Effect of Gd polarization on the large magnetocaloric effect of GdCrO4 in a broad temperature range

E. Palacios,1 C. Tomasi,1,2 R. Sáez-Puche,3 A. J. Dos santos-Garcı́a,4 F. Fernández-Martı́nez,4 and R. Burriel1 1Instituto de Ciencia de Materiales de Aragón (ICMA) and Departamento de Fı́sica de la Materia Condensada, CSIC–Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain 2IENI-CNR, Corso Promessi Sposi 29, 23900 Lecco, Italy 3Departamento Quı́mica Inorgánica, Facultad Quı́micas, Universidad Complutense Madrid, 28040 Madrid, Spain 4Departamento de Ingenierı́a Mecánica, Quı́mica y Diseño Industrial, ETSIDI, Universidad Politécnica de Madrid, 28012 Madrid, Spain (Received 12 November 2015; revised manuscript received 27 January 2016; published 19 February 2016)


I. INTRODUCTION
The research on magnetocaloric materials has attracted worldwide interest in recent years due to their high potential use for magnetic refrigeration processes [1,2]. Gadolinium metal presents a magnetocaloric effect (MCE) through a second-order phase transition from paramagnetic (PM) to ferromagnetic (FM) at room temperature (T C = 293 K), and it has been used as a cooling material in magnetic refrigeration prototypes since the 1970's, starting with the Brown refrigerator [3]. However, since 1997 when the so-called "giant MCE" was discovered by Pecharsky et al. in Gd 5 Si 2 Ge 2 [4], many studies have been done mainly in intermetallic compounds containing rare-earth elements [5]. In oxides, similar examples are abundant, especially for rare-earth transition metal oxides. Gd 3 Ga 5 O 12 (GGG) [6], RMnO 3 [7], RMn 2 O 5 [8], and EuR 2 O 4 [9], where R = rare earth, present high thermal and chemical stabilities and display large MCEs in the low-temperature region. New families without rare earths in their compositions have also been investigated as materials for refrigeration purposes close to room temperature [10].
Magnetic refrigeration is an alternative technology to traditional cooling systems based on cycles of compression and expansion of a gas. The efficiency can reach values up to 60% of a theoretical Carnot cycle and the absence of polluting gases constitutes a great advantage. It reduces expenses and improves profitability for liquefying fuel gases before transportation. Matsumoto et al. described a hydrogen liquefier based on the Dy 3 (Ga,Al) 5 O 12 garnet [11,12]. Therefore, magnetocaloric materials can be used as efficient refrigerants in the lowtemperature range with the aim of producing the liquefaction of He, H 2 , or natural gas [13].
It has been reported very recently that RCrO 4 (R = Ho, Gd, and Dy) oxides crystallizing with the zircon-type structure (space group I 4 1 /amd) show large values of the MC parameters [14,15]. Recently, we have succeeded in synthesizing RCrO 4 scheelite polymorphs (space group I 4 1 /a) by treating the zircon phases at high temperatures and high pressures [16].
