Most books on solid state physics and/or crystallography state that the reciprocal lattice is the Fourier transform of the crystalline lattice in real space. I have not, however, come across a demonstration of this.
I attempted the following:
(1) Represent the lattice with a collection of delta functions (Dirac masses), one at each lattice point.
(2) Express the lattice as f(x,y,z), where x,y,z are spatial coordinates in real space.
(3) Find the Fourier transform of f.
My confusion is related to the complex terms in the transform- what do they mean? How do we plot them to generate the reciprocal lattice?
Do we ignore imaginary terms, take absolute values etc?
Dear Aasharanth,
I perfectly see your point, as sometimes textbooks go pretty fast on mathematical aspects. Anyway, I don't see where those imaginary terms would come from. A lattice function is a summation of Dirac Delta functions covering the direct space in 3D, and its Fourier transform is another summation of Dirac Delta functions covering, now, the reciprocal space. No imaginary terms are present concerning the Lattice Function and its transformation as far as I know. Given the diffraction from a crystal considered as convolution of Lattice function and electron density of the unit cell, in its FT (the product of the FT of both terms) the part containing an imaginary component is the last one, the FT of the electron density in the unit cell (See "Fundamentals of Crystallography" by Giacovazzo et al., Chap.3). Going back to the FT of the lattice function and the demonstration I did not provide, this link might be useful: https://courses.cit.cornell.edu/mse5470/handout5.pdf .
I hope I've been of some help.
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