X-rays of wavelength λ gets reflected from parallel planes of atoms in a crystal with spacing d between two planes as shown in the figure. If the two reflected beams interfere constructively, then the condition for maxima will be, (n is the order of interference fringe)
2d sin θ = nλ
This is Bragg reflection. Each atomic plane reflects part of the X-ray beam, and the beams reflected from neighbouring planes interfere. In the figure the angle θ is measured between the incident ray and the atomic plane (the glancing angle).
Step 1 — The ray reflected from the lower plane travels an extra distance before and after reflection; each part equals \(d\sin\theta\), so the path difference is \(\Delta = 2d\sin\theta\).
Step 2 — Constructive interference needs the path difference to be a whole number of wavelengths: \(2d\sin\theta = n\lambda\), n = 1, 2, 3, ….
The expressions d sin θ = nλ and d tan θ = nλ miss the factor 2 from the two legs of the extra path (and tan has no geometric basis here), while 2d cos θ = nλ would be the condition if θ were measured from the normal to the planes, not from the planes as drawn.
2d sin θ = nλ option is correct.
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