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We know that the Bloch theorem about electrons in a perfect crystal help us to investigate desired properties of solids.

So as we know, because of the periodicity of the lattice, we use the Fourier series (for periodic and non-periodic) to go to the wave-vector space.

as far as I know, we can use the Fourier transform (for periodic and non-periodic functions) to go to the wave-vector space. But, why don't we do this and use the Fourier series? (What are the limitations that prohibit us of doing FT?)

Why not using FT in this problem?

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The continuous Fourier transform is for continuous functions. The discrete Fourier transform (Fourier series) is for discrete functions. A crystal lattice is inherently discrete, it begs for the Fourier series versus the continuous Fourier transform.

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    $\begingroup$ What you say is morally correct, though I feel that it is important to note that the Fourier transform is just a generalization of Fourier series. For example, one can still take the Fourier transform of periodic functions if we allow for distributions (eg we can take the FT of sin(x), which gives delta functions). Hence, you can always use the Fourier transform when all you need is a Fourier series, though it might be more of a headache to do so. $\endgroup$ – Aaron Nov 2 '16 at 17:04
  • $\begingroup$ @Aaron: I think of just something more than a headache... because what we earn in the bloch theory is upon the periodicity of lattice and the ability to make the transform... but if the periodicity is not important and we could again fourier transform, then maybe we could investigate the properties of the non-periodic solids in some way... $\endgroup$ – P.A.M Nov 2 '16 at 18:55
  • $\begingroup$ The utility of the Fourier transform is precisely the utility of the k-space picture, which clearly extends beyond crystal structures (eg free electron gas). The main issue with using the FT for aperiodic solids, to my knowledge, is that it is hard to find structure after performing an FT. The utility of Bloch theory is due to Bloch's theorem, in which the periodicity of the Hamiltonian enforces periodicity in the wavefunction, which means the k-space/Brillouin zone picture is very fruitful. $\endgroup$ – Aaron Nov 2 '16 at 19:08
  • $\begingroup$ @Aaron: Thanks for your explanation... and I wonder what you mean by the word "structure"? $\endgroup$ – P.A.M Nov 2 '16 at 19:37
  • $\begingroup$ @P.A.M By structure, I mean get useful information. Basically, the idea is that there is no guarantee that the k-space picture will give you any more information that the real-space picture. For example, in the simplest case where I take a diffraction image (which is an FT) of a crystal, I'll get a very ordered and easy to understand pattern. However, the less order the material has, the more the diffraction image will look like white noise. This is precisely the challenge of imaging proteins with x-ray diffraction methods, and why they try to freeze them prior to imaging. $\endgroup$ – Aaron Nov 2 '16 at 19:44

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