number of indices, crystallographers very often combine pairs of two indices using the following scheme:
11 →1 12→6 13→ 5
22 →2 23→ 4
33 → 3So if we got a third rank tensor elementg 36 it would beg 312 , if it’s a fourth rank tensor it would be
g 3312.
Another interesting consequence of those symmetry operations is, that in all crystals, which have
inversion symmetry, all rank three tensors equal zero, that means, those crystals don’t show effects
described by a rank three tensor. Examples are piezoelectricity, piezomagnetism and the piezo-caloric
effect.
8.4 Example - Birefringence^5
A fairly interesting example is an effect called birefringence. If we look on a piece of newspaper
through a calcite crystal, we will see all letters doubled. This effect occurs from the fact that different
polarisations of light have different velocities in calcite. To explain this, the index of refraction in
calcite must be a matrix. If we follow that clue, we find, that the electric susceptibility has to be a
matrix too. To look up the form of this second rank tensor, we first need to know the point group
of calcite. After some research we can get the point group in two ways. The first possibility is to
look up the crystal structure of calcite, the second one to explicitly look for the point group of it. In
both cases we can look it up in a table and find out the structure of second rank tensors in trigonal
structures (like calcite has).:
gij =
g 11 0 0
0 g 11 0
0 0 g 33
Which is exactly what we expected. There are two independent constants, so the index of refraction
has to be a matrix.
(^5) Birefringence = Doppelbrechung