Past exam of the condensed matter physics course of the University of Oxford / 2013 / 1

The sites form a face-centered cubic lattice decorated by an identical two-atom atomic basis of a crystal, at fractional displacements $0$ and $\mathbf{\delta }=(1/4,1/4,1/4)$. This is the diamond cubic structure; the eight atoms in its conventional cube are not eight independent Bravais-lattice points.

For identical silicon atomic scattering factors $f$, the conventional-cell structure factor separates into the FCC factor and the two-site motif factor:The first factor vanishes unless $h,k,l$ are all even or all odd. For all-even indices, the motif also vanishes if $h+k+l=4r+2$; it survives if the sum is divisible by four. All-odd indices survive, with motif modulus $\sqrt{2}$. Thus $(100)$ and $(110)$ are absent by centering, $(200)$ is absent by the diamond basis, and $(111)$ and $(220)$ are allowed.

For a cubic cell, $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$. At fixed X-ray wavelength, Bragg's law makes the smallest scattering angles correspond to the smallest allowed positive values of $h^2+k^2+l^2$. These start with $3$ and $8$, soThe braces denote whole powder families, including their symmetry-equivalent plane normals. The next family is $\{311\}$, with squared-index sum $11$; using the FCC rule alone would incorrectly insert a $(200)$ line before $(220)$.

Interpret the reported angles as total scattering angles $\psi =2\theta$. The photon energy givesCombining $2d\sin \theta =\lambda$ with the two index assignments yieldsThe slight difference is consistent with the rounded angular data, givingThe intensity estimate needs a stated convention. The geometric structure factors obey $|F_{111}{|}^2=32|f_{111}{|}^2$ and $|F_{220}{|}^2=64|f_{220}{|}^2$, so equal atomic factors give a single-directed-reflection ratio $1/2$. For a powder line, the powder diffraction multiplicities are eight for $\{111\}$ and twelve for $\{220\}$. Hence a multiplicity-and-structure-factor estimate isif other angle-dependent factors are neglected.

A measured integrated powder intensity also contains the Lorentz-polarization factor, atomic form factor, thermal-motion factor and instrument/sample corrections. For example, in the common unpolarized symmetric powder geometry,Including this factor but still taking equal atomic factors gives $I_{111}/I_{220}\simeq 0.9725$, rather than $1/3$. More generally,within that geometry, before other corrections. The actual form factors and measurement geometry are not supplied, so no unique precision experimental peak-height ratio follows. The values $1/2$, $1/3$ and $0.9725$ refer to distinct explicitly stated levels of approximation; a powder family must not be silently treated as one directed reflection.

At fixed radiation wavelength, differentiating Bragg's law gives $\cot \theta d\theta =-da/a$. With constant linear thermal expansion coefficient ${\alpha }_L$,Both scattering angles decrease as the lattice expands. Substitution givesThese are shifts of $2\theta$, not of the Bragg angle alone. As a finite-step check, using $a^{\prime }=a(1+0.0018)$ gives ${\psi }^{\prime }=2\arcsin [\sin (\psi /2)/(1.0018)]$. It produces shifts $-0.04151^{\circ }$ and $-0.07008^{\circ }$, consistent with the first-order estimate. A strictly constant differential coefficient would instead give $a^{\prime }/a=e^{{\alpha }_L\Delta T}$; its difference from the linear length approximation is second order in this small expansion.

Diffraction thermal-expansion thermometry needs a stable monochromatic X-ray beam, controlled collimation, an accurately calibrated angular scale, and an X-ray-sensitive detector with enough counts to fit reproducible line centroids. Crystal monochromators or appropriate energy selection are relevant here; an apparatus for visible-light interference does not directly measure these X-ray peaks. Sample displacement and angular-zero drift should be checked with a suitable reference standard.

The first-order angular sensitivities are about $6.93\times 10^{-5}$ and $1.17\times 10^{-4}$ degrees per kelvin, respectively. A one-kelvin inference therefore requires comparable centroid stability, with high count statistics and control of systematic errors; a centroid can be fitted more precisely than the line width. Fractional wavelength drift also mimics lattice expansion, so the desired temperature precision constrains $\Delta \lambda /\lambda$ relative to ${\alpha }_L\Delta T$.

The powder must be at a known uniform temperature and allowed to equilibrate. Mechanical stress, clamping, composition changes, oxidation and phase changes can also shift or broaden peaks. Accurate work requires a calibrated $a(T)$, including the actual temperature dependence of the thermal expansion coefficient, rather than assuming the supplied constant describes every temperature exactly. Using both lines provides a consistency check on a common cubic expansion.