Accuracy of ab initio methods in predicting the crystal...

Computer Coupling of Phase Diagrams and Thermochemistry 29 (2005) 163–211

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Accuracy ofab initio methods in predicting the crystal structures ofmetals: A review of 80 binary alloys

Stefano Curtaroloa,∗, Dane Morganb, Gerbrand Cederb

aDepartment of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, United StatesbDepartment of Materials Science and Engineering, MIT, Cambridge, MA 02139, United States

Accepted 21 January 2005Available online 22 February 2005

Abstract

Predicting and characterizing the crystal structure of materials is a key problem in materials research and development. Weresults ofab initioLDA/GGA computations for the following systems: AgAu, AgCd, AgMg, AgMo∗, AgNa, AgNb∗, AgPd, AgRh∗, AgRu∗,AgTc∗, AgTi, AgY, AgZr, AlSc, AuCd, AuMo∗, AuNb, AuPd, AuPt∗, AuRh∗, AuRu∗, AuSc, AuTc∗, AuTi, AuY, AuZr, CdMo∗, CdNb∗,CdPd, CdPt, CdRh, CdRu∗, CdTc∗, CdTi, CdY, CdZr, CrMg∗, MoNb, MoPd, MoPt, MoRh, MoRu, MoTc∗, MoTi, MoY∗, MoZr, NbPd,NbPt, NbRh, NbRu, NbTc, NbY∗, NbZr∗, PdPt, PdRh∗, PdRu∗, PdTc, PdTi, PdY, PdZr, PtRh, PtRu, PtY, PtTc, PtTi, PtZr, RhRu, RhRhTi, RhY, RhZr, RuTi, RuTc, RuY, RuZr, TcTi, TcY, TcZr, TiZr∗, YZr∗ (∗ = systems in which theab initio method predicts that nocompounds are stable). A detailed comparison to experimental data confirms the high accuracy with whichab initio methods can predictground states.© 2005 Elsevier Ltd. All rights reserved.

Keywords: Binary alloys; Ab initio; Intermetallics; Transition metals; Structure prediction; Phase stability; Aluminum; Cadmium; Gold; MagneMolybdenum; Niobium; Palladium; Platinum; Rhodium; Ruthenium; Scandium; Silver; Sodium; Titanium; Technetium; Yttrium; Zirconium

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1. Introduction

First principles computation, whereby the propertieof materials are predicted starting from the principles oquantum mechanics, is becoming well integrated with motraditional materials research. A list ofab initio studies onbinary and ternary alloy phase stability up to 1994 can bfound in Ref. [1]. Since the earliest, completelyab initiocomputation of a binary phase diagram [2], the approachesfor computing the total energy of a solid have significantlimproved, and computing resources have continuedbecome faster and less expensive. We believe that a pointbeen reached where, with a reasonable amount of resourhigh throughput first principles studies of a large numbof alloys can be performed [3–8]. In this paper we presentthe results of a first principles study of 14 080 compute

∗ Corresponding author.E-mail address:[email protected] (S. Curtarolo).

0364-5916/$ - see front matter © 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.calphad.2005.01.002

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total energies on 176 crystal structures in 80 binary alloAll energies were computed in the Local Density (LDAor Generalized Gradient Approximation (GGA) to DensiFunctional Theory, which are standard approaches for fiprinciples studies on solids. To our knowledge this is tlargest first principles study of its kind on alloys. As whave compared the results in every system to experimecompilations, this study also offers a statistical test on taccuracy of some currentab initio approaches in correctlypredicting the structure of materials.

For 89 compounds we find unambiguous agreembetween experiment and theab initio computation (Table 5),giving some indication of the predictive power of moderab initio electronic structure methods. For many systemverification of theab initio results is difficult, as the systemshave been poorly or incompletely characterized, or onhigh temperature information is available experimentalFor most of these system, we make predictions thatconsistent with the limited available information. Evethough our library of 176 crystal structures is, to ou

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164 S. Curtarolo et al. / Computer Coupling of Phase Diagrams and Thermochemistry 29 (2005) 163–211

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knowledge, the largest library ofab initio energies everproduced, there are still 27 compounds for which we cannverify the experimental structures as they are not in oulibrary. We have not included such prototypes because thare extremely rare and complicated (many atoms per ucell) (Table 8).

Overall, we find remarkably few significant discrepanciebetween theab initio predictions and the experimentalobservations (Table 9). On the basis of the experimental datain Refs. [9,10], we find only nine compounds for whichLDA/GGA seems to predict the ground state incorrectly. Fofour of these nine systems, the experimental ground statewithin less than 10 meV/atom of theab initio ground state.For the remaining five systems, there are at least twowhich further investigation indicates that the experimentstructure assignment is poorly justified, leaving threcompounds for which a significant disagreement betweeexperiment andab initio LDA/GGA is likely. Suchdisagreements are addressed inSection 4.

We believe that the low ratio of unambiguous error(3) to the number of unambiguous correct predictions (89is encouraging, and establishes clearly the potentialpredicting crystal structure correctly withab initio methods.

We also predict the stability of five new crystal structurewhich, to the best of our knowledge, have not yet beeobserved in any system: an AB3 superstructure of the fcclattice, stable for CdPt3, PdPt3 and Pd3Pt, an AB bccsuperstructure for MoTi, an AB3 bcc superstructure forMoTi3, Mo3Ti, Nb3Tc, RuTi3 and TcTi3, an A2B2 hcpsuperstructure for RhRu, and an A2B4 hcp superstructure forRhRu2 (Appendix A). In addition, we find two new crystalstructures which are not superstructures of fcc, bcc or hcMo5Ti (Mo5Ti and Nb5Ru) (Appendix B).

2. The library: alloys and structures

Binary alloys

Our calculated library contains 80 binary intermetallicalloys. The alloys include the binaries that can be madfrom row 5 transition metals, as well as some systems wialuminum, gold, magnesium, platinum, scandium, sodiumtitanium, and technetium. The alloys are: AgAu, AgCdAgMg, AgMo∗, AgNa, AgNb∗, AgPd, AgRh∗, AgRu∗,AgTc∗, AgTi, AgY, AgZr, AuCd, AuMo∗, AuNb, AuPd,AuPt∗, AuRh∗, AuRu∗, AuSc, AuTc∗, AuTi, AuY, AuZr,AlSc, CdMo∗, CdNb∗, CdPd, CdPt, CdRh, CdRu∗, CdTc∗,CdTi, CdY, CdZr, CrMg∗, MoNb, MoPd, MoPt, MoRh,MoRu, MoTc∗, MoTi, MoY∗, MoZr, NbPd, NbPt, NbRh,NbRu, NbTc, NbY∗, NbZr∗, PdPt, PdRh∗, PdRu∗, PdTc,PdTi, PdY, PtRh, PtRu, PtY, PtTc, PdZr, PtTi, PtZr, RhRuRhTc, RhTi, RhY, RhZr, RuTi, RuTc, RuY, RuZr, TcTi,TcY, TcZr, TiZr∗, YZr∗, where the superscript∗ indicatesthe systems in which thehigh throughput ab initiomethodpredicts that no compounds are stable: 57 alloys forcompounds and 23 are non-compound forming.

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Structures and their prototypes

The library contains 176 crystal structures. Many of theshave the same structure type but with different compositioof occupancies, for example, AB3 and A3B (also AB andBA if the point groups of atomic positions of A and B aredifferent), so the number of distinct prototypes is 101. Thvarious concentrations are listed inTable 1.

Table 1Compositions, concentrations and number of prototypes inside the librar

Compounds Concentration Number ofcomposition of B (%) prototypes

A & B 0 4

A5B & AB 5 16.66 3

A4B & AB 4 20 2

A3B & AB 3 25 27

A2B & AB 2 33.33 32

A5B3 & A 3B5 37.5 2

A3B2 & A 2B3 40 1

A4B3 & A 3B4 42.85 1

AB (& BA ∗) 50 29 (+3)

The library has 176 structures, and 101 distinct prototypes (∗ at compositionAB, three prototypes have different point groups in atomic positions A anB; therefore they represent distinct structure types).

Of such prototypes, 65 are chosen from the mocommon intermetallic binary structures in the CRYSTMETdatabase [11] and in the Pauling File [10]. Such prototypescan be described by their Strukturbericht designation andnatural prototypes [9,10]: A1, A2, A3, A4, A15, Bh, B1,B2, B3, B4, B81, B82, B10, B11, B19, B27, B32, B33(Bf ), Cc, C2, C6, C11b, C14, C15, C15b, C16, C18, C22,C32, C33, C37, C38, C49, D0a, D03, D09, D011, D019,D022, D023, D024, D13, D1a, D2d, D73, D88, L10, L11,L12, L60, CaIn2, CuTe, CuZr2, GdSi2 (1.4), MoPt2, NbAs(NbP), NbPd3, Ni2Si, Ω (with z = 1/4), Pu3Al (Co3V),Ti3Cu4, W5Si3, YCd3, ZrSi2, γ -Ir. The rest of the structures(36) are fcc, bcc or hcp superstructures. Twelve of thesuperstructures consist of stacking of pure A and B planalong some direction. Of such prototypes, 12 containstacking direction; therefore we can name them followinthe parent lattice and the stacking direction:

LATTICE[direction]stacking . (1)

For example the designation FCC[001]A2B2 indicates a structure

prototype with FCC parent lattice and A2B2 stacking alondirection [001]. The 12 prototypes of the library that can blabeled in this manner are: BCC[011]

AB2 , BCC[211]AB2 , FCC[001]

A2B2,

FCC[011]A2B2, FCC[111]

A2B2, FCC[311]A2B2, FCC[100]

AB2 , FCC[111]AB2 ,

FCC[001]AB3 , FCC[011]

AB3 , FCC[011]AB3 , FCC[111]

AB3 . For the FCC

S. Curtarolo et al. / Computer Coupling of Phase Diagrams and Thermochemistry 29 (2005) 163–211 165

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superstructures, a conversion table for converting betwethe lattice-stacking direction, the Sanchez–de Fontaine notion [12], and the Lu et al. designations [13,14] is includedin Table 2.

Table 2Conversion table for converting between the lattice-stacking direction,Sanchez–de Fontaine notation [12], and the Lu et al. designations [13,14]

Space group Sanchez–de Lu et al. HereFontaine [12] [13,14] and [15]

AB2 I4/mmm #139 AB2(c) β1/β2 FCC[100]AB2

AB2 P3m1 #164 α1/α2 FCC[111]AB2

AB3 P4/mmm #123 Z1/Z3 FCC[001]AB3

A2B2 P4/nmm #129 AB(a) Z2 FCC[001]A2B2

A2B2 C2/m #12 W2 FCC[311]A2B2

AB3 Pmmm #47 Y1/Y3 FCC[011]AB3

A2B2 Pmmn #59 AB(e) Y2 FCC[011]A2B2

A2B2 I41/amd #141 CH or “40” FCC[201]A2B2

AB3 R3m #166 V1/V3 FCC[111]AB3

A2B2 R3m #166 V2 FCC[111]A2B2

The prototypes of stable structures, that cannotdescribed by Strukturbericht designation, natural prototypor the lattice-stacking-direction convention, are describedAppendix A. The complete geometrical description of all thfcc, bcc, and hcp superstructures of the library can be fouin Ref. [15].

3. High throughput, first principles calculations

Ultra Soft Pseudopotential LDA calculations (US-LDA)

Most of the energy calculations of the library werperformed using density functional theory in the LocDensity Approximation (LDA), with the Ceperley–Alderform for the correlation energy as parameterizedPerdew and Zunger [18] with ultra soft Vanderbilt-typepseudopotentials [19], as implemented in VASP [20].Calculations are at zero temperature and pressure,without zero-point motion. The energy cutoff in an alloy waset to 1.5 times the larger of the suggested energy cutoffsthe pseudopotentials of the elements of the alloy (suggesenergy cutoffs are derived by the method described in [20]).Brillouin zone integrations were done using 2000/(numbof atoms in unit cell)k-points distributed as uniformlyas possible on a Monkhorst–Pack mesh [21,22]. For asample set of calculations, we verified that with these enecutoffs andk-point meshes the absolute energy is convergto better than 10 meV/atom. Energy differences betwee

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structures are expected to be converged to much smatolerances. Spin polarization was used only for magnealloys. For the non-magnetic ones, spin polarization wused only to verify the energies of the ground stastructures. All structures were fully relaxed. With sucmethodology, we consider degenerate structures the othat have energies closer than about 5 meV/atom.

We have decided to calculate the entire library within tLDA formalism instead of the PAW-GGA method (discussebelow) because of the faster execution speed of LDA.

PAW-GGA calculations

When several structures are in close competition forground state, we also performed calculations in the Genalized Gradient Approximation (GGA), with Projector Augmented Wave (PAW) pseudopotentials, as implementedVASP [20,23,24]. One would expect the PAW-GGA approach to be somewhat more accurate than the US-LDA.the GGA correlation energy, we used the Perdew–Wangrameterization (GGA-PW91) [25]. Similar to the US-LDAcase, all PAW-GGA calculations are performed at zero teperature and pressure, and without zero-point motion. Tenergy cutoff in each calculation was set to 1.75 timeslarger of the suggested energy cutoffs of the pseudopotials of the elements of the alloy. Brillouin zone integrations were done using at least∼3500/(number of atoms inunit cell) k-points distributed as uniformly as possible onMonkhorst–Pack mesh [21,22]. Spin polarization was usedin all calculations. We expect such calculations to be ato distinguish energies of structures previously degenerThis is because the increased density of thek-point mesh,the reduced radial cutoff of the PAW potentials versus tultrasoft pseudopotentials, and the GGA correlation eneimplementation. With such methodology, we consider dgenerate structures the ones that have energies closer1 meV/atom.

For some structures both the US-LDA and PAWGGA calculations are presented. To avoid confusion,specify the type of calculation in brackets: “(us-lda)” o“(paw-gga)”. In addition, all the results of the PAW-GGAcalculations are given in italics.

Symmetries of the pure elements

Both the US-LDA and PAW-GGA calculations reproducthe correct experimental crystal structures of the puelements at room temperature. The only exception iscase for sodium. In the two formalisms, Na-hcp is veslightly favored over Na-bcc and Na-fcc, in agreemewith other first principle calculations [26–30]. At roomtemperature sodium has the bcc structure, and undergomartensitic transformation below 35 K, to a close packstructure [31–38]. Therefore our results, Na-hcp stable anthe very small energy differences with Na-bcc and Na-fcare consistent with the behavior at low temperature only.


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High throughput computing

To perform the high number of calculations, we havimplemented a set of automatic tools to prepare initial daperform calculations and analyze results.

Preparation of input filesAll input files are prepared by starting from th

templates based on the prototype chemistry. The voluof the unit cells is determined as a linear combinatioof atomic volumes of pure elements with Vegardlaw [39]. The internal position of the atoms have beetaken from geometrical configurations (for fcc, bchcp superstructures) or have been extracted fromCRYSTMET and the Pauling File databases [11,10].

Ab initio calculationsThe calculations are performed by a high throughp

tool, called AFLOW (automatic flow), which searches insidthe library for potential input files and performs the propab initio calculations. AFLOW is able to balance thCPU loads in a multiprocessor and cluster environmemaximizing the total throughput of the process [15].AFLOW extracts the energy and initiates further relaxatiowith the atomic positions and unit cell geometries obtainfrom VASP. Therefore, all the structures are relaxedleast twice, and many (∼10% of the total) are relaxedthree or more times, until proper geometrical and energeconvergence is obtained. To conclude, for a library176 crystal structures in 80 binary alloys, with a totof 14 080 structures, AFLOW performed 32,402 VAScalculations. The total computing time used for this projewas∼9.05× 108 CPU seconds (∼28.7 CPU years), whichwas spread over a large set of computers [40]. Given thedifferent types of CPUs used, we can onlyestimatethe totalamount of computation for the project at roughly∼1.2×106

TERAFLOP.

Collection and error check of the resultsOnce each structure has been calculated, space g

symmetry, bond distances, coordination numbers and uvolumes are compared for all the structures of that alloy wthe same concentration in the library. Frequently, sevestructures, starting from different prototypes, relax to tsame final structure with the same energy. Therefore,identify the correct stable phase, it is mandatory to be ato determine the final relaxed structure with the highepossible reliability. Details of the high throughput errochecking tool can be found in Ref. [15]. A few stablestructures remain unidentified. For those we report the ucell and atomic positions inAppendix A. Such prototypesmight be new phases to be checked experimentally.

Calculation of the formation energies and the convex hullThe formation energy for each structure is determin

with respect to the most stable structure of the pure eleme

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To determine the ground states of a system one needs toas a function of composition, the ordered compounds thave an energy lower than any other structure or any lincombination of structures that gives the proper compositiThis set of ground state structures forms aconvex hull, asall other structures have an energy that falls above theof tie lines that connects the energy of the ground statesthermodynamical terms, theconvex hullrepresents the Gibbsfree energy of the alloy at zero temperature.

Comparison with experiments

To compare theab initio results to experimentalinformation, we relied on the information and referencin the Binary Alloy Phase Diagrams (Massalski [9]) and inthe Pauling File [10]. Though in some cases references nincluded in these were also used, no systematic approacgo beyond information in the Pauling File or Binary AlloPhase Diagrams was used.

4. Discussion and summary of results

In comparing the stable structures predicted by theabinitio computations with available experimental informatiowe have attempted to classify the results in a few distincategories.Table 5gives the compounds where theab initioresult and experiments are in unambiguous agreement.fact that there are a large number of compounds (8in Table 5 is a positive statement about the accuracyLDA/GGA in capturing the close energetic competitiobetween the 176 structures in our library. For one of tsystems inTable 5(PdTi3) we took the liberty of modifyingthe experimental result [9] which shows the A15 structure(stoichiometry AB3) as a line compound at compositioPdTi4. While off-stoichiometric compounds are obviouspossible, this usually goes together with significant widof the single-phase field. Hence, we concluded (mayerroneously) that the placement of A15 at composition Pd4in Ref. [9] is likely a typographical error.

Table 6 shows compositions which are experimentalknown to form compounds, but lack a complete identifiction of the structure type. Hence, theab initio predictionshould be seen as a likely crystal structure for the compouIn most cases, theab initio structure is consistent with theconditions imposed by the limited experimental data (ebcc ordering is seen for Ag–Cd, and we predict B19; or Cis speculated for Au–Sc, in agreement with what we predicHence, it is likely that with further experimental characterzation, many of these systems would move toTable 5.

Tables 7a and 7b contains compositions which areexperimentally characterized as solid solutions, hitemperature two-phase regions, or have not been studat all in a particular composition range. Because of tlack of low temperature information in these systeman unambiguous statement about the accuracy of


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LDA/GGA prediction cannot be made. For some systemtheab initio result is very consistent with the experimentallobserved behavior. For example, in Ag–Au we finmany fcc ordered superstructures with low formatioenergies, indicating a weak ordering interaction. Sit is likely that the ab initio predicted structuresare stable at low temperature, but disorder intosolid solution at elevated temperature, a rather commoccurrence for noble-metal alloys. For other systems in ttable the comparison between experiments andab initio ismore troubling. For example, while Mo–Ti is experimentalldescribed as a two-phase system with no known compoudown to 400C, we find very strong compound formationWhile it is possible that these compounds disappear throuperitectoid reactions below 400C, the large negativeformation energies make this unlikely. It is more likelthat either the experimental characterization of this systeor the ab initio result is significantly in error. A systemsuch as Mo–Nb, on the other hand, is not at all studiexperimentally, and no statement about the accuracy ofprediction in this system can be made.

Table 8contains the compounds whose crystal structuis not present in the library, and hence, it is not possibto make a comparison with theab initio data. In somecases (e.g. Au10Zr7), this is because the crystal structuris uncommon and has a large unit cell, making it leworthwhile to include it in the library. In other cases (e.gthat of Rh5Ti) the structure is simply unknown.Table 8alsoincludes structures that are only stable at off-stoichiometcompositions (e.g. B19 in Au–Ti), as we did not includdisorder in the library prototypes.

Finally, Table 9 contains the compositions for whichthere seems to be a clear disagreement betweenexperimental data in Refs. [9,10] and theab initio results.Because these systems can indicate either experimeerrors, or shortcomings of theab initio method, we discussthem here in more detail. Note thatab initio errors aremost likely due to the LDA or GGA approximations othe pseudopotentials being used. In general, one shoconsider that our calculations only produce the zerkelvin energy, whereas all the experimental results aat non-zero temperature. Given that entropic differencbetween structures can be of the order 0.1–1.0kB peratom [375], very small energy differences between thexperimentally observed structure and ourab initio resultscould be reversed at elevated temperature. In particuAuNb3-A15, NbRh3-L12, NbRu3-L12 and NbRu′-L10 maybe in this category. This phenomenon is quite commoAs an example, Wolverton and Ozilins have shown thin the case of the aluminum–copper system, the vibratioentropy difference is responsible for stabilizing theθ phaseAl2Cu (tetragonal C16) over the competing Al2Cu-θ ′ phase(distortion of θc-C1, the cubic fluorite phase with CaF2prototype), which has the lowest energy, and, thereforethe stable structure at zero kelvins [376]. At temperatureshigher thanT ∼ 150–200C, Al2Cu-θ is more favorable.

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4.1. Discussion of disagreement between experimentalab initio results

AuNb3

The structure AuNb3-A15 has been reported a numbeof times [9,10,126–140]. In particular, Röschel et al. [131]observed the A15 structure to grow in quenched sampSamples of the material ball-milled to form disordered brevert upon heat treatment in an exothermic reactionA15, suggesting that A15 could be the ground state for tcomposition at low temperature. The relatively small enerdifference between A15 and the AuNb2 ↔ Nb tie-linein our calculation could lead to finite temperature stabiliof A15. Alternatively, the∼7 meV/atom energy difference(US-LDA and PAW-GGA) could simply be an error of thab initio approach.

AuYWhile we predict the B33 structure to be stab

by 26 meV/atom below the B2 structure, this systeis experimentally listed as having a B2 structure. Thexperimental classification is based on a paper by Chet al. [169,10]. Chao’s work mentioned that they onlysee B2 when performing a rapid quench. In further woKusma [171] and Dwight [157] did not observe the B2structure. The experimental results in this systemtherefore somewhat suspect. It is possible that B2 is a htemperature structure while B33 is the low temperatuform. However, it should also be pointed out that the mixinenergies in this system are very large (∼1 eV), and thatthe potential error of∼25 meV/atom (US-LDA and PAW-GGA) is a small fraction of this.

Cd3NbThe structure of this compound gives one of the mo

interesting discrepancies betweenab initio and experimentsto emerge from our study. On the basis of experimental wof Von Holleck [177], a single L12 compound is claimedto be stable in this system. Computationally, no compounare stable at all, and we find this system to be immiscibThe compound with lowest energy is a bcc superstructat composition CdNb, and energy∼55 meV/atom abovethe Cd ↔ Nb tie-line (∼62 meV/atom with PAW-GGA).The Cd3Nb-L12 is 70 meV/atom above that same tie-line(>100 meV/atom with PAW-GGA). It is not common forLDA/GGA to make such large and qualitative errorsmetals, and further investigation of this system is suggesas it will likely lead to a reconsideration of the experimentclassification, or to some novel understanding of the errthatab initio methods can produce in metals.

CdPt3Experimentally this compound is claimed to form i

the L12 structure, while we predict it to form in a newstructure for which no prototype yet exists (CdPtproto

3 inAppendix VIII). We believe the experimental assignment


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be uncertain. It is based on a single paper by Nowotny [180]which actually does not observe the L12 structure in thissystem, but assigns it by chemical similarity with the Pt3Znsystem.

NbRh3This compound is characterized as an L12 structure,

whereas ourab initio results find the Al3Pu prototypeto be 8 meV/atom lower in energy(5.3 meV/atom withPAW-GGA). This small energy difference could very welbe an LDA/GGA error, or could be reversed at elevatetemperature by entropic effects.

NbRu3NbRu3 shows a similar discrepancy to the compoun

NbRh3. Experimentally it is found to be an L12 structure,whereas we find a D024 structure that is 8 meV/atombelow the L12. With PAW pseudopotentials and theGGA exchange–correlation corrections the D024-L12 energydifference reduces to 2.5 meV/atom. Because this numberis so small, extremely largek-point sets and high energycutoff were used to converge it. The results indicate that L2and D024 are all but degenerate in this system. Since D024and L12 are very similar (D024 is a periodically antiphasedversion of L12) subtle entropic effects can easily reverse thstability between these two.

