# Fileset

[diff.pdf](https://mdr.nims.go.jp/filesets/3f57e27b-24cd-4974-a79d-55e25dc484f3/download)

## Creator

Satoru Enomoto, Sonju Kou, [Taichi Abe](https://orcid.org/0000-0002-5065-0939), Yoshihiro Gohda

## Rights

[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

## Other metadata

[Subphase exploration for SmFe12-based permanent magnets by Gibbs energies obtained with first-principles cluster-expansion method](https://mdr.nims.go.jp/datasets/c3383a01-3b8e-4ea2-a88e-e8aad8063f1e)

## Fulltext

Subphase exploration for SmFe12-based permanent magnets by Gibbs energiesobtained with first-principles cluster-expansion methodSatoru Enomotoa, Sonju Koua,1, Taichi Abeb, Yoshihiro Gohdaa,∗aDepartment of Materials Science and Engineering, Tokyo Institute of Technology, Yokohama 226-8502, JapanbNational Institute for Materials Science, Tsukuba 305-0047, JapanAbstractIn the development of SmFe12-based rare-earth permanent magnets, it is yet to establish a pathway to liquid-phasesintering producing nonmagnetic grain-boundary subphases. Here, phase equilibria of the Sm-Fe-Cu ternary system areexamined computationally to explore such a pathway and grain-boundary subphases of SmFe12-based magnets throughGibbs free energies for various phases. A key quantity, the mixing enthalpy is examined from first principles. Weshow that the B2 Sm-Cu-Fe is a candidate for a nonmagnetic subphase, where the solubility of Fe into Sm-Cu is betterdescribed by the cluster-expansion method finding attractive Cu-Fe interaction within the B2 sublattice. In addition, itis found that the liquid phase of Sm-Cu slightly including Fe directly equilibrates with SmFe12, which is consistent withexperiments. We point out the possibility of liquid-phase sintering of the SmFe12-based magnets using this liquid phasewithout significant composition changes by the solidification. The mechanism of attractive interaction between Fe andCu in B2 Sm(Cu, Fe) is clarified from the viewpoint of electron theory.Keywords: Permanent magnets; Phase diagrams; Cluster-expansion method; First-principles electron theory;Materials Design1. IntroductionAs attempts to develop new rare-earth permanent mag-nets beyond Nd2Fe14B-based magnets, SmFe12 have beenstudied intensively [1–16]. Even though magnetic prop-erties such as the magnetization and the anisotropy fieldof SmFe12 are superior to those of Nd2Fe14B, insufficientstructural stability has been one of the most severe prob-lems, which has been becoming gradually overcome:as::::::::discussed::::::below [6–9]. However, the design of SmFe12-based magnets also suffers from identifying an appro-priate grain-boundary subphase, a crucial component ofmicrostructures. As an additive element for the grain-boundary phase, a few elements including Cu have beenstudied [15, 16]. In particular, it has been demonstratedthat SmFe11Ti equilibrates with Cu-41at.% Sm at 1273K [16]. Even though this finding is promising, phase equi-libria for a wide temperature range are yet to be obtained.Such information should be helpful to design the sinteringprocess of magnets. In addition, details of atomic inter-actions and atomic structures within the subphase are ex-pected to be helpful for further optimization of microstruc-tures.:::::Since:::::::SmFe12:::is::a:::::::::::metastable::::::phase:::in:::::the::::::Sm-Fe::::::binary:::::::system,:::it:::::must:::be:::::::::stabilized:::by:::::::adding:::::::various∗Corresponding authorEmail address: gohda.y.ab@m.titech.ac.jp (Yoshihiro Gohda)1Present address: Institute for Solid State Physics, The Univer-sity of Tokyo, Kashiwa 277-8581, Japan::::::::elements::::such:::as:::Ti,:::V,::::Mo,::::Al,::::and:::Ga::::::::through::::::Fe-site::::::::::::substitutions,::::and::::Zr,:::Y,:::::and:::Gd::::for:::::::::::substituting::::Sm::::sites:::::::::[9, 17–21]:.::::::::::Although:::Ti::is:::an::::::::effective::::::::element::to:::::::stabilize::::the:::::::::::compound,::::the::::::::::processing::::::::::::temperature::is:::::higher:::::than:::::1273:::K:::::::[22, 23].::::::From::::the:::::::::viewpoint:::of:::the:::::::::::liquid-phase:::::::sintering:::to::::::achieve:::::::::optimum::::::::::::::microstructures,:it::::::::requires::::the:::::::::two-phase:::::::::equilibria::::::::between::::the:::::::SmFe12:::and:::::::liquid::::::::phases.::::::::The::::::::::two-phase::::::::::equilibria::::are::::::::::interrupted:::by::::the::::::::::formation:::of::::::other:::::::::::compounds:::in::::::ternary:::::and:::::::binary:::::::::::::sub-systems,:::::e.g.,:::::::Fe2Ti:::in::::the:::::Fe-Ti::::::binary::::::::system.:::::Due:::to::::the::::::Fe2Ti:::::::::::::precipitations,:::::::SmFe12 :::::does::::not::::::have::::::direct::::::::tie-lines::::to::::the::::::liquid:::::phase::::::::::[16, 22, 23]:.:::::::Among::::::::additive::::::::elements,:::Ti,:::::Mo,:::and::Zr:::::have::::::binary:::::::::::compounds:::::with:::Fe,:::::while:::V,:::Al::::and:::Cu:::are::::free::::from:::::such::::::::::compounds:::in:::the:::::::Fe-rich::::::region::of:::the::::::binary:::::phase:::::::::diagram.::::As:::::::::::fundamental:::::::::::knowledge,::it::is::of::::::::::importance::to::::::::examine::if:::::::SmFe12::::can:::be::in::::::::::equilibrium::::with:::the::::::liquid::::::phase::in::::::::ternary:::::phase:::::::::diagrams:::::such::as:::::::::Sm-Fe-Cu,:::::::before::::::::::proceeding:::to:::::::::::::::multicomponent::::::phase::::::::diagrams:::of::::::::::quaternary:::or:::::more:::in:::::::::::::consideration::of:::all:::::::possible::::::factors:::at:::::once.:Even if main-phase single