Non-trivial critical behavior at the magnetic transitions: A case study of Sm$_7$Pd$_3$

TL;DR

Sm$_7$Pd$_3$ experiences a second-order PM–FM magnetoelastic transition near $T_{ m c} \approx 173$ K. The authors apply standard convergence (SCP) and average normalized slope (ANS) analyses to extract critical exponents, but find that $β$ and $γ$ diverge near $T_{ m c}$ and do not fit any known universality class, indicating strong spin–lattice coupling effects. Kouvel–Fisher scaling yields $β=0.325(5)$ and $γ=1.068(5)$, while iterative MAP analysis reveals a drift in exponents, highlighting two-sided criticality with distinct behavior above and below $T_{ m c}$; ANS further requires independent extraction of $β$ and $γ$ from $T \le T_{ m c}$ and $T \ge T_{ m c}$ data, respectively. The study advocates revised theoretical frameworks that incorporate magnetoelastic coupling and provides methodological guidance for robust critical-behavior analysis in systems with strong spin–lattice interactions.

Abstract

We present a comprehensive analysis of the critical behavior of Sm$_7$Pd$_3$ in the vicinity of its second-order magnetoelastic transition at $T_ {\rm c} = 173$ K. The critical exponents (CEs) $β$ and $γ$, determined using both the standard convergence procedure and the average normalized slope (ANS) method, diverge at $T_{\rm c}$: a characteristic typically associated with first-order transitions. Notably, none of the established universality classes satisfactorily describe the critical behavior of Sm$_7$Pd$_3$, and we discuss the possible origins of this deviation in the context of the strong spin-lattice coupling intrinsic to the sample. We emphasize the importance of accurately selecting the critical temperature and magnetic field ranges to ensure robust critical behavior analysis and propose a quantitative approach to assess the reliability of the extracted CEs. Additionally, we demonstrate that in the ANS method, the critical exponents $β$ and $γ$ should be calculated separately using data for $T \leqslant T_{\rm c}$ and $T \geqslant T_{\rm c}$, respectively. Our findings underscore the need for a revised theoretical framework to accurately describe second-order magnetoelastic transitions.

Figures (5)

• Figure 1: (a) Temperature-dependent magnetization of Sm$_7$Pd$_3$ measured in the field-cooled warming (FCW) mode at 100 Oe. (b) Zero field cooled (ZFC) isothermal magnetization ($M$--$H$) curve at 20 K; inset shows the ZFC $M$--$H$ at 150 K. (c) Virgin magnetization isotherms recorded at various temperatures across the magnetic transition. (d) Temperature-dependent magnetization at different magnetic fields derived from the virgin isotherms. (e) Temperature derivative of magnetization at different fields; inset shows the shift in $T_{\rm c}$ as a function of external field, where the solid black curve represents the power-law fit. (f) Temperature-dependent magnetic entropy change ($\Delta S_{\rm M}$) at various magnetic fields; inset shows the maximum $\Delta S_{\rm M}$ as a function of field on a log--log scale. (g) Local field exponent $n$ as a function of temperature at different fields; inset shows the field dependence of $n$ at $T_{\rm c}$.
• Figure 2: (a, b) Normalized slopes (NS) and $y$-intercepts of the modified Arrott plots (MAPs) for different universality classes. (c--e) Variation of the critical exponents $\beta$, $\gamma$, and the transition temperature with the number of iterations, starting from different universality classes.
• Figure 3: (a) The critical isotherm at $T_{\rm c}$ = 173 K on a log-log scale. The black solid line represents the straight fit for $H=20-70$ kOe. The inset shows the dependence of $\delta$ on the minimum value of the magnetic field ($H_{\rm min}$) used to fit the critical isotherm for $H_{\rm max}$ = 70 kOe. (b) The average normalized slope (NS$_{\rm avg}$) as a function of $\beta$ for different $T_{cri}$, both above and below $T_{\rm c}$ = 173 K. The horizontal dashed line represents NS$_{\rm avg}$ = 1. The inset shows the dependence of $\beta$ on the $T_{cri}$ for $T \leqslant T_{\rm c}$ and $T \geqslant T_{\rm c}$. (c) The critical exponent $\beta$ as a function of $T_{cri}$ for different ranges of magnetic fields. The dashed curves are the rough extrapolation of data for $H=60-70$ kOe down to $T_{cri}$ = 0 K, both above and below $T_{\rm c}$.
• Figure 4: (a) Modified Arrott plot constructed using the extrapolated values of the critical exponents. The black circles and red lines show the experimental data points and the linear fit for $H \geqslant$ 20 kOe, respectively, while the blue line represents the extrapolation of the linear fit at $\approx T_{\rm c}$ down to $H=0$. (b) Temperature-dependent spontaneous magnetization $M_{\rm SP}$ (on the left axis) and inverse initial susceptibility $\chi_0^{-1}$ (on the right axis), where the black solid curves represent the best fit using Eqs. (\ref{['c1']}) and (\ref{['c2']}), respectively. (c) Temperature-dependent $M_{\rm SP}(T)/[{\rm d}M_{\rm SP}(T)/{\rm d}T]$ (on the left axis) and $\chi_0^{-1}/[({\rm d}\chi_0^{-1})/{\rm d}T]$ (on the right axis) plot, where the black solid lines represent the linear fit using Eqs. (\ref{['KF1']}) and (\ref{['KF2']}), respectively. (d) The reduced magnetization, $m$ vs. reduced field $h$, and (e) $m^2$ vs. $h/m$ plot, where the insets show their respective log-log plots. (f) $MH^{-1/\delta}$ vs. $\epsilon H^{-1/(\beta \delta)}$ plot from 165-180 K range, where the vertical dashed line locates the $T_{\rm c}$.
• Figure 5: (a) The critical exponent $\beta$ as a function of $T_{cri}$ above and below $T_{\rm c}$ for $H = 60-70$ kOe. The shaded curves are the guide to the eyes. (b) The dependence of critical exponents $\beta$ (on the left axis) and $\gamma$ (on the right axis) on the number of iterations. The inset shows the variation in $T_{\rm c}$ with $n$.