NbRuExperimentally this system is classified as having th

L10 structure, whereas theab initio results indicate anunusual two-phase region between Nb3Ru (D03) and NbRu2(C37) covering this composition [9,10,227,231]. The mostrecent experimental result is based on the assignmentChen [231] who observes a disordered bcc solution at higtemperature and a transformation to a tetragonal structnear 1000C. Between 720 and 920C they see a furthersymmetry reduction to an orthorhombic phase. Hence, evif the tetragonal structure below 1000C were to be L10(which is only speculated by Chen) its further transformatioto an orthorhombic structure would indicate that L10 is notthe ground state structure. On the basis of a diffusion coustudy, Hurley [229] believes that a two-phase region existnear 50% Ru, though the results seem to be irreproducibBesides the difficulty with interpreting the experimentadata, significant problems also exist on theab initio sidewith this compound. The PAW-GGA result is substantialldifferent from the US-LDA result. GGA gives L10 only4 meV/atom above the tie-line (versus 20 meV/atom iLDA), whereas B19 seems to be>100 meV/atom above thetie-line (versus 13 meV/atom with LDA).

PtYExperimental assignment of the structure of th

compound as B27 is based on two papers [271,310]. Theab initio work finds B33 to be lowest in energy with B27

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higher by 50 meV/atom (60 meV/atom with PAW-GGA).Given that the experimental results seem sound, thisprobably a trueab initio error.

Pt3Zr5

The work of Schubert et al. [317] establishes Pt3Zr5 as aD88 structure, whereas ourab initio (LDA) results put theD88 structure 36 meV/atom above the tie-line formed byPtZr2-C16 and PtZr-B33. The GGA result is 26 meV/atomabove the tie-line. In each case the W5Si3 structure is lowerin energy than the D88. Given the large mixing energies inthis system (of the order of 1 eV/atom), theseab initio errorsseem plausible.

5. Conclusions

Overall, the comparison between experimental data aab initio results is encouraging. A large number of thground states are predicted correctly, even though significcompetition exists between various structures, indicatithat relative energy differences are well reproducedLDA/GGA. In many cases, the direct comparison betweab initio and experiments is difficult, as systems have oftnot been studied completely, or have not been studiedlow enough temperature to make a reliable statement abthe ground state structures. For only a small numbercases, there is a clear discrepancy betweenab initio andexperiments (Table 9). On further investigations, we find thasome of the experimental assignments are poorly justifi(e.g. for CdPt3). In a few of the “error systems”, the energdifference between the observed structure and theab initiostructure is small, which makes it fall within the uncertainone would expect from LDA/GGA or from a lack of entropicconsiderations.

A few systems display, in our opinion, such a significadiscrepancy between available experimental data aab initio results as to make them worthwhile for furthestudy. These include Cd3Nb (L12 observed, but nocompounds fromab initio study), Nb–Ru and Pt3Zr5(respectively L10 and D88 observed, but no stablecompounds at that composition inab initio study), PtY(significant error between B33 and B27) and the compleMo–Ti system (which from experiments appears to havemiscibility gap, but in theab initio results displays a numberof novel and strongly stabilized compounds). These systeshould be investigated further, either to find experimenerrors, or to shed new light onab initio problems.

Our findings are summarized inTable 3. By comparingall the availableab initio calculations with the experimen-tal results [9,10], we obtain 284 potential structure comparisons. A subset of these (48) are not available duethe lack of the proper prototype or the unknown expemental compound. The rest of the structure compariso(236) can be divided into 89agreementsin ordered sys-tems (Nac), 21 agreementsin immiscible systems(Nai ),


obtain

Table 3Summary of the 284 structure comparisons between experimental andab initio calculations

Total experiment comparisons: 284

Available comparisons: 236

Experimental Ab initio # Table

Agreements Compound Compound Nac 89 5

Agreements Immiscibility Immiscibility Nai 21 3

Disagreements Comp./ Immisc. Incorrect Nd 9 9

Good predictions Comp. unknown, Compound Ngp 21 6

uncertain, guessed prediction

Possible predictions Not studied, Possible Npp 92 7aand7b

solid solution, compound

two-phase region

Unavailable comparisons: 48

Reason # Table

Unavailableab initio prototype 21 8

Unknown experimental compound 27 8

When available, we use the PAW-GGA calculations; otherwise we use the US-LDA ones. If only US-LDA calculations were used, we wouldNLDA

ac = 84 agreements andNLDAd = 14 disagreements.

vean

ing

tit.

thth

l

or

ef

ssth

d

9 disagreements(Nd), 21 good predictions(Ngp), and92 possible predictions(Npp). This division is summarizedin Table 3.

The reliability of ourab initio method can be definedas the fraction of correct compounds (agreements) othe number of accessible compounds (agreementsdisagreements):

ηc ≡ Nac

Nac + Nd≈ 90.8%. (2)

Such a quantity,ηc, contains the reliability of theexperimental results, which can be removed by consideronly thetrue unambiguous disagreementsof Table 9(N

d =3). Hence the reliability of the method becomes

ηc ≡ Nac

Nac + Nd

≈ 96.7%. (3)

Such quantities,ηc and ηc, can be considered as the

probabilities that the most stableab initio compoundsreproduce the correct experimental compound. The quanη

c should be used if the experimental compound is certainIf we extend the argument to the agreements wi

experimental compounds and immiscible systems, thenreliabilities become:

η ≡ Nac + Nai

Nac + Nai + Nd≈ 92.4%, (4)

η ≡ Nac + Nai

Nac + Nai + Nd

≈ 97.3% (5)

rd

y

e

where the quantityηc should be used if the experimenta

compound or immiscibility is certain. Such quantities,η

andη, can be considered as the probabilities thatab initioresults reproduce the correct experimental compoundsimmiscibilities.

Given the large number of calculations of this project, wbelieveη, η, ηc, andη

c to be good estimations of state othe art pseudopotentialab initio accuracies in phase stabilityprediction.

6. Alloys without ab initio compounds

Table 4gives the alloys for which we find no compoundwith negative formation energy, and the structure with loweformation energy in the system. All of these agree witexperiments except for the ones described below.

Cd–Nb

One compound, Cd3Nb-L12, has been reported forthe system Cd–Nb [9,177]. However, we did not findany stable phase (Cd3Nb-L12 has formation energy of∼70 meV/atom). The disagreement for the compounCd3Nb-L12 is further discussed inSection 4.


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Table 4Systems without intermetallic compounds

System Structure E f (us-lda) References

with lowestE f (meV/atom) [9–11] and

Ag–Mo AgMo3-BCC 208 [80,81]

proto. #72 in [16]

Ag–Nb AgNb3-BCC 62 [85,86]

proto. #72 in [16]

Ag–Rh AgRh3-FCC[111]AB3 116 [49,93–95]

Ag–Ru AgRu3-HEX 199 [41,42,95–97]

proto. #127 in [16]

Ag–Tc AgTc-HEX 147 [98]

proto. #126 in [16]

Au–Mo AuMo3-D019 75 [41,45,117–121]

Au–Pt Au2Pt-FCC[111]AB2 9 [41,116,118,145–147]

Au–Rh AuRh3-FCC[001]AB3 91 [115,118,148,149]

Au–Ru AuRu3-HEX 165 [115,118,150,151]

proto. #125 in [16]

Au–Tc AuTc-HEX 74 [115,158]

proto. #126 in [16]

Cd–Mo Cd5Mo-HEX 173 [9]

proto. #128 in [16]

Cd–Nb CdNb-BCC 55 (us-lda) [177]

proto. #71 in [16] 58 (paw-gga) See note “Cd–Nb”

Cd–Ru CdRu-FCC[001]A2B2 88 See note “Cd–Ru”

Cd–Tc CdTc-B11 92 [181]

Cr–Mg CrMg-B11 251 [44,190]

Mo–Tc Mo5Tc-A15 53 [43,211]

See note “Mo–Tc”

Mo–Y Mo2Y-C15 40 [216,217]

Nb–Y Nb2Y-C49 132 [238–242]

Nb–Zr NbZr2-HEX 22 [243–247,89]

proto. #129 in [16] [232–234,248–252]

Pd–Rh PdRh3-FCC[111]AB3 35 [13,257–264]

Pd–Ru PdRu-HEX 44 [265–268]

proto. #126 in [16]

Ti–Zr Ti2Zr-B82 18 [47,48,365–368]

Y–Zr YZr5-HEX 35 [48,369–374]

proto. #141 in [16]

The second and third columns give, for each system, the structure with thlowest formation energyE f (US-LDA calculations). The table contains23 entries. 21 of these are in agreement with experiments. Referencinclude both experimental andab initio studies.

e

s

Cd–Ru

The authors are not aware of any experimental result fthe Cd–Ru system.

Mo–Tc

Two compounds have been suggested on the basisthermodynamic models:βMo2Tc3-A15 andσMo3Tc7-D8b

[9,43]. However, we did not find any stable phase fothe system, nor can we check the suggested compounsince our library does not have theσ phase or the off-stoichiometry A15.

7. Alloys with ab initio compounds

Ag–Au (silver–gold)

The system Ag–Au has not been studied in gredetail at low temperatures, and no intermetallic compounhave been reported [9,10,51–54,118]. The solid is reportedto be short-range ordered fcc, though it is suggestthat long-range order might exist at low temperatureAt low temperature we find several stable compoundAg4Au, Ag3Au, Ag2Au, AgAu-L10, AgAu2 and AgAu3.In our electronic structure approach, the ground statare degenerate for Ag3Au, Ag2Au, AgAu2 and AgAu3.Our best candidates are L12, D023, Al3Pu, NbPd3, D022and D024 for Ag3Au, C37 and MoPt2 for Ag2Au andAgAu2, and L12, D022 and D023 for AgAu3, as shown inFig. 1. The considerable degeneracy of the ground statindicates that only very small effective interactions exisbetween Ag and Ag, consistent with the experimentalobserved complete solid solubility. For composition Ag4Au,the structure D1a is degenerate with the tie-line of the two-phase region Ag↔ Ag3Au. We conclude that furtherexperimental and theoretical investigations are necessfor determining the behavior of AgAu.To address thedegenerate structures, we further investigate Ag3Au, Ag2Au,AgAu2 and AgAu3 with PAW-GGA potentials, as described inSection 3. With PAW, for composition Ag3Au, L12 is the moststable compound and D022, D023, Al3Pu, NbPd3, D024 arehigher by 0.9, 1.6, 2.2, 2.4, and 3.0 meV/ atom, respectively.For both compositions Ag2Au and AgAu2, C37 is the moststable compound and MoPt2 is higher by 1.9 meV/atomand 4.3 meV/atom, respectively. For composition AgAu3,L12 is the most stable compound and D023 andD022 are higher by 3.4 meV/atom and 3.9 meV/atom,respectively.

Otherab initio studies, relevant for this system, can bfound in Refs. [55–65].


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in-teeon

nd

eng

Ag–Au system

Low temperature phases comparison chart

Composition Experimental Ab initio% Au (Massalski [9]) result

20 Short-range order950C

Ag4Au-D1a/tie-line

25 Short-range order Ag3Au950C L12/D023/Al3Pu/ (us-lda)

/NbPd3/D022/D024 (us-lda)L12 stable (paw-gga)D022∼0.9 meV/atomD023∼1.6 meV/atomAl3Pu∼2.2 meV/atomNbPd3∼2.4 meV/atomD024∼3.0 meV/atomabove L12 (paw-gga).

33.3 Short-range order Ag2Au-C37/MoPt2 (us-lda)950C C37 stable (paw-gga)

MoPt2∼1.9 meV/atomabove C37 (paw-gga).

50 Short-range order AgAu-L10950C

66.6 Short-range order AgAu2-C37/MoPt2 (us-lda)950C C37 stable (paw-gga)

MoPt2∼4.3 meV/atomabove C37 (paw-gga).

75.0 Short-range order AgAu3950C L12/D022/D023 (us-lda)

L12 stable (paw-gga)D023∼3.4 meV/atomD022∼3.9 meV/atomabove L12 (paw-gga).

Fig. 1. AgAu (silver–gold) ground state convex hull.

Ag–Cd (silver–cadmium)

At composition AgCd3 we find D019 stable, while onlydisordered hcp solid solution has been reported atcomposition. Hence, D019, an ordered hcp superstructurprobably is stable at low temperature in this systeAt composition AgCd2 we find a previously unknown

s

compound AgCd2–ZrSi2. The low temperatureβ ′, nearAgCd stoichiometry, is reported to be ordered bccMassalski [9]. In particular, the Pauling File reports AgCdB2 [10,66–69]. We find B2, B19 and B27 to have degeneraenergies (B19 is quite common for Cd alloys). Thexperimental phase diagram only displays solid solution the Ag-rich side [9,10]. We find C37 at Ag2Cd and astable phase with stoichiometry Ag3Cd. The ground stateof Ag3Cd is degenerate: three distinct structure (D019,D022, D024) have similar energy.To address the degeneratestructures, we further investigate AgCd and Ag3Cd withPAW-GGA potentials, as described inSection 3. With PAW,for compound AgCd, B19 is the most stable structure, aB27 and B2 are higher by 2.8 meV/atom and 3.4 meV/atom,respectively. For compound Ag3Cd, D022 and D024 arestill degenerate ground states and D019 is higher by only2.2 meV/atom. It is possible that these compounds havnot yet been observed, or that they have low orderitemperatures. (SeeFig. 2.)

Ag–Cd system


Composition Experimental Ab initio% Cd (Massalski [9]) result

25.0 fcc solid solution Ag3CdD019/D022/D024 (us-lda)D022/D024 (paw-gga)D019 ∼ 2.2 meV/atomabove D022/D024 (paw-gga)

33.3 fcc solid solution Ag2Cd-C37

48.5–50 β′-bcc AgCd-B2/B19/B27 (us-lda).B2 [10] B19 stable (paw-gga)

B27∼2.8 meV/atomB2∼3.4 meV/atomabove B19 (paw-gga).

66.6 none AgCd2–ZrSi2

64.5–81 hcp solidsolution

AgCd3-D019

Fig. 2. AgCd (silver–cadmium) ground state convex hull.


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rgythe

Ag–Mg (silver–magnesium)

The phase diagram of the system Ag–Mg is knowwith reasonable accuracy [9,10,70,71,74,76]. Our ab initiomethod confirms the stability of AgMg-B2. In the Ag-richside of the phase diagram, at stoichiometry Ag3Mg, wefind D023 and D024 to be degenerate. While Massalski [9]indicates Ag3Mg to be L12, it is known from detailed HighResolution Electronic Microscopy (HREM) that Ag3Mgforms Long-Period Superstructure (LPS) modulationsL12 [72,73] (LPS D023 is present in our library). Fora detailed discussion of this system see Kulik et a[75]. L12 is 15.2 meV/atom above D023/D024. Onthe Mg-rich side, our library does not have prototypeat 80% Mg concentration, so we are not able tfind any hexagonal AgMg4 (with unknown prototype[74,76]). However we find a stable compound AgMg3. Twophases, AgMg3-D0a and AgMg3-D019, have degenerateenergies. This is important, since experiments suggthat AgMg4 was probably identified as hexagonal AgMg3in early studies. Unfortunately, we cannot determine thcorrect ab initio solution since we lack the hexagonaAgMg4 prototypes in our library.To address the degeneratestructures, we further investigate Ag3Mg and AgMg3with PAW-GGA potentials, as described inSection 3. Forcompound Ag3Mg, with PAW, D023 is the most stablecompound (in agreement with[72,73,75]), D024 and L12are 1.1 meV/atom and 11.3 meV/atom higher than D023,respectively. In addition, for compound AgMg3, D019 is themost stable structure andD0a is ∼6.8 meV/atom higherthan D019.

Otherab initio studies, relevant for this system, can bfound in Refs. [77–79]. (SeeFig. 3.)

Ag–Mg system


Composition Experimental Ab initio

% Mg (Massalski [9]) result

25.0 L12 Ag3Mg-D023/D024 (us-lda)

D023 [72,73,75] L12∼15 meV/atom above.

D023 stable (paw-gga)

D024∼1.1 meV/atom

L12∼11.3 meV/atom

above D023 (paw-gga)

35.5–65.4 B2 AgMg-B2

75.8–78.2 cF∗ (unknown) AgMg3D019 (hP8)/D0a (oP8) (us)


D0a∼6.8 meV/atom

aboveD019 (paw-gga)

80.0 hP∗ (unknown) Unavailable (see text)

t

Fig. 3. AgMg (silver–magnesium) ground state convex hull.

Ag–Na (silver–sodium)

The phase diagram of the system Ag–Na is knowith reasonable accuracy and has only one intermetacompound [9,10,82–84]. We confirm the stability ofAg2Na-C15 and find almost all the other compounto have positive formation energies. Both with US-LDand PAW-GGA, Na-hcp is very slightly favored over Nabcc and Na-fcc, in agreement with other first principcalculations [26–30]. At room temperature sodium has thbcc structure, and undergoes a martensitic transformabelow 35 K, to a close packed structure [31–38]. Thereforeour results, Na-hcp stable and the very small enedifferences from Na-bcc and Na-fcc, are consistent withbehavior at low temperature. (SeeFig. 4.)

Ag–Na system


Composition Experimental Ab initio% Na (Massalski [9]) result

33.0 C15 Ag2Na-C15

Fig. 4. AgNa (silver–sodium) ground state convex hull.


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d

ra-te.

ee

Ag–Pd (silver–palladium)

The system Ag–Pd has not been studied in gredetail and no intermetallic compounds have been repor[9,10,41,87]. The solid is reported to be disordered fccAt low temperature we find three stable compoundAgPd-L11, Ag2Pd and Ag3Pd. In our computations, forAg2Pd and Ag3Pd, the ground states are degenerate: obest candidates are C49 or C37 for Ag2Pd, and L12 or D022for Ag3Pd, as shown inFig. 5. To address the degeneratestructures, we further investigate Ag2Pd and Ag3Pd withPAW-GGA potentials, as described inSection 3. With PAW,for composition Ag2Pd, C37 is the most stable compounand C49 is higher by 4 meV/atom. For composition Ag3Pd,L12 and D022 remain degenerate. In fact, D022 has anenergy 0.4 meV/atom greater than L12, too small comparedto the numerical accuracy of the ab initio calculation.

Otherab initio studies, relevant for this system, can bfound in Refs. [88–92].

Ag–Pd system


Composition Experimental Ab initio% Ag (Massalski [9]) result

50 Solid solution AgPd-L11>900C

66.6 Solid solution Ag2Pd>900C C37/C49 (us-lda)

C37 stable (paw-gga)C49∼4 meV/atomabove C37 (paw-gga)

75.0 Solid solution Ag3Pd>900C L12/D022 (us & paw-gga)

Fig. 5. AgPd (silver–palladium) ground state convex hull.

Ag–Ti (silver–titanium)

The stability of AgTi2-C11b and AgTi-B11 is con-firmed [9,10,99,100], and no other stable phases are foun

td

:

r

computationally. Hence, we conclude that the low tempeture part of the phase diagram of AgTi is probably accura(SeeFig. 6.)

Ag–Ti system



33.3 C11b AgTi2-C11b

50 B11 AgTi-B11(γ CuTi)

Fig. 6. AgTi (silver–titanium) ground state convex hull.

Ag–Y (silver–yttrium)

The stability of AgY-B2 and Ag2Y-C11b is confirmed[9,10,101]. In the Ag-rich region, we do not determine thstability of Ag51Y14, since we do not have the prototypin our library [102,103]. Instead, we find Ag3Y with D0a

structure stable with an energy∼22 meV/atom below theenergy of a two-phase region Ag↔ Ag2Y. In the realsystem, the presence of Ag51Y14 probably makes structureD0a metastable. In the Y-rich region we find AgY2-C37degenerate with the two-phase region Y↔ AgY (within∼1.7 meV/atom) as shown inFig. 7.

Ag–Y system



33.3 Two-phase region AgY2-C37/tie-lineabove 200C

50 B2 AgY-B2

66.6 C11b Ag2Y-C11b

75 Two-phase region Ag3Y-D0aabove 200C (uncertain)

78.5 Ag51Gd14 Unavailable


d[

s,

at

w

e

Fig. 7. AgY (silver–yttrium) ground state convex hull.

Ag–Zr (silver–zirconium)

The phase diagram of Ag–Zr is known partially, anit has been estimated from thermodynamic properties9,10,104,105]. The stability of the two known phases AgZr-B11 (γ -CuTi prototype) and AgZr2-C11b is confirmed byour calculation. In addition to the known intermetalliccompounds, we find a stable phase Ag2Zr, which isnot present in Massalski [9]: the C6 and C32 structuresare degenerate. To conclude, we find AgZr3-FCC[001]

AB3degenerate with the two-phase region Zr↔ AgZr2 (within∼1.3 meV/atom). To address the degenerate structurewe further investigate Ag2Zr with PAW-GGA potentials,as described inSection 3. With PAW, C32 is the moststable compound and C6 is 2 meV/atom above C32. (SeeFig. 8.)

Ag–Zr system



% Ag (Massalski [9]) result

25.0 Two-phase region AgZr3

estimated FCC[001]AB3 /tie-line

above 700C

33.3 C11b AgZr2-C11b

50 B11 AgZr-B11(γ CuTi)

66.6 Two-phase region Ag2Zr

estimated C6/C32 (us-lda)

above 700C C32 stable (paw-gga)

C6∼2 meV/atom

above C32 (paw-gga)

Fig. 8. AgZr (silver–zirconium) ground state convex hull.

Al–Sc (aluminum–scandium)

The experimental phase diagram has compoundscompositions: Al3Sc, Al2Sc, AlSc, and AlSc2 [9,10,106–108]. The ab initio technique confirms the stabilityof Al3Sc-L12, Al2Sc-C15, AlSc-B2, and AlSc2-B82. Inthe Sc-rich region of the phase diagram, we find a nehexagonal stable phase AlSc3-D019, which is not presentin Massalski [9]. Only very limited experimental data isavailable for this side of the phase diagram.


Al–Sc system


Composition Experimental Ab initio% Al (Massalski [9]) result

25.0 Two-phase region AlSc3-D019above 0C

33.3 B82 AlSc2-B82

50 B2 AlSc-B2

66.6 C15 Al2Sc-C15

75 L12 Al3Sc-L12

Fig. 9. AlSc (aluminum–scandium) ground state convex hull.


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Au–Cd (gold–cadmium)

Several compounds have been reported for the systAu–Cd, and the phase diagram is known with reasonaaccuracy [9,10,41,116,118,122–125]. We confirmβ ′-AuCd-B19 and the high temperature phaseβ-AuCd-B2 (withenergy∼6 meV/atom higher than B19). At 25% Cd composition, we do not find the compoundα1Au3Cd–Ag3Mgbecause we do not have the proper prototype in our libraInstead, we find structures D024, D019, and Al3Pu (Co3V)with degenerate energies. Such structures have hexanal lattices and they are candidates for the phase fieldα2which has been reported to be a long-period superstrture with hP lattice [9]. At 33% Cd composition, we findthree phases Au2Cd-C37, Au2Cd–MoPt2, and Au2Cd-C49with energies 4, 14 and 16 meV/atom above the tie-lineAu3Cd ↔ AuCd, respectively. At 50% Cd compositionthe prototype of the phaseβ ′′-AuCd is not known, but it isreported to have a hexagonal structure. Forβ ′′, we suggesttwo candidates: AuCd-L10 and AuCd-FCC[201]

A2B2 (CH “40” inRef. [256]), which have energies higher by 5.4 meV/atomand 5.5 meV/atom with respect to B19. None of thesstructures has hexagonal lattice type, and all our hexagoprototypes have much higher energies.

Au–Cd system



∼25 α1-Ag3Mg Unavailable

∼25.5–35.5 α2-hP? (unknown) Au3CdD024/D019/Al3PuhP candidates forα2

33.3 Two-phase region Au3Cd ↔ AuCdAu2Cd-C37, MoPt2, C49above tie-line

∼47.5 β′-B19 AuCd-B19

∼50 β′′ unknown L10/FCC[201]A2B2 ∼5 meV/atom

43–57 β-B2 (high T) B2∼6 meV/atomabove B19

61.6–62.1 Au3Cd5-D8m Unavailable

72–76 ε′ unknown AuCd3-L60tentative phase field

∼82 η′ unknown Nothing stabletentative phase field at 83.3%

At concentration∼75% Cd, a stable phaseε′, withunknown structure, is believed to exist [9]. We confirm theexistence of a stable compound and our best predictis AuCd3-L60. At concentration∼82% Cd another stablephaseη′ is believed to exist. We are not able to confirmη′. In fact at 83.3% Cd, the phase with lowest energy∼60 meV/atom higher than the two tie-line AuCd3 ↔ Cd.

me

.

o-

-

al

n

To address the degenerate structures, we further investigAu3Cd with PAW-GGA potentials, as described inSection 3.Also with PAW, D024, Al3Pu and D019 remain degenerate.(SeeFig. 10.)

Fig. 10. AuCd (gold–cadmium) ground state convex hull.

