Designing magnetocaloric materials for hydrogen liquefaction with light rare-earth Laves phases

Magnetocaloric hydrogen liquefaction could be a"game-changer"for liquid hydrogen industry. Although heavy rare-earth-based magnetocaloric materials show strong magnetocaloric effects in the temperature range required by hydrogen liquefaction (77 ~ 20 K), the high resource criticality of the heavy rare-earth elements is a major obstacle for upscaling this emerging liquefaction technology. In contrast, the higher abundances of the light rare-earth elements make their alloys highly appealing for magnetocaloric hydrogen liquefaction. Via a mean-field approach, it is demonstrated that tuning the Curie temperature ($T_C$) of an idealized light rare-earth-based magnetocaloric material towards lower cryogenic temperatures leads to larger maximum magnetic and adiabatic temperature changes ($\Delta S_T$ and $\Delta T_{ad}$). Especially in the vicinity of the condensation point of hydrogen (20 K), $\Delta S_T$ and $\Delta T_{ad}$ of the optimized light rare-earth-based material are predicted to show significantly large values. Following the mean-field approach and taking the chemical and physical similarities of the light rare-earth elements into consideration, a method of designing light rare-earth intermetallic compounds for hydrogen liquefaction is proposed: tunning $T_C$ of a rare-earth alloy to approach 20 K by mixing light rare-earth elements with different de Gennes factors. By mixing Nd and Pr in Laves phase $(Nd,Pr)Al_2$, and Pr and Ce in Laves phase $(Pr,Ce)Al_2$, a fully light rare-earth intermetallic series with large magnetocaloric effects covering the temperature range required by hydrogen liquefaction is developed, demonstrating a competitive maximum effect compared to the heavy rare-earth compound $DyAl_2$.


Introduction
Discovered in 1917 by Weiss and Picard, magnetocaloric effect is a cooling or warming effect of a magnetic material being exposed to a magnetic field [1]. Soon after its discovery, magnetic cooling has been successfully applied to attaining extremely low temperature [2,3]. In 1949, the Nobel prize in chemistry was awarded to Giauque, who developed a magnetic refrigeration device to approach absolute zero [4].
If pre-cooled by liquid nitrogen, the temperature range required by magnetocaloric hydrogen liquefaction is from 77 (condensation point of nitrogen) to 20 K (condensation point of hydrogen). For the success of a practical application of magnetocaloric hydrogen liquefaction on an industrial scale, affordable magnetocaloric materials with large isothermal magnetic entropy and adiabatic temperature changes (Δ and Δ ) in the target temperature regime under affordable magnetic fields are needed [33][34][35][36][37][38]. In this work, we focus on the criticality of raw elements, the two physical quantities Δ and Δ in temperature range of 77 ~ 20 K under magnetic field changes such as 2 T, which can be realized by Nd-Fe-B permanent magnets [5], or 5 T, which can be generated by commercial superconducting magnets [32].
Rare-earth based magnetocaloric materials are one big family of the magnetocaloric materials for hydrogen liquefaction [37,39,40]. Lanthanide rare-earth elements can be divided into two major groups: light rare-earth elements (La, Ce, Pr, Nd, Sm) and heavy rare-earth elements (Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu). Putting aside the criticality of the raw materials for a moment, heavy rare-earth based magnetocaloric materials would be strong contenders for magnetocaloric hydrogen liquefaction in the context of performance. Materials such as HoB2 [15,41,42], ErCo2 [43,44], DyAl2 [45], and ErAl2 [45,46] show large Δ and Δ due to the large magnetic moments of the heavy rare-earth ions, and they are often proposed to be used in an active magnetic regenerator for hydrogen liquefaction. Because of the excellent magnetocaloric properties, heavy rare-earth based materials are intensively studied.
