Piezomagnetism of $\alpha$-${\mathrm{Fe}}_2$${\mathrm{O}}_3$ and the Magnetoelastic Tensor of ${\mathrm{Fe}}^{3+}$ in ${\mathrm{Al}}_2$${\mathrm{O}}_3$

A theory is developed for the piezomagnetism of insulating antiferromagnets based on a single-ion Hamiltonian and a molecular-field model of exchange. The piezomagnetic constants are found in terms of the single-ion magnetoelastic tensors of the magnetic ions. In contrast to magnetostriction, where the macroscopic magnetoelastic tensors are obtained by summing microscopic magnetoelastic tensors, the piezomagnetic tensor involves the difference of the microscopic magnetoelastic tensors of the ions on the two antiferromagnetic sublattices. The theory is then applied to the piezomagnetism of $\alpha$-${\mathrm{Fe}}_2$ ${\mathrm{O}}_3$. The single-ion magnetoelastic tensor is measured for ${\mathrm{Fe}}^{3+}$ in ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$ using electron paramagnetic resonance under uniaxial strain. The piezomagnetic constants of $\alpha$-${\mathrm{Fe}}_2$ ${\mathrm{O}}_3$ in the low-temperature ($\overline{3}m$) phase are predicted on the single-ion model to be $P_{11}=4.5\mathrm{}\times {10}^{-12\mathrm{}}$ emu/cc per dyn/${\mathrm{cm}}^2$, $P_{14}=8.5\mathrm{}\times {10}^{-12\mathrm{}}$ emu/cc per dyn/${\mathrm{cm}}^2$. The single-ion contribution to the piezomagnetic constants of $\alpha$-${\mathrm{Fe}}_2$ ${\mathrm{O}}_3$ in the high-temperature ($\frac{2}{m}$) phase are predicted to be $P_{14}=P_{36}=7.9\mathrm{}\times {10}^{-12\mathrm{}}$ emu/cc per dyn/${\mathrm{cm}}^2$, $P_{15}=-2\mathrm{}{P}_{31}=2\mathrm{}{P}_{33}=-1.8\mathrm{}\times {10}^{-12\mathrm{}}$ emu/cc per dyn/${\mathrm{cm}}^2$, $P_{24}=8.5\mathrm{}\times {10}^{-12\mathrm{}}$ emu/cc per dyn/${\mathrm{cm}}^2$, $P_{24}=9.0\mathrm{}\times {10}^{-12\mathrm{}}$ emu/cc per dyn/