crystals have high magneti-zations, high anisotropy fields, and high Curie tempera-tures, permanent magnets do not exhibit high performancewithout having appropriate microstructures, because mi-crostructure interfaces play significant roles in preventingthe magnetic domain-wall motion. While microstructureinterfaces have been examined on the atomic scale obtain-ing insights into magnetic couplings between constituentPreprint submitted to Elsevier March 13, 2023phases [24–26], identification of phase equilibria requiresquantification of the Gibbs free energy. Even though therehave been successful attempts to evaluate the Gibbs freeenergy from first principles [27–36], such evaluations forgeneral cases with sufficient precision can be computa-tionally demanding due to, e.g., effects of the positionalentropy due to displacements [37], and effects of anhar-monic lattice vibrations such as the lift of the dynamicalinstability at finite temperatures. In contrast, the mod-eling of the Gibbs free energy to reproduce experimentalphase equilibria, known as the CALPHAD (CALculationof PHAse Diagrams) approach [38], has been successfulin providing thermodynamical database for multicompo-nent phase diagrams of, e.g., elements relevant to Nd-Fe-B permanent magnets [39]. Since Gibbs-energy functionsare required for phases with experimentally unavailablecompositions, the CALPHAD approach combined withfirst-principles calculations is effective [40–44]. Thus, suchan approach should be promising also in the explorationof grain-boundary subphases of SmFe12-based permanentmagnets by examining phase equilibria through phase di-agrams.In this study, the ternary phase diagram for Sm-Fe-Cu is studied by the CALPHAD approach together withfirst-principles calculations. For compounds relevant toSmFe12-based permanent magnets, CALPHAD Gibbs freeenergies are constructed with mixing enthalpies from firstprinciples. Various phases including B2 Sm-Cu-Fe andSmFe12 are examined using dilute alloys that is slightlyoff-stoichiometric. In addition, mixing enthalpies for theB2 phase are evaluated by the first-principles cluster-expansion method [45, 46], where the calculated Gibbsenergy results in phase equilibria more consistent with ex-periments [15, 16]. We show that the liquid phase withcompositions in the vicinity of Cu-45at.%Sm-1at.%Fe di-rectly equilibrates with SmFe12 above 1053 K opening upthe possibility of the liquid-phase sintering. At lower tem-peratures, the equilibrating phase becomes nonmagneticB2 Sm-Cu-Fe that is suitable as the grain-boundary phase.Since B2 Sm-Cu-Fe is much more stable than the amor-phous phase, the grain-boundary phase may be crystallinedepending on the cooling rate. Finite solubility of Fe intothe B2 Sm-Cu phase comes from attractive interaction be-tween Cu and Fe within the B2 sublattice, which is incontrast with repulsive one in binary Cu-Fe. The mech-anism of the attractive interaction is the stabilization ofFe minority-spin states with lower single-electron energiescompared with the case of binary SmFe.2. Computational DetailsWe performed spin-polarized first-principles calculationsto obtain total energies of the endmembers with the struc-tures shown in Fig. 1 and their substituted solid solu-tions. The total energy of density functional theory (DFT)was self-consistently minimized using the frozen-core all-electron projector-augmented wave [47] (PAW) method as(a) (b)(c)(d) (e)(a)(f)Sm Fe CuFigure 1: Atomic configurations of ordered binary Sm-Cu and Sm-Fe alloys: (a) B2 SmCu (CsCl type), (b) SmCu2 (CeCu2 type), (c)SmFe12 (ThMn12 type), (d) Sm2Fe17 (Th2Zn17 type), (e) SmFe3(PuNi3 type), and (f) C15 SmFe2 (MgCu2 type).implemented in the VASP code [48–50]. The exchange-correlation energy functional are treated using the gen-eralized gradient approximation [51] parametrized as theform of Perdew–Burke–Ernzerhof [52]. For Sm::It::::::should::be::::::::::reasonable::to:::::::assume:::::that::::::::strongly::::::::localized:::4f:::::states::::::hardly::::::::hybridize:::::with:::::other:::::::::electronic::::::states,:::::even::::::though::::some:::of:::Sm::::4f:::::states:::::have::::::::::::::single-electron:::::::energies:::::close::to:::the::::::Fermi:::::level::::[53].::::::Thus, an open-core PAW poten-tial were used:::was:::::used:::for::::Sm, where partially-occupied4f electrons are kept frozen in the core. During:::the struc-ture optimization, plane wave cutoff energy of 390 eV wereused,:and k-grid spacing were set as 0.1 Å−1×0.1 Å−1×0.1 Å−1. The full relaxed structures were obtained onceatomic forces were smaller than 0.01 eV/Å with the total-energy convergence criterion of 10−7 eV. As initial spinconfigurations, Sm and Fe spins set as antiparallel, whilespins within Sm and Fe were aligned as parallel.Lattice models for mixing enthalpies Hmixij were con-structed by the cluster-expansion method as implementedin the ATAT code [54, 55]. We considered binary mixingwithin one B2 sublattice for X(Y1−yZy) with the sublat-tice concentration y, where all possible 9 combinations ofSm, Cu, and Fe were examined for elements X, Y , andZ. In the cluster-expansion lattice model for the binary2mixing, Hmixij is described asHmixij =∑αJα⟨∏I∈α′σI⟩α, (1)where α is the cluster type, Jα is the effective cluster in-teraction, and⟨∏I∈α′ σI⟩αis the correlation function ofthe cluster α. At each lattice point I, the variable σI inthe binary model has the value of +1 or −1 depending onelements. We considered up to tetrahedron clusters. Theeffective cluster interaction was determined by fittings toreproduce the mixing enthalpy obtained