An assessment of the Al 50 Cr 21-x Mn 17 + x Co 12 (x = 0, 4, 8) high-entropy alloys for magnetocaloric refrigeration application

This study investigates the magnetocaloric potential of the Al


Introduction
The development of new magnetic refrigeration based on the magnetocaloric effect (MCE) has been expanding as a new competitor of conventional vapor compression refrigerators nowadays.The MCE is the reversible temperature change of a magnetic material under the variation of an applied magnetic field and explored for the generation of highly efficient refrigeration technologies [1][2][3].A large magnetic entropy change (ΔS M ), large adiabatic temperature change (ΔT ad ), large cooling power (RCP), and proper magnetic transition temperature (T C ) are the most important criteria to select refrigerant materials [4,5].
Rare-earth metals, e.g., gadolinium (Gd) have been considered the most active magnetic refrigerant with large MCE [6].However, geopolitical issues and their price have commercially limited their usage and motivated the search towards transition metal systems with giant MCE, good mechanical properties, and excellent corrosion resistance [7,8].
High-entropy alloys (HEAs) are a group of metal alloys composed of multi-principal elements in relatively high concentrations frequently forming solid solutions with face-centered cubic (FCC) structure, hexagonal close-packed (HCP) structure, or body-centered cubic (BCC) structure [9][10][11].The concept of HEAs was formed and developed by Yeh et al. [12] and Cantor et al. [13] bringing a whole new perspective and opportunities to material design as they exhibit exciting chemical and physical properties, such as high strength and toughness, good wear and corrosion resistance, high thermal stability, and good electrical and magnetic properties [14][15][16].
These days, the study of the functional properties of HEAs, especially MCE, has received considerable attention due to the significant magnetic entropy change of the 3d-metal-based HEAs near room temperature [17,18].For example, Na et al. [19] studied the structural and magnetocaloric properties in the FeCoNiCrAl HEA.They changed the Ni to Cr ratio in this alloy and found that the structure changed from single BCC phase for 1:1 ratio to a dual phase containing BCC and FCC phases for 1.5:0.5 ratio.They reported a ΔS M of 0.674 Jkg − 1 K − 1 and large RCP of 242.6 Jkg − 1 for FeCoNiCrAl alloy at 290 K and 7 T, demonstrating their promise for room temperature magnetic refrigerators.Law et al. [20] studied FeMnNiGe x Si 1-x HEAs and found the magneto-structural first-order phase transition with a ΔS M as large as 13 Jkg − 1 K − 1 (for 2.5 T) at 203 K for x=0.45, which is the largest reported number to date for magnetocaloric HEAs.Huang et al. [18] reported a ΔS M of 0.458 Jkg − 1 K − 1 and a ΔT ad of 0.28 K for Al 0.44 Cr 0.25 MnFeCo 0.25 Ni 0.2 alloy at ~340 K and ~0.82 T. The study of Zuo et al. [21] on CoFeMnNiX (X= Al and Cr) HEAs showed that addition of Al makes a dual phase structure containing B2 and BCC phases with a M S of 148 Am 2 kg − 1 .
Recently, the present authors [22] reported that the high fraction of Al (=50 at%) in Al 50 V x (Cr 0.33 Mn 0.33 Co 0.33 ) (50-x) (x = 12.5, 6.5, 3.5, and 0.5 at%) HEAs enables the formation of the B2 and BCC phases with large differences in magnetization and magnetic transition temperature.We suggested that a good magnetocaloric material with a T C around room temperature may be achieved by omitting V from this alloy system.In this work, we employ experiments, ab initio simulations and thermodynamic calculations to investigate the structure, magnetization behavior, and magnetic transition temperature of the Al 50 Cr 21-x Mn 17+x Co 12 (x=0, 4, and 8 at%) alloys with different Mn and Cr fractions.The experimental results show that increasing Mn and decreasing Cr increases magnetization and T C .At the same time, the magnetic entropy change and adiabatic temperature change improve with increasing Mn and decreasing Cr, underscoring the potential of the HEAs in magnetocaloric applications.

