Past exam of the physics A course of the University of Cambridge / 2016 / Paper 2 / Section C / C11

The reciprocal lattice consists of all wavevectors $\mathbf{G}$ whose plane-wave phase is unchanged under every direct-lattice translation $\mathbf{R}$:It expresses periodicity in wavevector space, with dimensions of inverse length.

Choose incident field convention $e^{i{\mathbf{k}}_i\cdot \mathbf{r}-i\omega t}$. The wave at the second atom acquires incident phase ${\mathbf{k}}_i\cdot {\mathbf{r}}_n$ relative to the atom at the origin. Its path to a distant detector is shorter by approximately ${\hat{\mathbf{k}}}_f\cdot {\mathbf{r}}_n$, contributing outgoing phase $-{\mathbf{k}}_f\cdot {\mathbf{r}}_n$. ThereforeNeglecting variation of detector distance across the specimen, identical atoms have a common scattering amplitude factor andThe original geometry diagram shows the incident direction, outgoing direction and direct-space displacement separately; ${\mathbf{k}}_s$ is a wavevector difference, not an extra ray path.

The primitive reciprocal vectors obeywith cyclic versions for $\mathbf{B},\mathbf{C}$. If ${\mathbf{k}}_s=h\mathbf{A}+k\mathbf{B}+l\mathbf{C}$, every lattice term has phase $2\pi$ times an integer, so all terms add constructively. Conversely, full translational reinforcement requires ${\mathbf{k}}_s\cdot \mathbf{a}$, ${\mathbf{k}}_s\cdot \mathbf{b}$ and ${\mathbf{k}}_s\cdot \mathbf{c}$ to be integer multiples of $2\pi$, hence ${\mathbf{k}}_s$ is reciprocal. These are the Laue equations. A finite crystal gives finite-width maxima rather than ideal reciprocal-space delta peaks; elastic scattering must also satisfy $|{\mathbf{k}}_i|=|{\mathbf{k}}_f|$.

For the two-dimensional Bravais lattice interpretation, with one atom per primitive cell, takeSolving the duality conditions givessoThe reciprocal primitive vectors have equal length $4\pi /(\sqrt{3}a)$ and angle $120^{\circ }$; the array itself is triangular. A different primitive basis can use a $60^{\circ }$ angle without changing the lattice. The direct primitive-cell area is $\sqrt{3}{a}^2/2$ and its reciprocal-cell area is $(2\pi {)}^2$ divided by that area.

For nonzero integer pairs, $h^2-hk+k^2$ has minimum one, attained for six shortest vectors. Therefore the first nonforward maximum has $|{\mathbf{k}}_s|=4\pi /(\sqrt{3}a)$. If the forward transmitted maximum is included literally, the smallest value is $|{\mathbf{k}}_s|=0$; the nonzero value is the first diffraction shell. Reaching that shell in elastic scattering additionally requires $4\pi /(\sqrt{3}a)\le 2|{\mathbf{k}}_i|$.