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ID: past-exam-of-the-condensed-matter-physics-course-of-the-university-of-oxford/2016/1/c/solution

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Past exam of the condensed matter physics course of the University of Oxford / 2016 / 1 / c / Solution by

Codex  0 2026-10-25

The isothermal volume compressibility is

κ = - (1/ V) (\partial V / \partial p)_{T}

. For isotropic cubic compression

V \propto a^{3}

and

d_{hk l} = a / h^{2} + k^{2} + l^{2}

, so

\frac{1}{d _{hk l}} (\frac{\partial d _{hk l}}{\partial p})_{T} = \frac{1}{a} (\frac{\partial a}{\partial p})_{T} = - \frac{κ}{3} .

Differentiate Bragg's law

2 d_{hk l} sin θ = λ

at fixed incident wavelength and fixed reflection:

0 = sin θ \frac{\partial d}{\partial p} + d cos θ \frac{\partial θ}{\partial p}, (\frac{\partial θ}{\partial p})_{T} = \frac{κ}{3} tan θ, α = \frac{1}{3} .

This derivative uses radians. It remains locally valid with a pressure-dependent compressibility, provided the cubic isotropic phase is retained. For a constant stipulated

κ

, integration gives

a (p) = a_{0} e^{- κ p /3}, sin θ (p) = sin θ_{0} e^{κ p /3} .

Thus

a_{0} (1 - κ p /3)

is only the first-order approximation, not the exact consequence of a constant logarithmic compressibility. The finite-pressure calculation also assumes that the reflection still exists, the crystal remains in the same structure and

sin θ_{0} e^{κ p /3} \leq 1

. These limitations matter for a large extrapolation.

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