'low' temperature most nuclei will be parallel to the B-field, in the (m=+1) state. - [Answer to questions](https://www.hyperfinecourse.org/forums/topic/answer-to-questions/) - From the I=1 state, we got three different magnetic hyperfine levels, with state ml = +1 the lowest > ml = 0 > ml = -1. From given picture, we also see that each quantumstate ml has a different orientation. For a low temperature, the lowest energy levels are most occupied (so for ml = - [temperature and radiation](https://www.hyperfinecourse.org/forums/topic/temperature-and-radiation-3/) - 1) We know that the nucleus has spin I=1, therefore there are three posible magnetic quantum states: ml= -1, 0 ,+1. Each of those corresponding to a specific orientation of the spin with quantized energy levels. 2) Low temperature means means low energy, thus most nuclei will be at the lowest energy state, corresponding to - [Ferry nvm magnetic wheel](https://www.hyperfinecourse.org/forums/topic/ferry-nvm-magnetic-wheel/) - In this case the children would be the electron moving around the center of the ferris wheel, where I would place the magnetometer. The bar magnet can be used to simulate the spin. While the charged ball can be used to create a current-> Magnetic field. All the fields can be detected by the magentic - [task 2](https://www.hyperfinecourse.org/forums/topic/task-2-17/) - By measuring again the different energy levels, one can divide the first result by the known µ and then look which µ is now required to match the different levels. Best, Hannes - [L, S and J](https://www.hyperfinecourse.org/forums/topic/l-s-and-j/) - When L and S are parallel J = 2, perpendicular J = 1, opposing J = 0. This is easy seeable when thinking of vector addition. - [Paper](https://www.hyperfinecourse.org/forums/topic/paper/) - The graph in figure 10 I don't quite get how to interpret it. - [my answer](https://www.hyperfinecourse.org/forums/topic/my-answer-10/) - Because I = 1, the nucleus is in an excited state. I = 1 result in 3 different magnetic hyperfine levels. I believe that an ensemble of these nuclei at a low temperature will be an aligned ensemble with a different probability of pointing up or down, because at a low temperature, the lowest energy - [the two questions](https://www.hyperfinecourse.org/forums/topic/the-two-questions/) - I did not read the questions before watching the video. The video asked what we can tell about the nucleus and the environment from the given picture. I did not think about spin orientation. I thought only about what the energy splitting could tell me. Given I=1, and no degeneracy between +1 and -1, that - [EPR](https://www.hyperfinecourse.org/forums/topic/epr-2/) - To create these kinds of spectra, you need to spin the sample at different angles inside the spectrometer's magnets while using the right microwave frequency. Then gradually increase the strength of the magnetic field applied to the sample. Peaks show up in the spectra when the magnetic field reaches a certain strength, causing an EPR - [Nuclear Resonant Scattering](https://www.hyperfinecourse.org/forums/topic/nuclear-resonant-scattering/) - Nuclear Resonant Scattering is a hyperfine spectroscopy method utilizing synchrotron radiation for material analysis. This method takes advantage of the Mössbauer effect, which can be summarized as the absorption and emittance of photons by atoms within a crystalline structure without energy lost to recoil. The photons of the synchrotron radiation can be tuned to very - [NRS wikipedia](https://www.hyperfinecourse.org/forums/topic/nrs-wikipedia/) - Nuclear resonant scattering (NRS) is a specialized spectroscopic technique used to study the properties of atomic nuclei within materials. By exploiting the resonant absorption of synchrotron radiation by specific isotopes, NRS provides unique insights into the local environment, dynamics, and electronic structure of the target nuclei. This method finds application in various scientific disciplines, including - [Description of 1 s electron over time](https://www.hyperfinecourse.org/forums/topic/description-of-1-s-electron-over-time/) - Looking from the nucleus, is moving/ jumping around the nucleus with about 10% the speed of light. In average without any further spin spin interactions the electron will stay at a certain distance away from you, as nothing is affecting the electron. Almost like the moon rotates around the earth. However, if you average the - [NRS article](https://www.hyperfinecourse.org/forums/topic/nrs-article/) - Synchrotron radiation has a wide energy spectrum. The hyperfine levels of an exposed nuclei will excite at the same time. These hyperfine levels then decay at slightly different rates, because of their different frequencies. The superposition of these frequencies result in quantum beats. By taking the Fourier transform of these quantum beats, it is possible - [NRS](https://www.hyperfinecourse.org/forums/topic/nrs-2/) - Nuclear resonant scattering uses synchrotron radiation to study materials' properties by analyzing interactions with atomic nuclei. It's based on the Mössbauer effect, probing hyperfine splitting of nuclear levels via gamma radiation absorption. Synchrotron radiation broadens energy spectra, allowing simultaneous excitation of resonances. This produces a beat pattern during decay, revealing nuclei-sample interaction parameters. Synchrotron brilliance - [General shape nucleus PT](https://www.hyperfinecourse.org/forums/topic/general-shape-nucleus-pt/) - I think that the Hamiltonian describing such a system would be highly complex because the expression to describe interactions between electrons and nucleus (potential energy term) doesn't simplify at all if the multiple expansion is not applied. Also, in that regard, the complexity of the Hamiltonian doesnt allow us to discern which effect is more - [Minimization of the energy](https://www.hyperfinecourse.org/forums/topic/minimization-of-the-energy/) - For alpha > 0, the prefactor is positive and the orientation that minimizes the energy is with the dumbbell along the z-axis. For alpha < 0, the situation is the opposite, being the prefactor now negative, the orientation lying over the x-y plane minimizes the energy. For alpha = 0, the prefactor being 0, vanishes - [MP Expansion](https://www.hyperfinecourse.org/forums/topic/mp-expansion/) - Describe the complications you would run into if you would try to use perturbation theory to study a system with a nucleus of general shape without having made a multipole expansion first. The multipole expansion simplifys the complicated shape of the distribution. Otherwise it would not be possible to solve for any given distribution the - [LO Orientation](https://www.hyperfinecourse.org/forums/topic/lo-orientation/) - If α=0 all the orientation does not matter as the Quadrupole vanishes. If α>0 -> θ=0,180° minimizing the term If α θ= 90, 270° minimizing the term - [second task](https://www.hyperfinecourse.org/forums/topic/second-task-4/) - In the absence of hyperfine interactions, the energy levels of a lanthanum atom under an external magnetic field can be calculated straightforwardly. Each energy level E(mJ) depends on the magnetic quantum number mJ​ according to the formula E=−gμNB0mJE=−gμN​B0​mJ​, where g is the Landé g-factor, μN​ is the nuclear magneton, and B0​ is the external magnetic - [EPR La](https://www.hyperfinecourse.org/forums/topic/epr-la/) - When an atom is irradiated and absorbs a photon, this energy is stored in an opposing spin wrt the magnetic field. So the electron will go from a lower to a higher energy state (e.g. Mj: 1/2 -> Mj:-1/2) - [answer](https://www.hyperfinecourse.org/forums/topic/answer-19/) - If the magnetic field strength at the nucleus increases similarly to the zeeman effect, I expect the level splitting to increase. It doesn't quite seem to explain why the existing peaks at 50K get deeper at 18K because one would expect them to shift. Why the field strength at the nucleus increases at lower temperatures - [nuclei on a train](https://www.hyperfinecourse.org/forums/topic/nuclei-on-a-train/) - 1. So taken literally I imagine a vessel containin a gas with our nuclei of interest on a track moving at the speed of sound. This seems rather cumbersome given that the world record fastest train is 460km/h (maglev China) < 1234.8 km/h speed of sound . One could argue that's a stationary target, however - [Temperature change between 50 and 18 K](https://www.hyperfinecourse.org/forums/topic/temperature-change-between-50-and-18-k/) - The hyperfine field between the two temperatures looks to be from the same magnitude, so temperature will not affect a change in the hyperfine field. However, due to lowering the temperature, the magnetic moments of the nuclei will become more aligned to the magnetic field, resulting in a more noticeable mossbauer effect. - [Hyperfine Field Interaction](https://www.hyperfinecourse.org/forums/topic/hyperfine-field-interaction/) - It looks like less scattering is occurring at 50K than at 18K, so maybe there is more degeneracy at 50K than at 18K. I would think the hyperfine interaction must be stronger then, at 18K, so the field must be stronger. Are the electrons moving faster? The charge currents in the nucleus? The nucleus itself - [spectrum](https://www.hyperfinecourse.org/forums/topic/spectrum-2/) - As the temperature drops from 50 K to 18 K, more nuclei within the sample start aligning with the magnetic field. This alignment makes the hyperfine lines in the spectra stronger because it increases the magnetic field strength within the sample. Interestingly, even though the magnetic field strength changes, the appearance of the hyperfine lines - [train](https://www.hyperfinecourse.org/forums/topic/train/) - Conducting a resonant scattering experiment on a fast-moving train presents several practical challenges. The train's motion can cause vibrations, making it difficult to maintain stability in the experimental setup, while aligning the source, target, and detectors becomes more challenging with rapid movement.To overcome these challenges, innovative solutions such as advanced stability technology, remote control operation, - [Source on fast train](https://www.hyperfinecourse.org/forums/topic/source-on-fast-train/) - I think one of the main difficulties in this problem is accelerating the source to a speed similar to the speed of sound in air. For this, it is useful to first use charged ions during the acceleration. - [g-factor](https://www.hyperfinecourse.org/forums/topic/g-factor-6/) - online calculator doesn't work - [Train Experiment](https://www.hyperfinecourse.org/forums/topic/train-experiment/) - My first thought is that the distances would be quite limiting. I would think you would need a large 'run-up' distance compared to the distances involved in the measurement, but I don't really have any familiarity with these types of measurements and the maximum effective distances. And you wouldn't have very much data collection time, - [PAC first task](https://www.hyperfinecourse.org/forums/topic/pac-first-task/) - 1) The decay is exponential because it stems from the energy levels lifetime, which is an exponential decay 2) The different count rate is the result form the other orientation of the nucleus, thus a different emission probability - [NMR/ON](https://www.hyperfinecourse.org/forums/topic/nmr-on-4/) - The higher the field, the more isotropic the B-ray's are emmited. This means that the hyperfine field works against the direction of the magnetic field, making it negative - [First task PAC spectroscopy](https://www.hyperfinecourse.org/forums/topic/first-task-pac-spectroscopy/) - Why do you observe an exponential decay ? Because this is the behavior of the lifetime of the excited states. Radioactive decay appears in an exponential behavior in time. Why is the number of events/counts for detector combination 1-2 smaller than for 1-3 ? Because the nuclei detector 1 sees are oriented parallel to this - [NMR/ON](https://www.hyperfinecourse.org/forums/topic/nmr-on-3/) - So in iron, one Fe is replaced by a Sr. With NMR/ON, one can see what the hyperfine field is that this Sr nucleus feels in iron by looking at where we see a reduction, in other words, where we see a change in orientation of the ensemble. I think the sign of the hyperfine - [Paper “short-short introduction to EPR spectroscopy“](https://www.hyperfinecourse.org/forums/topic/paper-short-short-introduction-to-epr-spectroscopy/) - At the end of p7 and begin of p8 I start to be a little lost. - [EPR experiment](https://www.hyperfinecourse.org/forums/topic/epr-experiment/) - To get a spectrum like this, you need an experimental set-up with a generator of microwaves. The externally applied magnetic field is tuned and the microwaves stay the same frequency. You then search for the magnetic field that gives you the EPR transitions. This is then done for several angles over which you turn your - [Change in photon energy](https://www.hyperfinecourse.org/forums/topic/change-in-photon-energy/) - Without hyperfine interactions we have: E=-g*mu_N*B_0*m_J with mu_N=3.15*10^-8eV/T so we have E(-3/2)=-1*3.15*10^-8 eV/T * 2 T * -3/2 = 9.45*10^-8 eV E(-1/2)=-1*3.15*10^-8 eV/T * 2 T * -1/2 = 3.15*10^-8 eV E(+1/2)=-1*3.15*10^-8 eV/T * 2 T * +1/2 = -3.15*10^-8 eV so E(m=-3/2)-E(m=-1/2)= 6.3 * 10^-8 eV and E(m=-1/2)-E(m=+1/2)= 6.3 * 10^-8 eV With hyperfine - [EPR of free La atom](https://www.hyperfinecourse.org/forums/topic/epr-of-free-la-atom/) - When an external magnetic field is applied to a free atom and we radiate it with a microwave photon that satisfies the EPR resonance condition, we cause a transition in the Zeeman multiple. The absorption thus changes the spin orientation of the free atom so the system is brought to a higher level. - [my answer](https://www.hyperfinecourse.org/forums/topic/my-answer-9/) - I believe that the field stays the same, but because of the lower temperature, the mössbauer effect causes an increase in absorption for the absorber, so that the hyperfine field is more visible. - [answer to train problem](https://www.hyperfinecourse.org/forums/topic/answer-to-train-problem/) - I think the most impact on the train acceleration would be the air resistance, so the train should be in a vacuum chambre so that it can accelerate more easily. However, this is assuming that the track is linear, it would be more easy to put the train on a centrifuge in my oppinion, because - [Experimental EPR pictures](https://www.hyperfinecourse.org/forums/topic/experimental-epr-pictures-2/) - To achieve spectra like these, researchers must rotate the sample at various angles while positioned within the spectrometer's magnets and exposed to a suitable microwave frequency. The measurement involves applying a magnetic field to the sample and then gradually increasing its strength across a range of values. Peaks appear in the resulting spectra whenever the - [Photon energy](https://www.hyperfinecourse.org/forums/topic/photon-energy-3/) - mu_N = 3.15 *10^-8 eV/T B = 2T The transition energie delta E is calculated as: E(mJ = -3/2) – E(mJ= -1/2) With: E(-3/2)= -g * mu_N * B * (-3/2) = -1 * 3.15 * 10^-8 eV/T * 2T * (-3/2) = 9.45 * 10^-8 eV E(-1/2)= -g * mu_N * B * (-1/2) - [EPR](https://www.hyperfinecourse.org/forums/topic/epr/) - When a free Lanthanum (La) atom is placed in an external magnetic field, its electrons can absorb microwave photons of a specific frequency.The absorption of the microwave photon will cause the electron to change its spin orientation and move to a higher energy level. - [answer](https://www.hyperfinecourse.org/forums/topic/answer-18/) - (1) the lowest energy state corresponds to m=+1, indicating a nucleus with a spin orientation aligned in the positive z-direction. For m=0, the orientation lies in the z=0 plane, while for m=-1, the nuclear spin points in the negative z-direction. (2) In low-temperature conditions, the system primarily occupies low-energy states. Consequently, most nuclei will be - [NRS wiki](https://www.hyperfinecourse.org/forums/topic/nrs-wiki/) - In contrast to conventional MS methods like Conversion Electron Mössbauer Spectroscopy (CEMS), which measure hyperfine interactions via incoherent processes like resonance fluorescence or internal conversion, NRS harnesses the coherent superposition of scattering probability amplitudes from an ensemble of nuclei simultaneously excited by a synchrotron radiation pulse. NRS exploits synchrotron radiation to effectively perform the Fourier - [orientation: temperature and radiation](https://www.hyperfinecourse.org/forums/topic/orientation-temperature-and-radiation/) - (1) The hyperfine interaction tells us that the lowest energy state is m=+1 and this will correspond to a nucleus with a spin orientation pointing in the positive z-direction. The m=0 state will give an orientation in the z=0 plane and a m=-1 will give a nuclear spin in the negative z-direction. (2) At low - [answers](https://www.hyperfinecourse.org/forums/topic/answers-2/) - There are 3 hyperfine levels, m = +- 1 and m = 0. From this we also know that the nucleus has a spin of 1. At low temp the lowest energy level will be taken, so the orientation will be in the negative z direction. - [NRS](https://www.hyperfinecourse.org/forums/topic/nrs/) - NRS works because of the hyperfine splitting just like mossbauer spectroscopy, but is different with respect to it in that the synchrotron radiation contains a very broad range of energies. This will then in turn make it so that al transitions between nuclear levels are excited simulaneously by the pulses at t=0. The subsecuent decay - [spectra temperature](https://www.hyperfinecourse.org/forums/topic/spectra-temperature/) - Inside the sample, as the temperature decreases from 50 K to 18 K, more and more magnetic moments align in the direction of the magnetic field. This alignment leads to an increase in the overall magnetic field strength within the sample, as evidenced by the increase in hyperfine field observed in the Mössbauer spectra. - [resonant scattering on a train](https://www.hyperfinecourse.org/forums/topic/resonant-scattering-on-a-train/) - When you'd want to do this experiment it will be difficult to have a calibrated and aligned setup since everything is moving and vibrating a lot. Maybe using a smaller setup with a very fast moving object, maybe from a railgun so you can control the speed, that pushes the source out at a contstant - [spectrum](https://www.hyperfinecourse.org/forums/topic/spectrum/) - I think the magnetic field is smaller at 50K, when lowering the temperature the atoms become more and more aligned resulting in a higher overal field at 18K. I do also notice that the big peak at 50K splits when lowering the temp. - [Wikipedia NRS](https://www.hyperfinecourse.org/forums/topic/wikipedia-nrs/) - Nuclear Resonant Scattering (NRS) is an experimental method in material research. With NRS one can perform spectroscopy but by looking at the time domain of light pulses instead of the energy domain like the Mössbauer spectroscopy. As coherent light falls onto an atom (or crystal), all the hyperfine resonances get excited at the same time. - [fast source](https://www.hyperfinecourse.org/forums/topic/fast-source/) - Accelerating the source could be a problem, one would need to use ions. - [Mossbauer spectrum](https://www.hyperfinecourse.org/forums/topic/mossbauer-spectrum/) - I think the hyperfine field you see beginning at 50 K is smaller than the one at 18K. I think that if you lower the temperature from 50 to 18 K, you see the magnetic fields of the atoms in the material are becoming more and more aligned and that is why the hyperfine peaks - [Speed of sound source](https://www.hyperfinecourse.org/forums/topic/speed-of-sound-source/) - If you want the source to travel at a speed like the speed of sound in the air, you would have difficulty accelerating this source since you are dealing with neutral atoms (I think). One can maybe overcome this by accelerating ions, letting them collide with a certain material so they pick up a certain - [Axes](https://www.hyperfinecourse.org/forums/topic/axes/) - I do not understand what is meant by #-fold axes - [A Nearly-Classical Quantum Model](https://www.hyperfinecourse.org/forums/topic/a-nearly-classical-quantum-model/) - An EFG that is negative along the z-axis is needed Other than this, I believe the only difference is that quantum states have a limited number of specific possible values. I suppose if a nucleus has a very large spin number, there are more possible orientations, which would look more and more like the continuous - [PAS](https://www.hyperfinecourse.org/forums/topic/pas-2/) - For the Fe-I atoms, there are three 4-fold rotational axis. Thus, by theorem 1, any of these three rotational axis are the z-axis of a PAS with η=0. By theorem 2, since there are more than two 4-fold rotational axis, the EFG tensor must be zero. For the Fe-II atoms, we have two 2-fold rotational - [Quantum to Classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-16/) - For a classical system, there is a continuous set of states that can describe the quadrupole interaction. However, in a quantum state, we only have a discrete set of states to work with. In the case of l=1, we have 3 states: 1, 0, and -1. If we increase l to some arbitrary value n, - [Axial Symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-17/) - From the equation, the EFG tensor given in the video has axial symmetry in the PAS because V_xx and V_yy terms in the tensor are equal to each other (i.e. η = 0). From the visual inspection of the toy model, the only part that matters is the electrons. Since the electrons lie on the - [From toy model to quantum](https://www.hyperfinecourse.org/forums/topic/from-toy-model-to-quantum-7/) - Observing the vertical axis, as the nucleus grows smaller, the quadrupole term becomes smaller as well. - [Task 2](https://www.hyperfinecourse.org/forums/topic/task-2-16/) - FeI has three 4-fold rotation axes. The EFG tensor is zero. FeII has one 4-fold and two 2-fold rotation axes. The z-axis can be axis of the PAS and eta=0. - [Quantum to Classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-15/) - In classical system we have a continuous distribution however in a quantum system we have quantization. If we split the system into many levels, we can achieve a continuum which is very similar to the classical system. - [Quadrupole Operator](https://www.hyperfinecourse.org/forums/topic/quadrupole-operator-3/) - From equation: With only diagonal term in the matrix Vxx and Vyy are equal. From Picture : The system is doesn't change under rotations of the x-y plane, there exists axial symmetry. - [Toy Model to Quantum](https://www.hyperfinecourse.org/forums/topic/toy-model-to-quantum-3/) - As the nucleus becomes smaller the quadrupole term should also decrease. - [Symmetry axes](https://www.hyperfinecourse.org/forums/topic/symmetry-axes-3/) - Fe-I: 3 times fourfold symmetry. Hence, the EFG is zero for this atom. Fe-IIa: 3 times twofold symmetry. Eta will be nonzero and there is no axial symmetry in the EFG. Fe-IIb: 1 time fourfold symmetry, and 2 times twofold. The PAS has a z-axis parallel to the 4 fourfold symmetry axis, pointing in the - [Quantum approaches classical](https://www.hyperfinecourse.org/forums/topic/quantum-approaches-classical/) - We would like to go to a continuum of state, instead of the discrete levels in the quantum situation. Hence, we need many different levels that are spaced closely to each other. This could be reached by having a nucleus with a very large spin I (and many m_z, preferably integer), having a small, nonzero - [classical](https://www.hyperfinecourse.org/forums/topic/classical/) - For a classical interpretation the energy is not degenerated, thus the state spacing has to be zero or very small. - [axial](https://www.hyperfinecourse.org/forums/topic/axial-2/) - In the equation, since Vxx and Vyy terms exhibit identical behavior, the symmetry parameter eta is determined to be zero. Observing the picture, as the charges inducing the EFG lie along the principal axis, the behavior within the xy-plane remains consistent regardless of location within it. - [toy model to quantum](https://www.hyperfinecourse.org/forums/topic/toy-model-to-quantum-2/) - Quadrupole term becomes smaller as the size of the nucleus decreases - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-16/) - 1) If the tensor is diagonalized, this implies V_xx=V_yy and V_xy=0, so clearly symmetry around the z-axis. Both the x and y components are 'equal'. 2) By inspecting rotation along the z-axis, it is clear that the energy of the system is not altered since the distances between all charges remain the same. - [Rotation axes Fe4N](https://www.hyperfinecourse.org/forums/topic/rotation-axes-fe4n/) - Fe1-atom: We see 3 4-fold rotation axes. Due to the first theorem one of these axes (x, y, z) can be chosen as the z-axis of the PAS, with eta 0. From the second theorem, we get that the EFG tensor is 0. At Fe2-atom, we see a 4-fold rotation axis and 2 2-fold rotational - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-14/) - If we want to go from a quantum situation, where the energy is quantized, to a classical situation, with a continuum energy spectrum, the quantized energy levels should be 'infinitely' close to eachother. This way, the quantized energy levels appear as a continuum energy spectrum. - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-15/) - -In the equation, Vxx and Vyy are the same, indicating an axial symmetry. -In the figure, a symmetry in the xy-plane is indicated. - [From toy model to quantum task](https://www.hyperfinecourse.org/forums/topic/from-toy-model-to-quantum-task/) - When the size of the nucleus decreases in size, the quadrupole term will also decrease. Hence the multipole expansion converges faster with a decrease in size. - [Fe rotation axes](https://www.hyperfinecourse.org/forums/topic/fe-rotation-axes/) - FeI: This Fe atom has 3 4fold rotation axes. FeII: This Fe atom has 1 4fold rotation axis. - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-13/) - If the energy levels are very close to each other, the energy distribution will start to look continuous, you then have a classical system! - [Question about transition toy model to quantum](https://www.hyperfinecourse.org/forums/topic/question-about-transition-toy-model-to-quantum/) - In the slides about the toy model and the question in that subchapter, there is concluded that the absolute value of the quadrupole energy splitting becomes smaller if the nucleus becomes smaller w.r.t. the electron cloud, so the splitting also gets bigger if the nucleus becomes bigger with respect to the same electron cloud. However, - [Quadrupole term](https://www.hyperfinecourse.org/forums/topic/quadrupole-term-7/) - If the nucleus becomes smaller compared to the distance, then the size of the quadrupole term decreases. Hence, it is quite obvious that the truncation after the quadrupole term is satisfactory, since the series does simple converge faster. - [Answer to task 2](https://www.hyperfinecourse.org/forums/topic/answer-to-task-2/) - Fe-I) I see 3 4-fold rotation axes for this, which results in the EFG tensor being zero and eta beign zero in all three cases. Fe-II) I see 1 4-fold rotation axis and 1 2-fold rotation axis, resulting in eta being zero in the 4-fold case. - [Task 2: Fe4N](https://www.hyperfinecourse.org/forums/topic/task-2-fe4n/) - - FeII has 3 4-fold rotational axes ==> eta = 0 ==> axial symmetry in PAS - FeI has 1 4-fold rotational axis and 2 2-fold rotational axes ==> eta = 0 ==> also axial symmetry in PAS - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-14/) - 1) Since Vxx = Vyy, we have eta = 0 and therefore axial symmetry 2) The charge distribution will remain the same for rotation around its own axis - [answer to task 1](https://www.hyperfinecourse.org/forums/topic/answer-to-task-1/) - You would need to make the quantum picture continues by connecting the discrete energy levels with sigmoid functions I think. - [task 1: quantum to classical](https://www.hyperfinecourse.org/forums/topic/task-1-quantum-to-classical/) - Maybe if you have a large ensemble with differently oriented nuclei, you can get an almost continuous distribution of the energy on a macroscopic scale? - [Task 1: axial symmetry](https://www.hyperfinecourse.org/forums/topic/task-1-axial-symmetry-2/) - - The Vxx and Vyy components in the matrix are the same, so eta = 0. - The electrons are symmetric w.r.t. rotations around the z-axis, so there is axial symmetry of the electron cloud around the z-axis. - [my answer](https://www.hyperfinecourse.org/forums/topic/my-answer-8/) - 1) From the equation, you will know if it has axial symmetry because the matrix is diagonal. 