by first-principlescalculations. In order to consider only structures that canbe recognized as B2, structures with lattice vectors devi-ating from those of the ideal B2 lattice significantly wereexcluded from the fitting with the criterion R of more than9%, where R is defined asR =√∑i,jε2ij , (2)where {εij} are the eigenstrains induced by the relaxationfrom the B2 structure, i.e., other than the isotropic scalingand the rigid rotation [56]. In addition, structures withlarge distortions by atomic displacements were excluded sothat the cross-validation score for the fitting of Jα becomessmaller than 65 meV per atom.In the CALPHAD approach, the sub-regular solutionmodel is used to describe Gibbs energies of phases thatare considered in phase diagrams of interest. In the sub-regular solution model, the Gibbs energy is decomposedinto three terms: (i) linear combination of Gibbs ener-gies of endmembers, (ii) the configurational-entropy termwithin the Bragg-Williams approximation, and (iii) theexcess Gibbs energy as all effects other than the formertwo terms. We used the two-sublattice model for in-termetallic compounds, where the Gibbs energy of a bi-nary stoichiometric compound, e.g., the B2 phase is de-scribed as an endmember by the two-sublattice model.The Gibbs energy of endmembers with the magnetiza-tion at low temperatures was described by adding a mag-netic term based on the Inden model [57, 58]. In thisstudy, we constructed Gibbs-energy functions of SmFe12as well as experimentally unavailable counterparts of Sm-Fe and Sm-Cu compounds, while preexisting Gibbs-energydatabases were used for all other endmembers [59–62].The Kopp-Neumann rule for the specific heat indicatesthat the temperature dependence of excess Gibbs energiescan be linear. Assuming this linear dependence is smallin the present study, mixing enthalpies obtained from firstprinciples were used for phases shown in Fig. 1 as excessGibbs energies in the present study, details of which aregiven in the following. Using Gibbs energies as inputs,phase diagrams were calculated by the PANDAT code [63].3. Results and Discussion3.1. Mixing enthalpyFigure 2 shows mixing enthalpies Hmixij between B2 Sm-Cu-Fe stoichiometric compounds as endmembers in thetwo-sublattice model. Even though the ground-state struc-ture of SmCu is the B27 (FeB-b) structure [24], we focuson the B2 structure in the present study, because high-temperature phases are of our primary interest. FromFigs. 2(a) and (b), it is clear that the interaction be-tween Sm and Cu is repulsive within each sublattice ofB2 SmCu. This means that each sublattice of binary B2SmCu does not allow the alloying of Sm-Cu at zero tem-perature. In addition, substitutional Fe impurities in B2SmCu are more stable at Cu sites than at Sm sites asseen in Figs. 2 (g) and (i). Other mixing enthalpies arealso used to construct the Gibbs-energy landscape in thewhole Sm-Fe-Cu compositions.The mixing enthalpy is of importance to evaluate theGibbs energy beyond the ideal-solution model through theRedlich-Kister polynomial Lk :i,j . In the present CAL-PHAD two-sublattice model, Hmixij and Lk :i,j has the re-lation, Hmixij = yj(1 − yj)Lk:i,j , where yi the sublatticeconcentration of the element i. Since the mixing enthalpydepends on specific atomic configuration, we adopted thecase of the disorder limit. Curves for this disorder limitshown in Fig. 2 are obtained from the cluster-expansionmodel asHmixij (yj) =∑αJα∏I∈α′⟨σI⟩α=∑αJα(2yj − 1)kα , (3)where kα is the order of a cluster, e.g., kα = 3 for tri-angle clusters. Figure 2 also includes results obtainedby fitting from the dilute alloys that are slightly off-stoichiometric from the endmembers. In some cases, thecluster-expansion model significantly improves the mixingenthalpy compared with the dilute-alloy approach. Thus,mixing enthalpy from the cluster-expansion model is usedfor the B2 phase unless otherwise stated, while the dilutealloys are used to evaluate mixing enthalpies of the rest ofthe phases shown in Fig. 1 that are found to be of less im-portance considering roughly-estimated phase equilibria.These mixing enthalpies are added to CALPHAD Gibbsenergies as an approximation of the excess Gibbs energy.:::::Other:::::::phases:::::::::including::::the::::::liquid::::::phase:::are::::::::::considered:::::using:::::::existing::::::::::::Gibbs-energy:::::::::functions::::::::[60–62]:to::::::::examine:::::phase:::::::::equilibria::::::::reported::::::below.:In the evaluation of the mixing enthalpy from first prin-ciples, it was assumed that one of the sublattice is alwaysoccupied by one element. On the other hand, this as-sumption is lifted in examining phase equilibria by usingthe condition∂G∂y(n)i= 0 , (4)3(a) (b) (c)(d) (e) (f)(g) (h) (i)Mixing Enthalpy [eV]Mixing Enthalpy [eV]Mixing Enthalpy [eV]disorder limit of CE CE DFT fitting from dilute alloysySmySmySmySmyFeyFeySmySmyFeFigure 2: The mixing enthalpyHmixij between B2 stoichiometric compounds that is related to the Redlich-Kister polynomial Lk :i,j in the presentCALPHAD two-sublattice model through Hmixij = yj(1− yj)Lk:i,j : (a) HmixCuSm of (SmySmCu1−ySm )Cu, (b) HmixCuSm of Sm(Cu1−ySmSmySm ),(c) HmixFeSm of (SmySmFe1−ySm )Fe, (d) HmixFeSm of Sm(Fe1−ySmSmySm ), (e) HmixCuFe of (FeyFeCu1−yFe )Cu, (f) HmixCuFe of Fe(Cu1−yFeFeyFe ), (g)HmixFeSm of (SmySmFe1−ySm )Cu, (h) HmixCuSm of (SmySmCu1−ySm )Fe, (i) HmixCuFe of Sm(Cu1−yFeFeyFe ). Note that the first and second sublatticesare equivalent, i.e., Lk:i,j = Li,j:k. Values directly obtained by first-principles DFT calculations are represented by crosses. Circles are