In contrast, light rare-earth based magnetocaloric materials for hydrogen liquefaction are less investigated since their magnetocaloric effects are usually weaker due to their smaller magnetic moments [45]. Figure 1 (a) shows the theoretical effective magnetic moment and the de Gennes factor ( ) of the rare-earth ions [47]. The rare-earth ions are divided into three categories, namely the light rareearth ions, Eu 3+ , and the heavy rare-earth ions. Pr 3+ has the highest theoretical of 3.58 among the light rare-earth ions, whereas the heavy rare-earth ions from Gd 3+ to Tm 3+ have a theoretical larger than 7.5 . However, the criticality of the raw materials cannot be ignored for a viable industrial-scale application. The consumption of H2 in EU is predicted to reach 2250 TWh/year by 2050 according to Hydrogen Roadmap Europe [48]. If about one-third of H2 needs to be transported and stored in its liquid state, one would require about 13,000 small-scale liquefaction plants with a production capacity of 5 tons per day. Providing a potential hydrogen liquefier using   [47]. (b) relative abundance of Ni, Co, Cu, and the rare-earth elements which are categorized into non-heavy rare-earth elements with zero magnetic moments (Sc, Y, La), light rare-earth elements with non-zero magnetic moments (Ce, Pr, Nd, Sm), Eu, and heavy rare-earth elements with non-zero magnetic moments (Gd, Tb, Dy, Ho, Er, Tm, Yb), and zero magnetic moments (Lu). Data are taken from Ref. [53]. (c) Prices of the rare-earth oxides (Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy) in year 2018 and 2021. Data are taken from Ref. [54,55]. magnetocaloric material HoAl2 as the refrigerant for the final cooling state operating at 20 K at a frequency of 1 Hz in fields of 7 T, about 1 ton of holmium would be needed [49]. This means a total of 13,000 tons for the EU alone, and the total holmium production is a measly 10 tons per year [50].
Heavy rare-earth elements belong to the highly critical elements [51,52]. One contribution to the high criticalities is their poor abundances on the earth's crust. Figure 1 (b) shows the relative abundances of Ni, Co, Cu, and the rare-earth elements on the earth's crust [53]. Heavy rare-earth elements such as Tb, Ho, Tm, and Lu are not abundant, whereas light rare-earth metal Ce is even more abundant than Cu, and light rare-earth elements La and Nd, and Y are more abundant than Co. The total abundances of the heavy rare-earth elements combined are not as rich as that of Nd alone. Figure 1 (c) plots the prices (always volatile) of the rare-earth oxides in year 2018 and 2021 for Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, and Dy [54,55]. The prices of the heavy rare-earth oxides, namely for Tb, Dy, are over 400 $/kg in 2021, whereas the prices of the Pr and Nd oxides are around 50 $/kg, and the prices of Y, La, Ce, Sm oxides are just several dollars per kg. Eu and Gd oxides are also relatively cheap in comparison with Tb and Dy oxides. This is because they are not as largely used in industry as Tb and Dy.
It needs to be emphasized that criticality is much more than just simple geological abundances. Factors such as mining, beneficiation, hazardous by-products, separation (and their social and ecological consequences along this value chain), geopolitics, trade restrictions and monopolistic supply in terms of demand vs. supply, need to be understood and quantified in terms of LCA (Life-Cycle-Analysis) and LCC (Life-Cycle-Costing) [56]. Nevertheless, the discussion above points out that the high criticality of the heavy rare-earth elements questions the feasibility of using heavy rare-earth based magnetocaloric materials for hydrogen liquefaction in a viable industrial scale [57,58]. Taking the dominant abundance of the light rare-earth elements over the heavy rareearth elements into consideration, it is more feasible to use light rare-earth based magnetocaloric materials for hydrogen liquefaction. In this work, we aim at developing a light rareearth based material system with sufficient magnetocaloric effects covering a full temperature range (77 ~ 20 K) required by magnetocaloric hydrogen liquefaction.

Mean-field Approach
Inspired by the sharply increasing trends of Δ and Δ with decreasing in the vicinity of the condensation point of hydrogen (20 K) for heavy rare-earth based magnetocaloric materials demonstrated by a mean-field approach from our previous work (Ref. [45] ), we focus in this work on the light rare-earth based magnetocaloric materials. We continue to develop the mean-field approach used in the aforementioned work, where the detailed description can be found [45]. It is worth mentioning that using the mean-field model to describe the magnetocaloric properties is a wellknown method, as used in Ref. [65][66][67][68][69]. In some studies, also effects related to the crystalline electric field are taken into consideration [70][71][72]. In this work, we aimed at providing a simple way to understand the sharply increasing feature of magnetocaloric effect of the light rare-earth magnetocaloric materials with a Curie temperature in the vicinity of hydrogen condensation point.
