Table Of ContentINTERNATIONAL UNION OF CRYSTALLOGRAPHY
BOOK SERIES
IUCr BOOK SERIES COMMITTEE
Ch.Baerlocher,Switzerland
G.Chapuis,Switzerland
P.Colman,Australia
J.R.Helliwell,UK
K.A.Kantardjieff,USA
T.Mak,China
P.Müller,USA
Y.Ohashi,Japan
A.Pietraszko,Poland
D.Viterbo(Chairman),Italy
IUCrMonographsonCrystallography
1 Accuratemolecularstructures
A.Domenicano,I.Hargittai,editors
2 P.P.EwaldandhisdynamicaltheoryofX-raydiffraction
D.W.J.Cruickshank,H.J.Juretschke,N.Kato,editors
3 Electrondiffractiontechniques,Vol.1
J.M.Cowley,editor
4 Electrondiffractiontechniques,Vol.2
J.M.Cowley,editor
5 TheRietveldmethod
R.A.Young,editor
6 Introductiontocrystallographicstatistics
U.Shmueli,G.H.Weiss
7 Crystallographicinstrumentation
L.A.Aslanov,G.V.Fetisov,J.A.K.Howard
8 Directphasingincrystallography
C.Giacovazzo
9 Theweakhydrogenbond
G.R.Desiraju,T.Steiner
10 Defectandmicrostructureanalysisbydiffraction
R.L.Snyder,J.FialaandH.J.Bunge
11 DynamicaltheoryofX-raydiffraction
A.Authier
12 Thechemicalbondininorganicchemistry
I.D.Brown
13 Structuredeterminationfrompowderdiffractiondata
W.I.F.David,K.Shankland,L.B.McCusker,Ch.Baerlocher,editors
14 Polymorphisminmolecularcrystals
J.Bernstein
15 Crystallographyofmodularmaterials
G.Ferraris,E.Makovicky,S.Merlino
16 DiffuseX-rayscatteringandmodelsofdisorder
T.R.Welberry
17 Crystallographyofthepolymethylenechain:aninquiryintothestructureofwaxes
D.L.Dorset
18 Crystallinemolecularcomplexesandcompounds:structureandprinciples
F.H.Herbstein
19 Molecularaggregation:structureanalysisandmolecularsimulationofcrystalsandliquids
A.Gavezzotti
20 Aperiodiccrystals:frommodulatedphasestoquasicrystals
T.Janssen,G.Chapuis,M.deBoissieu
21 Incommensuratecrystallography
S.vanSmaalen
22 Structuralcrystallographyofinorganicoxysalts
S.V.Krivovichev
23 Thenatureofthehydrogenbond:outlineofacomprehensivehydrogenbondtheory
G.Gilli,P.Gilli
24 Macromolecularcrystallizationandcrystalperfection
N.E.Chayen,J.R.Helliwell,E.H.Snell
25 Neutronproteincrystallography:hydrogen,protons,andhydrationinbio-macromolecules
N.Niimura,A.Podjarny
26 Intermetallics:structures,properties,andstatistics
W.Steurer,J.Dshemuchadse
IUCrTextsonCrystallography
1 Thesolidstate
A.Guinier,R.Julien
4 X-raychargedensitiesandchemicalbonding
P.Coppens
8 Crystalstructurerefinement:acrystallographer’sguidetoSHELXL
P.Müller,editor
9 Theoriesandtechniquesofcrystalstructuredetermination
U.Shmueli
10 Advancedstructuralinorganicchemistry
Wai-KeeLi,Gong-DuZhou,ThomasMak
11 Diffusescatteringanddefectstructuresimulations:acookbookusingtheprogramDISCUS
R.B.Neder,T.Proffen
13 Crystalstructureanalysis:principlesandpractice,secondedition
W.Clegg,editor
14 Crystalstructureanalysis:aprimer,thirdedition
J.P.Glusker,K.N.Trueblood
15 Fundamentalsofcrystallography,thirdedition
C.Giacovazzo,editor
16 Electroncrystallography:electronmicroscopyandelectrondiffraction
X.Zou,S.Hovmöller,P.Oleynikov
17 Symmetryincrystallography:understandingtheInternationalTables
P.G.Radaelli
18 Symmetry relationships between crystal structures: applications of crystallographic group theory
incrystalchemistry
U.Müller
19 SmallangleX-rayandneutronscatteringfromsolutionsofbiologicalmacromolecules
D.I.Svergun,M.H.J.Koch,P.A.Timmins,R.P.May
20 Phasingincrystallography:amodernperspective
C.Giacovazzo
21 Thebasicsofcrystallographyanddiffraction,fourthedition
C.Hammond
Intermetallics
Structures, Properties, and Statistics
Walter Steurer
Julia Dshemuchadse
LaboratoryofCrystallography,DepartmentofMaterials,ETHZürich,
Vladimir-Prelog-Weg5,8093Zürich,Switzerland
3
3
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Preface
It is a fascinating world, the world of intermetallics, and it is largely unexplored.