Hmixijfrom cluster-expansion (CE) model for configurations same as crosses. Red solid lines indicate the disorder limit of the CE model that areused in the CALPHAD model through Lk :i,j . Blue dashed lines are fitted curves from results of first-principles calculations for dilute alloysthat are slightly off-stoichiometric from the endmembers.where y(n)i is the sublattice concentration of the elementi in the sublattice n. With the above constraint, the ele-ments can occupy on both sublattices of the B2 structureas (Sm,Cu,Fe)1(Cu,Fe,Sm)1 . in the two-sublattice model.In this study, the Gibbs-energy function per atom of theB2 phase within the two-sublattice model is given asGB2 =∑ijy(1)i y(2)j Gij +12kBT2∑n=1∑iy(n)i log y(n)i+∑ik∑j>i[y(1)k y(2)i y(2)j Lk:i,j + y(2)k y(1)i y(1)j Li,j:k],(5)where {Gij} = {Gbcc−Fe, Gbcc−Sm, Gbcc−Cu, GB2−SmCu,GB2−SmFe, GB2−FeCu, }, kB is the Boltzmann constant, Tis the temperature, andLk:i,j =2∑κ=0L(κ)k:i,j(y(2)i − y(2)j)κ. (6)Parameters {L(κ)k:i,j} are obtained from {Jα}. Numeri-cal values of parameters appearing in the above formu-lae are provided in the unit of J /mol:::per:::::mole::of::::::atomsas Supplemental Material. Other phases including theliquid phase are considered using existing Gibbs-energyfunctions [60–62] to examine phase equilibria reportedbelow.4SmCuFe0 0.2 0.4 0.6 0.8 100.20.40.60.81(a)Sm2Fe17SmFe3bcc-FeSmFe2α-SmB2-SmCuSmCu2773 KSmCuFe0 0.2 0.4 0.6 0.8 100.20.40.60.81(b)SmCuFe773 Kw/o Sm2Fe17 SmFe12SmFe3bcc-FeSmFe2α-SmB2-SmCuSmCu20 0.2 0.4 0.6 0.8 100.20.40.60.81(c)SmFe12SmFe3bcc-FeSmFe2β-SmLiquidB2SmCu2Liquid1053 Kw/o Sm2Fe17SmFe12+Liquidx Fex Fex Fex FexSmx Fex Fex Fex FexSmx Fex Fex Fex FexSmFigure 3: (a) Ternary phase diagram of the Sm-Fe-Cu system at 773K. (b) Phase diagram excluding the Sm2Fe17 phase from (a) assum-ing the stabilization of the SmFe12 phase due to possible additiveelements. (c) Ternary phase diagram of the the Sm-Fe-Cu systemexcluding the Sm2Fe17 phase at 1053 K. In all cases, the CALPHADGibbs energy obtained with Redlich-Kister polynomials Lk :i,j fromthe cluster-expansion method is used for the B2 phase.3.2. Phase equilibriaThe Cu-Fe, Sm-Cu, and Sm-Fe binary system have beencritically assessed by Turchanin et al. [60], Zhuang etal. [61], and Chen et al. [62], respectively. We adoptedthese assessments for the Sm-Fe-Cu ternary system. SinceSmFe12 in the Sm-Fe binary system is metastable, and,thus, the formation energy of SmFe12 was estimated fromfirst principles. As a finite-temperature effect, the for-mation entropy was given so as SmFe12 became stablewhen Sm2Fe17 was excluded from the equilibrium calcu-lations. The magnetic excess Gibbs energy was given byInden model [57, 58] with the following two parameters:the Curie temperature of the compound was set as 555 Kmeasured approximately in experiments [4], and the ther-modynamic effective magnetic moment was set as 1 µB [4],where µB is the Bohr magneton. The obtained Gibbs en-ergy of the SmFe12 as an endmember is given in Appendix.For the calculations of the phase equilibria in the ternarysystem, solubility of the third element to the binary com-pounds were considered by the cluster-expansion formal-ism for the B2 phase, while dilute-alloy models were usedfor other structures shown in Fig. 1. Even though thecluster-expansion formalism obviously provides more reli-able results, changes by the use of dilute-alloy models forother phases are expected to be minor in phase equilibria,due to their limited solubility (approximately 1 at.%) inthe phases evaluated by dilute-alloy models. The Gibbsenergy of the phases considered are written in the TDBformat, which is accepted for various thermodynamic soft-ware packages such as PANDAT [63] and Thermo-Calc[64]. The TDB file is provided as Supplemental Material.The isothermal section at 773 K is presented as a func-tion of the compositions {xi} in Fig. 3(a). The calcu-lated phase diagram agrees well with the experimentallyreported one [15]. In the present calculations, the prefer-ential site of Fe in the B2 structure is the Cu site as ex-pected from the mixing enthalpy mentioned above, whichis in good agreement with the experiments [15]. The solu-bility of Fe in the B2 SmCu is estimated to be 2 at.% thatqualitatively agrees with 5.8 at.% reported experimentallywith the annealing at 723 K for relatively short time of48 h [15].::It:::has::::::::recently::::been::::::found:::::using:::::::::::::first-principles::::::::::::::thermodynamics:::::that:::the::::::::::theoretical:::Fe:::::::::solubility::::::should:::::::increase:::by::::::taking:::the::::::Gibbs::::::energy::of::::the::::B27:::::phase::::into:::::::account::::[65].:Even though the stability of Sm2Fe17 overwhelms thatof SmFe12 in the Sm-Fe-Cu ternary system, the relativestability can be modulated by doping of other elementsinto SmFe12. Assuming such stabilization of pseudobinarySmFe12 compared with Sm2Fe17, we exclude the Gibbs en-ergy of Sm2Fe17 from the phase diagram. As presented inFig. 3(b), SmFe12 becomes stable appearing in the phasediagram instead of Sm2Fe17. At elevated temperatures,SmFe12 can be directly in equilibrium with the liquid phasein alloy compositions around Cu-45at.%Sm-1at.%Fe, asfar as the precipitation of Sm2Fe17 is suppressed. Theisothermal section at 1053 K is shown in Fig. 3(c), wherethe liquid-SmFe12 two-phase regions are indicated withred arrows. This two-phase equilibrium suggests that theliquid-phase sintering of the SmFe12-based magnets shouldbe possible without inducing the precipitation of bcc Fe.For the liquid phase with Cu compositions of at least 50at.