E =
1 0 0
0 1 0
0 0 1
, C 4 =
0 −1 0
1 0 0
0 0 1
, C 2 =
− 1 0 0
0 −1 0
0 0 1
, C 43 =
0 1 0
−1 0 0
0 0 1
,C4 contains the operations for rotations by 90°, 180°, 270°and the identity. The next step is to find
out how these symmetries affect the properties of the tensors.
As an example we want to find out how a 180°rotational symmetry affects the polarisation tensor.
Since both the electric field and the polarisation can be described as vectors, this is a rank two tensor,
a matrix:
Px
Py
Pz
=
χxx χxy χxz
χyx χyy χyz
χzx χzy χzz
Ex
Ey
Ez
If we perform a crystal symmetry operation on both the electric field and the polarisation, the equation
must hold:
UP = χUE withU =
− 1 0 0
0 −1 0
0 0 1
Now we multiply with the inverse ofUfrom the left
U−^1 UP = U−^1 χUEThis can only be true if the susceptibility matrix is invariant under the symmetry operation
χ = U−^1 χUwhich would mean
χ =
χxx χxy −χxz
χyx χyy −χyz
−χzx −χzy χzz
This is fulfilled ifχxz,χyz,χzxandχzyare equal to zero. Further we know from the intrinsic
symmetries explained above, thatχij = χji. Thus the susceptability tensor of a crystal with only
the 180°rotational symmetry looks like this:
χ =
χxx χxy 0
χxy χyy 0
0 0 χzz
Those symmetries and the point- and space groups can be looked up in tables^4. Since it would be
far too complicated to write all symmetry operations of one group in such a table, these tables often
only contain the generating matrices. Those matrices describe the symmetry operations with which all
other symmetry operations of this group can be constructed. Another quite important thing is that es-
pecially in those tables, tensors with a rank higher than two have to be displayed. To simplify the high
(^4) i.e. the very own of this course at http://lamp.tu-graz.ac.at/ hadley/ss2/crystalphysics/crystalclasses/crystalclasses.html