Experimental procedures
Ingots with the nominal compositions of Al 50 Cr 21-x Mn 17+x Co 12 (x=0, 4, and 8 at%) were prepared by arc melter under a high-purity argon atmosphere.The melting process was repeated three times to improve the homogeneity of the ingots.Structural analysis of the samples was carried out by X-ray diffraction (XRD, Siemens D5000) using Cu Kα radiation with the angle of diffraction in 30 • ≤2θ≤100 • at a scanning rate of 0.02 • per 2 s with adopting a graphite monochromator.Phase composition was determined by a scanning electron microscope (SEM, Hitachi S-3700 N) equipped with an energy dispersive spectrometer (EDS, Oxford Instruments, X-Max, England) operating at 15 kV.
Magnetic hysteresis loops were determined with a vibrating sample magnetometer (VSM, EG&G model 155) in an applied maximum field of ~650 kAm − 1 (0.82 T) at room temperature and at the boiling point of liquid nitrogen.Isothermal magnetization curves were measured in the temperature range of 261-369 K with a step of 4 K.
The temperature dependence of the normalized magnetization was measured by magneto-thermo-gravimetry (MTG, TGS2, Perkin Elmer) in a magnetic field of ca 3 kA/m and a heating rate of 40 K/min in an argon atmosphere.
The Curie temperature of one of the samples was determined by AC susceptibility measurements at 181 Hz and a magnetic field of 800 Am − 1 .
The measurement of the direct adiabatic temperature change, ΔT ad , was realized by slowly heating the sample at a constant rate in an evacuated tube from room temperature to maximum temperature.During this heating event, lasting several hours, the sample is subjected to a magnetic field alternating between 650 kAm − 1 and zero field.The period of the alternating field is about 10 s.The thermometer consisted of a spot welded thermoelement of type E.
The volume fraction of phases in a sample was estimated from their SEM area fractions and the Image J software [23].

Theoretical modeling
The Curie temperatures were calculated by mapping calculated magnetic properties of the alloys on a classical Heisenberg model, where the J ab ij are the magnetic exchange interactions between pairs of vector spin variables s i and s j associated with alloy constituents a and b occupying lattice sites i and j, respectively.The respective magnetic moment magnitudes m a i and m b j are included in the J ab ij , i.e., |s i | = 1.Furthermore, a,b ∈ Al,Cr,Mn,Co, and i,j = 1…N, where N is the number of lattice sites.The occupation number χ a i equals 1 if constituent a occupies site i and is zero otherwise.
The exchange interactions were determined from scalar-relativistic, spin-collinear density-functional theory calculations.Exchangecorrelation effects were treated within the generalized gradient approximation parameterized by Perdew-Burke-Ernzerhof [24].Substitutional disorder was treated with the coherent-potential approximation (CPA) [25] describing random alloys.Being a single-site approximation, the use of the CPA implies neglect of local-environment dependences of the J ab ij and m a i .The exchange interactions were determined in the paramagnetic state by the generalized perturbation method implemented in the Lyngby version [26] of the EMTO code [27].The paramagnetic state was modeled by the disordered local magnetic moment (DLM) model [28] and solved within the random alloy analogue as described by the CPA.For instance, the paramagnetic state of Al 50 Cr To check the sensitivity of the J ab ij and derived Curie temperature on the reference magnetic state, we also calculated the exchange interactions in a long-range magnetically ordered state using the magnetic force theorem [29].
The J ab ij calculations were carried out for theoretical lattice parameters determined with EMTO-CPA using a very fine k-point mesh (45×45×45 division).The magnetic transition temperature of the Heisenberg model Eq. ( 1) was estimated from classical Monte Carlo (MC) simulations as implemented in the UppASD code [30] by localizing the maximum of susceptibility versus temperature.Simulation cells modelling Al 50 Cr 21-x Mn 17+x Co 12 random alloys were constructed from 40×40×40 repetitions of the conventional unit cell, and periodic boundary conditions were employed.The spin system was equilibrated by 20,000 MC steps, measurements were performed during 80,000 MC steps, and averages over five ensembles were made.
Calculations of phase diagrams were based on thermodynamic equilibrium and used the Thermo-Calc software [31] and the TCHEA5 database [32,33].