%, the two-phase equilibrium appears in a temperaturerange of 1049–1163 K. In addition, the grain-boundarysubphase appearing with the solidification of this liquidphase is expected to have compositions similar to that ofthe liquid phase. Thus, the subphase becomes nonmag-netic as confirmed by first-principles calculations. Thenonmagnetic behavior of the subphase is preferable be-cause it hinders magnetic domain walls from propagat-ing.::In::::::::::::experiments,::::::::::::::::Sm12Fe74V12Cu2::::::alloys:::::have::::high:::::::::::::concentrations::of:::Sm::::and:::Cu::at::::::::::::intergranular:::::::regions,::::even::::::though:::the:::::alloy:::::::::::composition::::may:::be:::too:::Sm::::rich::::[17].::::The:::::::::theoretical::::::::findings::in:::::this:::::study::::can:::be::::::::::compared::::with::::these::::::::::::experimental::::::::::::observations::::and::::::should:::be:::::::utilized::to:::::design::::the::::::::::::::microstructures::in::::::more::::::details.:The Sm0.5Cu0.5-Sm0.5Fe0.5 transverse sections of theternary system are presented in Fig. 4 to compare effectsof the excess Gibbs energies estimated from the cluster-expansion method and from the dilute alloys. For the casewhere the Gibbs energy of mixing is estimated from the di-lute alloys, Fe has a negligibly limited solubility to the B2structure as seen in Fig. 4 (b), whereas the Gibbs energyof mixing estimated from the cluster-expansion method5(a)(b)T  [K]T  [K]xFeB2+SmFe2+α-SmB2+SmFe2+α-SmB2+SmFe2+LiquidB2+SmFe3+LiquidLiquidxFeSmFe2+LiquidSmFe3+LiquidSm2Fe17+LiquidLiquidB2+SmFe2+LiquidB2+SmFe3+LiquidSmFe2+LiquidSmFe3+LiquidSm2Fe17+LiquidB2B2+Liquid0 0.1 0.2 0.3 0.4 0.50 0.1 0.2 0.3 0.4 0.53005007009001100130030050070090011001300Figure 4: Pseudobinary phase diagram of Sm0.5Cu0.5−xFeFexFe .The Gibbs energy of the B2 phase, i.e., Sm(Cu1−yFeFeyFe ), is ob-tained with Redlich-Kister polynomials Lk :i,j (a) from the cluster-expansion method and (b) from dilute alloys.results in a much wider area of the B2 single phase as in-dicated with the red arrows in Fig. 4(a). The use of thecluster-expansion method improves the description of theB2 phase substantially, because this wider solubility of Feis consistent with experimental result [15] as was alreadymentioned. The discrepancy in the dilute-alloy approachcomes from significant underestimation of the magnitudeof the mixing enthalpy as shown in Fig. 2(i).3.3. Interaction between Cu and FeIt is well established that Cu and Fe are repulsive witheach other in the Cu-Fe binary system. This fact is seenin the Cu-Fe binary phase diagram as the phase separa-tion. The repulsive interaction makes the mixing enthalpypositive that can be seen for the A2 and B2 phases inFig. 5(a). The result for the Sm(Cu1−yFeFeyFe) B2 phasemakes remarkable contrast with the Cu-Fe binary caseas seen in Fig. 2(i), where the interaction between Cuand Fe becomes attractive within the B2 sublattices ofSm(Cu1−yFeFeyFe). This attractive interaction is the ori-gin of the Fe solubility into the B2 phase that is clear fromthe pseudobinary phase diagram in Fig. 4(a).Differences in the attractive interaction in ternary Sm-Cu-Fe and the repulsive one in binary Cu-Fe can be un-derstood by the local density of states (LDOS) shown inFigs. 5(b), (c), and (d). Here, we chose ordered B2 andMixing Enthalpy [eV]xFeCu Fe(a)(b) (d) Energy relative to εF [eV] Energy relative to εF [eV]LDOS [eV−1]LDOS [eV−1](b) Energy relative to εF [eV]LDOS [eV−1]Figure 5: (a) Mixing enthalpy of Cu1−xFeFexFe in the A2 bcc phaseand the B2 phase obtained by first-principles calculations. Atom-projected local density of states (LDOS) for (b) B2 CuFe, (c) L21Sm2CuFe, and (d) B2 SmFe.L21 structures as representative atomic configurations forconvenience. It is clear from Fig. 5(b) that 3d states of Cuand Fe strongly hybridize with each other in binary Cu-Fecreating broad 3d bands. Energies of minority-spin Cu 3dstates become in average higher by the hybridization com-pared with the case of the Cu single phase, which makesthe Cu-Fe interaction repulsive. In::::::::addition,::::::::::::minority-spin::Fe:::3d::::::states::::have::a:::::high:::::peak::at::::the::::::Fermi::::level:::εF::::also:::::::::::contributing::to::::::::repulsive:::::::::::interaction.:::::Note::::that:::the::::::LDOS::of:::the::::::::majority::::spin:::at::εF::is::::::::nonzero:::due:::to::4s::::::bands::::with::::wide:::::::::::dispersions.:::In:contrast, overlaps between 3d bandsof Cu and Fe are considerably small in ternary Sm-Cu-Feas is seen in Fig. 5(c), hence repulsive interaction betweenCu and Fe is absent. Compared with the phase-separatedcase of SmFe and SmCu, Fe in Sm-Cu-Fe is stabilized by anavoided peak of the minority-spin Fe LDOS at the Fermilevel εF that differs significantly from the case of SmFe inFig. 