g-factor calculations

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  • #3594 Score: 0
    Brecht.Biesmans
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    We know from the lecture video that µ = gµ_NI/hbar
    Since in the expression above we have operators, we need to look at an expression where we can use the size of the nuclear magnetic moment and the value of the spin. Thus we use, in accordance with the former expression, that
    µ_z = gµ_NI
    with µ_z the z component of the nuclear magnetic moment, and I the value of the spin (note that the spin values are quantised in terms of hbar*I, thus the hbar disappears)
    Now we can calculate g as
    g = µ_z/µ_NI
    Now given the groundstate of the 111Cd isotope we can find the tabulated values of -0.5940 nm for µ_z and 1/2 for I. The tabulated values for µ_z are often presented in units of the nuclear magneton µ_N, thus g = µ_z/I (unit-less), which gives us the results:
    (1.a)
    ground state 111Cd -> g = -0.5948861/(1/2) = -1.1897722
    245 keV level 111Cd -> g = -0.766/(5/2) = -0.3064
    (1.b)
    For an electron we have to use the Bohr magneton µ_B instead of µ_N (mass difference), but the rest of the expression is similar (instead of nuclear spin I, use electron spin, i.e. 1/2)
    g = 1/(1/2) = 2
    which is as we have seen in textbooks (or close to 2.002319)
    (1.c)
    Since the mass of a neutron is nearly equal to the mass of a proton, we can again use the first equation we had for µ, thus using the nuclear mangeton
    µ_z = g*1/2 = -1.913

    #3595 Score: 0
    Brecht.Biesmans
    Participant

    Note that the g-factors are dimensionless and the the magnetic moment values are in units of µ_N (unless it is an electron, there it is µ_B)

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