Au–Nb (gold–niobium)

The phase diagram of the system Au–Nb is basedthree known compounds (Au2Nb, Au2Nb3, AuNb3) [9,10,46,116,118,126–140]. We do not confirm the stability ofphase Au2Nb-C32 [9,10,46]. Instead of C32, we find astable compound Au2Nb-C6 and C32 with energy higherby ∼12 meV/atom with respect to C6. We cannot concludanything about stability of Au2Nb3 since our library doesnot contain any A2B3 prototype. The Nb-rich side of theexperimental phase diagram is drawn with the assumptthat AuNb3-A15 is stable at low temperature (∼500C) [9].Our calculations suggest that AuNb3-A15 might not bestable at low temperature as it is∼7 meV/atom above thetwo-phase field AuNb2 ↔ Nb. In addition, we find a stablephase AuNb2 at 66.6% Nb composition, with bcc parenlattice, and space group Fmmm #69. The compound, labeas BCC[011]

AB2 , has AB2 stacking along the [011] direction. I

the prototype BCC[011]AB2 were not included in the calculation

the phase AuNb3-A15 would be stable, as shown inFig. 11.To conclude, we think it might be worthwhile to reconsidethe thermodynamic modeling of the Nb-rich side of thphase diagram with the inclusion of AuNb2-BCC[011]

AB2 , toobtain the proper temperature and concentration ranof phase A15.To address the disagreements with thexperimental results, we further investigate Au2Nb andAuNb3 with PAW-GGA potentials, as described inSection 3.For the compound Au2Nb C32 is the most stable compounand C6 is higher by 3.2 meV/atom. For composition AuNb3,A15 is still unstable, being∼6.5 meV/atom above thetie-line AuNb2 ↔ Nb. The disagreement at compositioAuNb3 is further discussed inSection 4.


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Au–Nb system


Composition Experimental Ab initio% Nb (Massalski [9]) result

33.3 C32 Au2Nb-C6 (us-lda).C32∼12 meV/atomabove C6 (us-lda).C32 stable (paw-gga)C6 ∼ 3.2 meV/atomabove C32 (paw-gga).

60 Au2Nb3 (unknown) UnavailabletI10 I4/mmm

66.6 Two-phase region AuNb2-BCC[011]AB2

(calculated)

73–83 A15 Two-phase regionAuNb2↔ Nb.AuNb3-A15 is∼7 meV/atomabove tie-line (us-lda).A15 is∼6.5 meV/atomabove tie-line (paw-gga).SeeSection 4.

Fig. 11. AuNb (gold–niobium) ground state convex hull.

Au–Pd (gold–palladium)

Two ordered compounds have been suggested forsystem Au–Pd in the fcc solid solution (Au, Pd): Au3Pdand AuPd3 [141–144]. The low temperature part of thephase diagram has been constructed with the additionphase field AuPd that is believed to exist below∼100 C[9,10,116,118]. In the Au-rich part of the phase diagram, wconfirm the stability of a compound Au3Pd, but we find threestructures, D023, D022 and L12, with degenerate energyAt composition Au4Pd, we find the structure D1a to bedegenerate with the tie-line Au↔ Au3Pd. At composition33.3% Pd, we find a stable compound Au2Pd, but again,two structures C49 and C37 are degenerate. At composi50%, we confirm the existence of a stable phase AuPd.

e

a

ne

prototype is not known. Our best guess is AuPd-FCC[201]A2B2

(CH “40” in Ref. [256]), and L10 with an energy higher by8 meV/atom with respect to FCC[201]

A2B2. Finally, in the Pd-richpart of the phase diagram, we find a compound AuPd3 to bedegenerate with the tie-line of the two-phase region AuPd↔Pd. For this compound, three structures, D023, D022 and L12,have degenerate energies. The considerable degenerathe ground states is indicative of weak effective interactioand consistent with the significant miscibility of thetwo elements inside each other. The near degeneracystructures such as D023, D022 and L12, which are relatedto each other by periodic antiphase boundaries, indicathat more complicated Long-Period Superstructures (LPmight be present.To address the small energy differencebetween structures, we further investigate Au3Pd, Au2Pd,AuPd, and AuPd3 with PAW-GGA potentials, as described inSection 3. For the compound Au3Pd, D023 is the most stablecompound and D022 and L12 are higher by 9.4 meV/atomand 12.7 meV/atom, respectively. For the compound AuP3,D023 is the most stable compound and L12 and D022 arehigher by 1.8 meV/atom and 4.8 meV/atom, respectiveAlso with PAW, for the compound Au2Pd, C37 and C49remain degenerate. For the compound AuPd, FCC[201]

A2B2 is themost stable compound and L10 is higher by 10.8 meV/atom.

Anotherab initio study, relevant for this system, can bfound in Ref. [59]. (SeeFig. 12.)

Au–Pd system



% Pd (Massalski [9]) result

12–32 Au3Pd-L12 Au3Pd

(guessed) D023/D022/L12 (us-lda)


D022∼9.4 meV/atom

L12∼12.7 meV/atom


20 Au4Pd-D1a/ tie-line (us-lda)

33.3 Solid solution Au2Pd

C49/C37 (us-lda/paw-gga)

50 AuPd (guessed) AuPd-FCC[201]A2B2(us-lda)

unknown L10 ∼ 8 meV/atom

prototype above FCC[201]A2B2 (us-lda)

FCC[201]A2B2stable (paw-gga)

L10∼10.8 meV/atom

above FCC[201]A2B2 (paw-gga).

68–90 L12 AuPd3-D023/D022

(guessed) /L12/tie − line (us-lda)


L12∼1.8 meV/atom

D022∼4.8 meV/atom



A

li cc[A nwlta eA deg

er

s,

drdsein

the

,

te

ry.

se.

Fig. 12. AuPd (gold–palladium) ground state convex hull.

u–Sc (gold–scandium)

The phase diagram of the system Au–Sc is known onn the Au-rich region, and a total of three intermetalliompounds have been reported: Au4Sc, Au3Sc, and AuSc9,10,118,152–157]. At 20% Sc, we confirm Au4Sc-D1a.t 25% Sc, we find a two-phase region, in agreemeith experiments. In fact, Au3Sc-D0a, the structure with

owest energy, is higher by∼7 meV/atom with respecto the tie-line Au4Sc-D1a ↔ Au2Sc-C11b. At 33% andt 50% concentrations of Sc, we find two stable phasu2Sc and AuSc, but in both cases two structures areenerate C11b and MoPt2 for Au2Sc, and B2 and B19

Au–Sc system



% Sc (Massalski [9]) result

20 D1a Au4Sc-D1a

25 Two-phase region Two-phase region

Au3Sc-D0a ∼ 7 meV/atom

above Au4Sc↔ Au2Sc.

33.3 C11b Au2Sc-C11b/MoPt2 (us-lda)

C11b stable (paw-gga)

MoPt2∼1.0 meV/atom

above C11b (paw-gga).

50 B2 AuSc-B2/B19 (us-lda)

B2 stable (paw-gga)

B19∼2.2 meV/atom

above B2 (paw-gga).

66.6 C37 (Co2Si) AuSc2-C37

(guessed)

y

t

s-

for AuSc. The experimental results indicate Au2Sc-C11band AuSc-B2. The AuSc2-C37 (Co2Si prototype) compoundhas been speculated to exist by similarity with othAu-(RareEarth)2 systems [152]. Our ab initio methodconfirms AuSc2-C37.To address the degenerate structurewe further investigate Au2Sc and AuSc with PAW-GGApotentials, as described inSection 3. For compound Au2Sc,with PAW, Au2Sc-C11b is the most stable compound anAu2Sc–MoPt2 is higher by 1.0 meV/atom. In addition, focompound AuSc, Au2Sc-B2 is the most stable compounand Au2Sc-B19 is higher by 2.2 meV/atom. While theenergy differences are very small, both calculations areagreement with experiments. (SeeFig. 13.)

Fig. 13. AuSc (gold–scandium) ground state convex hull.

Au–Ti (gold–titanium)

Five ordered compounds have been reported forsystem Au–Ti at low temperature: Au4Ti, Au2Ti, αAuTi,βAuTi, and AuT3 [9,10,47,100,159–168,277,292,305,315].We confirm the stability ofαAuTi-B11, Au4Ti-D1a andAuTi3-A15. At 25% Ti, we find a two-phase regionin agreement with experiments. In fact, Au3Ti-D0a, thestructure with lowest energy, is higher by∼11 meV/atomwith respect to the tie-line Au4Ti ↔ Au2Ti. Forcompound Au2Ti, the experimental phase is C11b. Atsuch composition we find two structures with degeneraenergies: Au2Ti-C11b and Au2Ti–MoPt2. We do notconfirm an off-stoichiometry stable phaseβAuTi-B19,because we do not have such a prototype in our libraThe stoichiometric B19 has an energy∼50 meV/atomaboveαAuTi-B11. At 42.8% concentration of Ti, we finda previously unknown compound Au4Ti3–Cu4Ti3. Thiscompound is degenerate with the tie-line of the two-phafield Au2Ti ↔ AuTi; therefore its existence is uncertainTo address the degenerate structures Au2Ti-C11b/MoPt2


ia

or

n

r

and the energy difference betweenαAuTi-B11 andβAuTi-B19 phases, we further investigate Au2Ti and AuTi withPAW-GGA potentials, as described inSection 3. With PAW,Au2Ti-C11b is the most stable compound and Au2Ti–MoPt2is higher by 3.7 meV/atom. For phase AuTi, AuTi-B11 is themost stable compound (as in the ultrasoft pseudopotentcase), and AuTi-B19 is higher by 53.6 meV/atom. (SeeFig. 14.)

Au–Ti system


Composition Experimental Ab initio% Ti (Massalski [9]) result

18–21 D1a Au4Ti-D1a

25 Two-phase region Two-phase regionabove 500C Au3Ti-D0a ∼ 11 meV/atom

above Au4Ti ↔ Au2Ti.

33.3 C11b Au2TiC11b/MoPt2 (us-lda)C11b stable (paw-gga)MoPt2∼3.7 meV/atomabove C11b (paw-gga).

42.8 Two-phase region Au4Ti3above 500C Cu4Ti3/tie

50 αAuTi-B11 AuTi-B1150–51 βAuTi-B19 B19∼ 50 meV/atom

B19∼ 53.6meV/atom (us-lda)

above B11 (paw-gga).

38–52 γ AuTi-B2 B2 ∼ 135 meV/atomhigh temperature above B11 (us-lda).

75 A15 AuTi3-A15

Fig. 14. AuTi (gold–titanium) ground state convex hull.

l

Au–Y (gold–yttrium)

Four intermetallic compounds have been reported fthe system Au–Y: Au3Y, Au2Y, AuY, and AuY3 [9,10,118,154–157,169–172]. With our ab initio method, weconfirm the stability of Au3Y-D0a and Au2Y-C11b. Atequal composition, AuY, we do not confirm B2: we finda stable B33 and phase B2 higher by∼26 meV/atom.No experimental compound is reported for compositioAuY2, though we obtain AuY2-C37. At 75% compositionof Y, we do not find a stable AuY3 compound reportedexperimentally [10,155]: the structure with lowest energy isAuY3-D011 with an energy higher by∼30 meV/atom withrespect to the tie-line.We further investigate AuY with PAW-GGA potentials, as described inSection 3. With PAW, B33 isthe most stable compound and B2 is∼25 meV/atom higherthan B33. The disagreement at composition AuY is furthediscussed inSection 4. (SeeFig. 15.)

Au–Y system


Composition Experimental Ab initio% Y (Massalski [9]) result

25 D0a Au3Y-D0a

33.3 C11b Au2Y-C11b

50 B2 AuY-B33 (us-lda)B2 ∼ 26 meV/atomabove B33 (us-lda).B33 stable (paw-gga)B2∼ 25 meV/atomabove B33 (paw-gga).SeeSection 4.

66.6 Not studied AuY2-C37

75 hyp [10] Nothing stable(uncertain prototype) D011 ∼ 30 meV/atom

above tie-line

Fig. 15. AuY (gold–yttrium) ground state convex hull.


th

th

y

e

to

e

on

io.

f

ot

end

nd

weA

s

Au–Zr (gold–zirconium)

Several ordered compounds have been reported forsystem Au–Zr: Au4Zr, Au3Zr, Au2Zr, Au10Zr7, Au4Zr5,AuZr2, and AuZr3 [9,10,162,173–176]. Our ab initiomethod confirms the stability of Au3Zr-D0a, Au2Zr-C11b

and AuZr3-A15. At composition AuZr2, Massalski andthe Pauling File report C11b- and CuZr2-type compounds,respectively [9,10]. C11b (MoSi2 prototype) and CuZr2are very similar structures, both tetragonal (tI6) and withe same space group #139. C11b belongs to the bccsuperstructure family while CuZr2, a slightly distortedversion of C11b, is a close packed structure [10]; therefore,within the formalism of the atomic environments, therepresent two different prototypes [10]. Our library containsboth prototypes. However, in our calculations, AuZr2-C11b

and AuZr2–CuZr2 are both unstable with degeneratenergies higher by∼20 meV/atom with respect to the tie-line AuZr ↔ AuZr3. We cannot say anything about Au4Zr,Au10Zr7 and Au4Zr5, because our library lacks prototypes athe proper concentrations. At AuZr composition we find twstructures with degenerate energies: B11 and FCC[001]

A2B2. Thisprediction is unreliable because such compounds appin the two-phase region Au10Zr7 ↔ Au4Zr5, that weare not able to describe properly. At 42.8% concentratiof Zr, we find Au4Zr3–Cu4Ti3. As before, this predictionis unreliable because it appears in the two-phase regAu2Zr ↔ Au10Zr7, that we are not able to describeTo conclude, the inclusion of prototypes of Au10Zr7 andAu4Zr5 in our calculations, might change the stability othe predicted AuZr and Au4Zr3 phases.To address thedegenerate structures, we further investigate AuZr2 withPAW-GGA potentials, as described inSection 3. With PAW,AuZr2–CuZr2 and AuZr2-C11b are still degenerate, butbecome stable with an energy 6.6 meV/atom lower than thetie-line AuZr↔ AuZr3. (SeeFig. 16.)

Fig. 16. AuZr (gold–zirconium) ground state convex hull.

e

ar

n

Au–Zr system



% Zr (Massalski [9]) result

20 Au4Zr Unavailable

oP20–Pnma

25 D0a Au3Zr-D0a

33.3 C11b Au2Zr-C11b

42.8 Two-phase region Au4Zr3–Cu4Ti3Au2Zr ↔ Au10Zr7 unreliable, see text.

45 Au10Zr7 Unavailable

tI34

50 Two-phase region AuZr-B11/FCC[001]A2B2

Au10Zr7 ↔ Au4Zr5 unreliable, see text.

55.5 Au4Zr5 Unavailable

unknown

prototype

66.6 C11b [9] Two-phase region (us-lda)

CuZr2 [10] AuZr2–CuZr2/C11b

∼20 meV/atom above tie-line

AuZr ↔ AuZr3 (us-lda).

CuZr2/C11b stable

∼6.6meV/atom below

AuZr ↔ AuZr3(paw-gga).

75 A15 AuZr3-A15

Cd–Nb (cadmium–niobium)

The phase diagram for the system Cd–Nb is nknown [9]. It has been reported that Cd3Nb has L12prototype [177]. However, we are not able to find any stablcompound, so we propose that the system is not compouforming, but will display low temperature immiscibility.In our calculations, Cd3Nb-L12 is ∼70 meV/atom higherin energy than the phase separation of Cd↔ Nb.In addition, the structure with the lowest formationenergy is a bcc superstructure at composition CdNb aenergy ∼55 meV/atom higher then Cd ↔ Nb. Toaddress the disagreement with the experimental result,further investigate the relevant compounds with PAW-GGpotentials, as described inSection 3. With PAW, Cd3Nb-L12is >100 meV/atom higher in energy than the tie-line Cd↔Nb. The structure with the lowest formation energy ia bcc superstructure at composition CdNb3, and energy∼58 meV/atom above Cd↔ Nb. It is unlikely that the abinitio calculations have such large errors.The disagreementat composition Cd3Nb is further discussed inSection 4.


r

on-

,

n

e

lly-

-

nd

per-

dd

Cd–Nb system



% Nb (Massalski [9]) result

25 Cd3Nb-L12 ImmiscibleSeeSection 4.

Cd–Pd (cadmium–palladium)

Four Cd–Pd compounds have been identified expementally at low temperature (γ , γ1, γ ′ and β1 [178,179,9]). However, the phase boundaries are unknown. We cfirm β1-CdPd-L10. In the Cd-rich part of the phase diagram, we find Cd3Pd (nearγ ′ at 74% Cd). We are notable to determine the exact structure of Cd3Pd but ourbest candidates are D019, D024, NbPd3-type, and Al3Pu-type (Co3V). In the Pd-rich part of the phase diagramwe find a stable phase CdPd3. As before, we cannot de-termine its structure, precisely. Our guesses are D022 andNbPd3-type, which have degenerate energies. At concetration 33.3% Cd, we find a new compound CdPd2-C37which is not present in Massalski [9]. To address thedegenerate structures, we further investigate CdPd3 andCd3Pd with PAW-GGA potentials, as described inSection 3.Compounds CdPd3-D022 and CdPd3–NbPd3 remain degen-erate with PAW (the energy difference is 1.1 meV/atom). Inaddition, for Cd3Pd composition, NbPd3/Al3Pu remain de-generate compounds, and D022, D019, D024 are higher by4.7 meV/atom, 5.6 meV/atom, 6.7 meV/atom, respectiv.(SeeFig. 17.)

Fig. 17. CdPd (cadmium–palladium) ground state convex hull.

i-

-

-

ly

Cd–Pd system



% Cd (Massalski [9]) result

25 Pd phase CdPd3above 100C D022/NbPd3 (us-lda)

D022/NbPd3 (paw-gga)

33.3 Two-phase region CdPd2-C37above 100C,betweenβ1-Pd

37–55 β1-L10 CdPd-L10

74 γ ′-(unknown) Cd3Pd-D019 orD024/NbPd3/Al3Pu (us-lda)NbPd3/Al3Pu stable (paw-gga)

D022 ∼ 4.7 meV/atom,D019 ∼ 5.6 meV/atom,D024 ∼ 6.7 meV/atom

above NbPd3/Al3Pu (paw-gga)

77–80 γ1-(unknown) Unavailable

81–83 γ -D83 Unavailable

Cd–Pt (cadmium–platinum)

The phase diagram of the system Cd–Pt is partiaknown [180,9], with several compounds of unknown structure. We confirm the stability of phaseα′

1-CdPt-L10. In theCd-rich part of the phase diagram, the prototypes of Cd2Ptandγ1 are unknown. For Cd2Pt, our best guess is two degenerate structures, Cd2Pt-C37 and Cd2Pt-C16, which arealso degenerate with the hull. Forγ1, at stoichiometry Cd3Pt,we find two degenerate structures D011, D0a and D022.In the Pt-rich part of the phase diagram, we do not fia stable compoundα′-CdPt3-L12. Instead, we find a sta-ble orthorhombic CdPt3 with fcc superstructure and Cmmm#65 space group. The prototype, labeled as CdPtproto

3 , isdescribed inAppendix A. In addition, CdPt3-L12 is foundwith an energy∼25 meV/atom above CdPtproto

3 . To addressthe degenerate structures and the disagreement with eximental results for compound CdPt3, we further investigateCdPt3, Cd2Pt and Cd3Pt with PAW-GGA potentials, as de-scribed inSection 3. With PAW, CdPt3–CdPtproto

3 is the moststable structure and CdPt3-L12 higher by∼10.4 meV/atom.For compound Cd2Pt, C16 is the most stable compounand C37 is∼33 meV/atom higher than C16. For compounCd3Pt, D011 is the most stable compound and D0a andD022 are higher by∼52 meV/atom and∼60 meV/atom, re-spectively.The disagreement at composition CdPt3 is furtherdiscussed inSection 4. (SeeFig. 18.)


,

ti-

Ti

4)thed

y

Cd–Pt system


Composition Experimental Ab initio% Pt (Massalski [9]) result

∼13–17 γ -(unknown) Two-phase region

24–26 γ1-(unknown) Cd3PtD011/D0a/D022 (us-lda)D011 stable (paw-gga)D0a∼52 meV/atomD022∼60 meV/atomabove D011 (paw-gga)

∼26–28 γ2-(unknown) Unavailable

∼31–38 Cd2Pt-(unknown) Cd2PtC37/C16/tie-line (us-lda)C16 stable (paw-gga)C37∼33 meV/atomabove C16 (paw-gga)

∼49–51 α′1-L10 CdPt-L10

∼75 α′-L12 CdPtproto3

Appendix AL12 ∼ 25 meV/atomabove CdPtproto

3 .

CdPtproto3 stable (paw-gga)

L12∼10.4 meV/atom aboveCdPtproto

3 (paw-gga).SeeSection 4.

Fig. 18. CdPt (cadmium–Platinum) ground state convex hull.

Cd–Rh (cadmium–rhodium)

No experimental phase diagram is available [9]. Withour ab initio technique we find two stable compoundsCd2Rh-C37, and Cd3Rh. Our best guesses for Cd3Rh areCd3Rh-D024 and Cd3Rh–Al3Pu-type (Co3V), which havedegenerate energies (2 meV/atom difference). Interestingmetastable phases are Cd3Rh-D019, (11 meV/atom aboveCd3Rh), and Cd2Rh-C49 (8 meV/atom above Cd2Rh-C37).

To address the degenerate structures, we further invesgate Cd3Rh with PAW-GGA potentials, as described inSection 3. With PAW, Cd3Rh–Al3Pu is the ground state,and Cd3Rh-D024 has an energy 5 meV/atom higher. (SeeFig. 19.)

Cd–Rh system



66.6 Not studied Cd2Rh-C37C49 metastable

75 Not studied Cd3RhD024/Al3Pu (us-lda)D019 ∼ 11 meV/atomabove D024/Al3Pu (us-lda)Al3Pu stable (paw-gga)D024∼5 meV/atomabove Al3Pu (paw-gga)

Fig. 19. CdRh (cadmium–rhodium) ground state convex hull.

Cd–Ti (cadmium–titanium)

Little is known for the system Cd–Ti at hightemperature [9,10,182,183]. At low temperature only twostable intermetallic compounds have been reported [183].We confirm the stability of Ti2Cd-C11b. At compositionCdTi, Massalski and the Pauling File report B11 and Cdcompounds, respectively [9,10]. B11 (γ CuTi prototype)and CdTi are very similar structures, both tetragonal (tPand with the same space group #129. B11 belongs tobcc superstructure family while CdTi, a slightly distorteversion of B11, is a close packed structure [10]. Therefore,within the formalism of the atomic environments, therepresent two different prototypes [10]. The ab initio


erInn-rloth

In

o-

e

me

ns

s,

hccn

y

r,

calculation is able to relax one structure into the otheasily. For CdTi we confirm the experimental results.addition, at concentration 42.8%Cd, we find a compouCd3Ti4–Cu4Ti3, degenerate with the tie-line of the twophase region CdTi2 ↔ CdTi. We have not found any othestable or metastable compound: we conclude that thetemperature part of the phase diagram is complete. Notethis system is very similar to Ag–Ti. (SeeFig. 20.)

Cd–Ti system



33.3 C11b CdTi2-C11b

42.8 Two-phase region Cd3Ti4above 200C Cu4Ti3/tie-line

50 B11 CdTi-B11 (CdTi)CdTi [10]

Fig. 20. CdTi (cadmium–titanium) ground state convex hull.

Cd–Y (cadmium–yttrium)

The stability of compounds YCd-B2, YCd2-C6 andYCd3 (Cd3Er prototype) [9,10,184–186] is confirmed. Theyttrium-rich side of the phase diagram is poorly known.that region we find two degenerate phases Y2Cd-C49 andY2Cd-C37, both of which are close in energy to the twphase region Y↔ CdY (Fig. 21). We do not find anyother stable or metastable phase.To address the degeneratstructures, we further investigate Y2Cd with PAW-GGApotentials, as described inSection 3. With PAW, C37 is themost stable compound and C49 is higher by 7.7 meV/ato.


,

d

wat

Cd–Y system



33.3 Not studied/ CdY2two-phase region C37/C49 (us-lda)

C37 stable (paw-gga)C49∼ 7.7meV/atomabove C37 (paw-gga)

50 B2 CdY-B2

66.6 C6 Cd2Y-C6

75 Cd3Er Cd3Y-Cd3Er

80.4 Cd45Y11–Cd45Sm11 Unavailable

81.7 Cd58Y13–Pu13Zn58 Unavailable

85.7 Cd6Y Unavailable

Fig. 21. CdY (cadmium–yttrium) ground state convex hull.