5(d). The attractive interaction between Cu and Fewithin the B2 phase makes the B2 phase more stable thanthe amorphous phase. This result indicates that the grain-boundary subphase in SmFe12-based sintered magnets canbe crystalline depending on the cooling rate in the sinter-ing process. The stability of the crystalline B2 phase evenin equilibrium can be considered as a potential advantage,because a crystalline grain-boundary subphase might con-tribute to alignment of the main-phase grains improvingthe coercivity significantly.::In::::::::addition:::to:::::L21,:::we:::::::::examine::::the::::::::stability:::of:::the:::::::so-called:::::XA:::::::::structure:::::that:::is::a:::::::variant:::::::::structure:::of:::::::Heusler::::::alloys::::::::relevant::::to::::::::::spintronics::::::::[66–68]:.::::::The:::XA:::::::::Sm2CuFe:::::was::::::found:::to:::be::::::::unstable:::::::against::::::phase::::::::::separations:::::::Hmix:::::::relative:::to:::B2::::::SmCu::::and::::B2:::::SmFe::is:::172:::::::::::meV/atom,:::::::whereas::::::Hmix:::::::relative::to::::bcc:::Sm::::and:::B2:::::CuFe::is::85:::::::::::meV/atom.:::::The::::::values:::for::::L21:::::::::Sm2CuFe:::are::::−69::::::::::meV/atom::::and:::::−155:::::::::::meV/atom,:::::::::::respectively.:64. ConclusionWe applied first-principles calculations to the CAL-PHADmethod and calculated the Sm-Fe-Cu ternary phasediagram. Formation energies of metastable stoichiometricendmembers including SmFe12 were calculated by DFT toconstruct Gibbs energies within the so-called compoundenergy formalism. Mixing enthalpies of dilute alloys werealso calculated to determine the interaction parametersof substituted solid solutions. Moreover, the Gibbs en-ergy for the B2 phase, which is a candidate for the grain-boundary subphase of SmFe12-based magnets, was eval-uated by the first-principles cluster-expansion method toimprove the accuracy. Mixing enthalpies among B2 end-members were converted to the Redlich-Kister polynomialthat represents the effect of second-nearest-neighbor inter-actions in the B2 structure. The thermodynamic databaseof this study reproduces the finite solid solution of Fe inB2 SmCu in agreement with experiments. We have foundthat this solubility is attributed to the attractive interac-tion between Fe and Cu in B2 Sm(Cu,Fe) solid solutions,which is in contrast with the repulsive one in binary Cu-Fe. Furthermore, phase equilibria of the SmFe12 phase andthe Sm-Cu-Fe alloys were investigated. We identified thatthe liquid phase with compositions about Cu-45at.% Sm-1at.%Fe directly equilibrates with SmFe12 above 1053 K.At lower temperatures, the equilibrating phase becomesB2 Sm-Cu-Fe. Since B2 Sm-Cu-Fe is much more stablethan the amorphous phase, the grain-boundary phase maybe crystalline depending on the cooling rate. In the devel-opment of SmFe12-based magnets, these phases are ex-pected to suppress the precipitation of bcc Fe associatedwith thermal decomposition of 1-12 grain surfaces as wellas the magnetic interaction between 1-12 grains.AcknowledgementsThe authors thank Dr. Arkapol Saengdeejing for fruit-ful discussions. This work was partly supported by MEXTas Fugaku-DPMSD (Grant No. JPMXP1020200307) andas DXMag (Grant No. JPMXP1122715503). The calcu-lations were partly carried out by using supercomputersat ISSP, The University of Tokyo, and TSUBAME, TokyoInstitute of Technology, as well as the supercomputer Fu-gaku, RIKEN (Project No. hp220175).Appendix A. Gibbs-energy function of SmFe12For the stoichiometric SmFe12 intermetallic compoundas an endmember, the assessed Gibbs-energy function isgiven in the unit of J/mol as:GSmFe12 = a+ bT +1213Gbcc−Fe +112Gα−Sm+Gmag , (A.1)where a = −6270 J/mol, b = 4 J/molK, Gbcc−Fe andGα−Sm are the Gibbs energies of pure bcc Fe and pureα Sm, respectively [59], and Gmag is the magnetic excessGibbs energy [57, 58] with the parameters given in themain text. Full list of Gibbs energies used to examinephase equilibria in this study is summarized in Supple-mental Material as a TDB file.::::The:::file:::::::::extension::::::should::be::::::::changed::::from::::.txt::to::::.tdb:::for::::the:::use::of::::::::::::::thermodynamic::::::::software.:References[1] K. Ohashi, Y. Tawara, R. Osugi, and M. Shimao, Magneticproperties of Fe-rich rare-earth intermetallic compounds with aThMn12 structure, J. Appl. Phys. 64 (1988) 5714-5716.[2] D. Wang, S. H. Liou, P. He, D. J. Sellmyer, G. C. Hadjipanayis,and Y. Zhang, SmFe12 and SmFe12Nx films fabricated by sput-tering, J. Magn. Magn. Mater. 124 (1993) 62-68.[3] T. Saito, H. Miyoshi, and D. Nishio-Hamane, Magnetic proper-ties of Sm-Fe-Ti nanocomposite magnets with a ThMn12 struc-ture, J. Alloys Compd. 519 (2012) 144-148.[4] Y. Hirayama, Y. K. Takahashi, S. Hirosawa, and K. Hono, In-trinsic hard magnetic properties of Sm(Fe1−xCox)12 compoundwith the ThMn12 structure, Scr. Mater. 138 (2017) 62-65.[5] H. S. Li and J. M. D. Coey, Magnetic properties of ternary rare-earth transition-metal compounds, in Handb. Magn. Mater.,vol. 6, 1-83 (Elsevier, North Holland, 1991).[6] T. Kuno, S. Suzuki, K. Urushibata, K. Kobayashi,N. Sakuma, M. Yano, A. Kato, and A. Manabe,(Sm,Zr)(Fe,Co)11.0−11.5Ti1.0−0.5 compounds as new per-manent magnet materials, AIP Adv. 6 (2016) 025221.[7] K. Kobayashi, S. Suzuki, T. Kuno, K. Urushihara, N. Sakuma,M. Yano, T. Shouji, A. Kato, and A. Manabe, The stabilityof newly developed (R,Zr)(Fe,Co)12−xTix alloys for permanentmagnets, J. Alloys Compd. 694 (2017) 914-920.[8] P. Tozman, H. Sepehri-Amin, Y. K. Takahashi, S. Hirosawa, andK. Hono, Intrinsic magnetic properties of Sm(Fe1−xCox)11Tiand Zr-substituted Sm1−yZry(Fe0.8Co0.2)11.5Ti0.5 compoundswith ThMn12 structure toward the development of permanentmagnets, Acta Mater. 153 (2018) 354-363.[9] P. Tozman, T. Fukazawa, D. Ogawa, H. Sepehri-Amin, A. Boly-achkin, T. Miyake, S. Hirosawa, K. Hono, and Y.K. Taka-hashi, Peculiar behavior of V on the Curie temperature andanisotropy field of SmFe12−xVx compounds, Acta Mater. 232(2022) 117928.[10] I. Dirba, Y. Harashima, H. Sepehri-Amin, T. Ohkubo, T.Miyake, S. Hirosawa, and K.Hono, Thermal decomposition ofThMn12-type phase and its optimum stabilizing elements inSmFe12-based alloys, J. Alloys Compd. 813 (2020) 152224.[11] L. Zhao, R. Su, L. Wen, W. Li, X. L. Liu, Z. H. Zhang, R.Z. Zhao, Y. Y. Han, and X. F. Zhang, Intrinsically High Mag-netic Performance in Core-Shell Structural (Sm,Y)Fe12-BasedPermanent Magnets, Adv. Mater. 