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

The isothermal volume compressibility is $\kappa =-(1/V)(\partial V/\partial p{)}_T$. For isotropic cubic compression $V\propto a^3$ and $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$, soDifferentiate Bragg's law $2d_{hkl}\sin \theta =\lambda$ at fixed incident wavelength and fixed reflection:This derivative uses radians. It remains locally valid with a pressure-dependent compressibility, provided the cubic isotropic phase is retained. For a constant stipulated $\kappa$, integration givesThus $a_0(1-\kappa p/3)$ 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 $\sin {\theta }_0{e}^{\kappa p/3}\le 1$. These limitations matter for a large extrapolation.

The 110 plane spacing of the conventional cubic cell isFor the first-order Bragg condition,The quoted experimental deflection is twice the angle between the beam and the reflecting planes; reporting only $22.298^{\circ }$ would report the Bragg angle instead of the requested scattering angle. The 110 reflection is allowed for the identical-Fe body-centered lattice because $1+1+0$ is even. Wavelength and spacing have been expressed in the same units, so their ratio is dimensionless. Higher Bragg orders are conventionally represented by the corresponding higher-index reflection, rather than inserted as an unannounced extra order here.

Use the integrated constant-compressibility model from c, with $\kappa p=0.0059\times 100=0.59$:ThereforeCompression reduces the spacing and moves the same allowed reflection to larger deflection, as the positive pressure derivative predicts. The input compressibility corresponds to a constant bulk modulus $1/\kappa \simeq 169.5$ GPa in this ideal model.

This is the stipulated unchanged cubic-phase, constant-compressibility extrapolation, rather than a reliable prediction for real Fe at 100 GPa. Real Fe exhibits a pressure-induced body-centered-cubic to hexagonal-close-packed transformation, and compressibility can change with pressure.^[ref] A real high-pressure diffraction interpretation would require its actual phase and equation of state. The source gives neither, so its numerical exercise is answered within its explicitly stated idealization.