Temperature dependence of the optical birefringence of MnF2, MgF2, and ZnF2

Abstract

Measurements have been made of the temperature derivative of the optical birefringence $\frac{d\left(\Delta n\right)}{\mathrm{dT}}$ of (magnetic) Mn${\mathrm{F}}_2$ and the isostructural (nonmagnetic) Mg${\mathrm{F}}_2$ and Zn${\mathrm{F}}_2$. We show that the lattice contribution to $\frac{d\left(\Delta n\right)}{\mathrm{dT}}$ is not proportional to the anisotropy in the thermal expansion as had been previously conjectured. Instead, using a suitably weighted Debye plus Einstein phonon densities of states, a very good fit of the measured $\frac{d\left(\Delta n\right)}{\mathrm{dT}}$ to harmonic excitations of the lattice can be made for Mg${\mathrm{F}}_2$ and Zn${\mathrm{F}}_2$ from $5\le T\le 300$ K. For $T\ll {\Theta }_D$, $\frac{d\left(\Delta n\right)}{\mathrm{dT}}$ scales with ${\Theta }_D$ as does the lattice specific heat $C_L$. By using this scaling property, the small lattice conbribution to $\frac{d\left(\Delta n\right)}{\mathrm{dT}}$ Mn${\mathrm{F}}_2$ may be deduced. The remaining, much larger, magnetic contribution can then be compared with the magnetic specific heat $C_m$ over the range $5 \mathrm{K}\le T\le 100 \mathrm{K}$. The agreement found between the two is excellent. This definitive result establishes that the proportionality between $C_m$ and the magnetic part of $\frac{d\left(\Delta n\right)}{\mathrm{dT}}$ extends well outside the critical region.