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

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 $f_K-f_{Cl}$.