The total entropy change of a magnetocaloric material is contributed by three items, namely the magnetic entropy , the lattice entropy , and the electronic entropy : The equation to calculate the magnetic entropy is given as [73]: . is the magnetic field, the total angular momentum, the number of "magnetic atoms", the Boltzmann constant, the atomic magnetic moment, the Curie temperature, 0 the vacuum permeability, and ( ) the Brillouin function. A more detailed description on Equation (2) can be found in Ref. [73]. The lattice entropy is given as [73]: where is the Debye temperature, the total number of atoms, and can be regarded as a variable in the range of 0 ~ /T. The electronic entropy is given by [73] Journal XX (XXXX) XXXXXX W. Liu et al where is the Sommerfeld coefficient and is the electronic heat capacity ( = ). In the present work, is neglected out of simplification, since mostly is dominant only at sufficiently low temperatures [74].
We assume an idealized Nd-alloy family with the TC of its alloys varying from 300 K to 10 K and the other parameters, namely , , and , staying constant. For comparisons, we assume an idealized Dy-alloy family correspondingly. For the light rare-earth alloy series, and are taken to be 4.5 and 3.52 respectively, corresponding to the Nd 3+ ion. For the heavy rare-earth alloy series, and are taken to be 7.5 and 10 respectively, corresponding to the Dy 3+ ion.
of the light rare-earth alloy series is assumed to be 352 K, which corresponds to Laves phase LaAl2 [75]. For the heavy rare-earth alloy series, is assumed to be 384 K, which corresponds to Laves phase LuAl2 [75]. Δ ( , ) and Δ ( , ) can be calculated by Equation (5). Especially in the vicinity of the condensation point of hydrogen (20 K), "giant" values are observed, being almost two times larger than those of the heavy rare-earth alloys with a near room temperature. Figure 2 (b) shows the Δ (T) of the Nd-alloy family, and the maximum Δ of the Dyalloy family in magnetic fields of 5 T. Near 20 K, the light rare-earth alloy series shows a considerable maximum Δ , which is larger than or comparable to that of the heavy rareearth alloys with a near room temperature. Both Δ and Δ show an increasing maximum value with a decreasing in the temperature range of 77 ~ 20 K. As predicted by the calculations, we can see that the light rare-earth series also achieve a large Δ and Δ at low cryogenic temperatures, especially in the vicinity of the condensation point of hydrogen.
The calculations above point out a way of designing a fully light rare-earth based magnetocaloric materials for hydrogen liquefaction: tune the down to 20 K. Though there are no such idealized alloy families where only varies, the chemical and physical similarities of the light rareearth elements, namely Ce, Pr, and Nd, makes it easy to tune the of their alloys by mixing different rare-earth elements with different de Gennes factors on the rare-earth sublattices [76][77][78], as of rare-earth based alloys usually scales with de Gennes factor following the equation where Z is the nearest neighbors, is the Heisenberg exchange constant, and G is the de Gennes factor [47].
It has been reported that Laves phases NdAl2 with a near 77 K and PrAl2 with a near 30 K are two magnetocaloric materials with a large Δ and Δ [70]. However, these two compounds are unable to cover the full temperature range of 77 ~ 20 K. As shown in Figure 1 (a), the de Gennes factor of Nd, Pr, and Ce are significantly different, decreasing from 1.84 for Nd to 0.18 for Ce. Based on the analyses above, we predict that a light rare-earth RAl2 (R: rareearth elements) Laves phase series that covers the full temperature range (77 ~ 20 K) required by magnetocaloric hydrogen liquefaction can be realized by tuning the via mixing Pr and Nd in (Nd,Pr)Al2, and Pr and Ce in (Pr,Ce)Al2.