Microstructure
Fig. 1 shows the XRD patterns and representative SEM images of Al 50 Cr 21-x Mn 17+x Co 12 (x=0, 4, and 8 at%) alloys in the as-cast condition.The XRD analysis reveals the diffraction peaks of the B2 structure with the lattice constant of 2.9324, 2.9457, and 2.9477 Å for x= 0, 4, and 8 respectively.However, the SEM images demonstrate a dual-phase microstructure containing a dominant gray zone and a dark zone.For the x=0 alloy, a low fraction of a third phase with white image contrast is also visible in higher magnification.The chemical compositions of these phases presented in Table 2 demonstrate that the gray phase is enriched in Co, while the dark phase is depleted of Co.The volume fraction of the gray zone estimated from the SEM area fraction is also given in Table 2. Upon examining the SEM images, it is found that the characteristic grain size is at micrometer scale.Therefore, we expect that the grain boundaries have minor effect on the magnetic properties of the present alloys.
The chemical compositions of equilibrium phases at 1200 K predicted by Thermo-Calc for these alloys (Table 1) confirm that the melts are expected to solidify into a dual-phase microstructure containing a Co-depleted BCC phase and a Co-rich B2 phase.By matching the chemical compositions predicted from the thermodynamic calculations with those determined by SEM, we suggest that the gray zone and dark zone are the B2 phase and the BCC phase, respectively.The experimental phase compositions and phase fractions agree partially with the thermodynamic predictions.We attribute this to the assumption that thermal equilibrium has been reached in the calculations and to limitations of the available database for HEAs [34].

