Reciprocal Space: Symmetry Operations and Vector Components

[pinkuagarwal]

Hi,

I have a question regarding reciprocal space symmetry operations and would appreciate if somebody can answer that:

Lets say I have two k points, k1 and k2, in reciprocal space which are symmetric to each other. Since these two kpoints are symmetric to each other phonon scalar properties like frequency, lifetime, etc will be same for k1 and k2. Similarly, vectors like group velocity have same magnitude for k1 and k2. I am wondering how are vector components related at k1 and k2?

Thanks a lot in advace!

 

[gadong]

I am not sure if this will answer your general problem, but here is an illustration for phonons along high symmetry directions in monatomic lattices.

Phonons (lattice waves) that propagate along high symmetry directions such as [100] or [110] come in two forms: longitudinal and transverse.

Suppose we call the direction of propagation z. Then the longitudinal wave corresponds to atomic displacements (vibrations) parallel to the z-axis. There are two transverse vibrations that correspond to atomic displacements parallel to the x- and y-directions, respectively. These two "polarizations" are degenerate. Phonons that propagate along [111] directions also have two degenerate transverse modes (but not along the x- and y-axes, so I'll avoid them to keep the notation simple).

Thus, to address your question: "I am wondering how are vector components related at k1 and k2?"

(1) Symmetry between k1 and k2 might involve having k1 as a point in the kz direction, while k2 is a symmetrically equivalent point in the kx direction. Example, k1 = (0,0,q), k2 = (q,0,0).

(2) To specify a transverse phonon completely, you need to identify its polarization (e). For the pair of phonons in (1) the possibilities are for k1=kz, e1=kx or ky, and for k2=kx, e2=ky or kz.