Symmetry of solids/Topic 1

From WolfWikis

Groups

1. A group consists of a set of elements.
2. When combined ('multiplied') according to certain rules two elements always produce another element of the group.
3. A group always contains an identity element E that 'does nothing', i.e. E.A = A.E = A
4. For each element there is a reciprocal element such that A.A^-1= A^-1.A=E

The combination rules usually have the nature of an operator than a simple multiplication, but they can still be represented in multiplication table.

A good example is a mirror plane m. It is its own reciprocal and the table becomes:

Representations

Although groups are more general than symmetry alone, symmetry operators do form groups.

Representations can also be very general. Ignoring small differences, your left (L) and right (R) hands together form a representation of the above group m.

We can describe what the elements do to your hands:

We can summarize these results in matrix form:

E							The mirror symmetry applied to hands or functions
1	0		L		==>	L
0	1		R		==>	R
m
0	1		L		==>	R
1	0		R		==>	L

Irreps

The above representation is two dimensional (2D) as it involves 2x2 matrices. The reason for that is that m takes R into L and vice versa.

If we use other 'objects' as a basis for the rep, e.g. functions we can reduce the dimensionality of the representations by taking clever linear combinations.

Suppose L and R represent functions rather than your hands and the mirror operation flips around the argument, say mR(t) -> R(-t) = L(t)

As functions can be added we could take the linear combinations:

If we apply the mirror to F_sum= L+R, we get:

Likewise

This means that there is no element that takes F_sum into F_dif! In turn this means that we can write the corresponding matrices in diagonalized form and split them up:

This means we can split up the two-dimensional representations into two one-dimensional ones.

We can still write these new representations with a 'multiplication matrix', but these now reduce to dimension one, e.g. they become numbers (coefficients). The two irreps still differ in these coefficients and the functions F_dif and F_sum are said to transform as or be basis functions of two different representations of the group m.

These two representations are said to be irreducible because there is no way to repeat the splitting process on single function.

In general we can start with a group of N elements and a representation that has one object (hand, foot, elephant, matrix, iPod, function etc.) per element, i.e. an N-dimensional faithful representation. This is generally a reducible representation and when we write it out in matrices we can start diagonalizing them. (Actually that can be done on the matrices too, without defining basis functions.)

However, how far we can reduce the faithful rep depends on the structure of the multiplication table. If you reach a one dimensional rep it is of necessity an irreducible one, but the reverse is not always true. For points groups we may well get a few irreps, i.e. irreducible representations that are not of dimension 1. For cubic symmetries like O_h we get irreps of dimension 1 (A,B) but also 2 (E) and 3 (T). In such cases you still need more than one object (function, matrix) but no matter how you combine them you cannot reduce any further. We shall see why later.