# Reversibility of the Magnetocaloric Effect in the Bean-Rodbell Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

^{28}m

^{−3}and 2.5 × 10

^{−11}Pa

^{−1}, respectively), while $m$ (or $J$) and $\eta $ were varied to study their influence. These fixed values are in the range of those for typical MC materials at room temperature (e.g., $n$ = 3.1 × 10

^{28}m

^{−3}and $k$ = 2.6 × 10

^{−11}Pa

^{−1}for Gd [32] and $n$ = 1.6 × 10

^{28}m

^{−3}and $k$ = 0.9 × 10

^{−11}Pa

^{−1}[29] for La(Fe,Si)

_{13}).

## 3. Results and Discussion

_{13}can be technically saturated around 0.25 T [34]). On the other hand, it can also be ascribed to the abrupt (instantaneous) transformation between FM to PM states for FOPT in the model (e.g., see Figure 1b), while coexistence among both phases during transformations is experimentally observed. This can be solved by including the kinetic process specific to each material [35], although this would require the inclusion of different additional models besides the Bean-Rodbell one (which is the main focus of this work). Continuing the discussion of Figure 2a, the reversible response remains much smaller than both heating and cooling cases up to a certain magnetic field (denoted by ${H}_{I}$) at which a significant increase of the response is observed. Above that magnetic field, the reversible response is the same as the heating one. This ${H}_{I}$ shows how important the magnetic field is in overcoming the limitations of the hysteresis (and improving the reversible response). It should be noted that the value of ${H}_{I}$ is smaller than ${H}_{C}$ (0.58 T vs. 1.35 T). Analyzing the effect of the magnetic field on the transition temperature (inset of Figure 2a), it can be observed that ${H}_{I}$ corresponds to the magnetic field at which the transition temperature of the cooling branch reaches the value at zero field of the transition temperature of the heating branch, i.e.,:

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Temperature dependence of $M$ for an atomic magnetic moment of 7${\mu}_{B}$ for 0 and 2 T for (

**a**) $\eta $ = 0.5 (SOPT) and (

**b**) $\eta $ = 1.5 (FOPT). Corresponding $\u2206{S}_{iso}$ (from previous magnetization data) for (

**c**) $\eta $ = 0.5 and (

**d**) $\eta $ = 1.5.

**Figure 2.**(

**a**) Magnetic field dependence of the maximum value of $\u2206{S}_{iso}$ and the evolution of the transition temperatures (inset) for $\eta $ = 1.5 and $m$ = 7${\mu}_{B}$; (

**b**) Temperature dependence of $\u2206{S}_{iso}$ for $\eta $ = 1.5 at 0.5 T and (

**c**) 0.65 T; (

**d**) Magnetic field dependence of maximum reversible $\u2206{S}_{iso}$ for different $\eta $ values and $m$ = 7${\mu}_{B}$.

**Figure 3.**(

**a**) Magnetic field dependence of the reversible $TEC$ for $\eta $ = 1.5 and $m$ = 7${\mu}_{B}$ using a $\u2206{T}_{lift}$ of 3 and 10 K; (

**b**) ${H}_{I}$ as a function of $\eta $ for different atomic magnetic moments.

**Figure 4.**(

**a**) Thermal hysteresis at zero field and (

**b**) magnetic field evolution of the transition temperature (cooling branch) as a function of $\eta $ and $m$.

**Figure 5.**(

**a**) $TE{C}_{rev}$ for 2 T using $\mathsf{\Delta}{T}_{lift}$ of 3 K as a function of $\eta $ and $m$; (

**b**) optimal $\eta $ (left-y and circular symbols) and maximum $TE{C}_{rev}$ (right-y and triangular symbols) for 2 T using $\mathsf{\Delta}{T}_{lift}$ of 3 K (solid symbols) and 10 K (hollow symbols) as a function of $m$.

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Moreno-Ramírez, L.M.; Franco, V.
Reversibility of the Magnetocaloric Effect in the Bean-Rodbell Model. *Magnetochemistry* **2021**, *7*, 60.
https://doi.org/10.3390/magnetochemistry7050060

**AMA Style**

Moreno-Ramírez LM, Franco V.
Reversibility of the Magnetocaloric Effect in the Bean-Rodbell Model. *Magnetochemistry*. 2021; 7(5):60.
https://doi.org/10.3390/magnetochemistry7050060

**Chicago/Turabian Style**

Moreno-Ramírez, Luis M., and Victorino Franco.
2021. "Reversibility of the Magnetocaloric Effect in the Bean-Rodbell Model" *Magnetochemistry* 7, no. 5: 60.
https://doi.org/10.3390/magnetochemistry7050060