Section 3.1 summed equally spaced phases and found a peak of height at with its first…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction

Condensed Matter Physics, Ch Diffraction: how a structure is actually measured

Problem

Section 3.1 summed NN equally spaced phases and found a peak of height N2N^2 at δ=0\delta=0 with its first zero at δ=2π/N\delta=2\pi/N. Section 3.2 asserted that the lattice sum ∑Re−iΔk⋅R\sum_{\mathbf{R}}e^{-i\Delta\mathbf{k}\cdot\mathbf{R}} equals NN exactly, not merely of order NN, whenever Δk\Delta\mathbf{k} is any reciprocal lattice vector G\mathbf{G}, not only at Δk=0\Delta\mathbf{k}=0. For a row of NN atoms spaced aa apart, show that the peak at Δk=G\Delta\mathbf{k}=G, for any reciprocal lattice vector G=2πm/aG=2\pi m/a with mm an integer, is exactly as tall and exactly as wide, in Δk\Delta\mathbf{k}, as the peak at Δk=0\Delta\mathbf{k}=0.

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