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The Magnetic Moments of the
Even-Even Nuclides from
Tellurium to Uranium

Background

The magnetic moment of a nucleus is due to the spinning of its charges. One part comes from the net sum of the intrinsic spins of its nucleons. The other part is due to the rotation of the positively charged protons in the nuclear structure.

However nucleons form spin pairs with other nucleons of the same type but opposite spin. Therefore there should be no magnetic moment due to the intrinsic spins of its nucleons if it has an even number of protons and an even number of neutrons.

Analysis

The magnetic moment of a nucleus μ due to the rotation of its charges is proportional to ωr²Q, where ω is the rotation rate of the nucleus, Q is its total charge and r is an average radius of the charges' orbits. The angular momentum L of a nucleus is equal to ωr²M, where M is the total mass of the nucleus. The average radii could be different but they would be correlated. Thus the magnetic moment of a nucleus could be computed by dividing its angular momentum by its mass and multiplying by it charge; i.e.,

μ = α(L/M)Q = (αQ/M)L

where α is a constant of proportionality, possibly unity. Angular momentum may be quantized. This would make μ directly proportional to Q and inversely proportional to M. But Q and M can be expected to be proportional to each other. That means that if L is quantized then μ is quantized. This means means that μ should approximately be a constant independent of the scale of the nucleus.

There can be a variation in μ with the neutron number n beause of its effect on the charge/mass ratio (Q/M).

The Data

Here is the graph of the data.

Generally the magnetic moment is a small amount is the range of 0.3 to 1.2 magnetons independent of the number of protons. But there are a few cases outside of that range.

The Magnetic Moments of the Even-Even Nuclides
from Tellurium (52) to Uranium (92)
Proton
Number 		Neutron
Number 		Magnetic
Moment
(magnetons)
52 68 0.78
52 70 0.66
52 72 0.56
52 74 0.62
52 76 0.5
52 78 0.58
52 80 4.7
52 82 5.08
54 70 0.46
54 72 0.74
54 74 0.82
54 76 0.76
54 78 0.74
54 80 1.1
54 82 2.4
56 74 0.7
56 76 0.68
56 78 0.86
56 80 0.69
56 82 1.4
56 86 0.85
56 88 0.68
56 90 0.56
58 68 10
58 78 -1.8
58 80 -1.7
58 82 1.9
58 84 0.42
58 88 0.48
58 90 0.74
60 74 1.2
60 76 11
60 78 -1.74
60 80 -1.92
60 82 1.69
60 84 0.32
60 86 0.58
60 88 0.7
60 90 0.76
62 76 10
62 78 -1.8
62 82 1.5
62 86 0.51
62 88 0.77
62 90 0.8
62 92 0.78
64 80 12.76
64 82 2.1
64 84 -0.16
64 88 0.96
64 92 0.82
64 94 0.78
64 96 0.72
66 86 11.6
66 88 1
66 90 0.78
66 92 0.72
66 94 0.74
66 96 0.69
66 98 0.68
68 84 -0.6
68 86 0.169
68 88 0.8
68 90 0.72
68 92 1.28
68 96 0.697
68 98 0.649
68 100 0.62
68 102 0.633
70 88 0.2
70 90 1.9
70 92 0.24
70 100 0.674
70 104 0.676
72 94 0.19
72 96 0.07
72 100 0.14
72 102 5.42
72 104 0.63
72 106 0.48
72 108 0.61
74 104 0.5
74 106 0.51
74 108 0.52
74 110 0.578
74 112 0.62
76 106 10.6
76 108 0.56
76 112 0.58
76 114 0.7
76 116 0.79
78 106 0.56
78 108 0.54
78 110 0.58
78 112 0.57
78 114 0.57
78 116 0.6
78 118 0.59
78 120 0.63
80 108 -2.02
80 110 -2.5
80 114 -0.24
80 116 -0.29
80 118 0.76
80 120 0.65
80 122 0.78
80 124 0.9
80 126 5.45
82 110 2.08
82 112 -0.6
82 114 0.49
82 116 0.38
82 118 -0.21
82 120 0.008
82 122 0.02
82 124 0.03
82 126 1.9
82 128 -1.8
84 114 7.3
84 116 7.44
84 118 7.45
84 120 7.38
84 122 7.34
84 124 5.3
84 126 5.48
86 120 6.6
86 122 6.98
86 124 7.18
86 126 4
86 136 0.92
88 124 7.1
88 126 7.08
88 128 0.1
88 136 0.9
90 142 0.28
92 146 0.37

The Effect of the Neutron Number

The graph of magnetic moment versus neutron number reveals 82 and 126 as critical values.

Eighty two and 126 are of course a magic numbers in nuclear shell theory. They represent the filling of a nuclear shells. Apparently criticality only requires the neutron number to be near a magic number.

The equation μ=α(Q/M)L with L constant means that μ should be a function of (Q/M). For a nucleus with p protons and n neutrons Q is proportional to p and M is approximately proportional to (p+n). Thus Q/M is approximately proportional to p/(p+n).

A regression of the magnetic moment on p/(p+n) and variables indicating whether or not p and n are within 2 of the critical levels of 82 or 126 gives the following results.

μ = 46.92644(p/(p+n)) + 0.23497(p≅82) + 0.29929(p≅82) + 0.77689(n≅82) + 2.63400(≅126) − 18.04426
[2.9 ]    [0.4 ]    [0.4 ]    [2.9 ]    [-3.7 ]    

The coefficient of determination for this equation is only 0.1, but the t-ratio of 2.9 for p/(p+n) strongly confirms the result of the analysis that μ due to nuclear rotation depends upon the Q/M ratio of the nucleus. This also indicates that the angular momenta of nuclei are quantized.

Although it was not confirmed by the regression analysis the data shows the magnetic moment to be definitely higher for the neutron number equal to 82.

p n μ
52 82 5.08
54 82 2.4
56 82 1.4
58 82 1.9
60 82 1.69
62 82 1.5
64 82 2.1

Conclusions

The magnetic moments of the even-even nucludes between tellurium and uranium seem to be nearly constant except for some extreme values associated with the filling of nuclear shells. There is statistically significant variation in the magnetic moment with variation in the variable p/(p+n) which represents the charge to mass ratio Q/M ratio of the nuclide. This also indicates that the angular momenta of nuclei are quantized.

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