﻿ The Quantization of the Magnetic Moments and Angular Momenta of Nuclides
San José State University

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The Quantization of the Magnetic Moments
and Angular Momenta of Nuclides

## 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 of any nucleus due to the net intrinsic spin of its nucleons is due entirely to any singleton proton and/or singleton neutron which it might have.

The net magnetic dipole moments of a proton and a neutron, measured in magneton units, are 2.79285 and −1.913. respectively. Therefore the magnetic moment for a nuclide due to the intrinsic spins of its nucleons should be zero for even-even nuclides, 2.79285 magnetons for the nuclides with odd p and even n, −1.913 magnetons for nuclides with an even p and odd n and +0.87985 magnetons for odd-odd nuclides. ## 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 L by its mass M and multiplying by its charge Q; i.e.,

#### μ = α(L/M)Q = α(Q/M)L

where α is a constant of proportionality. 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 approximately proportional to each other. That means that if L is quantized then μ is quantized. This would mean that μ should approximately be a constant independent of the scale of the nucleus.

More precisely Q is proportional the proton number p. The mass of a nucleus is proportional to p+γn, where γ is the ratio of the mass of a neutron to that of a proton. Thus (Q/M) is proportional to p/(p+γn).

There could be a slight variation in μ with proton number p and neutron number n because of their effects on the ratio (Q/M).

The above relationship can be read the other way around; i.e., used to compute angular momentum from the magnetic moment, That is to say

## The Data

Here is the graph of the cumulative frequency data for the quantity (μ(1 + γ(n/p)) . The shape of the graph indicates some sort of centrality for the distribution function for angular momentum.

The distribution function for the quantity (μ(1 + γ(n/p)) is as follows The graph indicates concentrations at three levels but some peaks at other levels. The extreme values at either end of the range are left out of the analysis. The minor peaks in the distribution are better perceived in the table below.

As seen below the major peak in the distribution is at the value of zero. There are secondary peaks centered about the intervals with midpoint values of about ±0.65. There are lesser peaks at or near values of multiples of 0.7; i.e., at 1.55, 2.15, 2.75 and 3.45. This would represent a quantization of (μ(1 + γ(n/p)) and hence of angular momenta.

Distribution of Estimated
Angular Momenta of Nuclides
(μ(1 + γ(n/p))
Midpoint
of Interval
Frequency
-1.95 2
-1.85 1
-1.75 5
-1.65 3
-1.55 2
-1.45 4
-1.35 2
-1.25 7
-1.15 3
-1.05 7
-0.95 13
-0.85 18
-0.75 14
-0.65 48
-0.55 19
-0.45 14
-0.35 22
-0.25 17
-0.15 13
-0.05 8
0.05 80
0.15 24
0.25 16
0.35 14
0.45 22
0.55 51
0.65 55
0.75 50
0.85 37
0.95 23
1.05 17
1.15 8
1.25 11
1.35 10
1.45 13
1.55 16
1.65 15
1.75 8
1.85 5
1.95 11
2.05 15
2.15 9
2.25 10
2.35 13
2.45 5
2.55 8
2.65 6
2.75 16
2.85 13
2.95 4
3.05 7
3.15 6
3.25 4
3.35 5
3.45 8
3.55 5
3.65 5
3.75 6
3.85 12
3.95 2
4.05 10
4.15 2
4.25 6
4.35 6
4.45 5
4.55 2
4.65 8
4.75 5
4.85 4
4.95 4
5.05 3
5.15 0
5.25 1
5.35 3
5.45 7
5.55 11
5.65 4
5.75 0
5.85 1
5.95 3
6.05 1
6.15 5
6.25 3
6.35 0

(To be continued.)