This is a continuation of an extensive study the nuclear magnetic moments of nuclides as a means of measuring the rates of rotation of nuclei. 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 magnetic moment μ is given by iA, where A is the area of the loop which the current surrounds. That area is πr², where r is the radius of the orbit. 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. The angular momentum involves the mass of the particle rather than the term (Q/2).

Nuclear Dipole Moments

The magnetic moment of a nucleus is the sum of that which is due to the spin of its nucleons (protons and neutrons) and that due to the rotation of its charges. However when a nucleon is paired with a nucleon of the same type the spins are oppositely aligned and therefore cancel each other out. Thus the net magnetic moment of a nucleus is due to the presence of unpaired nucleons.

Let sP and sN denote the numbers of singleton protons and neutrons, respectively. Let P2s and N2s be the numbers of proton and neutron pairs, respectively, and let M be the magnetic moment of a nuclide measured in magneton units and U the moment due to the rotation of the nucleus. The most general formula for the magnetic moment is then

M = c₀ + a_P(sP) + a_N(sN) + b_P(P2s) + b_N(N2s) + U

where b_P captures the effect of the rotating positive charges of the protons in pairs. The coefficient a_P also captures the effect of the rotating nucleus as well as the spin of the proton.

The standard theory of magnetic moments is that b_N=0 and a_N is equal to the magnetic moment of a neutron and a_P is that of a proton plus the effect of the rotation of the charge of one proton. Also the constant c₀ is assumed in the simple theory to be zero. The magnetic moment U is assumed to be random with an expected value of zero.

The magnetic dipole moment of a proton, measured in magneton units, is +2.79285. That of a neutron is −1.9130. The ratio of these two numbers is −0.685, intriguingly close to −2/3.

The nuclear magnetic moments given below are from the compilation prepared by N.F. Stone. In Stone's compilation the sign of the moment is reported only if its value has been established by experiment.

The magnetic moments due to nucleonic spin are computed by the simple scheme:

Spin pairs of like nucleons account for zero moment
because nucleons pair with other nucleons of opposite spins.

The moment due to a singular proton is 2.79285 magnetrons
and −1.9130 magnetrons for a singular neutron.

The graph of measured moments versus the values predicted from the nucleonic spins shows a reasonably close relationship.

The regression equation is

MMM = 1.00818 +0.58923MMNS
      [5.3]      [5.8]

where MMM is the measured magnetic moment and MMNS is the magnetic moment due nucleonic spins.

The figures in the square brackets [ ] are the t-ratios for the coefficient above. In order for a variable to be statistically significant in explaining the variation in the dependent variable at the 95 percent level of confidence the magnitude of the t-ratio must be 2 or greater.

The coefficient of determination (R²) for this equation is 0.43. The regression coefficient for MMNS is statistically significantly different from 1.0 at the 95 percent level of confidence.

When all of the variables are included in the regression the result is

M = −1.6663 + 2.11873(sP) −0.8508(sN) + 0.43916(P2s) + 0.02757(N2s)
       [-1.9]     [5.5]     [-2.4]     [2.0]     [0.2]

The coefficient of determination (R²) for this equation is 0.540 and its standard error of the estimate is 1.125 magnetons.

The coefficients for sP, sN and P2s are statistically significantly different from zero at the 95 percent level of confidence. The coefficient for N2s is not significantly different from zero at the 95 percent level of confidence. This vindicates the standard theory.

Furthermore, the coefficient for sP is positive and on the order of 2 whereas the coefficient for sN is negative and on the order of −1. These results are compatible with the standard theory.

The value of 0.43916 magnetons per proton pair implies 0.21953 magnetons per proton. This means the effect strictly due to the spin of a singleton proton is 1.8992 magnetons. The effects of the spins of the singleton proton and neutron are significantly below the measured values for nucleonic spins but, as indicated above, they are the same order of magnitude.

The relationship between the measured magnetic moments and the estimates computed from the regression is shown below.

Size Effects

The magnetic moment due to the rotation of the positive charges of the protons is proportional to r²Q where r is the average radius of the rotating Q charges. The circumference of a ring containing P2s proton pairs would be proportional to 2πr and thus r would be proportional to P2s and r² to (P2s)².

When (P2s)² is included in the regression equation the results are:

M = −7.43118 + 2.17577(sP) −0.90499(sN) + 3.05474(P2s) −0.25983(P2s)² −0.06341(N2s)
       [-2.7]     [5.9]     [-2.7]     [2.5]     [−2.2]     [−0.5]

The coefficient of determination (R²) for this equation is 0.589 and its standard error of the estimate is 1.076 magnetons. So the inclusion of (P2s)² gives a small but definite improvement in the statistical performance of the regression analysis. Again the coefficient for N2s is not significantly different from zero at the 95 percent level of confidence.

Conclusion

The standard theory of the magnetic moment of nuclei is vindicated. That is to say the magnetic moment of a nucleus is equal to the net moment due to the spins of any unpaired nucleons plus that due to the rotation of the positive charges of its protons.

(To be continued.)