Inorganic Structural Chemistry

275 Pages·2006·11.02 MB·English

by  Ulrich M?ller

Inorganic
Structural Chemistry
Second Edition
Ulrich Mu¨ller
Philipps-Universita¨t Marburg, Germany
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 DescriptionofChemicalStructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 CoordinationNumbersandCoordinationPolyhedra . . . . . . . . . . . . . . . . . . . . . . 3
2.2 DescriptionofCrystalStructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 AtomicCoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Isotypism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1 SymmetryOperationsandSymmetryElements . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 PointGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 SpaceGroupsandSpace-GroupTypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 CrystalClassesandCrystalSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.6 AperiodicCrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.7 DisorderedCrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 PolymorphismandPhaseTransitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1 ThermodynamicStability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 KineticStability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Polymorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4 PhaseTransitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.5 PhaseDiagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 ChemicalBondingandLatticeEnergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.1 ChemicalBondingandStructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 LatticeEnergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6 TheEffectiveSizeofAtoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.1 VanderWaalsRadii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.2 AtomicRadiiinMetals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.3 CovalentRadii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.4 IonicRadii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
viii
7 IonicCompounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.1 RadiusRatios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.2 TernaryIonicCompounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.3 CompoundswithComplexIons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.4 TheRulesofPaulingandBaur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
8 MolecularStructuresI:CompoundsofMainGroupElements . . . . . . . . . . . . . . . . . 62
8.1 ValenceShellElectron-PairRepulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
8.2 StructureswithFiveValenceElectronPairs . . . . . . . . . . . . . . . . . . . . . . . . . . 71
8.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
9 MolecularStructuresII:CompoundsofTransitionMetals . . . . . . . . . . . . . . . . . . . 73
9.1 LigandFieldTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9.2 LigandFieldStabilizationEnergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
9.3 CoordinationPolyhedraforTransitionMetals . . . . . . . . . . . . . . . . . . . . . . . . . 80
9.4 Isomerism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
9.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
10 MolecularOrbitalTheoryandChemicalBondinginSolids . . . . . . . . . . . . . . . . . . . 85
10.1 MolecularOrbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
10.2 Hybridization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
10.3 TheElectronLocalizationFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10.4 BandTheory.TheLinearChainofHydrogenAtoms . . . . . . . . . . . . . . . . . . . . . 90
10.5 ThePeierlsDistortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.6 CrystalOrbitalOverlapPopulation(COOP) . . . . . . . . . . . . . . . . . . . . . . . . . . 96
10.7 BondsinTwoandThreeDimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
10.8 BondinginMetals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
10.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
11 TheElementStructuresoftheNonmetals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
11.1 HydrogenandtheHalogens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
11.2 Chalcogens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
11.3 ElementsoftheFifthMainGroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
11.4 ElementsoftheFifthandSixthMainGroupsunderPressure . . . . . . . . . . . . . . . . . 111
11.5 Carbon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
11.6 Boron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
12 Diamond-likeStructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
12.1 CubicandHexagonalDiamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
12.2 BinaryDiamond-likeCompounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
12.3 Diamond-likeCompoundsunderPressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
12.4 PolynaryDiamond-likeCompounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
12.5 WidenedDiamondLattices.SiO Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2
12.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
ix
13 PolyanionicandPolycationicCompounds.ZintlPhases . . . . . . . . . . . . . . . . . . . . . 128
13.1 TheGeneralized8 NRule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
(cid:0)
13.2 PolyanionicCompounds,ZintlPhases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
13.3 PolycationicCompounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
13.4 ClusterCompounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
13.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
14 PackingsofSpheres.MetalStructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
14.1 Closest-packingsofSpheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
14.2 Body-centeredCubicPackingofSpheres . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
14.3 OtherMetalStructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