The big picture of intermetallics still has many white spots, more than regions
already painted. Imagine the sheer number of the roughly three thousand bi-
nary and eighty thousand ternary intermetallic systems, which can be formed
by the about eighty metallic chemical elements. While in some of them ten or
more intermetallic phases have been observed, in others not even a single com-
poundhasbeenfound.Sofar,onlyabouttwentythousanddifferentintermetallic
phases are known from the hundreds of thousands that may exist. It is not clear
whether this rather small number of intermetallics represents just the tip of the
iceberg or already the iceberg itself; whether the just six thousand ternary in-
termetallic phase diagrams studied so far, and which were probably selected by
chemicalintuition,werealreadythemostrelevantandinterestingonesorwhether
the remaining seventy-four thousand will be as rich in intermetallic phases. Are
theremoresurprisestobeunveiled,suchasquasicrystals,theexistenceofwhich
nobody anticipated before 1982, or high-entropy alloys, which were introduced
twentyyearslater?
Furthermore, the parameter space has been only partially explored so far. In
particular,almostnothingisknownaboutthestructuresandpropertiesofbinary
and ternary intermetallics as functions of pressure, or of pressure and tempera-
ture. In contrast, a great many of allotropes of the chemical elements have been
discoveredundernon-ambientconditions,someofthemwithveryinterestingand
unexpected properties. We also know that properties of materials can drastically
change when their dimensions are reduced down to the nanoscale. Only a few
intermetallicphaseshavebeenstudiedinthiswaysofar.
Thinkoftheexcitingphysicalpropertiesandimportanttechnologicalapplica-
tionsofintermetallics,frommagnetismtosuperconductivity;perhapsmanymore
exciting materials might be waiting to be discovered. Keep in mind that not all
metalsareintermetallicphasesandnotallintermetallicphasesaremetals.Chem-
icalbondinginintermetallicsspansthefullrangebetweenmetallic,covalent,and
ionic,andinsomecluster-basedcomplexintermetallicsallofthemmaybepresent
atthesametime,leadingtopotentiallyinterestingproperties.
Thissaid,wewanttoemphasizethatthemostchallengingprobleminwriting
thisbookwasselectingtheintermetalliccrystalstructuretypestobediscussedout
of the more than two thousand ones known so far, and identifying the examples
that are suitable for illustrating the most common structural building principles,
and interesting structure/property relationships. While we think that for the me-
tallicelementsanencyclopedicdiscussionisfeasibleandappropriate,thiswould
not be possible and meaningful for all intermetallic compounds. Our goal was
vi Preface
to demonstrate in characteristic examples the fundamental structural building
principles underlying the large variety of intermetallics. In particular, the most
commoncrystalstructuretypeswillbediscussedaswellasthosethatunderlieand
governthepropertiesofintermetallicfunctionalandstructuralmaterials,respec-
tively. Of course, this attempt, and a particular focus on complex intermetallics,
reflectthepersonalinterestsandviewsoftheauthors.
The focus of this book is clearly on the statistics, topology, and geometry of
crystal structures and crystal structure types. The extensive use of data mining
gives a fresh view on the relatively old topic of intermetallics. For the first time,
the structure discussions follow largely the frequency of structure types, which
themselves are analyzed from different points of view. This allows us to uncover
important structural relationships and to illustrate the relative simplicity of most
ofthegeneralstructuralbuildingprinciples.Italsoallowsustoshowthatalarge
varietyofactualstructurescanberelatedtoarathersmallnumberofaristotypes.
