Subsections


14.16 Parity Data

The parity of a nucleus is even, or one, if its wave function stays the same if the positive direction of all three Cartesian axes is inverted. That replaces every ${\skew0\vec r}$ in the wave function by $\vphantom0\raisebox{1.5pt}{$-$}$${\skew0\vec r}$. The parity is odd, or minus one, if the wave function gets multiplied by $\vphantom0\raisebox{1.5pt}{$-$}$1 under axes inversion. Nuclei have definite parity, (as long as the weak force is not an active factor), so one of the two must be the case. It is an important quantity for what nuclear decays and reactions occur and at what rate.

This section provides an overview of the ground-state spins of nuclei. It will be seen that the shell model does a pretty good job of predicting them.


14.16.1 Even-even nuclei

For nuclei with both an even number of protons and an even number of neutrons, the odd-particle shell model predicts that the parity is even. This prediction is fully vindicated by the experimental data, figure 14.35. There are no known exceptions to this rule.

Figure 14.35: Parity of even-even nuclei. [pdf]
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14.16.2 Odd mass number nuclei

For nuclei with an odd mass number $A$, there is an odd proton or neutron. The odd-particle shell model says that the parity is that of the odd nucleon. To find it, the subshell that the last particle is in must be identified, section 14.12.2. This can be done with a fair amount of confidence based on the spin of the nuclei. Nuclei for which the parity is correctly predicted in this way are shown in green in figures 14.36 and 14.37. Failures are in red. Small grey signs are shell model values if the nucleons fill the shells in the normal order.

Figure 14.36: Parity of even-odd nuclei. [pdf]
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Figure 14.37: Parity of odd-even nuclei. [pdf]
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The failures above $Z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 50 and inside the $Z$ $\raisebox{.3pt}{$<$}$ 82, $N$ $\raisebox{.3pt}{$>$}$ 82 wedge are expected. The shell model does not apply in these regions, because the nuclei are known to be nonspherical there. Besides that, there are very few failures. Those near the $N$ $\vphantom0\raisebox{1.5pt}{$=$}$ 40 and $N$ $\vphantom0\raisebox{1.5pt}{$=$}$ 60 lines away from the stable line are presumably also due to nonspherical nuclei. The highly unstable nitrogen-11 and beryllium-11 mirror nuclei were discussed in section 14.12.6.


14.16.3 Odd-odd nuclei

For odd-odd nuclei, the odd-particle shell model predicts that the parity is the product of those of the surrounding even-odd and odd-even nuclei. The results are shown in figure 14.38. Hits are green, failures red, and unable-to-tell black. Small grey signs are shell model values.

Figure 14.38: Parity of odd-odd nuclei. [pdf]
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Failures for spherical nuclei indicate that sometimes the odd proton or neutron is in a different shell than in the corresponding odd-mass neighbors. A similar conclusion can be reached based on the spin data.

Note that the predictions also do a fairly good job in the regions in which the nuclei are not spherical. The reason is that the predictions make no assumptions about what sort of state, spherical or nonspherical, the odd nucleons are in. It merely assumes that they are in the same state as their neighbors.


14.16.4 Parity Summary

Figure 14.39 shows a summary of the parity of all nuclei together. To identify the type of nucleus more easily, the even-even nuclei have been shown as green check marks. The odd-odd nuclei are found on the same vertical lines as the check marks. The even-odd nuclei are on the same horizontal lines as the check marks, and the odd-even ones on the same diagonal lines.

Figure 14.39: Parity versus the shell model. [pdf]
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Parities that the shell model predicts correctly are in green, and those that it predicts incorrectly are in red. The parities were taken straight from section 14.12.2 with no tricks. Note that the shell model does get a large number of parities right straight off the bat. And much of the errors can be explained by promotion or nonspherical nuclei.

For parities in light green and light red, NUBASE 2003 expressed some reservation about the correct value. For parities shown as yellow crosses, no (unique) value was given.