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In a perfect periodic crystal, do lattice vibrational modes corresponding to different wave vectors (k-vectors) interact with each other? I'm asking with reference to anharmonic lattice-dynamics calculations of a time-independent nature.

For example: consider a linear, monatomic chain with a primitive basis of one atom. Build a supercell containing 3 primitive cells. Calculate lattice vibrations using periodic boundary conditions at the two edges of the supercell (lattice constant is now triple the primitive lattice constant). The primitive cell had 1 degree of freedom, and the supercell will have 3 (3-1 translation = 2 vibrational dof). One of the supercell's vibrational modes will correspond to wave vector [itex]k=\frac{1}{3}[/itex], and the other to [itex]k=\frac{2}{3}b[/itex], where [itex]b[/itex] is the reciprocal lattice vector for the chain. [itex]k[/itex] and [itex]b[/itex] are defined relative to the primitive cell. Note that in speaking of modes in the supercell which "correspond to" particular wave vectors, I could have more formally said that there is a set of wave vectors which are commensurate with the supercell.

Further clarification: if we had a linear, diatomic chain then the 3xsupercell would have 2 vibrational modes each for the wave vectors: [itex]0b, \frac{1}{3}b [/itex] & [itex]\frac{2}{3}b[/itex], and one could imagine mode-coupling between the two vibrational modes for [itex]k=\frac{1}{3}b[/itex], say. Does one also include coupling between modes corresponding to different wave vectors?

The Question: In a calculation including mode-coupling (evaluating the potential energy surface of a nucleus as a function of 2 vibrational mode coordinates [itex]V=V(Q_1,Q_2)[/itex] - [itex]Q_1,Q_2[/itex] are vibrational normal coordinates - rather than just [itex]V=V(Q_1)+V(Q_2)[/itex]), is it correct to couple the modes from different wave vectors?

If not, is there a physical/mathematical reason why one refrains from coupling different wave vectors? It seems reasonable to me, as I just imagine coupling modes of different wave vectors as superposing two plane waves. To be clear: I'm trying to avoid a discussion of multi-phonon scattering, because I'm only interested in an "equilibrium" snapshot in time / the "mean" vibrational state of the perfect crystal.

For example: consider a linear, monatomic chain with a primitive basis of one atom. Build a supercell containing 3 primitive cells. Calculate lattice vibrations using periodic boundary conditions at the two edges of the supercell (lattice constant is now triple the primitive lattice constant). The primitive cell had 1 degree of freedom, and the supercell will have 3 (3-1 translation = 2 vibrational dof). One of the supercell's vibrational modes will correspond to wave vector [itex]k=\frac{1}{3}[/itex], and the other to [itex]k=\frac{2}{3}b[/itex], where [itex]b[/itex] is the reciprocal lattice vector for the chain. [itex]k[/itex] and [itex]b[/itex] are defined relative to the primitive cell. Note that in speaking of modes in the supercell which "correspond to" particular wave vectors, I could have more formally said that there is a set of wave vectors which are commensurate with the supercell.

Further clarification: if we had a linear, diatomic chain then the 3xsupercell would have 2 vibrational modes each for the wave vectors: [itex]0b, \frac{1}{3}b [/itex] & [itex]\frac{2}{3}b[/itex], and one could imagine mode-coupling between the two vibrational modes for [itex]k=\frac{1}{3}b[/itex], say. Does one also include coupling between modes corresponding to different wave vectors?

The Question: In a calculation including mode-coupling (evaluating the potential energy surface of a nucleus as a function of 2 vibrational mode coordinates [itex]V=V(Q_1,Q_2)[/itex] - [itex]Q_1,Q_2[/itex] are vibrational normal coordinates - rather than just [itex]V=V(Q_1)+V(Q_2)[/itex]), is it correct to couple the modes from different wave vectors?

If not, is there a physical/mathematical reason why one refrains from coupling different wave vectors? It seems reasonable to me, as I just imagine coupling modes of different wave vectors as superposing two plane waves. To be clear: I'm trying to avoid a discussion of multi-phonon scattering, because I'm only interested in an "equilibrium" snapshot in time / the "mean" vibrational state of the perfect crystal.

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