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of the Nuclides of the Same Proton Number as a Function of their Neutron Number |
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. For background material from previous studies see Nuclides from Nitrogen through Silicon.
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.
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.
For an element of odd proton number the sequence of magnetic moments due to nucleonic spins starting from an even neutron number would be +2.79285, 0.87980464, +2.79285, 0.87980464, …. For an element of even proton number the sequence of magnetic moments due to nucleonic spins starting from an even neutron number would be 0, −1.9130, 0, −1.9130, ….
The measured magnetic moments contain the moments due to the rotation of the positive charges. Here is the graph of the magnetic moments of the isotopes of Sodium.
To show how much of the measured magnetic moment is accounted for by the spin of unpaired nucleons the two quantities are plotted on the same graph.
The apparent magnetic moment due to nuclear rotation is just the difference between the measured magnetic moment and that due to unpaired nucleonic spin. The values for the isotopes of sodium are shown below.
A positive value indicates rotation in the same direction as the spin of the unpaired proton. A negative value indicates rotation in the opposite direction from the spin of the unpaired proton.
The magnetic moments of the isotopes of Oxygen, Fluorine and Neon are shown in the following graph.
The values for Fluorine for 8 and 12 neutrons seem unrealistically high but they are apparently what the experimental measurement yielded.
A similar pattern occurs for Boron (P=5).
Now a different problem will be considered. Below are shown the magnetic moments for the isotopes of Carbon (P=6).
The reported magnetic moment for Carbon 13 (P=6 and N=7) is +0.7024118. Undoubtedly the correct value should be the negative of this. Below is shown what the profile for the Carbon isotopes would be with this correction made.
With this correction in the magnetic moment for Carbon 13 a regression analysis for the data for Carbon and below gives the following results.
where M is magnetic moment, sP is the number of singleton protons (0 or 1), sN the number of singleton neutrons, P the number of protons and N the number of neutrons. P^2 is P squared, a variable included to capture size effects.
The coefficient of determination for this equation is 0.968 and the standard error of the estimate is 0.411 magnetons. At the 95 percent level of confidence the regression coefficients for P, P^2 and N are not significantly different from zero.
For a fixed proton number its magnetic moments as a function of the neutron number show show a sharp fluctuation in values. This is consistent with the standard theory.
The standard theory explains 96.8 percent of the variation in the magnetic moments for carbon and below.
(To be continued.)
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