Experiment
Polycrystalline NdxPr1-xAl2 (x = 1, 0.75, 0.5, 0.25) and PrxCe1-xAl2 (x = 1, 0.75, 0.5) samples were synthesized by arc melting high-purity elements Ce (99.5 at. %), Pr (99.5 at%), Nd (99.5 at. %), and Al (99.998 at. %) under Ar atmosphere. We did not synthesize Pr0.25Ce0.75Al2 and CeAl2 since the later magnetization measurements show that the transition temperature of Pr0.5Ce0.5Al2 is already below 20 K. To ensure good homogeneity, all the samples were melted three times. The ingots were turned upside down before each melting step. Evaporation of the rare-earth elements was negligible. Powder X-ray diffraction (XRD) patterns were collected at room temperature with an x-ray diffractometer (Stadi P, Stoe & Cie GmbH) equipped with a Ge[111]-Monochromator using Mo-Kα-radiation in the Debye-Scherrer geometry. The XRD data were evaluated by Rietveld refinement with the FullProf software packages [79]. Backscatter electron (BSE) images were collected with a Tescan Vega 3 scanning electron microscope (SEM). A Physical Properties Measurement System (PPMS) from Quantum Design was used to measure the magnetization of the samples in magnetic fields up to 5 T. Heat capacity in magnetic fields of 0, 2, and 5 T was measured in the same PPMS with the 2τ approach.

Phase characterization
The Laves phases NdAl2 and PrAl2 crystallize in the MgCu2 cubic structure (space group: 227). Figure 3 (a) shows the XRD patterns of the (R1,R2)Al2 (R1: Nd, Pr, R2: Pr, Ce) samples. Detailed Rietveld refinements are included in the supplementary. The Rietveld refinements confirm that (Nd,Pr)Al2 and (Pr,Ce)Al2 does not change their crystal structures with the variation of Nd, Pr, or Ce content. Phase fraction analyses demonstrate the high quality of all the samples since the impurities are undetectable. The lattice constants of (Nd,Pr)Al2 and (Pr,Ce)Al2 samples are plotted in Figure 3 (b). The lattice constants increase almost linearly with increasing Pr content in (Nd,Pr)Al2 series and increasing Ce content in (Pr, Ce)Al2 series, respectively. This is coherent with the fact that CeAl2 has the largest and NdAl2 has the smallest lattice constant [78]. The quality of all the samples is further confirmed by SEM imaging.  magnetization measured in parallel. Under the same cooling and heating procedures, magnetization measurements in 0.1 T were done, the results are included in the supplementary. The saturated magnetization at 7 K of (Nd,Pr)Al2 and Pr0.75Ce0.25Al2 samples is rather close, in the range between 64 and 71 Am 2 /kg. Pr0.5Ce0.5Al2 is the only exception having a significantly lower magnetization of around 40 Am 2 /kg. Figure 4 (b) shows the Curie-Weiss fits of all the samples in magnetic fields of 1 T. To reduce the deviation of the reciprocal magnetic susceptibility from the Curie-Weiss behavior, which may be associated with intrinsic factors such as the van Vleck effect, or extrinsic factors such as impurities [80][81][82], we performed the Curie-Weiss fit in 1 T. The paramagnetic Curie temperature can be determined by the intercepts of the Curie-Weiss fit with the x-axis [45]. The total effective magnetic moment can be calculated by where is the molecular mass, the Avogadro constant, and the slope of the linear fitting for 0 −1 . [45]. Figure 4 (c) plots the of all the samples. The total effective magnetic moment of (Nd,Pr)Al2 increases roughly linearly with Pr content, whereas of (Pr,Ce)Al2 decreases with increasing Ce content. This observation can be explained by the fact that Pr 3+ has the largest magnetic moment of 3. 58 and Ce 3+ has the smallest magnetic moment of 2. 54 among the three light rare-earth ions. Figure 4 (d) plots the paramagnetic Curie temperature and the de Gennes factor of all the samples. The paramagnetic Curie temperature of (Nd,Pr)Al2 decreases almost linearly with increasing Pr content, from 78.5 K for NdAl2 to 32.6 K for PrAl2 , and it is the same case with (Pr,Ce)Al2, from 32.6 K for PrAl2 to 13.2 K for Pr0.5Ce0.5Al2. This agrees with the decreasing trend of the de Gennes factor from NdAl2 to Pr0.5Ce0.5Al2 since as Equation (6) indicates, smaller de Gennes factor, lower Curie temperature. The paramagnetic Curie temperature is in good agreement with the values of the Curie temperatures of NdAl2 and PrAl2 given by Ref. [70]. The reported Curie temperature for NdAl2 varies from 65 to 82 K, and that for PrAl2 varies from 31 to 38.5 K [70,78,[83][84][85].
