Symmetry of solids/Topic 4

From WolfWikis

Basis functions and frequencies

A general result of group theory is the concept of a basis, consisting of a set of basis functions. These are functions that are multiplied to give each other according to the matrices of a certain representation. For the one dimensional irreps or the abelian translation group we need one function for each irrep and it should have the property that when a certain translation is applied to it, the function is multiplied with the particular numbers (read: one dimensional matrices) of our character table.

Such a function is e.g. f(x) = exp(2πi.x/N) because when we shift x => x+1 this function will produce:

f(x+1) = exp(2πi.(x+1)/N)= exp(2πi.x/N).exp(2πi.1/N) =ε.f(x).

Clearly this function transforms (behaves) as irrep Γ₁

Likewise we could add another coefficient j to construct a basis for any irrep Γ_j:

f_j(x) = exp(2πi.j.x/N)

These functions are pretty ubiquitous in solid state science. Amongst others they are known as Bloch functions if we apply the translation symmetry to Cartesian coordinates like x. Of course in mathematics we could equally well use a different letter of the alphabet like t instead of x. Physically we now have a function of time that is usually known as the phase factor.

f_j(t) = exp(2πi.j.t/N). =exp(iωt)

The quantity 2π.j./N =ω is known as the frequency. It has a dimension of s^-1

If we keep thinking in x-values its equivalent is: 2π.j./N =q (or k). It is known under various names like the spatial frequency, the scattering vector, the reciprocal vector, the k-vector, the momentum vector, the resolution yardstick and a few more. Its dimension is m^-1.

Frequency or reciprocal space; some nomenclature

The Bloch function aka phase factor can be seen either as a function of t or of ω:

f_ω(t) = exp(iωt).

If we take t to be the variable we speak of the time domain or direct space.

If we take ω to be the variable we speak of the frequency domain or reciprocal space.

If we are dealing with position r (as a vector) rather than time t, the two spaces can be generalized to more than one dimension, e.g. 3D: r=(x,y,z) has a reciprocal q=(q_x,q_y,q_z). The latter is often given as (h,k,l) if the reciprocal vector is an integer.

Notational pitfalls

The notation varies a bit amongst disciplines. Physics likes to include the factor 2π into the frequency ω. Crystallography does not. They prefer to write: exp(2πi[hx+ky+lz])= exp(2πis.r). Thus q=2π.s.

Likewise in the 1D time domain: ω = 2π.ν. Both ω and ν are frequencies and have a dimension of reciprocal seconds. However ν is cycles per second, ω is radians per second. (Both cycles and radians are dimensionless)

Decomposing functions in irreps and normal modes

One major advantage of symmetry lore and its irreps is that it can be shown that the energy of a quantum state is specific for a given irrep and that basis functions of different irreps do not mix. If the Hamiltonian of a certain system is subjected to one of the symmetry elements, say R of the symmetry it possesses the energy of the system should not change:

HΨ = EΨ

R(HΨ) = R(EΨ)

HRΨ = ERΨ

Thus the energy of RΨ and Ψ should be the same. If we make sure that Ψ transforms according to one of the irreps then RΨ also belongs to the same irrep. There are no elements that can take RΨ to a different irrep than Ψ. Thus another function Ψ₂ that transforms according to a different irrep will in general have a different energy. This is why it is wise to make sure any function you try and solve Schrödinger's equation with is a basis for an irrep! It makes the calculation so much easier.

In general we could start with something that is not as basis, but by taking linear combinations we can always resolve it in a combination of bases of the irreps.

A good example is a hydrogen molecule. It has mirror symmetry m and we saw before that this group has two irreps. We could start with the s functions of the atoms and treat them like the L- and R-hand example and take a plus and a minus combination s(1)+s(2) and s(1)-s(2). To properly normalize we should divide by √2 and voila we have our bonding and antibonding MO's.

Decomposing things according to their irreps is a mighty way of making many complex problems more tractable. The easiest way to see that is to think of the motions of a molecule. Whichever way it is moving, we can always decompose that motion into a combination of normal modes