# Distance between planes in crystals

I'm working on a lab report on powder X-ray diffraction off of some relatively straight-forward crystals (Si, NaCl, CsCl) for an introductary course on modern physics.

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:

$$d_{hkl} = \frac{a}{\sqrt{h^2+k^2+l^2}}$$

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.

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Andrew Mason
Homework Helper
Stephan Hoyer said:
I'm working on a lab report on powder X-ray diffraction off of some relatively straight-forward crystals (Si, NaCl, CsCl) for an introductary course on modern physics.

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:

$$d_{hkl} = \frac{a}{\sqrt{h^2+k^2+l^2}}$$

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.

I am not sure if this helps you but have a look at http://www.eserc.stonybrook.edu/ProjectJava/Bragg/" [Broken]

AM

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Gokul43201
Staff Emeritus
Gold Member
I couldn't find it there with a quick look. So, anyway, it's short enough that I can write it down in a few lines.

Consider two adjacent planes, one of which goes through the origin. The second plane makes intercepts a/h, b/k, c/l (by definition of the Miller Indices). Let the point on this plane that's nearest the origin (O) be P. Then OP is the required d-spacing.

Let the line OP make angles A, B and C with each of the three axes. From trig, we have cos2(A)+cos2(B)+cos2(C)=1
But cos(A) = OP/OX = d/(a/h) = dh/a

Similarly, plug in for cos(B) and cos(C) and you will get the required result.

Thanks for you help. It looks like the general proof isn't actually so tedius after all, so I guess I'll include that instead.

Really Thanks. Thank you so much!