In conclusion, by tuning de Gennes factors via mixing different rare-earth elements, a fully light rare-earth based magnetocaloric material system with a paramagnetic Curie temperature covering the temperature range required for magnetocaloric hydrogen liquefaction (77 ~ 20 K) is developed. The large effective magnetic moments are retained from NdAl2 to Pr0.75Ce0.25Al2.  ]. An example of how the Δ is calculated by these two methods for PrAl2 is included in the supplementary. Both methods fit well, as the points of the Δ from magnetization measurements mostly lie on the lines of the Δ from heat capacity measurements. Besides, the temperatures where Δ peaks are close to the paramagnetic Curie temperatures, consistent with the feature that secondorder magnetocaloric materials show a maximum Δ near their Curie temperatures [36,65].
In agreement with the calculations in Figure 2 (a) above, the maximum Δ increases from 7.21 J kg -1 K -1 for NdAl2 to 18.53 J kg -1 K -1 for Pr0.75Ce0.25Al2 in magnetic fields of 5 T, and from 3.67 J kg -1 K -1 for NdAl2 to 10.48 J kg -1 K -1 for Pr0.75Ce0.25Al2 in magnetic fields of 2 T. We observe an exception in this material series that Pr0.5Ce0.5Al2 has a smaller maximum Δ than that of PrAl2 and Pr0.75Ce0.25Al2. Similar observations were reported in RNi2 (R: Gd, Tb, Dy, Ho, Er) [45,71] and RAl2 (R: Gd, Tb, Dy, Ho, Er, Tm) [72] series. One contribution to this decrease is the reduction of the effective magnetic moment . As revealed in Figure 4 (c), Pr0.5Ce0.5Al2 has the lowest in the series. Another contribution might be due to the fact that crystalline electric field has a considerable influence on the magnetocaloric effect in low temperatures [45]. In the case of Pr0.5Ce0.5Al2, the crystalline electric field may decrease the magnetic entropy change. However, this speculation needs to be validated. Figure 5 (h) and (i) compares the maximum Δ of the light rare-earth RAl2 series in this work to some of the other light rare-earth magnetocaloric materials [89][90][91][92][93][94] and the heavy rare-earth RAl2 (R: Tb, Dy, Ho, Er) and TbxHo1-xNi2 (x = 0.25, 0.5, 0.75, 1) series [45,95,96]. Form the plots, we see  [89,90]), RSi (Ref. [91,92]), and RGa (Ref. [93,94]) where R = Pr, Nd, the heavy rare-earth based RAl2 (Ref. [45,95]) where R = Tb, Dy, Ho and Er, and TbxHo1-xNi2 (x = 0.75, 0.25, 0.5, 0)(Ref. [61]) in magnetic fields of 2 and 5 T. The green shadows mark the temperature range required for magnetocaloric hydrogen liquefaction (77 K ~ 20 K) and the black dashed lines highlight the values of DyAl2. that the light rare-earth RAl2 series is highly competitive compared to the other light rare-earth magnetocaloric materials. Pr0.75Ce0.25Al2 exhibits the largest maximum Δ among all the light rare-earth magnetocaloric materials in Figure 5 (h) and (i). The rest light rare-earth RAl2 samples show a maximum Δ that is larger than or comparable to the other light rare-earth compounds with a similar ordering temperature (except PrSi which shows a Δ larger than PrAl2 and Pr0.75Nd0.25Al2 samples).