Cd–Zr (cadmium–zirconium)

The Cd–Zr system is poorly characterized [9,48,188],and compounds have been identified at four compositioCd3Zr, Cd2Zr, CdZr, and CdZr2. Massalski [9] reportsCd3Zr-L10–AuCu, which we consider a misprint forCd3Zr-L12–AuCu3, while the CRYSTMET and PaulingFile databases [10,11] report Cd3Zr-D0a–βCu3Ti fromRef. [189]. We confirm the stability of Cd3Zr-L12and CdZr2-C11b. At composition CdZr, Massalski andthe Pauling File report B11 and CdTi compoundrespectively [9,10]. B11 (γ CuTi prototype) and CdTi arevery similar structures, both tetragonal (tP4) and witthe same space group #129. B11 belongs to the bsuperstructure family while CdTi, a slightly distorted versioof B11, is a close packed structure [10]. Therefore,within the formalism of the atomic environments, therepresent two different prototypes [10]. The ab initiocalculation is able to relax one structure into the otheeasily. For CdZz, we find CdZr-L10 instead of CdZr-B11,


sd

u

te

e

la-

which has an energy∼18 meV/atom higher than L10.The prototype of Cd2Zr is not known: our best guesis Cd2Zr-C11b. In addition, we find a stable compounCdZr3-A15 not present in Massalski [9], and a metastableL12 which is 8.5 meV/atom higher than A15.To address thedisagreement with the experimental results for compoCdZr, we further investigate L10 and B11 with PAW-GGApotentials, as described inSection 3. With PAW, CdZr-B11is the most stable compound and CdZr-L10 is higher by13 meV/atom. (SeeFig. 22.)

Cd–Zr system



25 Two-phase region CdZr3-A15not studied L12 ∼ 8.5 meV/atom

above A15

33.3 C11b CdZr2-C11b

50 B11 CdZr-L10 (us-lda)B11∼ 18 meV/atomabove L10 (us-lda).B11 stable (paw-gga)L10 ∼ 13 meV/atomabove L10 (paw-gga).

66.6 Cubic (unknown) Cd2Zr-C11b

75 L12 [9] Cd3Zr-L12D0a [189,10,11]

Fig. 22. CdZr (cadmium–zirconium) ground state convex hull.

Mo–Nb (molybdenum–niobium)

The system MoNb has not been studied in great deand no experimental intermetallic compounds have bfound [9,10,46,191]. A bcc solid solution is reported from2400C up to the melt. We predict four stable compoundslow temperature: MoNb2-C11b, MoNb-B2, Mo2Nb-C11b,and Mo3Nb-D03, as shown inFig. 23.

nd

ailen

at

Otherab initio studies, relevant for this system, can bfound in Refs. [89,192–194]. (SeeFig. 23.)

Mo–Nb system


Composition Experimental Ab initio% Mo (Massalski [9]) result

33.3 Not studied MoNb2-C11b

50 Not studied MoNb-B2

66.6 Not studied Mo2Nb-C11b

75 Not studied Mo3Nb-D03

Fig. 23. MoNb (molybdenum–niobium) ground state convex hull.

Mo–Pd (molybdenum–palladium)

The phase diagram of this alloy is known fromexperimental investigations and thermodynamic calcutions [9,10,45,50,195]. The only compound is at com-position MoPd2 and is listed in Massalski as havingapproximatively the MoPt2 structure [9,10,195]. Our ab ini-tio method finds that the MoPt2 structure is 8 meV/atomhigher than MoPd2–ZrSi2. In addition, we find a stable phaseMoPd4-D1a (MoNi4 prototype), previously unknown. (SeeFig. 24.)

Mo–Pd system


Composition Experimental Ab initio% Pd (Massalski [9]) result

66–67 ∼MoPt2 MoPd2–ZrSi2MoPt2 ∼ 8 meV/atomZrSi2

80 Disorder fcc MoPd4-D1aPd-A1


i-

r-

5

ves

Fig. 24. MoPd (molybdenum–palladium) ground state convex hull.

Mo–Pt (molybdenum–platinum)

The phase diagram of this alloy is known from expermental investigations and thermodynamic modeling [9,10,195–204,271]. We confirm the stability of MoPt-B19 andMoPt2 (with MoPt2 prototype). In the Pt-rich part of thephase diagram, we find a compound MoPt4-D1a (MoNiaprototype), previously unknown. The observed high tempeature phase A15, reported as Mo6Pt in Massalski [9] andas Mo3.2Pt0.8 in the Pauling File [10], is off-stoichiometry.Our ab initio method finds a stoichiometric Mo3Pt-A15 tobe ∼45 meV/atom above the two-phase region Mo↔MoPt-B19. For a detailed experimental study of such an A1phase, see Ref. [204]. (SeeFig. 25.)

Mo–Pt system



% Pt (Massalski [9]) result

18.5 Mo6Pt-A15 [9] Mo3Pt-A15

20 Mo3.2Pt0.8-A15 [10] ∼45 meV/atom above

above∼1200C Mo ↔ MoPt-B19

31.5–45 ε′-D019 Unavailable

above 1000C at such composition

50 B19 MoPt-B19

66.6 MoPt2 MoPt2

80 Two-phase region MoPt4-D1a

MoPt2 ↔ Pt

Fig. 25. MoPt (molybdenum–platinum) ground state convex hull.

Mo–Rh (molybdenum–rhodium)

The phase diagram of the system Mo–Rh is known abo∼900 C and it is based on thermodynamic calculationand experimental results [9,10,50,205–208]. We confirmthe stability of the two known compounds MoRh-B19 [9]and MoRh3-CdMg3 [10,207]. At concentration 66.6% Rh,we find the stable phase MoRh2-C37 (prototype Co2Si),previously not identified [9]. (SeeFig. 26.)

Mo–Rh system


Composition Experimental Ab initio% Rh (Massalski [9]) result

∼50 B19 MoRh-B19

66.6 Two-phase region Mo2Rh-C37above 900C

∼75 CdMg3 [10,207] MoRh3–CdMg3

Fig. 26. MoRh (molybdenum–rhodium) ground state convex hull.


m

phi

le

e

A

h

e

tlic

gnb

t

d

Mo–Ru (molybdenum–ruthenium)

The phase diagram of this system is known at mediuand high temperature [9,10,209]. There is aσ phase around38% Ru, which is quite common when bcc (Mo) and hc(Ru) elements are mixed. We are not able to confirm tσ phase since we do not have the proper prototypeour library. At concentration 75% Ru, we find the stabphase MoRu3-D019 (formation energy∼60 meV/atom), notpresent in Massalski [9], indicating that it might be a lowtemperature ordered structure.


Mo–Ru system


Composition Experimental Ab initio% Ru (Massalski [9]) result

75 Disorder hcp MoRu3-D019Ru-A3 above 800C

Fig. 27. MoRu (molybdenum–ruthenium) ground state convex hull.

Mo–Ti (molybdenum–titanium)

We studied this systems completely in both the LDand GGA approximation as theab initio results differsignificantly from the experimental assesssment of tsystem. We report the PAW-GGA results inFig. 28 andthe Table below. The LDA results are similar so that thdifference betweenab initio and experiments in this systemcan not be attributed to a LDA or GGA problem, bumay be more profound. To our knowledge, no intermetalcompounds have been reported for the system Mo–Ti [9,10,47,212–214], and it is considered a non-compound forminsystem [10,11]. In contrast to the experimental assessmewe find seven stable compounds. Four of these havesuperstructures: orthorhombic MoTi3, degenerate with thetwo-phase region Ti↔ MoTi2, with space group Immm

en

e

t,cc

#71 and prototype MoTiproto3 (Appendix A); orthorhombic

Mo3Ti with space group Immm #71 and prototypeMo3Tiproto (Appendix A); trigonal MoTi2-BCC[211]

AB2 ; andorthorhombic MoTi with space group Imma #74 andprototype MoTiproto (Appendix A). In addition, we find astable compound Mo2Ti-C11b, a compound Mo4Ti-D1a

degenerate with the tie-line, and finally monoclinic Mo5Tiwhich has space group C2/m #12 and prototype Mo5Tiproto

(Appendix B). Given the relatively large value of theformation energy, the experimental miscibility gap issurprising. This system warrants further study to sort outhe apparent discrepancy between experiments andab initiocomputation.

Anotherab initio study, relevant for this system, can befound in Ref. [215]. (SeeFig. 28.)

Mo–Ti system



% Mo (Massalski [9]) result (paw-gga)

25 Two-phase region, MoTiproto3 /tie-line

(βTi, Mo)-A2 ↔ (αTi)-A3, Appendix A

above 400C

33.3 Same as above MoTi2-BCC[211]AB2

50 Same as above MoTiproto

(Appendix A)

66.6 Same as above Mo2Ti-C11b

75 Same as above Mo3Tiproto

Appendix A

80 Same as above Mo4Ti-D1a/tie-line

83.3 Same as above Mo5Tiproto

Appendix B

Fig. 28. MoTi (molybdenum–titanium) ground state convex hull, calculatewith PAW-GGA approach.


tythne

o

ld

e

ll,

l

%

n

ed

e

Mo–Zr (molybdenum–zirconium)

Because of the discrepancy betweenab initio computa-tions and experiments in Mo–Ti and the chemical similaribetween Zr and Ti, we have studied the Mo–Zr also wiboth LDA and GGA. The PAW-GGA results are shown iFig. 29 and in the Table below. Not much is known of thsystem Mo–Zr for producing a reliable phase diagram [9,10,48]. We confirm the stability of the only known compoundMo2Zr–C15. Given this good agreement for Mo–Zr, one alsexpects theab initio approach to perform well for Mo–Ti,giving further evidence that the experimental Mo–Ti shoube reevaluated.


Mo–Zr system


Composition Experimental Ab initio% Mo (Massalski [9]) result (paw-gga)

60–67 C15 Mo2Zr-C15

Fig. 29. MoZr (molybdenum–zirconium) ground state convex hucalculated with PAW-GGA approach.

Nb–Pd (niobium–palladium)

The Pd–Nb phase diagram is known with reasonabaccuracy in the Pd-rich region [9,10,46,253]. The stabilityof the experimental phase NbPd2–MoPt2 andαNbPd3-D022is confirmed. In the Nb-rich region, at concentration 33Nb, we find an orthorhombic compound Nb2Pd-BCC[011]

AB2 ,not present in Massalski [9]. The energy of this phase is∼11 meV/atom below the tie-line of the two-phase regioNb ↔ NbPd2.

e

Anotherab initio study, relevant for this system, can befound in Ref. [218]. (SeeFig. 30.)

Nb–Pd system




33.3 Two-phase region Nb2Pd-BCC[011]AB2

above 700C ∼11 meV/atom belowNb ↔ NbPd2

66.6 MoPt2 NbPd2–MoPt2

75 α-D022 NbPd3-D022

Fig. 30. NbPd (niobium–palladium) ground state convex hull.

Nb–Pt (niobium–platinum)

Several intermetallic compounds have been reportfor the system Nb–Pt [9,10,46,219–223]. We confirm theexperimental phases Nb3Pt-A15 andαNbPt3-D0a (βCu3Tiprototype). The prototype of NbPt2 is not reported inMassalski [9] (orthorhombic oI6, with space group Immm).We find NbPt2–MoPt2, in agreement with Refs. [10,221,222]. At 50% concentration we do not confirm the stabilityof NbPt-B19. Instead of B19 we find NbPt-L10, and B19 tobe higher by∼11 meV/atom above L10. It is possible thatL10 is a ground state and B19 a high temperature state. Wcannot say anything about theσ phase D8b since we do nothave such prototype in our library.We further investigatedNbPt-L10 and NbPt-B19 with PAW-GGA potentials, asdescribed inSection 3. With PAW, L10 is the most stablecompound and B19 is∼18.5 meV/atom higher than L10.(SeeFig. 31.)


ge

is

l-d-

er

e

Nb–Pt system




19–28 A15 Nb3Pt-A15

31–38 D8b Unavailable

σ phase

∼50 Nb1−xPt1+x-B19 NbPt-L10B19 ∼11 meV/atom

above L10 (us-lda)

L10 stable (paw-gga)

B19∼18.5meV/atom

above L10 (paw-gga)

∼67 MoPt2 [10,221,222] NbPt2–MoPt2

∼76 D0a NbPt3-D0a

Fig. 31. NbPt (niobium–platinum) ground state convex hull.

Nb–Rh (niobium–rhodium)

The system Nb–Rh is poorly characterized in the ranof concentration 50∼80% Rh [9,10,46,50,220,224,225]. Weconfirm the stability of Nb3Rh-A15, NbRh-L10. NbRh-B19,which is observed as a high temperature state,∼27 meV/atom above L10, and it is possibly stabilizedby entropic effects. Note: in Massalski and in the Pauing File [9,10] there is no phase at 50% composition, anNb0.96Rh1.04-L10 appears off-stoichiometry. At concentration 75% Rh, we find the stable phaseη-Al3Pu (Co3V)andκNbRh3-L12 to be higher by 8 meV/atom. Hence, wethink thatη prevails at low temperature overκ , in contrastwith the sketched phase diagram of Ref. [9]. We cannot sayanything about D8b and ξ(Nb2Rh3) since we do not have

the σ phase D8b or any A2B3 prototypes in our library.To address the structures with similar energy, we furthinvestigate NbRh3 with PAW-GGA potentials, as describedin Section 3. With PAW, NbRh3–Al3Pu is the most stablecompound and NbRh3-L12 is higher by 5.3 meV/atom. Thedisagreement at composition NbRh3 is further discussed inSection 4.


Nb–Rh system



% Rh (Massalski [9]) result

25 α′(Nb3Rh)-A15 Nb3Rh-A15

28.5–39.5 σ(Nb13Rh7)-D8b Unavailable

∼51.5 to 52 Nb0.96Rh1.04-L10 NbRh-L10

( 1337C)

∼56–62 ε(Nb2Rh3)-B19 NbRh-B19

(high temperature ∼27 meV/atom

1330C) above L10 (us-lda)

59 to∼64 ξ(Nb2Rh3)–Nb2Rh3 Unavailable

∼67–70 η(Nb13Rh27)–Al3Pu See below

(≡Co3V in [9]) for Al 3Pu

∼71–79 κ(NbRh3)-L12 NbRh3–Al3Pu (us-lda)

L12 ∼ 8 meV/atom

above Al3Pu (us-lda)

Al3Pu stable (paw-gga)

L12 ∼ 5.3 meV/atom

above Al3Pu (paw-gga)

SeeSection 4.

Fig. 32. NbRh (niobium–rhodium) ground state convex hull.


n

se

t

on

ers

R

m

e

e

n

,e

pe

Nb–Ru (niobium–ruthenium)

Very little is known for the alloy Nb–Ru, especially atlow temperature [9,10,46,227–231]. We do not confirm thestability of any low temperature compound at compositioNbRu, but instead find a two-phase field between Nb3Ruand NbRu2, even though a high and low temperature phaof NbRu has been observed [227,231]. In the Ru-rich side ofthe phase diagram, we find NbRu3-D024 to be 8 meV/atomlower than L12 which is suggested experimentally. A66% Ru we find NbRu2-C37 (with oP12–Co2Si prototype).With respect to the two-phase field Nb3Ru ↔ NbRu2,the structures closest to the tie-line at NbRu compositiare B19 (∼13 meV/atom), L10 (∼20 meV/atom),B27 (∼23 meV/atom), B33 (∼39 meV/atom), and B2(∼45 meV/atom). Experiments have not found any othstable compound. However, in the Nb-rich side of the phadiagram, we find Nb3Ru-D03 and a monoclinic Nb5Ruwith C2/m #12 space group and prototype Mo5Tiproto. Thestructure Nb5Ru–Mo5Tiproto is described inAppendix B. Toaddress the disagreements at compositions NbRu and Nb3we further investigate the relevant compounds with PAWGGA potentials, as described inSection 3. For compositionNbRu, the PAW-GGA result is substantially different frothe US-LDA result. GGA gives L10 only 4 meV/atom abovethe tie-line (versus 20 meV/atom in LDA), whereas B19was difficult to converge numerically, and seems to b>100meV/atom above the tie-line. For composition NbRu3,also with PAW, D024 is the most stable compound andL12 is higher by 2.5 meV/atom. Because this number isso small, extremely largek-point sets and high energycutoff were used to converge it. The disagreements atcompositions NbRu and NbRu3 are further discussed inSection 4.


Fig. 33. NbRu (niobium–ruthenium) ground state convex hull.

e

u-

Nb–Ru system



16.6 Disorder Nb-A2 Nb5Ru–Mo5Tiproto

Appendix B

25 Disorder Nb-A2 Nb3Ru-D03

∼50 NbRu′-L10 Two-phase region(room temperature) Nb3Ru ↔ NbRu2.

L10 ∼ 4meV/atom (paw-gga)B19> 100 meV/atomabove tie-line (paw-gga).

∼50 NbRu-B2 B19∼ 13 meV/atom (us-lda)(high temperature) L10 ∼ 20 meV/atom

B2 ∼ 45 meV/atomabove the tie-lineSeeSection 4.

66.6 Two-phase region NbRu2-C37above∼700C

75 L12 NbRu3-D024 (us-lda)L12 ∼ 8 meV/atomabove D024 (us-lda)D024 stable (paw-gga)L12 ∼ 2.5 meV/atomabove D024 (paw-gga)SeeSection 4.

Nb–Tc (niobium–technetium)

The phase diagram of the system Nb–Tc is not knowand only one intermetallic compound, NbTc3 (metallic andsuperconductor), has been reported withαMn structure [9,10,46,235–237,359]. The compound NbTc3 is classified asphase Nb0.15Tc0.85 in the Pauling File [10]. We do not haveαMn in our library of prototypes, and, at such stoichiometrywe do not find any stable compound. In the Tc-rich sidof the phase diagram, we find a two-phase field NbTc↔Tc, as shown inFig. 34. At 50% concentration, we obtainNbTc-B2, and in the Nb-rich side of the diagram, wefind Nb2Tc-C11b and an orthorhombic phase Nb3Tc withspace group Immm #71, bcc superstructure, and prototyNb3Tcproto described inAppendix A. Trends for Tc alloysare further discussed inSection 8. (SeeFig. 34.)

Nb–Tc system


Composition Experimental Ab initio% Tc (Massalski [9]) result

25 Not studied, Nb3Tcproto

unknown Appendix A

33.3 Same as above Nb2Tc-C11b

50 Same as above NbTc-B2

75 NbTc3–αMn [9] Nothing stable85 Nb0.15Tc0.85–αMn [10]


io

nean

lds

on

as

t

e

ed

ned

re

Fig. 34. NbTc (niobium–technetium) ground state convex hull.

Pd–Pt (palladium–platinum)

The low temperature part of the phase diagrambelieved to have a miscibility gap at a temperatureabout 770C. This miscibility gap is predicted on thebasis of the difference of melting points between Pd aPt [9,10,42,254,255]. Instead of such a gap, we find threunknown stable compounds with fcc superstructures (PdPt are both fcc). We find PdPt-L11, and two orthorhombicphases, Pd3Pt and PdPt3, with space group Cmmm #65and prototypes described inAppendix A. The prototypesare labeled as Pd3Ptproto and PdPtproto

3 . The compoundPd3Pt is degenerate with respect to the two-phase fiePd-A1↔ PdPt-L11; therefore its existence is uncertain. Ashown inFig. 35, all the stable phases have small formatienergy (<50 meV/atom) making them difficult to determineexperimentally. However, we believe the experimental phdiagram to be in error.We further investigate PdPt withPAW-GGA potentials, as described inSection 3. With PAW,PdPt-L11 is the most stable compound and PdPt-L10 is

Fig. 35. PdPt (palladium–platinum) ground state convex hull.

sf

d

d

s

e

higher by 5.5 meV/atom.Our results are in disagreemenwith previous FLAPW-LDA calculations, whereL10 isfound to be the most stable compound [256].

Otherab initio studies, relevant for this system, can bfound in Refs. [13,61]. (SeeFig. 35.)

Pd–Pt system




25 Predicted Pd3Ptproto/tie-line

two-phase region (uncertain)

Pd-A1↔ Pt-A1 Appendix A

50 Same as above PdPt-L11

L10 ∼ 11 meV/atom

higher than L11 (us-lda)

L11 stable

L10 ∼ 5.5 meV/atom

higher than L11 (paw-gga)

75 Same as above PdPtproto3

Appendix A

Pd–Tc (palladium–technetium)

The phase diagram for the system Pd–Tc is sketchon the basis of the solid solubility data [9,10,42,269,340,359,362]. Experimental results report two solid solutions(fcc Pd-rich) and (hcp Tc-rich) with a two-phase region ibetween. No intermetallic compounds have been report[9,10]. However, we find one stable phases PdTc3-D019. Inaddition, at 50% concentration, we find a hcp superstructu(trigonal lattice, hP4, space group P3m1) which has anenergy∼3 meV/atom higher than the tie-line of the two-phase region PdTc3 ↔ Pd. Trends for Tc alloys are furtherdiscussed inSection 8. (SeeFig. 36.)

Pd–Tc system




25 Two-phase region PdTc3-D019

above∼ 1000C

Pd-A1↔ Tc-A3

50 Two-phase region Two-phase region

above∼1000C PdTc3 ↔ Pd

Pd-A1↔ Tc-A3 hcp superstr.∼3 meV/atom

above tie-line


fo

-

d

an

-

ythei

e

nds

een

in

ld

e

Fig. 36. PdTc (palladium–technetium) ground state convex hull.

Pd–Ti (palladium–titanium)

This system has been subject of conflicting resultsseveral years [9,10,168,270–285]. Our ab initio methodconfirms the stability of the compounds PdTi2-C11b,PdTi3-A15, and Pd3Ti-D024. Near 80% Pd, Long-Period Superstructure (LPS) modulations of L12 are observed [273].While we do not have such off-stoichiometric LPS, we finL12 to be only 6 meV/atom above D024 at Pd3Ti composi-tion. The low energy difference between L12 and D024 indi-cates that it may be easy to form antiphase boundariesLPS near this composition. We find Pd2Ti–MoPt2 which isan orthorhombic distortion of C11b (MoPt2 is orthorhom-bic, while C11b is tetragonal). We find Pd2Ti-C49 andPd2Ti-C11b to be higher by 3 meV/atom and 14 meV/atomabove MoPt2, respectively. At low temperature, at concentration 50%, we find a stable compoundαTiPd, but two pro-totypes are degenerate: L10 and B19 (which is reported ex-perimentally). We cannot find the reported phase PdTi4-A15(off-stoichiometry) [9], which we think should appear in-side the two-phase region of Ti∼ PdTi3, or at compositionPdTi3. While off-stoichiometric compounds are obviouslpossible, this usually goes together with significant widof the single-phase field. Hence, we concluded (mayberoneously) that the placement of A15 at composition PdT4

in Ref. [9] is likely a typographical error.To address the de-generate structures, we further investigateαTiPd with PAW-GGA potentials, as described inSection 3. With PAW, PdTi-B19 is the most stable compound and PdTi-L10 is higher by10 meV/atom.


r

d

r-

Pd–Ti system



20 and 25 A15 at 20% PdTi3-A15 at 25%

33.3 C11b PdTi2-C11b

47–53 α(TiPd)-B19 PdTi-B19/L10 (us-lda)B19 stable (paw-gga)L10 ∼ 10 meV/atomabove B19 (paw-gga)

60 Pd3Ti2 ∼ Au2V Unavailable

62.5 Pd5Ti3 ∼ C11b Unavailable

66.6 Orthorhombic Pd2Ti–MoPt2distortion (distortion of C11b)of C11b C49∼ 3 meV/atom

C11b ∼ 14 mev/atomabove MoPt2

75 D024 Pd3Ti-D024L12 ∼ 6 meV/atomabove D024

80 L12 L12 metastableat 75% (see text)

Fig. 37. PdTi (palladium–titanium) ground state convex hull.

Pd–Y (palladium–yttrium)

The phase diagram for the system Pd–Y is knowwith reasonable accuracy. Several intermetallic compounhave been reported, but not all the structures have bdetermined experimentally [9,10,287–290]. We confirm thestability of the compounds Pd3Y-L12 and PdY3-D011. Theprototype ofαPdY is not known [287]. Our best guessesare PdY-B27 and PdY-B33 (CrB), which are degeneratethe calculation. We also find PdY2-C37, which occurs in aconcentration between two known compounds Pd2Y3 andPd2Y5 and is very close to the tie-line of the two-phase fiePdY3 ↔ PdY (5 meV/atom). Hence, if Pd2Y3 and Pd2Y5were to be included, C37 would likely not be stable. W


it

h

Z

e

he

nd,

ly.ous

ndn,

cannot check this prediction since we do not have Pd2Y3and Pd2Y5 prototypes in the set of calculations.To addressthe degenerate structures, we further investigate PdY wPAW-GGA potentials, as described inSection 3. With PAW,PdY-B27 is the most stable compound and PdY-B33 is higby 3.1 meV/atom.(SeeFig. 38.)

Pd–Y system



25 D011 PdY3-D011

28.6 Pd2Y5 unknown Unavailable

33.3 Two-phase region PdY2-C37 (uncertain)

40 Pd2Y3-hR15 Unavailableunknown

50 αPdY unknown PdY-B27/B33 (us-lda)B27 stable (paw-gga)B33∼ 3.1 meV/atomabove B27 (paw-gga)

57.1 Pd4Y3-hR14 Unavailableunknown

60 αPd3Y2 unknown Unavailable

66.6 Unknown Two phase region

79.5–75 L12 Pd3Y-L12

87.5 Unknown Unavailable

Fig. 38. PdY (palladium–yttrium) ground state convex hull.