The crystal structure, magnetic properties, and high pressure phase transformations of RCrO 4 zircon oxides have been previously reported [16][17][18][19][20]. Different mechanisms have been proposed to explain this first-order phase transition induced by pressure from zircon to scheelite and a reconstructive mechanism appears to account for it [21]. Reference [22] reports the MCE of a combination of these zircon phases with Gd and Er, derived from isothermal magnetization measurements. The coexistence of two magnetic ions, Cr 5+ and R 3+ , constitutes an interesting scenario to study 3d-4f magnetic interactions. Most of the zircon RCrO 4 oxides behave as ferromagnets, while the scheelite polymorphs are antiferromagnets. The change in the sign of the magnetic interaction can be explained considering the notable differences in the interatomic distances and Cr-O-R angles (see Fig. 1). Although the four-and eightfold coordinations remain in going from the zircon to the scheelite polymorph, there are some remarkable differences when these two structural types are compared. In the case of zircon, the structure can be described as formed by zigzag chains of edge-sharing RO 8 bisdisphenoids along the a axis of the tetragonal unit cell, connected to each other by sharing edges with CrO 4 tetrahedral units [ Fig. 1(a)]. However, in the scheelite form, the CrO 4 tetrahedra are aligned along the a axis, while the RO 8 bisdisphenoids of different chains are forming dimers of R 2 O 14 composition and there is no edge sharing between RO 8 and CrO 4 polyhedra, as in the case of zircon [ Fig. 1(b)]. Neutron diffraction studies reported that both rare-earth and chromium sublattices become ordered simultaneously at the same temperature [23] and the Cr 5+ and R 3+ ions will contribute to the MCE. In addition to the fundamental physical and chemical properties, the following features make these oxides suitable for potential refrigerants in comparison with other rare-earth transition metal oxides. First, a large magnetic moment, typical of many rare-earth ions, makes its polarization easier by a moderate external magnetic field. Second, a weak anisotropy and a weak crystal field favor the thermal population of all magnetic states (i.e., with a random distribution of the magnetic moment directions above T N or T C but still at low temperatures) at zero field. Third, the exchange interaction between 3d ions, usually much higher than the R-R exchange interaction, provides an internal field, increasing the polarization of the R ions below the ferromagnetic ordering temperature. The Cr 5+ ion plays an important role as the promoter of the interactions in the rare-earth sublattice, increasing its ordering temperatures at least one order of magnitude in comparison with the analogous RXO 4 (X = P, As, V), where X is a diamagnetic element [24,25]. The case R = Gd is easier to analyze because, for this spin-only atom, the anisotropy and crystal-field energies are of the order of 1 K or less [24], therefore, a higher MCE is expected. Actually, GdPO 4 showed the strongest MCE ever observed with an isothermal entropy increment − S T = 62.1 J/kg K at 2 K on application of a field of 7 T [24].
In this paper we report the magnetic and magnetocaloric properties of the zircon polymorph of GdCrO 4 oxide deduced from isothermal magnetization, heat capacity under field, and direct measurement of S T . The data have been analyzed assuming a stronger Cr-Cr ferromagnetic exchange interaction and a weaker Gd-Cr one. This Gd-Cr interaction is crucial to get a large MCE and gives a clue as to how to design materials for efficient magnetic refrigeration.

II. SAMPLE PREPARATION AND CHARACTERIZATION
The zircon-type GdCrO 4 powdered sample was prepared by heating stoichiometric amounts of Gd(NO 3 ) 3 · 6H 2 O and Cr(NO 3 ) 3 · 9H 2 O in an oxygen flow according to the following thermal process: 30 min at 433 K, 30 min at 473 K, and final treatment at 853 K for 12 h. Rietveld refinement of the x-ray diffraction data reveal that the Bragg reflections can be indexed according the tetragonal I 4 1 /amd space group. A small amount of the secondary phase GdCrO 3 perovskite has been taken into account in the refinement.
The magnetization M has been measured in a standard Quantum Design superconducting quantum interference device (SQUID) magnetometer at several constant temperatures, from 4 to 40 K for fields between 0 and 9 T.
The heat capacity has been obtained by the conventional relaxation method in a Quantum Design physical properties measurement system (PPMS) setup. The calorimeter consists of a sapphire plate with a resistance thermometer and a heater. The sample is attached to the upper side of the plate. In addition to electrical connections, the wires supply mechanical support and a weak thermal contact to a bath at constant temperature T 0 . For the determination of the heat capacity, a constant power is supplied to the heater. The relaxation time τ to the new equilibrium temperature gives the heat capacity C, knowing the sample mass m and the thermal conductance of the wires k, since in a rough approximation τ = mC/k. Nevertheless, more elaborated models are used to fit the relaxation, including the finite thermal conductivity of the sample.
For the direct measurement of S T , a magnetic field is applied at a typical rate of 0.01 T/s while the temperature is recorded. During the field increase period the sample temperature T is above T 0 , but relaxes again to thermal equilibrium with the bath when the field stabilizes at the final value. Knowing k, the entropy increment from the initial field B 1 at t 1 to the final value B 2 at t 2 is given by Details of this procedure will be given elsewhere.

