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The Magnetic Dipole Moments
of Nuclei and Their Implications
Concerning Nuclear Structure

A classical and macroscopic definition of magnetic dipole moment, such as for a bar magnet, would be the product of the pole strength and their separation distance, μ=ms.

If the pole strengths are not equal in magnitude and opposite in direction then the dipole is (m1−m2)s.

The torque exerted on a bar magnet by an external magnetic field would be proportional to this magnetic moment and also to the strength of the external magnetic field B.

On a microscopic level the magnetism of a particle or nucleus is better represented as an electromagnet which results from an electrical current traveling in a circular orbit.

For a particle with a net charge of Q that is spinning at a rate of ω (radians per second) or ν (turns per second) the effective current is i=Qν=Qω/(2π). The area of the loop which the current surrounds is πr². Thus

μ = iA = Qνπr²
or, equivalently
μ = Q(ω/(2π))πr² = (Q/2)ωr² = (Q/2)vr

where v is the tangential velocity of the charge. This is analogous to the angular momentum of a particle, but angular momentum involves the mass of the particle rather than the term (Q/2).

When the charge is not concentrated at a specific radius but has a radial distribution ρ(r) then the moment is given by

μ = ∫dQνπr² = ∫0(2πrρ(r))(ω/(2π))πr²dr = ωπ∫r³ρ(r)dr
with the total charge being given by
0(2πrρ(r))dr = Q

The radial distributions of charge for a proton and a neutron have been found to be

Although the net charge of the neutron is zero, because the negative charge is located at a greater distance from the center of rotation than the positive charge, the neutron has a dipole moment and it is negative. Note that in this case the sign of the dipole moment has to do with the sign of the charge rather than the direction of rotation.

Nuclear Dipole Moments

The magnetic dipole moment of a proton, measured in magneton units, is +2.79285. That of a neutron is −1.9130. The measured dipole moments indicate that protons and neutrons spin in the same direction. The ratio of these two numbers is −0.685, intriguingly close to −2/3.

The sum of the dipole moments of a neutron and proton is 0.86242 magnetons, intriguingly close to the magnetic dipole moment of a deuteron, which is 0.8574. The difference is only 0.6 of 1 percent of that value. The difference may be due to a rotation of the deuteron about its center of mass, as depicted diagrammatically below.

In such a system only the proton would generate a dipole moment. The small value due to the deuteron rotation could come from that rotation being at a much slower rate than the spins of the nucleons.

The dipole moments of composite nuclei are a function only of the numbers of neutrons and protons. It may be a complicated function but there are no other variables upon which it can depend. However the dependence can be decomposed into a number of functions of the nucleon numbers. For example, the magnetic dipole moment of a nucleus may depend upon whether the numbers of neutrons and protons are odd or even; or, if both are even or both are odd. The simple theory is that a nucleon pairs with another of the opposite spin so the pair has a net spin of zero and thus has no magnetic dipole moment. For a composite nucleus the dipole moment would then come from the odd (unpaired) nucleon(s). An alternate theory is that a charge uniformly distributed over a sphere would have zero dipole moment. If the shape of the charge distribution is distorted from a sphere there would be a dipole moment with positive values for a prolate spheroid and negative values for an oblate spheroid.

A number of functions of the neutron and proton numbers were selected and the dipole moments regressed upon these to see how well the dipole moment can be estimated from these variables for the database of 345 measured magnetic dipole moments. The results were:

DPM = -0.00492p + 0.00221n
+ 0.68234e(p) − 1.3277e(n)
+ 2.28505e(p)e(n) + 0.42459spin
− 0.00695p*e(p) + 0.00078n*e(n)

where e(k) is equal to 1 if k is even and zero otherwise. The coefficient of determination (R²) for this equation is 0.700. Not all of the regression coefficients are significantly different from zero at the 95 percent level of confidence.

This regression equation was used to compute estimated values of the dipole moment. The actual values were then plotted against the estimated values. One purpose of this operation is to look for nonlinearities in the relationship.

A regression was run for the dipole moment as a bent line function of the regression estimate. This regression had a coefficient of determination of 0.76, a small but significant improve over the value of 0.70 found for the original regression.

Partitioning of the Database

Of the 345 observed dipole moments nearly two hundred had values of zero. Essentially all of these were cases in which both the proton and neutron numbers are even. There were 13 cases in which the neutron numbers were not known. These were for the created elements such as Francium. There is only one case in which the proton and neutron numbers are both even but the dipole moment is not zero. That is Carbon 14. Also there is only one case in which the dipole moment is zero but the proton number is not even. That is for Tantalum 253, atomic number 73, and with 180 neutrons.

This means that the estimation of the magnetic dipole moment can proceed from a check on the odd-or-evenness of the nucleon numbers. If both are even then the predicted dipole moment is zero. The statistical analysis can be limited to the other cases. It is worthwhile to consider the cases of negative and positive dipole moments separately.

The scatter diagrams for the negative and positive magnetic dipole moments are shown below.

For the case of the negative dipole moments there seems to be a nonlinearity. For the cases in which the dipole moments is between 0 and −0.5 the values are uncorrelated with the regression estimates, which means those cases might be uncorrelated with the variables that went into the regression estimate.

For the more negative values there is a correlation but it is weak.

For the cases of positive dipole moments there is a correlation but the coefficient of determination is only about 0.41. For the negative dipole moments that coefficient is only 0.51.

However when the cases in which the dipole moments are between −6 and 0 are selected and a new regression run the coefficient of determination is 0.88.

Here is the scatter diagram of the dipole moments for those cases and their regression estimates.

If the case in which the dipole moment is −0.566 (this is the outlier located in the lower right corner of the above diagram) is left out of the analysis the coefficient of determination rises to 0.95.

Here is the scatter diagram for the actual dipole moments versus the revised regression estimates.

When the same procedure is carried out for the cases in which the dipole moment is less than -0.6 the coefficient of determination is also 0.88. Here is the scatter diagram for those cases.

However if the outlier for which the dipole moment is −4.25 is left out the coefficient of determination rises to 0.98.

The Distribution of Magnetic Dipole Moments

The cumulative distribution of the dipole moments is given below.

A smoothed version of the distribution is shown below.

Conclusions

The nuclear dipole moments are predictable on the basis of the proton and neutron numbers, but there is a definite nonlinearity in the relationship. There is a positive relationship for the cases of positive dipole moments but a near zero relationship for the cases of negative dipole moments. The theory that a nucleon pairs with another nucleon of the same type but opposite spin to produce a zero moment is born out.

The statistical predictability of nuclear dipole moments is strongly influenced by a few anomalous outliers. When these are left out of the statistical analysis the regression equations have coefficients of determination of near 100 percent.

(To be continued.)


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