﻿ The Magnetic Moments of the Odd-Odd Nuclides from Iodine (53) to Einsteinium (99)
San José State University

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Thayer Watkins
Silicon Valley
USA

The Magnetic Momentsof the
Odd-Odd Nuclides from
Iodine (53) to Einsteinium (99)

## 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 odd-odd nucleus there should be the net magnetic moment due to the intrinsic spins of one proton and one neutron. The magnetic dipole moment of a proton, measured in magneton units, is +2.79285. That of a neutron is −1.9130. The ratio of these two numbers is −0.685, intriguingly close to −2/3. The sum of the moments of a proton and a neutron is 0.87980464 magnetons.

## 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 beause of its affect on the ratio (Q/M).

## The Data

Here is the graph of the data for the magnetic moments of the odd-odd nuclides from to Antimony (p=51).

The data themselves are:

Magnetic Moments of the Odd-Odd Nuclides
from Iodine (53) to Einsteinium (99)
p n μ
(magnetons)
53 65 2
53 67 1.23
53 69 0.94
53 71 1.446
53 73 1.438
53 75 -0.72
53 77 3.349
53 79 3.088
55 63 3.876
55 65 3.87
55 67 -0.1333
55 69 0.673
55 71 0.777
55 73 0.974
55 75 1.46
55 77 2.222
55 79 2.9937
55 81 3.711
55 83 0.7
55 85 0.1338953
55 89 -0.546
55 91 -0.515
57 81 3.713646
57 83 0.73
59 77 2.3
59 83 0.234
59 85 -1.2
61 77 3.2
61 83 1.69
61 87 2.1
63 75 5.3
63 77 1.365
63 79 1.54
63 81 1.893
63 83 1.421
63 85 2.34
63 87 2.708
63 89 -1.9401
63 91 -2.005
63 95 1.44
65 83 -1.75
65 85 -0.9
65 87 -0.58
65 89 1.6
65 91 1.7
65 93 1.758
65 95 1.79
67 85 -1.02
67 87 -0.643
67 89 2.99
67 91 3.77
67 93 3.71
67 95 3.6
67 99 3.6
69 87 0.4
69 89 0.04
69 91 0.16
69 93 0.068
69 95 2.83
69 97 0.092
69 99 0.227
69 101 0.246
71 101 2.893
71 103 1.9
71 105 3.169
73 105 2.74
73 107 4.825
73 109 3.02
75 105 1.6
75 107 2.84
75 109 2.53
75 111 1.739
75 113 1.788
77 103 2.2
77 105 1.91
77 107 0.696
77 109 3.88
77 111 0.302
77 113 0.04
77 115 1.924
77 117 0.39
79 103 1.3
79 105 2.07
79 107 -1.28
79 109 -0.07
79 111 -0.065
79 113 -0.0107
79 115 0.0763
79 117 0.58
79 119 0.64
79 121 5.9
81 107 0.483
81 109 0.254
81 111 0.2
81 113 0.14
81 115 0.072
81 117 0
81 119 0.04
81 121 0.06
81 113 0.09
81 115 4.27
81 117 0.292
83 119 4.9
83 121 4.322
83 123 4.631
83 125 4.633
83 127 -0.4451
83 129 0.41
85 123 2.69
85 125 9.8
85 127 5.94
87 121 4.75
87 123 4.4
87 125 4.62
87 127 5.62
87 133 -0.67
87 135 0.63
87 137 0.4
87 139 0.0712
87 141 -0.76
91 137 3.5
91 139 2
95 147 0.3879
99 155 2.9

## The Effect of the Neutron Number

The graph of magnetic moments versus neutron numbers reveals does not reveal strongly a critical value.

But the criticality of neutron numbers near 126 shows up in the data.

 p n μ(magnetons) 85 125 9.8 85 127 5.94 87 121 4.75 87 123 4.4 87 125 4.62 87 127 5.62

A similar criticality is found near a neutron number of 82.

 p n μ(magnetons) 63 79 1.54 63 81 1.893 63 83 1.421

There is not a criticality of proton numbers near 82 as can be in the previous table for proton numbers 81 and 83.

There however is no appearance of a fixed level of 0.87980464 magnetons for the magnetic moment for proton and neutron numbers not near a critical number. On the other hand, there is clearly no linear dependece of the magnetic moment on the numbers of protons or neutrons.

## Regression Analysis

Regression analysis was used to establish what variables the magnetic moment is dependent upon. The first equation estimated was

#### μ = 5.77116p/(p+n) + 3.72713n≅126 − 0.19687n≅82 − 0.52314&p≅82 −0.84948       [0.6]       [4.8]       [-0.3]       [-1.1]       [-0.2]

The coefficient of determination for this equation is only 0.166. However, from the t-ratios it is seen that the neutron number being close to 126 is statistically significantly different from zero at the 95 percent level of confidence. None of the t-ratios for the other variables are statistically significant.

A re-estimation with only n&conng;126 gives