Magnetic and magnetocaloric measurements
The room-temperature hysteresis loops presented in Fig. 2(a) show that the samples are magnetically soft with virtually vanishing coercivity.The magnetization at the maximum magnetic field increases with Mn, and the highest value amounts to 29 Am 2 kg − 1 for x=8.The hysteresis loops measured at the boiling point of liquid nitrogen presented in Fig. 2(b) show that the magnetization saturates at the maximum magnetic field.Numerical values of the magnetization at the maximum magnetic field measured at room temperature and liquid nitrogen temperature are presented in Table 2.As the phase fractions of the BCC and B2 phases in the three alloys are approximately similar (~65-70%), we attribute the trend of magnetization with composition primarily to the intrinsic properties of one phase rather than the BCC+B2 aggregate.With the present experimental techniques, we cannot determine whether it is the B2 phase or the BCC phase that gives rise to the measured magnetic properties.Based on the findings of our DFT calculations (Section 3.3), we associate the experimentally measured magnetic properties to the B2 phase rather than the BCC phase.
The saturation magnetization of the samples depends on the volume fraction of the nonmagnetic BCC phase.To estimate the intrinsic saturation magnetization of the B2 phase at liquid nitrogen temperature, we consider its volume fraction in the respective samples and obtain about 47, 62 and 72 Am 2 kg − 1 for x=0, 4, and 8, respectively.From the temperature dependences of magnetization M(T) presented in Fig. 3(b), we estimated the experimental Curie temperature T C by locating maxima in the change of magnetization with temperature (i.e., dM/dT).For the x=0 and x=4 alloys, we identified two local maxima corresponding to two Curie temperatures in the temperature range between 250 K and 400 K, whereas we found only a single maximum for the x=8 alloy.Numerical values of T C are given in Table 2. To confirm the existence of two Curie points (i.e., two T C ), an AC-susceptibility measurement vs temperature was performed for the x=0 alloy, Fig. 3 (b).The in-phase magnetic susceptibility (χ´) is qualitatively similar to the M(T) response in Fig. 3(a), and the out-of-phase susceptibility (χ´) shows two distinct peaks.These observations verify the existence of two T C .The required temperature range for the x=4 and 8 alloys is above our experimental capability.It should be noted that the magnetic transition temperatures determined from the MTG and AC-susceptibility are close but do not coincide due to methodological differences.
While we attribute both Curie points detected in the x=0 and 4 alloys to their respective B2 phases, we are uncertain about the origin of the two magnetic transitions.One possibility is that the B2 phases may not have reached equilibrium.It is conceivable that in each alloy, two phases with closely similar B2 chemistries have formed, which are indistinguishable in our XRD and SEM analysis.These two B2 chemistries are linked to different magnetic properties, particularly slightly different Curie temperatures.An additional observation is that these two hypothetical phases may have a magnitude of magnetization of similar size according to the MTG trace, Fig. 3(a).The clarification of this question is deferred to future work.
One of the main parameters to evaluate the MCE of a material is the magnetic entropy change, ΔS M which can be estimated by using Eq. ( 2) [35], and using the isothermal magnetization curves M(H,T), In this equation, S(T,H) and S(T,0) are the total entropies under an applied magnetic field, H and in the absence of the magnetic field at a fixed temperature T, respectively.
Fig. 4(a) demonstrates the magnetic entropy change (ΔS M ) as a function of temperature for Al 50 Cr 21-x Mn 17+x Co 12 (x=0, 4, and 8 at%) alloys, and Fig. 4(b) shows the M-H curves for the x=8 alloy (as an example) under the magnetic field of 0-0.82 T to show how ΔS M was calculated.It is observed in Fig. 4(b) that at low temperatures, the magnetization increases at low magnetic fields and tends to saturate with increasing magnetic field.However, at higher temperatures, the M-H curves gradually turns into straight lines especially above T C , indicating the transition from the ferromagnetic state to the paramagnetic state, along with a peak at this temperature in the ΔS M curve (Fig. 4(a)).However, it does not coincide completely with the T C presented in Table 2, attributed to sample inhomogeneities and inequivalent magnetic field of measurements [36,37].The peak value of the ΔS M curve (denoted by ΔS peak M ) and associated T peak corresponding to the magnetic transition at a magnetic field of 0.82 T are presented in Table 3.It was observed that ΔS peak M increases with increasing Mn/Cr ratio which can be explained by the larger magnetization at the maximum magnetic field.
Another parameter to evaluate the magnetocaloric performance of materials is the relative cooling power (RCP), which is given by [38]:  * W denotes white zone in the SEM image of alloy x=0 (Fig. 1(b)).

E. Dastanpour et al.
In this equation, we measure RCP value by taking the direct product of the peak magnetic entropy change (ΔS peak M ) and the full width at half maximum of the peak (δT FWHM ).
The adiabatic temperature change (ΔT ad ) is another parameter to evaluate the magnetocaloric materials.Here we measured ΔT ad directly by recording the temperature change of the insulated sample when the magnetic field and temperature were swept.The measured ΔT ad of Al 50 Cr 21-x Mn 17+x Co 12 (x=0, 4, and 8 at%) alloys as a function of temperature is presented in Fig. 5.The peak position observed in the ΔT ad curves (denoted ΔT peak ad ) corresponds to the ferromagnetic to paramagnetic transition temperature.The values of ΔS peak M , RCP, T peak , and ΔT peak ad for these alloys as well as Gd metal are presented in Table 3.As seen the MC properties of the studied alloys improve by Mn increasing and Cr decreasing.The largest ΔS peak M and ΔT peak ad of 0.30 Jkg − 1 K − 1 and 0.123 K are determined for the x=8 alloy with high Mn/Cr ratio, which is significantly less than Gd metal with ΔS peak M =2.8 Jkg − 1 K − 1 and ΔT peak ad =3 K.However, the broad magnetic entropy change centered near room temperature, leads to a significant refrigerant capacity of 18.9 Jkg − 1 , in comparison with Gd metal (63.40 Jkg − 1 ).It should be mentioned that this alloy is a composite (mixture of the B2 and BCC phases), and the beneficial magnetocaloric properties are associated with the ferromagnetic B2 phase rather than with the paramagnetic BCC phase.By characterization of the ΔS M and RCP for the B2 phase through the simple rule of mixture, ΔS = ∑ α V α ΔS α and RCP = ∑ α V α RCP α , where V α , ΔS α and RCP α are the volume fraction, the magnetic entropy change, and the refrigerant capacity of the α phase.By ignoring the entropy configuration of the BCC phase, the normalized magnetic entropy change and the relative cooling power for the B2 phase in the x=8 alloy is 0.46 Jkg − 1 K − 1 and 29 Jkg − 1 , which is significant in comparison with Gd metal.

