Past exam of the materials science course of the University of Cambridge / 2022 / Paper 1 / 7

A diamond anvil cell concentrates force onto tiny opposing culets, reaching very high pressure with small samples and useful optical access through diamond; its small volume, pressure gradients, alignment demands and difficulties with some heating or sample environments can limit experiments. A multi-anvil press supports a much larger sample and is useful for high-temperature synthesis and bulk measurements, but usually reaches lower ultimate pressures and has more restricted optical access. Hydrostaticity, attainable temperature and calibration depend on the actual apparatus and pressure medium; neither design guarantees uniform pressure.

Use mass density $\rho =ZM/(N_A{a}^3)$, with $M=74.55\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$. The rock salt structure has $Z=4$ formula units per conventional cell; the CsCl structure has $Z=1$. In pressure order the densities are approximatelyThe increase across the transition is approximately $14.5\%$, much greater than the smooth compression within either branch. Comparing $a^3$ directly without correcting the different $Z$ would create a fictitious volume discontinuity.

Within each cubic phase, increasing pressure decreases $a$, so $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$ decreases and Bragg's law moves a given reflection to higher $2\theta$. Near $2.55\mathrm{GPa}$, peaks of the rock salt structure disappear as those of the CsCl structure appear, potentially with a coexistence interval. The former has an FCC translation selection rule, with $h,k,l$ all even or all odd; mixed-parity reflections vanish kinematically. The latter has primitive-cubic translations and basis factor $f_K+(-1{)}^{h+k+l}{f}_{Cl}$, so it has no FCC extinction rule. Odd-sum reflections may nevertheless be weak because the two ionic scattering factors are similar. Peak spacings, relative intensities and the cell volume per formula unit change discontinuously. A simple uniform shift of the entire original pattern cannot describe the structural transition.

Use only the high-pressure CsCl structure branch to extrapolate its volume per formula unit $V=a^3$. A straight-line least-squares fit to those three measurements gives approximatelywith $P$ in GPa. Thus $V(10)\simeq 39.29{\mathring{\mathrm{A}}}^3$, $a(10)\simeq 3.400\mathring{\mathrm{A}}$ and $d_{111}=a/\sqrt{3}\simeq 1.963\mathring{\mathrm{A}}$. Taking Cu K$\alpha$ wavelength $\lambda \simeq 1.541\mathring{\mathrm{A}}$ in Bragg's law givesThe estimate is model-dependent: a straight-line extrapolation of $a(P)$ instead gives $2\theta \simeq 46.0^{\circ }$. The data stop at $3.81\mathrm{GPa}$, so the absence of further phase transitions does not by itself determine an exact equation of state to $10\mathrm{GPa}$. The $111$ peak can also be weak because its basis factor is $f_K-f_{Cl}$.

The coordination number rises from six in the rock salt structure to eight in the CsCl structure. In the elementary hard-sphere ionic bonding model, accommodating eight opposite neighbours favours a larger cation/anion radius ratio: the geometrical limiting ratios are $\sqrt{2}-1$ for sixfold and $\sqrt{3}-1$ for eightfold coordination. The high-pressure change is therefore consistent with $r_K/r_{Cl}$ increasing under compression. Sincethe chloride ion is inferred to be more compressible than the potassium ion in this radius-ratio interpretation. This is a qualitative structural argument; real ionic radii depend on coordination and electronic structure, and the transition does not uniquely determine separate microscopic compressibilities.