2) From a visual inspection of the picture, the dumbbell is always symmetric in the xy-plane, granting it axial symmetry. - [Task 1 toy model to quantum](https://www.hyperfinecourse.org/forums/topic/task-1-toy-model-to-quantum/) - When the ratio l/d decreases, the nucleus becomes smaller w.r.t. the electron cloud and in the figures the values on the y-axis decrease in absolute value. This means the energy difference between the electric quadrupole splitted levels (m = 0 or m = +- 1) becomes smaller. - [The quadrupole term and the size of the nucleus](https://www.hyperfinecourse.org/forums/topic/the-quadrupole-term-and-the-size-of-the-nucleus/) - The relative energy difference between the monopole and the exact solution (or between the monopole and the monopole and quadrupole solution) becomes smaller for a smaller size of the nucleus compared to the electron cloud size. This is to be expected since a smaller nucleus will naturally have a smaller quadrupole moment and therefore a - [Symmetry](https://www.hyperfinecourse.org/forums/topic/symmetry-4/) - 1. There is axial symmetry if eta = 0, so if Vxx and Vyy are equal 2. Only the electrons matter to determine axial symmetry of the EFG system. By that picture, the system is axially symmetric because the two electrons are positioned on the z-axis, equidistant from the y-axis - [Task 2](https://www.hyperfinecourse.org/forums/topic/task-2-15/) - Fe-I: There are 3 4-fold rotation axis (x,y and z). -Theorema 1 gives that every axis can be chosen as a z-axis and eta=0. -Theorema 2 gives that the EFG tensor is zero Fe-II (a and b): There are 2 2-fold rotation axis (x and y) and 1 4-fold rotation axis (z) -Theorema 1 gives - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-25/) - To go from a quantum system (quantized energy) to a classical system (continous energy), the energy steps, from the quantum system, needs to be so small that the system start to look like a continous energy system. - [Task 1: Axial Symmetry](https://www.hyperfinecourse.org/forums/topic/task-1-axial-symmetry/) - Equation: Vxx =Vyy so there is axial symmetry Figure: The electrons (except the dumbell at the center of the axis) are symmetric with each other on the z-axis - [From toy model to quantum](https://www.hyperfinecourse.org/forums/topic/from-toy-model-to-quantum-6/) - Comparison of the 3 pictures shows that a smaller l/d gives a larger max energy level and a smaller min energy level. This also gives a smaller energy difference between the 2 values. So a smaller nucleus gives a smaller quadrupole term. - [my answer](https://www.hyperfinecourse.org/forums/topic/my-answer-7/) - The quadrupole term becomes larger (less negative) for the minimum energy and is lowered (more negative) for the maximum energy as the nucleus becomes smaller, making the variation of the quadrupole term smaller. - [Quadrupole Term w.r.t. Nucleus Size](https://www.hyperfinecourse.org/forums/topic/quadrupole-term-w-r-t-nucleus-size/) - As the ratio between the nucleus size and the electron cloud diameter becomes smaller, or as the nucleus becomes smaller, the difference between the monopole term and the maximum and minimum energy levels becomes smaller. So, we can conclude that the quadrupole term gets smaller as the nucleus gets smaller. - [EFG symmetry](https://www.hyperfinecourse.org/forums/topic/efg-symmetry/) - For Fe-I i found 3 4-fold axis. which, even if there is more, suffices to know that there will be no EFG felt by these nuclei. For Fe-IIa I found there to be 3 2-fold axis so this would not allow for an axially symmetric EFG, so eta will be different from 0 and the - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-12/) - Truth be told, i found the question rather strange. So after pondering a bit about why i did not get it, i looked at some of the given answers. But i can honestly say that i'm still a bit confused, especially regarding the terminology used in the question. I understand that by causing some sort - [Axial symmetry in the PAS](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-in-the-pas/) - From the form of the equation it can be noted that for it's PAS the matrix of the EFG would be diagonalised and thus Vxz and Vyz would be 0. Since the matrix also has to be traceless one would only need to know Vzz and Vxx or Vyy. Visually it is clear that the - [Quadrupole term](https://www.hyperfinecourse.org/forums/topic/quadrupole-term-6/) - The axis of the different pictures are not the same, they are more "zoomed in" for the smaller nuclei. I think that we can conclude by looking at the exact solution that the quadrupole term increases with an increase in size of the nucleus, and conversely decreases when the nucleus tends more and more to - [L and S](https://www.hyperfinecourse.org/forums/topic/l-and-s-4/) - When S and L are parallel and point in the same direction J the total angular momentum is 2 for a system with S = 1 and L = 1. When L and S are perpendicular J = 1 -. Lastly when L and S are parallel but opposite in direction J = 0. - [answer](https://www.hyperfinecourse.org/forums/topic/answer-17/) - I think if we somehow can ensure an infinitly small energy difference as to make it look like it's is a continuum spectrum it would look like the classical model. How this can be achieved, i don't really know though. - [answer](https://www.hyperfinecourse.org/forums/topic/answer-16/) - I think if we somehow can ensure an infinitly small energy difference as to make it look like it's is a continuum spectrum it would look like the classical model. How this can be achieved, i don't really know though. - [answer](https://www.hyperfinecourse.org/forums/topic/answer-15/) - We can see it in the matrix because Vxx = Vyy so this implies axial symmetry Also this symmetry can be seen when looking at the electrons on the z axis, they are also symmetrical. (not the dumbbell though) - [quadrupole term](https://www.hyperfinecourse.org/forums/topic/quadrupole-term-5/) - The quadrupole term also becomes (orders of magnitude) smaller - [No-overlap overlap contribution](https://www.hyperfinecourse.org/forums/topic/no-overlap-overlap-contribution/) - It looks to me like the Bohr-Weisskopf effect does not require electrons in the nucleus, just a magnetic field created by the electron cloud acting on a nucleus with non-uniform magnetic moment distribution. But, since the nucleus in this case is a perfect sphere, I imagine there would be no hyperfine anomaly term. The energy - [Overlap Contribution](https://www.hyperfinecourse.org/forums/topic/overlap-contribution-6/) - Yes, due to the Bohr-Weisskopf effect, electrons have an extended magnetic field inside the nucleus. - [Carousel](https://www.hyperfinecourse.org/forums/topic/carousel-8/) - Magnetometer goes in the center of the Carousel, representing the nucleus. The child rides the carousel, representing the electron. Bar magnet held by the child would represent the spin dipolar contribution. If the child now holds an electrically charged ball, this would represent the orbital contribution. Now if the child somehow comes in contact with - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-24/) - 1.(a) Ground state 111Cd: g= -1.18806 245keV state 111Cd: g= -0.30652 1.(b) g=µ/(I*µ_B), I = 1/2 g=2 1.(c) µ=g*I*µ_N, I=1/2, g= -3.826 µ= -1.913 - [Task 2](https://www.hyperfinecourse.org/forums/topic/task-2-14/) - If J is constant, then we can use the expression a1/a2= mu1*I2/(mu2*I1), and we know all parameters besides mu2 so its easy to calculate. - [L and S](https://www.hyperfinecourse.org/forums/topic/l-and-s-3/) - J=0 when L and S are anti parallel J=1 when L and S are perpendicular to one another since there is a partial contribution to the spin. J=2 when L and S are parallel in same direction, as both the spins would contribute. - [g-factors](https://www.hyperfinecourse.org/forums/topic/g-factors-12/) - 1.a : 0kev: \mu = 0.5940(2) nm (units of nuclear magneton) I = 1/2 . g = /mu/I = -2x0.5940(3) ~ -1.118 254 KeV I = 5/2 ; g = -0.3064 1.b -2 1.c -1.93 nm - [energy corrections on sphere](https://www.hyperfinecourse.org/forums/topic/energy-corrections-on-sphere/) - Since it is a perfect sphere, it would not undergo any hyperfine interactions as that requires a non-spherical shape. Alongside the fact that if all the electrons stay outside the nucleus there is no overlap and thus no need for any correction terms to the magnetic moments. - [L, S, J](https://www.hyperfinecourse.org/forums/topic/l-s-j/) - First I want to remind myself that L,S can be thought of as 'vectors' in the 'vector model of orbital angular momentum' where L,S are depicted as cones around an axis of choice, by convention often the z-axis. J=2 : L and S parallel and same direction J= 0 : L and S parallel and - ["Hey hold this magnet real quick, its for science"](https://www.hyperfinecourse.org/forums/topic/hey-hold-this-magnet-real-quick-its-for-science/) - So we would have the child going around the carousal as that represents a classic electron going around an atom. If they are holding a bar magnet that represents the electron's spin-dipole field, if they were also holding the electrically charged ball that will generate a field as it is now a current since its - [G-factors and magnetic moments](https://www.hyperfinecourse.org/forums/topic/g-factors-and-magnetic-moments-2/) - 1.a) We see that through using mu = ((g*mu_N)/h-bar)*I, isolating g => g = (h-bar*mu)/(mu_N*I) mu_N = 5.05x10^{-27}, h-bar = 1.05x10^{-34} For the ground state: I = 1/2 (h-bar), mu = -0.594nm g_0 = (-0.594nm)/(mu_N*1/2) = -2.35x10^{17} For the first excited state (245KeV): I = 5/2, mu = -0.766nm g_1 = (-0.766nm)/(mu_N*5/2) = -0.607x10^{17} 1.b) - [Magnetic moments](https://www.hyperfinecourse.org/forums/topic/magnetic-moments-2/) - mu_N = 3.152*10^-8 eV/T the nuclear magneton mu_B = 5.788*10^-5 eV/T the Bohr magneton a) We have g = mu/(mu_N*I) for the g-factor (no hbar contained in I). For 111^Cd,0 keV, the spin is I = 1/2 and mu = -0.595 mu_N. The g-factor is g = -1.190 mu_N For 245 keV, it follows that - [Carousel](https://www.hyperfinecourse.org/forums/topic/carousel-7/) - By placing a child on one of the carousel seats, and having them hold the bar magnet and later the electrically charged ball, you could measure the magnetic fields at the center of the carousel that would mimic the dipole contribution and orbital contribution, respectively. - [g factor](https://www.hyperfinecourse.org/forums/topic/g-factor-5/) - It doesn't seem like there is enough information to do this. I've considered using the formula mu-hat = g*mu_N*I-hat / reduced planck but I don't know what mu-hat is for Cd-111 I've considered using the differences between energy levels, but I don't know what B_hf is. - [L and S](https://www.hyperfinecourse.org/forums/topic/l-and-s-2/) - J=0 corresponds to the orientation of L and S being parallel and opposite. J=1 corresponds to a perpendicular orientation of L and S. J=2 corresponds to the orientation of L and S being parallel and pointing in the same direction. - [Determine mu experimentally](https://www.hyperfinecourse.org/forums/topic/determine-mu-experimentally/) - Assumptions: "a" can be measured B would be the same and J would be the same, as these are independent of the size/radius of the nucleus. Then, mu can be calculated from the experimental data from the first isotope to establish B and J, and from the second isotope to then find mu_2. - [L and S Orientations](https://www.hyperfinecourse.org/forums/topic/l-and-s-orientations/) - I have not understood this well, despite doing a lot of background reading. I guess all of these vectors are being added only as projections along the same direction? So, a vector perpendicular to another would not change its value, parallel vectors would add, antiparallel vectors would subtract. I don't think I understand the concept - [g Factors](https://www.hyperfinecourse.org/forums/topic/g-factors-11/) - 1a) -0.595 mu_N for Cd111 ground level which has spin 1/2, and -0.766 mu_N for Cd11 at 245 keV with spin 5/2. Dividing the former by the latter produces the g factors -1.190 and -0.306. 1b) The free electron has only spin, as its L = 0, S = 1/2 and J = S. g_s - [Carousel as a Nuclear-Angular momentum problem](https://www.hyperfinecourse.org/forums/topic/carousel-as-a-nuclear-angular-momentum-problem/) - First, place the child on a point in the carousel and fasten the electrically charged ball someplace on the child, without loss of charge. Allow the child to spin on its own axis at a rate that is impossible for a solid object. Place the magnet at a non-origin location of stationary pillar of the - [Experimental Idea](https://www.hyperfinecourse.org/forums/topic/experimental-idea/) - We could assume that the electron cloud density and its energy levels stay consistent but there will be an "Isotope" shift in the monopole and thus energy levels. If we assume that this shift is negligible then when we measure and calculate the a_1 of the isotope we can compare this to the a_1 of - [Finite Size Nucleus Overlap Correction](https://www.hyperfinecourse.org/forums/topic/finite-size-nucleus-overlap-correction/) - The hyperfine magnetic field does not stay homogenous in space even for spherically symmetric nuclei, so there must still be a contribution due to the Bohr-Weisskopf effect. - [Energy correction](https://www.hyperfinecourse.org/forums/topic/energy-correction-5/) - Yes, there will still be a contribution. The Fermi-contribution is zero, since no electrons are in the nucleus. However, the small Bohr-Weisskopf distribution persists since the magnetic field of the electrons still penetrates inside the nucleus, and the contributions of the different regions in the nucleus will not be the same. - [Experimental procurement of new magnetic dipole moment](https://www.hyperfinecourse.org/forums/topic/experimental-procurement-of-new-magnetic-dipole-moment/) - A photoluminescence spectrometry of sufficient precision performed on the new isotope will reveal its hyperfine zero-field splitting energies which can be noted as corrections to the corresponding Zeeman lines. From these corrections must be ordered according to their J, F and I configurations, then factored accordingly to produce the common term containing mu_2*B_j/I_2J. Comparing to - [L and S Coupling](https://www.hyperfinecourse.org/forums/topic/l-and-s-coupling/) - So we have three options available with L=1, S=1. J can take the values of 1+1=2, 1+1-1=1, 1+1-1-1=0. The first option (J=2) makes the most sense when both L and S are parallel and pointing in the same direction as this takes more energy and a stronger coupling. Therefore, when they are parallel but pointing - [Describing the hyperfine field](https://www.hyperfinecourse.org/forums/topic/describing-the-hyperfine-field/) - We put a child with the magnetometer at the center of the carousel. On one of the horses, we attach the magnet and the charged ball. There are three contributions: 1) First, the electrons have a spin and we can therefore compare them to bar magnets that generate field. These fields reach to the position - [Orbital Momentum Numbers](https://www.hyperfinecourse.org/forums/topic/orbital-momentum-numbers/) - The question doesn't make sense because it convolutes multiple different properties. L and S are the quantum numbers for the electron orbital, the orientations are procured from the projections onto the z axis given by m_l and m_s which can take ranges of values between {-1, 0, 1}. The projections of J = 2 onto - [Determining mu2](https://www.hyperfinecourse.org/forums/topic/determining-mu2/) - For the different isotopes, we can assume the electron cloud to stay the same, and so do all parameters related to J (B_J, J, ..). If we thus measure a1 and a2 in both cases, then a1/a2= mu1*I2/(mu2*I1), in which all parameters are known except for mu2. We can thus easily calculate this value. - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-23/) - Yes, there still can be a small contribution from the Bohr-Weiskopf-effect, from the magnetic field due to the moving electrons. - [Carousel](https://www.hyperfinecourse.org/forums/topic/carousel-6/) - I think the magnetometer should be used in the middle of the carousel, representing the nucleus, while the child with the electrically charged ball and the bar magnet represents an electron moving around the nucleus. This magnetometer then senses the magnetic field of either the bar magnet (when the carousel is not rotating), or the - [Rotation axes and PAS](https://www.hyperfinecourse.org/forums/topic/rotation-axes-and-pas/) - For Fe-I we see that there are tree 4-fold rotation axis (x, y and z) and thus we get that a rotation axis is the z-axis of PAS and that eta=0 For Fe-IIa we see a 4-fold rotation axis and two 2-fold rotation axis. So a 2-fold rotation axis can be chosen as z-axis of - [Task 1:](https://www.hyperfinecourse.org/forums/topic/task-1-22/) - 1.a) To calculate the g-factor, we first need the magnetic dipole moments. For the ground state this is -0.5940 nm, and for the 245 keV level, we got -0.766 nm. Combining this in the formula gives: g = -0.5940 * hbar * 2 / mu_N, and g = -0.766 * hbar * 2 /( 5 - [Quantum to classics](https://www.hyperfinecourse.org/forums/topic/quantum-to-classics/) - To go from a quantum situation, where you have a few discrete levels, to a classical situation, where you have a continuum, we need a quantum situation where the levels are very close to each other so that it can be seen as a continuum. - [Monopole shift with Toy Model 0 and A](https://www.hyperfinecourse.org/forums/topic/monopole-shift-with-toy-model-0-and-a/) - So in toy model 0 we assume a very generic nucleus where there is no overlap in the electron density. However to introduce overlap in this physical example it lowers the coulomb interaction between the position charges and thus lowers the general energy of the toy model A as compared to toy model 0. This - [Task 1: overlap contribution](https://www.hyperfinecourse.org/forums/topic/task-1-overlap-contribution-2/) - Even if the electrons stay outside the nucleus, their generated magnetic field is non-zero at the position of the nucleus. There will thus be a small overlap contribution because of the Bohr-Weisskopf effect. - [Second task](https://www.hyperfinecourse.org/forums/topic/second-task-3/) - From mu2/mu1 = a2 *I2/(a1 *I1), where we expect only a change from the nucleus, we get: mu2 = mu1 *a2 *I2 / (a1 *I1). Since we got all the values in the right-hand side, it is now possible to calculate mu2. - [First Task](https://www.hyperfinecourse.org/forums/topic/first-task-2/) - For J = 0, L and S are substracted from eachother, and thus are parallel to eachother, but point in a different direction. For J = 2, L and S are added to eachother, which represents vectors parallel and in the same direction. For J = 1, L and S are perpendicular to eachother. - [axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-13/) - - From the equation, we see that Vxx=Vyy and this means we have axial symmetry. - From a visual inspection of the picture of this toy model: The EFG is only related to the electron cloud and we see in the picture that with this axis system defined, indeed there is axial symmetry. (Do not - [Task 2: Carousel](https://www.hyperfinecourse.org/forums/topic/task-2-carousel/) - Place the magentometer at the center of the carousel. If the child holds the magnet and the carousel is not turning yet, the magnet creates a magnetic dipole field. This is already one hyperfine field. If the child holds the electrically charged ball and the carousel is turning, a current loop around the magnetometer is - [task 1: g-factor](https://www.hyperfinecourse.org/forums/topic/task-1-g-factor/) - We know formula mu = (g * mu_N * I) / hbar: 1.a): need g = (mu * hbar) / (mu_N * I) - GS: mu = -0.5940 nm, I = 1/2+ ==> g = (-0.5940 [nm] * hbar) / (mu_N * 1/2) - 1st excited state: mu = -0.766 nm, I = 5/2+ ==> - [Quadrupole term](https://www.hyperfinecourse.org/forums/topic/quadrupole-term-4/) - After inspecting the vertical axis, it becomes clear that the size of the quadrupole term gets smaller when the nucleus becomes smaller. So I think this means that we are dealing with smaller corrections as l/d becomes smaller and thus the multipole expansion converges faster. - [rotation axes](https://www.hyperfinecourse.org/forums/topic/rotation-axes-5/) - The FeI atom has three 4-fold rotation axes: the x, y and z axes. From theorem 1 it follows that all these axes can be chosen as the z-axis of the PAS so that eta = 0. In addition, from theorem 2 it follows that the EFG tensor is zero. The FeII atom has two - [How to explain the position of an electron](https://www.hyperfinecourse.org/forums/topic/how-to-explain-the-position-of-an-electron/) - So to begin with, the position of an electron is usually never known exactly but the general radius and shape it has around the atom (nucleus) is known. There is a general area understood to be where the electron could exist in. This would be like guessing what you are having for dinner, maybe its - [quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-11/) - In the quantum system, the energy levels are quantized and discrete. In the classical system the energy is continuous. So in order to go from quantum to classical, we need to make the energy levels continuous instead of discrete. - [PAS](https://www.hyperfinecourse.org/forums/topic/pas/) - For Fe-I we have three 4-fold axes (x, y and z) these can thus be chosen as z-axis. using theorem 2 we then have that the EFG tensor is zero and from theorem 1 that n = 0 For Fe-II we have three 2-fold axes. - [quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-10/) - If we want to go from the quantum picture (quantized) to the classical picture (continous) we would need very small energy steps as to make it almost continous. - [Task 2: mu_2](https://www.hyperfinecourse.org/forums/topic/task-2-mu_2/) - In the first experiment you measure a_1 = mu_1*B / (I_1 * J). In the second experiment you measure a_2 = mu_2*B / (I_2 * J). Assuming B and J originating from the electron cloud of the same element do not change, you can do: B/J = a_1 * I_1 / mu_1 = a_2* I_2 - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-12/) - We can see the symmetry in 2 ways: 1) There are no non diagonal matrix elements and Vxx and Vyy are equal, so eta = 0 2) The figure indicates a symmetry in the xy plane - [Task 1: coupling L and S](https://www.hyperfinecourse.org/forums/topic/task-1-coupling-l-and-s/) - J = 0 corresponds to L antiparallel to S = lowest energy J = 2 corresponds to L parallel to S = highest energy J = 1 ==> intermediatie orientation: L perpendicular to S - [quadrupole term](https://www.hyperfinecourse.org/forums/topic/quadrupole-term-3/) - Looking at the y-axis in these figures, we see that the quadrupole term becomes smaller as the nuclei becomes smaller. - [axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-11/) - We notice the symmetry is present in two ways. - there are no non-diagonal elements present in the matrix element and both Vxx and Vyy are the same - the figure also indicates a symmetry - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-21/) - If the L and S vectors are parallel and pointing in the same direction, then we add their values to obtain the magnitude of their sum, so J=2. If the L and S vectors are parallel and pointing in the different directions, then we subtract their values to obtain the magnitude of their vectorial sum, - [answer](https://www.hyperfinecourse.org/forums/topic/answer-14/) - When the nucleus becomes smaller, we see that the quadripole term also becomes smaller. - [Energy correction](https://www.hyperfinecourse.org/forums/topic/energy-correction-4/) - Yes, there will be an energy correction due to the Bohr-Weisskopf effect. Even though the electrons themselves do not enter the nucleus themselves, their electric and magnetic fields will still have a non-zero effect within the nucleus. - [Overlap Contribution](https://www.hyperfinecourse.org/forums/topic/overlap-contribution-5/) - Yes, there will be corrections as due to the Bohr-Weisskopf effect, electrons will still provide a magnetic field at the location of the nucleus, thus contributing to the energy. - [my answer](https://www.hyperfinecourse.org/forums/topic/my-answer-6/) - Yes, the protons and neutrons in the nucleus will still move around, so the Bohr-Weisskopf effect will contribute to the energy correction. - [Carousel](https://www.hyperfinecourse.org/forums/topic/carousel-5/) - (No children were hurt in the making of this explanation) Suppose we have a carousel operator that is crazy enough to let this demonstration go through. We will place the magnetometer at the center of the carousel in order to measure the magnetic field of the "nucleus" our makeshift atom. We then place a child - [g-factor](https://www.hyperfinecourse.org/forums/topic/g-factor-4/) - 1.(a) γ = µ/(Iℏ) = (g*µ_N)/ℏ => g = (µ)/(I*µ_N) Taking into account that µ is in units of µ_N, we get, γ = µ/I Ground State Cd-111: µ = -0.59403 I = 1/2 g = -1.18806 245 keV State Cd-111: µ = -0.7663 I = 5/2 g = -0.30652 1.