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Crystal Structures of Intermetallic Compounds. Edited by J _ H. Westbrook and R. 1.Fleischer 1995, 2000 John Wiley &Sons Ltd

sections, liquidus and solidus projections, as well asvertical sections. For the literature up to the year1977, there also exists the very comprehensiveMulticomponent Alloy, Constitution Bibliography(Prince, 1956, 1978, 1981).Looking at the research activities in the last 20 years

with respect to the number of systems investigated.grouped according to binary. ternary. and quaternarysystems, a shift in the ongoing research from binary toternary and quaternary systems is clearly seen. Theresearch field of multinary systems is a huge field forpotential novel materials with optimized known physicalproperties (e.g, tP68 BFel4Nd2-typepermanent magnetsand, in the inorganic field, the new high-Z] superconductors) as well as 'new' physical properties, as therecent past has proven with the quasicrystals (seeChapter 20 by Kelton in this volume). All this wasach iev ed by moving from the binary to th e ternary andquaternary systems, but this trend confronts us withmany additional 'difficult-to-handle' problems. Wehave to be aware that in the future most research willbe done on rnultinary systems; therefore, in this chapterwe will first discuss the binary intermetallic compounds(systems), also called binaries, and whenever possibleextend the discussion to ternary intermetallic compounds (systems), called te rn ari es. W e will try to showthe main problems by extending to ternaries ourknowledge gained by investigation of binaries andpropose ideas which, according to the author, have ahigh probability of success in the quantitative extensionto multinaries of regularities found in binaries. Theexperimental variables we have are the selection of thechemical elements, their possible combinations, their

The fundamentals of the constitution of an alloyingsystem are determined by the crystal structure of itsintermetallic compounds and its phase diagram.Knowing these fundamentals enables scientists to solvemany problems in materials science, and therefore it isimportant to have easy access to the experimentallydetermined data, especially as the experimental workto determine such information is very time- and costintensive.Recently, scientists working in this field have had the

advantage of access to comprehensive, up-to-datehandbooks for crystal structures as well as for phasediagrams. Pearson's Handbook of CrystallographicData for Intermetallic Phases, second edition (Villarsand Calvert, 1991) contains critically evaluatedcrystallographic data for over 25000 distinctly differentintermeralhc compounds (over 50000 binary, ternary,etc., entries) covering the world literature from 1913to1989. The A tlas oj Crystal Structures jor IntermetallicPhases (Daams et al., 1991), a companion of Pearson'sHandbook, contains for most of the intermeta1liccompounds in Pearson's Handbook graphicalrepresentations of the crystal structures. The handbookBinary Alloy Phase Diagrams, second edition (Massalskiet 01., 1991) contains about 4000 phase diagrams, mostof them critically evaluated phase diagrams. In 1994ASM International willpublish the Handbook of TernaryAlloy Phase Diagrams (Villars et ai., 1993), whichcomprehensively covers the world literature from 1900to 1989 and will contain information for over 8800ternary systems including over 15 000 isothermal

1. Introduction

Pierre VillarsIntermetatllc Phases Data Bank (IPDB) and Materials Phases Data System (MPDS),Postal Box L, CH6354 Vitznau, Switzerland

Chapter 1Factors Governing Crystal Structures

Copyrighted MaterialsC opyright", 2000 J ohn W iley & S ons R etrfeved from YiYiv/.knovel.com


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When we taJk about factors governing crystal structureswe intend to reduce these to atomic property expressions(APEs) of the constituent chemical elements, so that theexisting experimental data can be systematized. Forthe purpose of this chapter, we consider that the onlyregularities of real practical value are those able tosystematize a large group of data with an accuracy inthe range of at least 9 5 0 1 0 . For the not yet experimentallyinvestigated systems, one goes in exactly the reverse way,assuming the validity of the regularity found for a welldefined group of data. The more data considered, themore trustworthy is a prediction based on suchregularities. To make predictions one starts fromthe chemical elements, their concentrations. and thetabulated atomic properties (APs) of the elements, andcalculates the APEs for a system of interest. Therefore,APs are only of practical interest where they are knownfor most chemical elements with adequate accuracy.From the APEs, in the context of the consideredregularity, one can predict how the systems of interestmost probably (with an accuracy of at least 95070)willbehave. Then scientists can decide which systems arepromising to investigate experimentally.

2. Strategy to Find the Factors GoverningCrystal Structure .

be checked with the available experimentally determineddata).To increase the efficiency in the successful search for

'new' interrnetallic compounds, the main effortsshould go toward creating an internationally accessibleinformation-prediction system incorporating all databases (experimentally determined facts) as well asgenerally valid principles and the 'highest-quality'regularities. The problems involved in such a projecthave been reviewed by Westbrook (1993). The combination of the experience and intuition of theexperimentalist together with easy access to the dataalready experimentally determined in the form of upto-date handbooks as well as access to the envisionedinformation-prediction system would very much helpto coordinate world research activities. Furthermore,it would reduce the number of unwanted duplicationsas well as increase the. probability of investigating firstthe most promising systems and not, as in the past,provide us with the systems in a random statisticalsequence. Otherwise wewill have to wait until the year5500 for the next 140 generations of scientists toinvestigate the ternary systems, assuming an activity ratesimilar to that in this century.

concentrations, the temperature, and the pressure. Ofenormous consequence from the practical point of viewis the variety of combination possibilities of the chemicalelements and the increase in the possible number ofcombinations on going from binaries to multinariesaccompanied by a much larger available concentrationrange. Unfortunately, the number of available experimentally determined data is in inverse proportion tothese opportunities.We have a relatively robust database for binaries, but

a sparse database for ternaries, and almost no data forquaternaries. This is quantitatively demonstrated bylooking at the relevant numbers of binary, ternary, andquaternary systems. Tn the binary systems, taking 100chemical elements into account, there exist (lOOx 99)/(1x 2)=4950 binary systems, a large number, but still,with united. coordinated international research efforts,all these systems could be experimentally investigatedby the end of this century. Therefore, one wouldinevitably have found among these the most interestingand economically important binary materials. Massalskiet 01 . (1991) and Villars and Calvert (1991) containinformation on about 4000 systems, so already we havea very robust database with about 8 0 0 1 0 of the possiblesystems fully or partly investigated experimentally. Withthe ternary systems the situation looks completelydifferen t. T here exis t (J 00 x 99 x 98)/(1 x 2 x 3) = 1 6 1 7 0 0systems. Villars and Calvert (1991) and Villars et at.(1994)contain experimentally determined data for onlyabout 8800systems, most of which have only been partlyinvestigated. So here we have a sparse database withonly about 5 0 1 0 of the potentially available systemshaving been partly investigated. In addition, in orderto establish structu re s and phase relationships, one hasto prepare and investigate at least 10times more samplesper system in the ternary case compared to the binarycase. In the quaternary case with (100 x 99 x 98 x 97)/(J x 2 x 3 x ,,) :::3 " .921225 potential systems, less than0.1010have been partly investigated. Without having verygood guidelines, it is a hopeless situation to searchsystematically for novel materials with an adequatesuccess rate in multinary systems. Therefore, the onlypracticable way to go is, with the help of the robustbinary database, to find regularities. such as laws. rules.principles, factors. tendencies, and patterns valid withinthe binaries, and then to extend these to the ternariesand quaternaries in such a way that the binaries andmultinaries can be treated together; otherwise theregularities would be based on too few data sets andtherefore would not be trustworthy. In addition. theregularities should show an accuracy clearly aboverandomness to be of practical use (these can easily

Crystal Structures of Intermetallic Compounds


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Figure 1. An atomic property, AP, versus a function of atomic number, AN, plot 0f the pseudo potential radii after Zunger,Rl ,representing an example of the size factor. In such diagrams the chemical elements are ordered by increasing group numbera~~tach group is ordered by increasing quantum number. This special atomic-number (AN) scale was used instead of a linearlyincreasing atomic-number scale because it gives units of comparable size for distance between the atoms within any group aswen as along any period. The scale chosen for the comparison, of course, does not affect the number of factors (groups), onlythe appearance of the patterns changes