34 (2022) 2203503.[12] H. Sepehri-Amin, Y. Tamazawa, M. Kambayashi, G. Saito, Y.K. Takahashi, D. Ogawa, T. Ohkubo, S. Hirosawa, M. Doi,T. Shima, and K. Hono, Achievement of high coercivity inSm(Fe0.8Co0.2)12 anisotropic magnetic thin film by boron dop-ing, Acta Mater. 194 (2020) 337-342.[13] T. Fukazawa, H. Akai, Y. Harashima, and T.Miyake, First-principles study of intersite magnetic couplings and curie tem-perature in RFe12−xCrx (R = Y, Nd, Sm), J. Phys. Soc. Jpn.87 (2018) 044706.[14] Y. Harashima, T. Fukazawa, H. Kino, and T. Miyake, Effect ofR-site substitution and the pressure on stability of RFe12: Afirst-principles study, J. Appl. Phys. 124 (2018) 163902.[15] C. Lu, X. Hong, X. Bao, X. Gao, and J. Zhu, Changing phaseequilibria: A method for microstructure optimization and prop-erties improvement in preparing anisotropic Sm2Fe17N3 pow-ders, J. Alloys Compd. 784 (2019) 980-989.[16] O. Takeda, K. Shimizu, H. Zhu, and S. Sugimoto, Phase equilib-ria of Sm-Fe-Ti-M (M=Cu, Ag) systems and wetting behavior7of M–Sm alloys on SmFe11Ti substrate, J. Magn. Magn. Mater.542 (2022) 168620.[17]::::A.M.:::Sch:̈o::nh:̈o:::bel,:::R.::::::::::Madugundo,::::J.M.:::::::::Barandiar:́a:n,:::::G.C.::::::::::Hadjipanayis,::D.::::::::::Palanisamy,::T.:::::::Schwarz,::B.::::::Gault,::D.::::::Raabe,::K.:::::::Skokov,:::O.:::::::::Gutfleisch,::J.::::::::::Fischbacher,::::and:::T.:::::::Schrefl,::::::::::::Nanocrystalline::::::::Sm-based::::1:12::::::::magnets,:::::Acta::::::Mater.::::200:::::(2020)::::::652-658.:[18]::::A.M.:::Sch:̈o::nh:̈o:::bel,:::R.::::::::::Madugundo,:::::A.M.::::::Gabay,::::and:::::J.M.:::::::Barandiaŕ:::an,:::The:::::::Sm-Fe-V:::::based::::1:12:::bulk::::::::magnets,::J.:::::Alloys::::::Compd.:::791:::::(2019):::::::1122-112.:[19]::::A.M.::::::Gabay,::C.::::Han,:::C.::Ni,::::and::::G.C.:::::::::::Hadjipanayis,:::::Effect::of::::::alloying::::with:::Sc,:::Nb::::and::Zr:::on::::::::::::::reduction-diffusion::::::::synthesis:of:::::::::::magnetically::::hard::::::::::::::::::Sm(Fe,Co,Ti)12-based:::::::::::::monocrystalline:::::::powders,:J.::::::Magn.:::::Magn.:::::Mater.::::541::::(2022)::::::168550.:[20]:::::::::A.M.Gabay:::and::::G.C.:::::::::::Hadjipanayis,::::::::::::Microstructure:::and:::::Hard:::::::Magnetic::::::::Properties::of:::::::::::::::::::::::::Sm1−xZrx(Fe,Co)11.3−yTi0.7By :::::Ingots:::and:::::Thick::::::::Melt-Spun:::::::Ribbons,:::::IEEE:::::Trans.::::::Magn.::58:::::(2022)::::::2100805.:[21]:::::::::A.M.Gabay:::and::::G.C.:::::::::::Hadjipanayis,::::::::Isotropic::::::::::::nanocrystalline::::::::::::::Sm(Fe,Co)11.3Ti0.7:::::::magnets:::::::modified::::with::B:::and:::Zr,::J.:::::Magn.:::::Magn.:::::Mater.::::529:::::(2021)::::::167867.:[22]::::A.M.::::::Gabay,::::C.:::::Han,:::C.::::Ni,::::and:::::G.C.::::::::::::Hadjipanayis,::::::::::::Re-investigation::of:::::::::::::high-temperature:::::phase:::::::equilibria::in::::::Fe-rich:::::::Sm–Fe–Ti::::::alloys,::J.:::::Alloys::::::Compd.::::947:::::(2023)::::::169520.:[23]::::A.M.:::::::Gabay::::and::::::G.C.::::::::::::Hadjipanayis,::::::::::::::High-coercivity::::::::::ThMn12-type::::::::::::::monocrystalline:::::::::::::::Sm–Zr–Fe–Co–Ti::::::::particles::by::::::::::::::high-temperature::::::::reduction::::::::diffusion,::::Scr.:::::::Mater.::::196:::::(2021)::::::113760.[24] Y. Ainai, T. Shiozawa, Y. Tatetsu, and Y. Gohda, First-principles study on surface stability and interface magneticproperties of SmFe12, Appl. Phys. Express 13 (2020) 045502.[25] Y. Gohda, First-principles determination of intergranularatomic arrangements and magnetic properties in rare-earth per-manent magnets, Sci. Technol. Adv. Mater. 22 (2021) 113-123.[26] Y. Gohda, Y. Tatetsu, and S. Tsuneyuki, Electron Theory onGrain-Boundary Structures and Local Magnetic Properties ofNeodymium Magnets, Mater. Trans. 59 (2018) 332-337.[27] T. Mohri, S. Takizawa, and K. Terakura, First-principles cal-culation of heats of formation for Au-Cu, Au-Pd and Au-Agalloys with thermal vibration effects, J. Phys.: Condens. Mat-ter 5 (1993) 1473-1480.[28] P. Pavone, S. Baroni, and S. de Gironcoli, α ↔ β phase tran-sition in tin: A theoretical study based on density-functionalperturbation theory, Phys. Rev. B 57 (1998) 10421-10423.[29] J.-C. Crivello and J.-M. Joubert, First principles calculations ofthe σ and χ phases in the Mo-Re and W-Re systems, J. Phys.:Condens. Matter 22 (2010) 035402.[30] F. Körmann, A. Dick, T. Hickel, and J. Neugebauer, Role ofspin quantization in determining the thermodynamic propertiesof magnetic transition metals, Phys. Rev. B 83 (2011) 165114.[31] A. van de Walle, Methods for First-Principles Alloy Thermody-namics, JOM 65 (2013) 1523–1532.[32] M. Palumbo, S. G. Fries, A. Dal Corso, F. Körmann, T. Hickel,and J. Neugebauer, Reliability evaluation of thermophysicalproperties from first-principles calculations, J. Phys.: Condens.Matter 26 (2014) 335401.[33] T. Mohri, Cluster Variation Method as a Theoretical Tool forthe Study of Phase Transformation, Metall. Mater. Trans. A 48(2017) 2753-2770.[34] T. Tanaka and Y. Gohda, Prediction of the Curie temperatureconsidering the dependence of the phonon free energy on mag-netic states, npj Comput. Mater. 6 (2020) 184.[35] T. Tanaka and Y. Gohda, First-Principles Study of Magnetism-Dependent Phonons Governed by Exchange Ligand Field, J.Phys. Soc. Jpn. 89 (2020) 093705.[36] A. Saengdeejing and Y. Chen, Improving Thermodynamic Sta-bility of SmFe12-Type Permanent Magnets from High EntropyEffect, J. Phase Equilib. Diffus. 42 (2021) 592-605.