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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 due to the net intrinsic spin of any singleton proton and/or singleton neutron. 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 by its mass and multiplying by it charge; i.e.,

μ = α(L/M)Q = α(Q/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 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 precisly 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)≅p/(p+γn).

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

The Data

Here are the graphs of the data for the magnetic moments of all nuclides .

What shows up in graph of magnetic moment (μ) versus the number of neutrons are vertical clusters above certain critical numbers. Those numbers are 82, 50 and slightly for 28. These numbers are nuclear magic numbers which represent filled shells. The relationships are clearer if the data is separated into the even-even, even-odd, odd-even and odd-odd categories.

For the number of neutrons:

 

 

For the number of protons:

 

Having a singleton proton results in greatly increased variability of the magnetic moment of the nuclides.

The Statistical Explanation of
Magnetic Moments

The results of the regression anaalysis are:

Magnetic Moments of All Nuclides
Variable Coefficient t-Ratio
p≅28 -0.01983 -0.05
p≅50 0.98845 3.7
p≅82 -0.69893 -2.5
n≅28 0.58688 -1.2
n≅50 2.94773 7.9
n≅82 0.31815 1.2
n≅126 2.75978 8.5
p/(p+γn) 3.82749 2.0
sn -1.29880 -10.3
sp 1.48552 11.7
C0 -0.42004 -0.5

At the 95 percent level of confidence the variables that μ depends upon are sp (the presence of a singleton proton), sn (the presence of a singleton neutron), p/(p+γn) (the Q/M ratio), n≅126, n≅50, p≅50 and p≅82. The coefficient of determination for this equation is only 0.297.

When the constant is eliminated from the regression equation the coefficients are generally modified only slightly. The notable exception is coefficient and t-ratio for p/(p+γn) representing the Q/M ratio. In this regression the coefficient for p/(p+γn) highly significantly different from zero, confirming the dependence of μ on the Q/M ratio.

Magnetic Moments of All Nuclides
Variable Coefficient t-Ratio
p≅28 0.00720 0.02
p≅50 0.98143 3.6
p≅82 -0.71665 -2.6
n≅28 0.60910 1.3
n≅50 2.96018 7.9
n≅82 0.30370 1.2
n≅126 2.73324 8.5
p/(p+γn) 2.86517 10.5
sn -1.30096 -10.3
sp 1.48134 11.7
C0 0 N/A

The real surprise from the regression analysis is that if the proton and neutron numbers p and n are included the results indicate that they have a statistically significant influence on magnetic moments above and beyond their influence on the charge/mass ratio Q/M and in relation to the critical nuclear magic numbers. Here are those results.

Magnetic Moments of All Nuclides
Variable Coefficient t-Ratio
n -0.05755 -4.1
p 0.09546 4.8
p≅28 0.094086 0.2
p≅50 0.95536 3.6
p≅82 -0.92506 -3.3
n≅28 0.832041.8
n≅50 2.86932 7.7
n≅82 0.19261 0.7
n≅126 2.76719 8.3
p/(p+γn) 0.37416 3.0
sn -1.29358 -0.1
sp 1.51788 0.1
C0 0.20352 1.4

The coefficient of determination for this regression is 0.320.

(To be continued.)


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