ANSi

CAI8

SeSA sP

PoTe

B iSb

PbSn

Ge

TIIn

GaCa ScMg

Be

A tIAa Ac Hf TaW Re Os Ir Pt Au HgSa La Zr NbMo Tc Au RhPd Ag CdSr Y Ti V Cr Mn Fe Co Ni Cu In

FrCsRbK

- 3: : : : : : I~-.+(/)Nct 2

only in the order of about 1-10eV/atom, one must havea method that is accurate to one part on 106, or better.'The other fact that greatly complicates evaluating thecohesive energy by theoretical methods is the numberof particles involved. Given that a macroscopic solidmay contain 10 23 nuclei and electrons, it is impossibleto determine the total energy of the crystal structurewithout some approximations. Within the last 15 years,two advances have made it possible to predict thecohesive energy of solids by numerical solutions ofthe quantum-mechanical equations of motion, e.g. theSchrodinger equation: the invention of high-speedcomputers and the device of one-electron potentials,which greatly simplifies many-body interactions.The accuracy of these computations is usually not at

the same level as experiment. Nonetheless, it is nowpossible, for chemical elements and simple intermetallic

3In principle, it would be sufficient to use as input onlythe atomic numbers (ANs) and the compositions of the

intermetallic compounds of the systems under consideration. Slater (I 956) once made a comment asfollows: 'Ion't understand why you metallurgists areso busy in working out experimentally the constitution[crystal structure and phase diagram] of multinarysystems. We know the structure of the atoms (needingonly the AN] , we have the laws of quantum mechanics,and we have electronic calculation machines, which cansolve the pertinent equation rather quickly!' Some 35years later Chelikowsky (1991) writes in an excellentreview the foUowing: 'Although the interactions inintermetallic compounds are well understood, it is notan easy task to evaluate the total energy of solids, evenat absolute zero. As the energy of an isolated atom isin the order of about io-ev, but the cohesive energy

Factors Governing Crystal Structures


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Straightlines with. negativeslopes

The first twoand the last twoare irregular;the Test arestraight lineswith positiveslopesFor every group, horizontal straight lines

Straightlines withnegativeslopes

The first twoand the lasttwoare irregular;the rest arestraight lineswith positiveslopes

Straight lineswith negativeslopes

Straight lines with positive slopes and adiscontinuity at CuValence-electronfactor

Electrochemicalfactor

p elementsStraight lines with negative slopessand d elementsLines with a maximum around V to Ni groupp elements Straight lines with positive slopes

Straight lines with positive slopesStraight lines with positive slopes

Straight lineswith negativeslopesaximum atthe Mn groupLine with a clear Line with a clearmaximum atthe V group

sand d elementsLine with twomaxima at theV group and theCo group

Cohesive-energyfactor

Size factor Straight lines with negative slopes and a slightmaximumaround the Ni and Cu group elementsAtomic-number Straight lines with positive slopesfactor

d elementshird p elementselementsecondirstactors (groups)Groupsong periods

Table 1. Idealized characteristics of the 'patterns' in the atomic property (AP) versus atomic number (AN) plots of the fivefactor (groups)

to describe the alloying behavior. Villars (1983)conducted a survey of 53 different APs as a functionof the AN (182 variables in all when the differentmethods of determination are taken into account), andit was found that there were only five main groups, herecalled factors. The results are best seen in AP versusAN plots. as shown in Figure 1 for the pseudopotentialradii R ; + after Zunger (1981) for s, p, and d elements.The fele~ents have been left out because in most casesonly incomplete data are available. In all but a fewcases (19 out of 182) very regular symmetric patternswereobtained insuch plots; this mean s a r eg ula r behaviorwithin a group with increasing quantum number (QN)as wen as along a period with increasing AN.In these diagrams the chemical elements are orderedby increasing group number and each group isordered by increasing quantum number. This special'AN' scale was used instead of a linearJy increasing ANscale because it gives units of comparable size fordistance between atoms within any group as well asalong any period. The scale chosen for comparison doesnot affect the number of factors (groups), of course;only the appearance of the patterns changes. It shouldbe stressed that the equivalence of APs belonging to thesame group is of a qualitative, not a quantitative nature.Adherence to one or another of the five factors is veryobvious. Table 1summarizes the idealized characteristics

compounds, to predict whether a given crystal structureis the most stable one at 0 K and 1 atm. We have to stressthat the crystal structures (and from Pearson's Handbook(Villars and Calvert, 1991) we know there are at least2750 different ones) have to be given as input for firstprinciples calculations, and this for each potential intermetallic compound. Assuming the potential intermetalliccompound crystallizes in one of the 2750 known crystalstructures and knowing its nominal stoichlometry=All,A~> etc.-it may still, in some cases, require a fewhundred first-principles calculations for each potentialintermetaUic compound; even with high-speedcomputers, this is not yet workable. But the largerproblem isthat the differences of the cohesive energiesof those few hundred calculations will be so small thatthe accuracy would have to be in the range of one partin 109 to determine the most stable crystal structure.The complexity of the above-mentioned problem showsthat one cannot expect that within the next decade theconstitution (crystal structure and phase diagram) ofmultinary systems will be calculated from first principles.Meanwhile it is therefore sensible to adopt semiempirical approaches based on the experimentallyknown data to search for the most reliable regularities.As at the moment it is impossible to start only from theatomic property AN, it is obvious to try to find outwhich other APs of the chemical elements are needed



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"Herm an and Skillman (1963).bde Boer et al, (1988).

23533

42213

323

34433

3422274654

6633263

15

Classical crossing points of the self-consistently screened. non-local atomicpseudopotentialsRadius of maximum radial electron density for outer orbitals fromHermann- Skillman calculations"Renorrnalized orbital radiusRadius calculated by Hartree-Fock-Slater methodIonic radiusSoftness parameterAtomic volumeMetallic radiusCovalent radiusReduced thermodynamic potential at 298KElectrochemical weight equivalentEntropy of solid elements at 298 KDensityAtomic numberAtomic weightPrincipal quantum numberAtomic electron scattering factorBond energy of deep-lying electronsSpecific beatWavelength of K and L seriesMaximum number of electrons in the solid elementMelting pointBoiling pointHeat of fusionHeat of vaporizationHeat of sublimationEnergy for atomization of I mol of the solid element at 0 KBulk modulusYoung's modulusTorsion modulusCompression modulusCrystal lattice energyDimer dissociation energySurface tensionLiquid-solid interfacial energyEnthalpy of formation or monovacanciesCohesive energySolubility parameterCompressibility modulusLinear coefficient of thermal expansion at 273 KElectronegativityChemical potential (after Miedema)"~3 (after Miedema)"s e'ectron binding energys-p parameterPositron annihilation rateElectron affinityHardnessNormal electrode potentialFirst ionization potentialTerm value (after Herman and Skillman)"Number of valence electrons (corresponding to group number)Number of vacancies or holes in the d bands above the Fermi levelValeace -e tect ron factor

Electrochemical factor

Cohesive-energy factor

Atomic-numberfactor

Size factorNumber ofdata setslement property determined experimentally or derived from a modelactors (groups)

5actors Governing Crystal StructuresTable 2. Atomic properties (APs) of the chemical elements grouped according to the five factors (groups)


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P and Q are constants for certain groups of systems,e.g. systems containing only transition elements. Theconcept that the energy effect upon alloying is generatedsolely at the contact surfaces between dissimilarWignerSeitz atomic cells results consequently in the fact thatA H or does not vary with the concentration as long asthe basis atomic cell A remains fully surrounded bydissimilar atomic cells B. For intermetallic compoundsthe elastic mismatch effect will be of no practicalimportance, since only crystal structures that arefavorable for the given constituent chemical elementsizes will be realized. Therefore, relation (1) directlyapplies to intermetallic compounds A B y if they aresufficiently rich in B such that A atoms are completely

3.1 Binary SystemThe most outstanding model for binary systems isMiedema's model, which is excellently explained in theseries Cohesion and Structure, volume Ide Boer et al.,1988). Two APs of the constituent chemical elementsenter the description of enthalpies of formation AJlfo r:the chemical potential for electronic charge (electronegativity) _ xM and the electron density at the boundaryof the Wigner-Seitz atomic cell, Nws. Both APs belongto the electrochemical factor group. Table 2.4of de Boeret al. (1988) gives the most recent recommended valuesfor _ xM and nws for most chemical elements. The keyexpression for the sign of tbe enthalpy of formationtJIfor of binary alloys is

A fundamental question that has to be answered firstbefore discussing crystal structures is: 'Which systemsform at least one "new" intermetallic compound?'

