﻿ The Magnetic Moments of the Odd-Odd Nuclides from Hydrogen (1) to Antimony (51)
San José State University

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Thayer Watkins
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The Magnetic Momentsof the
Odd-Odd Nuclides from
Hydrogen (1) to Antimony (51)

## 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 Hydrogen to Antimony (p=51).

The data themselves are:

Magnetic Moments of the Odd-Odd Nuclides
from hydrogen (1) to antimony (51)
p n μ
(magnetons)
1 1 0.857438228
3 3 0.8220473
3 5 1.6534
5 3 1.0355
5 5 1.80064478
5 7 1.00272
5 9 1.185
7 5 0.4573
7 7 0.403761
7 9 1.6
9 9 1.6
9 11 2.09335
11 9 0.3694
11 11 1.746
11 13 1.69
11 15 2.851
11 17 2.426
11 19 2.083
13 13 2.804
13 15 3.242
17 19 1.28547
17 21 2.05
19 17 0.548
19 19 1.371
19 21 -1.2981
19 23 -1.1425
19 25 -0.856
19 27 -1.051
21 23 2.56
21 25 3.03
23 23 1.64
23 25 2.012
23 27 3.3456889
25 27 3.0622
25 29 3.2819
25 31 3.2266
27 29 3.85
27 31 4.044
27 33 3.799
29 31 1.219
29 31 -0.38
29 33 -0.217
29 37 -0.282
31 35 1.01
31 37 0.01175
31 39 -0.26
31 41 -0.1322
33 35 0.23
33 37 2.1061
33 39 -2.1566
33 41 -1.597
33 43 -0.906
35 37 0.6
35 39 1.68
35 41 0.54821
35 43 0.13
35 45 0.514
35 47 1.627
35 49 1.9
37 39 -0.3726228
37 41 2.549
37 43 -0.0836
37 45 0.5545083
37 47 -1.324116
37 49 -1.692
37 51 0.508
37 53 1.616
37 57 1.498
37 59 1.466
39 49 4.87
39 51 -1.63
41 49 4.96
41 51 6.137
41 55 4.976
43 49 8.9
43 51 5.12
43 53 5.09
43 65 0.5
45 55 4.324
45 57 0.5
45 59 2
45 61 2.575
47 55 4.6
47 57 3.917
47 59 2.9
47 61 2.6884
47 63 2.727
47 65 0.0547
49 55 4.44
49 57 4.916
49 59 4.561
49 61 4.365
49 63 2.82
49 65 2.817
49 67 2.7876
49 69 4.231
49 71 4.295
49 73 4.318
49 75 4.043
49 77 4.034
51 61 2.192
51 63 1.72
51 65 2.715
51 67 2.47
51 69 2.3
51 71 1.9
51 73 1.2
51 75 1.28
51 77 1.3

## The Effect of the Neutron Number

The graph of magnetic moments versus neutron numbers reveals does reveal strongly a critical value.

But the criticality of neutron numbers near 50 shows up in the data.

 p n μ(magnetons) 35 49 1.9 37 49 -1.692 39 49 4.87 41 49 4.96 43 49 8.9 37 51 0.508 39 51 -1.63 41 51 6.137 43 51 5.12

A similar criticality is found near a neutron number of 28.

 p n μ(magnetons) 19 27 -1.051 23 27 3.3456889 25 27 3.0622 25 29 3.2819 27 29 3.85 25 31 3.2266 27 31 4.044 29 31 1.219 29 31 -0.38

There is also a criticality of proton numbers near 50 and 28 as can be in the previous table for proton numbers 49 and 27.

There however is no appearance of the constancy of the magnetic moment for proton and neutron numbers not near a critical number.