Adiabatic temperature change Δ is as important as Δ for magnetocaloric effect. Figure 6 (a)~(g) plot the Δ of the light rare-earth RAl2 samples determined by equation (5) via constructing the ( , ) curves from the heat capacity measurements in magnetic fields of 0, 2, and 5 T. In agreement with the theoretical calculations above, the maximum Δ of the light rare-earth RAl2 series is large in the vicinity of the condensation point of hydrogen (20 K), with all the three (Pr,Ce)Al2 samples showing a maximum Δ over 2 K in magnetic fields of 2 T. In 5 T, all the three (Pr,Ce)Al2 samples have a maximum adiabatic temperature change over 4 K, and Pr0.75Ce0.25Al2 even shows a value close to 5 K. Figure 6 (h)~(i) compares the maximum Δ of the light-and heavy rare-earth RAl2, and the Laves phases TbxHo1-xNi2. The light rare-earth RAl2 shows a maximum Δ which is about 1/3 ~ 1/2 of the maximum values of the heavy rare-earth RAl2 in the vicinity of their ordering temperatures. Compared to TbxHo1-xNi2 [45,95,96] in magnetic fields of 2 T, PrAl2 and Pr0.75Nd0.25Al2 show a maximum Δ that is about half of the values of HoNi2 and Tb0.25Ho0.75Ni2, but comparative to that of Tb0.5Ho0.5Ni2 and Tb0.75Ho0.25Ni2.
In summary, the light rare-earth RAl2 alloy series shows a large Δ being comparable to the other light rare-earth based materials in Figure 5   counterparts in the vicinity of their ordering temperatures, we see a high potential of the light rare-earth RAl2 series for magnetocaloric hydrogen liquefaction, especially near 20 K, the condensation point of hydrogen.

Conclusions
The relatively high abundance of light rare-earth elements in the earth's crust makes their alloys appealing for magnetocaloric hydrogen liquefaction on an industrial scale. In this work, we aimed at designing a fully light rare-earth based magnetocaloric material system covering the full temperature range from 77 to 20 K required by hydrogen liquefaction.
In order to formulate a strategy for alloy design, the mean-field approach shown in our previous work is further developed to be applied to the light rare-earth alloys. From the theoretical analysis, we see that if of a light rare-earth based magnetocaloric material is tuned towards lower cryogenic temperature, the magnetocaloric effect is supposed to become stronger. Especially in the vicinity of the condensation point of hydrogen, the mean-field approach predicts significantly large Δ and Δ . Based on these observations and taking the chemical and physical similarities of the light rare-earth elements, a design strategy for developing light rare-earth material series for hydrogen liquefaction is used: tune the by mixing the light rare-earth elements with different de Gennes factors.
Consequently, a light rare-earth RAl2 Laves phase series with the ordering temperatures covering the temperature range from 77 to 20 K is successfully developed. This material series exhibits large Δ . Especially near 20 K (condensation point of hydrogen), the (Pr x , Ce 1−x )Al 2 (x = 0, 0.75, 0.5) samples show a Δ that is larger than or comparable to that of DyAl2, a heavy rare-earth based magnetocaloric material which is often proposed to be used in an active regenerator for hydrogen liquefaction [32]. Large Δ in the vicinity of 20 K are achieved in the (Pr 1−x Ce x )Al 2 (x = 0, 0.75, 0.5) samples, which show a value that is more than two third of that of DyAl2.
This design strategy for designing the light rare-earth RAl2 Laves phase series for magnetocaloric hydrogen liquefaction may be applied to other light rare-earth alloys to tailor their magnetocaloric effects for the liquefaction of industrial gases, inclusive but not limited to hydrogen gas. In addition, our work is also helpful for designing magnetocaloric composites, since tuning the Curie temperature in layered structures with a constant Δ over a wide temperature range is important for applications [60,62].

Acknowledgement
We appreciate the financial supports from Helmholtz Association via the Helmholtz-RSF Joint Research Group (Project No. HRSF-0045), from the HLD at HZDR (member of the European Magnetic Field Laboratory (EMFL)). We further gratefully acknowledge supports from the DFG through the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter-ct.qmat (EXC 2147, Project ID 39085490), the CRC/TRR 270 (Project-ID 405553726) and the Project-ID 456263705, from the ERC under the European Union's Horizon 2020 research and innovation program (Grant No. 743116, Cool Innov), and the Clean Hydrogen Partnership and its members within the framework of the project HyLICAL (Grant No. 101101461).

Data availability statements
The data that support the findings of this study are available upon reasonable request from the authors.

References
[1] Smith   As shown in Figure S3, the total entropy ( ) curves of PrAl2 in magnetic fields of 0, 2, and 5 T were constructed by the heat capacity measurements via the equation ( ) = ∫ 0 . The Δ and Δ from the heat capacity measurements were calculated by the equation (5) shown in the main article.