14.15 Spin Data

The net inter­nal angular momentum of a nucleus is called the “nuclear spin.” It is an important quantity for appli­cations such as NMR and MRI, and it is also important for what nuclear decays and reactions occur and at what rate. One previous example was the categorical refusal of bismuth-209 to decay at the rate it was supposed to in section 14.11.3.

This section provides an overview of the ground-state spins of nuclei. According to the rules of quantum mechanics, the spin must be integer if the total number of nucleons is even, and half-integer if it is odd. The shell model can do a pretty good job of predicting actual values. Histori­cally, this was one of the major reasons for physicists to accept the validity of the shell model.

14.15.1 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 spin is zero. This prediction is fully vindicated by the experi­mental data, figure 14.29. There are no known exceptions to this rule.

Figure 14.29: Spin of even-even nuclei. [pdf]

14.15.2 Odd mass number nuclei

Nuclei with an odd mass number $A$ have either an odd number of protons or an odd number of neutrons. For such nuclei, the odd-particle shell model predicts that the nuclear spin is the net angular momentum (orbital plus spin) of the last odd nucleon. To find it, the subshell that the last particle is in must be identified. That can be done by assuming that the subshells fill in the order given in section 14.12.2. This ordering is indicated by the colored lines in figures 14.30 and 14.31. Nuclei for which the resulting nuclear spin prediction is correct are indicated with a check mark.

Figure 14.30: Spin of even-odd nuclei. [pdf]

Figure 14.31: Spin of odd-even nuclei. [pdf]

The prediction is correct right off the bat for a considerable number of nuclei. That is very much non­trivial. Still, there is an even larger number for which the prediction is not correct. One major reason is that many heavy nuclei are not spherical in shape. The shell model was derived assuming a spherical nuclear shape and simply does not apply for such nuclei. They are the rotational nuclei; roughly the red squares in figure 14.20, the smallest squares in figure 14.17. Their main regions are for atomic number above $Z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 82 and in the inter­ior of the $Z$ $\raisebox{.3pt}{$<$}$ 82, $N$ $\raisebox{.3pt}{$>$}$ 82 wedge.

For almost all remaining nuclei near the stable line, the spin can be explained in terms of the shell model using various reasonable excuses, [34, p. 224ff]. However, it seems more inter­esting to see how may spins can be correctly predicted, rather than justified after the fact. It may be noted that even if the wrong value is predicted, the true value is usually either another value in the same major shell, or one less than such a value.

One modifi­cation of the odd-particle shell model has been allowed for in the figures. It is that if the subshell being filled is above one of lower spin, a particle from the lower subshell may be promoted to the higher one; it can then pair up at higher spin. Since the odd nucleon is now in the lower shell, the spin of the nucleus is predicted to be the one of that shell. The spin is lowered. In a sense of course, this gives the theory a second shot at the right answer. However, promotion was only allowed for subshells immediately above one with lower spin in the same major shell, so the nucleon could only be promoted a single subshell. Also, no promotion was allowed if the nucleon number was below 32. Nuclei for which spin lowering due to promotion can explain the observed spin are indicated with an “L” or “l” in figures 14.30 and 14.31. For the nuclei marked with “L,” the odd nucleon cannot be in the normal subshell because the nucleus has the wrong parity for that. Therefore, for these nuclei there is a solid additional reason besides the spin to assume that promotion has occurred.

Promotion greatly increases the number of nuclei whose spin can be correctly predicted. Among the remaining failures, notable are nuclei with odd proton numbers just above 50. The 5g$_{7/2}$ and 5d$_{5/2}$ subshells are very close together and it depends on the details which one gets filled first. For some other nuclei, the basic shell model is valid but the odd-particle assumption fails. In particular, for subshells with at least three particles and three holes (empty spots for additional nucleons) the spin is often one unit less than that of the odd nucleon. This is evident, for example, for odd nucleon numbers 23 and 25. Fluorine-19 and its mirror twin neon-19 are rare outright failures of the shell model, as discussed in section 14.12.6.

In a few excep­tional cases, like the unstable nitrogen-11 and beryllium-11 mirror nuclei, the theoretical model predicted the right spin, but it was not counted as a hit because the nuclear parity was incon­sistent with the predicted subshell.

14.15.3 Odd-odd nuclei

If both the number of protons and the number of neutrons is odd, the nuclear spin becomes much more difficult to predict. According to the odd-particle shell model, the net nuclear spin $j$ comes from combining the net angular momenta $j^{\rm p}$ of the odd proton and $j^{\rm n}$ of the odd neutron. In particular, quantum mechanics allows any integer value for $j$ in the range

$$\vert j^{\rm p}-j^{\rm n}\vert \mathrel{\raisebox{-.7pt}{$... ...}j \mathrel{\raisebox{-.7pt}{$\leqslant$}}j^{\rm p}+j^{\rm n}$$ (14.25)

That gives a total of $2\min(j^{\rm p},j^{\rm n})+1$ different possi­bilities, two at the least. That is not very satis­factory of course. You would like to get a specific prediction for the spin, not a range.

