🔬 applied physics

Magnetic field induced modification of a first-order ferromagnetic transition in Eu2In

This study demonstrates that while magnetic fields induce a two-step transition process in Eu $_2$ In that alters its magnetocaloric response, the ferromagnetic-paramagnetic transition remains fundamentally first-order up to at least 70 kOe, as confirmed by specific heat measurements and the absence of local structural changes.

Original authors: Ajay Kumar, Anis Biswas, Trevor A. Tyson, Daniel Haskel, Christopher J. Pollock, Yaroslav Mudryk

Published 2026-03-31

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CC BY 4.0

Original authors: Ajay Kumar, Anis Biswas, Trevor A. Tyson, Daniel Haskel, Christopher J. Pollock, Yaroslav Mudryk

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Magnetic "Snap" That Won't Let Go

Imagine you have a magical material called Eu2In. At a specific temperature (about -218°C or 55 Kelvin), this material undergoes a dramatic personality change. It snaps from being a chaotic, disorganized crowd of atoms (paramagnetic) into a perfectly organized, marching army (ferromagnetic).

In physics, we call this a First-Order Phase Transition. Think of it like water freezing into ice. It happens suddenly, with a sharp "crack," and there is a bit of a "hangover" (hysteresis) where the material doesn't immediately snap back when you warm it up. This "snap" is actually very useful for magnetic refrigeration—a super-efficient way to cool things down without using harmful gases.

The scientists in this paper wanted to see what happens if you squeeze this material with a strong magnetic field. They had a hunch, based on computer simulations, that if you pushed hard enough (around 25,000 Oersteds), the "snap" would soften. They thought the material would stop acting like ice freezing and start acting more like water slowly turning into slush (a "second-order" transition), which is smoother and less "jumpy."

The Experiment: Testing the Theory

The team, led by Ajay Kumar, put Eu2In through a rigorous workout:

Magnetization: They measured how much the material wanted to align with a magnetic field as they heated and cooled it.
Specific Heat: They measured how much energy the material absorbed or released during the transition (like checking how much heat is needed to melt ice).
X-ray Eyes (EXAFS): They used powerful X-rays to look at the atomic structure, checking if the atoms were physically moving or rearranging themselves when the magnetic field was applied.

The Surprise: The "Two-Step" Dance

Here is where the story gets interesting. The computer models predicted that at 25,000 Oersteds, the transition would change its nature from a "snap" to a "smooth slide."

The Reality: The transition did not change its nature. It remained a sharp "snap" (first-order) all the way up to the highest field they tested (70,000 Oersteds).

However, the transition did change its behavior in a weird way.

The "Doublet" Mystery: When they looked at the heat data, the single sharp peak (the "snap") split into two distinct peaks as the magnetic field increased.
The Analogy: Imagine a crowd of people trying to leave a stadium through one gate.
- Zero Field: Everyone rushes out at once. Crash! (One big peak).
- High Field: The crowd splits into two groups. Group A leaves first, then Group B leaves a moment later. You see two smaller rushes instead of one big one.

The scientists found that one of these "rushes" was caused by the magnetic alignment (the main event), but the other "rush" seemed to be caused by the atoms in the crystal lattice getting slightly nervous or unstable, even though they didn't actually move far enough to be seen by the X-ray cameras.

The Detective Work: Why Didn't the Structure Change?

The team used X-ray absorption spectroscopy (EXAFS) to look at the local neighborhood of the Europium atoms. They were looking for evidence of the atoms stretching, twisting, or changing their positions (a structural distortion).

The Result: Nothing. The atoms looked exactly the same, whether the magnetic field was weak or super strong.

This led to a fascinating conclusion: The "splitting" of the transition wasn't because the atoms physically moved apart. Instead, the magnetic field forced the material to go through a two-step process.

Step 1: The magnetic spins align.
Step 2: The crystal lattice (the skeleton of the material) reacts to this new magnetic state, causing a tiny, subtle instability that shows up as a second heat peak.

