Past exam of the general relativity and cosmology and condensed matter physics paper of Part B of the physics course of the University of Oxford / 2011 / Section VI / 5

For the mutually orthogonal direct primitive lattice translation vectors ${\mathbf{a}}_i$, define the reciprocal lattice basis byThus $\mathbf{G}={\sum }_i{h}_i{\mathbf{b}}_i$, with integer $h_i$, satisfies $\exp (i\mathbf{G}\cdot \mathbf{R})=1$ for every lattice vector $\mathbf{R}$.

Write $\mathbf{r}={\sum }_i{s}_i{\mathbf{a}}_i$. The electron density is periodic with period one in each $s_i$, so its three-dimensional Fourier series has integer harmonics:If $\Omega$ is a primitive unit cell of volume ${\Omega }_c$, Fourier orthogonality givesConversely, translation by ${\mathbf{a}}_i$ multiplies a general plane wave by $e^{i\mathbf{q}\cdot {\mathbf{a}}_i}$. Periodicity permits only $\mathbf{q}\cdot {\mathbf{a}}_i\in 2\pi \mathbb{Z}$, precisely the reciprocal lattice vectors. For a physical bounded periodic electron density, the expansion exists in the usual square-integrable sense.

Use the scattering vector $\mathbf{q}={\mathbf{k}}_{\mathrm{in}}-{\mathbf{k}}_{\mathrm{out}}$; reversing its sign conjugates the following amplitudes without changing the intensities. In the kinematic approximation, the atomic scattering factor of atom $s$ isAtoms at $\mathbf{R}+{\mathbf{d}}_s$ therefore give the scattering amplitudeHere $A_e$ is the single-electron amplitude; $F$ is the unit cell structure factor, and $S_L$ is the geometrical lattice sum. If the crystal has $N_i$ repeats along ${\mathbf{a}}_i$, the latter factor is a product of geometric series, whose magnitudes containThe sharp constructive peaks occur when every $\mathbf{q}\cdot {\mathbf{a}}_i=2\pi h_i$. Thus a permitted reflection has wavevector change equal to a reciprocal lattice vector, with $F(\mathbf{G})\ne 0$ and the additional elastic condition $|{\mathbf{k}}_{\mathrm{out}}|=|{\mathbf{k}}_{\mathrm{in}}|$. A finite crystal broadens these peaks; it does not impose an exact delta-function selection on every scattered photon.

In the conventional cubic unit cell, the four face-centered cubic lattice positions contributeThe two-atom atomic basis of a crystal gives the zinc blende structure factorMixed-parity Miller indices have $S_{\mathrm{FCC}}=0$. With all indices even or all odd, $S_{\mathrm{FCC}}=4$. For real atomic scattering factors,Hence all same-parity reflections are generally permitted in GaAs, although the last class is weak when the two atomic scattering factors nearly coincide. These are the structural selection rules; a particular experiment must also satisfy the elastic Bragg scattering condition.

In the electron-count estimate $f_{\mathrm{Ga}}\simeq 31$, $f_{\mathrm{As}}\simeq 33$, the structure factors are $F_{111}=4(31-33i)$ and $F_{200}=4(31-33)$. Thus a single-directed-reflection estimate isA powder X-ray diffraction peak collects all equivalent reflections. The powder diffraction multiplicities are eight for $\{111\}$ and six for $\{200\}$, so the multiplicity-and-electron-count estimate isAt $(200)$ the two sublattice amplitudes are opposite in phase and nearly cancel; at $(111)$ their relative phase is a quarter cycle, so this cancellation does not occur. In silicon, the two atomic scattering factors are identical and the diamond cubic structure has an exactly forbidden $(200)$ reflection, while $(111)$ remains allowed in the kinematic approximation.

The numerical ratio neglects the angular dependence of the atomic scattering factors, the Lorentz-polarization factor and thermal-motion corrections. No wavelength or measurement geometry is supplied to refine those factors. This is therefore a stated structural estimate, rather than a unique precision ratio of measured peak areas.