We have studied theoretically the specific heat of a large number of nonfrustrated magnetic structures described by the Heisenberg model for systems with total angular momentum $J=7/2$, corresponding to the $4f^7$ configuration of ${\mathrm{Gd}}^{+3}$ and ${\mathrm{Eu}}^{+2}$. 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 governs the extent of thermal 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 gadolinium compounds and two europium 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.

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• Magnetic anisotropy
• Specific heat
• Magnetic systems
• Rare-earth ions
• Quantum Monte Carlo