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

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.