Selection Rules and Angular Dependence in Paramagnetic Acoustic Resonance

The problem of spin transitions between magnetic sublevels under ultrasonic excitation is treated by considering perturbation terms $\beta {\mathrm{H}}_0·h·\mathrm{S}$ (dipolar) and $\mathrm{S}·d·\mathrm{S}$ (quadrupolar). When the tensors $h$ and $d$ are expanded in acoustic strains, the resulting expansion coefficients form a magnetoelastic matrix which describes the spin-phonon coupling. For the quadrupolar term which is dominant for $S>\mathrm{}\frac{1}{2}$, the magnetoelastic matrices are obtained for all crystal classes with the assumption that they are not necessarily symmetric. By using the transformation of spin operators, which includes the case of anisotropic $g$ factors, general expressions for acoustic transition probabilities in dipolar and quadrupolar cases are derived in terms of $h_i$ and $d_i$. It is shown, on the basis of these expressions with magnetoelastic matrices and strain transformations to crystalline axes, how the acoustic absorption coefficients are obtained for any direction of polarization and wave propagation, for arbitary direction of the external magnetic field, and for all crystal classes. It turns out that the explicit angular dependence of absorption coefficients is different in the dipolar and quadrupolar cases for anisotropic $g$ factors, but often for isotropic $g$ factors the forms of angular dependences are similar. This treatment is valid for iron group and $S$ state ions and also is formally applicable to nuclear quadrupole transitions under acoustic excitation. The theory is compared to the results of our experiments, in which absorption coefficients at magnetic resonance were measured for 10 kMc/sec acoustic waves. The essential points of the theory are borne out by experimentally measured angular dependences of acoustic absorption.