3. Compound-Formation Diagrams

structures.

of those five factors in AP versus AN diagrams, andTable 2 lists the 53 APs grouped by those five factors.Figure 2 shows two APs belonging to the same factorplotted against each other. the cohesive energy versusmelting point T of the chemical elements. As a firstapproximation a linear dependence may be seen to exist.The five factors (groups) are:

S iz e factor Atomic-number factor Cohesive-energy factor Electrochemical factor Valence-elect ron factorIn Table 3 we have given, in periodic table representation for the chemical elements, an AP for each

factor for which accurate and complete data areavailable as wen as (and this is very important forpractical use) values that are independent of theconstituent chemical elements of a compound. Theseare: pseudopotential radius after Zunger (1981), ~+p;atomic number. AN; melting point, T; eleetronegativityafter Martynov and Batsanov (1980), XM&B ; and groupnumber = number of valence electrons, V. It is clearlyshown in Table 3 that there exist no overall tendenciesbetween those five factors. so they represent the mostindependent factors. In recent years some other APshave come to the attention of the author like, forexample. average electron distances in the structuresof the individua1 elements (Schubert, 1990), the

figure 1. Plot for the chemical elements of the relationshipof two atomic properties, APs, belonging to the cohesive-energyfactor (group), namely cohesive energy and melting point T(Chelikowsky, 1979)

6

107- -E 5

s a~>.!. 3:>-2'Q)c 2)Q)>'u,Q).c .0o0.7

Crystal Structures of Intermetallic Compoundsenergodynarnic potential (Volchenkova, (989), andvibrational frequencies and dissociation energies ofhomonuclear diatomic molecules (Suffczynski, 1987),etc.: however, all of them could be assigned to one ofthe five factors chosen.In Section 4 we will always check to determine towhich factor(s) the AP and respectively theAPE belong

that are successful in organizing large groups ofexperimentally determined data. At the end we will findout which of those five factors are governing crystal


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3_2 .Ternary SystemsAttempts to extend Miedema's model to ternary systemfailed. The extension of compound-formation maps tternary systems is described in Villars (1986). Ithe ternary systems we face two additional majorproblems: only very few ternary systems are fulldetermined, and for many of them only a part of aisothermal section has been experimentally determined.During an extensive literature search (Villars andCalvert, 1985) we found in 1984 information on 559

(3)

nw s diagram corresponds to a particular binary systemSince information on phase diagrams is easier to retrievthan numerical values of su= . in assigning' +' or '.'to the points in this plot, the following criteria are used'.' in the binary systems with one or more intermetalIicompounds that are stable at low temperature,indicating f).}lOf.. QIP the Il.}{'or value is negative, while in theopposite case the sn= value is positive. The analysisof the sign of the predicted and experimental sn=


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system with compound formation is known whosebinary systems are all of the type that do not formintermetallic compounds, we included all combinations(ternary systems) that are surrounded by the 583experimentallyestabhshed binary phase diagrams showingno compound formation and endedupwith 1602additionalternary systems where the absence of intermetalliccompounds is expected (assuming the correctness of theabove-mentioned statement). All together we thus know2152 ternary systems in which no < ~< ~~~~~~) C~~~)~ ~VI X I ~ IX IXE ~! X I ' \ ~~~>


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We have tried to summarize the most outstandingpublications about regularities and gave special preference to the ones which included in their investigationlarge groups of well-defined data sets, in the rangeof hundreds to thousands of data sets. Again, whenever possible, we discuss first the binaries followedby the ternaries (multinaries) in the subsequentapproaches.

4. Regularities in IntermetaUic Compoonds

where TA> Ts> Te. The values for R;+ ' V, and Tare Pgiven InTable 3.In one three-dimensional diagram we plotted the

2152 systems where the absence of intermetallic compounds is observed and 5048 systems where compoundformation is observed. For the user's convenience wemade our plots on 12 sections with various isovalenceelectron difference (IAVASI +IAVAd+l..iVsd)/3values. Figure 5 shows such a section for (I..i v A B I+ l..i V " c ! +!AVBc!)/3 = 2. The symbol 11 ' stands forno compound formation determined by experiment; '0stands for no compound formation by extrapolationfrom experimentally established binary phase diagrams,all showing absence of intermetallic compounds; and'.' stands for compound-forming systems. With arelatively simple demarcation surface it was possibleto separate these two groups satisfactorily. Figure 6shows the schematic three-dimensional formationdiagram. The separation surface is based on 2152+5048 ternary systems and is accurate to 940'/0, whichmight be in the range of experimental accuracy. Itis nicely seen from Figure 6 that the absence of'new' intermetallic compounds is found along thethree coordinates; this means that the magnitude of anyone of the factors, i.e. the pseudopotential radiidifference expression. or the valence-electron differenceexpression, the melting-point ratio expression of theconstituent chemical elements, has to be small. Inthe cases where the values of any two or all threeexpressions are small, we also have absence of intermetallic compounds.

(6)

and the sum of the ratios of the melting temperaturesin kelvin,

(IM;+Ll.ABI + IM~+P'ACI+IM~+P'BCI)l3 (4) the magnitude of the difference in the number ofvalence electrons,

assigned structure is sufficient. For all such ternarysystem s w e checked where the nearby binary intermetallic compounds have the same crystal structure.Where we found this situation we assumed a solidsolubility range between the binary and ternary intermetallic compounds and therefore excluded thosesystems from the new-compound-formation group. Byadjusting our coordinates used for the binary systems(Villars, ]985) to the ternary systems, we came to thefollowing coordinates:.the magnitude of Zunger's pseudopotentiaJ radiisums,

Fagure5. ViUars' three-dimensional compound-formation ploto f th e s ec tio n fOT VB = 2 for te rn a ry s ys tems. For the meaningof' 11 '. '0' and '.' symbols, see text; each symbol representsone ternary system

4

+al-cci+(/)Nct.-). . . . .::lt3:::I 1 0 0_.-(/)-0. . . . . ..!E:::IZ 50



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}992). The simplicity principle can also be shownby plotting the number of Bravais-type intermetalliccompounds, e.g, '(p' versus the number of atomsper unit cell. Figure 26 shows dearly that the vastmajority of all 'tP'-type intermetallic compoundshave less than 24 atoms per unit cell, although there arecrystal structures known with up to 1000atoms per unitce ll . Analogous behavior is observed for al l Bravaistypes.

Both the symmetry and simplicity principles canbe considered as a consequence of the angularvalence-orbital factor. In Figure 8, the 14 AETsmost often realized are very highly symmetrical. andbecause these AETs are not building units, butinterpenetrating, it is not likely that crystal structureswith many different AETs within a crystal structure arerealized.

considered the 650 most populous crystal structuresincluding 16500 intermetallic compounds. As thenumber of point sets per crystal structure is justthe upper limit of the number of possible differentAETs. Daams and Villars (1993) investigated thenumber of different AETs with in a crystal structure,which is the real measure for simplicity, for all cubicIntermetallic compounds. Figure 25 shows that thesituation is even simpler: of 5521 investigated intermetallic compounds, 5086 or 9 2 0 /0 crystallize in crystalstructures with three or fewer different AETs withina given crystal structure; 2561 lnterrnetallic compoundshave just one AE within the crystal structure. 2035belong to the two-environment types, and only 490to the three-environment types (see Figure 25). Asimilar behavior was also observed for the rhornbohedral and hexagonal crystal structures (Daams et al.,

ltigurelii. Number of intermetallic compounds versus their number of atoms per unit cell for the 'tP' Bravais-typecrystal structures

180 200 220 240 260 2R6040Number of atoms per unit cell

120000000oo

100

600

700-r-- 00~

,_"_o___

. _ . _ .

..