III. ISOTHERMAL MAGNETIZATION
The experimental magnetization as a function of field from 0 to 9 T, measured every 4 K between 4 and 40 K, have been plotted in Fig. 2(a). To clarify the discussion, the experimental M T (B) data have been plotted as functions of T at constant fields [ Fig. 2(b)] in units of Bohr magnetons per formula unit containing one Gd and one Cr atom. This compound does not have thermal hysteresis, so the isothermal procedure used in the experiment saves measuring time and does not introduce substantial differences with respect to true isofield measurements.
The magnetization curves are consistent with the reported data of M T (B) [23] and with the heat capacity C p,B (Sec. IV B) that show a ferromagnetic transition with a Curie temperature T C = 21.3 K, but present some unusual features, different than in a typical ferromagnet. We leave out of the analysis the data at low fields, B < 0.2 T, where the sample shape affects the magnetization values through the internal demagnetizing field. For higher fields, M does not saturate quickly below T C , but increases gradually on decreasing T , and the saturation is not reached even at 4 K, for fields as high as 9 T, where it reaches a value of M = 7.87μ B /f.u. The saturation magnetization M s can be inferred to be 8μ B /f.u., corresponding to the spinonly values for Gd 3+ (S = 7/2) and Cr 5+ (S = 1/2), in both cases with a gyromagnetic ratio g = 2. For Cr 5+ this is clearly shown in cases where Gd is replaced by a nonmagnetic atom as in YCrO 4 [26]. For Gd 3+ the spin-only value for the 4f 7 electrons results in a magnetic moment of 7μ B /atom. In other Gd compounds the saturation value is frequently higher than 7μ B /atom due to the polarization of the conduction 6s or the 5d electrons, which in turn transfer the exchange via a Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. As an example, highly pure Gd metal has a saturation moment of 7.63μ B /atom [27]. These electrons practically do not exist in GdCrO 4 , that is electrically an insulator, with Gd 3+ bonded ionically in a high percentage to the CrO 4 3− group. A very precise electron-paramagnetic resonance (EPR) determination of g in Gd 3+ -doped zircon-type orthophosphates gave g = 1.99 [28]. These bonding characteristics produce a low Gd-Gd exchange interaction, crucial to have a high magnetocaloric effect at low temperatures (see Sec.V), as already observed in GdPO 4 [24]. In a usual ferromagnet, the magnetization increases below T C and saturates rapidly. This is not the case for GdCrO 4 , since at 15 K (T = 0.7T C ) and a field as high as 1 T, M = 4.7μ B /f.u., only slightly higher than M s /2. Considering also the heat capacity and the magnetocaloric results, as discussed later, we interpret that the weak Gd-Cr exchange acts on the Gd sublattice polarizing the Gd 3+ ions, but is not strong enough to orient them completely. This interaction would also act on the Cr sublattice raising T C , which increases from 9.1 K for YCrO 4 to 21.3 K for GdCrO 4 . This also explains the low moment at the Gd sublattice observed by Mössbauer spectroscopy just below T C , at zero field [23].

