Reciprocal Space: Symmetry Operations and Vector Components

In summary, reciprocal space symmetry operations can result in symmetric k points with similar phonon scalar properties, but the vector components of transverse phonons at these k points may differ depending on their polarizations.
  • #1
pinkuagarwal
1
0
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!
 
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  • #2
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.
 

FAQ: Reciprocal Space: Symmetry Operations and Vector Components

1. What is reciprocal space?

Reciprocal space is a mathematical construct used to describe the diffraction patterns produced by crystals. It is a complementary representation of real space, where the positions of atoms are described.

2. What are symmetry operations in reciprocal space?

Symmetry operations in reciprocal space refer to the transformations that can be applied to a diffraction pattern without changing its overall appearance. These include translations, rotations, and reflections.

3. How are symmetry operations related to vector components in reciprocal space?

Vector components in reciprocal space are used to describe the intensity and direction of diffraction peaks. These components are directly related to the symmetry operations, as they represent the transformations that can be applied to the diffraction pattern.

4. What is the importance of reciprocal space in crystallography?

Reciprocal space is essential in crystallography because it allows for the determination of crystal structure from diffraction data. This is because the diffraction pattern in reciprocal space contains information about the arrangement of atoms in a crystal.

5. How is reciprocal space used in materials science?

In materials science, reciprocal space is used to study the properties of materials, such as their electronic band structure. This information is crucial for understanding the behavior of materials and designing new materials with specific properties.

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