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

D. J. Garcia, J. G. Sereni, A. A. Aligia

TL;DR

The paper develops a two-parameter description of the magnetic specific heat for non-frustrated 4f$^7$ systems (Gd$^{3+}$ and Eu$^{2+}$) using a Heisenberg model with a small axial anisotropy. Through extensive quantum Monte Carlo simulations, it shows that the specific heat is largely governed by the effective neighbor number $z$ and the anisotropy $K$, with a remarkable universality of $C(T)$ across lattices when $z$ is held fixed. This approach enables extraction of exchange-interaction information beyond mean-field theory by fitting $C(T)$ and provides a practical method to constrain underlying couplings from experimental data. The framework correctly captures Eu anisotropy effects and can guide interpretation of 4f$^7$ magnetic materials, while noting limitations in frustrated or long-range-dominated systems.

Abstract

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.

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

TL;DR

The paper develops a two-parameter description of the magnetic specific heat for non-frustrated 4f systems (Gd and Eu) using a Heisenberg model with a small axial anisotropy. Through extensive quantum Monte Carlo simulations, it shows that the specific heat is largely governed by the effective neighbor number and the anisotropy , with a remarkable universality of across lattices when is held fixed. This approach enables extraction of exchange-interaction information beyond mean-field theory by fitting and provides a practical method to constrain underlying couplings from experimental data. The framework correctly captures Eu anisotropy effects and can guide interpretation of 4f magnetic materials, while noting limitations in frustrated or long-range-dominated systems.

Abstract

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 , corresponding to the 4f configuration of Gd and Eu. 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 , which dictates the extent of spatial and quantum fluctuations, and the axial anisotropy . The universality of (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 systems.

Paper Structure

This paper contains 14 sections, 17 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The two main parameters
  3. The effective number of neighbors
  4. Axial anisotropy in 4f$^7$ ions
  5. Methods
  6. Role of the effective number of neighbors
  7. Effect of anisotropy
  8. Mean-field Approximation
  9. Beyond Mean-field
  10. Comparison of Different Structures with the Same Effective Number of Neighbors
  11. Getting information beyond mean field
  12. Fits of experimental data
  13. Summary
  14. Appendix

Figures (8)

  • Figure 1: Critical temperature $T_C$ in units of the nearest‑neighbour exchange as a function of effective number of neighbors $z$ for several ferromagnetic structures. Finite-size results ($N=10,20,30$) and extrapolated values are shown, with the mean-field value $T_C^\mathrm{MF}$ indicated by the straight line.
  • Figure 2: Specific heat $C/T$ versus reduced temperature $T/T_C$ for several ferromagnetic structures with different effective number of neighbors $z$. Top panel ($z \leq 12$) and bottom panel ($z \geq 12$) show distinct ranges. The curves for SC, FCC, and HCP structures coincide at $z=18$, while SC, BCC, and EuPdSn$_2$ coincide at $z=26$, explaining the single curves shown for each set. LHc(4) denotes the layered honeycomb lattice with $z=4$. The bottom inset shows finite-size effects for HCP(12) with $N=10$ (black), $N=20$ (red), and $N=30$ (green) in the range $0.5 < T/T_C < 1.5$.
  • Figure 3: Specific heat $C/T$ versus $T/T_C$ for the mean-field approximation with varying axial anisotropy $K$ (expressed in units of $k_B T_C$).
  • Figure 4: Specific heat $C/T$ versus $T/T_C$ for the EuPdSn$_2$(16) structure with varying axial anisotropy $K$ ($J_\delta = -1$). The critical temperatures for $K = -0.6, -0.3, 0, 0.3, 0.6$ are approximately $78, 74, 68.5, 71.5, 73$ respectively. To compare with Fig. \ref{['cstmf']} the ratio $K/{k_B T_c}$ for these cases are $-.0077$, $-0.0040$,$0$,$0.0042$ and $0.0082$ respectively.
  • Figure 5: Specific heat $C/T$ versus $T/T_C$ for diamond and FCC lattices with different values of effective number of neighbors $z$ [defined by Eq. (\ref{['Rdefinicion']})], with $J_1 = -1$. The critical temperatures $T_C$ and $z$ values are tabulated in Table \ref{['Table:CambioJs']}. Top panel: Diamond lattice with varying interactions with $|J_2| \leq |J_1|$ ($z = 4, 6, 8, 10, 12, 14, 16$ from bottom to top). Top panel inset: $z$ as a function of $J_2$ for diamond. Bottom panel: same results for FCC lattices with varying interactions with $|J_2| \geq |J_1|$ ($z = 8, 10, 12, 14, 16, 18$). Bottom panel inset: $z$ as a function of $J_2$ for FCC.
  • ...and 3 more figures