It's like a dance where the partners (magnetism and structure) are so tightly linked that when one steps forward, the other wants to step back, but they are stuck in a rhythm that forces them to do it in two separate beats instead of one.

The Conclusion: A "Softened" Snap, Not a Smooth Slide

So, did the magnetic field turn the "snap" into a "smooth slide"? No.

The paper concludes that the transition remains a sharp, first-order event (a "snap") even at very high fields. However, the magnetic field does modify how that snap happens. It breaks the single, violent "crack" into two smaller, closely spaced "cracks."

Why does this matter?
For engineers building magnetic refrigerators, this is great news. It means the material is robust. It keeps its powerful "cooling punch" (large magnetic entropy change) even under strong magnetic fields. While the transition didn't become "smooth" as the computers predicted, the fact that it splits into a two-step process suggests the material is very complex and responsive, offering new ways to tune it for better cooling efficiency without losing its power.

In a nutshell: The scientists expected the material to change its personality from "jumpy" to "calm" under pressure. Instead, they found it stayed "jumpy," but it learned to do its jump in two distinct, rhythmic steps.

1. Problem Statement

First-order magnetic phase transitions (FOMPTs) are highly desirable for magnetocaloric applications due to their large magnetic entropy changes ( $\Delta S_M$ ). However, they are often accompanied by thermal hysteresis, which leads to energy dissipation and reduced efficiency in refrigeration cycles.

The Specific Context: The compound $\text{Eu}_2\text{In}$ exhibits a giant magnetocaloric effect (GMCE) with negligible thermal hysteresis at its first-order ferromagnetic (FM) to paramagnetic (PM) transition ( $T_c \approx 55$ K).
The Theoretical Conflict: Previous theoretical calculations (Alho et al., 2020) predicted that applying a magnetic field of approximately $H_c \approx 25$ kOe would induce a crossover in $\text{Eu}_2\text{In}$ , changing the transition order from first-order to second-order. Such a change would theoretically reduce the latent heat and alter the magnetocaloric performance.
The Knowledge Gap: No experimental investigation had yet verified this predicted field-induced crossover or explained the anomalies observed in the magnetocaloric response near this critical field.

2. Methodology

The authors conducted a comprehensive, multi-probe study on polycrystalline $\text{Eu}_2\text{In}$ to investigate the nature of the phase transition under varying magnetic fields (0 to 70 kOe).

Synthesis & Structural Characterization:
- Synthesized via arc melting and annealing.
- Confirmed Co $_2$ Si-type orthorhombic structure (space group $Pnma$) via Room-Temperature X-ray Diffraction (XRD).
Magnetization Measurements:
- Performed using a SQUID magnetometer.
- Recorded isotherms and temperature-dependent susceptibility under Zero-Field Cooled (ZFC), Field Cooled Warming (FCW), and Field Cooled Cooling (FCC) protocols.
- Analyzed critical behavior using Modified Arrott Plots (MAPs) and the Banerjee criterion (slope of $H/M$ vs. $M^2$ ) to determine transition order.
Specific Heat ( $C_P$ ) Measurements:
- Utilized both standard $2\tau$ relaxation and long heat-pulse methods to accurately capture the sharp, $\delta$ -like anomalies associated with first-order transitions.
- Measured under various magnetic fields to track the evolution of the transition peak.
Local Structure Analysis (XAS/EXAFS):
- Performed Magnetic-field-dependent X-ray Absorption Spectroscopy (XAS) at the Eu $L_3$ edge at the Cornell High Energy Synchrotron Source (CHESS).
- Analyzed Extended X-ray Absorption Fine Structure (EXAFS) to detect any field-induced changes in local coordination (Eu-In and Eu-Eu bond lengths) and Debye-Waller factors.

