Past exam of the condensed matter physics course of the University of Oxford / 2019 / 1 / d / Solution

Separate the entire-system atom index into a cell translation $\mathbf{R}$ and an index $p$ within a single primitive unit cell. The potential becomesThe same $u_p$ is used in every translated cell because the atomic basis repeats. Normalize plane waves in volume $\Omega$ by $⟨\mathbf{x}|\mathbf{k}⟩={\Omega }^{-1/2}{e}^{i\mathbf{k}\cdot \mathbf{x}}$. For $\mathbf{G}={\mathbf{k}}^{\prime }-\mathbf{k}$, substitution and $\mathbf{s}=\mathbf{x}-\mathbf{R}-{\mathbf{r}}_p$ giveFor a real static scalar potential, define the atomic scattering factor in the positive-exponent convention of the question byThe second bracket in the matrix element is $S(\mathbf{G}{)}^{\ast }$. At a reciprocal lattice vector every term of the first bracket is one, giving the number of cells $N_c$. Therefore the stated transition-rate rule, equivalently the first Born approximation and Fermi's golden rule up to the final-state kinematic factors, yieldsThe harmless complex conjugation records the distinction between the source's positive-exponent Fourier transform and the usual positive-exponent incoming plane wave. Reversing every Fourier sign gives the equivalent amplitude convention. Only the cell-translation sum ranges over the full crystal; the crystal basis structure factor contains the atoms of one cell.