[37] R. Yamada, M. Ohno, and T. Mohri, Configurational kinet-ics studied by Path Probability Method, Prog. Mater. Sci. 120(2021) 100765.[38] N. Saunders and A. P. Miodownik, CALPHAD : calculationof phase diagrams : a comprehensive guide, (Elsevier, NorthHolland, 1998).[39] T. Abe, M. Morishita, Y. Chen, A. Saengdeejing, K. Hashimoto,Y. Kobayashi, I. Ohnuma, T. Koyama, and S. Hirosawa, Devel-opment of a prototype thermodynamic database for Nd-Fe-Bpermanent magnets, Sci. Technol. Adv. Mater. 22 (2021) 557-570.[40] P. E. A. Turchi, V. Drchal, J. Kudrnovsky, C. Colinet, L. Kauf-man, and Z. K. Liu, Application of ab initio and CALPHADthermodynamics to Mo-Ta-W alloys, Phys. Rev. B 71 (2005)094206.[41] A. Berche, J. C. Tédenac, and P. Jund, Ab-initio calculationsand CALPHAD description of Cr-Ge-Mn and Cr-Ge-Si, Cal-phad 49 (2015) 50-57.[42] A. Saengdeejing, Y. Chen, M. Matsuura, S. Sugimoto, First-principles study of stability of Cu in the Nd-rich and Nd oxidephases of Nd-Fe-B permanent magnet, J. Chin. Chem. Soc. 63(2016) 506-512.[43] A. Saengdeejing, Y. Chen, O. Takeda, M. Enoki, S. Sugimoto,H. Ohtani, and T. Abe, Sm-Ti binary thermodynamic databaseand phase diagram, Calphad 75 (2021) 102357.[44] W. Xiong, Q. Chen, P. A. Korzhavyi, and M. Selleby, An im-proved magnetic model for thermodynamic modeling, Calphad39 (2012) 11-20.[45] J. W. D. Connolly and A. R. Williams, Density-functional the-ory applied to phase transformations in transition-metal alloys,Phys. Rev. B 27 (1983) 5169-5172.[46] J. M. Sanchez, F. Ducastelle, and D. Gratias, Generalizedcluster description of multicomponent systems, Physica A 128(1984) 334-350.[47] P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B50 (1994) 17953-17979.[48] G. Kresse and J. Hafner, Ab initio molecular dynamics for open-shell transition metals Phys. Rev. B 48 (1993) 13115-13118.[49] G. Kresse and J. Furthmüller, Efficient iterative schemes forab initio total-energy calculations using a plane-wave basis set,Phys. Rev. B 54, (1996) 11169-11186.[50] G. Kresse and D. Joubert, From ultrasoft pseudopotentials tothe projector augmented-wave method, Phys. Rev. B 59 (1999)1758-1775.[51] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M.R. Pederson, D. J. Singh, and C. Fiolhais, Atoms, molecules,solids, and surfaces: Applications of the generalized gradientapproximation for exchange and correlation, Phys. Rev. B 46(1992) 6671-6687.[52] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized GradientApproximation Made Simple, Phys. Rev. Lett. 77 (1996) 3865-3868.[53]:::X.B.::::Liu:::and::Z.:::::::::Altounian,:::::::Magnetic::::::::moments:::and::::::::exchange::::::::interaction::in:::::::::Sm(Co,Fe)5::::from:::::::::::first-principles,:::::::Comput.::::::Mater.:::Sci.::50::::(2011):::::::841-846.:[54] A. van de Walle, M. Asta, and G. Ceder, The alloy theoreticautomated toolkit: A user guide, Calphad 26 (2002) 539-553.[55] A. van de Walle and M. Asta, Self-driven lattice-model MonteCarlo simulations of alloy thermodynamic properties and phasediagrams, Modell. Simul. Mater. Sci. Eng. 10 (2002) 521-538.[56] G. Ghosh, A. van de Walle, and M. Asta, First-principles calcu-lations of the structural and thermodynamic properties of bcc,fcc and hcp solid solutions in the Al-TM (TM=Ti, Zr and Hf)systems: A comparison of cluster expansion and supercell meth-ods Acta Mater. 56 (2008) 3202-3221.[57] G. Inden, The role of magnetism in the calculation of phasediagrams, Physica B+C 103 (1981) 82-100.[58] M. Hillert and M. Jarl, A model for alloying in ferromagneticmetals, Calphad 2 (1978) 227-238.[59] A. Dinsdale, SGTE data for pure elements, Calphad 15 (1991)317-425.[60] M. A. Turchanin, P. G. Agraval, and I. V. Nikolaenko, Ther-modynamics of alloys and phase equilibria in the copper-ironsystem, J. Phase Equilib. 24 (2003) 307-319.8[61] W. Zhuang, Z.-Y. Qiao, S. Wei, and J. Shen, Thermodynamicevaluation of the Cu-R (R: Ce, Pr, Nd, Sm) binary systems, J.Phase Equilib. 17 (1996) 508-521.[62] X. L. Chen, J. Wang, T. L. Chen, X. D. Lin, M. H. Rong, G.H.Rao, and H. Y. Zhou, Thermodynamic re-assessment of theFe-Gd and Fe-Sm binary systems, Calphad 58 (2017) 151-159.[63] PANDAT, CompuTherm LLC: https://computherm.com/ (ac-cessed on 2022.12.21)[64] Thermo-Calc, Thermo-Calc Software AB:https://thermocalc.com/ (accessed on 2022.12.21)[65]:S.:::::::Nishino:::and:::Y.::::::Gohda,::::::::Structures::of::::::Sm-Cu:::::::::::intermetallics:::with:::Fe::as::::::::subphase::::::::candidates:::in:::::::::::SmFe12-based:::::::::permanent::::::magnets::::::studied:::by::::::::::::first-principles:::::::::::::thermodynamics,:::::Jpn.::J.::::Appl.:::::Phys.,::in:::::press.:[66]:S.::::::Idrissi,::S.::::Ziti,::H.:::::::Labrim,:::and::L.:::::::Bahmad,:::::::::::::Half-Metallicity:::and:::::::::Magnetism::in:::the::::Full::::::Heusler:::::Alloy:::::::Fe2MnSn::::with::::L21:::and:::XA:::::::Stability:::::::Ordering:::::::Phases,::J.::::Low:::::Temp.:::::Phys.::::202:::::(2021):::::::343–359.[67]:S.:::::::Idrissi,::L.::::::::Bahmad,:::R.::::::::Khalladi,::I.:::El:::::::Housni,:::N.:::El::::::::Mekkaoui,::S.:::::::Mtougui,::H.::::::Labrim,::::and::S.:::Ziti,:::::Phase::::::::diagrams,:::::::electronic::::and:::::::magnetic::::::::properties::of::::the::::::::quaternary:::::::Heusler:::alloy::::::::::NbRhCrAl,::::::Chinese::J.::::Phys.:::60::::(2019):::::::549–563.:[68]:S.::::::Idrissi,:::H.::::::Labrim,::S.::::Ziti,::::and::L.:::::::Bahmad,:::A::::DFT:::::study:of:::the:::::::::equiatomic::::::::quaternary::::::Heusler:::::alloys::::::::ZnCdXMn:::::::(X=Pd,::Ni::or:::Pt),:::::Solid::::State::::::::Commun.:::331:::::(2021)::::::114292.9