﻿ The Statistical Explanation of the Magnetic Moments of Nuclides
San José State University

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Thayer Watkins
Silicon Valley
USA

The Statistical Explanation of the
Magnetic Moments 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 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 = αL(Q/M)

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 for which the magnetic moment is unusually. 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 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 are

#### μ = 1.74425sp −1.45842sn + 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 above, indicate that μ is definitely dependent upon sp, sn, p/(p+γn), p≅50 and n≅50 at the 95 percent level of confidence..

A re-regression with the variables whose coefficients were not significantly different from zero at the 95 percent level of confidence yields

#### μ = 1.74458sp −1.43806sn + 2.48749p/(p+γn) + 1.02813(p≅50) + 3.08987(n≅50) [9.2 ]                 [-7.7 ]                 [6.4 ]                 [3.7 ]                 [8.2 ]

The coefficient of determination (R²) for this equation is 0.555, essentially the same as the previous regression.