A. MCE from magnetization
The isothermal entropy change as a function of temperature can be evaluated from magnetization measurements at different temperatures and magnetic fields by numerical integration of the well-known Maxwell relation (2) Figure 2 shows the isothermal magnetization data as a function of the magnetic field up to 9 T at different temperatures above and below T C . The magnetization curves confirm the ferromagnetic behavior below 22 K, and the value of the magnetic moment obtained below T C indicates that both Gd 3+ and Cr 5+ ions contribute to the magnetic ordering. The isothermal entropy variation S T for different magnetic fields is shown in Fig. 3(a). A maximum is observed in the − S T values around 22 K, near the Curie temperature. | S T | decreases below T C but not as quickly as expected for a ferromagnetic arrangement. Indeed, there is a bump near 10 K. The entropy change increases at T C with increasing external field and reaches a value as high as 30 J/kg K for 9 T. Figure 4 shows the heat capacity of GdCrO 4 from 2 to 80 K. The peak at 21.3 ± 0.2 K can be ascribed to the ferromagnetic to paramagnetic transition at T C observed in magnetization. The phonon contribution C ph can be estimated from the heat-capacity values for another nonmagnetic zircontype compound C 0 (T ), applying a corresponding state law C ph (T ) C 0 (Tf ), which is exact for the theoretical Einstein and Debye models and approximately valid for real solids when the scale factor f is close to unity. For this purpose, the fully diamagnetic compound ZrSiO 4 is not the best choice since f = 1.6 due to its much lighter atoms [29]. A more adequate compound is LuCrO 4 [26]. The Cr 5+ ion introduces a magnetic component in its heat capacity, but this compound orders at T N = 9.1 K and there is a wide T range above 20 K in which the magnetic contribution is negligible. Using 064420-3 Nevertheless, the Lu compound does not provide any phonon baseline below 20 K, due to its magnetic contribution. Then, we can make some theoretical considerations in order to determine C ph . The zircon-type compounds have 18 phonon branches, but nine of them correspond to the internal modes of the CrO 4 tetrahedron, which have frequencies above 260 cm −1 (data for YCrO 4 from Raman spectroscopy) [30], giving Einstein temperatures above 370 K. These branches do not contribute appreciably up to 70 K. Among the nine remaining branches, three are acoustic and six are optical. C ph can be fairly described by a three-dimensional Debye function (as in the case of one atom per formula unit) and a sixfold Einstein function. We took the data of LuCrO 4 , affected by a scale factor, and fitted them to a sum of Einstein and Debye functions, resulting in their characteristic temperatures T E = 255 K and T D = 155 K, which matches the experimental data between 20 and 60 K (Fig. 4), providing a physically reasonable baseline for GdCrO 4 as the phonon contribution.

B. Heat capacity
To determine the absolute entropies at different fields, we have followed a procedure similar to the case used with GdPO 4 [24], using the direct measurements of S T for several fields. The heat-capacity measurements at B = 0 can be hardly extrapolated to T → 0 due to the existence of a peak at 4.8 K, but the results for 9 T are smooth and can be extrapolated with an approximate C p,9 T = AT 1.2 dependence. In any case, the entropy is small for 9 T at these low temperatures and any estimation errors are even smaller. Plotting the entropy for 9 T, the data extrapolate correctly to zero for T → 0, giving S(5.1 K,9 T)/R = 0.34. For higher temperatures, the absolute entropy can be obtained by integration of C p,9 T /T . For other fields, the directly measured S T with respect to 9 T at a single fixed temperature provides a proper integration constant, as explained in Sec. IV C.
With this estimation for the phonon part we find at 60.8 K the phonon entropy to be S ph /R = 2.10, while the total entropy is S/R = 4.89 at zero field, giving a magnetic contribution S m /R = 2.79, very close to the theoretical value for the spins 1/2 (Cr 5+ ) and 7/2 (Gd 3+ ), S m,th /R = ln 2 + ln 8 = 2.77.
The resulting "experimental" magnetic heat capacity C m,B (T ) = C p,B (T ) − C ph (T ) is plotted in Fig. 5.
