Past exam of the condensed matter physics course of the University of Oxford / 2012 / 1

The incident X-ray drives the electrons of an atom. Away from an absorption edge, and when the photon energy greatly exceeds the relevant electronic binding energies but is small compared with $m_e{c}^2$, each electron has approximately the same Thomson scattering amplitude. The much heavier nucleus makes a negligible contribution because its acceleration is much smaller.

For elastic scattering, the amplitudes from different bound electrons add coherently. In units of the single-electron amplitude, the atomic scattering factor is the Fourier transform of the bound-electron number density:Consequently, when the electron cloud is not well resolved by the scattering vector, its internal phase differences are small andThis is an approximation to the coherent scattering amplitude, not an assertion that independently scattered intensities add as $N_e$. At larger momentum transfer, destructive interference within the cloud reduces $f$; near absorption edges, the atomic scattering factor acquires dispersive corrections. Inelastic scattering is a separate contribution. In an ionic crystal, use the ion's bound-electron count rather than automatically using the neutral atom's atomic number.

Use the total scattering angle $\psi =2\theta$. For a cubic unit cell, Bragg's law and the Miller indices giveThe four measured values of ${\sin }^2\theta$, normalized to the first, are $1,1.330,2.655,3.649$. Multiplying by three gives approximately $3,4,8,11$, the initial allowed sequence for a face-centered cubic lattice:The FCC lattice factor is $1+(-1{)}^{h+k}+(-1{)}^{h+l}+(-1{)}^{k+l}$; it vanishes unless all three indices have the same parity. Thus the initial ratios differ from those of a primitive or body-centered cubic lattice. This is the conventional minimal cubic indexing; arbitrarily hiding additional weak reflections could make a short list alone ambiguous.

The low-pressure rock salt structure has an FCC Bravais lattice with one NaCl formula unit per primitive unit cell, or four per conventional cube. A possible atomic basis of a crystal is Na at zero and Cl at $(1/2,0,0)$ in conventional fractional coordinates. The two species have unequal atomic scattering factors, so both the odd and even FCC families can occur.

The four independent lattice constant estimates areTheir small scatter reflects the rounded angular data. The appropriate reported result is

The high-pressure atomic basis of a crystal has the caesium chloride structure. Its chemically labelled Bravais lattice is simple cubic lattice: exchanging Na and Cl is not a translation symmetry. Its structure factor isBecause ionic bonding gives approximately Na${}^{+}$ and Cl${}^{-}$, their bound-electron counts are $10$ and $18$. In the stipulated electron-count approximation, odd-index-sum reflections have $|F|=8$, while even-index-sum reflections have $|F|=28$. In particular, the first ring $\{100\}$ is weak but present; it must not be discarded using a body-centering extinction rule.

The next ring is $\{110\}$, and Bragg's law givesThe powder diffraction multiplicities are six and twelve. Counting all symmetry-equivalent reflections, while neglecting other angular factors, therefore givesThe corresponding single-directed-reflection ratio would be $12.25$. Neutral-atom counts would give the wrong contrast because the material is ionic.

For measured integrated photon counts, the geometry must also be specified. More generally the ratio is $2|F_{110}/F_{100}{|}^2$ times the ratio of Lorentz-polarization factors, thermal-motion factors and detection efficiencies. For example, the common unpolarized symmetric-powder convention hasand gives $24.5LP(21.2506^{\circ })/LP(14.85^{\circ })\simeq 11.18$ if all other corrections are suppressed. Thus 24.5 is the multiplicity-and-structure-factor estimate, rather than a geometry-independent precision count ratio; actual angle-dependent ionic form factors are also not supplied.

The high-pressure lattice constant follows from its first $(100)$ reflection:Conserve the number of NaCl formula units. The low-pressure conventional unit cell contains four, whereas the high-pressure cell contains one. The volume per formula unit changes from $a_0^3/4$ to $a_P^3$, soThe sample contracts by about $39.5\%$. Comparing $a_P^3$ directly with $a_0^3$ would compare different amounts of material.