Specific heat of Gd$^{3+}$ and Eu$^{2+}$-based magnetic compounds

We have studied theoretically the specific heat of a large number of non-frustrated magnetic structures described by the Heisenberg model for systems with total angular momentum $J=7/2$, corresponding to the 4f$^7$ configuration of Gd$^{+3}$ and Eu$^{+2}$. For a given critical temperature (determined by the magnitude of the exchange interactions), we find that, to a high degree of accuracy, the specific heat is governed by two primary parameters: the effective number of neighbors $z$, which dictates the extent of spatial and quantum fluctuations, and the axial anisotropy $K$. The universality of $z$ (its ability to describe specific heat across diverse lattices) holds robustly for systems where exchange interactions do not strongly increase with distance and in the absence of frustration. Otherwise, deviations from universality emerge. Using these two parameters we fit the specific heat of four Gd compounds and two Eu compounds, achieving a remarkable agreement. The present approach enables the extraction of magnetic interaction parameters not accessible through mean-field theory, offering a powerful tool for interpreting specific heat data in 4f$^7$ systems.