Generalised Longitudinal Susceptibility for Magnetic Monopoles in Spin Ice

The generalised longitudinal susceptibility $\chi({\bf q}, \omega)$ affords a sensitive measure of the spatial and temporal correlations of magnetic monopoles in spin ice. Starting with the monopole model, a mean field expression for $\chi({\bf q}, \omega)$ is derived as well as expressions for the mean square longitudinal field and induction at a point. Monopole motion is shown to be strongly correlated, and both spatial and temporal correlations are controlled by the dimensionless monopole density $x$ which defines the ratio of the magnetization relaxation rate and the monopole hop rate. Thermal effects and spin lattice relaxation are also considered. The derived equations are applicable in the temperature range where the Wien effect for magnetic monopoles is negligible. They are discussed in the context of existing theories of spin ice and the following experimental techniques: dc and ac-magnetization, neutron scattering, neutron spin echo, and longitudinal and transverse field $\mu$SR. The monopole theory is found to unify diverse experimental results, but several discrepancies between theory and experiment are identified. One of these, concerning the neutron scattering line shape, is explained by means of a phenomenological modification to the theory.