..

l-r-

II~ 1 . 1 I I I I I' I I J .J L( I


Vl"'0C 500:::l00.E0U'U 40U-T l~uE. . .c 300._0. . . .~E:::l 200Z

30


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Figure 27. Part of the concentration area (dark), showing where no 'new' ternary intermetallic compound(s) occur

Concentrationareawhere no 'ternaryintermetalliccompoundsOccur

B

4.7. J Binary Intermetallic CompoundsAs shown in Section 3 one is able to predict compoundformation or respectively its absence fairly accuratelyfor both ternary as well as binary systems. In the casewhere compound formation is predicted,onewould bevery much interested to know how many intermetalliccompound(s) occur and at which stoichiometries.Unfortunately, very little work has been done in thisfield. Miedema and co-workers (de Boer et al., 1988)

4. 7 Stoichiometric Restraint Approach

very few established exceptions to this rule. Nevertheless,in some cases it is very difficult to distinguish betweenternary solid solubility of the nearby binary intermetalliccompou nd and a 'n ew ' te rn ary intermetall ic compound.This rule has the following impact on our five factors:any APE will come into effect only when at least 5 at. 0 1 0of the second or third chemical element is added. The

same holds for the angular valence-orbital factor; thismeans an AE within a crystal structure will not change,rather it will buffer the produced 'mismatch' caused bythe additional chemical element, at least for the first5 at.0 7 0 .

c00 40A (at.%)80A

4.6.1 Binary and Ternary SystemsThere is a regularity that has been observed by theauthor to be valid for binary as well as ternary systems,namely that at least 5 at. % of a second or a thirdchemical element is needed to be alloyed to a unary orbinary system to get a 'new' intermetallic compound(s).There exist only a few exceptions, like e.g. B 6 6Y,BwGe, Ca33Ge, Zn22Zr, and AgMS16Zn3J' This condition is necessary but not sufficient to get 'new'interrnetallic compound(s). In the binary A-B systemsthis means that one will not usually find inter metalliccompounds ABy with y> 19. Figure 27 shows a part ofthe ternary concentration triangle. The dark area is theconcentration range where no 'new' ternary intermetaIliccompound(s) occur; it is seen that near the A, B, andC comers even 15 at. O J o of the other two chemicalelements are needed. This has quite a consequence forexperimentalists searching for 'new' intermetallic com-pounds as well for the determination of phase diagrams.In the binary systems 100/0and in the ternary systemseven 28.50/0of the available concentration area does notneed to be investigated for 'new' intermetallic compounds. To the knowledge of the author there exist but

31actors Governing Crystal Structures4.6 Active Concentration Range Approach


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(i) Space-group approach A crystal structure isdefined by the formula, the unit-cell dimension(s), thespace group, and its occupied point sets. As 230 spacegroups exist, one could generate within each space groupall possible point-set combinations assuming anarbitrary upper limit of e.g. 100 atoms per unit cell.Villars (1981)generated such point-set combinations forspace group P421m and got 67072 possiblecombinations (crystal structures). Knowing the experimental fact that the great majority of all daltonideintermetallic compounds have each point set occupiedby just one kind of chemical element, one can inprinciple also generate for all binary and ternaryintermetallic compounds all possible stoichiometricratios. We observed three restraints that are obeyed byall now known intermetallic compounds:

4.7.2 Ternary Intermetattic CompoundsIn ternaries the situation is even more restricted,although one would expect many more possiblestoichiometric ratios because of the much largeravailable concentration area. Figure 30 shows ananalogous plot for the ternary systems; again the sevensimplest stoichiometric ratios, I: 1: 1 , 1: 1 :2 , 1: I :3 , 1:2 :3 ,1:2 :2 , 1: 1 :4 and 1 :2 :4 , cover 90070 of all known ternarydalton ide intecmetallic compounds. It is very interestingthat almost all realized stoichiometric ratios are alongfive directions in the concentration plane. For example,two of these directions are AfiC. X= 1-5, and A)3yC,x=y= 1-5. The knowledge of this experimental factdrastically reduces the number of samples to beinvestigated per system from about 200 samples to25 samples (only 12 .5070 ), with the chance of finding atleast 90010 of a ll existing daltonide intermeta11iccompounds. The Ni-Si- Ti system determined byWestbrook et al. (1958)represents a nice example havingsix ternary intermetallic compounds, a11with simpleatomic ratios (Figure 31). No comprehensive approachuntil now has been able to define 'new' structure typesand therefore predict the most preferred stoichiometricratios of binary and ternary intermetalJic compoundsstarting from a simple approach. According to theauthor, two approaches might be able to predict mostof the unknown, yet possible, crystal structures, as wellas describe which stoichiometric ratios occur and therules by which those ratios are highly preferred.

found for binary systems a quantitative relation betweenthe average number of stable intermetallic compoundsand the enthalpy of formation AH~~~(;predicted forequiatomic intermetallic compounds. Figure 28 showsa histogram that summarizes the results from experimentally determined phase diagrams and the calculatedenthalpy of formation .Mf~~~for the equiatomic intermetallic compounds containing transition metals. Forequiatomic intermetallic compounds t l H ~ : r ccan be calculated for most binary systems with Miedema's model,essentially starting from XM and nws of the constituentchemical elements. No attempts have been made toextend this to ternary systems. Rodgers and Villars(1993) made some statistical investigations on 25000distincrly different intermetallic compounds with thehelp of CRYSTMET (1992) and found some interestingand useful results. Looking at daltonide intermetalliccompounds, one observes, not a random statisticaldistribution of the experimentally determinedintermetallic compounds over the available concentration range, but a strong preference for five particularstoichiometric ratios, 1:I, 1:2, 1:3, 2:3, and 3:5, asshown inFigure 29 (only those stoichiometric ratios areshown where at least five compounds are known to havethis composition). The other stoichiometric ratios

o 2 3 4 5Number of intermetalliccompounds per system

Figure 28. The relation between the average number of stableintermetailic compounds per system and the enthalpy offormal ion HJo~ predicted for equiatomic interrnetallicco mpounds. T h e his togra m summarizes the experimental phasediagram information for systems containing two transitionmetals

5 > llH fo f> 00 > ll H for> -4'-4 > ll .Hfor~-IO-10 ~ llH 'or> - 20- 20 ~ ll.Hfor> -40- 40 ~ llHfor> -75-75 > llHfO (

I

Crystal Structures of Intermetallic Compoundsare much less frequent, but are still not randomlydistributed over the whole concentration range. As themajority of all binary systems have been investigated,Figure 29, it is very representative.

32

c:0-~mE 0. . . .. $ 2 - ,- ~->.(/)0.-0- c:to :J. ; 0c a.Q) E"'0 0Q) ( . )um:o~Q) . . . .....a..-


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(ii) Atomic environment/rsimulated annealingapproach From the investigations of Daams andVillars (1992) and Daams et 01. (1992) of all cubic,rhombohedral, and hexagonal crystal structures, weknow that the number of atomic environment types(AETs) in realized intermetallic compounds is limitedto a total of 20 most-often occurring AETs and a groupof 2-3 times less-often occurring AETs. Preliminaryresults on the remaining crystal classes indicate that thenumbers will not be increased significantly by includingall crystal structures. This experimental fact togetherwith 'simulated annealing' gives us the possibility topredict as yet unknown crystal structures.

of the above empirical restraints reduces the number ofpotential structures to a level similar to that of theinvestigated space group) in comparison with the 2750experimentally found crystal structures.

Applying these three restraints to those 67072 pointset combinations of space group P421m reduces themto 499 remaining combinations, still too high a numberto be of practical use taking all 230 space groups intoaccount. Meanwhile Rodgers and Villars (1993) realizedthat over 900/0of all known intermetallic compoundscrystallize in one of the 11most common space groups(see Figure 22), This experimental fact brings thenumber to about 5000 potential crystal structures(assuming that in each of the 11space groups application

.The c/a (bla) value is smaller than 4.5. An intermetallic compound does not have any 4holes'within its crystal structure that are ]arger than thelargest atom witbin that intermetallic compound . No intermetallic compound is known to be built upof densely packed layers each lying exactly above theothers.

