Expanding atomic displacements in terms of all lattice wave modes?

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The discussion centers on the expansion of atomic displacements in terms of lattice wave modes, drawing parallels to Fourier series expansion. The expression referenced is found in a handout from Cornell University but lacks widespread citation in existing literature, with some sources, such as "Fundamentals of Semiconductors" by P.Y. Yu and M. Cardona, imposing limitations on long wavelengths. The validity of this expression in a 3D lattice context is affirmed, emphasizing the importance of normal mode decomposition in solid state theory.

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