The comparison with other RCrO 4 compounds, where R = nonmagnetic atom, can give an idea about the Cr-Cr exchange interactions. The Cr ions form a distorted diamond lattice [26] in which each Cr ion has four nearest neighbors. LuCrO 4 orders antiferromagnetically at T N = 9.1 K while YCrO 4 , with very similar Cr-Cr distances and O-Cr-O angles, orders at 9.2 K, but ferromagnetically [26]. This different behavior is probably due to the competition of the exchange interactions between nearest and next-nearest neighbors, whose distance is not much higher. Isostructural compounds of GdCrO 4 in which Cr has been replaced by a nonmagnetic 064420-4 atom provide information about the Gd-Gd interaction. Gd also has four nearest neighbors. GdVO 4 and GdAsO 4 order antiferromagnetically at T N = 2.5 and 1.26 K, respectively [31]. In all cases the behavior departs significantly from the mean-field model. GdCrO 4 is ferromagnetic below T C with the magnetic moments along the c axis (F z type in Bertaut's notation) [23]. The Gd-Cr exchange interaction seems to be responsible for the increase in the Curie temperature up to T C = 21.3 K, as found in the C p data (Figs. 4 and 5). This interaction not only increases T C but also the heat-capacity values well below T C . With regard to the weak Gd-Cr interaction, the value J 3 /k B = 10 K gives a good account of the experimental C p,B at low temperatures (see Fig. 5) and also of S T , as discussed later (see Fig. 6). The Cr sublattice is almost saturated at T T C and this interaction can be viewed as an external magnetic field of around 7 T (see Sec. V) that produces a Schottky-like anomaly or, more generally speaking, a shoulder in C p,0 at zero field.
The polarization of the rare-earth sublattice by a weaker exchange with the 3d atoms was already analyzed more than 40 years ago for the distorted perovskite NdCrO 3 [32]. In a detailed study of NdFeO 3 , the heat-capacity data matched very well with the result obtained for a paramagnetic Nd 3+ in an external effective field produced by an Fe sublattice above 1 K, where Nd orders due to the much weaker Nd-Nd interaction [33]. The Fe sublattice orders at very high temperature, therefore its magnetic moment is nearly constant below 20 K and so is the effective field acting on the Nd 3+ ions. As a result, the heat capacity agrees very precisely with the polarization model in an external field [33]. Nevertheless, these perovskites are antiferromagnetic and the staggered effective field does not help the external field in the polarization of the rare-earth ions. Moreover, Nd 3+ is not a good ion to produce a high magnetocaloric effect because of its low magnetic moment. Moreover, its relatively large anisotropy splits the ground free atom manifold in five Kramers doublets and, at low temperature, Nd 3+ behaves as a spin 1/2. The behavior of GdCrO 4 is much more favorable for magnetic refrigeration. The Gd-Cr interaction is weaker but comparable to the Cr-Cr one, consequently, the moments of the Cr and Gd sublattices grow below T C , but at a different rate.
A simple mean-field model can be applied in order to have a qualitative explanation of the experimental results. In any case, we cannot expect a precise quantitative agreement from this model. Considering two inequivalent magnetic atoms A and B with spins s 1 and s 2 , the Hamiltonian for 1 mol of the entire lattice is given by where N A is the Avogadro constant, z 1 is the number of nearest A neighbors of a given A atom, z 2 is similar to z 1 for B atoms, and z 3 is the number of nearest B atoms to a given A atom. Due to the Gd-Cr interaction, the peak at T C is considerably higher than for compounds with only Cr as a magnetic atom (i.e., LuCrO 4 , Fig. 4) because, when ordered ferromagnetically, the Cr sublattice polarizes Gd. In the mean-field model the exchange field acting on the Gd moment is proportional to the moment of Cr, due to the term of the Hamiltonian containing J 3 . This term is much higher than the term with J 2 , which we have neglected and in any case is zero for an antiferromagnetic Gd-Gd interaction (in such a case s 2 should be replaced by the staggered average of the x or y 064420-5 component of s 2 , zero above T N ). Neglecting J 2 , the terms with J 1 and J 3 contribute to the magnetic entropy of the peak. Moreover, the heat capacity does not drop sharply below T C , but has a shoulder at lower temperatures. The mean field is an oversimplification and predicts Curie temperatures that are too high, especially in this case of few nearest neighbors. The predicted Schottky-like contribution at low temperatures is evident and agrees with the experimental data for 5 and 9 T at temperatures far from the critical point, where the mean-field approximation is more precise.
