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The Magnetic Moments of
the Even p Even n Nuclides

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 for an even p, even n nucleus there should be zero magnetic moment due to the intrinsic spins of its nucleons.

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 could be a slight variation in μ with the neutron number n or proton number p because of their effects on the ratio (Q/M).

The Data

Here are the graphs of the data for the magnetic moments of the even p, even n nuclides .

The appearances of these graphs are extraordinary. First of all there are clusters of data that fit the notion that μ is roughly linear. In these cases the lines are slightly declining to the right. Second there is a scattering of data points separate from those clusters. In the case of μ versus the number of protons there does not seem to be any rhyme or reason to the scattering. But for the case of μ versus the number of neutrons it is a different story. Above the nuclear magic numbers of 126, 82, 50 and 28 there are clusters of data points showing unusual levels of μ near those magic numbers. The magic numbers represent the filling of nuclear shells.

The graph of μ versus proton number shows the significance of the magic numbers as well, but value of μ for a data point is determined by the neutron number as well as the proton number.

A better picture of the relationships comes with the presentation of μ as a two dimensional function of neutron number and proton number. Before such a presentation is made the results of a regression analysis will be considered. The previous analysis indicated that μ should be proportional to the charge/mass ratio, Q/M. The charge of a nuclide is proportional to its proton number p. The mass of a nuclide is proportional to (p+γn), where γ is the ratio of the mass of a neutron to that of a proton. So one variable determining μ is Q/M as p/(p+γn). The other variables are 0/1 variables such as n≅126 −representing whether or not the neutron number is near the magic number of 126. Nearness is taken to be "within two units."

The regression results are:

μ = −0.84216p≅28 + 0.13491n≅28 + 0.23662p≅50 + 2.73742n≅50 + 0.55415p≅82 + 0.83983n≅82 + 2.54899n≅126 + 18.65300Q/M −6.79217
       [-1.0]       [0.1]       [0.4]       [3.4]       [0.9]       [1.2]       [3.0]       [2.3]       [-2.0]      

The coefficient of determination for this regression equation is only 0.125 but the t-ratios shown in square brackets confirm that at the 95 percent level of confidence μ does depend upon Q/M and the nearness of the neutron number to the magic numbers of 126 and 50. What is needed for further analysis is a theory of what determines μ when a nuclear number is equal to a critical number.

The two dimensional display of the data is relevant for that issue. The data themselves are:

Magnetic Moments of the Even p, Even n Nuclides
protons →
neutrons ↓ 6 8 10 12 14
8 0.82 1.688
10 -0.57 1.08
12 -0.7 0.65
14 1.0 1.1
16 0.8

Magnetic Moments of the Even p, Even n Nuclides
protons →
neutrons ↓ 16 18 20 22 24
16 0.9
18 1.0
20 1.6
22 -0.2 -0.6
24 1.0
26 0.9 1.2
28 2.7 3.0
30 1.1

Magnetic Moments of the Even p, Even n Nuclides
protons →
neutrons ↓ 28 30 32 34
28 3.4
30 1.22 -0.1
32 0.9 0.2
34 0.68 0.8
36 0.9 0.5 2.4
38 1 0.94
40 0.6 0.8
42 0.87 0.8
44 0.84 0.8
46 0.8
48 0.9

Magnetic Moments of the Even p, Even n Nuclides
protons →
neutrons ↓ 36 38 40 42 44
42 1.08
44 -1.97 2
46 0.84 0.5 0.5
48 0.55 -1.51 5.5
50 2.3 6.25 11.3 8.12
52 0.76 -0.06 0.7 0.8
54 -0.52 0.7 1.02
56 -0.52 0.84 0.74
58 0.4 0.82

Magnetic Moments of the Even p, Even n Nuclides
protons →
neutrons ↓ 46 48 50 52 54
50 10.97
52 0.82 9.9
54 0.93 10.3 0.5
56 0.72 0.8 -1.51
58 0.62 0.7 6.25
60 0.57 -0.06
62 0.6 -0.52
64 0.58 -0.52
66 0.6
68 0.78
70 0.66 0.46
72 0.56 0.74
74 0.62 0.82
76 0.5 0.76
78 0.58 0.74
80 4.7 1.1
82 5.08 2.4

Magnetic Moments of the Even p, Even n Nuclides
protons →
neutrons ↓ 56 58 60 62 64
74 0.7 1.2
76 0.68 10 11 10
78 0.86 -1.8 -1.74 -1.8
80 0.69 -1.7 -1.92 1.5 12.76
82 1.4 1.9 1.69 0.51 2.1
84 0.85 0.42 0.32 0.77 -0.16
86 0.68 0.48 0.58 0.8 0.96
88 0.56 0.74 0.7 0.78 0.82
90 0.76 0.76 0.78
92 0.72

Magnetic Moments of the Even p, Even n Nuclides
protons →
neutrons ↓ 66 68 70 72 74
84 -0.6
86 11.6 0.169
88 1 0.8 0.2
90 0.78 0.72 1.9
92 0.72 1.28 0.24
94 0.74 0.697 0.674 0.19
96 0.69 0.649 0.676 0.07
98 0.68 0.62 0.14
100 0.56 0.633 5.42
102 0.76 0.63
104 0.48 0.5
106 0.61 0.51
108 0.52
110 0.578
112 0.62

Magnetic Moments of the Even p, Even n Nuclides
protons →
neutrons ↓ 76 78 80 82 84
106 10.6 0.56
108 0.56 0.54 -2.02
110 0.58 0.58 -2.5 2.08
112 0.7 0.57 -0.24 -0.6
114 0.79 0.57 -0.29 0.49 7.3
116 0.6 0.76 0.38 7.44
118 0.59 0.65 -0.21 7.45
120 0.63 0.78 0.008 7.38
122 0.9 0.02 7.34
124 5.45 0.03 5.3
126 1.9 5.48
128 -1.8 0.51
130 0.52
132 0.578
134 0.62

Magnetic Moments of the Even p, Even n Nuclides
protons →
neutrons ↓ 86 88 90 92 94
120 6.6
122 6.98
124 7.18 7.1
126 4 7.08
128 0.92 0.1
130 0.9
132
134
136
138
140
142 0.28
144
146 0.37

(To be continued.)

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