Figure 29. Number of daltonide intermetallic compounds versus available concentration range for binary systems (A..B)', x


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{i) Polymorphic modification combinations restraintVillars (1983) showed, in context with the structuremaps looking at around 1000 binary AB compoundscrystallizing in one of the 22 most populous crystalstructures, that from the possible (22 x 21)/( I x 2) = 231crystal-structure combinations only 13 are realized (seeTable 7). Figure 32 shows the structure map (Rodgersand Villars, 1993) of the three-dimensional plot: sumof valence electrons at r;v1\8 = 7 versus magnitude ofMartynov and Batsanov's electronegativity differenceI~BI versus magnitude of Zunger's pseudopotentialradii difference It&_R z Asl for binary 1; t interrnetallics+p,compounds. It is nicely seen that there exist domains

intermetallic compounds, systems or crystal structuresthere exist very strict structural relation restraints. Thiscan be done by investigation of the experimentallyknown polymorphic modifications within an intermetallic compound (normal condition, high temperature,and high pressure), investigation of the realized crystalstructure combinations within binary systems, andinvestigation of the experimentally found interrnetalliccompounds within a certain crystal structure.

4.8.1 Binary Intermeta/licCompoundsThere exist three main ways to show that, within

4.8 Structural Relation Approach

'Simulated annealing' (Pannier et al., 1990) is anattempt at solving the following problem: Given thechemical composition of an intermetallic compound andthe values of its unit-cell dimensions. predict its crystalstructure by optimizing the arrangement of the atomsin accordance with a set of prescribed rules. Theprocedure uses simple, empirical crystal-chemistryarguments (Pauling's principles for ionic compounds)and a powerful stochastic search procedure, known as'simulated annealing'. Combining the AET restraintwith 'simulated annealing' by varying the unit-celldimensions and chemical compositions in a systematicway might also result in the prediction of the mostprobable 'new' crystal structures as well as theirstoichiometric ratios. The stoichiometric restraint isdirectly connected with the occupation of the point setswithin a crystal structure. and therefore is a consequenceof the angular valence-orbital factor.

Figure 30. Number of daltonide intermetallic compounds versus available concentration range for ternary systems (A xB yCt,x


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AuTiDySi, EuSi, GdNi, GePr,HoNi, HoSi, LuNi. NiTmNiYb, PrPtAuCe, AuGd. AuNd, AuPr,AuSm, AuTbAgYb, AuPm, AuYbAuDy, AuEr, AuRo, AuTmRhSiPbS, PbSe, PbTe, SnTeAgTi, CuTiAuMn, JrMn, MnRhAgl, BrCu, CdS, CdSe,ClCu, CuI, OZn, PoZn,SZn, SeZnHgS, HgSePdSb, PdTeOsSi, RuSi

Representatives

HgS (hP6)-SZn (cP8)AsNi (hR6)-CINa (cFS)CICs (cP2)-FeSi (cPS )

BFe (oP8)-BCr (oC8)-CICs (cP2)BPe (oP8)-CICs (cP2)BCr (oC8)-CICs (cP2)MnP (oP8)-FeSi (cP8)GeS (oP8)-C1Na (cPS)AuCu (tP4)-CuTi (tP4)AuCu (tP4)-ClCs (cP2)SZn(hP4)-SZn (cPS)

AuCd (oP4)-CuTi (tP4)Bfe (oP8)-BCr (oC8)

Structure types relatedthrough polymorphism

Table 7. Binary AB compounds that areexperimentally foundto crystallize in two or three modificationswhere two modifications are realized and domains whereonly one modification is found. Exactly the same isshown in a chemical element versus chemical elementdiagram for AB compounds (see Figure 33). The crystalstructure{s) are indicated in each field with the notationused in Villars et al. (1989). This experimental factenables us to predict both the possible crystal-structuremodifications in polymorphic intermetallic compoundsand which intermetallic compounds are likely to showpolymorphism.(ii) Companion crystal-structure restraint Belov andco-workers (Smirnova et al., 1983) traced rulesgoverning the coexistence of crystal structures with eachother in binary sy st ems with the help of experimentallydetermined phase diagrams and then constructed asystem of interrelations of crystal structures. Anexample of such a crystal-structure sequence in binarysystems containing transition elements is: cP8 Cr3Si-rI6MoSi2-tP4 AuCu-hP3 AlB2

Figure 31. Isothermal section at 1273Kof the Ni-Si- Ti system showing six ternary intermetallics, all with simple atomic ratios(after Westbrook et al., 1958)

T i,\

,\ 10

_ 2030

50 SiTi

35

60

60

80

90

Si

4000i

NiSi



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All three restraints have only been partially investigatedand only for binaries. The most practical impact of thisapproach will be extension to ternaries. We are convinced

easily explained: the 4(a) and 8(h) positions have AETswith eN 10, whereas the 4(b) and 16(k) positions haveAETs with CN J4 and 15 respectively. It is well-knownthat pelements prefer low CN in their elemental state,whereas d or felements prefer a higher CN. Compoundswith 46 crystal structures are formed only by onecombination group (e.g. p with d elements), 34 crystalstructures are formed by two different combinationgroups (e.g, p-d and p-f), and the 10 structuresremaining are formed from three or more groups.

(iii)Chemical element/point-set restraint In 90 of the106investigated, most populous, binary crystal structures(Villars, 1981) we found a very strict regularity betweenthe position of the chemical element in the periodic table(s.p.d.f elements) and its point-set occupation withina crystal structure. This can be best explained with e.g.the tI32 Si, W s type. Its binary representatives are onlyfound in the following chemical element combinations:p elements with d elements, and p elements with felements. In addition. the ratio between p andd or f elements is always 3;5. This means in this examplethat pelements always occupy the point sets 4(a) and8{h), and d or f elements the point sets 4(b) and 16(k).Looking at the AETs of those positions, this can be

Figure 32. Villars' structure map for EVAO=7 of the three-dimensional plot of valence electrons EVAB versus magnitude ofMartynov and Batsanov's electronegativity difference IAXM&:B I versusmagnitude of Zunger's pseudopctential radii differenceI.6R~+pl. showing domains where two modifications can occur oP8 BFe/oeS BCr and oP8 BFe/cF8 CINa

4 cF8 CINacF8 CINa 0

0i 0i ogo () 003 ()

0 . . . . . . . . o (10;:, et" O o o()~)-) 0 () hP12 NaO oc( 0 0Q. 2

0

()+!/l eN '6 0 0ct< e0)

0

e0 cF6 CINa

0 \ < > 0e 00 o ~ 0""

~ 0 0a o. o~ e0 o ~ ,~000 0" < >0 00 o 0 0 00 2I~XMBABI

Crystal Structures of Intermetallic Compounds6


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oCD


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In the previous sections we showed which factors governcrystal structures and formulated some generally validprinciples. all this based on experimentally determineddata, with the main aim to .predict 'new' intermetalliccompounds. The question arises: "Why should we beinterested in predicting and verifying "new" ternary andquaternary intermetallic compounds']' Most scientists,in universities and especially in industry, are mainly

6. Quantibltive Relations Between CrystalStructures and Physical Properties of

Inlermetallic Compounds

Chemistry principle The vast majority of crystalstructures show a very strict regularity between theposition of the chemical elements in the periodic table(s, p. d, and felements) and its point-setfs) occupationwithin a crystal structure.

Stoichiometric restraint principle The vast majority ofdaltonide intermetaUiccompounds have each point set(s)occupied by just one kind of chemica) element. Thefollowing stoichiometric ratios are highly preferred: forbinary intermetallic compounds, 1:1, 1:2. 1:3, 2:3, and3:5; for ternary interrnetallic compounds, 1:1:1, 1:1:2,1:1:3, 1:2:2, 1:1:4, 1:2:3 and 1:2:4.

Active composition range principle In binary (ternary)systems at least 5 at. 0'/0is needed of the second (third)chemical element to form 'new' intermetallic com-pound(s). This reduces the available concentration rangefor the occurrence of 'new' intermetallic compoundsby 100'/0 fo r binary systems and by 28.5070 for ternarysystems.

