Expanding atomic displacements in terms of all lattice wave modes?

[zhanhai]

I find the expression:

"In general, one can expand the atomic displacements in terms of all the lattice wave
modes (resembles a Fourier series expansion)"

at:
https://courses.cit.cornell.edu/ece407/Lectures/handout17.pdf

But I have not found the expression in any other literature. (In fact, some literature, like P.Y. Yu and M. Cardona: Fundamentals of Semiconductors, a long wavelength limitation seems to have been imposed.) Is the above-cited expression correct (in 3D lattice)?

Thank for your assistance.

 

[DrDu]

Yes, that's true. Should be explained in any book on solid state theory.

 

[Dickfore]

Have you ever done normal mode decomposition of a system of coupled oscillators? This is the same, except that the system is infinite, but periodic boundary conditions are imposed.