Two-Magnon Scattering Amplitudes for a Pit of Arbitrary Shape in a Ferromagnet

The calculation by Sparks, Loudon, and Kittel of the amplitudes $F\left(\mathrm{k}\right)$ in the two-magnon scattering $\mathcal{H}=\Sigma \left[F\right(\mathrm{k})b_{\mathrm{k}}{b_0}^{†}+\mathrm{H}.\mathrm{c}.\mathrm{}]$ induced by a spherical pit in an infinite ferromagnetic crystal is extended to pits of arbitrary shape. By expressing the energy in terms of the interaction of magnetostatic charges, two terms of different types are identified. One, which is linear in pit distribution, is solved exactly and explicitly for any collection of pits of arbitrary shape and size. The second, which is nonlinear in pit distribution, is solved explicitly only for a general ellipsoidal pit. The approach presented here can also be used to solve the problem of scattering from modes other than the uniform precession.