(b) g = µ/(I*µ_B) µ - [my answer](https://www.hyperfinecourse.org/forums/topic/my-answer-5/) - A child is riding a carousel, while holding a bar magnet and an electrically charged ball, with a magnetometer at the centre of the carousel. When the carousel stands still, the spin dipolar contribution is represented. when the child only has an electrically charged ball and the carousel rotates, this is the orbital contribution. when - [Correction](https://www.hyperfinecourse.org/forums/topic/correction-2/) - There will still be a correction because of the Bohr-Weissekopf effect that will cause that the magnetic field of the electrons will enter the nucleus. - [Carousel](https://www.hyperfinecourse.org/forums/topic/carousel-4/) - If you asume that the center of the carousel is the nucleus than you can put there the magnetometer so it can measure the magnetic field of the different hyperfine interactions. A child is on a horse of the carousel and holding a bar magnet. The child can now be seen as an electron with - [my answer](https://www.hyperfinecourse.org/forums/topic/my-answer-4/) - The formula g = (mu * h_bar) / (mu_N * I) is used. 1a) mu = -0.5948861 nm for the ground state and mu = -0.766 for the 245 keV. g_ground = (- 0.5948861 nm * h_bar) / (0.5 * mu_N) and g_245keV = (-0.766 * h_bar) / (0.5 * mu_N) 1b) g = (mu - [g-factor](https://www.hyperfinecourse.org/forums/topic/g-factor-3/) - 1.a: g = (mu*hbar)/(mu_N*I) = (-0.5940*hbar)/(mu_N*(1/2)) (Ground state) g = (mu*hbar)/(mu_N*I) = (-0.766*hbar)/(mu_N*(5/2)) (245 keV) 1.b: g = (mu_B*hbar)/(mu_N*I) = (mu_B *hbar)/(mu_N*(1/2)) 1.c: mu_B = (g*I*mu_N)/hbar = (-3.826*(1/2)*mu_N)/hbar - [Task 2](https://www.hyperfinecourse.org/forums/topic/task-2-13/) - Since both isotopes are from the same element, J will be the same for both. This gives a_2/a_1 = (mu_2/I_2)*(I_1/mu_1). To find mu_2 this can be rewriten as mu_2 = mu_1*(a_2*I_2)/(a_1*I_1) - [Determining µ_2](https://www.hyperfinecourse.org/forums/topic/determining-µ_2/) - Once you measure a_1 and a_2 for the two isotopes, you can then take their ratio a_2/a_1 = (µ_2 * I_1)/(µ_1 * I_2). This equation stems from the assumption that adding extra neutrons to the nucleus does not affect the electron cloud. This assumption then comes from the fact that neutrons have zero charge and - [my answer](https://www.hyperfinecourse.org/forums/topic/my-answer-3/) - Because this is an isotope of the same element, J will be equal. This gives the formula a2/a1 = (nu2/I2)*(I1/nu1). The magnetic moment of the second isotope is then: nu2 = nu1*(a2/a1)*(I2/I1). - [Which J-value corresponds to which orientation?](https://www.hyperfinecourse.org/forums/topic/which-j-value-corresponds-to-which-orientation-2/) - J = 0 corresponds to L and S being parallel but opposite to each other, since then they would cancel each other out. J = 1 corresponds to L and S being perpendicular, as then there is a partial contribution to the spin, but one of the two terms (either S or L) is not - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-20/) - You will have a correction from the Fermi contact contribution which does not depend on the size of the nucleus, it only depends on the spin imbalance inside the nucleus. I do not think there is a contribution from the Bohr-Weisskopf effect since the electrons do not go inside the nucleus... - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-19/) - J = 2 = 1 + 1 = L + S so L and S are parallel with each other and having the same direction J = 1 lies between J = 0 and J = 2 so the 2 term most be perpendicular with each other and one of the terms give a value - [My answer](https://www.hyperfinecourse.org/forums/topic/my-answer-2/) - For J = 0, the orbital angular momentum L and the spin angular mommentum S point in opposite directions, cancelling each other out, resulting in the lowest energy. For J = 1, L and S are perpundicular to each other, giving some contribution to the energy. For J = 2, L and S point in - [Which J-value corresponds to which orientation?](https://www.hyperfinecourse.org/forums/topic/which-j-value-corresponds-to-which-orientation/) - J=0 means that J=L-S. So this means L and S are parallel and pointing in opposite directions. J=1 means that J and S are perpendicular to eachother. J=2 means that J=L+S so L and S are parallel and point in the same direction. - [contributions to the hyperfine field](https://www.hyperfinecourse.org/forums/topic/contributions-to-the-hyperfine-field-2/) - To illustrate the contributions to the hyperfine field, one can hold a magnetometer in the middle of a carousel. The hyperfine field can be described with 3 terms. - The dipole term can be seen as a child that is holding a bar magnet and is on the carousel but the carousel is still. You - [g-factors](https://www.hyperfinecourse.org/forums/topic/g-factors-10/) - (1.a) 111Cd in ground state has g= -0.594 * hbar / (1/2 * mu_N) and 111Cd at 245 keV has g=-0.766 * hbar / (1/2 * mu_N) (1.b) the electron is a spin 1/2 particle so the g-factor can be equal to g=+- 2 *hbar * mu_B/ mu_N (1.c) By rearranging the formula of the - [second task](https://www.hyperfinecourse.org/forums/topic/second-task-2/) - Since we have an experiment where we can measure the hyperfine splitting a, we can make a comparison between both isotopes. Since J is equal for both isotopes, we get from the comparison that: a2/a1=(mu2/I2)(I1/mu1) so from this we get mu2=mu1(a2/a1)(I2/I1) - [task](https://www.hyperfinecourse.org/forums/topic/task/) - there will still be a correction because of the finite size - [task 2](https://www.hyperfinecourse.org/forums/topic/task-2-12/) - We know that we are only looking at a difference in the nucleus. This means that all factors associated with the electron cloud will cancel out when taking a(1)/a(2). Doing this we then get: mu(1) I(2) / mu(2) I(1). We can then isolate mu(2) and then we have: mu(2) = mu(1) a(2)I(2)/a(1)I(1) - [L and J](https://www.hyperfinecourse.org/forums/topic/l-and-j/) - For J = 0, both the orbital angular momentum (L) and the spin angular momentum (S) are aligned head-to-tail but in opposite directions. This arrangement nullifies their effects, yielding a total angular momentum (J) of zero. In the case of J = 1, L and S are perpendicular, resulting in partial cancellation of their angular - [answer](https://www.hyperfinecourse.org/forums/topic/answer-13/) - Yes, there might still be a small magnetic field due to the electrons so the Bohr-Weisskopf effect has to be corrected for. - [answer](https://www.hyperfinecourse.org/forums/topic/answer-12/) - Picture a child on a carousel holding a bar magnet. The spinning magnet represents an atomic nucleus with its magnetic field. Imagine the child also holds a charged ball, like the electrons surrounding the nucleus. As the child walks around the carousel, the magnetometer (a device measuring magnetic fields) detects a fluctuating field, similar to - [answer](https://www.hyperfinecourse.org/forums/topic/answer-11/) - a. From mu = g*muN*I /(hbar) we get: g = mu*hbar/(mu_N*I) which we can fill in with the values -0.5940(3) = mu (GS) and -0.766(3) = mu(245keV) and a spin of 1/2 for GS and 5/2 for 245keV. Using the same formula we get b. g = 2*hbar*mu_B/mu_N c. mu = -3.826*(1/2)*mu_N/hbar - [answer](https://www.hyperfinecourse.org/forums/topic/answer-10/) - We know, a1,a2, mu1,I1,I2 and J is the same since only the amount of neutrons change. So we can find mu2 by following formula mu2 = (a2 mu1 I2) / (a1 I1) - [L and S](https://www.hyperfinecourse.org/forums/topic/l-and-s/) - J =0: both L and S lie head-to-tail with opposite directions. This configuration cancels out their contributions, resulting in a zero total angular momentum (J). J =1: L and S lie perpendicular, so there is a small cancellation of some of their angular momentum. But there is a total angular momentum J pointing inbetween L - [Toy model contradiction](https://www.hyperfinecourse.org/forums/topic/toy-model-contradiction-2/) - I might be wrong about this, but I think by adding a negative charge to the center of the "nucleus" lowers the energy as you are decreasing the total charge of the nucleus. Since we are working with the nucleus, you want to add a small positive charge instead of a negative charge. This would - [1s electron](https://www.hyperfinecourse.org/forums/topic/1s-electron-3/) - I might be very wrong. A diffuse sphere. - [Toy Model](https://www.hyperfinecourse.org/forums/topic/toy-model-7/) - In this case the negative shift is because of the negative charge of the electron, since there is no positive contribution from angular momentum. - [Time-Averaged position of electron](https://www.hyperfinecourse.org/forums/topic/time-averaged-position-of-electron-3/) - Since we are talking about the 1s electron, the electron would spend most of the time close to the nucleus. To explain a 15 year old child we can take example of a merry-go-round, we can take a 360-degree camera and place it inside the merry-go-round on time lapse mode and spin around the merry-go-round. - [toy model](https://www.hyperfinecourse.org/forums/topic/toy-model-6/) - the negative contribution is given by the charge of the electron - [Explanation to 15-year old (attempt 2)](https://www.hyperfinecourse.org/forums/topic/explanation-to-15-year-old-attempt-2/) - Imagine you are standing in the middle of a football pitch. If you now close your eyes, and the game goes on, you don't know the specific positions of the individual players. You can however make a good guess in which region specific players due to their roles, such as the striker, defender, midfielder,... are - [Loophole](https://www.hyperfinecourse.org/forums/topic/loophole-6/) - Due to the addition of a negative (small) epsilon in the dumbell, the "total dumbell charge" will be smaller, and will thus have a smaller coulomb energy, than in the model without the added negative charge. - [Toy Model](https://www.hyperfinecourse.org/forums/topic/toy-model-5/) - The issue with the toy model is that an additional electric charge is added to the system without also subtracting this charge from the fixed negative charges representing the electron shell, and this negative charge does not interact with the electron shell points. Naturally this would increase the binding of the system as it introduces - [Explaining the Hydrogen Nucleus](https://www.hyperfinecourse.org/forums/topic/explaining-the-hydrogen-nucleus/) - Unless the 15-year old already has the comprehensive notion of wave-particle duality or the uncertainty principle, the content of such an explanation would be meaningless because it would default to misinterpreting the situation as the familiar orbital motion of classical mechanics (or the Bohr model) without learning anything new. The idea of an observer being - [Contradiction](https://www.hyperfinecourse.org/forums/topic/contradiction-5/) - The toy model shows that if an additional charge is added in the center, that we then have a negative shift. I don't think that the extra charge is the reason for the opposite sign. We could subtract epsilon/2 charge from the two electrons and the effect would still be negative, since the electrons are - [1s electron](https://www.hyperfinecourse.org/forums/topic/1s-electron-2/) - Alright, imagine you are really tiny, like super tiny, and you sit right at the center of a hydrogen nucleus. You have a little friend, an electron, that's zooming around you. Now, this electron is quite quick and zips around really fast, so it's hard to pinpoint exactly where it is at any given moment. - [Hydrogen Nucleus](https://www.hyperfinecourse.org/forums/topic/hydrogen-nucleus/) - I would explain the time averaged position of an electron over time to a 15 year old as follows: If I were somehow small enough to sit on the nucleus of a Hydrogen atom in the 1s state, then the electron's time-averaged position in the would appear as if I was trapped inside a hollow - [Description of the 1s electron](https://www.hyperfinecourse.org/forums/topic/description-of-the-1s-electron/) - The electrons around atoms are not fixed in space. In fact, we cannot not predict the exact spot in which we will find an electron if we try to measure it. We can only assign a probability to each point. If we measure the position of this electron, we are testing this probability. Let's repeat - [Time-averaged position](https://www.hyperfinecourse.org/forums/topic/time-averaged-position-3/) - Imagine you are standing in the middle of a football pitch. If you now close your eyes, and the game goes on, you don't know the specific positions of the individual players. You can however make a good guess in which region specific players are located due to their roles, such as the striker, defender, - [Toy model comparision](https://www.hyperfinecourse.org/forums/topic/toy-model-comparision/) - By adding an additional charge epsilon to the center of the nucleus, the total amount of electrons is altered. The charge of the electron cloud should be lowered by epsilon to consider the same situation. I however tried to plot this on desmos with different constants for the different negative charges, but I did not - [Average position 1s electron](https://www.hyperfinecourse.org/forums/topic/average-position-1s-electron/) - The electron can be described by a wavefunction instead of a particle. This wavefunction tells us something about the probability that the electron is at some place or another. This probability of finding the electron inside the nucleus is nonzero. - [Energy correction spherical nucleus](https://www.hyperfinecourse.org/forums/topic/energy-correction-spherical-nucleus/) - I think that there would still be an energy correction as the magnetic hyperfine field is not a constant value within the nucleus but can vary a bit between infinitesimal volumes inside the nucleus. This would alter the value of the dipole interaction by a small amount. - [simulating contributions to the hyperfine field](https://www.hyperfinecourse.org/forums/topic/simulating-contributions-to-the-hyperfine-field/) - If the magnetometer gets placed in the middle of the carousel one can measure the effects that a nucleus would "feel" when different hyperfine contributions get simulated. To simulate the orbital cntribution the child could carry the charged ball around the carousel, which would induce a magnetic field. To simulate the spin dipolat contribution the - [magnetic moments](https://www.hyperfinecourse.org/forums/topic/magnetic-moments/) - To find the answers to all the questions, one only needs to use the formula µ = (g µN I) / hbar. For question 1, the g factor can be calculated by inserting µ = -0,5940 as found in the table for Cd111 aswell as the nuclear magneton µN = 5,050E(-27) and spin I = - [Loophole Toy model](https://www.hyperfinecourse.org/forums/topic/loophole-toy-model-2/) - The added negative charge (-epsilon), in model 0, will reduce the overall charge of the system, model A. This will, ithink, reduce the energy of the toy atom. - [my answer](https://www.hyperfinecourse.org/forums/topic/my-answer/) - I believe this is because the total charge of the system changes from model 0 to model A by introducing the -epsilon charge. - [measuring µ2](https://www.hyperfinecourse.org/forums/topic/measuring-µ2/) - After measuring the a(1) of isotope 1 of which we know the I(1) and µ(1) I would measure the a(2) of isotope 2 of which i know I(2). By looking at the ratio between the two measured a's, a(2)/a(1), the factors that are determined by the electron cloud would cancel out leaving a ratio µ(2)I(1)/µ(1)I(2). - [time-averaged position of the electron](https://www.hyperfinecourse.org/forums/topic/time-averaged-position-of-the-electron-2/) - Suppose you are in the nucleus of a hydrogen atom, namely the proton, you can then see that an elekron is moving around you. This is the 1s electron of the hydrogen and moves very randomly. Suppose you take several pictures of the electron after a certain times (say 10 minutes). If you compare these - [time-averaged position](https://www.hyperfinecourse.org/forums/topic/time-averaged-position-2/) - I would say that, in a 2d plane, the average position of the 1s electron could be represented as the water in a circular moat around you. - [An analogy to the hyperfine magnetic field](https://www.hyperfinecourse.org/forums/topic/an-analogy-to-the-hyperfine-magnetic-field/) - The nucleus can be imitated by placing the magnetometer on the rotational axis of the carousel. The child (that is riding one of the horses) then represents an electron orbiting the nucleus. The child is holding the bar magnet and the electrically charged ball. The electrically charged ball will then orbit the magnetometer, and thus - [Magnetic moments and g-factors](https://www.hyperfinecourse.org/forums/topic/magnetic-moments-and-g-factors/) - The g-factor can be calculated using the formula g = (µ hbar) / (µ_N I). a) For the ground state of Cd^(111) we have I = 1/2 and µ = 0.595 nm. The g-factor is thus given by: g = 0.595 nm / µ_N * hbar * 2 For the 245 keV excited state, we - [Measuring µ_2](https://www.hyperfinecourse.org/forums/topic/measuring-µ_2/) - Assuming that the only thing that changes in the experiment is the magnetic moment µ. We can then measure a for both the first nucleus with known µ_1 and the second nucleus with unknown µ_2. The ratio of a_1 / a_2 will then be equal to µ_1 / µ_2. Since only µ_2 is unknown at - [The values of J](https://www.hyperfinecourse.org/forums/topic/the-values-of-j/) - I unfortunately cannot see the picture. For three values of J, the minimal value will occur when L and S are parallel to eachother, but point in opposite directions. The maximal value for J has L and S parallel to eachother and pointing in the same direction. The "intermediate" value for J will occur when - [The position of the electron](https://www.hyperfinecourse.org/forums/topic/the-position-of-the-electron-2/) - On average, the electron in the ground state will be at a certain distance (the Bohr radius) from the nucleus. At different times, the electron might be a little closer to or furhter away from the nucleus. The electron is just as likely to be on top of the nucleus as beneath it, to the - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-18/) - There will still be energy corrections due to the small contribution of the Bohr-Weisskopf effect. The magnetic fields of the electrons can still extend to the inside of the nucleus. - [Task 2](https://www.hyperfinecourse.org/forums/topic/task-2-11/) - Place the magnetometer in the center of the carousel. This represents the nucleus. Place the child in the carousel. This represents an electron. When the child holds the bar magnet, the magnetometer records the dipole contribution. If the child is on the carousel, holding an electrically charged ball, the magnetometer registers the orbital contribution. - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-17/) - using the formula g = (hbar * mu)/(mu_N *I): a) g(GS) = (-0.5940 * 2 * hbar) / mu_N g(245 keV) = (-0.766 * hbar * 2/5) / mu_N b) g = 2* hbar c) mu = (-3.826 * 2 * mu_B) / hbar - [Logical Loophole](https://www.hyperfinecourse.org/forums/topic/logical-loophole/) - The addition of the small negative charge would reduce the overall charge of the nucleus, and so would reduce the magnitude of the monopole moment eZ, which therefore reduces the negative contribution of the term that would be hard to write here and I don't know the name of, but is proportional to the inverse - [Entry Ticket](https://www.hyperfinecourse.org/forums/topic/entry-ticket/) - I posted first from the assignment page. I think I can only see other posts if I post first from the forum page. - [15 year old quantum physicist](https://www.hyperfinecourse.org/forums/topic/15-year-old-quantum-physicist/) - I would show them the radial distribution plot and describe it as the amount of time the electron spends in the area at increasing distances from the nucleus. So, an electron in the 1s orbital of a hydrogen atom spends most of its time around 52.9 pm away from the center of the nucleus. Sometimes - [Rotation axes](https://www.hyperfinecourse.org/forums/topic/rotation-axes-4/) - For the FeI atom: It has three 4-fold rotation axes (x,y and z). All of these axes can be chosen as the z-axis of the PAS and eta = 0. (theorem 1) Here, theormem 2 also states that the EFG tensor is zero. For the FeII atom: The z-axis is a 4-fold rotation axis and - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-9/) - The difference is that the quantum case has quantisation in the energy levels, while in the classical case it is continuous. To go from the quantum case to the classical case, we would have to make the diffences very small, so that it resembles a continuous distribution. - [Loophole](https://www.hyperfinecourse.org/forums/topic/loophole-5/) - By going from charge distribution 0 to charge distribution A, an additional negative charge is added. My guess would be that if the negative charge epsilon was extracted from the electrons (so that they ended up with charges e - epsilon/2) would in total increase the energy of the system. Another thing that I noted - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-10/) - We can see that there is axial symmetry in two ways: - The Vxx and Vyy component are the same, and thus eta = 0 - We can see symmetry in the charge distribution. - [Loophole](https://www.hyperfinecourse.org/forums/topic/loophole-4/) - I think it was said before, but i too think that the important destinction here is that the total charge changed from toy model 0 to A. If in stead we took away some negative charge out of the two "electrons" and then put these charges in the middle the first term would change in - [from toy model to quantum](https://www.hyperfinecourse.org/forums/topic/from-toy-model-to-quantum-5/) - If the nucleus becomes smaller with respect to the electron cloud, the quadrupole contribution becomes smaller as well. - [Time average position of the electron](https://www.hyperfinecourse.org/forums/topic/time-average-position-of-the-electron/) - The time-averaged position of the 1s electron is like a cloud where the electron has a higher probability of being close to the center rather than further away. It is often with us in the nucleus or very close by and sometimes it drifts further away - [Loophole](https://www.hyperfinecourse.org/forums/topic/loophole-3/) - i think the reason there seems to be a mistake is because there actually is just a different situation. adding a charge to a certain configuration can in- or decrease the total value depending on the charge sign. The monopole shift for the situation where there is an electron in the nucleus could still be - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-16/) - I think there will still be corrections. The fermi-contact contribution will be zero, since the charges do not enter the nucleus, but the Bohr-Weisskopf term will not be zero. The charges don't enter the nucleus, but their magnetic fields can still extend into it. - [Second task](https://www.hyperfinecourse.org/forums/topic/second-task/) - Measure for both isotopes the constant a. Then we can use the ratio a1 / a2 = (mu1 I2 )/ (mu2 I1) to calculate mu2 = (a2 mu1 I2) / (a1 I1) - [First task](https://www.hyperfinecourse.org/forums/topic/first-task/) - Unfortunatly I cannot see the picture. But the lowest energy value will be for L and S parallel and pointing in opposite directions. The highest energy value will be for L and S parallel and pointing in the same direction. - [Loophole comparrison toy model 0 and toy model A](https://www.hyperfinecourse.org/forums/topic/loophole-comparrison-toy-model-0-and-toy-model-a/) - If the charges in toy model 0 are equal but opposite, then adding a -epsilon charge in toy model A will cause the total charge of the system to be negative. And thus the system in toy model A is in a lower energy state. The monopole shift still gives relatively positive distributions but inside - [Carousel](https://www.hyperfinecourse.org/forums/topic/carousel-3/) - The magnetometer should be placed in the middle of the carousel. This represents the nucleus. The child going around on the carousel represents an electron. If the child holds a bar magnet, then this can represent the spin of the electron, which will cause the spin-dipolar contribution. If the child is holding an electrically charged - [time averaged position](https://www.hyperfinecourse.org/forums/topic/time-averaged-position/) - The electron is really fast and zips around the nucleus in a flash. It's so quick that you can't really tell exactly where it is at any one moment because it's always moving. But if you sit there and watch it for a long time, say, for many zips around the nucleus, you'll start to - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-15/) - We know the formula: g = mu*hbar/(mu_N*I) a) g = -0.5940*2*hbar/mu_N g = -0.766*(2/5)*hbar/mu_N b) g = 2*hbar*mu_B/mu_N c) mu = -3.826*(1/2)*mu_N/hbar - [Time-average position of a 1s electron](https://www.hyperfinecourse.org/forums/topic/time-average-position-of-a-1s-electron/) - The electron moves a lot in time. It has however certain positions in the universe where it is more probable to see the electron. That is why the time-averaged picture of the 1s electron in hydrogen seems like a ball. The electron can most probably be found anywhere inside this ball at a certain time. - [Contradiction](https://www.hyperfinecourse.org/forums/topic/contradiction-4/) - Perhaps the contradiction appears because we are comparing two different systems. Also, maybe the extra term from the small charge will never be larger than the other term, resulting in an overall positive term? - [Task 2](https://www.hyperfinecourse.org/forums/topic/task-2-10/) - If J stays the same, we can use the following relation: a1/a2 = (u1/I1)/(u2/I2) as all the constants in the expression for a cancel out. All of the values in this expression are either known or can be measured except for u2, so we can calculate this value. - [task 1](https://www.hyperfinecourse.org/forums/topic/task-1-14/) - The picture doesn't work for me, so I took a guess of what should be on there. The J value corresponds to: J=L+S. If there are three energy levels and thus three values for J, the lowest value should correspond to the case where L and S are oriented parallel and opposite. The highest value - [observer in hydrogen nucleus](https://www.hyperfinecourse.org/forums/topic/observer-in-hydrogen-nucleus/) - Imagine you're in the nucleus of a hydrogen atom, watching the 1s electron's behavior over time. The electron moves around the nucleus very fast in a seemingly random pattern. The time-averaged position is where the electron spends most of its time on average. The s orbital is spherical in shape and centered around the nucleus. - [Toy model contradiction](https://www.hyperfinecourse.org/forums/topic/toy-model-contradiction/) - Maybe the contradiction lies in the fact that we are comparing two systems with a different charge to begin with? Maybe we have to compare the toy model A with a system which has a negative charge -2e -epsilon where all of the negative charge is outside of the nucleus instead of the system with - [time-averaged position to a 15-year old](https://www.hyperfinecourse.org/forums/topic/time-averaged-position-to-a-15-year-old/) - I would say to the child that the electron moves very fast in the atom and that we never exactly know where it is. Scientists can perform measurements and can take a lot of "pictures" at certain times. The "time-averaged position" is then like figuring out where the electron spends most of its time on - [Perturbation Theory and Multipole Expansions](https://www.hyperfinecourse.org/forums/topic/perturbation-theory-and-multipole-expansions/) - If you try to study a general shaped nucleus without making a multipole expansion first, you can run into complications with systems that can be unsolvable. Thus, you need to make a multipole expansion in order to take into account the part of the multipole that contributes the most to what you're calculating. This breaks - [Lowest energy configuration](https://www.hyperfinecourse.org/forums/topic/lowest-energy-configuration/) - For α0, - [The position of the electron](https://www.hyperfinecourse.org/forums/topic/the-position-of-the-electron/) - The electron is on average spaced at a certain distance (the Bohr radius) away from the nucleus. There is no preferred direction for the electron to be in, it is just as likely to be above you or underneath you, to the left or to the right... - [using perturbation theory prematurely](https://www.hyperfinecourse.org/forums/topic/using-perturbation-theory-prematurely/) - I don't know for sure. I need to read up on Hamiltonians, it appears. But, it looks like perturbation theory seeks to approximate an entire system/quantity by approximating increasingly small "perturbations" to a known system/quantity. Without doing an expansion on this system before trying to use the perturbation theory, we would not have set the - [lowest energy dumbbell orientation](https://www.hyperfinecourse.org/forums/topic/lowest-energy-dumbbell-orientation/) - For all alpha, the lowest energy orientation of the dumbbell would be that which maximizes negativity of the quadrupole contribution, since the quadrupole term will always be smaller in magnitude than the monopole term. As (2(cos(theta))^2-(sin(theta))^2) is equal to (2-3(sin(theta))^2), the extrema for this term are 2 and -1. So, if alpha < 0, then - [Complication without Monopole](https://www.hyperfinecourse.org/forums/topic/complication-without-monopole/) - Using multipole expansion simplifies calculations by isolating the nucleus and electron systems. Without this the calculations would be very complicated due to the fact that we have to calculate nucleus and electrons at once. - [Lowest energy orientation](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-4/) - alpha>0 implies theta is 0, dumb-bell lies parallel to the z axis. alpha - [Perturbation theory and Multiploe expansion](https://www.hyperfinecourse.org/forums/topic/perturbation-theory-and-multiploe-expansion/) - Like any complex system in physics, it is best to break down the system to its most basic part. Create a 'toy model' and add complexity to it. Of course the toy model should be mostly accurate otherwise the perturbation becomes larger than the original system which leads to the model becoming too complicated. Therefore, - [Complications](https://www.hyperfinecourse.org/forums/topic/complications-4/) - 1. Not so sure, the potential might not be analytically solvable , whereas supposedly a multipole expansion to every order is exactly solvable. 