San José State University

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Thayer Watkins
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The Quantization of the Magnetic Moments
and Angular Momenta of Odd-Odd 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 for an even p, even n nucleus there should be zero magnetic moment due to the intrinsic spins of its nucleons. However an odd-odd nucleus should have magnetic moment of 0.87985 magnetons which is the net sum of the 2.79285 magnetons of the singleton proton and the −1.913 magnetons of the singleton neutron.

Analysis

The magnetic moment of a nucleus ν due to the rotation of its charges is proportional to ωr² ;Q, 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 due strictly to its rotation could be computed by dividing its angular momentum by its mass and multiplying by it charge; i.e.,

ν = α(L/M)Q = (αQ/M)L

where α is a constant of proportionality. Angular momentum may be quantized. This would make ν 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 ν is quantized. This means means that ν should approximately be a constant independent of the scale of the nucleus.

There would be a variation in ν with the neutron number n or proton number p because of their effects on the ratio (Q/M). The charge of a nucleus is proportional to its proton number. Its mass is roughly proportional to the sum its proton number and its neutron number; more precisely to (p+gn) where g is the ratio of the mass of a neutron to the mass of a proton.

The quantization of angular momentum can be tested by inverting the above formula to get angular momentum as a function of magnetic moment and the mass/charge ratio; i.e.,

L = (1/α)ν(M/Q)
L = βν(p+gn)/p = β(μ−0.87985)(1 + gn/p)

The relationship between (1 + gn/p) and p for the odd-odd nuclides is shown below:

The Empirical Results

The quantity (μ−0.87985)(1 + gn/p) was computed for the odd-odd nuclides and a frquency distribution tabulated. Here is that frequency distribution.

Angular momentum appears to be quantized. Peaks symmetric about zero dominates the distribution. There are secondary peaks on the rght side which are centered at values roughly two, three and four times the value for the primary peaks.


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