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The nucleus is described in the periodic system of the elements with 2 numbers,
-the atomic number Z, the amount of protons in the nucleus
-the mass number A, the combined mass of the protons and neutrons of which the nucleus is build. We have A=Z+N with N the number of neutrons in the nucleus.
If nuclides have the same number Z, they are called isotopes.
The nucleons in every nucleus are bound to each other, with a binding energy B per nucleon (and thus, like atoms, we can say that nuclei have excited states). This binding energy is in the order of MeV, usually 8-9 MeV per nucleon. The mass of the nucleus can be determined by the semi-empirical mass formula or Weizsäcker formula With this formula we can determine the mass of an atom. Apart from the mass of the neutrons, protons and electrons it contains, we have a volume term, proportional to A due to the short range nucleon force (consider only pair interactions), a surface term proportional to A^(2/3) as the radius of the nucleus is proportional to A^(1/3), due to less surrounding nucleons at the surface, which lowers the binding energy, a Coulomb term due to proton-protonrepulsion, an asymmetry term (heavier nuclei have more neutrons than protons, to compensate for the coulomb term) and finally a pairing term. This last term is a consequence of the neutrons and protons to be coupled as pairs, which makes nuclei with an even number of protons and/or neutrons more stable than odd numbered nuclei.

Speaking of stability, this implies that nuclei can and will decay. This decay can happpen due to different processes. Most important are alpha and beta decay.

-beta decay:
if we plot the nuclear mass M in function of the atomic number Z, we have a parabole form. Isobars (same mass number) which are at the minimum of the parabola are most stable.
For odd mass of the nucleus:
Isobars not at the minimal mass will decay via beta-decay: a neutron decays into a proton, an electron and an anti-electron neutrino (excess of neutron), or beta+ decay: proton to neutron, positron and electron neutrino.
For even mass nuclei, we have due to the pairing term two parabolas instead of one. This allows for more than one stable isobar, due to double beta decay. Even/even isobars are therefor stable, while odd/odd are unstable (in general).
Another kind of beta decay is if the proton captures an electron, which becomes a neutron and an electron neutrino. This only happens if the electrons come close enough to the nucleus, so this proces is sometimes called K-capture (the K-shell is the innermost shell).

Alpha decay:
Sometimes (especially for heavy nuclei), the total binding energy of the nucleus can be lowered by emitting a group of nucleons, most likely alpha particles.

Nuclei can not only decay, but also fission to build heavier nuclei.
-spontaneous fission: only for very heavy nuclei, like uranium for example. We can see spntaneous fission as the moment when the coulomb forces by the protons are larger than the binding energies of the nucleons. The nucleus will deform until it splits.
-induced fission: by a stream of nucleons, the potential barriere that holds the nucleus together can be broken due to neutron capture, if the nucleus is heavy/large enough.

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