The existence of a peak at 4.8 K in C p,B=0 indicates another ordering process due to the much weaker Gd-Gd exchange interaction, |J 2 |/k B 0.1 K. The magnitude of this exchange can be estimated from the data for GdAsO 4 and GdVO 4 [31]. In these isostructural compounds, Cr is replaced by a nonmagnetic atom and orders antiferromagnetically with T N = 1.26 and 2.5 K, respectively. In GdCrO 4 the peak is much less pronounced and, consequently, the anomalous entropy of the peak at 4.8 K is much smaller since Gd is already partially ordered due to the Gd-Cr interaction. This low-temperature ordering has been described as a ferromagnetic component along the a crystal axis (F x type in Bertaut's notation) [23]. This behavior is surprising because, in cases with nonmagnetic atoms replacing Cr (e.g., GdVO 4 , GdAsO 4 ), the ordering takes place at similar low temperatures but in an antiferromagnetic configuration. In GdCrO 4 , the Gd-Cr exchange interaction acts as a constant external field. The final structure would minimize the energy in a way that is analogous to the effect of a strong external field applied to an antiferromagnet in the hard direction. The field produces a spin-flop transition in the case of weak anisotropy. The final order would not be ferromagnetic F z F x , but canted with the x component of the Gd moment antiferromagnetically ordered, probably F z G x . Moreover, a ferromagnetic Gd-Gd interaction would not produce any new transition but an increase of the order parameter, that is, the spontaneous magnetization below T C . The assumption of a F x component below T N was based on a neutron diffraction experiment made on a powder with a short wavelength [23]. No extra reflections were observed below 4.8 K apart from the nuclear ones, which was considered to correspond to ferromagnetic ordering. Nevertheless, the entropy content of the peak, S m (T N )/R 0.2 ln 8 = 2.08, is very small and, consequently, the x component of the Gd moment should be much lower than its total moment, surely less than 1μ B , due to the strong polarization induced by the Cr sublattice above T N . The magnetization data at weak fields do not give an idea because of the demagnetization field, but for B = 1.5 T (surely above the demagnetization field) give M = 6.8μ B /f.u. at 4 K, which means a difference of 1.2μ B at the Gd site, to be ordered at T N . This small component is difficult to be observed on powder neutron diffraction, especially in a Gd sample that is a strong absorber for thermal neutrons.

C. Magnetocaloric effect
The isothermal entropy change S T has been determined in three ways: (1) from magnetization, via the Maxwell relation (∂S/∂B) T = (∂M/∂T ) B , (2) from heat capacity, and (3) directly, via Eq. (1) [see Fig. 3(a)]. In this case there is no precise way to extrapolate the heat capacity to T = 0 to obtain the absolute entropy, which would lead to large uncertainties in the determination of S T for different fields. Instead, we used the direct determination of S T for the fields of interest, at a temperature T 0 well above T C , where this value is small. The measured S T 0 (B) become integration constants in the calculation of S T (B) as follows: and S T 0 (B) = S(T 0 ,B) − S (T 0 ,0). Then, for any other temperature, Proceeding in this way, the results agree very well with those directly measured in the full temperature range, which would not be the case if C p,B alone were used. This direct determination of S T 0 (B) also allows one to obtain the absolute entropy at every field B when the heat-capacity data can be extrapolated to T = 0 only for one of these fields. The extrapolation provides the entropy at every temperature for one field, and the direct measurements of S T 0 (B) make its calculation possible for the other fields. The extrapolation of C p,9 T (T → 0) was used to obtain the absolute entropy at 9 T, as discussed in Sec. IV B. The values deduced from field dependent magnetization measurements agree well with the results from heat capacity. Actually, the consideration of the real internal fields on each experimental sample gives a good account of the small differences in S T obtained from M T (B) and C p,B (T ) measurements. An estimation of the demagnetizing fields for the different sample shapes explains these differences. Moreover, the small temperature increments used for the numerical integration of Eq. (2) can give some errors in the calculations from M T (B).
