Internal energy versus moment for ferromagnets and pyroelectrics

Since the internal energy $E$ is a monotonically increasing function of the temperature $T$, the moment $M\left(T\right)\mathrm{}$ can be expressed as a function of internal energy $E$. The corresponding form of $E\left(M\right)\mathrm{}$ is focused on in this paper. For ferromagnets, the two-dimensional Ising model is used to obtain the exact form of $E\left(M\right)\mathrm{}$ and this is discussed both near $T=0\mathrm{}$ and near the critical temperature $T_c$. In three dimensions, we are only able to discuss the forms of $E\left(M\right)\mathrm{}$ near $T=0\mathrm{}$ and $T=T_c$. For low temperatures, spin-wave theory readily yields the result that, for insulating ferromagnets, $\Delta E=E-E\left(0\right)\mathrm{}$ is proportional to ${\left(\Delta M\right)}^{\frac{5}{3}}$ with $\Delta M=M\left(0\right)-M\left(T\right)\mathrm{}$ while for the metallic case $\Delta E\propto {\left(\Delta M\right)}^{\frac{4}{3}}$. Though the theory is much less complete than for ferromagnets, some results are very briefly discussed for pyroelectrics.