A tom ic- enviro nment principle The vast majority of aUatoms (point sets) in intermetallic compounds realizeas their atomic environment one of the 14 atomicenvironment types shown in Figure 8.

Simplicity principle The vast majority of all crystalstructures have three or fewer different atomic environment types (AETs) within the crystal structure (single-,two- or three-environment types). In addition, the vastmajority of al1intermetallic compounds have less than24 atoms per unit cell.

Symmetry principle The vast majority of all intermetal1ic compounds crystallize in one of the following11 space groups: nos. 12.62,63, 139, 166, 191,194,216,221, 225 , and 227.

Solid-solubility map principle The size, electrochemical,and valence-electron factors control solid solubility.Solid-solubility maps do quantitatively separate regionsof limited and extended solid solubility for a givenchemical element solvent.

Structure map principle The size, electrochemical,valence-electron, and atomic-number factors are thefactors governing crystal structures of intermetalliccompounds. Structure maps do quantitatively separateintermetallic compounds into distinct crystal-structuredomains. There exists a whole range of differentstructure maps for binary intermetallic compounds aslisted in Table 5 and for ternary intermetallic compounds (see Section 4.2.2, Ternary IntermetallicCompounds).

Compound-formation map principle The size, electrochemical, valence-electron, and cohesive-energy factorsare the factors governing compound formation. Thecompound-formation maps do quantitatively predictthe compound formation in binary systems (afterMiedema) and in ternary systems (after Villars).

Here we formulate nine principles which, among thepresently known regularities, are the most quantitativeones. The validity of these principles was testedon a fair number of compounds in a well-definedgroup of experimentally determined data sets andshowed accuracies in the range of 90-1000 '/0 . W e cantherefore rely on predictions based on those principleswith considerable confidence, and can say that theseapply 'for the vast majority'. Applying these nineprinciples will both drastically reduce the number ofsystems (or respectively, samples) to be investigated andremarkably increase the success rate of finding novelmaterials.

s. Nine Quantitative Principles

that most of these relations can be understood by considering the AETs. The comprehensive investigation ofthe connection between the above-mentioned restraintsand the AETs will reveal many new regularities andenable us to understand polymorphic transformations,closely related crystal structures, and the role of theposition of the chemical elements in the periodic table.Analogous to the stoichiometric restraints, the

structural restraints are a consequence of the angularvalence-orbital factor.



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~xPaulingFigure 34. Isoelectrical conductivity plot for the fractional number of bonds m versus Pauling's electronegativity differencesx: Each compound is represented by a symbol indicating its crystal structure (broken curves are numbered in units lif a cm")

2 .0.6

!> .

--.--,-cP2 CICs0 cl2 w6 cF4 Cu0. (Fe CINa. . hPi. SZn

. . ,v cFS SZnt. .. . V.. .

. .66 6 . .I ! > . oi.I ! > . 6,T

I a~ 6t : J . t.~

,,,

systematization of the occurrence of high-T, superconductors (see Figure 35) and high-Z; ferroelectrics(see Figure 36). They are presented in two-dimensionalL l, i ? '; +p versus MM&Bplots together with plots of Vversus Te. The superconductors with T, higher than10 K are clustered around three domains A, B, and C,each having a very limited number of crystal structures.The recently found high-T, compounds are locatedin C. Even more restricting are the conditions for highT, ferroelectrics with T;500K.These are located ina very small V versus a R ; - +P versus UM&B volume(about 1 0 / 0 of the available volume). Again, thisproperty is found in only about a dozen crystalstructures. The same QSD was also successful in thesystematization and prediction of stable ternary quasicrystals. Figure 37 (Villars et al., 1986) shows thedomain in which the known stable and metastableternary quasicrystals are located, together with the stableintermetallic compounds crystallizing in the cI162A~MgllZnll type. The experimentally known quasicrystals have in addition to fulfil the condition thatthey are located at the boundary between compoundformation and its absence in the compound-formationdiagram (also calledQFD, see Figure 6). The use of QSDand QFD in the context of the global structuralmultinary chemistry of high-T, superconductors,ferroelectrics, and stable quasicrystals 1 S very comptehensively and self-consistently discussed in a publicationby Rabe et at. (1992).

1 . 2

".';). ,."0.8.4

6\,'. .o , . .o,

a s '\r V 6v.v'"___ ~__ _ .4b.- - . . " '- V - t ~i ~" ..06 Z... " ~_v_ \E . -, v. ,.. ~__.-.. , 6 I

,. - ,. ~ '. V v ,\' t1 . .. Ie ,.,, . 'e..L." ' 6 -~ . . ' C "~~~'t...',_. ,.,'. ., ,02 '" ,,"6 ~6 6. .," .,'.

L O

interested in specific physical properties of intermetalliccompounds. So an obvious question is: 'Does a quantitative connect ion exist betw ee n intermetallic compoundscrystallizing ina certain crystal structure and a physicalproperty of interest?' Here we just mention briefly themost outstanding correlations, which should stimulatemotivation for future research in that area. The followingexamples show definitely that such relations exist. It isimportant to realize that the crystal structure of anintermetallic as well as the APEs of its constituentchemical elements are necessary conditions, but notnecessarily sufficient for prediction of a particularphysical property. Belowwe Jist some physical propertiesfor which ithas been possible to show such relations:electrical conductivity. high-T; superconductivity, highT; fe rroelectrici ty, stable quasicrystals, higb-weldabilitymaterials. high-melting low-density materials, andmaterials with tailored melting point.Kiang and Liu (1980)found a connection between the

electrical conductivity of a binary compound and itsfractional number of bonds m versus Pauling's electronegativity difference X" calculated from its constituentchemical elements. Figure 34 shows a projection of theelectricalconductivity (isoconrours). The symbols indicatethe crystal structures of those intermetallic compounds.Villars and Phillips (1988) and Phillips (1989) usedthe three-dimensional Vversus llR;+ p versus MM&Bstructure maps, also called quantum structure diagrams(QSD). and they have proven to be very successful for

39actors Governing Crystal Structures


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Fleischer (1987) constructed a whole series of specificgravity versus melting point plots for intermetalliccompounds, each plot for a certain crystal structure.Figure 39 shows that by using intermetallic compoundscrystallizing in a certain crystal structure one can tailorcompounds with e.g. highmelting point but low specific

Pettifor (l988a) shows a Mendeleev number MNversus Mendeleev number MN weld ability plot (Figure38) for metal pairs to be joined (differentiating onlybetween excellent, good, poor, and unsatisfactorywelds). Itis definitely seen that excellent weld ability isonly found in a specific domain.

Figure 36. Quantum structural diagram for high-T, ferroelectrics (Te>500K) in an analogous representation to Figure 35

-0.51.01.5

7v5

Structuretypes+ cPS CaTi03 hP:30 LuMn03..L oP20Gd~T oP24 Sb{Nb,T a)04c tI24 COW04 Others.to Quasiternariesand quaternaries

500 4

1000

1500

1.0

..-::ia s-~1.5tnNlet10K projected (with respect to V). The coordinates a X M 4 : B and tJi;~are explained in the text. (b) Matthias profiles of T, versus V for islands A, B, and C. The crystal structures a re designa tedon the extreme right

f- Structure typesB. . 1-2.5 20- ~- ...cF8 CINa. . ~. ~ . . . ... oP 16 B2LuRu . . ~ 15- .. .. . . .. . I-2.0 .to cI40 C 3P u2B 10 _j hP 2 CW, . .. . . -- ++ - 1.5 3996 +hR15 MOGPbSe- C~ +" ++ " cP 5 Ca03Ti- 20-C - 1.0 .-.. .L tl* (Lo,S r)2Cu04~._ . 'f cF56 At2,Mg04I-u15- rI++t : - tP 14 B02,Cu30SY- 0.5 - t - oP 14 B02Cu307Y~O . ~- -/V\ o cP 8 C r3S iI L _L J I I

-2.0 -1.5 -1.0


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80

Figure 38. Weldability property plot for metal pairs (after Pettifcr)