2.1 The usefullness of the multipole expansion is in cutting it off, if the particular charge distribution at hand is an n+1 pole or doesn't converge quickly I would think that this - [Lowest Energy Orientation](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-3/) - When Alpha is zero, it does not matter what the angles are (as they are multiplied by zero). When Alpha > 0 (separation between the wings is larger than the radius of the wings) the lowest energy state should be when theta equals to 0/180 degrees as that keeps the energy negative. When Alpha - [Multipole Expansion](https://www.hyperfinecourse.org/forums/topic/multipole-expansion-2/) - The perturbation theory without a previous multipole expansion would lead to very hard to solve calculations. - [Dumb-bell](https://www.hyperfinecourse.org/forums/topic/dumb-bell/) - a>0: When the dumb-bell is parallel to the z-axis it will have the lowest potential energy. a=0: the quadruple moment is zero, then the orientation doesn't matter. a - [Complications of multipole expansion](https://www.hyperfinecourse.org/forums/topic/complications-of-multipole-expansion/) - If you don’t use multipole expansion first than the calculations, for he interactions, will become very complicated when using the pertubation theory on the system. - [lowest-energy orientation of the dumb-bell](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-of-the-dumb-bell-4/) - alpha > 0: When the orientation of the dumbbell is parallel with the z-axis than the lowest energy is obtained. alpha = 0: The orientation of the dumbbell is not a factor to obtain the lowest energy alpha < 0: When the orientation of the dumbbell is perpendiculaire with the z-axis than the lowest energy - [Complications](https://www.hyperfinecourse.org/forums/topic/complications-3/) - I see two possible complications: If you assume a general nucleus where not r_n - [Lowest energy contribution.](https://www.hyperfinecourse.org/forums/topic/lowest-energy-contribution/) - I think the following for the lowest energy distribution of the dumbbell: In case α>0, the configuration with the dumbbell axis perpendicular on the plane of the rings (along the z-axis) is minimal. In case α=0, all configurations have minimal energy. In case α - [Use of multipole expansion](https://www.hyperfinecourse.org/forums/topic/use-of-multipole-expansion/) - The interaction term can become very complicated due to the shape of the nucleus, to combat the complication of this interaction term, the multipole expansion is introduced. - [Hierarchy and Perturbation](https://www.hyperfinecourse.org/forums/topic/hierarchy-and-perturbation/) - The reason why the perturbation treatment of quantum mechanics is aided by the multipole expansion in such an investigation of electrodynamics is because the multipole expansion allows for an immediate sense of energy scale hierarchy for the interactions between the charged particles and currents, meaning that the order of perturbation can be easily deduced from - [Double Ring Question](https://www.hyperfinecourse.org/forums/topic/double-ring-question/) - For a dumbell suspended between two rings, the configurations are given by the sign of alpha, a factor that is wholly dependent on the difference between the double of the square of the radius of the rings and the square of their distance from each other. For alpha positive, it is a ring radius sufficiently - [Quantum Multipole Expansion](https://www.hyperfinecourse.org/forums/topic/quantum-multipole-expansion-2/) - The interaction term between the nucleus and the electron cloud depends on the shape of the nucleus. This shape can be very complicated, because of this a multipole expansion is made of the interaction. - [Min. Energy of dumb-bell](https://www.hyperfinecourse.org/forums/topic/min-energy-of-dumb-bell/) - For alpha > 0, the dumb-bell is oriented along the z-axis at minimum energy. For alpha = 0, the minimal energy does not depend on the orientation. For alpha < 0, the dumb-bell is oriented perpendicular to the z-axis at minimum energy. - [quadrupole energy](https://www.hyperfinecourse.org/forums/topic/quadrupole-energy/) - For alpha = 0 there's seems to be no dependence on the angle theta For alpha =1 >0 we arrived at the theta =0 being the lowest quadrupole energy configuration For alpha - [Minimum energy of dumbell](https://www.hyperfinecourse.org/forums/topic/minimum-energy-of-dumbell/) - - When alpha > 0, the minimum energy configuration is obtained when the dumbell lies parallel to the z-axis (theta = 0). - When alpha < 0, the minimum energy configuration is the one where the dumbell lies perpendicular to the z-axis (alpha = 90). - When alpha = 0, there is no quadrupole term - [Use of the multipole expansion](https://www.hyperfinecourse.org/forums/topic/use-of-the-multipole-expansion/) - You will obtain a very complex interaction potential, which will be significantly simplified if you first expand the potential of the nucleus into a multipole expansion. The ground state also already requires the monopole moment of the nucleus, which is already a multipole moment of the nucleus. The multipole expansion is only valid if you - [Orientation for the lowest energy](https://www.hyperfinecourse.org/forums/topic/orientation-for-the-lowest-energy/) - For alpha >0 I expect the energy to reach a minimum when the dumbbell is oriented parallel to the z-axis. For alpha = 0 the enrgy will not depend on the orientation of the dumbbell. For alpha < 0 I think the energy will reach a minimum if the dumbbell is oriented perpendicular to the - [contradiction](https://www.hyperfinecourse.org/forums/topic/contradiction-3/) - Could it be that when adding the small negative charge inbetween the positive dumbell that the positive charges are attracted more tho the centrum thus putting them closer toghetter. This could then result in the outer negative charges getting closer to the centrum thus lowering the overal potential? - [explanation](https://www.hyperfinecourse.org/forums/topic/explanation-2/) - Imagine you're sitting inside a tiny room at the center of a hydrogen atom, watching the electron move around you. After a while, you notice that although the electron zips around unpredictably, it spends most of its time closer to you, near the center of the atom. So, if you were to average out its - [why use multipole expansion](https://www.hyperfinecourse.org/forums/topic/why-use-multipole-expansion/) - You would run into complications when not using a multipole expansion, because it would be very difficult to derive the answer. - [lowest energy answer](https://www.hyperfinecourse.org/forums/topic/lowest-energy-answer/) - For alpha > 0, the dumbbell has the lowest energy if it is oriented along the z-axis. For alpha = 0, the dumbbell energy is indepentent of its orientation. For alpha < 0, the dumbbell has the lowest energy if it is oriented along the xy-axis. - [nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-31/) - nuclear spin, quadrupole moment, mass, mass number, radius, energy - [multipole expansion](https://www.hyperfinecourse.org/forums/topic/multipole-expansion/) - Without using the multipole expansion, the calculations of the interaction term would be much more difficult. The multipole expansion makes it easier to calculate. - [lowest energy orientation](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-2/) - alpha > 0: dumbbell along z axis alpha < 0: dumbbell perpendicular to z axis alpha = 0: energy independent of orientation dumbbell - [why multipole expansion](https://www.hyperfinecourse.org/forums/topic/why-multipole-expansion/) - – for complex shapes of the nucleus perturbation theory would be very unaccurate because it normally just adds small perturbations to a solvable system. Thus first expanding the shape into many more solvable cases and then pertubating it, the error we would get should be way smaller. - [Lowest energy orientation of the dumbbell](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-of-the-dumbbell-3/) - for a > 0 most negative contribution if along z-axis For a < 0 most negative contribution if no disk in xy-plane for a = 0 orientation doesn't matter - [Complications without multipole expansion](https://www.hyperfinecourse.org/forums/topic/complications-without-multipole-expansion-2/) - The expression we get from the electron-nucleus interaction will be too complicated if we didn’t made a multipole expansion first. We would not be able to solve it analytically. One would get an expression that becomes very hard to handle I think. - [lowest-energy orientation of the dumbbell](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation-of-the-dumbbell-2/) - I think that in case: - alpha>0, the dumbbell must be oriented along the z-axis - alpha=0, the dumbbell's orientation can be something in between the z-axis and the xy-plane - alpha - [multipole expansion and perturbation](https://www.hyperfinecourse.org/forums/topic/multipole-expansion-and-perturbation/) - The multipole expansion makes it easy to know the general orders of energy that each term brings with it, which is very usefull for perturbation theory as this makes it clear which terms can be regarded as perturbations and which terms can not. One can only truncate when you are sure that the states that - [Quadrupole moment dependence on configuration](https://www.hyperfinecourse.org/forums/topic/quadrupole-moment-dependence-on-configuration/) - I expect that if the rings are in the alpha>0 configuration the dumbell would need to lie on the z axis for the lowest energy, seeing as this would put the masses of the dumbell closest to rings, if alpha < 0 I expect that te dumbell has to be oriented in the xy plane - [Perturbation questions](https://www.hyperfinecourse.org/forums/topic/perturbation-questions/) - - Without making a multipole expansion, the potential due to the general shape of the nucleus would be highly complex and difficult to handle mathematically. Perturbation theory relies on treating the perturbing potential as a small deviation from a known, solvable system. If the potential is too complex, it becomes challenging to apply perturbation theory - [Answer lowest energy](https://www.hyperfinecourse.org/forums/topic/answer-lowest-energy/) - If alpha > 0, like alpha =1 the energy is lowest when theta is 0 or 180°, the dumbbell is oriented parallel to the z axes. If alpha - [complications without multipole expansion](https://www.hyperfinecourse.org/forums/topic/complications-without-multipole-expansion/) - If we were to not use a multipole expansion, the interaction term would be very hard to calculate. The multipole expansion ensures that we can systems of the nucleus and the electrons, which makes the calculation easier. - [lowest-energy orientation](https://www.hyperfinecourse.org/forums/topic/lowest-energy-orientation/) - If alpha is larger than 0, the angle should be 0 degrees, so the dumb-bell should be parallel to the z-axis. If alpha is smaller than 0, the angle should be 90 degrees, so the dumb-bell should be in the xy-plane. If alpha is 0, the energy is independent of the orientation of the dumb-bell. - [Wave functions](https://www.hyperfinecourse.org/forums/topic/wave-functions/) - In the third case for the ellipsoid when calculating the monopole, dipole, and quadrupole moments, it was never explained how to obtain the charge density used to calculate these values. When I searched this online, the only thing that I found was that I need the nuclear wave function. This leads me to my question, - [Nuclear Wave Function](https://www.hyperfinecourse.org/forums/topic/nuclear-wave-function/) - I was wondering what the nuclear wave function was, explicitly. I keep seeing it mentioned in different texts, but it's never explicitly written out. - [I Feel I Misunderstood the Prerequisites](https://www.hyperfinecourse.org/forums/topic/i-feel-i-misunderstood-the-prerequisites/) - Reading the introductory materials, I understood this class to be suitable for anyone with a bachelors degree in a scientific or engineering major. With a Bachelors and Masters in Mechanical Engineering, I felt confident going into this, but after this section, I feel completely unprepared. I have no knowledge of these concepts and it seems - [Moments of Cd-111](https://www.hyperfinecourse.org/forums/topic/moments-of-cd-111-7/) - Magnetic Dipole Moment: -0.766 micro? Electric Quadrupole Moment: +0.64 ? - [Multipole Moment Example](https://www.hyperfinecourse.org/forums/topic/multipole-moment-example/) - I am somewhat embarrassed to submit an answer, as this was my first time hearing about electric or magnetic multipole moments at all. I think I am beginning to understand. The only example I can think of is on the molecular level, rather than the atomic level, with polar molecules. Water being a dipolar molecule. - [Moments of 111Cd](https://www.hyperfinecourse.org/forums/topic/moments-of-111cd/) - The magnetic dipole moment for 111Cd at 245 keV is -0.766(3) nm. The electric quadrupole moment for 111Cd at 245 keV is +0.64(3) b. - [How does the table get updated?](https://www.hyperfinecourse.org/forums/topic/how-does-the-table-get-updated/) - As the title says I am wondering how exactly the database gets new or updated data? Does the article/report with new data need to be peer-reviewed? Or is there a certain number of articles that can getting the same result confirm this new information before getting added on? - [Shape of Nuclei](https://www.hyperfinecourse.org/forums/topic/shape-of-nuclei/) - Multipoles play an important role in shaping how a nuclei is shaped based on how the multipoles impact the parameters for the shape. It can stretch or squash a nuclei in a particular way to create more prolate or oblate shapes. - [Multipole Moment in Daily Life](https://www.hyperfinecourse.org/forums/topic/multipole-moment-in-daily-life/) - MRI machines use multipole moment of hydrogen nuclei in human body to assess different tissues. - [Example of multipole](https://www.hyperfinecourse.org/forums/topic/example-of-multipole/) - Modeling of a dipole antenna - [My thoughts on Nuclear Properties](https://www.hyperfinecourse.org/forums/topic/my-thoughts-on-nuclear-properties/) - Some Nuclear properties I think of are: Mass (and by extension number of protons and neutrons) Lifetime (and if it is a isotope) Shape (is the nucleus close to a magic number and if it is even/even, even/odd or odd/odd) - [Hello, Anders here!](https://www.hyperfinecourse.org/forums/topic/hello-anders-here/) - I am Anders Rasmussen, I come from Denmark and did my bachelor in Physics in Groningen and wanted to continue to do a masters here in Leuven, Belgium as my interest is in Nuclear Physics. I would say all the typically stuff any Flemish person would say to visit Leuven, its got a great cultural - [Moments of Cd-111](https://www.hyperfinecourse.org/forums/topic/moments-of-cd-111-6/) - The magnetic dipole moment for the 245 keV level is -0.766(3) nm and the electric quadripole moment is: +0.64(3) b - [Technical Difficulties](https://www.hyperfinecourse.org/forums/topic/technical-difficulties/) - I posted a topic from the course materials page but now I can't see it or anyone else's. I will try to post from here to see if that works better. - [What Amelia knows about the atomic nucleus](https://www.hyperfinecourse.org/forums/topic/what-amelia-knows-about-the-atomic-nucleus/) - Composed of protons and neutrons, and therefore positively charged. Can have a mismatch in the number of protons vs. number of neutrons. The number of protons defines which element the atom is, and the various neutron counts are classified as isotopes. Some arrangements are unstable, and decay over time, emitting radiation. The protons and neutrons - [multipole moments](https://www.hyperfinecourse.org/forums/topic/multipole-moments-5/) - Chemical Reactivity: The arrangement of atoms within a molecule and the resulting multipole moments affect chemical reactivity. In chemical reactions, understanding the charge distribution helps predict how molecules will interact and form new compounds. - [properties of a nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-a-nucleus-4/) - number of protons and neutrons positive charge accounts for almost all of the mass of an atom spherical or distorted shape stability/radio activity nuclear spin ground and excited state - [Nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-30/) - - Mass number - Atomic number - Nuclear radius - Charge - Nuclear spin - [properties of a nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-a-nucleus-3/) - postively charged protons, neutrons with neutral charge, accounts for almost all of the mass of a atom, spherical or distorted shape, stability/radio activity, nuclear spin, ground and excited states - [Question about multipole radiation](https://www.hyperfinecourse.org/forums/topic/question-about-multipole-radiation/) - Is only gamma radiation a form of multipole radiation, or are the other types of radiation (alpha, beta) multipole radiation too? I am confused about this because the video mostly talks about electromagnetic radiation. - [Example multipole moment](https://www.hyperfinecourse.org/forums/topic/example-multipole-moment-2/) - The earth is (until first order) a magnetic dipole. - [nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-29/) - Watched the video first, my bad. Would have said, radius, charge, mass, spin. - [Example Multipole moment](https://www.hyperfinecourse.org/forums/topic/example-multipole-moment/) - When following the course 'Astrophysical simulations', last semester, there was a method which used multipole expansion to perform an approximation of the n-body problem: Barnes-Hut simulation. This method distributed the volume with all particles into different subvolumes (of different magnitude) with each one particle. This way, only the particles from such nearby cubic cells should - [Examples](https://www.hyperfinecourse.org/forums/topic/examples/) - Radio antennae operate under the rapid oscillation of dipole charges and many chemical systems are held together through intermolecular bonds that rely on the intrinsic dipole moments of those molecules. - [Properties of nuclei](https://www.hyperfinecourse.org/forums/topic/properties-of-nuclei-4/) - The nucleus is made of protons and neutrons, where protons are positively charged particles, and neutrons are particles without charge. Around this net-positive nucleus, a cloud of negatively charged electrons resides. The protons and neutrons are bound together through the nuclear force, a result of the strong interaction. - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-15/) - Hi, I'm Wannes Beirlant, or WannesB as username. I'm based in Ghent, Belgium, where I study Physics and Astronomy at UGent. A very good reason to visit Ghent in comparison to other cities are the well-known 'Gentse Feesten' in the summer. In this course I'm primarily interested in the implementation of hyperfine interactions in experimental - [Nuclear Properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-28/) - Properties of Nucleus: 1. Mass 2. Atomic Number and number of neutrons. 3. Size 4. Spin 5. Parity 6. Isospin 7. Shape - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-14/) - Hello, I am Himanshu. I will be taking this course from Leuven, Belgium. One of the reasons you should consider visiting my city is because it's extremely beautiful with rich history. I did my bachelor's from University of Mumbai, and I am currently doing my masters from KU Leuven. I hope to learn about intricacies - [Dipole and Cuadruple moment Cd-111](https://www.hyperfinecourse.org/forums/topic/dipole-and-cuadruple-moment-cd-111/) - According to the first website, the magnetic dipole moment for the isotope is -0.766 nm and the quadrupole moment is +0.64 b. - [Moments of Cd-111](https://www.hyperfinecourse.org/forums/topic/moments-of-cd-111-5/) - The magnetic dipole moment, µ, for Cd-111 at 245 keV is -0.766(3) nm. The electric quadrupole moment, Q, for Cd-111 at 245 keV is +0.64(3) b. - [Dipole and quadrupole moments for calculating London forces](https://www.hyperfinecourse.org/forums/topic/dipole-and-quadrupole-moments-for-calculating-london-forces/) - In order to calculate the London forces in molecules due to fluctuating electrons, it is useful to only consider terms with the highest order moment, since their magnitude falls off quickly with increased distance. In this way, an easy approximation for the London forces on neighboring molecules in e.g. a gas can be found. - [Properties of the nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-the-nucleus-8/) - - pos. charge - mass - binding energy - electromagnetic moments - spherical / deformed - protons + neutrons - decay - strong & coulomb force - fm - [Properties of the nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-the-nucleus-7/) - - Mass - Binding energy - build up from neutrons and protons - spin / isospin - can decay & emit radiation - electromagnetic moments - spherical / deformed - strong & coulomb force - fm - [Moments of Cd-111](https://www.hyperfinecourse.org/forums/topic/moments-of-cd-111-4/) - Properties of 245 KeV level in Cd-111 Magnetic dipole moment: -0.766(3) nm (INDC(NDS)-0816) Nuclear quadrupole moment: 0.64(3) b (indc-nds-0833) - [Multipole Moments in Daily Life](https://www.hyperfinecourse.org/forums/topic/multipole-moments-in-daily-life/) - Multipole expansions play a role in Newtonian gravity. You take the gravitational potential for a mass distribution from the reference frame where the origin is the center of mass of the object, and expand it into a monopole, dipole, quadrupole, etc.. This expansion shows that the elongation of a body causes a lower potential in - [Properties of nuclei](https://www.hyperfinecourse.org/forums/topic/properties-of-nuclei-3/) - - Mass - Isotope - Spin - Stability - Binding energy - [Properties of nuclei](https://www.hyperfinecourse.org/forums/topic/properties-of-nuclei-2/) - - Number of nucleons - Binding energy - Spin - Stability - [Properties of the Nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-the-nucleus-6/) - My list: The nucleus is held together by the strong force. From an isotope, you can get the mass of the nucleus, the number of protons and neutrons, whether the nucleus is stable or not, and the charge of the nucleus. Report: The properties of the nucleus, in total, are: the number of protons (Z) - [multipole moments](https://www.hyperfinecourse.org/forums/topic/multipole-moments-4/) - Multipole moments play a role in the interaction between the nucleus and electrons of a atom. This is because the multipole moments in the nucleus, will not having a exact sphericall form, will have influence on the electrons and so influence the electron density of the atom - [My multipole examples](https://www.hyperfinecourse.org/forums/topic/my-multipole-examples/) - For my bachelor project, I worked with gravitational multipoles. Here we showed that the shape of the Earth itself (and therefore its gravitational multipoles) have a non-negligible influence on the orbits of the satellites. It is therefore important to include those multipole effects in calculating the orbit of the satellite to ensure accurate GPS locations, - [My nuclear properties](https://www.hyperfinecourse.org/forums/topic/my-nuclear-properties/) - Nuclei contain both protons and neutrons. They have a proton number Z and a neutron number N that represent the number of protons and neutrons within the nucleus respectively. These protons and neutrons are held together by the strong nuclear force, while the protons repel eachother via the Coulomb interaction. Nuclei also gain properties from - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-13/) - Hi everybody! My name is Jef Vermeiren and I am currently enrolled in the master in theoretical physics in Leuven. Leuven is a relatively small city and is very clean compared to other cities. Most of the people that live here are students, so there are a lot of places that give discounts to young - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-12/) - I am Yannick from the city of Antwerp. Despite Antwerpers notorious arrogance I don't think there's particularly much to see in Antwerp compared to say Bruges or Ghent, drinking a coffee in front of the cathedral at night is nice though. I am doing a master's in physics at the VUB, I am interested in - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-11/) - Hello, I'm Cole. I'm taking this course in Leuven, Belgium. You should visit my city at least once in your life to take a tour of the Stella Artois Brewery and eat some Belgian fries. I'm currently doing a master in physics with a major profile of theoretical physics and a minor profile of condensed - [Post First Forum](https://www.hyperfinecourse.org/forums/topic/post-first-forum-2/) - Testing, part 2. - [First Forum](https://www.hyperfinecourse.org/forums/topic/first-forum-5/) - Testing. - [Lotte B](https://www.hyperfinecourse.org/forums/topic/lotte-b/) - Hi! I'm Lotte. I am a nuclear physics student at the KU Leuven in Belgium. I like Leuven because I think the city feels very welcoming. It is really a student city so the majority of people you meet on a street are younger people going to or coming back from their classes, going out - [Properties of nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-nucleus-3/) - (Equal) number of protons and neutrons Positive electrical charged spin - [Properties of a Nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-a-nucleus-2/) - Key points of the nucleus, in order of classical to quantum complexity Mass Charge Nucleon composition Mean radius Half-life (lifetime) Inertial tensor (deformation parameter) Magnetic dipole moment Electric quadrupole moment Energy (energy levels) Nuclear spin Parity Electric dipole moment (if ever found) - [Kirill Danilov](https://www.hyperfinecourse.org/forums/topic/kirill-danilov/) - Greetings, My name is Kirill Danilov. I am a PhD student working for Professor Lino da Costa Pereira in the Quantum Solid State Physics division of KU Leuven. I am taking this course primarily for my personal interest in the experimental work that can probe the energy levels of nuclear (or potentially even subnuclear) effects - [what](https://www.hyperfinecourse.org/forums/topic/what/) - what - [Nucleus Properties](https://www.hyperfinecourse.org/forums/topic/nucleus-properties-2/) - Charge, mass, spin, angular momentum - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-10/) - Hi everyone! I'm Carlos (CJ11). I'm greeting you from Darmstadt, Germany, a place that you should definitely have to visit. It's not only a university-centered city but it has beautiful places and at the same time avoids the big city environment. I may invite you a drink if you decide to come! I'm currently a - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-9/) - Hi, my name is Louis and i'm a master physics student. I'm following this course from Antwerp which you need to visit for its beautifull buildings and many good restaurants. I'm expecting from this course to understand more about hyperfine interactions sinds my knowledge about is limited. I hope this course can help me to - [The properties that come to mind](https://www.hyperfinecourse.org/forums/topic/the-properties-that-come-to-mind-2/) - the atomic number, number of neutrons, mass number and mass itself, magnetic moment - [The properties that come to mind](https://www.hyperfinecourse.org/forums/topic/the-properties-that-come-to-mind/) - atomic number, neutron number, mass number and the mass itself, lifetime, magnetic moment - [multipole moment](https://www.hyperfinecourse.org/forums/topic/multipole-moment/) - departure from sphericity in a nucleus means the electric quadrupole moment is nonzero - [properties](https://www.hyperfinecourse.org/forums/topic/properties-5/) - charge, mass, spin, parity, radius, quadrupole moment - [Amelia Sheffler](https://www.hyperfinecourse.org/forums/topic/amelia-sheffler/) - My name is Amelia Sheffler. I am in Los Alamos, New Mexico, in the United States. You should visit Los Alamos for the beautiful landscapes and history. My education is in mechanical engineering, specializing in controls. For many years I worked at a steel mill (located in Ghent, Kentucky!), which was very exciting, but I - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-8/) - Hello, I am Jana, I am taking this course from Leuven, Belgium. You should visit my city at least once in your life to try Belgian fries. I am currently doing a master in physics with a profile in nuclear physics and I am taking hyperfine interactions as part of my program. I expect this - [first forum](https://www.hyperfinecourse.org/forums/topic/first-forum-4/) - blablabla - [This is the one I want to test](https://www.hyperfinecourse.org/forums/topic/this-is-the-one-i-want-to-test/) - test - [Moments or Cd-111](https://www.hyperfinecourse.org/forums/topic/moments-or-cd-111/) - The magnetic dipole moment µ that i found is: -0.766(3) nm The electric quadripole moment Q that i found is: +0.64(3) b - [Moments of 245 keV 111Cd](https://www.hyperfinecourse.org/forums/topic/moments-of-245-kev-111cd/) - THe magnetic dipole moment of 111 Cd is -0.766(3) nm. The electric quadrupole moment of 111 Cd is +0.64(3) b. - [Example of multipole moments](https://www.hyperfinecourse.org/forums/topic/example-of-multipole-moments-5/) - I think drawing could be considered an example of multipole moments. You first start with a rough sketch, and then start adding increasingly more fine levels of detail. - [nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-27/) - Charge, mass number, spin, life-time, magnetic dipole moment - [My oppinion on the nucleus (2?)](https://www.hyperfinecourse.org/forums/topic/my-oppinion-on-the-nucleus-2/) - (I don't know if my first answer went through, so I will give it again to be sure. -Number of protons -Number of neutrons -Total number of nucleons -Wether there is an even or odd number of protons - [My oppinion on the nucleus](https://www.hyperfinecourse.org/forums/topic/my-oppinion-on-the-nucleus/) - -Number of protons -Number of neutrons -Total number of nucleons -Wether there is an even or odd number of protons - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-7/) - Hello! My name is Niels, I am taking this course in Gent. You should go to Gent to see a lot of impressive buildings, such as the Gravensteen. I am currently in my first year of being a master student in physics and astronomy. For this course, I hope to learn more about experimental techniques - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-6/) - Hi all I'm Ruben or RubenVanderBorght. I live in Leuven, Belgium and am following this course at the KU Leuven. Leuven is quite a picturesque city which makes it definitely worth visiting. Certainly visit the science campus at the more wooded side of Leuven. I did a bachelors in Physics and Mathematics at the KU - [First forum post-first](https://www.hyperfinecourse.org/forums/topic/first-forum-post-first/) - Try out - [First forum](https://www.hyperfinecourse.org/forums/topic/first-forum-3/) - blablabla - [nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-26/) - charge, mass number, spin, life-time, energy, excitation states, parity, shape, size/radius, magnetic dipole moment, electric quadrupole moment, - [Example where multipole moments play a role](https://www.hyperfinecourse.org/forums/topic/example-where-multipole-moments-play-a-role/) - One example of where multipole moments play a role: If I recall correctly, then multipole moments can tell us something about the temperature fluctuations of the cosmic microwave background. - [Properties of the nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-the-nucleus-5/) - - number of neutrons - number of protons - spin - life-time - charge (but one can get this from the number of protons) - [properties](https://www.hyperfinecourse.org/forums/topic/properties-4/) - Spin, number of protons and neutrons, charge, radius - [Hi, I'm Danté](https://www.hyperfinecourse.org/forums/topic/hi-im-dante/) - I'm Danté, a student from Ghent university. I live in Izegem, a nice city in West-Flanders. It used to be famous for the several shoe makers and there even is a Shoe museum, which i guess is kind of interesting. Also the birthplace of the famous T'hof van commerce. I study physics at Ghent and - [Example of usecase of a multipole moment](https://www.hyperfinecourse.org/forums/topic/example-of-usecase-of-a-multipole-moment/) - MRI (magnetic resonance imaging): MRI relies on the interaction between magnetic fields and the magnetic moments of atomic nuclei, particularly hydrogen nuclei (protons) in water. - [nucleus properties](https://www.hyperfinecourse.org/forums/topic/nucleus-properties/) - -Radius (root mean square) -energy -spin -Magnetic/electric moments -charge -Isospin -Mass -deformation (oblate, prolate) or (beta, gamma parameters) - [Example multipole moments](https://www.hyperfinecourse.org/forums/topic/example-multipole-moments-6/) - I think if we just look at magnets, then we have multiple moments. - [properties](https://www.hyperfinecourse.org/forums/topic/properties-3/) - mass spin magn dipole radius parity - [Properties of nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-nucleus-2/) - protons neutrons -> nucleons strong force coulomb force charge - [multipole moments](https://www.hyperfinecourse.org/forums/topic/multipole-moments-3/) - The electric dipole moment in H2O molecules leads to defining properties like surface tension. Small animals can then use this property to walk over water without sinking. - [Properties of nucleus](https://www.hyperfinecourse.org/forums/topic/properties-of-nucleus/) - Protons Neutrons -> Nucleons Strong force Coulomb force mass - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-5/) - My name is Ella. The city and country from where I take this course is Belgium, Leuven. One reason why you should visit my city at least once in your life is that there are a lot of good places to eat and a lot of nice people. My background is a bachelor in physics - [About me...](https://www.hyperfinecourse.org/forums/topic/about-me-4/) - Hi there. My name is Fien. I am a master student Physics & Astronomy from Beersel. When visiting Beersel, a few kilometers from the city is the Hallerbos. Here you can have a beautiful walk between the forest hyacinths in the spring. Currently, I am doing my master's thesis on the detection of cosmic muons - [nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-25/) - N = neutron number Z = proton number = charge A = N + Z = mass number radius, charge, quadrupole moment, spin, parity - [Dorien Vanbinst](https://www.hyperfinecourse.org/forums/topic/dorien-vanbinst/) - Hello, my name is Dorien and I will be following this course from Leuven. Leuven is a beautiful city with a lot of historical architecture and it is ofcourse the home of the stella artois brewery. I am currently in the first year of my master, specializing in nuclear physics. I hope to learn more - [post first forum](https://www.hyperfinecourse.org/forums/topic/post-first-forum/) - blablabla - [first forum](https://www.hyperfinecourse.org/forums/topic/first-forum-2/) - blablabla - [Bram Hendrickx](https://www.hyperfinecourse.org/forums/topic/bram-hendrickx/) - Hello all My name is Bram Hendrickx and I study this course from Belgium usually from home (in a small town called Bunsbeek) at the university of KU Leuven. I honestly don't think the small town i'm from has much worth noting as to get people to visit, however i do think that Leuven is - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-3/) - Hi, I'm Jakob Vermeulen. I'm from Markegem in West-Flanders, Belgium but have a dorm in Ghent from which I follow this course. Ghent is fun to visit because there is a lot to do and see! I'm in the first master year of physics and astonomy at Ghent University. In this course I expect to - [fourth forum](https://www.hyperfinecourse.org/forums/topic/fourth-forum-2/) - test. - [fourth forum](https://www.hyperfinecourse.org/forums/topic/fourth-forum/) - test. - [My first forum](https://www.hyperfinecourse.org/forums/topic/my-first-forum/) - This is a test. - [Cd in PAc](https://www.hyperfinecourse.org/forums/topic/cd-in-pac/) - Well in PAC for 111mCd they supposedly play a role. Suppose we understand better why later. - [Properties](https://www.hyperfinecourse.org/forums/topic/properties-2/) - Mass, Numbers Protons, Neutrons, deforamtion(oblate, prolate), lifetime, decay, Parity, quarks - [About me](https://www.hyperfinecourse.org/forums/topic/about-me-2/) - Hi I am Hannes, I take this course from Dresen. A beautiful city with a rich history. Many inventions were done here, as for example the bra. I am a physics master student, that is working on PAC. This course was recommended to me by my supervisor. I work on domain walls using PAC: To - [My Expectations](https://www.hyperfinecourse.org/forums/topic/my-expectations/) - Hi I am here to learn Hyperfine Interactions in detail. - [Properties](https://www.hyperfinecourse.org/forums/topic/properties/) - The nucleus has Z, N and A-numbers, a charge, a mass, a lifetime and a size. - [nuutti_postila](https://www.hyperfinecourse.org/forums/topic/nuutti_postila/) - Hello, my name is Nuutti Postila. I come from Jyväskylä, Finland. You should visit Jyväskylä if you are interested in Alvar Aalto's architecture. There are a lot of buildings designed by him. I am currently studying a BSc and MS in physics. My interests are material physics and science education. I except to learn more - [SG](https://www.hyperfinecourse.org/forums/topic/sg/) - Hey..I am new here to learn basics of this method - [PhZ](https://www.hyperfinecourse.org/forums/topic/phz/) - Hi, I'm Penghan, currently working in Bath, UK. Bath is a unique city in the UK, located in the valley of hills. I'm an analytical chemist, hoping to understand better on NMR by taking this course. Don't know if I will get a formal degree in physics in the future, but I will keep learning - [Pawel Butkiewicz](https://www.hyperfinecourse.org/forums/topic/pawel-butkiewicz/) - Hi My name is Pawel, Ia am from Bialysok in Polnad. You should visit Bialystok because you can find a good wife for you here. I graduated in experimental physics and continued my studies by enrolling in a doctoral school two years ago. this is what I expect from this course remind the basic konowledge - [third forum](https://www.hyperfinecourse.org/forums/topic/third-forum/) - Hi - [Nuclear Properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-24/) - Nuclear Mean squared charge radius, Spin, Number of protons/neutrons, mass of the nucleus, deformation - [nuclear properties](https://www.hyperfinecourse.org/forums/topic/nuclear-properties-23/) - Positive charge, concentrated mass, nuclear binding energy - [Introducing Myself](https://www.hyperfinecourse.org/forums/topic/introducing-myself-2/) - I am a PhD at GANIL working on Laser spectroscopy. I have come across this term "hyperfine" during my master thesis. So coming until this course is part of the curiosity to know more about this topic - [CC](https://www.hyperfinecourse.org/forums/topic/cc/) - Undergrad student - [Ivan Polcowñuk](https://www.hyperfinecourse.org/forums/topic/ivan-polcownuk/) - Hello! My name is Ivan, I am from Argentina and I am going to take this course in La Plata, Buenos Aires (Arg). I think you should visit my city because the people are very friendly, the streets of the city are particularly organized and it is also a beautiful place. I am a PhD - [Perturbed angular correction spectroscopy](https://www.hyperfinecourse.org/forums/topic/perturbed-angular-correction-spectroscopy/) - 1-a) The time difference measured between the detection moments of γ1 and γ2 stems from the nuclei’s intermediate state lifetime. Given that not every nucleus in an intermediate state “lives” for an equal amount of time, an exponential decrease in the intermediate states will be observed as part of the nuclear decay process, thereby progressively - [NMR/ON](https://www.hyperfinecourse.org/forums/topic/nmr-on-2/) - The sign of a hyperfine field indicates whether the field that the nucleus "feels" is in either the same (positive) or the opposite (negative) direction as the spin of the electron cloud. Should the hyperfine field be positive, then the spin of the nucleus would be in the same direction as that of the electron - [Perturbed angular correlation spectroscopy](https://www.hyperfinecourse.org/forums/topic/perturbed-angular-correlation-spectroscopy/) - a)As the number of radioactive nuclei decreases over time due to decay, the probability of decay also decreases. b)1-3 detects more events by summing up the signals from two detectors, while 1-2 only detects a fraction of the decay events that occurs in the sample. - [NMR/ON](https://www.hyperfinecourse.org/forums/topic/nmr-on/) - The applied field are used to orient the atomic magnetic moment. If the hyperfine field is positive, then the NMR/ON spectrum would show a shift of the spectral peak to higher magnetic field values. On the other hand, if the hyperfine field is negative, the peak would shift to lower magnetic field values. - [Experimental EPR pictures](https://www.hyperfinecourse.org/forums/topic/experimental-epr-pictures/) - In order to obtain similar spectra to the ones shown, performing the same measurement at different angles on a sample that is placed between the spectrometer’s magnets and suitable microwave frequency must be used. The spectra are then measured by applying the magnetic field and then tuning it across a spread of values. Peaks will - [Photon Energy](https://www.hyperfinecourse.org/forums/topic/photon-energy-2/) - mu_N = 3.15 *10^-8 eV/T, B = 2T Transition energie delta E = E(mJ = -3/2) - E(mJ= -1/2) E(mJ = -3/2)= -g * mu_N*B* (-3/2) = -1 * 3.15 *10^-8 eV/T * 2T * (-3/2) =9.45*10^-8 eV E(mJ= -1/2)= -g * mu_N*B* (-1/2) = -1 * 3.15 *10^-8 eV/T * 2T * (-1/2) = - [EPR - La atom](https://www.hyperfinecourse.org/forums/topic/epr-la-atom/) - When an atom in a magnetic field absorbs a photon it will induce a shift in the spin orientation of the electrons and this will move the system to a higher energy state. - [Introduction to Electron Paramagnetic Resonance](https://www.hyperfinecourse.org/forums/topic/introduction-to-electron-paramagnetic-resonance/) - p.6 "A radiation source for radar waves produces only a very limited spectral region. In EPR such a source is called a klystron. A so-called X-band klystron has a spectral band width of about 8.8-9.6 GHz. This makes it impossible to continuously vary the wavelength similarly to optical spectroscopy." Why are they using a klystron - [EPR on molecules and crystals](https://www.hyperfinecourse.org/forums/topic/epr-on-molecules-and-crystals/) - To obtain the G6 and G7 defect in an EPR experiment with spatial anisotropy, I would need to prepare a sample with anisotropic g-tensors, acquire its EPR spectrum at different orientations, analyze the spectra to obtain the g-tensor components and determine the PAS by diagonalizing the g-tensor matrix. - [Photon energy](https://www.hyperfinecourse.org/forums/topic/photon-energy/) - g = 1, I = 1/2, B_0 = 2T, mu = 3.15 * 10−8 eV/T energy level: E = -g * mu * m_j * B_0 E = -g * mu * m_j * B_0 + (A * m_I * m_J) Hyperfine constant: A = mu * B_hf / I * J = 3.15 * - [EPR on La atoms](https://www.hyperfinecourse.org/forums/topic/epr-on-la-atoms/) - When a free La atom absorbs a microwave photon that matches a specific energy difference determined by the external magnetic field, its electron moves to a higher energy level. The absorption of the microwave photon will cause the electron to change its spin orientation and move to a higher energy level. - [Temperature and radiation](https://www.hyperfinecourse.org/forums/topic/temperature-and-radiation-2/) - The nuclear moment is equal with 1 and the hyperfine field is splitting the energy level in three components. At low temperature the orientation is in the direction of the z axis. - [Temperature and radiation](https://www.hyperfinecourse.org/forums/topic/temperature-and-radiation/) - From the picture, we can say that the nucleus has a spin 1 with three magnetic levels m=1, m=0 and m=-1. At low temperature, the orientation of the nuclear ensemble will be with the lowest energy level m=1. - [symmetry](https://www.hyperfinecourse.org/forums/topic/symmetry-3/) - FE-I: the x, y and z-axes have 4 -fold rotation symmetry, so there the EFG will be zero. Also, there is a diagonal rotation and there is four 3-fold axes of symmetry. FE-II: the z-axis has 4 -fold rotation symmetry and the x and y axes only 2 -fold symmetry. - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-8/) - to go to classical system we should increase the value of spin. - [Axial Symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-9/) - From equation: the V_xx = V_yy =-1, so eta=0 and there is axial symmetry. From picture: the charge distribution is symmetric in the xy plane, which therefore has axial symmetry. - [answer](https://www.hyperfinecourse.org/forums/topic/answer-9/) - when the nucleus becomes smaller with respect to the electron cloud $l/d$ becomes smaller, so the quadrupole term decreases in absolute value. Then the multipole expansion converges more rapidly. - [answer](https://www.hyperfinecourse.org/forums/topic/answer-8/) - The field of electrons is not homogeneous, so Bohr-Weisskopf effect tells us that with different radii of nuclei we will have different energy of interaction in atom. - [answer](https://www.hyperfinecourse.org/forums/topic/answer-7/) - I have no idea. But maybe if the bar magnet is strong and if we put it to the center of the carousel and then we will shift it to the middle of the radius the differences in magnetometer that the child will hold will correspond to the hyperfine shift. - [answer](https://www.hyperfinecourse.org/forums/topic/answer-6/) - g = (mu*hbar)/(mu_N*I) a) E = 0 keV: g = -0.5940*hbar/(mu_N*1/2) E = 245 keV: g = -0.766*hbar/(mu_N*5/2) b) m_B*hbar/(mu_B*1/2) = 2*hbar c) mu = mu_N*g*I/hbar = -3.826*2*mu_N/hbar - [answer](https://www.hyperfinecourse.org/forums/topic/answer-5/) - We know that a \tilde \mu /(IJ), so from the proportion we find that \mu_2=\mu_1 \frac{a_1}{a_2} \frac{I_2}{I_1} - [Recoil energy with a source on a fast train](https://www.hyperfinecourse.org/forums/topic/recoil-energy-with-a-source-on-a-fast-train/) - The increase in momentum of the source is equivalent to a decrease in the momentum of the scatterer. So, the recoil energy of the scatterer is reduced if the source would move at the speed of light. There is some limit on how fast the train can go and the doppler effect could also affect - [Mössbauer spectroscopy](https://www.hyperfinecourse.org/forums/topic/mossbauer-spectroscopy/) - Lowering the temperature more nuclei are aligned and the hyperfine lines are stronger. If the field change the hyperfine splitting would change but they look the same in the picture so the field doesn't change with temperature. - [answer](https://www.hyperfinecourse.org/forums/topic/answer-4/) - J = 2 - Parallel, same direction J = 1 - Perpendicular J = 0 - Parallel, opposite direction - [Rotation axis](https://www.hyperfinecourse.org/forums/topic/rotation-axis/) - Fe-I 4 fold rotation symmetry over x,y and z axis. Fe-II 4 fold rotation symmetry over z axis and 2 fold rotation symmetry over x and y axis. - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-7/) - Infinite value for the spin for an infinite number of descrete orientations. - [Quadrupole operator-Axial symmetry](https://www.hyperfinecourse.org/forums/topic/quadrupole-operator-axial-symmetry/) - 1. The symmetry parameter eta = (V_xx - V_yy)/V_zz When eta = 0 (V_xx = V_yy) we have axial symmetry. 2. For any rotation around x and y axes we the figure stays the same. - [Symmetry axes](https://www.hyperfinecourse.org/forums/topic/symmetry-axes-2/) - For the Fe-I, there is a 4-fold rotation axis. For the Fe-II, there is also a 4-fold rotation axis. => Both theorems 1 and 2 are valid is this case, so the EFG tensor is zero. - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-6/) - We would need to have an inifite fold degeneracy in eta=0 states so that in the eta=/0 states, the energies would ressemble a continuous distribution. - [iron axis](https://www.hyperfinecourse.org/forums/topic/iron-axis/) - Fe-I: 4-fold rotation axis thus according to theorem 1: z-axis is rotation axis and theorem 2: EFG tensor 0 Fe-II; 6-fold rotation axis thus identical to Fe-I - [Quantum to classical picture](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-picture/) - A single particle has defined energy levels with bandgaps. When more and more particles are brought together, this bandgap will vanish and thus we get the classical picture where all energy levels are allowed. This is because of the many hyperfine interactions between the particles and thus many, many quadrupole energy shifts. - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-8/) - The two ways to determine if axial symmetry is present: 1. From the matrix. Only diagonal elements are non-zero and xx-yy values are equal. 