Direct and reciprocal effects
In the exact same way we can find relationships between direct and reciprocal effects. As an example
we look at the direct and the reverse piezoelectric effect. This effect describes how the electric polari-
sation of a crystal changes, when we apply stress to it.
(
∂Pk
∂σij
)
= dkijAgain following back to the derivatives of the Gibbs free energy leads us to an interesting solution:
−
(
∂^2 G
∂σij∂Ek)
=(
∂Pk
∂σij)
= −(
∂^2 G
∂Ek∂σij)
=(
∂ij
∂Ek)
= dkijThe same tensor elements used for the direct piezoelectric effect are also used in the tensor for the
reverse piezoelectric effect. An interesting use of this symmetry is to track down systematical errors
in a measurement. We could measure the piezoelecric effect by squeezing a crystal and measure the
polarisation as well as applying an electric field to it and measure the strain. If we perform both
measurements and don’t get the same solution there must be a systematic error in at least one of the
two measurements.
Groups
Crystals can have translational, rotational, inversion and reflectional symmetries. Those of them,
where one point of the crystal remains the same, can be written as a matrix. The matrices of those
symmetry operations to one crystal form the group of this crystal. The groups, which contain only
operations where one point of the crystal remains the same, are calledpoint groups. (There are 32
of them.) If we also include translational symmetries, where no point in the crystal remains at it’s
incident position, we get the 230space groups.
Fig. 57 shows the symmetries in a crystal with point group C2 and C4. For C2 we can define the
Figure 57: Illustration of the crystal symmetries in the point groups C2 and C4symmetry operations
E =
1 0 0
0 1 0
0 0 1
, C
2 =
− 1 0 0
0 −1 0
0 0 1
where the first one is the identity and the second one rotates the crystal by 180°. As we can easily
verify, those two operations form a group. C4 contains a few more operations.