75S 70M N A605

00 00 co 00 oooeo cococco 0 C 080 oc oc 00 00 OCOCO I:IJCOOOO 0 0

00 00 00 00 00000 0000000 07S00 00 00 !Xl 00000 00000008 8 8 8 00 00 00 .... 0000 00 8 8 51 0 00 00 00 . . . .00 8 8 00 00 I t -0 00 00 000000 00 00 CD 0000

c o 00 g g 8888 . 0. . .z 00 .08 8 00 81 " 00 0 00 o... Weldability00 00 o .. Excellent00 00 o. . . Good00 00 o. o Poor0 co

o Unsatisfactory00 g o00o

Figure 31. Quantum structural diagram for icosahedral intermetallic compounds and quasicrystals. The insert shows the domainspanned by dsp and psd combinations (s, p. and d elements) with average valence-electron concentration Vbetween 1.3 and 2.5

1 .0

41

-0.5 0.51 . 0

Stable quasicrystal cI 162AIsMg,1Zn,1o MetastablequasiCfYSlaI


-1.0

-:lm-


42/49

hexagonal crystal structures and some cross-checks, weknow that the gross features of the 2750cry sta l structures(based on the space-group theory) can be described withfewer th an 2 0 often-occurring AETs and so m e 2-3 timesIes s-frequently occurring A E T s, which simplifies thedescription of crystal structures drastically. Based onthose AETs one expects that the 2750 crystal structurescan be reduced to around 30 0 different coordinationtypes that are crystal structures with the same numberand kind of AETs, i.e. crystal structures having verysimilar gross features. Crystal structures belonging totwo different coordination types are therefore distinctlydifferent crystal structures; crystal structures belongingto the same coordination type are very similar. Thisconclusion is highly supported by the results revealedwith the symmetry and simplicity approaches; in otherwords this means that the short-range order of atomsis very dominant, and long-range order is of much lessimportance in crystal structures. Of great practicalimpact are the nine quantitative principles as well as therestricted number of crystal-structure combinationsfound in intermetallic compounds (polymorphism) andBelov's observed companion crystal-structure combinations within binary systems. One of the main goalsin the future will be to find quantitative relationsbetween the bonding types (metallic, ionic, and covalent)of intermetallic compounds, and the AETs and APE ofthe constituent chemical elements.rom the investigations of cubic, rhombohedral, and

7. Conclusion

gravity (aerospace industry). From Figure 39 it is clearlyseen that most cFS CINa-type intermetallic compoundsare better in that respect than cP4 AuCu} intermetalliccompounds.

Finally Hulliger and Villars (1993) show a chemicalelement ve rsu s chemical element melting point Tplot forAB intermetallic compounds. A part of plot (see Figure40) shows that the behavior is very complex, butdefinitely not a random statistical distribution. It is alsoobserved that there is no significant difference betweencongruent and incongruent melting compounds. Thecrystal structure is not of first priority for predictionof the melting point; much more relevant are theconstituent chemica1 elements.This very short summary shows that there exist quantitative connections between the physical properties of an

intermetallic compound. its crystal structure, and theAPE of its constituent chemical elements. Such connec tions enable the experimentalist to reduce the enormousnumber ofpossible systemsto the systemsmost similar tothe a1ready known intermetallic compounds having thephysical property of interest, and therefore to the systemswith the highest probability of finding novel materials.

Filure 39. Melting point T versus density plots (after Fleischer) (a) for cF8 CINa type compounds and (b) cP4 AuCu, typecompounds

3000500000000 1500T(OC )

I

- (b) cP4 A uC u)-

I

2500000500 I(a) cF8 CINa

4 b3 -

n

::>.~ 6cQ)o

IJo c :P 0 -- 0~~ ~DC 0 -0P o B OCbo0{! 0 -

- 10Eo 80)-

15 ~I0

Crystal Structures of IntermetaJlic Compounds2


43/49

with the help of AP versus AN plots. Petti forformulated an additional factor, the angular valenceelectron factor, which is fully supported by the resultsof the atomic-enviromnent, space, synunetry, simplicity.

With the help of the regularities found semiempirically within the experimentally established intermetallic compounds, four factors are shown to governcrystal structure from the initially suggested five factors

Figure 44). Part of a melting point T{ K) contour plot forAB intermetalliccompounds and alloysplotted as elementA vs elementB. Melting temperatures T D 1(K ). Lower limits (e.g. >9(0) are deduced from inf()f'mationabout sample preparation (annealingtemperatures. etc.). Crossed fields. nOD-existence accordingto experimental phase diagrams; half-crossedfields, non-existencepredicted by Miedema ('\, ) or Villars (/); u, existenceunknown; c, congruentmelting; (c), the givenTvalue is the solidus temperatureof an alloy;n, non-congruent melting (peritectic, peritectoidal); d. the given Tvalue istbe decomposition temperature at nonnalpressure; s, sublimation; ., decomposition at a lower temperature

IL' NG I( fIlo C. C. "lb. ~t St 80 $c Lv . . . . . . T. . .(, He 'I' Or TO G Il E ." 50ft r.d . . . ~ Lo lint Pv , . . IJ Ttl Ti Z, HI V ND To t< Mo 11 MIl Tt Rr' ~ rx [X x x x X IX " IX'YIX'XX IX'IX'IX'X IXXXX IXXX/ II ~ 0.. IX II~ [X[XX[XX L X ~ [X~ L X D < lip~ L X ~ XX L x : X 4 , . XX [ > < X[XXXXXVXX[XX r-,/ IIX-,[X[X[XxX - X lXXXX IX [> < IX/H.C II. . . :- I~ > < -,r-,X x vrx [> < -,[> < IX~ , , , - " " ~ X " " " lX :\,XX II l6:~ U II xX [XlX[XXX x [XL X V/r.Jo t ' I:')< K. -,X x VIXe - e x : ) < D < lX X X [) < i5 < II [X G elW lX.)'< x ) < I X D < x )< tx x I X I X . , J. . .I:. ; o r~: u X r-, ) < !X )< [) < )< II U ~C lID 1 1 1 : 1, . . . . . . , IIX i " . . .X . . . . . D < II IX)(X' )< X ") ( ')( " X ' II I X , . .C lEI :trl 1 1 :1. . ., . . . . U I X f'., [Xr:)< I X I X I X[XI> X )< )< IX III X c.It!. ., u i > < II II r a lXD < lXlXD < i)< X I)< X IX II I X~AS compound (I) . . . .I';:II UII II II II II .. II >< I> II II II U -. . . I'; IXIX I';;IXIXIX I X x L X. IXIX u l X1 1 :1 "". '. t ' : i ["\r'-- f"-, f"-,i"["\I'X rx U I X II II H9c ~"lX- : : , ,_ , I':!IX ,';;! [X IX L X I > < rx II r x IJ1 1 : 1 ''7I X 'oU IX'IX~ I X IXIXD < IX" " u n.-r~.I':;:t~1 - ; I':l ~ ,.~. . .":' r,:: T,t f C I " n1 " '" r:;l X r ' P ! I~IX XIX [X u IX z.t c16 )


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information system in material science. This systemshould always be extended by newly found 'highest'quality regularities. The benefits would be:(i) Refinement of the already known regularities with

exact definition of their validity and boundaryconditions.

(ii) Opportunity to search in a systematic way, witha high probability of success, for regularities untilnow 'missed'.

(iii) Verification of all regularities against the databasesand derivation of an 'accuracy level' for eachregularity.

(iv) Reduction of the number of unwanted duplicationsof experimental work.(v) Provision of a highly developed tool for the

experimentalist working in material science. Thistool, together with the intuition and experience ofthe scientists, should reduce the 'unlimited' largenumber of potential systems to a limited, thoughlarge, number of systems. The resultwould bethesynthesis of a n potential novel materials withdesired physical properties with the highest probability of success.(vi) The interaction of the results of first-principlescalculations would be very stimulating for theimprovement of such an information-predictionsystem and vice versa.

(vii) Enormous economic benefits to those wishing tomaintain their competitiveness in the area. Thisapproach, combined with effective materialsprocessing techniques, should show new results inmaterials science and engineering.

