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 g^' 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 g^' 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 (2{H₄H₀) }^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.