Pd–Zr (palladium–zirconium)

The experimental phase diagram for the system Pd–is based on limited information [9,10,276,291–298]. Weare able to confirm the stability of Pd3Zr-D024 andPdZr2-C11b. The stable phase of PdZr is reported to bCrB (B33) [10,298]. At that composition, we find twodegenerate structures: PdZr-B27 or PdZr-B33 (CrB). In tZr-rich side of the phase diagram, we find PdZr3-FCC[001]

AB3 .

However, the energy difference between PdZr3-FCC[001]AB3

h

er

r

and the two-phase region Zr↔ PdZr2-C11b is verysmall (∼6 meV/atom). Hence, the existence of compouPdZr3-FCC[001]

AB3 remains uncertain. At low temperatureinstead of a stable phase with stoichiometry Pd2Zr, we findthe two-phase field PdZr↔ Pd3Zr-D024. The least unstablephases are Pd2Zr-C49, Pd2Zr–MoPt2, Pd2Zr-C11b, withenergies∼18 meV/atom,∼24 meV/atom,∼26 meV/atomabove the tie-line of the two-phase region, respectiveTo address the degenerate structures and the erronetwo-phase region PdZr↔ Pd3Zr, we further investigatePdZr and Pd2Zr with PAW-GGA potentials, as described inSection 3. With PAW, PdZr-B33 is the most stable compouand PdZr-B27 is higher by 3.2 meV/atom. In additioPd2Zr-C11b is degenerate with the tie-line PdZr↔ Pd3Zr,and MoPt2 and C49 are higher by∼1.3 and∼7.1 meV/atomwith respect to C11b (the tie-line PdZr↔ Pd3Zr has beenrecalculated with PAW-GGA potentials). (SeeFig. 39.)

Pd–Zr system



25 Two-phase PdZr3-FCC[001]AB3

region (uncertain)

33.3 C11b PdZr2-C11b

50 Disorder fcc [9] PdZr-B27/B33 (us-lda)CrB(B33-TlI) B33 stable (paw-gga)[10,298] B27∼ 3.2 meV/atom

above B33 (paw-gga)

66.6 C11b Two-phase region (us-lda).C49∼ 18 meV/atomMoPt2 ∼ 24 meV/atomC11b ∼ 26 meV/atomabove the tie-line (us-lda).C11b/tie-line (paw-gga).MoPt2 ∼ 1.3 meV/atom,C49∼ 7.1 meV/atomabove tie-line (paw-gga).

75 D024 Pd3Zr-D024

Fig. 39. PdZr (palladium–zirconium) ground state convex hull.


Roa

ccu

ll

e

Ru

ren

c

he

en

n:

c-

Pt–Rh (platinum–rhodium)

The experimental phase diagram of the system Pt–is similar the system Pd–Pt. The low temperature partthe phase diagram is believed to have a miscibility gapa temperature of about 760C [9,10,42,257,299]. Insteadof the gap, we find several stable phases, all with fsuperstructure (Pt and Rh are both fcc), similarly to previoFLAPW-LDA calculations [256]. We find Pt4Rh-D1a,Pt3Rh-D022, PtRh2-C49, PtRh3-D022, PtRh4-D1a, and, at50% concentration, PtRh-FCC[201]

A2B2 (CH “40” in Ref. [256]).As shown inFig. 40, all the stable phases have very smaformation energy (<30 meV/atom) indicating that they maydisorder at relatively low temperature.


Pt–Rh system



Theoretical [256]

20 Two-phase region [9] Pt4Rh-D1aD1a [256]

25 Two-phase region [9] Pt3Rh-D022Two-phase region [256]

50 Two-phase region [9] PtRh-FCC[201]A2B2

FCC[201]A2B2 [256]

66.6 Two-phase region [9] PtRh2-C49

71.4 Two-phase region [9] UnavailableX2 [256]

75 Two-phase region [9] PtRh3-D022D022 [256]

80 Two-phase region [9] PtRh4-D1aD1a [256]

Fig. 40. PtRh (platinum–rhodium) ground state convex hull.

hft

s

Pt–Ru (platinum–ruthenium)

Only one compound has been found for the system Pt–[9,10,300,301]. At low temperature, the phase diagramreported in Massalski [9] has platinum-rich and ruthenium-rich solid solutions with large solubilities of the otheelement, and a two-phase region for concentration betwe∼70% to ∼80% of ruthenium. However, recent X-raydiffraction experimental work reported the existence of a fcphase at 50% composition, with unknown prototype [301].For PtRu, our prediction is PtRu-FCC[001]

A2B2. At 25%

Ru composition, we find a stable phase Pt3Ru-FCC[001]AB3 ,

degenerate with the two-phase field Pt↔ PtRu. Hence, theexistence of compound Pt3Ru-FCC[001]

AB3 remains uncertain.To our knowledge, PtRu is the first known system where tprototype structure FCC[001]

A2B2 would be stable. (SeeFig. 41.)

Pt–Ru system



25 Disorder Pt-A1 [9] Pt3Ru

FCC[001]AB3 /tie-line

(uncertain)

50 Disorder Pt-A1 [9] PtRu-FCC[001]A2B2

PtRu-fcc [301]

Fig. 41. PtRu (platinum–ruthenium) ground state convex hull.

Pt–Tc (platinum–technetium)

The phase diagram for the system Pt–Tc has bedetermined from experimental solid solubility data [9,10,42,359,362]. No intermetallic compounds have beereported [9]. However, we find two stable phasesPt3Tc-FCC[001]

AB3 and PtTc3-D019. PtTc3-D019 appears in thecomposition range of a two-phase region Pt-A1 and TA3, that is present at temperatures higher than∼1000C.


,

e

f-s

e

hasumen

d37

eY,

Trends for Tc alloys are further discussed inSection 8. (SeeFig. 42.)

Pt–Tc system



25 Disorder Pt-A1 Pt3Tc-FCC[001]AB3

75 Two-phase region PtTc3-D019above∼1000CPt-A1 ↔ Tc-A3

Fig. 42. PtTc (platinum–technetium) ground state convex hull.

Pt–Ti (platinum–titanium)

Not much is known of Pt–Ti system for producinga precise phase diagram [9,10,47,277,279,284,302–306].Some intermetallic compounds are reported [277,282,302,305,306]. We confirm the stability of phases PtTi3-A15,αPtTi-B19, Pt3Ti-D024. B19 is stable at 50% compositionand L10 and B33 are higher by∼20 meV/atom and∼30 meV/atom, respectively. At concentration 25% Ti, wconfirm the existence ofγ -L12, which has an energy that is∼5 meV/atom higher than Pt3Ti-D024. However, L12 hasbeen reported to be stable away from stoichiometry<25%Ti [9]. At composition 33% Ti, Pt2Ti, Massalski reports atwo-phase region above 600C [9,10]. Instead of the two-phase field, we find a stable compound Pt2Ti: two structures,C49 and C37, are degenerate. We cannot say anythabout Pt8Ti-D1a, because our library does not contain ofstoichiometry D1a. To address the degenerate structurewe further investigate Pt2Ti with PAW-GGA potentials, asdescribed inSection 3. With PAW, Pt2Ti-C49 is the moststable compound and Pt2Ti-C37 is higher by 1.9 meV/atom.


ing

,

Pt–Ti system



% Ti (Massalski [9]) result

1–12 Pt8Ti-D1a Unavailable

20–27 D024 (>25% Ti) Pt3Ti-D024

γ -L12 (<25% Ti) L12 ∼ 5 meV/atom

above D024

33.3 Two-phase region Pt2Ti-C49/C37 (us-lda)

above 600C [9] C49 stable

C37∼ 1.9 meV/atom

above C49 (paw-gga)

46–54 αPtTi-B19 PtTi-B19

L10 ∼ 20 meV/atom

above B19

71–78 A15 PtTi3-A15

Fig. 43. PtTi (platinum–titanium) ground state convex hull.

Pt–Y (platinum–yttrium)

The experimental phase diagram of the system Pt–Ybeen sketched by analogy with other rare earth–platindiagrams [9]. Several intermetallic compounds have bereported [9,10,42,155,271,279,307–314,318]. The stabilityof the compounds Pt3Y-L12, Pt2Y-C15, and PtY3-D011is confirmed. At 50% concentration, we do not finany stable PtY-B27. Instead of B27, we find PtY-B3(CrB prototype) to have lowest energy with B2 and B2having energies∼50 meV/atom above B33. At 66.6% Yconcentration, we find PtY2-C37 which is the prototypeof Co2Si and Ni2Si. Hence our calculations confirm thcorrect experimental structure. At concentration 62.5%


s

un

ve

lin

h

-

in

-

s

we do not find Pt3Y5-D88, but find the two-phase fieldPtY ↔ PtY2. D88 and W5Si3 are higher by∼24 and∼80 meV/atom with respect to the tie-line PtY↔ PtY2

(this problem is solved with PAW-GGA potentials). Ashown in Fig. 44, at concentration 16.6% Y, we find atwo-phase region instead of the reported stable compoPt5Y with unknown structure [9]. Our best guess isPt5Y-D2d which is the least metastable structure we haat such composition (∼18 meV/atom above the tie-linePt ↔ Pt3Y). We conclude that further experimentaand theoretical investigations are necessary to determthe behavior of PtY.To address the disagreement witthe experimental results for compounds PtY and Pt3Y5,we further investigate the relevant structures with PAWGGA potentials, as described inSection 3. With PAW, PtY-B33 is stable and PtY-B27 is higher by 60 meV/atom.For compound Pt3Y5, D88 is stable with an energy6.9 meV/atom lower than the tie-line PtY↔ PtY2, whichhas also been recalculated with PAW-GGA potentials. Thedisagreement at composition PtY is further discussedSection 4.

Pt–Y system


Composition Experimental Ab initio% Y (Massalski [9]) result

16.6 Unknown Pt5Y-D2dmetastable.D2d ∼ 18 meV/atom.above the tie-line

25 L12 Pt3Y-L12

33.3 C15 Pt2Y-C15

42.9 Pt4Y3–Pd4Pu3 Unavailable

50 B27 PtY-B33B2/B27∼ 50 meV/atomabove B33B33 stable (paw-gga)B27∼ 60 meV/atomabove B33. (paw-gga)SeeSection 4.

55.6 Pt4Y5–Pu5Rh4 Unavailable

62.5 Pt3Y5-D88 Two-phase regionPtY ↔ PtY2. (us-lda)D88 ∼ 24 meV/atomabove the tie-line (us-lda)D88 stable (paw-gga).∼6.9 meV/atom belowPtY ↔ PtY2(paw-gga).

66.6 PtY2–Ni2Si [9] PtY2-C37(Co2Si) (Co2Si)

70 Pt3Y7-D102 Unavailable

75 D011 PtY3-D011

d

e

Fig. 44. PtY (platinum–yttrium) ground state convex hull.

Pt–Zr (platinum–zirconium)

The system Pt–Zr is quite interesting. Ourab initiomethod confirms the stability ofαPtZr-B33 (CrB prototype).Two crystal structures have been reported for Pt3Zr: D024and L12 [9,10,279,296,315–318]. We confirm the stability ofPt3Zr-D024 and we find L12 to be higher by 10 meV/atomwith respect to D024. In the Zr-rich part of the phase dia-gram, we find two stable phases PtZr2-C16 and PtZr3-A15.At concentration 62.5% Zr, we do not find Pt3Zr5-D88, butfind the two-phase field PtZr↔ PtZr2. W5Si3 and D88 arehigher by∼26 meV/atom and∼36 meV/atom with respectto the tie-line PtZr↔ PtZr2. To address the phase instability at composition Pt3Zr5, we further investigate the rele-vant compounds with PAW-GGA potentials, as described inSection 3. With PAW, at concentration Pt3Zr5, there is a two-phase region PtZr↔ PtZr2. In addition, Pt3Zr5–W5Si3 andPt3Zr5-D88 have energies higher by 23 and 26 meV/atomwith respect to the tie-line PtZr↔ PtZr2, respectively. Thedisagreement at composition Pt3Zr5 is further discussed inSection 4. (SeeFig. 45.)

Pt–Zr system


Composition Experimental Ab initio% Zr (Massalski [9]) result

25 D024 in Pt3Zr-D024[316,296,318,279] L12 ∼ 10 meV/atomL12 in [317] above D024

50 αPtZr-B33 (CrB) PtZr-B33

62.5 Pt3Zr5-D88 Two-phase region (us-lda)W5Si3 ∼ 26 meV/atomD88 ∼ 36 meV/atomabove tie-line (us-lda).two-phase region (paw-gga)W5Si3 ∼ 23 meV/atomD88 ∼ 26 meV/atomabove tie-line (paw-gga).SeeSection 4.

66.6 Two-phase region PtZr2-C16above 600C (uncertain)

75 Two-phase region PtZr3-A15above 600C (uncertain)


o

)%s

n

i

e-

on

c-lic

in

Fig. 45. PtZr (platinum–zirconium) ground state convex hull.

Rh–Ru (rhodium–ruthenium)

The phase diagram for the system Rh–Ru is basedthe solid solubility data [9,10,42,320]. Experimental resultsreport two solid solutions (fcc Rh-rich) and (hcp Ru-richwith a two-phase region in between (from 34.5% to 40atomic percent ruthenium). No intermetallic compoundhave been reported [319–321] and the system is consideredto be non-compound forming [10]. However, we findtwo stable phases: an orthorhombic oC12 RhRuproto

2 , withhcp superstructure and Cmcm #63 space group, aa trigonal hP4 RhRuproto, with hcp superstructure andP3m1 #164 space group. Both prototypes are describedAppendix A. As shown in Fig. 46, all the stable phaseshave small formation energy (<10 meV/atom) making themdifficult to determine experimentally.To better describethe stability of this system, we further investigate all thstructures with negative formation energies with PAWGGA potentials, as described inSection 3. Also with PAW,RhRuproto

2 and RhRuproto are the most stable structures, withformation energies of−8.3 meV/atom and−8.8 meV/atom,respectively. (SeeFig. 46.)

Rh–Ru system



33 Solid solution RhRuproto2 stable

Ru-A3 both us-lda andpaw-gga,E f ∼ −8.3 meV/atom(paw-gga).

50 Solid solution RhRuproto stableRu-A3 both us-lda andpaw-gga,

E f ∼ −8.8 meV/atom(paw-gga).

n

d

n

Fig. 46. RhRu (rhodium–ruthenium) ground state convex hull.

Rh–Tc (rhodium–technetium)

The phase diagram for the system Rh–Tc is basedthe solid solubility data [9,10,42,360,362]. Experimentalresults report two solid solutions (fcc Rh-rich) and (hcp Trich) with a two-phase region in between. No intermetalcompounds have been reported [360,362]. However, wefind three stable phases: RhTc3-D019, RhTc-B19, andRu2Tc–ZrSi2. Trends for Tc alloys are further discussedSection 8. (SeeFig. 47.)

Rh–Tc system



25 Solid solution Tc-A3 RhTc3-D019

50 Solid solution Tc-A3 RhTc-B19B27 ∼ 29 meV/atomabove B19 (us-lda).

66.6 Two-phase region Rh2Tc–ZrSi2above∼1000CPt-A1 ↔ Tc-A3

Fig. 47. RhTc (rhodium–technetium) ground state convex hull.


as

al

c

r

ld

s

e

em

e

le

o

cy

.o-

Rh–Ti (rhodium–titanium)

There are qualitative disagreements about the phdiagram of the Rh–Ti system [9,10,47,316,322–326].We confirm the stable phases that where found byinvestigators [322–325]: RhTi2, αRhTi-L10, and Rh3Ti-L12.Refs. [10,316,326] report RhTi2–CuZr2 instead of RhTi2-C11b. The prototype CuZr2 is a distortion of C11b andhas the same lattice type (tetragonal, tI6) and spagroup (I4/mmm #139) [10]. We find RhTi2-C11b andRhTi2–CuZr2 to have degenerate energy. In the Rh-rich paof the phase diagram we find a stable phase Rh2Ti-C37.However, we do not have Rh5Ti3 in our library, so C37might likely be unstable with respect to the two-phase fieRh5Ti3 ↔ L12. At concentration≈84% Rh, we do notfind any stable compound, in agreement with [324] and incontrast with [322]. To address the degenerate structureRhTi2-C11b/CuZr2, we further investigate RhTi2 with PAW-GGA potentials, as described inSection 3. Also with PAW,C11b and CuZr2 remain degenerate.

Otherab initio studies, relevant for this system, can bfound in Refs. [226,286]. (SeeFig. 48.)

Rh–Ti system



33.3 C11b[9]/CuZr2[10] RhTi2C11b/CuZr2 (us-lda,)C11b/CuZr2 (paw-gga)

∼38–58 αRhTi-L10 RhTi-L10

62.5 Rh5Ti3–Ge3Rh5 Unavailable

66.6 Two-phase region Rh2Ti-C37above 600C

73–78 L12 Rh3Ti-L12

∼83.8 Rh5Ti (unknown) Nothing stable

Fig. 48. RhTi (rhodium–titanium) ground state convex hull.

e

l

e

t

Rh–Y (rhodium–yttrium)

Several compounds have been reported for the systRh–Y at low temperature [9,10,42,271,327–329,331]. Thestability of Rh2Y-C15, RhY-B2, RhY3-D011 is confirmed.At composition Rh5Y, where a D2d structure has been seenat high temperature (but not stable at low temperature), wfind the D2d to be the lowest energy structure (of all thestructures at that composition), even though it is metastabwith respect to the phase separation into Rh2Y-C15 ↔ Rh.At composition Rh3Y one compound has been reported tbe stable with prototype CeNi3 and space group P63/mmc#194 [10,330,332]. We do not have such prototype in ourlibrary and we do not find any stable phase: Rh3Y-D019is the least metastable prototype we obtain. Rh3Y-D019 ishigher by 130 meV/atom with respect the tie-line, whichis at least one order of magnitude bigger than the accuraof the calculations. We also find RhY2-C37, which appearsin a concentration between two known compounds, Rh3Y5and Rh3Y7, that are not present in our set of calculationsTherefore C37 might be unstable with respect to the twphase region Rh3Y5 ↔ Rh3Y7. (SeeFig. 49.)

Rh–Y system



% Rh (Massalski [9]) result

25 D011 RhY3-D011

30 Rh3Y7-D102 Unavailable

33.3 Two-phase region RhY2-C37

above 0C (uncertain)

37.5 Rh3Y5 (unknown) Unavailable

40 Rh2Y3 (unknown) Unavailable

tI140–I4/mcm

50 B2 RhY-B2

66.6 C15 Rh2Y-C15

75 Rh3Y (unknown) D019 ∼ 130 meV/atom.

hP24–P63/mmc above tie-line

83.5 Rh5Y-D2d D2d ∼ 110 meV/atom.

(high temperature) above tie-line


u-

in

a

dd

se

be

redat

leTc.

Fig. 49. RhY (rhodium–yttrium) ground state convex hull.

Rh–Zr (rhodium–zirconium)

Although the system Zr–Rh is well known for its sperconducting phases [9,10,333,336], further investigationsare needed to clarify the stability and presence oftermediate phases [334,335]. Our ab initio method con-firms the stability of RhZr2-C16 and Rh3Zr-L12. Massalskidoes not report the prototype of the low temperature phαRhZr [9]. References [336–338] report αRhZr-B27. Weconfirm the stability ofαRhZr-B27 (BFe prototype) with noother metastable compounds with similar energy. In ation, we find three new phases Rh2Zr-C37, RhZr4-D1a andRhZr3-FCC[001]

AB3 , which are degenerate with the two-phafields RhZr↔ Rh3Zr, Zr ↔ RhZr, and Zr↔ RhZr, respec-tively.

Otherab initio studies, relevant for this system, canfound in Refs. [226,339]. (SeeFig. 50.)

Rh–Zr system



20 Two-phase region RhZr4-D1a/tie-lineabove 0C

25 Two-phase region RhZr3-FCC[001]AB3 /tie-line

above 0C

33.3 C16 RhZr2-C16

50 to ? αRhZr RhZr-B27unknown [9]B27 [336–338]

57.1 Rh4Zr3 (unknown) unavailable

62.5 Rh5Zr3–Pd3Pu3 unavailable

66.6 Two-phase region Rh2Zr-C37/tie-lineabove 0C

72–82 L12 Rh3Zr-L12

-

se

i-

Fig. 50. RhZr (rhodium–zirconium) ground state convex hull.

Ru–Tc (ruthenium–technetium)

The phase diagram for the system Ru–Tc is consideto have a continuous disordered hcp solid solutionlow temperature [9,10,340,359,362]. We find three stableordered phases Ru3Tc-D019, RuTc-B19, and RuTc3-D019,all of which are hcp superstructures. Hence, it is possibthat they are low temperature phases of the system Ru–Trends for Tc alloys are further discussed inSection 8. (SeeFig. 51.)

Ru–Tc system



25 Disorder solution RuTc3-D019(Ru,Tc)-A3

50 Same as above RuTc-B19

75 Same as above Ru3Tc-D019

Fig. 51. RuTc (ruthenium–technetium) ground state convex hull.


yn

y

e

Y

ith

onse

d,

Ru–Ti (ruthenium–titanium)

The phase diagram of Ru–Ti is well determined bseveral investigators. A single low temperature compouRuTi-B2 has been reported [9,10,47,279,341–348], and ourab initio method confirms its stability. In addition, we findtwo stable phases: RuTi2-C49, and orthorhombic RuTi3 withspace group Immm #71, bcc superstructure, and prototRuTiproto

3 described inAppendix A. Such compounds areclose to the tie-line Ti ↔ RuTi. In fact, for C49 andRuTiproto

3 , the formation energies are lower by∼34 and∼37 meV/atom with respect to the two-phase field Ti↔RuTi. (SeeFig. 52.)

Ru–Ti system



25 Two-phase region RuTiproto3

above 600C Appendix A∼37 meV/atombelow Ti↔ RuTi

33.3 Same as above RuTi2-C49∼34 meV/atombelow Ti↔ RuTi

45 to 52±1 B2 RuTi-B2

Fig. 52. RuTi (ruthenium–titanium) ground state convex hull.

Ru–Y (ruthenium–yttrium)

Several compounds have been reported for the systRu–Y [9,10,42,327,349–353]. We confirm the stability ofRuY3-D011 and Ru2Y-C14. Also, we find RuY2-C16,which appears in a concentration between two knowcompounds Ru25Y44 and Ru2Y5 that are not in our library ofcalculations. Hence, the existence of RuY2-C16 is uncertain.At concentration 50%, we do not find any stable Rucompound, in agreement with [352]. (SeeFig. 53.)

d

pe

m

n

Ru–Y system



25 D011 RuY3-D011

28.6 Ru2Y5–C2Mn5 UnavailablemS28 C12/c1[10,349,350,352]

33.3 Two-phase region RuY2-C16above 0C (uncertain)

36.2 Ru25Y44 (unknown) UnavailableoP276 Pnna[10,349,353]

40 Ru2Y3–Er3Ru2 UnavailablehP10 P63/m [354]

66.6 Ru2Y-C14 Ru2Y-C14

Fig. 53. RuY (ruthenium–yttrium) ground state convex hull.

Ru–Zr (ruthenium–zirconium)

The phase diagram of RuZr is known accurately [9,10,355,356]. Our ab initio method confirms the stability ofthe low temperature phase RuZr-B2. In agreement wexperiments, we find no ground state at composition Ru2Zr,though the lowest energy structure at that compositiin our calculations is C14 which appears in the phadiagram at high temperature. Ru2Zr-C14 is higher by∼60 meV/atom with respect to the two-phase field Ru↔RuZr. In the Zr-rich region we find one stable compounRuZr4-D1a, previously unknown. In addition, we find threemetastable phases: RuZr5 (monoclinic with space groupC2/m #12), RuZr3, and RuZr2-C49, with energies higherby 21 meV/atom, 16.6 meV/atom, and 14 meV/atom, withrespect to the two-phase fields RuZr4 ↔ Zr, RuZr ↔RuZr4, and RuZr ↔ RuZr4, respectively. The structureof RuZr5 is similar to Mo5Tiproto (Appendix B), while thestructure of RuZr3 is similar to RuTiproto

3 (Appendix A).


e

es

ne

ti

a

d

Otherab initio studies, relevant for this system, can bfound in Refs. [232–234,357]. (SeeFig. 54.)

Ru–Zr system



16.6 Two-phase region RuZr5 ≈ Mo5Tiproto

RuZr ↔ Zr ∼21 meV/atomabove 400C above RuZr4 ↔ Zr

20 Same as above RuZr4-D1a

25 Same as above RuZr3 ≈ RuTiproto3∼16.6 meV/atom

above RuZr↔ RuZr4

33.3 Same as above RuZr2-C49∼14 meV/atomabove RuZr↔ RuZr4

48–52 B2 RuZr-B2

66–68 C14 (high T) Ru2Zr-C14∼60 meV/atomabove Ru↔ RuZr

Fig. 54. RuZr (ruthenium–zirconium) ground state convex hull.

