10. Appendix

Solution to Problem 3.5 603
Similarly, we expect the stress tensor of the zincblende crystal to be reducible
to three irreducible representations:
Xij(°1) X11 X22 X33; Xij(°3) X11 X22, X33 (X22 X11)/2 and
Xij(°4) X12, X23, X31.
It is easily seen that the °1 component of the stress tensor is a hydrostatic
stress. The two remaining °3 and °4 representations can be shown to correspond
to shear stresses applied along the [100] and [111] axes, respectively.
If we apply stresses represented by these irreducible representations to a
zincblende crystal, the resulting strain tensors will also be irreducible and belong
to the same irreducible representations as the stress tensors. In other
words, the stress and strain tensors of the same irreducible representation
will be related by a scalar. As an example, we apply a hydrostatic stress to
a zincblende crystal. From Problem 3.2 this stress has the form:
⎛
P
X ⎝ 0
0
0
P
0
⎛ ⎞
P
⎞ ⎜ P ⎟
0 ⎜ ⎟
0 ⎠ ⎜ P ⎟
⎜ ⎟
⎜ 0 ⎟
P ⎝
0
⎠
0
where P is the pressure. The resultant strain tensor is given by:
⎛ ⎞ ⎛
P S11 S12 S12
⎜ P ⎟ ⎜ S12
⎜ ⎟ ⎜
⎜ P ⎟ ⎜ S12
e ⎜ ⎟ ⎜
⎜ 0 ⎟ ⎜
⎝
0
⎠ ⎝
S11
S12
S12
S11
S44
0
⎛ ⎞
1
⎜ 1 ⎟
⎜ ⎟
⎜ 1 ⎟
(P)(S11 2S12) ⎜ ⎟
⎜ 0 ⎟
⎝
0
⎠
0
⎛
1
(P)(S11 2S12) ⎝ 0
0
1
⎞
0
0⎠
0 0 1
which belongs to the °1 irreducible representation, as we expect. Thus, the
strain tensor is identical in form to the stress tensor, except for the difference
of a factor of (S11 2S12). If we had started with a strain tensor of the above
form we would have ended with the stress tensor:
S44
S44
⎞
⎟
⎠