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.
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 14.31. There are no known exceptions to this rule.
Nuclei with an odd mass number
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. These are the nonspherical,
rotational nuclei, the ones that are easily excited; the nonsmall
squares in 14.45, the red (or yellow) big
squares in figure 14.22, the small squares in figure
14.19. Their main regions are for neutron number
For almost all remaining nuclei near the stable line, the spin can be explained in terms of the shell model using various reasonable excuses, [35, p. 224ff]. Some excuses are marked.
In particular, if the subshell with the odd neutron is above a subshell of lower spin, a particle from the lower subshell may be promoted to the higher one. This particle can then pair up at a higher spin, which is believed to be energetically favorable. Since an odd nucleon occurs now in the lower shell, the spin of the nucleus is predicted to be the one of that shell. So the nuclear spin is lowered compared to the bare shell model.
Nuclei for which such spin lowering due to promotion can explain the
observed spin are indicated with an
l in figures 14.32 and 14.33.
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 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. The idea is that the gained pairing energy should not be big enough to make major modifications to the shell model.
For some nuclei, the basic shell model is valid but the odd-particle assumption fails. The odd particle assumption implies that all nucleons except the odd one pair up in states of zero spin. But sometimes at least three nucleons combine in a state with one unit less spin than the single odd particle would have. This is only possible for subshells with at least three particles and three holes (empty spots for additional nucleons). Nuclei for which this unit spin reduction might to be the case are marked with a minus sign. It 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 already discussed in section 14.12.6.
Among the remaining failures, notable are nuclei with odd proton
numbers just above 50. The 5
In a few exceptional cases, like the highly unstable nitrogen-11 and
halo 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. In all
cases, it was demanded that the nuclear parity, if known, did not
conflict with the proposed shell model verification (or explanation)
of the spin.
For nuclei marked with a cross, no explanation for the spin using the above rules could be found. In general, you might not want to take nuclei well away from the stable line that serious. For yellow squares, NUBASE 2003 gave either no value for the spin or more than one possible value.
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
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 odd proton and odd
neutron, just like the deuterium nucleus does.
To describe the rules, forget about quantum mechanics for now. Just
think of the spin and orbital angular momenta involved as simple
vectors that can either point up or down. And take
to be the direction that the aligned proton and neutron spin vectors
Conversely, if both orbital momenta point downwards (and are nonzero),
then spin and orbital angular momenta are in opposite directions and
subtract. Then proton and neutron have net angular momenta of
However, if for one nucleon the orbital angular momentum is zero or
points in the direction of the spin and the other is nonzero and
points in the direction opposite to the spin, then one of
That then gives the Nordheim rules as, [35, p. 239]:
Of course, the real quantum rules for angular momentum are a lot more
complicated than the simplified picture above. Note in particular
from the Clebsch-Gordan coefficients in figure 12.5 that if
To check those rules is not trivial, because it requires the values of
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
N the Nordheim rules were again really verified,
with no possibility of violation within the now larger context. For
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
Preston and Bhaduri [35, p. 239] suggest that the
proton and neutron angular momenta be taken from the neighboring pairs
of nuclei of odd mass number. Figure 14.35 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 14.35, incorrect ones by an
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 [35, 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
The nuclei marked with
E in figure 14.35
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
It may be noted that the unmodified Nordheim rules imply that there
cannot be any odd-odd nuclei with