The adiabatic temperature increment T S [ Fig. 3(b)] has been deduced in two ways: (1) from heat capacity through the entropy derived functions [Eq. (4)] and (2) in a semidirect way, using the basic equation Here, S T and C p,B (T ) are experimental values, and the integral is numerically solved for every S T data to find the upper limit of the integral and, then, T S (T ,B). This second method uses only the heat capacity with field and does not require one to determine any integration constant. To obtain the integration constants at different fields in the first method, a single S T measurement was used for each field at T = 30.9 K. Both methods gave the same results within the experimental precision. The magnetocaloric effect is a consequence of the magnetic properties of Cr 5+ and Gd 3+ , and their exchange interactions. On one hand, Cr provides a relatively high T C in comparison with compounds with a nonmagnetic atom instead of Cr. On the other hand, the isotropic Gd allows all its 2s + 1 magnetic levels to be occupied at zero field. This fact produces high entropy values and a large magnetic moment, which is easy to be polarized by an external field. The Gd-Cr interaction acts as an external field polarizing the Gd atom. As a result, the MCE reaches high and wide curves with maximum values − S T ,max = 18.8 and 29.0 J/kg K, and T S,max = 7.9 and 12.1 K for 5 and 9 T, respectively. The MCE keeps high values in a wide temperature range between 5 and 35 K, ideal for hydrogen liquefaction. The peak values are similar to those of several conventional materials, such as ErCo 2 [6], but in this compound the MCE drops to very small values just below T C = 35 K.
Finally, it is worth realizing that the magnetic behavior of the scheelite polymorph of GdCrO 4 is completely different, i.e., antiferromagnetic, with T N = 21 K [17]. The Cr-O-Cr and Cr-O-Gd angles are different in the scheelite and zircon polymorths, and the sign of the superexchange interaction depends critically on this angle. The, probably, accidental coincidence of T C (zircon) with T N (scheelite) gives similar heat-capacity peaks at zero field but different field dependences of magnetization [17] from which we expect very different MCEs. Midya et al. expected to find a significantly larger MCE for these high pressure scheelite phases [14], assuming that a larger magnetization would produce a higher MCE. However, the MCE does not depend on the absolute magnetization but rather on its temperature derivative along the field change. For the compounds studied in Ref. [14], DyCrO 4 and HoCrO 4 , the magnetization depends strongly on the crystal field, which is different in each phase. For GdCrO 4 , the spin-only Gd magnetic moment is nearly the same in both phases, but the antiferromagnetism strongly hinders the MCE, since the internal field is staggered. Actually, the MCE for low fields in an antiferromagnet is inverse ( S T > 0) below T N when the external field is applied in the hard direction.

V. DISCUSSION
Usually, the MCE of a ferromagnet decreases strongly below T C , as happens for ErCo 2 [6]. In the zircon phase of GdCrO 4 , the MCE reaches its maximum value − S T (9 T) = 29 J/kg K near T C = 21.3 K and retains high values below this temperature down to T N = 4.8 K. This is an interesting feature in regard to applications in magnetic refrigeration. Assuming J 1 J 3 J 2 , the Cr 5+ sublattice orders at a higher temperature and partially polarizes the Gd 3+ ions. Well below T C , the moment of the Cr 5+ ions is nearly saturated and the weaker Gd-Cr exchange acts on the Gd moments as a constant field in the z direction. The much weaker Gd-Gd exchange would order antiferromagnetically the x or y component. Actually, the lowest-temperature phase, below 4.8 K, has been described as a ferromagnetic F z F x , supported by weak experimental evidence (see Sec. IV B), but it is probably a canted ferromagnetic F z G x . Therefore, between 4.8 and 21.3 K, the Gd sublattice behaves approximately as a paramagnet in an effective constant magnetic field.
