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I thought it would be useful to include a partial derivation of the formula relating the distance between parallel planes, d, the length of a cell edge, a, and the miller indices (hkl) for a cubic lattice:

[tex]d_{hkl} = \frac{a}{\sqrt{h^2+k^2+l^2}}[/tex]

I would be happy (and it would be sufficient for my purposes) to do a basic derivation of the spacing between lines in a hypothetical two dimensional square lattice. I've thought a lot about this problem, however, and what I thought would be a clear geometrical fact is turning out to be not so obvious.

Does anyone have any hints or links to a derivation? I got several texts on X-ray diffraction from my college's library, including a text, "Interpretation of x-ray powder diffraction patterns" but none of them include a clear derivation. What I've found online seems to be generally cursory, as well. I've drawn out a two dimensional square lattice and sample parallel lines going through it and I can see that the equation holds, but I'd like a simple proof, from first principles if possible.

Thanks for your help.