Subatomic Physics

528 The Shell Model
The splitting between states l + 1
2
1 and l − 2 is now known to be caused primarily
by the interaction between the nucleon spin and its orbital angular momentum.
Such a spin–orbit force is well known in atomic physics, (6) but it was not expected
that it would be so strong in nuclei. Since the orbital angular momentum increases
with A, so does the importance of the spin–orbit force. We return to the spin–orbit
force in the next section but show here that the magic numbers can be explained if
its effects are taken into account. A nucleon, moving in the central potential of the
nucleus with orbital angular momentum l, spins, and total angular momentum j,
acquires an additional energy
j = l + s, (17.6)
Vls = Clsl·s. (17.7)
We must find the effect of this potential-energy operator on a state |α; j, l, s〉. Here
α denotes all quantum numbers other than j, l, ands. (The reason that j, l, and
s can be specified simultaneously is that states of l = j ± 1
2 have opposite parities,
and parity is conserved in the hadronic force.) With the square of Eq. (17.6), the
operator l·s is written as
l·s = 1
2 (j2 − l 2 − s 2 ). (17.8)
Theactionsoftheoperatorsj 2 ,l 2 ,ands 2 on |α; j, l, s〉 are given by Eq. (5.7) so
that
l·s|α; j, l, s〉 = 1
2 �2 {j(j +1)− l(l +1)− s(s +1)}|α : j, l, s〉. (17.9)
For a nucleon, with spin s = 1
2
j = l − 1
2
, and for these Eq. (17.9) yields
l·s|α; j, l, 1
2 〉 =
� 1
, only two possibilities exist, namely j = l + 1
2 and
2 �2 l|α; j, l, 1
2
− 1
2�2 (l +1)|α; j, l, 1
2
〉 for j = l + 1
2
〉 for j = l − 1
2 .
The energy splitting ∆Els, shown in Fig. 17.8, is proportional to l + 1
2 :
(17.10)
∆Els =(l + 1
2 )�2 Cls. (17.11)
The spin–orbit splitting increases with increasing orbital angular momentum l. It
consequently becomes more important for heavier nuclei, where higher l values
appear. For a given value of l, the level with higher total angular momentum,
j = l + 1
2
with j = l − 1
2
, lies lower, and it has a degeneracy of 2j +1=2l + 2. The upper level,
,is2l-fold degenerate.
6 Tipler-Llewellyn, Chapter 7; H. A. Bethe and R. Jackiw, Intermediate Quantum Mechanics,
2nd ed. Benjamin, Reading, Mass., 1968, Chapter 8; Park, Chapter 14; G. P. Fisher, Am. J.
Phys. 39, 1528 (1971).