10. Appendix

A Short Cut to the Calculation of the Raman tensor for the °6 mode
Solution to Problem 7.5 653
For readers who have worked out Problem 3.7(b) to derive the symmetry
properties of the stiffness tensor for the wurtzite crystal, the above derivation
should look very familiar. The reason is that the stiffness component Cijkl
can be defined as:
Cijkl Xij/ekl where Xij and ekl are, respectively, the second rank, symmetric
stress and strain tensors.
The two functions f1 xy and f2 x2 y2 happen to be related to the
components of the second rank tensor:
⎛
⎞
xx xy xz
⎝ yx yy yz ⎠
zx zy zz
We should note that the above second rank tensor is not symmetric. However,
as we have seen in the case of the strain tensor, we can always symmetrize
(xy yx)/2. As long as we consider
this tensor by defining a new function f ′ 1
symmetry operations within the C6 point group f1 and f ′ 1
will have the same
symmetry properties. This observation allows us to map f ′ 1 into exy or e6 and
f2 into exx eyy or e1 e2. Now we apply this mapping to the Raman tensor
components. For example, for the mode f1 the Raman tensor should be given
by:
⎛ ¯11 ¯12 ¯13
⎞
⎜ f1
⎜ ¯21
Rij(f1) ⎜ f1
⎝
f1
¯22
f1
f1 ⎟
¯23 ⎟
f1 ⎟
⎠
¯31
¯32
¯33
f1 f1 f1
Next we can apply the mapping: ¯ij/f1 ⇔ ¯ij/f ′ 1 ⇔ Xij/e6 Ci6 where i
1...6. From the results of Problem 3.7 we see that the only non-zero element
of the form Ci6 is C66. Thus, based on the results of Problem 3.7 we conclude
that the Raman tensor for the f1 mode has the form:
⎛ ⎞
0 d 0
Rij(°6(xy)) ⎝ d 0 0⎠
0 0 0
To obtain the Raman tensor for the f2 mode we map ¯ij/f2 into (Xij/exx
Xij/eyy) (Xij/e1 Xij/e2). Again using the results of Problem 3.7 we
conclude that all the off-diagonal elements of the Raman tensor vanish. The
only non-vanishing elements are of the form: (X11/e1 X11/e2) C11 C12
and (X22/e1 X22/e2) C21 C22 (C11 C12). Thus, the Raman tensor
for the f2 mode is of the form:
Rij(°6(x 2 y 2 ⎛ ⎞
e 0 0
)) ⎝ 0 e 0 ⎠
0 0 0
In Problem 3.7 the stiffness tensor components C66 (C11 C12)/2. Similarly,
one can show that d e.