2. This can be seen from the picture - [Symmetry](https://www.hyperfinecourse.org/forums/topic/symmetry-2/) - For Fe-1, there are 3 4-fold rotation axis thus EFG tensor = 0 For Fe-2, there are 6 4-fold rotation axis thus EFG tensor = 0 - [task 2](https://www.hyperfinecourse.org/forums/topic/task-2-9/) - Fe-II, there are 2 two fold rotations, so the z-axis is the rotation axis which is part of the PAS Fe-I, 3(for each axes) 4-fold rotation axes - [nuclear quadrupole interaction: from toy model to quantum](https://www.hyperfinecourse.org/forums/topic/nuclear-quadrupole-interaction-from-toy-model-to-quantum-2/) - When the nucleus becomes smaller, the size of the quadrupole will also become smaller. - [nuclear quadrupole interaction: from toy model to quantum](https://www.hyperfinecourse.org/forums/topic/nuclear-quadrupole-interaction-from-toy-model-to-quantum/) - When the size of the nucleus decrease also the amplitude of the quadrupole moment decrease. - [“case studies / symmetry – task 1”](https://www.hyperfinecourse.org/forums/topic/case-studies-symmetry-task-1/) - In the quantum discrete the energy, angular momentum,... are all discrete, to go to the classical case (which is continuous), the energy spacing (or angular momentum,...) must become very small/go to zero to make it approach continuity. - [Rotation axes](https://www.hyperfinecourse.org/forums/topic/rotation-axes-3/) - The rotation axis of Fe-IIb atoms is a 4 fold rotation axis => z-axis of PAS and eta = 0 (Theorem 1). Two 2-fold rotation axes of Fe-IIa atoms => the z-axis of PAS (Theorem 1). - [Quantum to statistical](https://www.hyperfinecourse.org/forums/topic/quantum-to-statistical/) - The continuum distribution of spins from classical picture is achieved in the quantum description when a lot of spin I = 1 particles have a very narrow energy splitting. The discrete levels can than be approximated by continuous energy. - [Quantum to Classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-5/) - In quantum model, angular momentum occupied is quantized only for several values compared to the continuos distribution of classical model. In order to get the similar trend as classical model, the quantized angular momentum values can be reproduced for more values so that it has a wider distribution - [Task2](https://www.hyperfinecourse.org/forums/topic/task2/) - For FeI there will be a 4-fold rotation symmetry in the vertical axis, and so eta is zero. For FeII the vertical axis has a 4-fold rotation symmetry as well so in this case, eta is also zero. - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-4/) - Since starting from a quantum state, we know that the energy levels as well as the angular momentum is quantized and so discrete. In a classical system the energy is continuous so to go from quantum to classical we need to go from discretized energy to continuous energy. For an l=1 the splitting of the - [task 2](https://www.hyperfinecourse.org/forums/topic/task-2-8/) - Fe-1 : multiple 4-fold rotation axes --> theorem 2: EFG tensor is zero --> theorem 1: rotation axes can be the z-axis of PAS (eta=0) Fe-2a : two 2-fold rotation axes --> theorem 1: can be chosen as z-axis of PAS Fe-2b : 4-fold rotation axis --> rotation axis is z-axis of PAS and eta=0 - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-7/) - From equation: the V_xx component is equal to the V_yy component (both -1), therefore eta is 0 and there is axial symmetry. From picture: the charge distribution is symmetric in the xy plane, which therefore has axial symmetry. - [quadrupole operator](https://www.hyperfinecourse.org/forums/topic/quadrupole-operator-2/) - From the equation: V_xx = -1 = V_yy; non-diagonal elements are zero From the picture: charge distribution of the toy model is symmetric in the xy-plane - [Axial Symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-6/) - 1) From equation: The term V11 and V22 are the same => Vxx = Vyy => axial symmetry in PAS 2) From drawing: Symmetric w.r.t. xy-plane => axial symmetry (along z) - [task 1](https://www.hyperfinecourse.org/forums/topic/task-1-13/) - Take the quantum situation with many energy splitting, this can be considered as a continuum (almost the same as the classical result), since we have to go from a system with quantized states to one with a continuum of states. - [Presence of axial symmetry](https://www.hyperfinecourse.org/forums/topic/presence-of-axial-symmetry-2/) - 1) Under rotation around the z-axis, the system must remain unchanged 2) The tensor must be diagonal with xx and yy component the same - [from toy model to quantum](https://www.hyperfinecourse.org/forums/topic/from-toy-model-to-quantum-4/) - When the nucleus becomes smaller with respect to the electroncloud, the energy fluctuations due to the quadrupole moment become smaller too - [From toy to quantum](https://www.hyperfinecourse.org/forums/topic/from-toy-to-quantum-2/) - If the nucleus becomes smaller, the absolute value of the quadrupole term also becomes smaller. - [Axial Symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-5/) - From the picture, we can see that there is no difference between the charge along the x-axis. As from the equation, we can calculate the asymmetric parameter eta as (V_xx-V_yy)/V_zz. By inputting V_xx=V_yy=-1, we have eta=0, so there is no asymmetric thus it is axially symmetric - [quadrupole operator](https://www.hyperfinecourse.org/forums/topic/quadrupole-operator/) - From the equation, we see that V_xx = -1 = V_yy and the elements that are not on the diagonal are zero. This indicates the axial symmetry. From the picture, we see that the charge distribution of the toy model is symmetric in the xy plane, which also indicates axial symmetry. - [from toy model to quantum](https://www.hyperfinecourse.org/forums/topic/from-toy-model-to-quantum-3/) - As the nucleus becomes smaller, the quadrupole term also becomes smaller. - [Rotation axes](https://www.hyperfinecourse.org/forums/topic/rotation-axes-2/) - There's a 4-fold rotation axis going through the Fe-IIb's. Theorem 1 says that this axis can be chosen as the z-axis in the PAS, with eta=0 There are also two 2-fold rotation axes going through opposite Fe-IIa's. Both of these can be chosen as z-axis in the PAS - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-3/) - the classical case allowed all energy values as a continuum, while the quantum case only allows discrete values. A system which would have a very high-order degeneracy would result in many little hyperfine energy-levels which would approximate the continuous classical model. - [From toy to quantum](https://www.hyperfinecourse.org/forums/topic/from-toy-to-quantum/) - For a smaller radius, the range of y-axis of the quadrupole term will converge. For a higher radius, the amplitude of the term will be much broader as well as the max lay at much lower values, together with reaching a much lower minimu. - [From Toy Model to Quantum](https://www.hyperfinecourse.org/forums/topic/from-toy-model-to-quantum-2/) - As l becomes smaller than d: The absolute value of the quadrupole term becomes smaller. The quadrupole term approaches the exact value better. The range of values for the quadrupole term at different angles becomes narrower (smaller range). - [Rotation axes](https://www.hyperfinecourse.org/forums/topic/rotation-axes/) - The Fe-I atoms have 3 perpendicular 4-fold rotation axes so each of them can be chosen as the z-axis in the PAS and η=0 according to theorem 1. According to theorem 2 the EFG tensor is then zero. The Fe-II atoms have 1 4-fold rotation axis, the z-axis and 2 2-fold axes, x- and y-axis. - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-2/) - The total spin of the system needs to be higher so that you have much more different available states due to the quadrupole splitting. If the amount of available states becomes much higher the difference between the levels becomes negligible and indistinguishable from the classical case. - [Quadrupole Moment](https://www.hyperfinecourse.org/forums/topic/quadrupole-moment-4/) - If the nucleus becomes smaller, we can see that the amplitude of the energy will be higher relative to the absolute value, thus result in the small difference between them - [questions on this chapter](https://www.hyperfinecourse.org/forums/topic/questions-on-this-chapter/) - I have a couple of questions about this chapter (I thought I'd post them here). I have put 2 of them in my answer to the tasks of the miscellaneous topics question. So my first is that something goes wrong with the calculation of the shift in the energy levels for the EFG contribution and - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-4/) - It can be seen that the EFG tensor has axial symmetry from the η parameter which is the deviation from axial symmetry. This is zero for the given equation since Vxx = Vyy. You can also see from the toy model that it is completely symmetric under rotations around the z-axis. - [symmetry](https://www.hyperfinecourse.org/forums/topic/symmetry/) - for FE2, there are 2 two fold rotations, so the z-axis is the rotation axis which is part of the PAS (theorem 1) (similar to slide case) for Fe1, there are 3 symmetry axes (x,y,z), but I don't understand what you mean with 2-fold and 3-fold rotation axis. - [quantum classical](https://www.hyperfinecourse.org/forums/topic/quantum-classical/) - In classical case, the states are continuous and in quantum discrete. So the energy spacing must go to zero in quantum case to make it somewhat continuous. - [Rotation axes and symmetries](https://www.hyperfinecourse.org/forums/topic/rotation-axes-and-symmetries/) - The rotation axis through the two (orange) Fe-IIb atoms is a 4 fold rotation axis, which implies that this rotation axis is the z-axis of PAS and eta = 0 (Theorem 1). There are also two 2 fold rotation axes, going through two opposite (green) Fe-IIa atoms. This implies that these two 2 fold rotation - [Classical situation vs quantum situation](https://www.hyperfinecourse.org/forums/topic/classical-situation-vs-quantum-situation/) - To go from the quantum situation to the classical situation, we would need to go from a system with (a few) discrete states to a system with a continuum of states. So if we have a quantum situation with many narrow energy splitting which can approximately be described by a continuum of states, we would - [Axial](https://www.hyperfinecourse.org/forums/topic/axial/) - 1: The matrix is diagonal and V_xx = V_yy 2: Rotations around the z-axis does not change the distribution (V_xx + V_yy =0) 3: One positive charge not in the origin - [Quadrupole moment](https://www.hyperfinecourse.org/forums/topic/quadrupole-moment-3/) - The size of the quadrupole moment contribution becomes significantly smaller (2 orders of magnitude) as the size of the nucleus decreases. - [toy](https://www.hyperfinecourse.org/forums/topic/toy/) - The magnitude becomes smaller if the nucleus becomes smaller - [PAS for Fe4N](https://www.hyperfinecourse.org/forums/topic/pas-for-fe4n/) - Fe-I: The vertical axis through N and the two perpendicular axes are all 4-fold rotation axes, so the EFG tensor is zero. Fe-II: Just like Fe-I: The vertical axis through N and the two perpendicular axes are all 4-fold rotation axes, so the EFG tensor is zero. - [Quantum situation to classical](https://www.hyperfinecourse.org/forums/topic/quantum-situation-to-classical/) - If one would consider a quantum system that consists of a large number of I=1 particles, the resulting energy scheme would strongly resemble the classical one. - [Axial symmetry of EFG tensor](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-of-efg-tensor/) - The axial symmetry is indicated in the equation by the fact that the tensor is diagonal and that Vxx and Vyy are equal. From the picture, we can see that the charge distribution is symmetric in the x-y plane, which indicates axial symmetry. - [Quadrupole term](https://www.hyperfinecourse.org/forums/topic/quadrupole-term-2/) - The size of the quadrupole term decreases as the nucleus becomes smaller - [symmetry axes](https://www.hyperfinecourse.org/forums/topic/symmetry-axes/) - Fe-I has 4-fold rotation symmetry in the z-axis (and 2-fold symmetry in x and y) and therefore eta is zero. I think the same is true for the Fe-II and so also eta is zero. - [quantum and classical system](https://www.hyperfinecourse.org/forums/topic/quantum-and-classical-system/) - As with many quantum systems, once you consider a large ensemble of them together, they can behave like a classical system. This case is no different, when you have a large ensemble of these quantum systems, orientations can be considered pretty much continuous, while still being quantized for each quantum system. - [axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-3/) - 1) the axial symmetry in the equation for V(2) can be seen in the diagonal character of the tensor. 2) I think the picture shows that not much changes for rotations around the z-axis. A system that is not axially symmetric could maybe be one where the nucleus is no longer in the origin. - [size of quadrupole moment](https://www.hyperfinecourse.org/forums/topic/size-of-quadrupole-moment/) - There seems to be less variation (smaller range) in the quadrupole moment when the nucleus gets smaller. - [Quadrupole term](https://www.hyperfinecourse.org/forums/topic/quadrupole-term/) - Quadrupole term should become smaller as the size of the nucleus decreases - [Presence Of Axial Symmetry](https://www.hyperfinecourse.org/forums/topic/presence-of-axial-symmetry/) - From the equation: We see that the matrix is a diagonal matrix and that V_xx = V_yy, which implies the axial symmetry. From the picture: The axial symmetry follows from the picture because the charge distribution is symmetric in the xy-plane. This implies the axial symmetry as well. - [From Toy Model To Quantum](https://www.hyperfinecourse.org/forums/topic/from-toy-model-to-quantum/) - If the size of the nucleus decreases, the magnitude of the quadrupole decreases as well. - [EFG tensor symmetry](https://www.hyperfinecourse.org/forums/topic/efg-tensor-symmetry/) - From the equation: the terms Vxx and Vyy are identical, so the symmetry parameter eta is zero. From the picture: the charges that cause the EFG are on the principle axis, so the behaviour in the xy-plane will look the same wherever you are in the xy-plane. - [PAS in Iron system](https://www.hyperfinecourse.org/forums/topic/pas-in-iron-system/) - Fe-I: I think the z-axis (arrow pointing up, I imagine the x and y axis being perpendicular to this arrow) is a 4 fold rotation axis, while the x and y axis are both a two fold rotation axis. This means that the z-axis can be the z-axis of the PAS and that there is - [Quantum to classical](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical/) - In the classical model, the nucleus can have any orientation, while in the quantum model, these states are quantised. Maybe if you would increase in the quantum model the number of allowed orientation so that experimentally it cannot be distinguished from a continuum of orientations you get something resembling the classical model? - [EFG tensor axial symmetry](https://www.hyperfinecourse.org/forums/topic/efg-tensor-axial-symmetry/) - 1. From equation: in the equation of V(2), there are only diagonal terms in the matrix and V_xx and V_yy are the same. 2. Visually: in the picture of the toy model you can imagine the dumbell "spinning" around the z-axis without it changing the electric field gradient. The system is invariant under rotations of - [Toy model to quantum](https://www.hyperfinecourse.org/forums/topic/toy-model-to-quantum/) - The absolute value of the quadrupole moment decreases as the size of the nucleus increases. - [Rotation axis of Fe-sites in Fe4N](https://www.hyperfinecourse.org/forums/topic/rotation-axis-of-fe-sites-in-fe4n/) - Fe-I: relevant rotation axis is the base axis of the crystal (z-axis). So the PAS symmetries are around the z-axis for the four-fold symmetry of the crystal. Fe-II: relevant rotation axis is the base axis and y-axis. So the PAS symmetries are around the z and y-axis. - [case studies task 2](https://www.hyperfinecourse.org/forums/topic/case-studies-task-2/) - 1. Fe-I :- x,y,z-axes have 4-fold rotation symmetry. EFG is zero in this case. 2. Fe-II :- z-axis has a 4-fold rotation symmetry; this can be the z-axis for the PAS and eta is zero. x and y- axes have 2-fold rotation symmetry; these two axes can also be chosen as the z axis of - [Quantum to classical situation](https://www.hyperfinecourse.org/forums/topic/quantum-to-classical-situation/) - To evolve from a quantum situation to a classical one, the number of particle has to increase. The quadrupole interaction of a particle has discrete level of energy but when there is multiple particle, it appears continuous. It can then be treated as a classical situation. - [case studies task 1](https://www.hyperfinecourse.org/forums/topic/case-studies-task-1/) - we can have infinetely many spin states in order to create a continuum of these spin states for the classical picture. - [Axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry-2/) - 1. from equation: if two of the three eigenvalues are equal and the off-diagonal values are 0, then the EFG tensor has axial symmetry in its PAS. 2. visual: if the toy model has rotational symmetries around its axis, the EFG tensor has axial symmetry in its PAS. - [Size of quadrupole moment vs size of nucleus](https://www.hyperfinecourse.org/forums/topic/size-of-quadrupole-moment-vs-size-of-nucleus/) - If the nucleus becomes smaller, the quadrupole moment also becomes smaller. As the nucleus becomes smaller, the distribution of protons and neutrons becomes more symmetric, resulting in a smaller quadrupole moment. - [Quadrupole moment](https://www.hyperfinecourse.org/forums/topic/quadrupole-moment-2/) - I would say that the "misleading" part is that even though the quadrupole moment becomes a better approximation for l - [axial symmetry](https://www.hyperfinecourse.org/forums/topic/axial-symmetry/) - 1. V_xx = V_yy 2. sum of the 5 numbers of the matrix is zero. - [symmetry task 2](https://www.hyperfinecourse.org/forums/topic/symmetry-task-2/) - For the Fe-I site the relevant rotation axis is the z-axis. For the Fe-II site, the relevant rotation axis is the x-axis. The electric field gradient tensor for each site will depend on this symmetry and will give the information about the orientation of the PAS with respect to the crystal axes. The EFG tensor - [symmetry task 1](https://www.hyperfinecourse.org/forums/topic/symmetry-task-1/) - If we consider a large number of I=1 particles we can approximate this with a classical model. A large number of spins can be treated as a continuous distribution of magnetic moments, which can be described using a classical vector field. The magnitude and direction of the vector field represent the spin of the particles. - [determining presence of axial symmetry](https://www.hyperfinecourse.org/forums/topic/determining-presence-of-axial-symmetry/) - 1. A tensor (of rank 2) has axial symmetry when it remains unchanged under the rotation operation. 2. The toy model depicted is perfectly symmetrical in the x-y, y-z and x-z plane so it remains unchanged when rotated around any axis. - [Energy Correction](https://www.hyperfinecourse.org/forums/topic/energy-correction-3/) - Yes, the hyperfine field is not constant (Bohr-Weisskopf). - [g factor](https://www.hyperfinecourse.org/forums/topic/g-factor-2/) - Ground state : g = h*µ / µ_N * I => g = (-0.594 nm*h_bar *2)/µ_N = -1.188 h_bar/µ_N 245 Kev : g = h*µ / µ_N * I => g = (-0.766 nm*h_bar *2)/5*µ_N = -0.30653 h_bar/µ_N - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-12/) - Yes, because of fermi contact contribution and bohr-weisskopf effect - [Task 1](https://www.hyperfinecourse.org/forums/topic/task-1-11/) - We can use the formula g = \mu/(\mu_N I) a. g_gs = -0.5940/(\mu_N 1/2); g_(e=245) = -0.766/(\mu_N 5/2) b. - [Task 2](https://www.hyperfinecourse.org/forums/topic/task-2-7/) - Measure all possible values of a with I_1, find the B_j from the gradient result in the linear regression of a and I_1 with known \mu_1 Measure all possible values of a with I_2, find the \mu_2 from the gradient result in the linear regression of a and I_1 with known B_j from the previous - [Task 2](https://www.hyperfinecourse.org/forums/topic/task-2-6/) - A child in the carousel is like an electron spinning over a nucleus. The movement of the child in a carousel relative to the core is the same as an electrically charged ball. The difference temperature between the child and the carousel is the same as a bar magnet representation of electron over a nucleus. - [Determining µ2 task 2](https://www.hyperfinecourse.org/forums/topic/determining-µ2-task-2/) - We know (mu_1*B_j)/(I_1*J)=-a_1 (mu_2*B_j)/(I_2*J)=-a_2 having the same nuclear charge and electron configuration we can divide away the dipole field (mu_1/I_1)/(mu_2/I_2)=a_1/a_2. ## Replies - [](https://www.hyperfinecourse.org/forums/reply/6936/) - Correction: the orientation of the magnetic moments has nothing to do with the strength of the magnetic hyperfine field. This field arises due to the presence of electrons around (or inside) the atomic nucleus. This does not change with temperature, so the magnetic hyperfine field will be the same for all temperatures. - [](https://www.hyperfinecourse.org/forums/reply/6794/) - Hi Jeroen, it's nice to meet a fellow physics enthusiast from Ghent University. I looked up that church and it looks like it is definitely worth a visit! - [](https://www.hyperfinecourse.org/forums/reply/6792/) - Hi Sander, nice to see that I'm not the only one following this course this year! - [](https://www.hyperfinecourse.org/forums/reply/6784/) - Hi, I am a first year master student from Gent university. Glad to see that this course has an international outreach! - [](https://www.hyperfinecourse.org/forums/reply/6771/) - hey, I am a first year master student too, at University of Havana, and I also need to deepen my knowledge in solid-state physics - [](https://www.hyperfinecourse.org/forums/reply/6446/) - Adding on that : This means that it's not more accurate but just smaller : while on the graphs the smaller one looks more accurate. - [](https://www.hyperfinecourse.org/forums/reply/6432/) - Me too, I don't really understand how to find the PAS of the magnetic or electric interaction, and how these are related to the placement of the atom in the molecule. - [](https://www.hyperfinecourse.org/forums/reply/6385/) - I'm not sure whether we can consider Fe-IIa and b as the same. - [](https://www.hyperfinecourse.org/forums/reply/6092/) - hey, i am happy someone from ghent university is also taking this course!!! - [](https://www.hyperfinecourse.org/forums/reply/6085/) - Hello Ward, I appreciate the suggestions and will keep them in mind! - [](https://www.hyperfinecourse.org/forums/reply/6067/) - Hi Diego, As a Belgian inhabitant I can also recommend visiting Brussels and Antwerp (mabye Bruges aswell as it is close to Ghent). These are beautiful cities and have nice culture. I hope you enjoy your study in Belgium. - [](https://www.hyperfinecourse.org/forums/reply/6058/) - Hi Ester, I'm right there with you! No current plans on how I will apply the knowledge gained during the course but I believe the knowledge itself to be already valuable enough. - [](https://www.hyperfinecourse.org/forums/reply/6034/) - Hi Guillaume, I don't know, I never really thought about that. For me it was a given that I would study in Leuven because the university is so well known and it has so good connections everywhere. Also just because a majority of my family has studied there, so not only was I sure it - [](https://www.hyperfinecourse.org/forums/reply/6009/) - I can find my responses on the test forum by manually going to "https://www.hyperfinecourse.org/topics/" but the nucleus still seems locked. - [](https://www.hyperfinecourse.org/forums/reply/6006/) - Hello Victor, Considering is there a particular reason why you went to study in Leuven? I was under the impressions the VUB also has a good physics program. - [](https://www.hyperfinecourse.org/forums/reply/5994/) - Hi Ester, Can you maybe recommend me a nice place to go eat near the Central station in Antwerpen? I'm going there tomorrow (Friday 14/02) and I'm too lazy to make myself a sandwich haha. Kind regards, Victor Navarro de la Torre - [](https://www.hyperfinecourse.org/forums/reply/5868/) - Correction to answer 2, for some nuclei this anisotropy is particularly pronounced. - [](https://www.hyperfinecourse.org/forums/reply/5866/) - At low temperatures I would expcect something like Bose-Einstein condensation. Where the state with the lowest energy is the most populated but others also appear. - [](https://www.hyperfinecourse.org/forums/reply/5862/) - I agree with this. Having read 5 other comments, I think this one suits from the scientific approach nicely. Maybe a bit too complicated for Wikipedia though! - [](https://www.hyperfinecourse.org/forums/reply/5849/) - OK, fixed (they had moved to a different site) - [](https://www.hyperfinecourse.org/forums/reply/5726/) - I agree to this. - [](https://www.hyperfinecourse.org/forums/reply/5696/) - Centrifuge! I didn't think of that at all. Good idea. - [](https://www.hyperfinecourse.org/forums/reply/5552/) - After having watched the next video, it turns out that both Fe-II's are equivalent without magnetic interactions taken into consideration. Hence, all Fe-II sites will have 1 time fourfold and 2 times twofold symmetry. - [](https://www.hyperfinecourse.org/forums/reply/5547/) - *diagonalized=> diagonal - [](https://www.hyperfinecourse.org/forums/reply/5546/) - quantized* (not degenerated) - [](https://www.hyperfinecourse.org/forums/reply/5470/) - Oops i seem to have written this in the wrong forum; this is my answer: In the cube, there is a 4-fold rotation axis through both Fe-IIb's. So this can be the z-axis. There are also two 2-fold rotation axis through the other Fe-IIa. any of these can be chosen as z-axis for the Fe-I. - [](https://www.hyperfinecourse.org/forums/reply/5424/) - I am Temitope kalejaye from Nigeria. I have been looking a place to learn and acquire more wisdom regarding Architecture until I get to find this. - [](https://www.hyperfinecourse.org/forums/reply/5016/) - Hey Wannes, I was wondering what a 'Gentse Feesten' is? I have heard of other reasons to visit Gent but not this. Why did you choose Physics and Astronomy? How does your study combine the two? - [](https://www.hyperfinecourse.org/forums/reply/5006/) - Hi Ricardo, did you study physics in Aveiro? - [](https://www.hyperfinecourse.org/forums/reply/5004/) - Hello Himanshu, It sounds like you are enjoying Leuven. I'll make sure I visit it next time I'm in this part of Europe. - [](https://www.hyperfinecourse.org/forums/reply/4988/) - Hi! Finland is definitely on my travel bucket list, and I do can appreciate some modern architecture, so I'll try to remember checking out some buildings of Alvar Aalto in Finland, when the opportunity arises! - [](https://www.hyperfinecourse.org/forums/reply/4985/) - Hey! I hope you enjoy the course. - [](https://www.hyperfinecourse.org/forums/reply/4968/) - Hi Both are indeed related. In physics, we mainly look at the phenomena that happen in solids, and you will probably use them to make working devices. It's the same stuff, but we look at it differently :). - [](https://www.hyperfinecourse.org/forums/reply/4961/) - I should have known about Izegem's shoemaking prowess before throwing away my nikes for having a hole in them. - [](https://www.hyperfinecourse.org/forums/reply/4956/) - Hi Carlos, I wish you good luck with your PhD! Darmstadt sounds like a nice city, I'll definitely try to visit it someday. All the best, Lotte - [](https://www.hyperfinecourse.org/forums/reply/4949/) - Antwerp is a beautiful city! Belgium in general is quite a nice place :D Welcome, hope you enjoy the course! - [](https://www.hyperfinecourse.org/forums/reply/4947/) - Hi Niels, Sounds indeed interesting to follow this course! Are you studying general experimental techniques in a specific field or general in physics? - [](https://www.hyperfinecourse.org/forums/reply/4939/) - Thanks for greeting me! I'd love to try Belgian fries someday. - [](https://www.hyperfinecourse.org/forums/reply/4938/) - Hello Ruben. Thank you for sharing. I am a mechanical engineer, and aspiring electrician, not a physicist. Is solid state physics directly related to solid state electrical/computing devices, or are the names only a coincidence? - [](https://www.hyperfinecourse.org/forums/reply/4937/) - Hi Amelia, nice that you are following this course from the other side of the globe! -Jana - [](https://www.hyperfinecourse.org/forums/reply/4920/) - Hi Fien At which university are you enrolled? Ruben - [](https://www.hyperfinecourse.org/forums/reply/4906/) - Wow that's so cool! I'm also from Ghent maybe we could follow this course together. I hope this course seems as interesting to you as it does to me - [](https://www.hyperfinecourse.org/forums/reply/4886/) - Hey! Nice that you are also taking this course! - [](https://www.hyperfinecourse.org/forums/reply/4882/) - Hi Hannes, Dresen sure sounds like a city worth a visit with that history! I hope you reach your goal of happiness in life! - [](https://www.hyperfinecourse.org/forums/reply/4866/) - Hi From all the posts i found, this one speaks to my soul the most! I'm sure that we will learn a lot about Hyperfine interactions. with kind regards - [](https://www.hyperfinecourse.org/forums/reply/4855/) - Hey Jin, nice to see that you already particitated in the course! Best, Hannes - [](https://www.hyperfinecourse.org/forums/reply/4841/) - Great - [](https://www.hyperfinecourse.org/forums/reply/4829/) - Hello - [](https://www.hyperfinecourse.org/forums/reply/4803/) - Hi Ricardo, what's your work about? - [](https://www.hyperfinecourse.org/forums/reply/4227/) - approx due to BW-effect - [](https://www.hyperfinecourse.org/forums/reply/4002/) - Hi Elisa, Bois de la Cambre sounds very nice, I will definitely check it out this summer! - [](https://www.hyperfinecourse.org/forums/reply/3826/) - Hi Jin, thanks for your recommendations. - [](https://www.hyperfinecourse.org/forums/reply/3824/) - Hi Ricardo, I will definitely visit - [](https://www.hyperfinecourse.org/forums/reply/3780/) - Hi Jeroen, I see we live in the same city. I have to agree it is a city scattered with cultural heritage. It is nice to meet you! - [](https://www.hyperfinecourse.org/forums/reply/3771/) - I have never been to Eastern Europe so thank you for the suggestion! - [](https://www.hyperfinecourse.org/forums/reply/3767/) - Hi, I have once been to Krakow, it is indeed a very nice city and I hope to be able to visit it again sometime. Best regards - [](https://www.hyperfinecourse.org/forums/reply/3764/) - Hi Elise, I have never been to Leuven so thank you for the recommendation. I will try it when I will go visit Leuven! - [](https://www.hyperfinecourse.org/forums/reply/3700/) - I will visit the Oude Markt this week then, thanks for the recommendation. We have similar answers I see. - [](https://www.hyperfinecourse.org/forums/reply/3670/) - Hi, I would love to visit Krakow at some point, would you have any recommendations to go? ;) How did actually find this course, I would think it can be quite interesting to find courses from other institutes? Kind regards - [](https://www.hyperfinecourse.org/forums/reply/3643/) - There seem to be a lot of nuclear physicists on this course! Nice to meet you Ferdinando! - [](https://www.hyperfinecourse.org/forums/reply/3640/) - That's cool, nuclear physics is hard and interesting, I had some parts of it in my Bachelour's. - [](https://www.hyperfinecourse.org/forums/reply/3633/) - My primary responsibilities includes upgrading a certain digital spectrometer in CERN to enhance its data acquisition abilities and to reduce overheads, which is made possible by the tremendous advances in technology over the last decade. - [](https://www.hyperfinecourse.org/forums/reply/3623/) - Hi Ricardo, I have been to Aveiro once on vacation and it is indeed a very beautiful city to visit ! - [](https://www.hyperfinecourse.org/forums/reply/3621/) - Hi Ricardo, it is nice to hear that there are also international students taking this course. ## My Templates - [Default Kit](https://www.hyperfinecourse.org/?elementor_library=default-kit-2-2) - [coming soon!](https://www.hyperfinecourse.org/?elementor_library=coming-soon) - [intro text short summary VIP2](https://www.hyperfinecourse.org/?elementor_library=intro-text-short-summary-vip2) - All experimental methods based upon hyperfine interactions can be classified into 3 different categories according to the kind of energy transition that is involved. If you take this course in a linear way, you have met the full explanation in the module ‘classification of methods‘. If you study only specific modules of this course, it - [short summary VIP2](https://www.hyperfinecourse.org/?elementor_library=short-summary-vip2) - [VIP1 summary framework video](https://www.hyperfinecourse.org/?elementor_library=vip1-summary-framework-video) - [VIP1 summary framework transcription button](https://www.hyperfinecourse.org/?elementor_library=vip1-summary-framework-transcription-button) - transcription - [account page](https://www.hyperfinecourse.org/?elementor_library=account-page) - Username or Email Address Password Remember Me Log In Lost your password? | Register Proceed to course! - [start onderdeel](https://www.hyperfinecourse.org/?elementor_library=start-onderdeel) - 5 TOPICS 3:00 Duration The third and last category of hyperfine interactions, is the electric quadrupole interaction. Mathematically spoken, this interaction involves tensors of rank 2. This makes it less intuitive then the magnetic hyperfine interaction. You will be given, however, several handles to establish the similarity between the electric quadrupole interaction and the magnetic - [space](https://www.hyperfinecourse.org/?elementor_library=space) - [IP](https://www.hyperfinecourse.org/?elementor_library=ip) - A free and open online course on Density Functional Theory, including hands-on applications.