Shape Dependence of the Thermodynamic Properties of Magnetic Systems

To investigate the dependence of the thermodynamic properties of a magnetic system in an applied field on the shape of the external boundary, we have studied the shape dependence of the free energy. Shape enters the thermodynamic properties through the dipolar sum, $\Phi =-\left(\frac{1}{N}\right)\Sigma j^{}\frac{(1-3\mathrm{}{cos}^2{\theta }_{\mathrm{ij}})}{{r_{\mathrm{ij}}}^3}$. The dependence of the free energy on shape has been ascertained for ellipsoids by using a linked-cluster diagrammatic expansion of the free energy. We have considered the full dipole-dipole interaction between the spins in materials for which the magnetization is parallel to the magnetic field, when applied along a principal axis for the sample, and have found that if we write a pseudo-free-energy in terms of the local field $L=H_0+\Phi M$, or the internal field $H_i=H_0-DM$, the resulting function is independent of the shape of the sample. Here $H_0$ represents the applied field, $M$ the magnetization, and $D$ is the demagnetization factor of the sample. Also, for a magnetic system in an applied field, the specific heat at constant local (internal) field is the same function of local (internal) field and temperature for all ellipsoidal shapes of the same material. The same is true of the susceptibility to within a demagnetization factor.