Title: The role of Debye temperature in achieving large adiabatic temperature changes at cryogenic temperatures: a case study on $Pr_2In$

Abstract: The excellent magnetic entropy change ($\Delta S_T$) in the temperature range of 20 $\sim$ 77 K due to the first-order phase transition makes $Pr_2In$ an intriguing candidate for magnetocaloric hydrogen liquefaction. As an equally important magnetocaloric parameter, the adiabatic temperature change ($\Delta T_{ad}$) of $Pr_2In$ associated with the first-order phase transition has not yet been reported. In this work, the $\Delta T_{ad}$ of $Pr_2In$ is obtained from heat capacity measurements: 2 K in fields of 2 T and 4.3 K in fields of 5 T. While demonstrating a $\Delta T_{ad}$ that is not as impressive as its remarkable $\Delta S_T$, $Pr_2In$ exhibits an unusual low Debye temperature ($T_D$) of around 110 K. Based on these two observations, an approach that combines the mean-field and Debye models is developed to study the correlation between $\Delta T_{ad}$ and $T_D$. The role of $T_D$ in achieving large $\Delta T_{ad}$ is revealed: materials with higher $T_D$ tend to exhibit larger $\Delta T_{ad}$, particularly in the cryogenic temperature range. This discovery explains the absence of an outstanding $\Delta T_{ad}$ in $Pr_2In$ and can serve as a tool for designing or searching materials with both a large $\Delta S_T$ and a $\Delta T_{ad}$.