Tc–Ti (technetium–titanium)

The phase diagram of the system TcTi has beconstructed by analogy with chemically related system[9,10,47,358,359]. Two intermetallic compounds, TcTi-B2andχ , are reported [358,359]. We confirm the stability ofTcTi-B2, but we cannot say anything aboutχ since we donot have prototypes at composition 85% Ru. In additiowe find Tc2Ti-C11b, TcTi2-C49, and an orthorhombic phasTcTi3 with space group Immm #71 and prototype TcTiproto

3described inAppendix A. These intermetallics have largenegative formation energy; therefore they are expectedbe very stable. Trends for Tc alloys are further discussedSection 8. (SeeFig. 55.)

n

,

on

Tc–Ti system



25 Disorder TcTiproto3

βTi-A2 Appendix A

33.3 Disorder TcTi2-C49βTi-A2

∼50 B2 TcTi-B2

66.6 Two-phase region Tc2Ti-C11bTcTi ↔ χ

∼85 χ -A12 Unavailable

Fig. 55. TcTi (technetium–titanium) ground state convex hull.

Tc–Y (technetium–yttrium)

Not enough information exists for constructingphase diagram for the system Tc–Y [9,10]. Only oneintermetallic compound has been reported: Tc2Y-C14(Friauf–Laves/Frank–Kasper phase) [361,363]. We confirmthe stability of Tc2Y-C14. In addition, we find another stablephase TcY3-D011. Trends for Tc alloys are further discussein Section 8. (SeeFig. 56.)

Tc–Y system



% Tc (Massalski [9]) result

25 No information TcY3-D011

66.6 C14 Tc2Y-C14

[361,363] (Laves phase)


etn

eryin

s

:

io

d

nd

by

er

ofo.n.

s,r,

Fig. 56. TcY (technetium–yttrium) ground state convex hull.

Tc–Zr (technetium–zirconium)

Not enough information exists for constructing aphase diagram for the system Tc–Zr [9,10]. Only twointermetallic compounds have been reported: Tc2Zr-C14(Friauf–Laves/Frank–Kasper phase) and Tc6Z-A12 [9,10,359,364]. We confirm the stability of Tc2Zr-C14, but wecannot determine A12 since we do not have the propprototype in our library. At 50% composition, Miedema eal. reported the existence of a TcZr compound with unknowstructure [10,279]. At such composition we find TcZr-B2. In addition, we find other stable phases: TcZr4-D1a

and TcZr2-C49. These intermetallics have large negativformation energy; therefore they are expected to be vestable. Trends for Tc alloys are further discussedSection 8. (SeeFig. 57.)

Tc–Zr system



20 No information TcZr4-D1a

33.3 No information TcZr2-C49

50 TcZr (unknown) TcZr-B2[279]

66.6 C14 Tc2Zr-C14[359,364] (Laves phase)

85.7 Tc6Z-A12 Unavailable

8. Trend for technetium alloys

In our set of calculations, we have noticed that the phaD019 appears in systems MTc3 where M is a transition metalin the columns on the right of Tc (Tc is in column 7B) whileD019 is not present if M is in the columns on the left of TcD019 is stable in PdTc3, PtTc3, RhTc3, RuTc3, and unstablein NbTc, TcTi, TcY, and TcZr.

r

e

Fig. 57. TcZr (technetium–zirconium) ground state convex hull.

9. Tables of results

9.1. Experimental compounds in agreement with ab initsolutions

SeeTable 5.

9.2. Experimentally unknown, non-identified or speculatecompounds and ab initio predictions

SeeTable 6.

9.3. Experimental solid solutions, two-phase systems a“not studied” regions and possible ab initio predictions

SeeTables 7aand7b.

9.4. Experimental compounds that could not be checkedour calculations

SeeTable 8

9.5. Experimental compounds in disagreement with othab initio compounds or two-phase regions

SeeTable 9.

Acknowledgments

The research was supported by the DepartmentEnergy, Office of Basic Energy Science under Contract NDE-FG02-96ER45571 and National Science FoundatioInformation Technology Research (NSF-ITR) Grant NoDMR-0312537.

It has benefited from discussion with John RodgerKristin Persson, Frank Hadley Cocks, Chris FischeAleksey Kolmogorov, Vincent Crespi and Zi-Kui Liu.


Table 5Experimental compounds in agreement withab initio solutions (89 entries)

Experimental compound⇔ ab initio compound

System Experimental result Ab initio result System Experimental result Ab initio result

Ag–Cd AgCd-B2 [10,66–69] AgCd-B2/B19/B27 Nb–Pd NbPd3-D022 NbPd3-D022

Ag–Mg Ag3Mg-D023 [75] AgMg-D023/D024 Nb–Pt Nb3Pt-A15 Nb3Pt-A15

Ag–Mg AgMg-B2 AgMg-B2 Nb–Pt NbPt2-MoPt2 NbPt2-MoPt2

Ag–Na Ag2Na-C15 Ag2Na-C15 Nb–Pt NbPt3-D0α NbPt3-D0α

Ag–Ti AgTi2-C11b AgTi2-C11b Nb–Rh Nb3Rh-α′-A15 Nb3Rh-A15

Ag–Ti AgTi-B11 AgTi-B11 (γ CuTi) Nb–Rh NbRh-L10 NbRh-L10

Ag–Y AgY-B2 AgY-B2 Nb–Rh ε(Nb2Rh3)-B19 (highT) NbRh-B19

Ag–Y Ag2Y-C11b Ag2Y-C11b Nb–Rh η(Nb13Rh27)-Al3Pu NbRh3–Al3Pu

Ag–Zr AgZr2-C11b AgZr2-C11b Pd–Ti PdTi3-A15 at 20% PdTi3-A15 at 25%

Ag–Zr AgZr-B11 AgZr-B11(γ CuTi) Pd–Ti PdTi2-C11b PdTi2-C11b

Al–Sc AlSc2-B82 AlSc2-B82 Pd–Ti α(TiPd)-B19 PdTi-B19

Al–Sc AlSc-B2 AlSc-B2 Pd–Ti Pd3Ti-D024 Pd3Ti-D024

Al–Sc Al2Sc-C15 Al2Sc-C15 Pd–Y PdY3-D011 PdY3-D011

Al–Sc Al3Sc-L12 Al3Sc-L12 Pd–Y Pd3Y-L12 Pd3Y-L12

Au–Cd β′AuCd-B19 AuCd-B19 Pd–Zr PdZr2-C11b PdZr2-C11b

Au–Cd βAuCd-B2 (highT) AuCd-B2 Pd–Zr PdZr-B33 PdZr-B33

Au–Nb Au2Nb-C32 Au2Nb-C32 (paw-gga) Pd–Zr Pd2Zr-C11b Pd2Zr-C11b/tie-line (paw-gga)

Au–Sc Au4Sc-D1a Au4Sc-D1a Pd–Zr Pd3Zr-D024 Pd3Zr-D024

Au–Sc Au2Sc-C11b Au2Sc-C11b/MoPt2 Pt–Ti Pt3Ti-D024 Pt3Ti-D024

Au–Sc AuSc-B2 AuSc-B2/B19 Pt–Ti αPtTi-B19 PtTi-B19

Au–Ti Au4Ti-D1a Au4Ti-D1a Pt–Ti PtTi3-A15 PtTi3-A15

Au–Ti Au2Ti-C11b Au2Ti-C11b/MoPt2 Pt–Y Pt3Y5-D88 Pt3Y5-D88 (paw-gga)

Au–Ti αAuTi-B11 AuTi-B11 Pt–Y Pt3Y-L12 Pt3Y-L12

Au–Ti AuTi3-A15 AuTi3-A15 Pt–Y Pt2Y-C15 Pt2Y-C15

Au–Y Au3Y-D0a Au3Y-D0a Pt–Y PtY2-C37 PtY2-C37

Au–Y Au2Y-C11b Au2Y-C11b Pt–Y PtY3-D011 PtY3-D011

Au–Zr Au3Zr-D0a Au3Zr-D0a Pt–Zr Pt3Zr-D024 Pt3Zr-D024

Au–Zr Au2Zr-C11b Au2Zr-C11b Pt–Zr PtZr-B33 PtZr-B33

Au–Zr AuZr2–CuZr2/C11b AuZr2–CuZr2 (paw-gga) Rh–Ti RhTi2-C11b [9]/CuZr2 [10] RhTi2-C11b/CuZr2

Au–Zr AuZr3-A15 AuZr3-A15 Rh–Ti αRhTi-L10 RhTi-L10

Cd–Pd CdPd-β1-L10 CdPd-L10 Rh–Ti Rh3Ti-L12 Rh3Ti-L12

Cd–Pt CdPt-α′1-L10 CdPt-L10 Rh–Y RhY3-D011 RhY3-D011

Cd–Ti CdTi2-C11b CdTi2-C11b Rh–Y RhY-B2 RhY-B2

Cd–Ti CdTi-B11 CdTi-B11(γ CuTi) Rh–Y Rh2Y-C15 Rh2Y-C15

Cd–Y CdY-B2 CdY-B2 Rh–Zr RhZr2-C16 RhZr2-C16

Cd–Y Cd2Y-C6 Cd2Y-C6 Rh–Zr αRhZr-B27 RhZr-B27

Cd–Y Cd3Y-Cd3Er Cd3Y-Cd3Er Rh–Zr Rh3Zr-L12 Rh3Zr-L12

Cd–Zr CdZr2-C11b CdZr2-C11b Ru–Ti RuTi-B2 RuTi-B2

Cd–Zr CdZr-B11 CdZr-B11 (paw-gga) Ru–Y RuY3-D011 RuY3-D011

Cd–Zr Cd3Zr-L12 Cd3Zr-L12 Ru–Y Ru2Y-C14 Ru2Y-C14

Mo–Pt MoPt-B19 MoPt-B19 Ru–Zr RuZr-B2 RuZr-B2

Mo–Pt MoPt2 MoPt2 Tc–Ti TcTi-B2 TcTi-B2

Mo–Rh MoRh-B19 MoRh-B19 Tc–Y Tc2Y-C14 Tc2Y-C14

Mo–Zr Mo2Zr-C15 Mo2Zr-C15 Tc–Zr Tc2Zr-C14 Tc2Zr-C14

Nb–Pd NbPd2–MoPt2 NbPd2–MoPt2


Table 6Experimentally unknown, non-identified or speculated compounds andab initio predictions (21 entries)

Experimental unknown/speculated compound⇔ ab initio compound

System Experimental result Ab initio result

Ag–Cd β′-bcc ordered AgCd-B2/B19/B27

Ag–Mg cF∗ (unknown) AgMg3-D019 (hP8)/D0a (oP8)

Au–Cd Au3Cd-α2-hP? (unknown) Au3Cd-D024/D019/Al3Pu

Au–Cd α′′ ∼ AuCd (unknown) AuCd-L10/FCC[201]A2B2

Au–Cd ε′ ∼ AuCd3 (unknown) AuCd3-L60

Au–Pd Au3Pd-L12 (speculated) Au3Pd-D023/D022/L12(wide concentration range) Au4Pd-D1a/tie-line

Au–Pd AuPd (speculated and unknown) AuPd-FCC[201]A2B2

Au–Pd AuPd3-L12 (speculated) AuPd3-D023/D022/L12/tie-line

Au–Sc AuSc2-C37 (speculated) AuSc2-C37

Cd–Pd γ ′-(unknown) Cd3Pd-D019/D024/NbPd3/Al3Pu

Cd–Pt γ1-(unknown) Cd3Pt-D011/D0a/D022

Cd–Pt Cd2Pt-(unknown) Cd2Pt-C37/C16/tie-line

Cd–Zr cubic (unknown) Cd2Zr-C11b

Mo–Pd MoPd2 ∼ MoPt2 MoPd2–ZrSi2

Mo–Rh MoRh3 (unknown) MoRh3-CdMg3

Nb–Pt Nb two-phase region1−xPt1+x-B19 NbPt-L10

Pd–Ti orthorhombic distortion of Pd2Ti-C11b Pd2Ti-MoPt2 (distortion of C11b)

Pd–Y αPdY (unknown) PdY-B27

Pt–Ru PtRu-FCC (unknown) PtRu-FCC[001]A2B2

Pt–Y Pt5Y (unknown) Pt5Y-D2d metastable

Tc–Zr TcZr (unknown) TcZr-B2


Table 7aExperimental solid solutions, two-phase systems and “not studied” regions and possibleab initio predictions

Experimental non-compound⇔ ab initio compound


Ag–Au short-range order 950C Ag4Au-D1a/tie-line

Ag–Au short-range order 950C Ag3Au-L12/D023/Al3Pu/NbPd3/D022/D024

Ag–Au short-range order 950C Ag2Au-C37/MoPt2Ag–Au short-range order 950C AgAu-L10

Ag–Au short-range order 950C AgAu2-C37/MoPt2Ag–Au short-range order 950C AgAu3-L12/D022/D023

Ag–Cd fcc solid solution Ag3Cd-D022/D024

Ag–Cd fcc solid solution Ag2Cd-C37

Ag–Cd none AgCd2–ZrSi2Ag–Cd hcp solid solution AgCd3-D019

Ag–Pd solid solution 900C AgPd-L11Ag–Pd solid solution 900C Ag2Pd-C37

Ag–Pd solid solution 900C Ag3Pd-L12/D022

Ag–Y two-phase region above 200C AgY2-C37/tie-line (uncertain)

Ag–Y two-phase region above 200C Ag3Y-D0a (uncertain)

Ag–Zr two-phase region above 700C AgZr3-FCC[001]AB3 /tie-line (uncertain)

Ag–Zr two-phase region above 700C Ag2Zr-C32

Al–Sc two-phase region above 0C AlSc3-D019

Au–Nb two-phase region (calculated) AuNb2-BCC[011]AB2

Au–Pd solid solution Au2Pd-C49/C37

Au–Ti two-phase region above 500C Au4Ti3-Cu4Ti3/tie

Au–Y not studied AuY2-C37

Au–Zr two-phase region Au4Zr3-Cu4Ti3 (unreliable)

Au–Zr two-phase region AuZr-B11/FCC[001]A2B2 (unreliable)

Cd–Pd Pd phase above 100C CdPd3-D022/NbPd3Cd–Pd two-phase region above 100C CdPd2-C37

Cd–Rh not studied Cd2Rh-C37

Cd–Rh not studied Cd3Rh–Al3Pu

Cd–Ti two-phase region above 200C Cu4Ti3/tie-line

Cd–Y not studied/two-phase region CdY2-C37

Cd–Zr not studied/two-phase region CdZr3-A15

Mo–Nb not studied MoNb2-C11b

Mo–Nb not studied MoNb-B2

Mo–Nb not studied Mo2Nb-C11bMo–Nb not studied Mo3Nb-D03

Mo–Pd disorder fcc Pd-A1 MoPd4-D1a

Mo–Pt two-phase region MoPt2↔ Pt MoPt4-D1a

Mo–Rh two-phase region above 900C Mo2Rh-C37

Mo–Ru disorder hcp Ru-A3 above 800C MoRu3-D019

Mo–Ti not studied/two-phase region above 400C MoTiproto3 /tie-line (uncertain),Appendix A

Mo–Ti not studied/two-phase region above 400C MoTi2-BCC[211]AB2

Mo–Ti not studied/two-phase region above 400C MoTiproto, Appendix A

Mo–Ti not studied/two-phase region above 400C Mo2Ti-C11b

Mo–Ti not studied/two-phase region above 400C Mo3Tiproto, Appendix A

Mo–Ti not studied/two-phase region above 400C Mo4Ti-D1a/tie-line

Mo–Ti not studied/two-phase region above 400C Mo5Tiproto, Appendix BThe table continues on the next page.


Table 7bExperimental solid solutions, two-phase and “not studied” regions, and possibleab initio predictions (Table 7a+ Table 7b= 96 entries)

Experimental non-compound⇔ ab initio compound


Nb–Pd two-phase region above 700C Nb2Pd-BCC[011]AB2

Nb–Ru disorder Nb-A2 Nb5Ruproto, Appendix B

Nb–Ru disorder Nb-A2 Nb3Ru-D03

Nb–Ru two-phase region above 700C NbRu2-C37

Nb–Tc not studied/unknown Nb3Tcproto, Appendix A

Nb–Tc not studied/unknown Nb2Tc-C11bNb–Tc not studied/unknown NbTc-B2

Pd–Pt two-phase region Pd↔ Pt Pd3Ptproto/tie-line (uncertain),Appendix A

Pd–Pt two-phase region Pd↔ Pt PdPt-L11Pd–Pt two-phase region Pd↔ Pt PdPtproto

3 , Appendix A

Pd–Tc two-phase region above∼1000C PdTc3-D019

Pd–Y two-phase region PdY2-C37 (uncertain)

Pd–Zr two-phase region PdZr3-FCC[001]AB3 (uncertain)

Pt–Rh two-phase region Pt↔ Rh [9], D1a [256] Pt4Rh-D1a

Pt–Rh two-phase region Pt↔ Rh Pt3Rh-D022

Pt–Rh two-phase region Pt↔ Rh PtRh–NbP

Pt–Rh two-phase region Pt↔ Rh PtRh2-C49

Pt–Rh two-phase region Pt↔ Rh PtRh3-D022

Pt–Rh two-phase region Pt↔ Rh [9], D1a [256] PtRh4-D1a

Pt–Ru disorder Pt-A1 Pt3Ru-FCC[001]AB3 /tie-line (uncertain)

Pt–Tc disorder Pt-A1 Pt3Tc-FCC[001]AB3

Pt–Tc two-phase region above∼1000C PtTc3-D019

Pt–Zr two-phase region above 600C PtZr2-C16 (uncertain)

Pt–Zr two-phase region above 600C PtZr3-A15 (uncertain)

Rh–Ru solid solution Ru-A3 RhRuproto2

Rh–Ru solid solution Ru-A3 RhRuproto

Rh–Tc solid solution Tc-A3 RhTc3-D019

Rh–Tc solid solution Tc-A3 RhTc-B19

Rh–Tc two-phase region above 1000C Rh2Tc2–ZrSi2Rh–Ti two-phase region above 600C Rh2Ti-C37

Rh–Y two-phase region above 0C RhY2-C37 (uncertain)

Rh–Zr two-phase region above 0C RhZr4-D1a/tie-line

Rh–Zr two-phase region above 0C RhZr3-FCC[001]AB3 /tie-line

Rh–Zr two-phase region above 0C Rh2Zr-C37/tie-line

Ru–Tc disorder solution (Ru,Tc)-A3 RuTc3-D019

Ru–Tc disorder solution (Ru,Tc)-A3 RuTc-B19

Ru–Tc disorder solution (Ru,Tc)-A3 Ru3Tc-D019

Ru–Ti two-phase region above 600C RuTiproto3 , Appendix A

Ru–Ti two-phase region above 600C RuTi2-C49

Ru–Y two-phase region above 0C RuY2-C16 (uncertain)

Ru–Zr two-phase region above 400C RuZr4-D1a

Tc–Ti disorderβTi-A2 TcTiproto3 , Appendix A

Tc–Ti disorderβTi-A2 TcTi2-C49

Tc–Ti two-phase region TcTi↔ χ Tc2Ti-C11b

Tc–Y no information TcY3-D011

Tc–Zr no information TcZr4-D1a

Tc–Zr no information TcZr2-C49The table starts in the previous page.


not in o
Table 8Experimental compounds that could not be checked by our calculations, because the proper structure prototype or concentration is not known orurlibrary
Experimental compound⇔ unavailable compound


Ag–Mg AgMg4-hP∗ (unknown) unknown experimental compound

Ag–Y Ag51Y14–Ag51Gd14 unavailable prototype

Au–Cd η′ (unknown) unknown experimental compound

Au–Cd α1Au3Cd–Ag3Mg unavailable prototype

Au–Cd Au3Cd5-D8m unknown experimental compound

Au–Nb Au2Nb3 (unknown) unknown experimental compound

Au–Ti βAuTi-B19 unavailable prototype off-stoichiometry B19

Au–Y AuY3 (unknown) unknown experimental compound

Au–Zr Au4Zr (oP20–Pnma) unavailable prototype

Au–Zr Au10Zr7 (tI34) unavailable prototype

Au–Zr Au4Zr5 (unknown prototype) unknown experimental compound

Cd–Pd γ1-(unknown) unknown experimental compound

Cd–Pd γ -D83 unavailable prototype

Cd–Pt γ -(unknown) unknown experimental compound

Cd–Pt γ2-(unknown) unknown experimental compound

Cd–Y Cd45Y11–Cd45Sm11 unavailable prototype

Cd–Y Cd58Y13–Pu13Zn58 unavailable prototype

Cd–Y Cd6Y unavailable prototype

Mo–Pt ε′-D019 above 1000C unavailable prototype off-stoichiometry D019

Nb–Pt D8b unavailable prototypeσ phase

Nb–Rh σ(Nb13Rh7)-D8b unavailable prototype

Nb–Rh ξ(Nb2Rh3)–Nb2Rh3 unavailable prototype

Nb–Tc NbTc3, Nb0.15Tc0.85–αMn unavailable prototype

Pd–Ti Pd3Ti2 ∼ Au2V unavailable prototype

Pd–Ti Pd5Ti3 ∼ C11b unavailable prototype

Pd–Y unknown at 66.6% Pd unknown experimental compound

Pd–Y unknown at 87.5% Pd unknown experimental compound

Pd–Y Pd2Y5 (unknown) unknown experimental compound

Pd–Y Pd2Y3-hR15 (unknown) unknown experimental compound

Pd–Y Pd4Y3-hR14 (unknown) unknown experimental compound

Pd–Y αPd3Y2 (unknown) unknown experimental compound

Pt–Ti Pt8Ti-D1α unavailable prototype off-stoichiometry D1α

Pt–Ti Pt2Ti–ReSi2 unavailable prototype

Pt–Y Pt4Y3–Pd4Pu3 unavailable prototype

Pt–Y Pt4Y5–Pu5Rh4 unavailable prototype

Pt–Y Pt3Y7-D102 unavailable prototype

Rh–Ti Rh5Ti (unknown) unknown experimental compound

Rh–Ti Rh5Ti3–Ge3Rh5 unavailable prototype

Rh–Y Rh3Y (unknown), hP24–P63/mmc unknown experimental compound

Rh–Y Rh3Y7-D102 unavailable prototype

Rh–Y Rh3Y5 (unknown) unknown experimental compound

Rh–Y Rh2Y3 (unknown), tI140–I4/mcm unknown experimental compound

Rh–Zr Rh4Zr3 (unknown) unknown experimental compound

Rh–Zr Rh5Zr3–Pd3Pu3 unavailable prototype

Ru–Y Ru2Y5–C2Mn5 unavailable prototype

Ru–Y Ru25Y44 (unknown) unknown experimental compound

Tc–Ti χ -A12 unavailable prototype

Tc–Zr Tc6Zr-A12 unavailable prototypeThe table contains 21 “unknown experimental compounds” and 27 “unavailable prototypes” (48 entries total).


Note (a)ertain

Table 9Experimental compounds (which we have in the library as prototypes) in disagreement with otherab initio compounds or two-phase regions

Experimental compound⇔ ab initio wrong compound or two-phase systems

System Experimental result Ab initio result E f (us-lda) E f (paw-gga) Note(meV/atom) (meV/atom)

Au–Nb AuNb3-A15 two-phase region 7 6.5 (a)

Au–Y AuY-B2 [169] AuY-B33 26 25 (b)

Cd–Nb Cd3Nb-L12 immiscible system 70 >100 (c)

Cd–Pt CdPt3-α′-L12 [180] CdPtproto3 , Appendix A 25 10.4 (b)

Nb–Rh κ(NbRh3)-L12 NbRh3–Al3Pu 8 5.3 (a)

Nb–Ru NbRu3-L12 NbRu3-D024 8 2.5 (a)

Nb–Ru NbRu′-L10 two-phase region 20 4 (a)

Pt–Y PtY-B27 PtY-B33 50 60 (c)

Pt–Zr Pt3Zr5-D88 two-phase region 36 23 (c)

We include the differences between the formation energies of the phases in disagreement, both for US-LDA and PAW-GGA potentials (9 entries).:theab initio ground state is within less than 10 meV/atom of the experimental ground state (4). Note (b): the assigned experimental structures are unc(2) [169,180]. Note (c): unambiguous significant disagreement (3).

