10. Appendix

664 Appendix B
higher in energy than the J 3/2 quadruplet. However, if one includes spin in
the split °4 state, it splits further into °8 and °7, the former being below the
latter.
Readers may want to consult:
K. Shindo, A. Morita and H. Kamimura: Spin-orbit coupling in ionic crystals with zincblende
and wurtzite structures. Proc. Phys Soc Japan 20, 2054 (1965), and errata in
Proc. Phys Soc Japan 21, 2748 (1966)
For further discussions.
Another exercise will be to show that the spin-orbit interaction does not split the °3
orbital doublet which becomes a °8 quadruplet when spin is included.
Solution to Problem 8.11
The problem is of interest in connection with the calculation of core level
shifts discussed in pages 453 and 454.
We consider a uniformly charged spherical shell with outer radius rm and
inner radius °rm with 0 ° 1. We impose the boundary condition on the
potential V → 0 for r (the distance from the center of the shell) →∞.We
use Gauss’s theorem: the electric field E is directed towards (or away from)
the center of the shell and has, at the distance r from that center and outside
of the shell (r rm) the magnitude E q/r 2 where q is the total charge in
the shell, taken to be negative for an electron (note that cgs units are used in
this problem). For r rm the magnitude of the field is q ∗ /r 2 , where q ∗ is the
charge within a sphere of radius r∗ rm.
q ∗ q[r 3 (°rm) 3 ]/[r 3 m (°rm) 3 ].
In order to calculate V(r) we integrate the field E(r) from infinity to rm and
add to the resulting V(rm) q/rm, the integral of E(r) between rm and °rm.
The result is:
V(0) q
rm
q
[r3 m (°rm) 3
°rm
r
] rm
q q
rm [r3 m (°rm) 3 ]
(°rm) 2
2
3 (°rm)
r2
dr
(°rm) 2 r2 m
2 r2 m °3
It is of interest to check the value of V(0) for the two extreme cases, ° 0
(uniformly charged sphere) and ° 1 (infinitesimal thickness of the shell.
In the former case we find V(0) q/rm. In the latter case we find V(0)
(3/2)q/rm.