The Magnetic Moments of the Even p Odd n Nuclides from Tellurium (52) to Curium (98)
San José State University

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The Magnetic Moments of
the Even p Odd n Nuclides
from Tellurium (52) to Curium (98)

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 the net magnetic moment in magnetons of a nucleus due to the intrinsic spins of its nucleons should be: 0.0 .for an even-even nucleus, 0.87985 for an odd-odd nucleus, 2.79285 for an odd p and even n nucleus and −1.9130 for an even p and odd n nucleus.

Analysis of the Magnetic Moment
Due to the Rotation of a Nucleus

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. More specifically μ should be proportional to Q/M. The charge is proportional to p and M is proportional to (p+γn), where γ is the ratio of the mass of a neutron to that of a proton; i.e., 1.001375. . Thus Q/M is proportional to p/(p+γn).

Previous Results

Previous studies found that there are critical values of proton or neutron numbers for which the magnetic moment is unusually high. The primary critical value is 50, a so-called magic number indicating the filling of a nuclear shell. To a much lesser extent 28, another nuclear magic number, is a critical value.

The Data

Here is the graph of the data.

Generally the magnetic moment is a small amount is the range of −2 to +2 magnetons independent of the number of protons. But there are a few cases outside of that range and their magnitudes might be related to the proton number.

The Magnetic Moments of the Even p, Odd n Nuclides
Proton
Number
Neutron
Number
Magnetic
Moment
(magnetons)
52 67 0.25
52 69 0.895
52 71 -0.7369478
52 73 -0.888505
52 75 0.635
52 77 0.702
52 79 0.696
52 83 -3.8
54 63 -0.5938
54 65 -0.6542
54 67 -0.701
54 69 -0.15
54 71 -0.269
54 73 -0.5033
54 75 -0.777976
54 77 0.6915
54 79 0.8129
54 81 0.9032
54 83 -0.968
54 85 -0.304
54 87 0.01
54 89 -0.4599
56 65 0.66
56 67 -0.68
56 69 0.177
56 71 0.0834
56 73 -0.4
56 75 0.708113
56 77 0.77167
56 79 0.83794
56 81 0.93737
56 83 -0.973
56 85 0.337
56 87 0.443
56 89 -0.285
58 77 -0.66
58 79 0.96
58 81 1.06
58 83 1.09
58 85 0.43
60 75 -0.78
60 77 -0.633
60 79 0.907
60 81 1.012
60 83 -1.065
60 85 -0.656
60 87 0.578
60 89 0.351
62 77 -0.53
62 79 -0.74
62 81 1.01
62 83 -1.11
62 85 -0.812
62 87 -0.6677
62 89 -0.3611
62 91 -0.021
64 83 1.02
64 85 0.88
64 87 0.77
64 89 0.38
64 91 -0.2572
64 93 -0.3398
64 95 -0.44
66 81 -0.915
66 83 -0.119
66 85 -0.945
66 87 -0.782
66 89 -0.385
66 91 -0.301
66 93 -0.354
66 95 -0.48
66 97 0.673
66 99 -0.52
68 75 -0.934
68 77 -0.669
68 79 -0.412
68 81 -0.304
68 83 -0.365
68 85 0.557
68 87 0.643
68 89 -0.56385
68 91 0.52
68 93 0.659
70 85 -0.84
70 87 -0.639
70 89 -0.368
70 91 -0.327
70 93 -0.374
70 95 0.478
70 97 0.623
70 99 -0.635
70 101 0.49367
70 103 -0.648
70 105 0.768
72 93 0.14
72 101 6.6
72 103 -0.62
72 105 0.7935
72 107 -0.6409
74 109 0.11778476
74 113 0.621
76 107 -0.794
76 111 0.06465189
76 113 0.659933
76 115 0.96
76 117 0.73
78 115 0.51
78 117 0.774
78 119 0.408
78 121 -0.421
78 123 -0.501
78 125 0.603
78 127 0.60952
78 129 0.51
80 101 0.507
80 103 0.524
80 105 0.509
80 107 -1.044
80 109 -0.6086
80 111 -0.618
80 113 -0.6276
80 115 0.5414749
80 117 0.5273744
80 121 -0.5602257
80 123 0.84895
80 125 0.60089
82 109 -1.172
82 111 -1.15
82 113 -1.128
82 115 -1.075
82 117 -1.0742
82 119 0.6753
82 121 0.6864
82 123 0.7117
82 125 0.592583
82 127 -1.4735
82 129 -1.4037
84 115 0.99
84 117 0.94
84 119 0.74
84 121 0.76
84 123 0.79
84 125 0.68
84 127 -0.38
86 117 -0.96
86 119 0.802
86 121 0.816
86 123 0.83884
86 125 0.601
86 127 4.73
86 133 -0.44
86 135 -0.02
86 137 -0.776
86 139 -0.696
88 121 0.865
88 123 0.878
88 125 0.613
88 127 15.78
88 133 -0.18
88 135 0.271
88 137 -0.7338
88 139 -0.404
88 141 0.503
90 139 0.46
92 141 1.5604
92 143 -0.38
94 145 -0.68
94 147 -0.683
96 147 0.4
96 149 0.5
96 151 0.36
98 155 4.1

The Effect of the Neutron Number

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

The Regression Equation and its Estimate

Let μ be the measured magnetic moment, sp and sn be the presence or absence (1 or 0) of a singleton proton or neutron, respectively. The variables like p≅82 represent 1 or 0 depending upon whether 81≤p≤83. The regression results were

μ = −0.73467p/(p+γn) −0.29198(n≅82) + 2.17673(n≅126) − 0.52393(p≅28) + 0.44073
    [-0.13 ]     [-0.75 ]    [4.6 ]    [-1.7 ]     [0.2 ]     

The coefficient of determination (R²) for this equation is only 0.125. The t-ratios for the coefficients, shown in square brackets indicate that μ is definitely dependent only upon n≅126 at the 95 percent level of confidence. Otherwise μ is a constant.


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