DFT and MC simulations
To study the intrinsic magnetic properties of the BCC and B2 phase components of the x = 0, 4, and 8 alloys, we adopted the experimentally determined chemical compositions (Table 2).The nominal compositions of the B2 phases suggest chemical disorder on both of their sublattices, but the preferential occupation of Al, Cr, Mn, and Co over the two sublattices cannot be inferred from the present experiments and thus required modelling.Al and Co were approximated to reside at different sublattices as justified by our total-energy calculations, which showed that Co and Al strongly energetically prefer to occupy different sublattices rather than the Cr-Al and Mn-Al pairings.This approximation significantly reduced computational complexity as, for substitutionally random alloys without short-range order, the remaining degree of freedom is a one-dimensional one, i.e., the distribution of Cr and Mn over the two sublattices.In the following, we assume that Al and Co occupy sublattice I and sublattice II, respectively, and introduce the concentration variable y to describe the sublattice occupation in the B2 phase.To exemplify, we have for the B2 phase of the x=0 alloy (nominal experimental composition Al 43.2 Cr 21.9 Mn 18.2 Co 16.7 ) the composition Al 43.2 Cr y Mn 6.8-y on sublattice I and the composition Cr 21.9- y Mn 11.4+y Co 16.7 on sublattice II, where y ≤ 6.8 at%.For brevity, we introduce the notation B2(x = 0, y) to denote the previous sublattice occupancy for the x = 0 alloy, and similar notation is used for the other two alloys.To determine how Cr and Mn preferably occupy both sublattices, we considered the paramagnetic, i.e., high-temperature, magnetic state on account of the rapid cooling of the samples from the melt to temperatures below which reaching thermodynamic equilibrium is kinetically strongly constrained.We then determined the minimum of the Gibbs energy difference at vanishing pressure, where E is the total energy in the paramagnetic state and S the entropy.G ref refers to the Gibbs energy of a reference state, which is independent on y and thus arbitrary as we are only interested in the concentration y = y(x) that minimizes ΔG.An approximation in this work is that S only considers the configurational entropy, described by the sublattice model [40], and the magnetic entropy assuming randomly disordered, localized magnetic moments [41], whereas vibrational and electronic degrees of freedom were neglected.We found that Cr rather than Mn preferentially occupies the same sublattice as Al up to temperatures of at least 1550 K (the approximate, calculated melting temperature), and that this chemical effect is driven by energy (i.e., E) rather than entropy.The concentrations that minimize Eq. ( 4) are y = 6.8 at%, 6.7 at%, and 5.5 at% for x = 0, 4, and 8, respectively, i.e., Cr occupies sublattice I together with Al.The theoretical equilibrium lattice parameters a are equal to 2.919 Å for these three alloys (variation of a over the alloys is less than 0.001 Å).These lattice parameters and sublattice concentrations were further employed to determine the magnetic properties of the B2(x, y) phases and are presented next.Table 4 shows the local magnetic moment magnitudes of Cr (m Cr I/II ) and Mn (m Mn I/II ) in the paramagnetic state of the B2(x, y) phases for the alloys with x = 0, 4, and 8. Finite local magnetic moments on Co and Al are not supported in the DLM state.m Cr I on sublattice I was found to be identical zero.m Cr II and m Mn I/II on sublattice II are of similar magnitude and show minor changes with x.Effectively, the B2(x, y) phases consist of the spin-polarized Cr-Mn-Co sublattice II on an underlying simple cubic lattice and the non-spin-polarized Al-Cr sublattice I on a second simple cubic lattice.