Appendix A. New structure prototypes which are superstructures of FCC, BCC, HCP (Table 10)

Table 10Geometry of FCC, BCC, HCP superstructures which appear as new prototypes in our study

System CdPt3, PdPt3, MoTi, MoTi3, Mo3Ti, Nb3Tc, RhRu RhRu2Pd3Pt RuTi3, TcTi3

Superlattice FCC AB3 BCC A2B2 BCC AB3 HCP A2B2 HCP A2B4

Lattice Orthorhombic Orthorhombic Orthorhombic Trigonal Orthorhombic

Space group Cmmm #65 Imma #74 Immm #71 P3m1 #164 Cmcm #63

Pearson symbol oC8 oI8 oI8 hP4 oC12

Primitive vectors(cart.)a1/a (1,−1/2, 1/2) (3/2, 1/2,−1/2) (3/2, 1/2,−1/2) (1/2, −√

3/2, 0) (−1/2, 3√

3/2, 0)

a2/a (−1,−1/2, 1/2) (1/2, 3/2, 1/2) (1/2, 3/2, 1/2) (1/2,√

3/2, 0) (−1/2,−3√

3/2, 0)

a3/a (0,−1/2,−1/2) (−1/2,−3/2, 1/2) (−1/2,−3/2, 1/2) (0, 0, 2√

8/3) (0, 0,√

8/3)

Atomic positions(fract.)A1 (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0)

A2 − (1/4, 3/4, 1/2) − (1/3, 2/3, 1/4) (5/9, 4/9, 1/2)

B1 (0, 1/2, 1/2) (1/2, 1/2, 0) (1/4, 3/4, 1/2) (0, 0, 1/2) (2/9, 7/9, 1/2)

B2 (1/2, 0, 1/2) (3/4, 1/4, 1/2) (1/2, 1/2, 0) (1/3, 2/3, 3/4) (1/3, 2/3, 0)

B3 (1/2, 1/2, 0) − (3/4, 1/4, 1/2) − (2/3, 1/3, 0)

B4 − − − − (8/9, 1/9, 1/2)

Positions are given as unrelaxed positions in the parent lattice.


Appendix B. Relaxed structure prototypes (Table 11)

Table 11Geometry of relaxed structures which appear as new prototypes in our study. Positions are given as fractional positions of the primitive vectors

System Mo5Ti∗ (AB5) Nb5Ru∗ (AB5)

Lattice Monoclinic Monoclinic

Space group C2/m #12 C2/m #12

Prototype Mo5Ti see note∗ Mo5Ti see note∗

Primitive vectors(a, b, c) (Å) (5.254, 5.254, 10.004) (5.339, 5.339, 10.178)(α, β, γ ) degrees (139.7, 139.7, 35.1) (139.8, 139.8, 35.1)

Atomic positions (fract.)A1 (0, 0, 0) (0, 0, 0)

B1 (0.165, 0.165, 0.833) (0.170, 0.170, 0.836)B2 (0.333, 0.333, 0.664) (0.335, 0.335, 0.672)B3 (0.500, 0.500, 0.500) (0.501, 0.501, 0.500)B4 (0.667, 0.667, 0.335) (0.666, 0.666, 0.328)B5 (0.836, 0.836, 0.167) (0.831, 0.831, 0.164)

Note ∗: Mo5Ti and Nb5Ru are slight distortions of the prototype Mo5Tiproto, monoclinic lattice, space group C2/m #12, with atoms in the followingWyckoff positions: Ti in 2a, Mo(1) in 2c, Mo(2) in 4i(x = 1/3, z = 1/6), Mo(3) in 4i (x = 2/3, z = 1/3), and primitive vectorsa1/a = (3.171, 1, 0),a2/a = (−0.315, 1, 0), a3/a = (−0.638, −0.201, 1.987) [17]. The Mo5Tiproto prototype has been calculated with the PAW-GGA approach.

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References

[1] D. de Fontaine, in: H. Ehrenreich, D. Turnbull (Eds.), Solid StaPhysics, vol. 47, Academic Press, 1994, pp. 33–176.

[2] M. Sluiter, P. Turchi, F. Zezhong, D. de Fontaine, Phys. Rev. Lett.(1988) 716.

[3] S. Curtarolo, D. Morgan, K. Persson, J. Rodgers, G. Ceder, PhRev. Lett. 91 (2003) 135503.

[4] S. Curtarolo, A.N. Kolmogorov, F.H. Cocks, CALPHAD 29 (2005155–161.

[5] A.N. Kolmogorov, S. Curtarolo, Stability of metal borides, Internareport, 2005.

[6] D. Morgan, G. Ceder, S. Curtarolo, Mat. Res. Soc. Symp. Proc. 8(2004) JJ9.25.

[7] D. Morgan, G. Ceder, S. Curtarolo, Combinatorial and HighThroughput Materials Research, Meas. Sci. Technol. 16 (20296–301 (special feature).

[8] Y. Wang, S. Curtarolo, C. Jiang, R. Arroyave, T. Wang, G. CedeL.-Q. Chen, Z.-K. Liu, CALPHAD 28 (2004) 79.

[9] T.B. Massalski (Ed.), Binary Alloy Phase Diagrams, ASMInternational, Metals Park, OH, 1992.

[10] P. Villars, K. Cenzual, J.L.C. Daams, F. Hulliger, T.B. MassalskH. Okamoto, K. Osaki, A. Prince, S. Iwata, Pauling File. InorganMaterials Database and Design System, Binaries edition, ASInternational, Metal Park, OH, 2003.

[11] P.S. White, J.R. Rodgers, Y. Le Page, Acta Cryst. B 58 (2002) 34[12] J.M. Sanchez, D. de Fontaine, Theoretical Prediction of Orde

Superstructures in Metallic Alloys, in: Structure and BondingCrystals, vol. II, Academic Press, 1981.

[13] Z.W. Lu, S.-H. Wei, A. Zunger, S. Frota-Pessoa, L.G. Ferreira, PhRev. B 44 (1991) 512.

[14] C. Wolverton, A. Zunger, Phys. Rev. B 50 (1994) 10548.[15] S. Curtarolo, Coarse-Graining and Data Mining Approaches to

Prediction of Structures and their Dynamics, Ph.D. Thesis, M2003. Download:burgaz.mit.eduandalpha.mems.duke.edu.

[16] Appendix B, pp. 196–210 of reference [15].[17] T. Hahn (Ed.), International Table of Crystallography, Kluwe

Academic Publishers, 2002.

.

)

.

[18] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048.[19] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892.[20] G. Kresse, J. Furthmuller, Comput. Mater. Sci. 6 (1996) 15.[21] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188.[22] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 16 (1977) 1748.[23] P.E. Blöchl, Phys. Rev. B 50 (1994) 17953.[24] G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758.[25] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244.[26] J.B. Neaton, N.W. Ashcroft, Phys. Rev. Lett. 86 (2001) 2830.[27] A.K. McMahan, J.A. Moriarty, Phys. Rev. B 27 (1983) 3235.[28] J.E. Jaffe, Z. Lin, A.C. Hess, Phys. Rev. B 57 (1998) 11834.[29] M.M. Dacorogna, M.L. Cohen, Phys. Rev. B 34 (1986) 4996.[30] M.I. Katsnelson, G.V. Sinko, N.A. Smirnov, A.V. Trefilov, K.Yu.

Khromov, Phys. Rev. B 61 (2000) 14420.[31] C.S. Barrett, O.R. Trautz, Trans. Am. Inst. Min. Metall. Pet. Eng

175 (1948) 579.[32] S. Barrett, Acta Crystallogr. 9 (1956) 671.[33] A.W. Overhauser, Phys. Rev. Lett. 53 (1984) 64.[34] G. Ernst, C. Artner, O. Blaschko, G. Krexner, Phys. Rev. B 33 (198

6465.[35] H.G. Smith, R. Berliner, J.D. Jorgensen, M. Nielsen, J. Trivisonn

Phys. Rev. B 41 (1990) 1231.[36] W. Schwarz, O. Blaschko, Phys. Rev. Lett. 65 (1990) 3144.[37] W. Schwarz, O. Blaschko, I. Gorgas, Phys. Rev. B 44 (1991) 6785[38] O. Blaschko, V. Dmitriev, G. Krexner, P. Tolèdano, Phys. Rev. B 5

(1999) 9095.[39] L. Vegard, Z. Phys. 5 (1921) 17.[40] The reference CPU is Compaq/HP - Alpha 21264 EV67 667MHz.[41] M. Hansen, K. Anderko, Constitution of Binary Alloys, McGraw-

Hill, New-York, 1958.[42] W.G. Moffatt (Ed.), Handbook of Binary Phase Diagrams, Busines

Growth Services, General Electric Co., Schenectady, NY, 1976.[43] F.A. Shunk, Constitution of Binary Alloys, Second Supplemen

McGraw-Hill, New-York, 1969 (General Electric Co., BusinessGrowth Services, Schenectady, New York).

[44] M. Venkatraman, J.P. Neumann, Phase Diagrams of Binary TitaniuMagnesium, ASM International, 1988.

mailto:burgaz.mit.edu

mailto:alpha.mems.duke.edu


-ica,

y,,

es,).

y,:

in:y,

C

6

6

ll.

. 5

)

.

.l.

.

1.an

oc

)

7..

,

2.

9.

)

,

,

)

sc.

)

esn,

)

.

e.

ll.

e-

a

es

)

)

l.

t.

,

)

)

s

ry

an

4

[45] L. Brewer, in: O. Kubaschewski (Ed.), Molybdenum: PhysicoChemical Properties of its Compounds and Alloys, in: AtomEnergy Review No. 7, International Atomic Energy Agency, Vienn1980 (special issue).

[46] V.I. Lavrentev, Ya.I. Gerassimov, P. Feschotte, D.T. LiveO. von Goldbeck, H. Nowotny, K. Seifert, R. Ferro, A.L. Dragooin: O Kubaschewski (Ed.), Niobium: Physico-Chemical Propertiof its Compounds and Alloys, in: Atomic Energy Review No. 2International Atomic Energy Agency, Vienna, 1968 (special issue

[47] J.L. Murray, Phase Diagrams of Binary Titanium Alloys, ASMInternational, 1987.

[48] C.B. Alcock, K.T. Jacob, S. Zador, O. von Goldbeck, H. NowotnK. Seifert, O. Kubaschewski, in: O. Kubaschewski (Ed.), ZirconiumPhysico-Chemical Properties of its Compounds and Alloys,Atomic Energy Review No. 6, International Atomic Energy AgencVienna, 1976 (special issue).

[49] D.R. Lide (Ed.), CRC Handbook of Chemistry and Physics, CRPress Inc., Cleveland, OH, 2003.

[50] L. Kaufman, CALPHAD 14 (1990) 163.[51] C. Wagner, Acta Metall. 2 (2) (1954) 242.[52] J.L. White, R.L. Orr, R. Hultgren, Acta Metall. 5 (12) (1957) 747.[53] J.L. White, Trans. AIME 215 (1) (1959) 178.[54] C.J. Cooke, W. Hume-Rothery, Acta Metall. 9 (10) (1961) 982.[55] S.-H. Wei, A.A. Mbaye, L.G. Ferreira, A. Zunger, Phys. Rev. B 3

(1987) 4163.[56] T. Mohri, K. Terakura, T. Oguchi, K. Watanabe, Acta Metall. 3

(1988) 547.[57] R.A. Johnson, Phys. Rev. B 41 (1990) 9717.[58] T. Mohri, K. Terakura, S. Takizawa, J.M. Sanchez, Acta. Meta

Mater. 39 (1991) 493.[59] T. Mohri, S. Takizawa, K. Terakura, J. Phys.: Condens. Matter

(1993) 1473.[60] Z.W. Lu, B.M. Klein, A. Zunger, Superlatt. Microstruct. 18 (1995

161.[61] Z.W. Lu, B.M. Klein, A. Zunger, J. Phase Equilib. 16 (1995) 36.[62] Z.W. Lu, B.M. Klein, A. Zunger, Modelling Simul. Mater. Sci. 3

(1995) 753.[63] V. Ozolins, C. Wolverton, A. Zunger, Phys. Rev. B 57 (1998) 4816[64] V. Ozolins, C. Wolverton, A. Zunger, Phys. Rev. B 57 (1998) 6427[65] A. Zunger, L.G. Wang, G.L.W. Hart, M. Sanati, Modelling Simu

Mater. Sci. 10 (2002) 685.[66] A. Tonejc, A.M. Tonejc, A. Bonefacic, Phys. Lett. A 49 (1974) 145[67] O.S. Zarechnyuk, V.V. Kinzhibalo, Russ. Metall. 6 (1978) 153.[68] Y. Matsuo, S. Minamigawa, K. Katada, Script. Metall. 12 (1978) 82[69] Y. Matsuo, T. Makita, T. Suzuki, A. Nagasawa, J. Phys. Soc. Jap

50 (1981) 1209.[70] W. Hume-Rothery, E. Butchers, J. Inst. Met. 60 (1937) 345.[71] R.J.M. Payne, J.L. Haughton, J. Inst. Met. 60 (1937) 351.[72] K. Schubert, B. Kiefer, M. Wilkens, R. Haufler, Z. Metallkd. 46

(1955) 1692.[73] K. Fujiwara, M. Hirabayashi, D. Watanabe, S. Ogawa, J. Phys. S

Japan 13 (1958) 167.[74] M.V. Prokof’ev, V.E. Kolesnichenko, Inorg. Mater. 21 (8) (1985

1168.[75] J. Kulik, S. Takeda, D. de Fontaine, Acta Metall. 35 (5) (1987) 113[76] V.E. Kolesnichenko, V.V. Karonik, S.N. Tsyganova, T.A

Kupriyanova, L.N. Sysoeva, Russ. Metall. (5) (1988) 199.[77] Y. Liu, R.G. Jordan, S.L. Qiu, Phys. Rev. B 49 (1994) 4478.[78] O.I. Velikokhatnyi, S.V. Eremeev, I.I. Naumov, A.I. Potekaev

J. Phys.: Condens. Matter. 12 (2000) 8825.[79] N.M. Rosengaard, H.L. Skriver, Phys. Rev. B 49 (1994) 14666.[80] F.V. Linen, Trans. Metall, Soc. AIME 175 (1948) 878.[81] X.D. Dai, H.R. Gong, B.X. Liu, J. Phys. Soc. Japan 73 (2004) 122[82] G.J. Lamprecht, P. Crowther, Trans. AIME 242 (1968) 2169.[83] J.R. Weeks, Trans. ASM 62 (1969) 304.[84] M.W. Chase, Bull. Alloy Phase Diagrams 4 (1983) 124.[85] R. Kieffer, H. Nowotny, Metallwissenschaft Technick 17 (1963) 66

.

[86] M.R. Baren, Bull. Alloy Phase Diagrams 10 (6) (1989).[87] I. Karakaya, W.T. Thompson, Bull. Alloy Phase Diagrams 9 (3

(1988).[88] N.V. Skorodumova, S.I. Simak, E.A. Smirnova, Yu.Kh. Vekilov

Phys. Lett. A 208 (1995) 157–160.[89] E. Bruno, B. Ginatempo, E.S. Guiliano, A.V. Ruban, Yu.Kh. Vekilov

Phys. Rep. 249 (1994) 353.[90] E. Bruno, B. Ginatempo, E.S. Giuliano, Phys. Rev. B 52 (1995

14544.[91] E. Bruno, B. Ginatempo, E.S. Giuliano, Nuovo Cimento D 20D

(1998) 1367–1376.[92] W. Olovsson, I.A. Abrikosova, B. Johansson, J. Electron Spectro

Relat. Phenom. 127 (2002) 65–69.[93] A.A. Rudnitskii, A.N. Khotinskaya, Russ. J. Inorg. Chem. 4 (1959

1053.[94] I. Barin, O. Knacke, O. Kubaschewski, Thermodynamical Properti

of Inorganic Substances (Suppl.), vol. 590, Springer-Verlag, Berli1977.

[95] I. Karakaya, W.T. Thompson, Bull. Alloy Phase Diagrams 7 (4(1986) 259.

[96] A.A. Rudnitskii, O.A. Novikova, Russ. J. Inorg. Chem. 4 (1959) 719[97] J.H. Li, L.T. Kong, B.X. Liu, J. Phys. Chem. B 108 (2004)

16071–16076.[98] B.B. Gulyaev, G.F. Dvorshkaya, in: E.M. Savitskii (Ed.), Phas

Diagrams of Metallic Systems, Nauka Pub., Moscow, 1968, p. 267[99] V.N. Emerenko, Y.I. Buyanov, N.M. Panchenko, Sov. Powder Meta

Met. Cer. 8 (7) (1969) 562.[100] M.R. Plichta, H.I. Aaronson, Acta Metall. 26 (1978) 1293.[101] E. Gebhardt, M. von Erdberg, U. Lüty, Nucl. Met. 10 (1964) 303.[102] K.A. Gschneidner, O.D. McMasters, D.G. Alexander, R.F. Vent

icher, Metall. Trans. 1 (1970) 1961.[103] O.D. McMasters, K.A. Gschneidner, R.F. Venteicher, Act

Crystallogr. B 26 (1970) 1224.[104] I. Barin, O. Knacke, O. Kubaschewski, Thermochemical Properti

of Inorganic Substances, Springer-Verlag, Berlin, 1977.[105] I. Karakaya, W.T. Thompson, Bull. Alloy Phase Diagrams 8 (4

(1987) 326.[106] O.P. Naumkin, V.F. Terekhova, E.M. Savitskii, Russ. Metall. (4

(1965) 176.[107] M.E. Drits, E.S. Kadaner, T.V. Dobatkina, N.I. Turkina, Russ. Metal

(4) (1973) 152.[108] S. Fujikawa, M. Sugaya, H. Takei, K. Hirano, J. Less-Common Me

63 (1979) 87.[109] H. Anton, P.C. Schmidt, Intermetallics 5 (1997) 449–465.[110] M. Asta, S.M. Foiles, A.A. Quong, Phys. Rev. B 57 (1998) 11265.[111] M. Asta, V. Ozolins, Phys. Rev. B 64 (2001) 094104.[112] V. Ozolins, M. Asta, Phys. Rev. Lett. 86 (2001) 448.[113] B. Mayer, H. Anton, E. Bott, M. Methfessel, J. Sticht, J. Harris

P.C. Schmidt, P.C. Intermetallics 11 (2003) 23–32.[114] E. Clouet, M. Nastar, C. Sigli, Phys. Rev. B 69 (2004) 64109.[115] H. Okamoto, T.B. Massalski, Bull. Alloy Phase Diagrams 5 (4

(1984).[116] H. Okamoto, T.B. Massalski, Bull. Alloy Phase Diagrams 6 (1

(1985).[117] T.B. Massalski, H. Okamoto, L. Brewer, Bull. Alloy Phase Diagram

7 (5) (1986).[118] T.B. Massalski, H. Okamoto, L. Brewer, Phase Diagrams of Bina

Gold Alloys, ASM International, 1987.[119] L. Dreibholz, Z. Phys. Chem. 108 (1924) 1.[120] G.A. Geach, E. Summersmith, J. Inst. Met. 82 (1953) 471.[121] J.A. Rodriguez, M. Kuhn, Surf. Sci. 330 (1995) L657–L664.[122] H. Hirabayashi, S. Ogawa, Acta Metall. 9 (4) (1961) 264.[123] J.D. Filby, J.N. Pratt, Trans. Faraday Soc. 60 (1964) 1934.[124] M. Hirabayashi, K. Hiraga, S. Yamaguchi, N. Ino, J. Phys. Soc. Jap

27 (1) (1969) 80.[125] K.L. Komarek, E. Reiffenstein, G. Stummerer, Monatsh. Chem. 10

(6) (1973) 1570.


te

n.

)

,

1

,

0

2

1

1)

3

,

e,

.r,

)

14

z,

)

3

et.

E

8

0

5

ls,

B

r.

5)

er,6)

,ten

.8)

e

,

)

5.9.10

90)

[126] E.A. Wood, B.T. Matthias, Acta Crystallogr. 9 (1956) 534.[127] A.E. Dwight, Metall. Soc. Conf. Proc. 10 (1961) 383.[128] R. Flükiger, P. Spitzli, F. Helniger, J. Muller, Phys. Lett. A 29 (1969)

407.[129] M. Bernasson, P. Descouts, R. Flükiger, A. Treyvaud, Solid Sta

Commun. 8 (1970) 837.[130] E. Ehrenfreund, A.C. Gossard, J.H. Wernick, Solid State Commu

8 (1970) 1925.[131] E. Röschel, C.J. Raub, O. Loebich Jr., Z. Metallkd. 64 (5) (1973

359.[132] H.R. Khan, E. Röschel, C.J. Raub, Z. Physik 262 (1973) 279.[133] E.Z. Kurmaev, F. Werfel, O. Brümmer, R. Flükiger, Solid State

Commun. 21 (1977) 239.[134] R. Flükiger, S. Foner, E.J. McNiff Jr., O. Fischer, Solid State

Commun. 30 (1979) 723.[135] M.S. Wire, G.W. Webb, J. Phys. Chem. Solids 42 (1981) 233.[136] N. Savvides, C.M. Hurd, S.P. McAlister, Solid State Commun. 41

(1982) 735.[137] M.S. Wire, G.W. Webb, D.E. Cox, C.L. Snead Jr., A.R. Sweedler

Solid State Commun. 48 (1983) 125.[138] S. Ramakrishnan, G. Chandra, M.C. Saxena, Physica B & C 14

(1986) 185.[139] L.M. Di, H. Bakker, J. Phys.: Condens. Matter. 3 (1991) 9319.[140] M. Baier, R. Wordel, F.E. Wagner, T.E. Antonova, aV.E. Antonov

J. Less-Common Met. 172–174 (1991) 358.[141] R. Ruer, Z. Anorg. Chem. 51 (1906) 391.[142] A. Nagasawa, Y. Matsuo, J. Kakinoki, J. Phys. Soc. Japan 20 (1

(1965) 1881.[143] Y. Matsuo, A. Nagasawa, J. Kakinoki, J. Phys. Soc. Japan 21 (1

(1966) 2633.[144] Y. Kawasaki, S. Ino, S. Ogawa, J. Phys. Soc. Japan 30 (6) (197

1758.[145] F. Doerinckel, Z. Anorg. Allg. Chem. 34 (1907) 333.[146] C.H. Johnsson, J.O. Linde, Ann. Phys. 5 (6) (1930) 762.[147] A.S. Darling, R.A. Mintern, J.C. Chaston, J. Inst. Met. 81

(1952–1953) 125.[148] A.A. Rudnitskii, A.N. Khotinskaya, Zh. Neorg. Khim. 4 (1959)

2518. TR: Russ. J. Inorg. Chem. 4 (1969) 1160.[149] E. Raub, G. Falkenburgh, Z. Metallkd. 55 (7) (1964) 392.[150] A.A. Rudnitskii, A.N. Khotinskaya, Zh. Neorg. Khim. 4 (1959)

1601. TR: Russ. J. Inorg. Chem. 4 (1969) 722.[151] M. Kuhn, J.A. Rodriguez, J. Hrbek, A. Bzowski, T.K. Sham, Surf.

Sci. 341 (1995) L1011–L1018.[152] A.T. Aldred, Trans. AIME 224 (1962) 1082.[153] H. Reule, S. Steeb, C. Donolato, J. Less-Common Met. 24 (197

108.[154] P.E. Rider, K.A. Gschneidner Jr., O.D. McMasters, Trans. AIME 23

(1965) 1488.[155] A.R. Miedema, J. Less-Common Met. 46 (1976) 67.[156] B.T. Matthias, E. Corenzwit, J.M. Vandenberg, H. Barz, M.B. Maple

R.N. Shelton, J. Less-Common Met. 46 (1976) 339.[157] A.E. Dwight, J.W. Downey, R.A. Conner Jr., Acta Crystallogr. 22

(1967) 745.[158] H.J.M. van Rongen, B. Knook, G.G. van den Berg, Low-Temperatur

Physics LT9, in: Proc. IXth Int. Conf. on Low-Temperature PhysicsColumbus, OH, B1041, 1964.

[159] E. Raub, P. Walter, M. Engel, Z. Metallkd. 43 (1952) 112.[160] M.K. McQuillan, J. Inst. Met. 82 (1954) 511.[161] P. Pietrokowsky, E.P. Frink, P. Duwez, Trans. AIME 206 (1956) 930[162] K. Schubert, H.G. Meissner, M. Pötzschke, W. Rossteutsche

E. Stolz, Naturwissenschaften 49 (1962) 57.[163] P. Pietrokowsky, J. Inst. Met. 90 (1961–62) 434.[164] H. Vonphilipsborn, F. Laves, Acta Crystallogr. 17 (1964) 213.[165] E. Vielhaber, H.L. Luom, Solid State Commun. 5 (1967) 221.[166] J.B. Vetrano, G.L. Guthrie, H.E. Kissinger, Phys. Lett. A 26 (1967

45.[167] P. Pietrokowsky, Scr. Metall. 2 (1968) 379.

)

)

)

[168] S.A. Shabalovskaya, Solid State Commun. 70 (1989) 23–26.[169] C.C. Chao, H.L. Luo, P. Dumez, J. Appl. Phys. 34 (1963) 1971.[170] H.R. Kirchmayr, Z. Metallkd. 56 (1965) 767.[171] J.B. Kusma, E. Laube, Monatsh. Chem. 96 (1965) 1496.[172] Y. Sadagopan, B.C. Giessen, N.J. Grant, J. Less-Common Met.

