Subatomic Physics

530 The Shell Model
Here L = 1
2 (x1 − x2) × (p1 − p2) is the relative orbital angular momentum of the
two nucleons and S = s1 + s2 = 1
2 (σ1 + σ2) is the sum of the spins.
Such a term in the nucleon–nucleon force will produce
aterm
Vls = Clsl · s
Figure 17.10: Nucleon with
orbital angular momentum l
and spin s moving in the nuclear
potential.
in the nuclear potential, where l is the orbital angular
momentum of the nucleon that moves in the nuclear
potential and s is its spin. To see the connection, we
consider an orbit as shown in Fig. 17.10. In the interior
of the nucleus, where the nuclear density is constant,
there are an equal number of nucleons on either side of
the orbit within reach of the nuclear force. The spin–
orbit interaction consequently averages out.
Near the surface, however, nucleons are only on the interior side of the orbit, the
relative orbital angular momentum L in Eq. (17.12) always points in the same
direction, and the two-body spin–orbit interaction gives rise to a term of the form
of Eq. (17.7). To make this argument more precise, the spin–orbit interaction energy
[Eq. (17.12)] between two nucleons, 1 and 2, is written as
V (1, 2) = 1
2VLS(r12)(x1 − x2) × (p1 − p2) · (s1 + s2). (17.13)
If particle 1 is the nucleon under consideration, an estimate of the nuclear spin–orbit
potential can be obtained by averaging V (1, 2) over nucleon 2,
�
Vls(1) = Av d 3 x2ρ(x2)V (1, 2), (17.14)
where Av indicates that we must average over the spin and the momentum of
nucleon 2, and where ρ(x2) is the probability density of nucleon 2. After inserting
V (1, 2) from Eq. (17.13), Vls(1) becomes
�
Vls(1) = 1
2
d 3 x2ρ(x2)VLS(r12)(x1 − x2) × p 1 · s1; (17.15)
the average of all other terms is zero. The nuclear density at position x2 can be
expanded in a Taylor series about x1 because of the short range of the spin–orbit
force:
ρ(x2) =ρ(x1)+(x2 − x1) · ∇ρ(x1)+··· . (17.16)
After inserting the expansion into Vls(1), the integral containing the factor ρ(x1)
vanishes. The remaining integral can be computed; under the assumption that the
range of the nucleon spin–orbit interaction is small compared to the nuclear surface
thickness, which is the only region wherein ∇ρ is appreciable, it gives
Vls(1) = C 1 ∂ρ(r1)
l1 · s1, (17.17)
r1 ∂r1