The role of Debye temperature in achieving large adiabatic temperature changes at cryogenic temperatures: a case study on $P{r}_{2}In$

The excellent magnetic entropy change ($\mathrm{\Delta }{S}_T$) in the temperature range of 20 $\sim$ 77 K due to the first-order phase transition makes $P{r}_{2}In$ an intriguing candidate for magnetocaloric hydrogen liquefaction. As an equally important magnetocaloric parameter, the adiabatic temperature change ($\mathrm{\Delta }{T}_{ad}$) of $P{r}_{2}In$ associated with the first-order phase transition has not yet been reported. In this work, the $\mathrm{\Delta }{T}_{ad}$ of $P{r}_{2}In$ 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 $\mathrm{\Delta }{T}_{ad}$ that is not as impressive as its remarkable $\mathrm{\Delta }{S}_T$, $P{r}_{2}In$ 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 $\mathrm{\Delta }{T}_{ad}$ and $T_D$. The role of $T_D$ in achieving large $\mathrm{\Delta }{T}_{ad}$ is revealed: materials with higher $T_D$ tend to exhibit larger $\mathrm{\Delta }{T}_{ad}$, particularly in the cryogenic temperature range. This discovery explains the absence of an outstanding $\mathrm{\Delta }{T}_{ad}$ in $P{r}_{2}In$ and can serve as a tool for designing or searching materials with both a large $\mathrm{\Delta }{S}_T$ and a $\mathrm{\Delta }{T}_{ad}$.