(1968) 279.[173] E. Raub, M. Engel, Z. Metallkd. 39 (1948) 172.[174] K. Schubert, M. Balk, S. Bhan, M. Breimer, P. Esslinger, E. Stol

Naturwissenschaften 49 (1962) 647.[175] M.V. Nevitt, J.W. Downey, Trans. AIME 224 (1962) 194.[176] E. Stolz, K. Schubert, Z. Metallkd. 53 (1962) 433.[177] H. Holleck, Acta Crystallogr. 21 (3) (1966) 451.[178] l. Arnberg, Acta Crystallogr. B 36 (1980) 527.[179] J.P. Neumann, A. Mikula, Y.A. Chang, Metall. Trans. A 13 (7) (1982

1123.[180] H. Nowotny, E. Bauer, A. Stampfl, H. Bittner, Monatsh. Chem. 8

(1952) 221.[181] M.G. Chasanov, I. Johnson, R.V. Schablaske, J. Less-Common M

7 (1964) 1127.[182] M.G. Chasanov, P.D. Hunt, I. Johnson, H.M. Feder, Trans. AIM

224 (1962) 935.[183] W.M. Robertson, Metall. Trans. 3 (1972) 1443.[184] R. Elmendorf, E. Ryba, Acta Crystallogr. B 24 (1968) 462.[185] E. Ryba, P.K. Kejriwal, R. Elmendorf, J. Less-Common Met. 1

(1969) 419.[186] G. Bruzzone, M.L. Fornasini, F. Merlo, J. Less-Common Met. 3

(1973) 361.[187] Y. Ishii, K. Nozawa, T. Fujiwara, Mat. Res. Soc. Symp. Proc. 80

(2003) 129–134.[188] W. Rossteutscher, K. Schubert, Z. Metallkd. 56 (1965) 730.[189] Arunsingh, B. Dayal, Acta Crystallogr. B 25 (1969) 1010.[190] E. Montignie, Bull. Soc. Chim. Fr. Mem. 5 (1938) 567.[191] E. Rudy, Compendium of Phase Diagram Data, Air Force Materia

Laboratory, Wright-Patterson AFB, OH, Rep. No. AFML-TR-65-2Part V, 1969.

[192] J. Chao, C. Wolverton, J. Sofo, L.-Q. Chen, Z.-K. Liu, Phys. Rev.69 (2004) 214202.

[193] S.S. Rajput, R. Prasad, R.M. Singru, S. Kaprzyk, A. Bansil, MateSci. Forum 175–178 (1995) 925.

[194] C. MoBecker, J. Hafner, J. Phys.: Condens. Matter. 7 (1995843–5856.

[195] A. Maldonado, K. Shubert, Z. Metallkd. 55 (1964) 619.[196] E. Raub, Z. Metallkd. 45 (1954) 23–30.[197] K. Schubert, W. Burkhardt, P. Esslinger, E. Günzel, H.G. Meissn

W. Schütt, J. Wegst, M. Wilkens, Naturwissenschaften 43 (195248–249.

[198] A.G. Knapton, Planseeber. Pulvermetall. 7 (1959) 2–3.[199] K. Schubert, K. Frank, R. Gohle, A. Maldonado

H.G. Meissner, A. Raman, W. Rossteutscher, Naturwissenschaf50 (1963) 41.

[200] H.P. Rooksby, B. Lewis, J. Less-Common Met. 6 (1964) 451–460[201] H. Ocken, J.H.N. Van Vucht, J. Less-Common Met. 15 (196

193–199.[202] R. Flükiger, A. Paoli, R. Roggen, K. Yvon, J. Müller, Solid Stat

Commun. 11 (1972) 61–63.[203] R. Flükiger, K. Yvon, C. Susz, R. Roggen, A. Paoli, J. Müller

J. Less-Common Met. 32 (1973) 207.[204] R. Flükiger, A. Paoli, J. Müller, Solid State Commun. 14 (1974

443–447.[205] C.W. Haworth, W. Hume-Rothery, J. Inst. Met. 87 (1958–1959) 26[206] E. Anderson, W. Hume-Rothery, J. Less-Common Met. 2 (1960) 1[207] B.C. Giessen, U. Jaehnigen, N.J. Grant, J. Less-Common Met.

(1966) 147.[208] R. Gurler, J.N. Pratt, J. Alloys Compounds 177 (1991) 321.[209] H. Kleykamp, J. Less-Common Met. 136 (1988) 271.[210] S. Ferreira, J. Duarte Jr., S. Frota-Pessoa, Phys. Rev. B 41 (19

5627.


n,

2.n,

)

9

3.0.9.4

5

93

oc

,

.

)

0

)

)

f

s-

.

94

)

.

3.ki

ndls,

,

r.

2

93)

v,1

;

)

ar0,

)

8

8)

.0

1

n

t.)

,

89)

48

8)

. 72

2

al

[211] P. Souvatzis, M.I. Katsnelson, S. Simak, R. Ahuja, O. ErikssoP. Mohn, Phys. Rev. B 70 (2004) 12201.

[212] G.N. Ronami, Krist. Tech. 7 (6) (1972) 615.[213] R.G. Baggerly, Metallography 8 (1975) 361.[214] D.L. Moffatt, U.R. Kattner, Metall. Trans. 19 (1988) 2389.[215] G. Rubin, A. Finel, J. Phys.: Condens. Matter. 7 (1995) 3139–315[216] K.A. Gschneidner Jr., Rare Earth Alloys, Van Nostrand, Princeto

NJ, 1961, p. 221.[217] L.T. Kong, J.B. Liu, B.X. Liu, J. Mater. Res. 17 (2002) 528–531.[218] R.E. Watson, M. Weinert, M. Alatalo, Phys. Rev. B 65 (2002

014103.[219] B.T. Matthias, V.B. Compton, E. Corenzwit, Phys. Chem. Solids 1

(1961) 130.[220] B.C. Giessen, N.J. Grant, Acta Crystallogr. 17 (1964) 615.[221] A. Maldonado, K. Shubert, Z. Metallkd. 55 (1964) 619.[222] B.C. Giessen, N.J. Grant, J. Less-Common Met. 8 (1965) 114.[223] R.M. Waterstrat, B.C. Giessen, Metall. Trans. A 16 (11) (1985) 194[224] D.L. Ritter, B.C. Giessen, N.J. Grant, Trans. AIME 230 (1964) 125[225] D.L. Ritter, B.C. Giessen, N.J. Grant, Trans. AIME 230 (1964) 125[226] M. Rajagopalan, M. Sundareswari, J. Alloys Compounds 379 (200

8–15.[227] D. Bender, E. Bucher, J. Muller, Phys. Kondens. Mater. 1 (1963) 22[228] E. Raub, W. Fritzsche, Z. Metallkd. 54 (1963) 317.[229] G.F. Hurley, J.H. Brophy, J. Less-Common Met. 7 (1964) 267.[230] T. Tsukamoto, K. Koyama, A. Oota, S. Noguchi, Nippon Kinzoku

Gakkai-shi 53 (3) (1989) 253.[231] B.H. Chen, H.F. Franzen, J. Less-Common Met. 153 (1989) L13.[232] J.M. Sanchez, J.D. Becker, Mat. Res. Soc. Symp. Proc. 291 (19

115–127.[233] J.D. Becker, J.M. Sanchez, J. Mater. Sci. A 170 (1993) 161–167.[234] J.D. Becker, J.M. Sanchez, J.K. Tien, Mat. Res. Soc. Symp. Pr

213 (1991) 113–118.[235] V.B. Compton, E. Corenzwit, J.P. Maita, B.T. Matthias, F.J. Morin

Phys. Rev. 123 (1961) 1567.[236] D.O. Van Ostenburg, D.J. Lam, M. Shimizu, A. Katsuki, J. Phys

Soc. Japan 18 (1963) 1744–1754.[237] A.L. Giorgi, E.G. Szklarz, J. Less-Common Met. 20 (2) (1970

173–175.[238] O.N. Carlson, V.C. Marcotte, D.E. Williams, USAEC Rep. ISC-105

(1959) 53.[239] O.N. Carlson, W.L. Larsen, V.C. Marcotte, D.E. Williams, USAEC

Rep. IS-17 (1959) 76.[240] C.E. Lundin, D.T. Klodt, J. Inst. Met. 90 (1961) 341.[241] A. Taylor, W.M. Hickam, N.J. Doyle, J. Less-Common Met. 9 (1965

214.[242] L.T. Kong, J.B. Liu, B.X. Liu, J. Phys. Soc. Japan 71 (2002

141–143.[243] A.G. Rabinkin, L.A. Klishanova, L.N. Pronina, Problems o

Superconducting Materials, vol. 141, 1970.[244] P. Van Effenterre, Report CEA0R04330, Atomic Energy Commi

sion, Saclay, France, 1972.[245] P.E.J. Flewitt, J. Appl. Crystallogr. 5 (1972) 423.[246] P.E.J. Flewitt, Acta Metall. 22 (1974) 47.[247] J.P. Abriata, J.C. Bolcich, Bull. Alloy Phase Diagrams 3 (1) (1982)[248] H.v. Leuken, A. Lodder, M.T. Czyzyk, F. Springelkamp, R.A. de

Groot, Phys. Rev. B 41 (1990) 5613.[249] J.M. Sanchez, J.D. Becker, Prog. Theor. Phys. Suppl. 115 (19

131–145.[250] G.B. Grad, A.F. Guillermet, J.R. Granada, Z. Metallkd. 87 (1996

726–731.[251] G.B. Grad, G.M. Benites, G. Aurelio, A.F. Guillermet, Mat. Sci. Eng

A-Struct. A 273–275 (1999) 175–180.[252] J. Kudrnovsky, V. Drchal, Z. Phys. B Cond. Mat. 73 (1989) 489–49[253] B.C. Giessen, N.J. Grant, D.P. Parker, R.C. Manuszews

R.M. Waterstrat, Metall. Trans. A 11 (1980) 709.[254] E. Raub, J. Less-Common Met. 1 (1) (1959) 3.

)

.

)

.

)

,

[255] D.T. Hawkins, Metals Handbook, Metallography, Structures aPhase Diagrams, 8th edition, vol. 8, American Society for MetaMetal Park, OH, 1973.

[256] Z.W. Lu, S.-H. Wei, A. Zunger, Phys. Rev. Lett. 66 (1991) 1753.[257] E. Raub, J. Less-Common Met. 1 (1) (1959) 3.[258] E. Raub, H. Beeskow, D. Menzel, Z. Metallkd. 50 (1959) 428.[259] P.E.A. Turchi, G.M. Stocks, W.H. Butler, D.M. Nicholson, A. Gonis

Phys. Rev. B 37 (1988) 5982.[260] J.E. Shield, R.K. Williams, Scr. Metall. 21 (1987) 1475.[261] M. Asato, T. Hoshino, K. Masuda-Jindo, J. Magn. Magn. Mate

226–230 (2001) 1051–1052.[262] M. Asato, T. Mizuno, T. Hoshino, H. Sawada, Mater. Trans. 4

(2001) 2216–2224.[263] C. Wolverton, D. de Fontaine, Mat. Res. Soc. Symp. Proc. 291 (19

431–436.[264] F.M. Marquez, C. Cienfuegos, B.K. Pongsai, M.Yu. Lavrentie

N.L. Allan, J.A. Purton, G.D. Barrera, Model. Simul. Mater. Sci. 1(2003) 115–126.

[265] A.A. Rudnitskii, R.S. Plyakova, Zh. Neorg. Khim. 4 (1959) 1014Russ. J. Inorg. Chem. 4 (1959) 631.

[266] A.S. Darling, J.M. Yorke, Platinum Met. Rev. 4 (1960) 104.[267] H. Kleykamp, J. Nucl. Mater. 167 (1989) 49.[268] H.B. Guo, X.Y. Li, B.X. Liu, J. Phys. Soc. Japan 71 (2002

2933–2935.[269] H.R. Haines, P.E. Potter, M.H. Rand, Thermodynamics of Nucle

Materials, Proceedings of a Symposium, 4th, vol. 1, Vienna 198pp. 471–501.

[270] A.A. Rudnitskii, N.A. Birun, Russ. J. Inorg. Chem. 5 (19601169–1173.

[271] A.E. Dwight, R.A. Conner Jr., J.W. Downey, Acta Crystallogr. 1(1965) 835–839.

[272] H.W. Rosenberg, D.B. Hunter, Trans. AIME 233 (1965) 681–685.[273] E. Raub, E. Röschel, Z. Metallkd. 59 (1968) 112–114.[274] P. Krautwasser, S. Bhan, K. Schubert, Z. Metallkd. 59 (196

724–729.[275] H. Schulz, K. Ritapal, W. Bronger, W. Klemm, Z. Anorg. Allg.

Chem. 358 (1968) 299–313.[276] I.R. Harris, M. Norman, J. Less-Common Met. 22 (1970) 127–130[277] H.C. Donkersloot, J.H.N. Van Vucht, J. Less-Common Met. 2

(1970) 83–91.[278] V.N. Eremenko, T.D. Shtepa, Sov. Powder Metall. Metal Cer. 1

(1972) 228–233.[279] A.R. Miedema, K.H.J. Buschow, H.H. Van Mal, J. Less-Commo

Met. 49 (1976) 463–472.[280] V.P. Sivokha, A.S. Savvinov, V.P. Voronin, V.N. Khachin, Phys. Me

Metallogr.; Translated from Fizika Metallov, Metallovedenie 56 (3(1983) 112–116.

[281] S.A. Shabalovskaya, A.I. Lotkov, A.G. Narmonev, A.I. ZakharovSolid State Commun. 62 (1987) 93–95.

[282] N. Selhaoui, J.C. Gachon, J. Hertz, J. Less-Common Met. 154 (19137–147.

[283] S. Pick, P. Mikusik, Solid State Commun. 80 (1991) 897–899.[284] P. Mikusik, S. Pick, Solid State Commun. 86 (1993) 467–469.[285] J.M. Zhang, G.Y. Guo, J. Phys. Cont. Mat. 7 (1995) 6001–60017.[286] C. Wolverton, G. Ceder, D. de Fontaine, H. Dreysse, Phys. Rev. B

(1993) 726.[287] O. Loebich, E. Raub, J. Less-Common Met. 30 (1) (1973) 47.[288] J. Le Roy, J.M. Moreau, D. Paccard, Acta Crystallogr. B 33 (

(1977) 2414.[289] R. Sanjines-Zeballos, R. Chabot, E. Parthé, J. Less-Common Met

(1) (1980) P17.[290] K. Takao, Y. Sakamoto, M. Yoshida, J. Less-Common Met. 15

(1989) 115.[291] K. Anderko, Z. Metallkd. 50 (1959) 681.[292] E. Stolz, K. Schubert, Z. Metallkd. 53 (1962) 433.[293] I.R. Harris, M. Norman, J. Less-Common Met. 18 (1969) 333.[294] E.M. Savistky, V.P. Polyakova, N. Gorina, N. Roshan, Physic

Metallurgy of Platinum Metals, MIR, Moscow, 1978.


ed

s-

s

3

l.

)

.

,

B

y,

41

et

ll.

5

.

,

6

(2

iy

l,

.4

.

.

,

2

)

5)

7)

72

)

t.

2)

)

n,

on

5

6

)

7)

et.

c.

.ic

[295] M.S. Chandrasekharaiah, M.J. Stickney, K.A. Gingerich, J.A. SpeJ. Alloy Phase Diagrams 6 (1990) 59.

[296] V.N. Kuznetsov, G.P. Zhmurko, E.M. Sokolovskaya, J. LesCommon Met. 163 (1990) 1.

[297] S. Stolen, T. Matsui, J. Nucl. Mater. 186 (1992) 242.[298] L.A. Bendersky, J.K. Stalick, R. Portier, R.M. Waterstrat, J. Alloy

Compounds 236 (1996) 19.[299] E. Gaganidze, P. Esquinazi, R. König, Europhys. Lett. 31 (1995) 1[300] J.M. Hutchinson, Platinum Met. Rev. 16 (1972) 88.[301] G.A. Camara, M.J. Giz, V.A. Paganin, E.A. Ticianelli, J. Electrona

Chem. 537 (2002) 21.[302] N. Nishimura, T. Hiramatsu, Nippon Kinzoku Gakkaishi 21 (1957

469.[303] T. Sato, S. Hukai, Y.C. Huang, J. Aust. Inst. Met. 5 (2) (1960) 149[304] P. Pietrokowsky, Nature 206 (1965) 291.[305] A.K. Sinha, Trans. AIME 245 (1969) 237.[306] P.J. Meschter, W.L. Worrell, Metall. Trans. A 7 (1976) 299.[307] V.B. Compton, B.T. Matthias, Acta Crystallogr. 12 (1959) 651.[308] T.H. Geballe, B.T. Matthias, V.B. Compton, E. Corenzwit

G.W. Hull, L.D. Longinotti, Phys. Rev. 137 (1965) 119.[309] W. Bronger, J. Less-Common Met. 12 (1967) 63.[310] N.H. Krikorian, J. Less-Common Met. 23 (1971) 271.[311] A. Raman, J. Less-Common Met. 26 (1972) 199.[312] B. Erdmann, C. Keller, J. Solid State Chem. 7 (1973) 40.[313] J. Le Roy, J.M. Moreau, D. Paccard, E. Parthé, Acta Crystallogr.

34 (1978) 3315.[314] X. Yifen, O. Loebich, J. Less-Common Met. 144 (1988) 301.[315] H.J. Wallbaum, Naturwissenschaften 31 (1943) 91.[316] A. Raman, K. Schubert, Z. Metallkd. 55 (1964) 704.[317] K. Shubert, S. Bhan, T.K. Biswas, K. Frank, P.K. Pand

Naturwissenschaften 55 (1968) 542.[318] A.R. Miedema, R. Boom, F.R. De Boer, J. Less-Common Met.

(1975) 283.[319] A. Hellawell, W. Hume Rothery, Phil. Mag. 45 (1954) 797.[320] J.O.A. Paschoal, H. Kleykamp, F. Thümmler, J. Less-Common M

98 (1984) 279.[321] R. Gurler, J. Alloys Compounds 191 (1993) 31.[322] E. Raub, E. Roeschel, Z. Metallkd. 57 (1966) 546.[323] V.N. Eremenko, T.D. Shtepa, V.G. Sirotenko, Sov. Powder Meta

Metal Cer. 5 (1966) 487.[324] V.N. Eremenko, T.D. Shtepa, Colloq. Int. CNRS 205 (1972) 403.[325] T.D. Shetpa, Fiz. Khim. Kondens. Faz. Sverkhtverd. Mater. (197

175.[326] R. Kuentzler, R.M. Waterstrat, Solid State Commun. 68 (1988) 85[327] V.B. Compton, B.T. Matthias, Acta Crystallogr. 12 (1959) 651.[328] T.H. Geballe, B.T. Matthias, V.B. Compton, E. Corenzwit

G.W. Hull, L.D. Longinotti, Phys. Rev. A 137 (1965) 119.[329] A. Raman, J. Less-Common Met. 26 (2) (1972) 199.[330] H. Ghassem, A. Raman, Z. Metallkd. 64 (3) (1973) 197.[331] J.M. Moreau, D. Paccard, E. Parthé, Acta Crystallogr. B 32 (197

1767.[332] O. Loebich, E. Raub, J. Less-Common Met. 46 (1976) 1.[333] S.T. Zegler, J. Phys. Chem. Solids 26 (1965) 1347.[334] V.N. Eremenko, E.L. Semenova, T.D. Shtepa, Russ. Metall.

(1978) 158.[335] V.N. Eremenko, E.L. Semenova, T.D. Shtepa, Diag. Sostoyan

Tugoplavk Sistem Kiev (1980) 119.[336] J.L. Jorda, T. Graf, L. Schellenberg, J. Müller, K. Cenzua

J.C. Gachon, J. Hertz, J. Less-Common Met. 136 (1988) 313.[337] R. Kuentzler, R.M. Waterstrat, Solid State Commun. 68 (1988) 85[338] E.L. Semenova, Y.V. Kudryavtsev, J. Alloys Compounds 203 (199

165.

,

.

.

)

)

)

a

)

[339] A. Szajek, A. Jezierski, Solid State Commun. 71 (1989) 917–922.[340] N.Y. Alekseyevskiy, O.A. Balakhovskii, I.V. Kirillov, Phys. Met.

Metallogr.; Translated from Fizika Metallov, Metallovedenie 40(1975) 38.

[341] C.B. Jordan, Trans. Am. Inst. Mining, Metallurgical Petroleum Eng203 (1955) 832–833.

[342] A.E. Dwight, Trans. Am. Inst. Mining, Metallurgical Petroleum Eng215 (1959) 283–286.

[343] D.D. Gulamova, M.V. Raevskaya, I.G. Sokolova, F.S. LitvakE.M. Sokolovskaya, Moscow Univ. Chem. Bull. 27 (4) (1972) 96.

[344] E. Raub, E. Roeschel, Z. Metallkd. 54 (8) (1963) 455.[345] Y. Tamminga, B. Barkman, F.R. De Boer, Solid State Comm. 1

(1973) 731–735.[346] V.N. Eremenko, T.D. Shtepa, V.T. Khoruzhaya, Russ. Metall. (2

(1973) 155–156.[347] N.G. Boriskina, I.I. Kornilov, Russ. Metall. 2 (162) (1976).[348] J.M. Zhang, G.Y. Guo, J. Phys.: Condens. Matter. 7 (199

6001–6017.[349] O. Loebich, E. Raub, J. Less-Common Met. 46 (1976) 7.[350] K. Cenzual, A. Palenzona, E. Parthé, Acta Crystallogr. B 36 (

(1980) 1631.[351] R. Sanjines-Zaballos, B. Chabot, E. Parthé, J. Less-Common Met.

(1980) P17.[352] P. Sharifrazi, R.C. Mohanty, A. Raman, Z. Metallkd. 75 (10) (1984

801.[353] M.L. Fornasini, A. Mugnoli, A. Palenzona, J. Less-Common Me

154 (1989) 149.[354] A. Palenzona, E. Canepa, J. Less-Common Met. 162 (1990) 267.[355] V.N. Eremenko, E.L. Semenova, T.D. Shtepa, Russ. Metall. (

(1980) 177.[356] V.N. Eremenko, V.T. Khoruzhaya, T.D. Shtepa, Russ. Metall. (1

(1988) 194.[357] M.J. Mehl, J.E. Osburn, D.A. Papaconstantopoulos, B.M. Klei

Mat. Res. Soc. Symp. Proc. 186 (1991) 277–282.[358] C.C. Koch, J. Less-Common Met. 6 (1976) 177.[359] J.B. Darby Jr., D.J. Lam, L.J. Norton, J.W. Downey, J. Less-Comm

Met. 4 (1962) 558.[360] J.B. Darby Jr., L.J. Norton, J.W. Downey, J. Less-Common Met.

(1963) 397.[361] J.B. Darby Jr., L.J. Norton, J.W. Downey, J. Less-Common Met.

(1964) 165.[362] J. Niemiec, Bull. Acad. Pol. Sci. Ser. Sci. Chem. 11 (1963) 665.[363] E.G. Szklarz, A.L. Giorgi, J. Less-Common Met. 81 (1981) 2349.[364] A.L. Giorgi, E.G. Szklarz, J. Less-Common Met. 22 (1970) 246.[365] P.A. Farrar, S. Adler, Trans. AIME 236 (1966) 1061.[366] D. Chatterji, M.T. Hepworth, S.J. Hruska, Metall. Trans. 2 (1971

1271.[367] J. Etchessahar, J. Debuigne, Mem. Sci. Rev. Metall. 74 (3) (197

195.[368] J.P. Auffredic, E. Etchessahar, J. Debuigne, J. Less-Common M

84 (1982) 49.[369] C.E. Lundin, Rare Earth Symposium, Annual Meeting, Am. So

Metals, Chicago, 1959.[370] J.C. Uy, D.J. Lam, L.C. Ianiello, R.A. Proebstle, A.P. Lee, B.T.M

Loh, A.A. Burr, Final Report AT(30-1)-2159, Rensselaer PolytechnInstitute, 1961.

[371] C.E. Lundin, D.T. Klodt, Trans. AIME 224 (1962) 367.[372] R. Wang, Metall. Trans. 3 (1972) 1213.[373] R. Wang, Y.B. Kim, Metall. Trans. 5 (1974) 1973.[374] R. Wang, Mater. Sci. Eng. 23 (1976) 135.[375] A. van de Walle, G. Ceder, Rev. Mod. Phys. 74 (1) (2002) 11.[376] C. Wolverton, V. Ozolins, Phys. Rev. Lett. 86 (2001) 5518.

Accuracy of ab initio methods in predicting the crystal...

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