The analysis of the predictions of the mean-field model for S T is straightforward but, in the approach of J 2 /k B = 0, needs the numerical solution of two equations of state. In these equations the reduced average of s 1 and s 2 are Brillouin functions that depend on the external field and two exchange fields deduced from the Hamiltonian given in Eq. (3). For an easy modeling and understanding of the main features, and especially to give a simple rule for the search for new compounds, instead of analyzing in detail the predictions of the mean field, we will consider the even more simplified case in which a constant s 1 = 1/2 corresponds to a fully ordered Cr 3+ ion. The mean-field predictions do not differ much from these results except for T > T C , as discussed below. In this case, the total field acting on each Gd atom can be considered as the sum of two terms, the externally applied field plus the exchange field: B = B app + B exc .
The entropy of a paramagnet for low fields or high temperatures is given by [6] S/R = ln(2s where C is the Curie constant. That is, S T of a paramagnet is proportional to the increment of B 2 , where B i,f are the effective initial and final fields. Therefore, the MCE, for a given increment of the external field, increases with the average field. But at low temperatures the MCE decreases on increasing the average field due to the saturation of the magnetic moment. In a usual ferromagnet the mean field is proportional to the magnetization and saturates very fast below T C , even without an external field. This is the reason why the MCE drops very fast below T C . In the present case, due to the internal molecular field, the magnetic moments of the Cr sublattice saturate quickly below T C , but the weaker Gd-Cr interaction does not saturate the moments of the Gd 3+ ions down to T N , which leaves room for the additional action of the external field. Figure 6 shows some calculations related to GdCrO 4 , along with experimental results. The solid colored lines are the computed − S T values for a system of paramagnetic spins 7/2 (as Gd 3+ ) for a field change B = 5 T, starting at different initial fields B i . These initial fields would represent the exchange field produced by the Cr sublattice. The results, − S T /R, are the molar entropy variations in units of the gas constant R, which is dimensionless and independent on the concentration of the magnetic atoms. For low B i values, the maximum | S T | occurs at very low temperatures (orange line, calculated for B i = 0). On increasing B i , the maximum decreases and occurs at higher temperatures, as shown in Fig. 6 for B i = 3, 5, and 7 T. In the high-temperature range one can observe the increase in the − S T /R curves with the average field, resulting in quite lower values for a zero initial field above 15 K (orange line). In GdCrO 4 this entropy change experienced by the Gd 3+ ions must be added to the corresponding change coming from the Cr 5+ ions. This last contribution has been computed via the mean-field Bean-Rodbell model [34] for spin 1/2. In this model the free parameters are T C and the variation of the exchange interaction with the volume. This variation is summarized by the η parameter. For η = 0 there is no dependence of T C with volume and values of η > 1 give rise to a first-order transition. Our chosen value η = 0.1 results in a total − S T /R contribution that agrees very well with the experimental data below T C for B = 5 T, with B i = 7 T as the exchange field acting on the Gd 3+ ions. This approach is not valid above T C where the Cr sublattice is not spontaneously ordered and, consequently, its magnetization is neither constant nor saturated. The exchange field acting on 064420-7 the Gd 3+ ions is proportional to the magnetization of the Cr sublattice, which in this T range is polarized by the external field. This effect is stronger near T C , where the susceptibility has a maximum. Above T C , the effective initial B i is zero, but B f has a value which depends on the polarization of the Cr sublattice with the external field, resulting in an effective B higher than the externally applied field of 5 T. Consequently, | S T | decreases with increasing T but remains much higher than the contribution of the Cr sublattice alone (dotted line in Fig. 6), and still higher than expected for the Gd sublattice with an internal constant field plus the contribution of Cr (black line in Fig. 6).
In summary, this simple mean-field model explains the qualitative behavior of C p,B and S T , although it does not allow to extract precise predictions. The Gd-Cr exchange is responsible for a large entropy change under the magnetic field action due to the Gd polarization below T C . The entropy change is also large above this temperature, in a wide temperature range. This feature becomes an important improvement in the use of magnetic refrigeration for the liquefaction of gases. It suggests an interest in searching for other ferromagnets having T C near the boiling point of the gas to be liquified and being composed of a transition metal and a rare-earth atom, with a relatively weak exchange between both sublattices. The exchange should be strong enough to produce polarization of the rare-earth atoms but not so large as to induce saturation of this sublattice.