The so-called “Nordheim rules” attempt to do so. The underlying idea is that nuclei like to align the spins of the two odd nucleons, just like the deuterium nucleus does. To describe the rules, the net angular momentum $j$ of an odd nucleon and its spin $s$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac12$ will be called “parallel” if $j$ $\vphantom0\raisebox{1.5pt}{$=$}$ $l+s$, with $l$ the orbital angular momentum. The idea is that then the spin acts to increase $j$, so it must be in the same direction as $j$. (Of course, this is a simplistic one-di­mensional picture; angular momentum and spin are really both three-di­mensional vectors with uncertainty.) Similarly, the total angular momentum and spin are called “opposite” if $j$ $\vphantom0\raisebox{1.5pt}{$=$}$ $l-s$. Following this picture, now make the assumption that the spins of proton and neutron are parallel. Then:

1. The angular momenta $j^{\rm p}$ and $j^{\rm n}$ of the odd proton and neutron will be opposite to each other if one is parallel to its spin, and the other is opposite to its spin. In that case the prediction is that the total spin of the nucleus is $j$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vert j^{\rm p}-j^{\rm n}\vert$.
2. Otherwise the angular momenta will be parallel to each other and the total spin of the nucleus will be $j$ $\vphantom0\raisebox{1.5pt}{$=$}$ $j^{\rm p}+j^{\rm n}$. New and improved version: if that fails, assume that the angular momenta are opposite anyway and the spin is $j$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vert j^{\rm p}-j^{\rm n}\vert$.

Figure 14.32: Spin of odd-odd nuclei. [pdf]

To check those rules is not trivial, because it requires the values of $l$ for the odd proton and neutron. Who will say in what shells the odd proton and odd neutron really are? The simplest solution is to simply take the shells to be the ones that the shell model predicts, assuming the subshell ordering from section 14.12.2. The nuclei that satisfy the Nordheim rules under that assumption are indicated with a check mark in figure 14.32. A blue check mark means that the new and improved version has been used. It is seen that the rules get a number of nuclei right.

An “L” or “l” indicates that it has been assumed that the spin of at least one odd nucleon has been lowered due to promotion. The rules are the same as in the previous subsection. In case of “L,” the Nordheim rules were really verified. More specifi­cally, for these nuclei there was no possi­bility consistent with nuclear spin and parity to violate the rules. For nuclei with an “l” there was, and the case that satisfied the Nordheim rules was cherry-picked among other otherwise valid possi­bilities that did not.

A further weakening of standards applies to nuclei marked with “N” or “n.” For those, one or two subshells of the odd nucleons were taken based on the spins of the immediately neighboring nuclei of odd mass number. For nuclei marked with “N” the Nordheim rules were again really verified, with no possi­bility of violation within the now larger context. For nuclei marked “n,” other possi­bilities violated the rules; obviously, for these nuclei the standards have become miserably low. Note how many “correct” predictions there are in the regions of non­spherical nuclei in which the shell model is quite meaningless.

Figure 14.33: Selected odd-odd spins predicted using the neighbors. [pdf]

Preston and Bhaduri [34, p. 239] suggest that the proton and neutron angular momenta be taken from the neighboring pairs of nuclei of odd mass number. Figure 14.33 shows results according to that approach. To minimize failures due to other causes than the Nordheim rules, it was demanded that both spin and parity of the odd-odd nucleus were solidly established. For the two pairs of odd mass nuclei, it was demanded that both spin and parity were known, and that the two members of each pair agreed on the values. It was also demanded that the orbital momenta of the pairs could be confidently predicted from the spins and parities. Correct predictions for these super­clean cases are indicated by check marks in figure 14.33, incorrect ones by an “E” or cross. Light check marks indicate cases in which the spin of a pair of odd mass nuclei is not the spin of the odd nucleon.

Preston and Bhaduri [34, p. 239] write: “When confronted with experi­mental data, Nordheim’s rules are found to work quite well, most of the exceptions being for light nuclei.” So be it. The results are definitely better than chance. Below $Z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 50, the rules get 43 right out of 71. It may be noted that if you simply take the shells directly from theory with no promotion, like in figure 14.34, you get only 41 right, so using the spins of the neighbors seems to help. The “Nuclear Data Sheets” policies assume that the (unimproved) Nordheim rules may be helpful if there is supporting evidence.

Figure 14.34: Selected odd-odd spins predicted from theory. [pdf]

The nuclei marked with “E” in figure 14.33 are particularly inter­esting. For these nuclei spin or parity show that it is impossible for the odd proton and neutron to be in the same shells as their neighbors. In four cases, the discrepancy is in parity, which is particularly clear. It shows that for an odd proton, having an odd neutron is not necessarily inter­mediate between having no odd neutron and having one additional neutron besides the odd one. Or vice-versa for an odd neutron. Proton and neutron shells inter­act non­trivially.

It may be noted that the unmodified Nordheim rules imply that there cannot be any odd-odd nuclei with 0$\POW9,{+}$ or 1$\POW9,{-}$ ground states. However, some do exist, as is seen in figure 14.32 from the nuclei with spin zero (grey) and blue check marks.