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The Magnetic Moments of
the Odd p Even n 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 even n, odd p nucleus there should be the net magnetic moment due to the intrinsic spins of one proton. The magnetic dipole moment of a proton, measured in magneton units, is 2.79285.

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 because of its effect on the ratio (Q/M).

The Data

Here is the graph of the data for the magnetic moments of the even n, odd p nuclides from Hydrogen (1) to Antimony (51) .

The data themselves are:

Magnetic Moments of the Even n, Odd p Nuclides
from Hydrogen (1) to Antimony (51)
p n mu
1 0 2.79285
1 2 2.97896244
3 4 3.256427
3 6 3.4391
3 8 3.668
5 6 2.6886489
5 8 3.1778
5 12 2.55
7 6 0.3222
7 8 -0.28318884
7 10 -0.352
9 8 4.7213
9 10 2.628868
9 12 3.93
11 10 2.8363
11 12 2.217522
11 14 3.683
11 16 3.895
11 18 2.449
11 20 2.305
13 12 3.6455
13 14 3.6415069
15 14 1.2349
15 16 1.1316
17 16 0.752
17 18 0.8218743
17 20 0.6841236
19 18 0.20321
19 20 0.39147
19 22 0.2148701
19 24 0.1633
19 26 0.1734
19 28 1.933
21 20 5.431
21 22 4.62
21 24 4.756487
21 26 5.34
23 26 4.47
23 28 5.1487057
25 26 3.5863
25 28 5.024
25 20 3.4532
27 28 4.822
27 30 4.72
27 32 4.627
29 32 2.14
29 34 2.227206
29 36 2.3817
31 36 1.8507
31 38 2.01659
31 40 2.56227
33 36 1.58
33 38 1.674
33 40 1.63
33 42 1.43948
33 44 0.74
35 38 1.97
35 40 0.76
35 42 0.92
35 44 2.1064
35 46 2.270562
37 40 0.654468
37 42 3.3579
37 46 1.4249
37 48 1.35298
37 50 2.75131
37 52 2.3836
37 54 2.1815
37 56 1.41
37 58 1.334
37 60 1.841
39 44 2.1
39 46 6.2
39 48 6.06
39 50 -0.1374154
39 52 0.1641
41 46 7
41 48 6.216
41 50 9.14
41 52 6.1705
41 54 6.141
41 56 6.153
43 50 6.32
43 52 5.94
43 56 5.6847
45 50 10.9
45 54 5.62
45 56 5.43
45 58 -0.884
45 60 4.41
47 54 5.7
47 56 4.47
47 58 0.1014
47 60 -0.11357
47 62 0.13056
47 64 -0.146
47 66 0.159
49 56 5.675
49 58 5.585
49 60 5.538
49 62 5.503
49 64 5.5289
49 66 5.5408
49 68 5.519
49 70 5.515
49 72 5.502
49 74 5.491
49 76 5.502
49 78 5.522
51 64 3.46
51 66 3.43
51 68 3.45
51 70 3.3634
51 72 2.5498
51 74 2.63
51 76 2.697
51 78 2.79
51 80 2.89
51 82 3

The Effect of the Neutron Number

The graph of magnetic moments versus neutron numbers does reveal critical values at 50 and 28.

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

p n μ
(magnetons)
41 50 9.14
43 50 6.32
45 50 10.9

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

There is also a criticality of proton numbers near 50 and 28.

There is an appearance of the constancy of the magnetic moment for proton and neutron numbers not near a critical number.

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