LiHoF4 : Cuboidal demagnetizing factor in an Ising ferromagnet

The demagnetizing factor can have an important effect on physical properties, yet its role in determining the behavior of nonellipsoidal samples remains to be fully explored. We present a detailed study of the role of spin symmetry in determining the demagnetizing factor of cuboids, focusing, as a model example, on the Ising dipolar ferromagnet ${\mathrm{LiHoF}}₄$. We distinguish two different functions: the demagnetizing factor as a function of intrinsic susceptibility $N(\ensuremath{\chi})$ and the demagnetizing factor as a function of temperature $N(T)$. For a given nonellipsoidal sample, the function $N(\ensuremath{\chi})$ depends only on dipolar terms in the spin Hamiltonian, but apart from in the limits $\ensuremath{\chi}\ensuremath{\rightarrow}0$ and $\ensuremath{\chi}\ensuremath{\rightarrow}\ensuremath{\infty}$, it is a different function for different spin symmetries. The function $N(T)$ is less universal, depending on exchange terms and other details of the spin Hamiltonian. We apply a recent theory to calculate these functions for spherical and cuboidal samples of ${\mathrm{LiHoF}}₄$. The theoretical results are compared with $N(\ensuremath{\chi})$ and $N(T)$ derived from experimental measurements of the magnetic susceptibility of corresponding samples of ${\mathrm{LiHoF}}₄$, both above and below its ferromagnetic transition at ${T}_{\mathrm{c}}=1.53$ K. Close agreement between theory and experiment is demonstrated, showing that the intrinsic susceptibility of ${\mathrm{LiHoF}}₄$ and other strongly magnetic systems can be accurately estimated from measurements on cuboidal samples. Our results further show that for cuboids, and implicitly for any sample shape, $N(\ensuremath{\chi})$ below the ordering transition takes the value $N(\ensuremath{\infty})$. This confirms and extends the scope of earlier observations that the intrinsic susceptibility of ferromagnets remains divergent below the transition, in contradiction to the implications of broken symmetry. We discuss the topological and microscopic origins of this result.