Fig. 6 shows that strong magnetic exchange interactions between Cr-Cr, Mn-Mn, and Mn-Cr pairs appear in several of the first six coordination shells in the B2(x, y) phases.For higher coordination shells, the magnitude of J ab ij is generally small.Interestingly, magnetic interactions give rise to magnetic frustration in the ground state.To see this, we inspect the J ab ij for the first two coordination shells: the nearest neighbor antiferromagnetic (J ab ij < 0) and next-nearest neighbor ferromagnetic (J ab ij > 0) interactions between Cr spins can be arranged on a simple cubic lattice without conflicting interactions, and so can the nearest neighbor ferromagnetic interactions between Mn spins (nextnearest J ab ij practically vanish and do not impose conflicting interactions).However, the inter-species Cr-Mn interactions, i.e., ferromagnetic interaction between nearest neighbors and next-nearest neighbors, cannot be satisfied simultaneously with the intra-species alignments preferred by Cr.In a picture where spins are arranged in a collinear way (Ising type magnet), we expect that most of the Cr spin moments are forced into parallel alignment with Mn as the energy gain for nearest neighbor ferromagnetic alignment between Cr-Mn is more than a factor of two larger than the penalty for parallel alignment of a pair of Cr spins.In local chemical environments poor in Mn mutually  antiparallel alignments of Cr spins may, however, occur.If spins are allowed to arrange in a non-collinear way, as in the present MC simulations, this analysis still largely applies, but the conflicting interactions introduce a slight degree of transverse disorder of both Mn and Cr spins with respect to a global magnetization axis (this occurs already at the lowest temperatures).Our MC simulations considered magnetic exchange interactions up to the ninth coordination shell (of which the first six shells were shown in Fig. 6), and analysis of the static structure factor confirmed a magnetically long-range ordered ground state without any detectable tendency towards antiferromagnetic long-range order.Fig. 7 presents the calculated magnetization versus temperature curves of the B2(x, y) phases for the x = 0, 4, and 8 alloys in the temperature interval T ≤ 250K.The magnetization is normalized per atom, thus includes the fraction of non-spin-polarized chemical species, and shown at dense temperature intervals of 5 K near the critical temperature.Numerical values of the theoretical Curie temperature and lowtemperature saturation magnetization are provided in Table 4.The theoretical Curie temperatures are about 190-200 K and thus lower than the experimental ones.The saturation magnetization and theoretical Curie temperature vary non-monotonically with x and peak at x = 4.This is in contrast to the experimental trends, where both the Curie temperature and the low-temperature saturation magnetization were found to increase monotonically with x.The main reason for the limited predictive accuracy is likely the coexistence of localized and itinerant degrees of freedom in the present magnetic alloys with d electrons, which are not fully accounted for in the local-moment picture of the Heisenberg model [Eq.( 1)].
Next, we turn to the properties of the BCC phases adopting the experimentally determined chemical compositions given in Table 2. Their theoretical equilibrium lattice parameters in the paramagnetic state are 2.943 Å, 2.942 Å, and 2.946 Å, respectively, for the x = 0, 4, and 8 alloys.Finite local magnetic moments in the paramagnetic state are only supported on Co (~0.8μB ) and Mn (~1.3μB ).The MC simulations, based on J ab ij derived in the paramagnetic reference state, revealed a virtually vanishing magnetization near zero temperature and a structure factor that peaks at a reciprocal lattice vector of ( 1 2 , 1 2 , 1 2 )π/a.Both results indicate that the magnetic ground state is antiferromagnetic rather than ferro-or ferrimagnetically ordered.As ferromagnetic order is not supported by the present MC calculations, no ferromagnetic to paramagnetic phase transition is found for the BCC phases.
Last, we briefly discuss the sensitivity of the estimated Curie temperatures on the reference magnetic state.Besides the paramagnetic state discussed above, we considered as reference state for the J ab ij calculations a long-range magnetically ordered state.Specifically, we assumed that the magnetic structure of the ordered state corresponds to the BCC/B2 crystal structure of the CPA effective medium (this assumption is compatible with the ordered state having a magnetic wave vector of (0, 0, 0)π/a).The predicted Curie temperatures of the B2(x, y) phases are 165 K, 165 K, and 170 K, for the x = 0, 4, and 8 alloys, respectively, somewhat lower than those derived from the paramagnetic reference state.Similar to the paramagnetic reference state, the MC calculations did not support a ferromagnetic ground state for the BCC phases.

