﻿ The Magnetic Moments of the Even p Odd n Nuclides from Helium (2) to Tin (50)
San José State University

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Thayer Watkins
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USA

The Magnetic Moments of
the Even p Odd n Nuclides
from Helium (2) to Tin (50)

## 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)
2 3 -2.12749772
4 5 -1.1778
6 3 -1.3914
6 5 -0.964
6 7 0.70241
6 9 1.32
8 5 1.3891
8 7 0.71951
8 9 -1.819379
8 11 1.53195
10 9 -1.88542
10 11 -0.661797
10 13 -1.08
12 11 0.5364
12 13 -0.85545
14 13 -0.8554
14 15 -0.55529
14 19 1.21
16 15 0.48793
16 17 0.64382112
16 19 1
18 15 -0.723
18 17 0.633
18 19 0.8
18 21 -1.588
20 19 1.02168
20 21 -1.594781
20 23 -1.3173
20 25 -1.3274
20 27 -1.38
22 21 0.85
22 23 0.095
22 25 -0.78848
22 27 -1.10417
24 25 0.476
24 27 -0.934
24 29 -0.47454
26 27 -0.386
26 29 2.7
26 31 0.09044
26 33 -0.3358
28 29 -0.7975
28 31 0.35
28 33 -0.75002
28 35 0.752
28 37 0.69
30 33 -0.28164
30 35 0.769
30 37 0.875479
30 39 1.157
30 41 1.052
32 35 -0.849
32 37 0.735
32 39 0.547
32 41 -0.8794677
32 43 0.51
34 39 0.87
34 41 0.67
34 43 0.5350422
34 45 -1.018
36 39 -0.531
36 41 -0.583
36 43 -0.536
36 45 -0.908
36 47 -0.970669
36 49 -1.005
36 51 -1.018
36 53 -0.33
36 55 -0.583
36 57 -0.413
36 59 -0.41
38 39 -0.348
38 41 -0.474
38 43 0.543
38 45 -0.829
38 47 -1
38 49 -1.0928
38 51 -1.147
38 53 -0.885
38 55 -0.793
38 57 -0.537
38 59 -0.498
38 61 -0.261
40 49 -1.08
40 51 -1.30362
40 55 1.13
40 57 1.37
40 59 0.42
42 47 8.3
42 49 8.81
42 51 9.93
42 53 -0.9142
42 55 -0.9335
42 57 0.375
42 59 0.375
42 65 -0.92
44 49 8.97
44 51 0.861
44 53 -0.787
44 55 -0.641
44 57 -0.719
44 59 -0.206
44 61 -0.32
44 65 -0.22
46 55 -0.66
46 57 -1.05
46 59 -0.642
48 55 -0.81
48 57 -0.7393
48 59 -0.6150554
48 61 -0.8278461
48 63 -0.5948861
48 65 -0.6223
48 67 -0.6484259
50 59 -1.079
50 61 0.608
50 63 -0.8791
50 65 -0.91883
50 67 -1.00104
50 69 -1.04728
50 71 0.6978
50 73 -1.37
50 75 -1.348

## The Effect of the Neutron Number

The graph of magnetic moment versus neutron number reveals 50 as a critical value.

## 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≅50 represent 1 or 0 depending upon whether 49≤p≤51. The regression results were

#### μ = 1.74425sp −1.45842 + 2.39179p/(p+γn) + 0.12082(p≅28) + 1.08106(p≅50) + 0.87698(n≅28) + 3.14748(n≅50)     [9.2 ]     [-7.8 ]    [6.1 ]    [0.3 ]    [3.9 ]     [1.9 ]    [8.4 ]

The coefficient of determination (R²) for this equation is only 0.56, but the t-ratios for the coefficients, shown in square brackets indicate that μ is definitely dependent upon sp, sn, p/(p+γn), p≅50 and n≅50 at the 95 percent level of confidence..