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Nuclear Dipole Moments and
the Spatial Structure of Nuclei

Much is known about nuclei but little about the spatial arrangements of their neutrons and protons. Dipole and quadrupole moments are unique in providing information that depends, in part, upon the spatial arrangements of the nucleons. Generally so far the information from those moments has been limited to whether the shape of a nucleus is spherical, prolate ellipsoid (watermelon shaped) or oblate ellipsoid (pumpkin shaped).

 

The spin axis is vertical in all three illustrations.

Measurements are available for the electric and magnet dipole and quadrupole moments. Because nucleons form spin pairs the moments for nuclides in which the numbers of neutrons and protons are both even are zero.

The Qualitative Explanation of the
Magnetic Dipole Moments of Nuclides

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 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.

Theory suggests that nuclides having an even of neutrons and an even number of protons should have a magnetic dipole moment of zero. This is because the nucleons pair with another nucleon of the same type and opposite spin. The opposite spins in a nucleonic pair cancel out their dipole moments. This is confirmed by empirical measurement.

Consider a triteron. It has one proton and two neutrons. The neutrons pair and the only contributions to its dipole moments is from its proton. The spin of the proton would contribute +2.79285 and the revolution of the proton around the center of mass of the triteron would contribute some more. The measured dipole moment of a triteron is +2.97896. That means the revolution of the proton about the center of mass of the triteron contributes 0.28611.

The Helium3 nuclide has two protons and one neutron. The protons pair up leaving only the neutron's spin as a contribution to the nuclide's dipole moment. The measured dipole moment for He3 is −2.12750. This means that there is −0.21446 that must be accounted for in the rotation of the nuclide He3.

The next most complicated nuclide is Lithium7, composed of three protons and four neutrons. Its neutrons all pair up and thus contribute nothing to the dipole moment. That leaves the unpaired proton's dipole moment of +2.79285. to account for the major part of the 3.25643 dipole moment of Li7. That leaves 0.46358 to be attributed to the rotation of the nuclide. For the triteron there is 0.28611 due to rotation and −0.21446 for the rotation of the He3 nuclide.

In a nuclide with more nucleons any unpaired proton would be farther from the center of rotation and thus have a greater dipole moment from that rotation.

The General Model for the
Magnetic Dipole Moment

Let D(n, p) denote the magnetic dipole moment for a nuclide with n neutrons and p protons. Let o(k) be a function which is equal to 1 if k is odd and 0 if it is even. The equation for the magnet dipole moment is then of the form

D(n, p) = o(n)*D(1, 0) + o(p)*D(0, 1) + R(n, p)

where R is the magnetic dipole moment due to the rotation of the nuclide.

The form of R(n, p) for an unpaired nucleon is bq(πr²)ω, where b is a constant, q is the charge of the nucleon, r is the orbit radius for the unpaired nucleon and ω is the rate of rotation of the nucleus. The effective current is qω/(2π) and the area circumscribed by the current is πr².

The simplistic theory suggests that there would be a contribution to the dipole moment due to R(n, p) only from an unpaired proton. In any case there should be a clustering of the dipole moments for the four cases: 1. Only an unpaired proton, 2. Only an unpaired neutron, 3. An unpaired proton and an unpaired neutron, 4. No unpaired nucleon.

A depiction of what might be expected for these four cases is shown below.

The Cases of Nuclides with
only an Unpaired Proton

In the graph below the magnetic dipole moments for the 18 nuclides in the database which have only an unpaired proton are shown ranked from lowest to highest. This graph shows the same sort of information as a histogram without arbitrarily specifying the numerical ranges.

The graph shows that there is a relatively tight clustering over the range of about 1.75 to 2.00. There is a looser clustering from about 2.8 to 4.1. There are five cases spread over the range from about 0 to 1.3.

The data for these cases can be converted into a histogram, as shown below.

The scatter diagram for these cases shows a weak correlation of the magnetic dipole moment with the number of protons in the nuclide. The number of protons would determine the distance of the unpaired proton from the center of mass of the nuclide.

The Cases of Nuclides with
only an Unpaired Neutron

The ranking of the moments for these cases shows a relatively tight clustering in the range from about −1.2 to −0.5 and in the range from about 0.5 to 1.0.

The histogram for these case is:

The scatter diagram below indicates that there are two regimes. One for the moments in the range −1.2 to −0.5 where there is definite correlation and another for the range 0.5 to 1.0 where there is no apparent correleation.

For the data in the range −1.2 to −0.5 the correlation is 0.45. The regression equation parameters are:

NDM = −1.06162 + 0.0021488n

The t-ratio for the slope is [2.95], so the slope is significantly different from zero at the 95 percent level of confidence.

In contrast the regression equation for the data in the range of 0.35 to 1.00 is:

NDM = 0.81155 − 0.000967n

The correlation coefficient is only 0.3.and the t-ratio for the slope is only [−1.5] and hence the slope is not significantly different from zero at the 95 percent level of confidence.

The Cases of Nuclides with
an Unpaired Neutron and
an Unpaired Proton

The ranking of the moments for these cases do not show any tight clustering but there seems to be three regimes. In one the values range from negative levels to about +0.2. In another the values range from about 3.5 and up. In the middle there is a curved regime running from 0.2 to 3.5.

The histogram for these data is

As seen above there are two clusters.

The scatter diagrams show no correlations.

The Cases of Nuclides with
No Unpaired Nucleons

For these cases the magnetic dipole moments are all zero.

Conclusion

The data for the magnetic dipole moments do show a clustering but the situation is more complicated than the simple theory suggests. Except for the case of no unpaired nucleons there is a spread of the values over a range not accounted for. In two of the cases there are two clusters instead of one. This indicates that there is another process involved besides the one indicated by the simple theory.



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