Spin-Phonon Interaction in Paramagnetic Crystals

A general theory of the spin-phonon interaction, which is applicable to any iron group spin not in an $S$ state, is developed. The theory employs a perturbation treatment that has a more direct physical meaning than techniques previously used and that leads to more accurate results. These results are presented in the form of an equivalent spin-phonon interaction Hamiltonian involving sums over products of spin operators and phonon creation-annihilation operators. The interaction between any two spin levels can then be calculated by using the spin wave functions associated with the usual "spin Hamiltonian." It is shown that, owing to the dominant role played by the quadratic term in the above interaction, odd half-integer iron group spins ($S>\mathrm{}\frac{1}{2}$) obey quadrupole selection rules. A formula is derived for order-of-magnitude calculations of the interaction strength. It is shown that acoustic experiments should provide the ideal way to test this theory in detail, and two methods of checking the quadrupole rule are proposed. Experimental results are reported on observed acoustic saturation in MgO doped with ${\mathrm{Cr}}^{+++}$, on the absence of saturation between low-field Kramers doublets in ruby, and an apparent saturation effect in $F$_center quartz.