© Stefaan Cottenier - [3kolom](https://www.hyperfinecourse.org/?elementor_library=3kolom) - I have a question I have a suggestion I have a problem - [footer](https://www.hyperfinecourse.org/?elementor_library=footer) - [home](https://www.hyperfinecourse.org/?elementor_library=home) - Computational Materials Physics It is possible to predict properties of materials “from scratch” or “ab initio”: by applying the laws of quantum physics to the atoms that make up the material. The methods for doing this have been developed by solid state physicists, and are now sufficiently mature to tackle materials engineering problems. These so-called - [shortcode forum](https://www.hyperfinecourse.org/?elementor_library=shortcode-forum) - [webinar](https://www.hyperfinecourse.org/?elementor_library=webinar) - Here you have the feedback webinar on the framework. If you have questions or comments, or want to discuss something about this webinar, then please use the chapter forum of this chapter, or use the question form at the side. - [wikipedia](https://www.hyperfinecourse.org/?elementor_library=wikipedia) - It may be interesting to stroll through related Wikipedia pages (optional). go to Wikipedia - [hi](https://www.hyperfinecourse.org/?elementor_library=hi) - Hi! Welcome to this online course on Computational Materials Physics! Ready to get started? Click on the modules below! - [about you](https://www.hyperfinecourse.org/?elementor_library=about-you) - You don’t find yourself browsing this course site for no reason, do you? We would like to get to know you! Your expectationsPlease tell us what you expect from this course (you will be asked to fill out your username) Present yourselfWherever you are in the world, you’re not studying alone in this course. At - [gewone pagina-hyperfine](https://www.hyperfinecourse.org/?elementor_library=gewone-pagina-hyperfine) - TEKST H5P INTERACTIEVE VIDEO download printable slides (3) download printable slides (6) TEKST FORUM FORM QUIZ Expected time: 20 minutes (report)Code - [header_module](https://www.hyperfinecourse.org/?elementor_library=header_module) - 4 TOPICS 2:00h Duration - [exit_finaal](https://www.hyperfinecourse.org/?elementor_library=exit_finaal) - Please use the form underneath to rate your perception of your knowledge about the essential topics of this week. This is a rapid detection system to find out whether specific topics need special attention during the upcoming feedback webinar. Simultaneously, you may consider this as a list that is relevant for exam knowledge. FORM This - [form shortcode](https://www.hyperfinecourse.org/?elementor_library=form-shortcode) - [Default Kit](https://www.hyperfinecourse.org/?elementor_library=default-kit-2) - [Default Kit](https://www.hyperfinecourse.org/?elementor_library=default-kit) ## Courses - [Hyperfine course](https://www.hyperfinecourse.org/courses/e-commerce-course/) - Hi! Welcome to this online course on Hyperfine! Ready to get started? Click on the modules below! Course Content Expand All Lessons quick start 2 Topics Expand quick start Lesson Content 0% Complete 0/2 Steps practical info about you the nucleus 7 Topics Expand the nucleus Lesson Content 0% Complete 0/7 Steps nuclear properties multipole ## Lessons - [magnetic hyperfine interaction](https://www.hyperfinecourse.org/lessons/magnetic-hyperfine-interaction/) - Hyperfine course magnetic hyperfine interaction 4 TOPICS 4:00h Duration You’re done with the general framework of hyperfine interactions. The next task ahead, is to examine specific manifestations of hyperfine interactions. These are the features that later on will play a role in experimental measurements. In this chapter you deal with the first one, the electric - [electric monopole shift](https://www.hyperfinecourse.org/lessons/electric-monopole-shift/) - Hyperfine course electric monopole shift 3 TOPICS 4:30h Duration You’re done with the general framework of hyperfine interactions. The next task ahead, is to examine specific manifestations of hyperfine interactions. These are the features that later on will play a role in experimental measurements. In this chapter you deal with the first one, the electric - [the nucleus](https://www.hyperfinecourse.org/lessons/the-nucleus-2/) - Hyperfine course the nucleus 4 TOPICS 2:00h Duration We cannot talk about hyperfine interactions without discussing some properties of nuclei that go beyond the point charge. That’s what this first chapter will be about. There are a few optional resources that can be useful. These are really optional resources: they do not belong to the - [EPR](https://www.hyperfinecourse.org/lessons/epr/) - Hyperfine course EPR 4 TOPICS 3:00 Duration This chapter assumes you are familiar with the g-factor. If you aren’t, have a look at the Refresher’s section first. If you take this course in sync with the spring edition at Ghent University, then please observe the due date for the activities of this week. Lesson Content 0% Complete - [synchrotron methods](https://www.hyperfinecourse.org/lessons/synchrotron-methods/) - Hyperfine course synchrotron methods 5 TOPICS 2:45 Duration This chapter assumes you are familiar with the double slit experiment and to some extent with the concepts of amplitude, probability and intensity. If you aren’t, have a look at the Refresher’s section first. If you studied the module on Mössbauer spectroscopy first, you have the necessary - [Mössbauer spectroscopy](https://www.hyperfinecourse.org/lessons/mossbauer-spectroscopy/) - Hyperfine course Mössbauer spectroscopy 5 TOPICS 3:00 Duration This chapter assumes you are familiar with the double slit experiment and with the concepts of amplitude, probability and intensity. If you aren’t, have a look at the Refresher’s section first. If you take this course in sync with the spring edition at Ghent University, then please - [μSR (optional)](https://www.hyperfinecourse.org/lessons/μsr-optional/) - Hyperfine course μSR (optional) This is an optional part of this course, and still under construction. μSR, or muon spin rotation is another method that makes use of hyperfine interactions. It is quite widely used, and therefore it deserves its place in this overview. There is no dedicated lecture material foreseen so far, but as - [PAC](https://www.hyperfinecourse.org/lessons/pac/) - Hyperfine course PAC 7 TOPICS 3:00 Duration Note that the present chapter on PAC is covered in the same week as the previous chapter on LTNO & NMR/ON. There will be only one feedback webinar on both chapters. Be sure to cover both chapters this week. The AYOB-form for this week is available at the - [LTNO & NMR/ON](https://www.hyperfinecourse.org/lessons/ltno-nmr-on/) - Hyperfine course LTNO & NMR/ON 7 TOPICS 3:00 Duration Note that the present chapter on LTNO & NMR/ON is covered in the same week as the next chapter on PAC. There will be only one feedback webinar on both chapters. Be sure to cover both chapters this week. The AYOB-form for this week is available - [NMR & NQR](https://www.hyperfinecourse.org/lessons/nmr-nqr/) - Hyperfine course NMR & NQR 5 TOPICS 2:45 Duration If you take this course in sync with the spring edition at Ghent University, then please observe the due date for the activities of this week. Lesson Content 0% Complete 0/7 Steps short overview classification orientation of a nuclear ensemble orientation: temperature and radiation NMR and - [laser spectroscopy](https://www.hyperfinecourse.org/lessons/laser-spectroscopy/) - Hyperfine course laser spectroscopy 2 TOPICS 1:45 Duration Now, you’ll tackle the first of our list of hyperine interaction methods: laser spectroscopy.(don’t forget: the previous short chapter on the classification of methods does belong to the material of this same week) If you take this course in sync with the spring edition at Ghent University, - [classification of methods](https://www.hyperfinecourse.org/lessons/classification-of-methods/) - Hyperfine course classification of methods 1 TOPICS 1:45 Duration Before tackling the first hyperfine interaction method, you’ll construct a roadmap to classify these methods in a systematic way.There is only a single topic in this short chapter. Afterwards, please move on to the chapter on laser spectroscopy, which belongs to the material for this same - [electric quadrupole interaction](https://www.hyperfinecourse.org/lessons/electric-quadrupole-interaction/) - Hyperfine course electric quadrupole interaction 5 TOPICS 3:00 Duration The third and last category of hyperfine interactions, is the electric quadrupole interaction. Mathematically spoken, this interaction involves tensors of rank 2. This makes it less intuitive then the magnetic hyperfine interaction. You will be given, however, several handles to establish the similarity between the electric - [framework](https://www.hyperfinecourse.org/lessons/framework/) - Hyperfine course framework 4 TOPICS 3:30h Duration Where can you fit hyperfine interactions into the science framework you know? That’s what you will examine in this chapter, starting from innocent situations that you can mentally imagine.There is an optional written summary available for this chapter (‘the framework’). Thanks to former student Jeffrey De Rycke for - [refreshers](https://www.hyperfinecourse.org/lessons/refreshers/) - [quick start](https://www.hyperfinecourse.org/lessons/quick-start/) - Hyperfine course quick start Lesson Content 0% Complete 0/2 Steps practical info about you Back to Course Next Topic - [The Nucleus](https://www.hyperfinecourse.org/lessons/the-nucleus/) - The Nucleus Let’s get started! The nucleus Lorem ipsum dolor sit amet, consectetur adipiscing elit. Etiam maximus tortor at diam gravida posuere. Curabitur et malesuada mi. Let’s get started! Back to - [Lesson](https://www.hyperfinecourse.org/lessons/lesson-7/) - [Lesson](https://www.hyperfinecourse.org/lessons/lesson-4/) - [Lesson](https://www.hyperfinecourse.org/lessons/lesson-5/) - [Lesson](https://www.hyperfinecourse.org/lessons/lesson-6/) - [Lesson](https://www.hyperfinecourse.org/lessons/lesson-2/) - [Lesson](https://www.hyperfinecourse.org/lessons/lesson-3/) - [Lesson](https://www.hyperfinecourse.org/lessons/lesson/) ## Topics - [webinar (w11)](https://www.hyperfinecourse.org/topics/webinar-w11/) - Hyperfine course PAC webinar (w11) This is a recording of the feedback webinar of this week. Previous Topic Back to Lesson Next Lesson - [NMR/ON](https://www.hyperfinecourse.org/topics/nmr-on/) - Hyperfine course LTNO & NMR/ON NMR/ON We studied about the NMR method in one of the preceding chapters. The crucial point there was the very weak yet essential orientation of the nuclear ensemble. But what would happen if you do NMR on an ensemble of nuclei that is at very low temperature and therefore strongly - [perturbed angular correlation spectroscopy](https://www.hyperfinecourse.org/topics/perturbed-angular-correlation-spectroscopy/) - Hyperfine course PAC perturbed angular correlation spectroscopy The need of having to go to milli-Kelvin temperatures is a disadvantage of LTNO and NMR/ON. Can we exploit the advantages of a strongly oriented ensemble of nuclei while staying at room temperature (or even above, when heating)? The answer is ‘yes’, and an experimental method doing this - [webinar (w10)](https://www.hyperfinecourse.org/topics/webinar-w10/) - Hyperfine course EPR webinar (w10) This is a recording of the feedback webinar of this week. Something went wrong with the audio of the video hereabove. Therefore, the feedback webinar of the previous year is provided as well (underneath). You may chose the one that is most convenient for you. Previous Topic Back to Lesson - [webinar (w9)](https://www.hyperfinecourse.org/topics/webinar-w9/) - Hyperfine course NMR & NQR webinar (w9) This is a recording of the feedback webinar of this week. Previous Topic Back to Lesson Next Lesson - [webinar (w8)](https://www.hyperfinecourse.org/topics/webinar-w8/) - Hyperfine course synchrotron methods webinar (w8) This is a recording of the feedback webinar of this week. Previous Topic Back to Lesson Next Lesson - [webinar (w7)](https://www.hyperfinecourse.org/topics/webinar-w7/) - Hyperfine course Mössbauer spectroscopy webinar (w7) This is a recording of the feedback webinar of this week. Previous Topic Back to Lesson Next Lesson - [webinar (w6)](https://www.hyperfinecourse.org/topics/webinar-w6/) - Hyperfine course laser spectroscopy webinar (w6) This is a recording of the feedback webinar of this week. Previous Topic Back to Lesson Next Lesson - [webinar (w5)](https://www.hyperfinecourse.org/topics/webinar-w5-2/) - Hyperfine course electric quadrupole interaction webinar (w5) This is a recording of the feedback webinar of this week. Previous Topic Back to Lesson Next Lesson - [case studies / symmetry](https://www.hyperfinecourse.org/topics/case-studies-symmetry/) - Hyperfine course electric quadrupole interaction case studies / symmetry In courses about hyperfine interactions, one often “defines” the hyperfine interaction by an energy level splitting scheme. In this course, this topic has been postponed until now. For a good reason: now you have the background to understand what this level splitting scheme means. Additionally, we’ll - [webinar (w4)](https://www.hyperfinecourse.org/topics/webinar-w4-2/) - Hyperfine course magnetic hyperfine interaction webinar (w4) This is a recording of the feedback webinar of this week. Previous Topic Back to Lesson Next Lesson - [in solids](https://www.hyperfinecourse.org/topics/in-solids/) - Hyperfine course magnetic hyperfine interaction in solids We turn now to a (crystalline) solid, with nuclei that are more than just a point charges: the nuclei have a magnetic moment. How does this change the energy levels of the solid ?You will understand how the crystal symmetry affects the way how we use perturbation theory - [webinar (w3)](https://www.hyperfinecourse.org/topics/webinar-w3-2/) - Hyperfine course electric monopole shift webinar (w3) This is a recording of the feedback webinar of this week. Previous Topic Back to Lesson Next Lesson - [webinar (w2)](https://www.hyperfinecourse.org/topics/webinar-w2/) - Hyperfine course framework webinar (w2) This is a recording of the feedback webinar of this week. Previous Topic Back to Lesson Next Lesson - [webinar (w1)](https://www.hyperfinecourse.org/topics/webinar-w1/) - Hyperfine course the nucleus webinar (w1) This is a recording of the feedback webinar of this week. Previous Topic Back to Lesson Next Lesson - [practical info](https://www.hyperfinecourse.org/topics/practical-info/) - Hyperfine course quick start practical info The different topics in this section help you to decide whether or not this course is suitable/useful/interesting for you. You learn how the course is organized, and you can try out the different tools that we will be using. You can set up accounts where needed, and register to - [Dates](https://www.hyperfinecourse.org/topics/dates/) - Due dates for the academic year 2025-2026Be aware that times are given in UTC (use this site to convert UTC to your local time zone). Mind the change after a few weeks, which is due to the transition from Winter Time to Daylight Saving Time in Europe.All due dates are on Tuesday early morning (UTC).All - [(optional) applications of PAC in biochemistry](https://www.hyperfinecourse.org/topics/optional-applications-of-pac-in-biochemistry/) - Hyperfine course PAC (optional) applications of PAC in biochemistry This is a 30-minute talk by Lars Hemmingsen (University of Copenhagen) on the use of PAC in a biochemistry context : Expected time: 35 minutes (report)B11-07 Previous Topic Back to Lesson Next Topic - [(optional) applications of PAC in biochemistry](https://www.hyperfinecourse.org/topics/optional-an-application-of-pac-2/) - This is a 30-minute talk by Lars Hemmingsen (University of Copenhagen) on the use of PAC in a biochemistry context : Expected time: 35 minutes (report)B11-07 - [(optional) an application of PAC in surface science](https://www.hyperfinecourse.org/topics/optional-an-application-of-pac/) - Hyperfine course PAC (optional) an application of PAC in surface science We’ll examine an example from the research literature where PAC has been used (and was crucial). After this video, you will understand better which features of hyperfine interaction methods (here PAC) make them sometimes the only tools available to obtain particular information. download printable - [(optional) the dilution refridgerator](https://www.hyperfinecourse.org/topics/optional-the-dilution-refridgerator-2/) - Hyperfine course LTNO & NMR/ON (optional) the dilution refridgerator (optional topic) Did you ever wonder by which devices one can cool a sample down to the milli-kelvin range? Dilution refridgerators do this job. If you’re interested, you can learn the basic principles either from this video, or from the Wikipedia page underneath download printable slides - [(optional) two conference talks](https://www.hyperfinecourse.org/topics/optional-two-conference-talks/) - Hyperfine course LTNO & NMR/ON (optional) two conference talks Here are two optional videos.One is a summary of what you just learned about LTNO and NMR/ON, formulated by a researcher who actually uses these methods. The other is meant for digging deeper on two related methods: beta-NMR and beta-NQR. In the first six minutes of - [EPR on free atoms](https://www.hyperfinecourse.org/topics/epr-on-free-atoms/) - Hyperfine course EPR EPR on free atoms The concept of EPR is introduced with the case of free atoms. After this video, you will be able to draw qualitatively and quantitatively an energy level scheme for a free atom in an external magnetic field, for which you can then find the possible EPR transitions. download - [the g-factor](https://www.hyperfinecourse.org/topics/the-g-factor/) - Hyperfine course refreshers the g-factor The g-factor is used in nuclear physics as a way to express the magnetic moment of a nucleus. But g-factors are used for objects that are no nuclei too, and in some cases their value can be even computed from other properties of the system. In this video, you learn - [EPR on molecules and crystals](https://www.hyperfinecourse.org/topics/epr-on-molecules-and-crystals/) - Hyperfine course EPR EPR on molecules and crystals There is one essential difference between free atoms at one hand, and molecules or crystalline solids at the other hand: the spherical symmetry of the former. You will understand how spatial anisotropy leads to direction-sensitive EPR-measurements, and you’ll get to know how this can be used to - [orientation: temperature and radiation](https://www.hyperfinecourse.org/topics/orientation-temperature-and-radiation/) - Hyperfine course NMR & NQR orientation: temperature and radiation How does radiation (photons) interacts with an oriented ensemble of nuclei? And how does temperature affects this interaction? In this video, you’ll learn how increasing the temperature leads to more isotropic ensembles, and you’ll understand how you can notice this change by observing the absorption of - [(optional) quadrupole shifts](https://www.hyperfinecourse.org/topics/optional-quadrupole-shifts/) - Hyperfine course electric quadrupole interaction (optional) quadrupole shifts In one of the previous chapters, you were pointed as optional reading to the first few sections of Rose (2012) (toy model paper). As an optional task, you are encouraged to read now sections 4 and 5. These sections show how anisotropic electron penetration in the nucleus - [in free atoms](https://www.hyperfinecourse.org/topics/in-free-atoms/) - Hyperfine course magnetic hyperfine interaction in free atoms How does that tiny bar magnet at the nucleus make itself felt? We will examine this first for a single atom, with a nucleus that is more than just a point charge: the nucleus has a magnetic moment. What does this mean for the energy levels of - [toy model](https://www.hyperfinecourse.org/topics/toy-model/) - Hyperfine course electric monopole shift toy model The essential ideas behind the monopole shift can be captured in a simple toy model. You will understand how the phenomenon that we call ‘monopole shift’ can arise in simple mechanical systems as well. It is not restricted to atoms or other quantum systems. The content of this video - [overview framework](https://www.hyperfinecourse.org/topics/overview-framework-3/) - Hyperfine course electric quadrupole interaction overview framework If you have studied the module “the framework”, then you are optimally prepared for the present module. You can skip this page and move on to the actual content of this module. Should you not have studied “the framework” yet, then either do so first, or digest at - [overview framework](https://www.hyperfinecourse.org/topics/overview-framework-2/) - Hyperfine course magnetic hyperfine interaction overview framework If you have studied the module “the framework”, then you are optimally prepared for the present module. You can skip this page and move on to the actual content of this module. Should you not have studied “the framework” yet, then either do so first, or digest at - [overview framework](https://www.hyperfinecourse.org/topics/overview-framework/) - Hyperfine course electric monopole shift overview framework If you have studied the module “the framework”, then you are optimally prepared for the present module. You can skip this page and move on to the actual content of this module. Should you not have studied “the framework” yet, then either do so first, or digest at - [miscellaneous topics](https://www.hyperfinecourse.org/topics/miscellaneous-topics/) - Hyperfine course electric quadrupole interaction miscellaneous topics We finish the quadrupole interaction topic with a series of small topics which are touched in a descriptive way only.You will get an impressionabout how to calculated electric-field gradients from first principles (the old and the new ways),about the effect of temperature on the quadrupole interaction,about energy level - [spin : an intuitive picture](https://www.hyperfinecourse.org/topics/spin-an-intuitive-picture/) - Hyperfine course refreshers spin : an intuitive picture Spin… It’s one of those topics that can drive anyone crazy. It all makes sense in the mathematical formalism, but unless someone has a spontaneously abstract mind (think Dirac) it doesn’t help for most of us to ‘understand’ what spin is. Rather than textbooks, Youtube is the - [tensors : several introductions](https://www.hyperfinecourse.org/topics/tensors-several-introductions/) - Hyperfine course refreshers tensors : several introductions Tensors inevitably play a role in the world of hyperfine interactions. Quite some people struggle with this concept. If that applies to you, you may want to refresh your knowledge about tensors, or even finally trying to get it all straight. Here are some resources that may help: - [gravitational analogue](https://www.hyperfinecourse.org/topics/gravitational-analogue/) - Hyperfine course framework gravitational analogue The key idea behind hyperfine interactions is introduced first in a context that might be more familiar to you than the quantum world of nuclei and electrons: the world of gravitational interaction between massive objects. After this video, you will understand how the gravitational quadrupole moment of a mass distribution - [short overview: orientation - temperature and radiation](https://www.hyperfinecourse.org/topics/short-overview-orientation-temperature-and-radiation/) - Hyperfine course LTNO & NMR/ON short overview: orientation – temperature and radiation In order to start the topic on LTNO and NMR/ON, you need to have some notion on the relations between orientation, temperature and radiation. This has been explained in the module on NMR. If you didn’t study that module (yet), then first have - [short overview: orientation of a nuclear ensemble](https://www.hyperfinecourse.org/topics/short-overview-orientation-of-a-nuclear-ensemble/) - Hyperfine course LTNO & NMR/ON short overview: orientation of a nuclear ensemble In order to start the topic on LTNO and NMR/ON, you need to know the concept of the orientation of a nuclear ensemble. This concept has been explained in the module on NMR. If you didn’t study that module (yet), then first have a - [short overview classification](https://www.hyperfinecourse.org/topics/short-overview-classification-5/) - Hyperfine course EPR short overview classification All experimental methods based upon hyperfine interactions can be classified into 3 different categories according to the kind of energy transition that is involved. If you take this course in