Conclusions
The structural and magnetic properties of Al 50 Cr 21-x Mn 17+x Co 12 (x = 0, 4, and 8 at%) HEAs for magnetocaloric applications were investigated by various experimental techniques and simulation approaches.The alloys synthesized by arc melting consisted of a mixture of B2 and BCC phases.Substituting Mn for Cr increased the saturation magnetization and Curie temperature in the alloy series, with values equal to 29 Am 2 kg − 1 (at room temperature) and T C =367 K, respectively, for the x=8 alloy.The Curie temperatures calculated by Monte Carlo simulations with exchange interactions determined from constrained spindensity approximation calculations suggest that the measured magnetic properties originate from the intrinsic properties of the B2 phase.Further, the Mn substitution improved the magnetocaloric performance.

Fig. 1 .
Fig. 1.(a) The XRD patterns for Al 50 Cr 21-x Mn 17+x Co 12 (x=0, 4, and 8 at%) alloys show the B2 structure for all alloys, and the SEM images of (b) x=0, (c) x=4, (d) x=8 alloys show a dual B2/BCC phase microstructure.For the x=0 alloy, in addition to the gray zone (G:B2) and dark zone (D:BCC) phases, white zone (W) can be seen.

Fig. 3 .
Fig. 3. (a) The normalized magnetization (ratio of magnetic weight to sample weight) as a function of temperature measured by MTG for Al 50 Cr 21-x Mn 17+x Co 12 (x=0, 4, and 8 at%) alloys, (b) in-phase magnetic susceptibility (χ´) and out-of-phase magnetic susceptibility (χ´) as a function of temperature at 800 Am − 1 for the x=0 alloy.

Fig. 4 .
Fig. 4. (a) The magnetic entropy change (ΔS M ) of the Al 50 Cr 21-x Mn 17+x Co 12 (x=0, 4, and 8 at%) alloys as a function of temperature under an applied magnetic field of 0.82 T, (b) Isothermal magnetization M-H curves for the x=8 alloy at different temperatures.ΔT between M-H curves is 4 K.

Fig. 6 .
Fig. 6.Intra-species and inter-species magnetic exchange interactions for the first six coordination shells between the species that support finite local magnetic moments in the paramagnetic B2(x, y) phases for the x = 0, 4, and 8 alloys.The J ab ij are plotted versus pair distance normalized by lattice parameter, d/a, and the alternative abscissa shows the multiplicity of each coordination shell.Markers connected by dashed lines are for x = 4, and the colored areas indicate the variation of J ab ij with composition (x = 0 or 8, not distinguished in the plot).Lines guide the eye.

Fig. 7 .
Fig. 7. Calculated magnetization versus temperature behavior from classical MC simulations for the B2(x, y) phases of the x = 0, 4, and 8 alloys.Lines guide the eye.

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.Dastanpour et al.

Table 4
Local magnetic moment magnitudes of Cr (m Cr I/II ) and Mn (m Mn I/II ) in the paramagnetic state of the B2(x, y) phases for the x = 0, 4, and 8 alloys from the present DFT calculations.Moments on Al and Co are zero and were omitted.The Curie temperature and saturation magnetization at 1 K determined from the present MC simulations are listed in the last two columns.