Savitskii (1976) was one of the first to stress the needfor information-prediction systems inmaterials science.He was also the first (Savltskii et al., 1984) to developin that context a database on the physical properties ofternary inorganic phases. The logical structure of hisfile is shown in Figure 41. Kiselyova (1993) uses thatdatabase together with cybernetic-statistical methods todevelop an information-prediction system. This is adevelopment of the idea of machine learning torecognize patterns for prediction, first published bySavitskii (1976). Figure 42 shows the scheme of theirsystem. Zhou et 01 . (1989) developed such a system forretrieval and prediction of binary intermetalliccompounds. and the scheme of this system is given inFigure 43.The need for an international information-predictionsystem has been strongly evident for the last 10 yearsas most research activities moved from binariesto ternaries and multinaries, opening an unlimited

stoichiometric restraint, and structural relation restraintapproaches. So we end up with the following five factorsgoverning crystal structures:.Size factor Atomic-number factor Electrochemical factor Valence-electron factor Angular valence-orbital factorThe cohesive-energy factor seems not to be governingcrystal structure, but does determine compound formation or respectively its absence. The size, electrochemical, and valence-electron factors govern bothcrystal structure as well as compound formation orrespectively its absence.The relations between these factors and the crystal

structures is very complex, but the many regularitiesfound prove that these relations are not only of aqualitative nature, but already in many cases exhibita quantitative nature with an acceptable accuracy, It isvery interesting that by comparing the present situationwith the 'classical' regularities found SOor so years agoby Hume-Rothery, Pauling. Laves. Goldschmid, Kasper,and Mooser and Pearson, only one of the six factorsmentioned is new: the cohesive-energy factor. From thenine quantitative principles the following are alsocompletely new: compound-formation principle. atomicenvironment principle. structure map principle, simplicity principle. active composition range principle,stoichiometric restraint principle, and chemistryprinciple.One great improvement made in the last 50 years istbe optimization of APEs of the constituent chemicalelements within those factors in choosing the optimalAP for our five groups of distinctly different APs. Amajor improvement was made for the size factor inusing pseudopotential radii, which are independent ofthe constituent chemical elements, in contrast to theclassical metallic, ionic, and covalent radii. Anothermajor improvement is the beginning of the extensionof regularities from binary to ternary (multinary) intermetallic compounds/systems with an acceptable quantitative accuracy.Forced by the almost unlimited number of availablechemical element combinations in multinary systems,the author is completely convinced that, if in the futureone wishes to find, efficiently and systematically, novelternary (multinary) materials, thi.swill only be possibleby linking databases (CRYSTMET, 1992; CRYSTIN,1992;AET. thermodynamic databases, physical propertydatabases. AP databases, etc.) in a clever way with thenine quantitative principlesand thus creating a prediction-



45/49

FigUAl 42. Scheme of the Russian system

I User I+ fI Conversational processor I+ ttf LMonitor l _L " ;J

There is no There are data aboutregularity 1 the phase ,r - - i Compounds database IElements J. . . . . . , database + C0There is . . . . . _ _ _ Knowledge mregularity - - . . . . . pbase . . ut. . . ePrediction system rI Learning II subsystem

I Predicting ,--I subsystem

Figure 41. Logical structure of Savitskii's file of inorganic phases

Com

. Syst

Atomicnumbers ofoomponents

em l ~ l lNumber of Degree of Temperature Composition QuaS ibinary Refer~compounds previous of isothermal of oompounds sections numbersstudy sections Ipounds ~ ~ ~

Type of Melting Types of Temperature of Boilingmelting point crystal dissociation or pointstructure decomposition

rtiesi t iPQ4ymorphous PolymorphOus Crystal Symmetry Number of ~uperoonducting Upper criticaltransition transition system group formula transition magnetictemperature pressure per unit temperature field at 4.2 KPrope



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Intermetallic compound Intermetallic compounds arbinary. ternary. quaternary, etc., compounds containingan y chemical elements except oxygen, the halides, andnoble gases. Also excluded are such compounds wittypical inorganic 'groups' like -NH, -NH2 -N2 etcThis definition has been used for Pearson's Handbook(Villars and Calvert. 1991) and the Atlas of Daams eaJ. (1991). This definition includes therefore alssulfides, selenides, carbides, and nitrides, which mos

8.1 Glossary8. Appendix

Interface

'reservoir' of potential novel materials. Only if such along-term project on a large scale could be started soon,would it be available 10 years from now. Without suchan information-prediction system the number of unwanted duplications will increase to an unaffordable magnitude, and the probability of finding novelmaterials with highest-quality physical properties willremain very low. Unfortunately, to the knowledgeof the author, such an activity does not yet exist .despite the fact that costs of such an informationprediction system would be negligible compared to themoney spent worldwide by the different countries forunwanted duplications.

Figure 43. Scheme of the Chinese systemQuery module

Graphic display

ClassificatiOn module

Reasoning and predictingmoduleEnd

Dynamic databasemonitormodule

Knowledge base

Knowledge acquiringand processing module

Explanation module

Conversation module



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m Fractional number of bondsMN Mendeleev numberPN Periodic numberQFD Quantum formation diagramQN Quantum numberQSD Quantum structural diagramR, Covalent radius (subscript: chemical elementdesignation; superscript: radii set after e.g.Zunger, Pauling, etc.)R, Ionic radius (see Rc)s; Metallic radius (see Rc)Rnp Pseudopotential radius (see Rc)T Melting point (see Rc)V Valence-electron number (see Rc)X Electronegativity (see R ,;;)

N

surrounding atomsEnthalpy of formationEnthalpy of solutionNumber of surrounding atomsElectron density at the boundary of theWigner-Seitz atomic cellNumber

tJ{orAl-PInnws

AP Atomic propertyAPE Atomic property expressionCN Coordination numberd Distance between atomsdm in Shortest distance between central atom and

AEAETAN

a.c Unit-cell dimension(s)AJ3yCz Chemical formula (A,B,C=hemical elements,

x,y .z= stoichiometric ratio)Atomic environmentAtomic environment typeAtomic number

8.2 Abbreviations and Symbols Used

Two- (three-} environment types Two- (three-)environment types are crystal structures in which thereexists two (three) different AETs.

Single-environment types Single-environment typesare crystal structures in which each atom has thesame AET.

Coordination type Coordination types are crystalstructures w ith the same number and kind of distinctlydifferent AETs.

Atomic environment type The atomic environment types (AETs) are the distinctly differentidealized atomic environments realized in intermetalliccompounds.

Definitions for ternary compounds are more complex.

Atomic environment The atomic environment (AE) isdefined by the atoms surrounding a centra) atom. Thesurrounding atoms to the left of the maximum gap ina plot of the number of surrounding atoms (n) versusthe distance (d) normalized relative to the shortestdistance (dmin) belong to the AE (see Figure 7). In thecases where the atomic environment can be describedby a convex polyhedron, it is often also called thecoordination polyhedron.

Berthollide inte rm etallic compounds Binary berthollideintermetallic compounds" are intermetallic compoundsthat allow deviations from stoichiometry, which meansthat at least one point set of such a crystal structure hasto be occupied by two or more chemical elements, orin some exceptional cases at least one point set is notfully occupied, and the degree of occupancy varies.

Daltonide intermetallic compounds Binary daltonideintermetaUic compounds" are intermetalIic compoundsthat occur only at a fixed stoichiometry, which meansthat each point set(s) of such a crystal structure can onlybe occupied by one chemical element.

Crystal structure (also ca lle d structure type or prototype)Based on the space-group theory, a crystal structure iscompletely determined by the following data: Chemical formula Crystal system and unit-cell dimension(s) Space group Occupation number and coordinates of the occupiedpoint setsCrystal structures are named by the first intermetalliccompound found to be unique in respect to the third andfourth items and are represented by the Pearson symbolfollowed by the formula of the prototype, e.g. hP3 AlB2The first two letters of the Pearson symbol are identicalto the Bravais type, and the digits give the number ofatoms per unit cell,


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Crystal Structures of Interrnetallic Compounds9. References

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This chapter was originally published in 1995as Chapter 11 in IntermetallicCompounds, Vol. 1:Principles, edited by 1. H. Westbrook and R. L.Fleischer.

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Villars, P., Mathis, K., and Hulliger , F. (1989). In TheStructures of Binary Compounds (Cohesion and Structure,


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