3. Key Results

A. Magnetocaloric Anomalies near 25 kOe

Entropy Change ( $\Delta S_M$ ): The maximum magnetic entropy change ( $\Delta S_M^{max}$ ) deviates from the expected power-law behavior ( $\Delta S_M^{max} \propto H^n$ ) for fields below 25 kOe but follows a linear power-law fit ( $n \approx 0.24$ ) for $H > 25$ kOe.
Local Exponent ( $n$ ): The local field exponent $n$ (derived from $\Delta S_M$ ) peaks at $\sim 6$ near 25 kOe and then decreases, suggesting a weakening of the first-order character.
Universal Master Curve (UMC): The scaling behavior of normalized entropy shows a distinct change in dispersion around 25 kOe, further hinting at a modification in the transition mechanism.

B. Persistence of First-Order Character

Despite the anomalies suggesting a crossover, rigorous analysis confirms the transition remains first-order up to the maximum tested field of 70 kOe:

Modified Arrott Plots: The plots for various universality classes (Mean Field, 3D Heisenberg, etc.) exhibit pronounced curvature in the high-field region (50–70 kOe) rather than the straight lines expected for second-order transitions.
Banerjee Criterion: The slopes of the MAPs remain negative (indicative of first-order) in the relevant temperature range, confirming the robustness of the first-order nature.

C. Specific Heat Splitting (The "Doublet")

Field-Induced Splitting: While the zero-field specific heat shows a single sharp peak, increasing the magnetic field causes this peak to split into a well-defined doublet (Low-Temperature and High-Temperature peaks).
Decoupling: The separation between the two peaks increases rapidly up to $\sim 25$ kOe and then saturates.
Origin:
- The High-Temperature (HT) peak correlates directly with the magnetic transition temperature ( $T_c$ ) derived from magnetization.
- The Low-Temperature (LT) peak shows no corresponding feature in magnetization data and its magnitude is nearly field-independent, suggesting a non-magnetic (likely lattice-related) origin.
- However, the systematic shift of both peaks to higher temperatures with increasing field implies a finite magnetoelastic coupling, ruling out a purely structural transition independent of magnetism.

D. Local Structure Stability (EXAFS)

No Local Distortion: EXAFS measurements revealed no detectable changes in Eu-In or Eu-Eu bond lengths, coordination numbers, or Debye-Waller factors up to 70 kOe.
Valence Stability: XANES spectra confirmed that the Eu valence state remains divalent ( $\text{Eu}^{2+}$ ) and invariant under magnetic fields.
Implication: The splitting of the specific heat peak is not caused by a measurable local structural distortion or a change in the electronic valence state within the experimental resolution.

4. Conclusion and Significance

Conclusion:
The study concludes that the theoretical prediction of a field-induced crossover from a first-order to a second-order transition in $\text{Eu}_2\text{In}$ at 25 kOe is incorrect. The transition retains its first-order character up to at least 70 kOe.
Instead, the observed anomalies (change in scaling exponents, UMC dispersion, and specific heat splitting) are attributed to a field-driven two-step transition process. In this scenario, the external magnetic field induces a subtle decoupling of the magnetic and lattice degrees of freedom, causing the single sharp transition to split into two closely spaced events (one magnetic, one likely magneto-elastic) without a fundamental change in the thermodynamic order of the transition.

Significance:

Refinement of Theory: It corrects the theoretical understanding of $\text{Eu}_2\text{In}$ , demonstrating that the "crossover" is actually a modification of the transition pathway (one-step to two-step) rather than a change in order (first to second).
Magnetocaloric Optimization: Understanding that the transition remains first-order but splits under high fields is crucial for optimizing $\text{Eu}_2\text{In}$ for magnetic refrigeration. It suggests that while the sharpness of the transition is modified, the large latent heat (and thus cooling capacity) is preserved, provided the two-step nature is accounted for in device design.
Methodological Insight: The work highlights the necessity of combining macroscopic thermodynamic measurements (specific heat, magnetization) with local structural probes (EXAFS) to distinguish between true phase-order crossovers and complex multi-step transition mechanisms in magnetocaloric materials.