Our main concern was writing a book that is readable and beneficial in one
way or another for everyone interested in intermetallic phases—from graduate
studentstoexpertsinsolidstatechemistry/physics/materialsscience.Forthatpur-
poseweavoidedusinganenigmaticabstractterminologyfortheclassificationof
structures. Our focus on the statistical analysis of structures and structure types
shouldbeseenasanattempttodrawthebackgroundofthebigpictureofinter-
metallics, and to point to the white spots in it, which could be worthwhile to be
explored. We also want to emphasize that this book was not planned as a text-
book;itshouldratherbeareferenceandguidethroughtheincrediblyrichworld
ofintermetallicphases.
Finally, we want to add our usual caveat. Structural subunits of intermetallic
phases (atomic environment types, coordination polyhedra, clusters, structure
modules, etc., however they may be named) must not be seen as entities that
could be stable outside their respective structures. The interactions between
atoms within such subunits can, but in most cases do not, differ from the
interactionsbetweentheatomsofthesubunitsandoftheiratomicenvironments.
Thus,inmostcases,thesestructuralsubunitsshouldbejustconsideredaspurely
geometrical objects, being quite useful for the description of complex structures
and/or for the illustration of structural relationships. Furthermore, ball-and-stick
structure models of intermetallics show the distribution of atomic sites and
shortestinteratomicdistances,butusuallytheydonotindicatechemicalbondsas
isthecasefororganicmolecules.Thechoiceofaparticularcrystallographicunit
cell, which is always a parallelepiped by definition, is based on conventions and
justallowsasimple,lattice-baseddescriptionofcrystalstructures.
Zurich WalterSteurer
November2015 JuliaDshemuchadse
Contents
PartI Conceptsandstatistics
1 Introduction 3
1.1 Generalnotationandterminology 3
1.1.1 Theroleofsymmetry 5
1.1.2 Wyckoffsetsandlatticecomplexes 9
1.1.3 Structuresandstructuretypes 10
2 Factorsgoverningstructureandstabilityofintermetallics 13
2.1 Quantumchemistry 14
2.1.1 Densityfunctionaltheory 14
2.1.2 ExtendedHückelmethod 15
2.2 Electronicstructure 16
2.2.1 Electrondensity 16
2.2.2 Electronlocalization 17
2.2.3 Electronlocalizability 18
2.3 Crystalstructureinterpretation 18
2.3.1 Conceptofelectronegativity 19
2.3.2 Zintl-Klemmconcept 20
2.3.3 Electroncounting 21
2.3.4 Hume-Rotheryelectronconcentrationrule 22
2.3.5 μ -acidsand-bases 23
3
2.3.6 μ -chemicalpressure 24
2
2.3.7 Topologicalanalysis 27
2.4 Crystalstructureprediction(CSP) 29
3 Crystallographicdescriptionofcrystalstructures 31
3.1 Coordinationpolyhedra,atomicenvironmenttypes,andclusters 31
3.2 Tilings(packings)andcoverings 36
3.2.1 1Dtilings:thequasiperiodicFibonaccisequence
anditsperiodicapproximants 37
3.2.2 2DArchimedean(Kepler)tilings 40
3.2.3 2DquasiperiodicPenrosetilingsandtheGummelt
decagoncovering 41
3.2.4 Spherepackingsandpolytypism 47
3.3 Polyhedraandpackings 50
3.3.1 Platonic,Archimedean,andCatalansolids 50
3.3.2 FullerenesandFrank-Kasperpolyhedra 56
viii Contents
3.4 Complexityinintermetallics 57
3.4.1 UnaryphasesA 58
3.4.2 BinaryphasesA–B 59
