Diagonalization of the Antiferromagnetic Magnon-Phonon Interaction

A technique is developed for the diagonalization of quadratic forms consisting of operators whose commutators are $c$ numbers. In particular, it is shown that the transformation matrix S which diagonalizes such quadratic forms, must satisfy S ${\mathbf{g}}^{\prime }$ S†=g, where g is a matrix whose elements are $c$ numbers depending upon the commutation relations of the original variables which constitute the quadratic form, and ${\mathbf{g}}^{\prime }$ is similarly defined by the new variables. A perturbation expression is then derived for the elements of S. These results are applied to the magnetoelastic interaction in antiferromagnets. It is found that a magnetic field oscillating at a frequency $\omega \ll \gamma {\left(2\mathrm{}{H}_E{H}_A\right)}^{\frac{1}{2}}$ applied transverse to the $z$ axis can parametrically excite phonons at half-frequency when the amplitude of the field exceeds a certain critical value.