a linear way, you have met the full explanation in the module ‘classification of methods‘. If you study only - [nuclear resonant scattering](https://www.hyperfinecourse.org/topics/nuclear-resonant-scattering/) - Hyperfine course synchrotron methods nuclear resonant scattering The photon beams produced by a synchrotron radiation facility can be used for many purposes. To name an obvious one: X-ray diffraction (XRD), just as you would do it in a regular lab. At a synchrotron, however, a XRD-measurement that takes hours in a lab, can be done - [reading task NMR/NQR](https://www.hyperfinecourse.org/topics/reading-task-nmr-nqr/) - Hyperfine course NMR & NQR reading task NMR/NQR On this page, you find two assignments related to NMR and NQR. One contains a research paper making use of NMR, the other contains a paper about NQR that is meant for a broad audience. Read one of them (the choice is up to you), and answer - [orientation of a nuclear ensemble](https://www.hyperfinecourse.org/topics/orientation-of-a-nuclear-ensemble/) - Hyperfine course NMR & NQR orientation of a nuclear ensemble Some experimental methods exploit properties of a collection of nuclei, rather than properties of just one nucleus. We’ll examine the concept ‘orientation of an ensemble of nuclei’ as such a collective property. This will introduce you to soccer balls, rugby balls, pears and pan cakes - [NMR and NQR](https://www.hyperfinecourse.org/topics/nmr-and-nqr/) - Hyperfine course NMR & NQR NMR and NQR Equiped with your new knowledge about orientation, temperature and radiation absorption, you are ready now to understand NMR and NQR — two methods that make use of these properties: download printable slides (6) If you want to read more/other explanations about NMR/NQR, then these are two optional - [short overview Mössbauer](https://www.hyperfinecourse.org/topics/short-overview-mossbauer/) - Hyperfine course synchrotron methods short overview Mössbauer To be completed — while awaiting the short version, feel free to study the complete module on Mössbauer spectroscopy. Previous Topic Back to Lesson Next Topic - [short overview classification](https://www.hyperfinecourse.org/topics/short-overview-classification-2/) - Hyperfine course Mössbauer spectroscopy short overview classification All experimental methods based upon hyperfine interactions can be classified into 3 different categories according to the kind of energy transition that is involved. If you take this course in a linear way, you have met the full explanation in the module ‘classification of methods‘. If you study - [short overview classification](https://www.hyperfinecourse.org/topics/short-overview-classification/) - Hyperfine course laser spectroscopy short overview classification All experimental methods based upon hyperfine interactions can be classified into 3 different categories according to the kind of energy transition that is involved. If you take this course in a linear way, you have met the full explanation in the module ‘classification of methods‘. If you study - [short overview classification](https://www.hyperfinecourse.org/topics/short-overview-classification-8/) - Hyperfine course μSR (optional) short overview classification All experimental methods based upon hyperfine interactions can be classified into 3 different categories according to the kind of energy transition that is involved. If you take this course in a linear way, you have met the full explanation in the module ‘classification of methods‘. If you study - [short overview classification](https://www.hyperfinecourse.org/topics/short-overview-classification-7/) - Hyperfine course PAC short overview classification All experimental methods based upon hyperfine interactions can be classified into 3 different categories according to the kind of energy transition that is involved. If you take this course in a linear way, you have met the full explanation in the module ‘classification of methods‘. If you study only - [short overview: classification](https://www.hyperfinecourse.org/topics/short-overview-classification-6/) - Hyperfine course LTNO & NMR/ON short overview: classification All experimental methods based upon hyperfine interactions can be classified into 3 different categories according to the kind of energy transition that is involved. If you take this course in a linear way, you have met the full explanation in the module ‘classification of methods‘. If you - [short overview classification](https://www.hyperfinecourse.org/topics/short-overview-classification-4/) - Hyperfine course NMR & NQR short overview classification All experimental methods based upon hyperfine interactions can be classified into 3 different categories according to the kind of energy transition that is involved. If you take this course in a linear way, you have met the full explanation in the module ‘classification of methods‘. If you - [short overview classification](https://www.hyperfinecourse.org/topics/short-overview-classification-3/) - Hyperfine course synchrotron methods short overview classification All experimental methods based upon hyperfine interactions can be classified into 3 different categories according to the kind of energy transition that is involved. If you take this course in a linear way, you have met the full explanation in the module ‘classification of methods‘. If you study - [(optional) synchrotron Mössbauer spectroscopy at extreme conditions](https://www.hyperfinecourse.org/topics/optional-synchrotron-mossbauer-spectroscopy-at-extreme-conditions/) - Hyperfine course synchrotron methods (optional) synchrotron Mössbauer spectroscopy at extreme conditions As an optional piece of fun, you might want to watch this conference talk of C. McCammon. It combines topics from two modules of this course: the present one (synchrotron) and the module on Mössbauer spectroscopy. The talk shows how scientists use a synchrotron - [nuclear inelastic scattering](https://www.hyperfinecourse.org/topics/nuclear-inelastic-scattering/) - Hyperfine course synchrotron methods nuclear inelastic scattering Another synchrotron-based method is Nuclear Inelastic Scattering (NIS), also known as Nuclear Resonant Vibrational Spectroscopy (NRVS). As the latter name indicates, its primary goal is to measure the vibrational properties (phonons) of solids or molecules. The basic information one wants to know in this respect is the density - [synchrotron radiation facilities](https://www.hyperfinecourse.org/topics/synchrotron-radiation-facilities/) - Hyperfine course synchrotron methods synchrotron radiation facilities Your goal for this topic is to understand the basics about synchrotron radiation: What is it? How is it produced? What can you do with it? More specifically: there is a form at the bottom of this page. It contains several concepts and questions. Try to answer these, - [expression and physics](https://www.hyperfinecourse.org/topics/expression-and-physics/) - Hyperfine course electric monopole shift expression and physics Here you will meet the first kind of experimentally observable hyperfine effects, which are due to the electric monopole shift. You will understand the physics behind the electric monopole shift and you can put it in relation to the general framework. You will understand how it gives - [laser spectroscopy](https://www.hyperfinecourse.org/topics/laser-spectroscopy/) - Hyperfine course laser spectroscopy laser spectroscopy Lasers with visible light (or at least in that frequency range) can be used to detect hyperfine interactions. In this video you’ll learn how. At the bottom of this page, you find an optional video of a conference presentation about laser spectroscopy. download printable slides (3) When the video - [course summaries](https://www.hyperfinecourse.org/topics/course-summaries/) - Hyperfine course refreshers course summaries If you are done studying (parts of) this course, you may wish to check whether you understood the essentials. Or perhaps you resume studying after a break, and want to refresh what came before. Or after having gone through all the details, you now want to see the big picture. - [summary of what came before](https://www.hyperfinecourse.org/topics/summary-of-what-came-before/) - Hyperfine course classification of methods summary of what came before If you did not (yet) study the first part of this course on the physics of hyperfine interactions, then it may be useful to watch the video on this page. It is a summary of the physics of hyperfine interactions (the video mentions at a - [overlap contribution](https://www.hyperfinecourse.org/topics/overlap-contribution/) - Hyperfine course magnetic hyperfine interaction overlap contribution In the monopole term of the Coulomb interaction between nuclei and electrons, there was an additional effect due to electron penetration in the nucleus (monopole shift). Now, we’ll examine what happens to the magnetic dipole term when electrons penetrate the nucleus.You will understand similarities (in principle) and differences - [VIP-1](https://www.hyperfinecourse.org/topics/vip-1/) - Hyperfine course framework VIP-1 How do hyperfine interactions relate to other interactions you know about in atoms, molecules or solids? After this video, you will understand the relation between hyperfine interactions and other interactions in atoms, and you will understand the origin of their name. The video culminates in the Very Important Picture nr. 1 - [quantum multipole expansion](https://www.hyperfinecourse.org/topics/quantum-multipole-expersion/) - Hyperfine course framework quantum multipole expansion You’ll need to know what perturbation theory is before you can tackle this video. If perturbation theory is not familiar to you (any more), you can brush up your knowledge first in the Refreshers section. For a quantum system, you cannot make a multipole expansion in the same way as - [(optional) why are odd electric moments zero?](https://www.hyperfinecourse.org/topics/optional-why-are-odd-electric-moments-zero/) - Hyperfine course the nucleus (optional) why are odd electric moments zero? Not all nuclear multipole moments are different from zero. All odd nuclear electric multipole moments (dipole, octupole, …) are zero, and all even nuclear magnetic multipole moments (monopole, quadrupole, …) are zero as well. This video explains why: transcription download slides (3) download slides - [double ring](https://www.hyperfinecourse.org/topics/double-ring/) - Hyperfine course framework double ring What you learned in the general derivation of the gravitational multipole expansion, can now be applied to this specific example of a a double ring system. You will understand better how the details of the shape of a dumb-bell relative to a double ring system determine the corrections to the - [multipole radiation](https://www.hyperfinecourse.org/topics/multipole-radiation/) - Hyperfine course the nucleus multipole radiation We can interact with nuclei by the radiation they emit or absorb. Just as we describe the shape of the nucleus by multipole components, the properties of the emitted radiation can be described in a multipole framework as well. We will need this to some extent in hyperfinecourse B, - [multipole moments](https://www.hyperfinecourse.org/topics/multipole-moments/) - Hyperfine course the nucleus multipole moments transcription printable slides (3) printable slides(6) There are two tasks related to this video:1. Find one example (in daily life or in science) where multipole moments of any kind play a role. Write down that example in the forum underneath. Comment on the examples of your colleagues if you - [nuclear properties](https://www.hyperfinecourse.org/topics/nuclear-properties/) - Hyperfine course the nucleus nuclear properties We’ll need nuclei a lot in this course. Let us refresh our knowledge on what they are. After this video, you’ll spontaneously think about more than the obvious properties when you’ll hear the word nucleus. transcription printable slides (3) printable slides (6) You will be asked in the video - [μSR](https://www.hyperfinecourse.org/topics/μsr/) - Hyperfine course μSR (optional) μSR To get you started on this topicsyou can read this article on Wikipedia.you find below a few school or conference talks by Stephen Blundell, a physicist who is well-known for his applications of the methods. The first video is an introduction to the method, given at the first day of - [from VIP1 to VIP2](https://www.hyperfinecourse.org/topics/from-vip1-to-vip2/) - Hyperfine course classification of methods from VIP1 to VIP2 The physics of hyperfine interactions can be summarized in one picture, which we baptized the “Very Important Picture nr. 1” (or VIP1) in hyperfinecourse A. If there is anything from hyperfinecourse A worth remembering for all of your life, it’s VIP1. In the same way, there - [perturbation theory](https://www.hyperfinecourse.org/topics/perturbation-theory/) - Hyperfine course refreshers perturbation theory Perturbation theory will come in handy for constructing the equations that describe the hyperfine structure. If you learned about perturbation theory before, you will brush up the main concepts. If you didn’t meet perturbation theory before, you will understand its aim and the global procedure.Go through this video, and solve - [from toy model to quantum](https://www.hyperfinecourse.org/topics/from-toy-model-to-quantum/) - Hyperfine course electric quadrupole interaction from toy model to quantum By studying the quadrupole term for a toy model, you’ll distinguish the essential features of the quadrupole interaction in atoms. And you’ll see sharply the major difference between the classical and the quantum case. After having studied this video, you will understand how the quadrupole - [amplitude, probability, intensity](https://www.hyperfinecourse.org/topics/amplitude-probability-intensity/) - Hyperfine course refreshers amplitude, probability, intensity In quantum physics, the way how properties of a system are measured, and the way how the evolution of a system from an initial to a final state are described, are unfamiliarly different from how this is done in classical physics. We need to understand how quantum physics deals - [exit (w11)](https://www.hyperfinecourse.org/topics/exit-w11/) - Hyperfine course PAC exit (w11) Please use the form underneath to rate your perception of your knowledge about the essential topics of this week. This is a rapid detection system to find out whether specific topics need special attention during the upcoming feedback webinar. Simultaneously, you may consider this as a list that is relevant - [exit (w9)](https://www.hyperfinecourse.org/topics/exit-w9-2/) - Please use the form underneath to rate your perception of your knowledge about the essential topics of this week. This is a rapid detection system to find out whether specific topics need special attention during the upcoming feedback webinar. Simultaneously, you may consider this as a list that is relevant for exam knowledge. FORM This - [(optional) the dilution refridgerator](https://www.hyperfinecourse.org/topics/optional-the-dilution-refridgerator/) - Here are two optional videos. One is a summary of what you just learned about LTNO and NMR/ON, formulated by a researcher who actually uses these methods. The other is meant for digging deeper on two related methods: beta-NMR and beta-NQR. In the first six minutes of this conference talk, C. Gaulard summarizes the LTNO and - [low-temperature nuclear orientation](https://www.hyperfinecourse.org/topics/low-temperature-nuclear-orientation/) - Hyperfine course LTNO & NMR/ON low-temperature nuclear orientation An ensemble of nuclei at low temperature, subject to a hyperfine interaction, spontaneously orients. If these are radioactive nuclei, how will this affect the radiation they emit? This is exploited by a method called Low-Temperature Nuclear Orientation (LTNO). download printable slides (3) In the video, a LTNO - [exit (w10)](https://www.hyperfinecourse.org/topics/exit-w10/) - Hyperfine course EPR exit (w10) Please use the form underneath to rate your perception of your knowledge about the essential topics of this week. This is a rapid detection system to find out whether specific topics need special attention during the upcoming feedback webinar. Simultaneously, you may consider this as a list that is relevant - [(optional) conference talk](https://www.hyperfinecourse.org/topics/optional-conference-talk/) - Hyperfine course EPR (optional) conference talk If you want to hear another voice on ENDOR, you may watch this optional conference lecture by J. Telser. It discusses ENDOR itself, as well as a variant of it, called PESTRE. The most relevant part for this course starts at 11m40, and takes somewhat less than seven minutes.Another - [ENDOR](https://www.hyperfinecourse.org/topics/endor/) - Hyperfine course EPR ENDOR EPR and NMR have both their advantages and disadvantages. Can we make a combination with the best of both worlds? You will understand the physics behind the advantages and disadvantages of NMR and EPR, and you’ll understand qualitatively how ENDOR succeeds in combining these advantages: download printable slides (3) download printable - [exit (w9)](https://www.hyperfinecourse.org/topics/exit-w9/) - Hyperfine course NMR & NQR exit (w9) Please use the form underneath to rate your perception of your knowledge about the essential topics of this week. This is a rapid detection system to find out whether specific topics need special attention during the upcoming feedback webinar. Simultaneously, you may consider this as a list that - [exit (w8)](https://www.hyperfinecourse.org/topics/exit-w8/) - Hyperfine course synchrotron methods exit (w8) Please use the form underneath to rate your perception of your knowledge about the essential topics of this week. This is a rapid detection system to find out whether specific topics need special attention during the upcoming feedback webinar. Simultaneously, you may consider this as a list that is - [Perusall: use and try-out](https://www.hyperfinecourse.org/topics/perusall-use-and-try-out/) - We will read in this course a few research papers. Typically, you will understand a lot of them, but not immediately everything. Wouldn’t it be helpful if you could point to the sentence you don’t understand, and ask somebody for help? Well, you can. You will read these papers via the social annotation tool Perusall. - [exit (w5)](https://www.hyperfinecourse.org/topics/exit-w5-2/) - Hyperfine course electric quadrupole interaction exit (w5) Please use the form underneath to rate your perception of your knowledge about the essential topics of this week. This is a rapid detection system to find out whether specific topics need special attention during the upcoming feedback webinar. Simultaneously, you may consider this as a list that - [exit (w6)](https://www.hyperfinecourse.org/topics/exit-w6/) - Hyperfine course laser spectroscopy exit (w6) Please use the form underneath to rate your perception of your knowledge about the essential topics of this week. This is a rapid detection system to find out whether specific topics need special attention during the upcoming feedback webinar. Simultaneously, you may consider this as a list that is - [exit (w7)](https://www.hyperfinecourse.org/topics/exit-w7/) - Hyperfine course Mössbauer spectroscopy exit (w7) Please use the form underneath to rate your perception of your knowledge about the essential topics of this week. This is a rapid detection system to find out whether specific topics need special attention during the upcoming feedback webinar. Simultaneously, you may consider this as a list that is - [miscellaneous Mössbauer topics](https://www.hyperfinecourse.org/topics/miscellaneous-mossbauer-topics/) - Hyperfine course Mössbauer spectroscopy miscellaneous Mössbauer topics To finish this topic, let us look at a few concepts, reasonings and examples that you will inevitably meet when reading about Mössbauer spectroscopy. You will learn what makes an isotope suitable for Mössbauer spectroscopy, what are the advantages of using Mössbauer spectroscopy in a radioactive beam facility, - [Mössbauer spectroscopy](https://www.hyperfinecourse.org/topics/mossbauer-spectroscopy/) - Hyperfine course Mössbauer spectroscopy Mössbauer spectroscopy With everything you learned in the previous videos, it is now straightforward to turn nuclear resonant scattering in a useful spectroscopy. You will understand how the different types of hyperfine interactions lead to a characteristic Mössbauer spectrum. And conversely, you’ll understand how to interpret a given (experimental) Mössbauer spectrum. - [realizing nuclear resonant scattering](https://www.hyperfinecourse.org/topics/realizing-nuclear-resonant-scattering/) - Hyperfine course Mössbauer spectroscopy realizing nuclear resonant scattering Scientists were hunting to realize nuclear resonant scattering a few decades ago. As it turned out, they were searching in the wrong direction… After this video, you will understand how high temperatures combined with high-speed drives can lead to a small number of resonant scattering events. And - [recoil, linewidth, resonant scattering](https://www.hyperfinecourse.org/topics/recoil-linewidth-resonant-scattering/) - Hyperfine course Mössbauer spectroscopy recoil, linewidth, resonant scattering The recoil phenomenon can spoil resonant scattering processes. However, if you’ve ever studied resonant scattering of light on atoms, you did probably not examine recoil. Here, you will see why. And you will appreciate why recoil is important when γ-rays scatter on nuclei. download printable slides (3) - [quadrupole operator](https://www.hyperfinecourse.org/topics/quadrupole-operator/) - Hyperfine course electric quadrupole interaction quadrupole operator You probably were familiar to the hamiltonian for the magnetic interaction. You met it many times in your education, in classical physics and in quantum physics. But do you know how the hamiltonian of the quadrupole interaction looks like? In this video you will learn how to construct - [exit (w4)](https://www.hyperfinecourse.org/topics/exit-w4-2/) - Hyperfine course magnetic hyperfine interaction exit (w4) Please use the form underneath to rate your perception of your knowledge about the essential topics of this week. This is a rapid detection system to find out whether specific topics need special attention during the upcoming feedback webinar. Simultaneously, you may consider this as a list that - [exit (w3)](https://www.hyperfinecourse.org/topics/exit-w3-2/) - Hyperfine course electric monopole shift exit (w3) Please use the form underneath to rate your perception of your knowledge about the essential topics of this week. This is a rapid detection system to find out whether specific topics need special attention during the upcoming feedback webinar. Simultaneously, you may consider this as a list that - [exit (w2)](https://www.hyperfinecourse.org/topics/exit-w2/) - Hyperfine course framework exit (w2) Please use the form underneath to rate your perception of your knowledge about the essential topics of this week. This is a rapid detection system to find out whether specific topics need special attention during the upcoming feedback webinar. Simultaneously, you may consider this as a list that is relevant - [exit (w1)](https://www.hyperfinecourse.org/topics/exit-w1/) - Hyperfine course the nucleus exit (w1) Please use the form underneath to rate your perception of your knowledge about the essential topics of this week. This is a rapid detection system to find out whether specific topics need special attention during the upcoming feedback webinar. Simultaneously, you may consider this as a list that is - [nuclear moment tabulation](https://www.hyperfinecourse.org/topics/nuclear-moment-tabulation/) - Hyperfine course the nucleus nuclear moment tabulation All knowledge of mankind about nuclear moments, is tabulated in this online database (the same data in a more dynamic presentation are here). Feel free to browse through it. You may, for instance, try to find in the database the nuclear quadrupole moment and the magnetic dipole moment - [about you](https://www.hyperfinecourse.org/topics/about-you/) - Hyperfine course quick start about you You don’t find yourself browsing this course site for no reason, do you? We would like to get to know you! Your expectationsPlease tell us what you expect from this course (you will be asked to fill out your username) Present yourselfWherever you are in the world, you’re not - [double split experiment](https://www.hyperfinecourse.org/topics/double-split-experiment/) - Hyperfine course refreshers double split experiment You might have learned about the double slit experiment before. In that case, this video is an entertaining reminder about the key aspects. If you didn’t meet the double slit experiment before, then this will be your first confrontation with quantum interference — a concept that plays a role - [webinar (w10)](https://www.hyperfinecourse.org/topics/webinar-w10-2/) - [b](https://www.hyperfinecourse.org/topics/b/) - [(optional) optional resources](https://www.hyperfinecourse.org/topics/optional-optional-resources/) - [webinar (w5)](https://www.hyperfinecourse.org/topics/webinar-w5/) - [exit (w5)](https://www.hyperfinecourse.org/topics/exit-w5/) - [(optional) quadrupole shift](https://www.hyperfinecourse.org/topics/optional-quadrupole-shift/) - [reading task - monopole shift???](https://www.hyperfinecourse.org/topics/reading-task-monopole-shift/) - [reading task - magnetic hyperfine interaction???](https://www.hyperfinecourse.org/topics/reading-task-magnetic-hyperfine-interaction/) - [webinar (w4)](https://www.hyperfinecourse.org/topics/webinar-w4/) - [exit (w4)](https://www.hyperfinecourse.org/topics/exit-w4/) - [webinar (w3)](https://www.hyperfinecourse.org/topics/webinar-w3/) - [exit (w3)](https://www.hyperfinecourse.org/topics/exit-w3/) - [Topic](https://www.hyperfinecourse.org/topics/topic-2/) - [Topic](https://www.hyperfinecourse.org/topics/topic/) ## Quizzes - [ENDOR paper](https://www.hyperfinecourse.org/quizzes/sfwd-quiz-635175a28954d8-44477165/) - [C11 calculation](https://www.hyperfinecourse.org/quizzes/sfwd-quiz-635175a2843655.69843102/) - [Forrest level scheme](https://www.hyperfinecourse.org/quizzes/sfwd-quiz-635175a28d9eb5.99236311/) - [gravitational analogue](https://www.hyperfinecourse.org/quizzes/sfwd-quiz-635175a291b763.49890000/) - [monopole shift](https://www.hyperfinecourse.org/quizzes/sfwd-quiz-635175a296e022.90873026/) - [multipole moments](https://www.hyperfinecourse.org/quizzes/sfwd-quiz-635175a29bde85.04638844/) - [NMR and NQR](https://www.hyperfinecourse.org/quizzes/sfwd-quiz-635175a29fee48.68392689/) - [no axial symmetry](https://www.hyperfinecourse.org/quizzes/sfwd-quiz-635175a2a3cb92.02631537/) - [orientation patterns](https://www.hyperfinecourse.org/quizzes/sfwd-quiz-635175a2a79e43.29068120/) - [PAC visual](https://www.hyperfinecourse.org/quizzes/sfwd-quiz-635175a2abb829.69717529/) - [perturbation theory](https://www.hyperfinecourse.org/quizzes/sfwd-quiz-635175a2b035b5.13205819/) - [quadrupole miscellaneous](https://www.hyperfinecourse.org/quizzes/sfwd-quiz-635175a2b4b179.66689703/) ## Elementor Header & Footer Builder - [Subscribe Section](https://www.hyperfinecourse.org/elementor-hf/subscribe-section/) - I have a question I have a suggestion I have a problem A free and open online course on the physics of hyperfine interactions and the experimental methods based thereupon.© Stefaan Cottenier