3.4.3 TernaryphasesA–B–C 59
3.4.4 Multinaryphases 60
3.4.5 Definitionanddescriptionofcomplexstructures 60
3.4.6 Layers,clusters,andinterfaces 64
3.4.7 Clusterdecompositionofcomplexstructures 67
4 Higher-dimensionalapproach 68
4.1 2Ddescriptionofthe1DquasiperiodicFibonaccisequence
bythestrip-projectionmethod 69
4.2 5Ddescriptionofthequasiperiodic2DPenrosetilingby
thestrip-projectionmethod 71
4.3 6Ddescriptionofthequasiperiodic3DAmmanntilingin
thestrip-projectionmethod 73
4.4 Example: Periodic complex intermetallics resulting in
projectionfromthe4D600-cellpolytope 74
4.5 AperiodiccrystalstructuresinthenDsectionmethod 75
4.6 WhatisthephysicsunderlyingthenDapproach? 81
5 Statisticaldescriptionandstructuralcorrelations 83
5.1 TheMendeleevnumbers 83
5.2 MiningthedatabasePearson’sCrystalData(PCD) 87
5.3 Stabilitymapsandcompositiondiagramsofintermetallics 91
5.3.1 Symmetry and size distributions, fractions
ofuniquestructuretypes 91
5.3.2 Structure types with very low or very high
symmetry 101
5.3.3 Structureswithoneatomperprimitiveunitcell 106
5.3.4 Mostcommoncrystalstructuretypes 108
5.4 Stabilitymapsandcompositiondiagramsofbinaryintermetallics 109
5.4.1 A B/AB 115
2 2
5.4.2 AB 117
5.4.3 A B/AB 122
3 3
5.4.4 A B /A B 123
5 3 3 5
5.4.5 A B /A B 125
3 2 2 3
5.4.6 A B/AB 128
5 5
5.4.7 A B /A B 128
17 2 2 17
5.4.8 A B/AB 133
4 4
5.4.9 A B /A B 133
4 3 3 4
5.4.10 A B /A B 134
7 3 3 7
5.4.11 A B /A B 134
5 4 4 5
Contents ix
5.5 Stabilitymapsandcompositiondiagramsofternaryintermetallics 137
5.5.1 Stoichiometriesofternaryintermetallics 141
5.5.2 Quasicrystals 146
5.5.3 Fourlinesofternarystoichiometries 151
5.6 Statisticsofcrystalstructuretypes 155
5.6.1 Commonstoichiometriesofternaryintermetallics 157
5.6.2 Symmetryvs.composition 160
5.6.3 Statistics based on atomic distances
andenvironmenttypes 178
PartII Structuresandproperties
6 Crystalstructuresofthemetallicelements 183
6.1 Groups1and2:Alkaliandalkalineearthmetals 189
6.1.1 Group1:AlkalimetalsLi,Na,K,Rb,andCs 189
6.1.2 Group2:AlkalineearthmetalsBe,Mg,Ca,Sr,Ba,
andRa 196
6.2 Groups3to12:Transitionmetals(TM) 198
6.2.1 Group3:Sc,Y,La,AcandGroup4:Ti,Zr,Hf 198
6.2.2 Group5:V,Nb,TaandGroup6:Cr,Mo,W 202
6.2.3 Group7:Mn,Tc,ReandGroup8:Fe,Ru,Os 203
6.2.4 Group9:Co,Rh,IrandGroup10:Ni,Pd,Pt 205
6.2.5 Group11:Cu,Ag,AuandGroup12:Zn,Cd,Hg 206
6.3 Groups13to16:(Semi)metallicmaingroupelements 207
6.3.1 Group13:Al,Ga,In,TlandGroup14:Ge,Sn,Pb 207
6.3.2 Group15metallicpnictogens:Sb,Bi,andGroup
16metallicchalcogens:Po 212
6.4 Lanthanoidsandactinoids 214
6.4.1 LanthanoidsLa–Lu 217
6.4.2 ActinoidsAc–Lr 222
7 Crystalstructuresofintermetalliccompounds 227
7.1 Statisticsofstructuresderivedfromsimplespherepackings 229
7.2 Close-packedstructuresandtheirderivatives 231
7.3 cI2-Wbasedstructuresandtheirderivatives:Heuslerand
Hume-Rotheryphases 238
7.3.1 Hume-RotheryphasesandthesystemCu–Zn 240
7.3.2 Heuslerphases 243
7.4 Frank-Kasperphases:σ-,M-,P-,R-phases,andLavesphases 245
7.4.1 Frank-Kasperpolyhedraandtheirpackings 246
7.4.2 σ-,M-,P-,andR-phases 250
7.4.3 Lavesphasesandrelatedpolytypes 252
7.4.4 Complex cluster-based superstructures: the
Al–Cu–Tastructurefamily 259
