10. Appendix

608 Appendix B
and third terms containing the deformation potentials b and d in (3.23) in the
first and second editions, depending on definition].
Since the first term in (3.23) contains the trace of the strain tensor,
the traceless shear strain tensor ˜eshear does not contribute to this term and
we obtain from the hydrostatic strain tensor ˜ehydrostatic a term of the form:
a(S11 2S12)X. Similarly, the hydrostatic strain tensor will not contribute to
the shear part of the Hamiltonian HPB. On the other hand, the contribution
of the shear strain tensor to HPB is simply:
(b)(S11 S12)(X/3)[(J 2 x J2 /3)2 (J 2 y J2 /3) (J 2 z J2 /3)]
(b/3)(S11 S12)X[2J 2 x J2 y J2 z ]
b(S11 S12)X[J 2 x (J2 /3)]
Combining the above results we find that the form of the strain Hamiltonian
for uniaxial stress along the [100] direction is:
HPB(X) a(S11 2S12)X b(S11 S12)X[J 2 x (J2 /3)] .
(c) To calculate the splitting in the J 3/2 states due to the uniaxial stress
X along the [100] direction we note that the hydrostatic component of the
stress Hamiltonian: a(S11 2S12)X will shift all the J 3/2 states by the same
amount so we can neglect this term in calculating the splitting. From the form
of the shear component of the stress Hamiltonian: b(S11 S12)X[J 2 x (J2 /3)]
it is clear that the two mJ ±3/2 states will remain degenerate while the
two mJ ±1/2 states also will not be split by the stress. Thus, we need only
calculate the eigenvalues of b(S11 S12)X[J 2 x (J2 /3)] for the mJ 3/2 and 1/2
states. Noting that:
while
〈3/2, 3/2|J 2 x|3/2, 3/2〉 (3/2) 2 9/4 and 〈3/2, 1/2|J 2 x|3/2, 1/2〉 (1/2) 2 1/4
〈3/2, 3/2|J 2 |3/2, 3/2〉 (3/2)(5/2) 15/4 〈3/2, 1/2|J 2 |3/2, 1/2〉
we obtain the following results:
〈3/2, 3/2|b(S11 S12)X[J 2 x (J2 /3)]|3/2, 3/2〉
b(S11 S12)X〈3/2, 3/2|J 2 x (J2 /3)|3/2, 3/2〉
b(S11 S12)X[(9/4) (5/4)]
b(S11 S12)X
and similarly
b(S11 S12)X〈3/2, 1/2|J 2 x (J2 /3)|3/2, 1/2〉
b(S11 S12)X[(1/4) (5/4)] b(S11 S12)X .
Thus the splitting between the mJ ±3/2 states and mJ ±1/2 states induced
by the [100] uniaxial stress is: 2b(S11 S12)X.
The results for the [111] uniaxial stress can be obtained similarly.