14.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
15 TheSphere-packingPrincipleforCompounds . . . . . . . . . . . . . . . . . . . . . . . . . . 157
15.1 OrderedandDisorderedAlloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
15.2 CompoundswithClose-packedAtoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
15.3 StructuresDerivedofBody-centeredCubicPacking(CsClType) . . . . . . . . . . . . . . . 160
15.4 Hume–RotheryPhases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
15.5 LavesPhases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
15.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
16 LinkedPolyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
16.1 Vertex-sharingOctahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
16.2 Edge-sharingOctahedra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
16.3 Face-sharingOctahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
16.4 OctahedraSharingVerticesandEdges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
16.5 OctahedraSharingEdgesandFaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
16.6 LinkedTrigonalPrisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
16.7 Vertex-sharingTetrahedra.Silicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
16.8 Edge-sharingTetrahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
16.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
17 PackingsofSphereswithOccupiedInterstices . . . . . . . . . . . . . . . . . . . . . . . . . . 190
17.1 TheIntersticesinClosest-packingsofSpheres . . . . . . . . . . . . . . . . . . . . . . . . . 190
17.2 InterstitialCompounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
17.3 StructureTypeswithOccupiedOctahedralIntersticesinClosest-packingsofSpheres . . . . 195
17.4 Perovskites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
17.5 OccupationofTetrahedralIntersticesinClosest-packingsofSpheres . . . . . . . . . . . . . 206
17.6 Spinels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
17.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
18 SymmetryastheOrganizingPrincipleforCrystalStructures . . . . . . . . . . . . . . . . . 212
18.1 CrystallographicGroup–SubgroupRelations . . . . . . . . . . . . . . . . . . . . . . . . . . 212
18.2 TheSymmetryPrincipleinCrystalChemistry . . . . . . . . . . . . . . . . . . . . . . . . . 214
18.3 StructuralRelationshipsbyGroup–SubgroupRelations . . . . . . . . . . . . . . . . . . . . 215
18.4 SymmetryRelationsatPhaseTransitions.TwinnedCrystals . . . . . . . . . . . . . . . . . 221
18.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
x
19 PhysicalPropertiesofSolids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
19.1 MechanicalProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
19.2 PiezoelectricandFerroelectricProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
19.3 MagneticProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
20 Nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
21 PitfallsandLinguisticAberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
AnswerstotheProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Preface
Giventhe increasingquantityof knowledgein allareasof science,the impartingofthis
knowledgemustnecessarilyconcentrateongeneralprinciplesandlawswhiledetailsmust
be restricted to important examples. A textbook should be reasonably small, but essen-
tial aspectsof the subjectmay notbe neglected,traditionalfoundationsmustbe consid-
ered,andmoderndevelopmentsshouldbeincluded.Thisintroductorytextisanattemptto
presentinorganicstructuralchemistryinthisway.Compromisescannotbeavoided;some
sections maybe shorter,while othersmaybe longerthansome expertsin this area may
deemappropriate.
Chemistspredominantlythinkin illustrativemodels:theylike to “see” structuresand
bonds.Modernbondtheoryhaswonitsplaceinchemistry,andisgivenproperattention
in Chapter 10. However, with its extensive calculations it correspondsmore to the way
ofthinkingofphysicists.Furthermore,albeitthecomputationalresultshavebecomequite
reliable,itoftenremainsdifficulttounderstandstructuraldetails.Foreverydayuse,simple
modelssuchasthosetreatedinChapters8,9and13areusuallymoreusefultoachemist:
“Thepeasantwhowantstoharvestin hislifetimecannotwaitfortheabinitiotheoryof
weather.Chemists,likepeasants,believeinrules,butcunninglymanagetointerpretthem
asoccasiondemands”(H.G. VON SCHNERING[112]).
This book is mainly addressed to advanced students of chemistry. Basic chemical
knowledge concerning atomic structure, chemical bond theory and structural aspects is
required.Parts of the text are based on a course on inorganiccrystalchemistry by Prof.
H. Ba¨rnighausen at the University of Karlsruhe. I am grateful to him for permission to
use the manuscriptof hiscourse,for numeroussuggestions,and for hisencouragement.
FordiscussionsandsuggestionsIalsothankProf.D.Babel,Prof.K.Dehnicke,Prof.C.
Elschenbroich,Prof.D.ReinenandProf.G.Weiser.IthankProf.T.Fa¨sslerforsupplying
figuresoftheelectronlocalizationfunctionandforreviewingthecorrespondingsection.
I thank Prof. S. Schlecht for providing figures and for reviewing the chapter on nano-
structures.IthankMs.J.GregoryandMr.P.C.Westonforreviewingandcorrectingthe
Englishversionofthemanuscript.
In thissecondedition the texthasbeenrevised andnew scientific findingshavebeen
takenintoconsideration.Forexample,manyrecentlydiscoveredmodificationsoftheele-
mentshavebeenincluded,mostofwhichoccurathighpressures.Thetreatmentofsym-
metryhasbeenshiftedtothethirdchapterandtheaspectofsymmetryisgivenmoreatten-
tioninthefollowingchapters.Newsectionsdealwithquasicrystalsandothernotstrictly
crystallinesolids,withphasetransitionsandwiththeelectronlocalizationfunction.There
isanewchapteronnanostructures.Nearlyallfigureshavebeenredrawn.
UlrichMu¨ller Marburg,Germany,April2006
1
1 Introduction
Structuralchemistryorstereochemistryisthescienceofthestructuresofchemicalcom-
pounds,thelattertermbeingusedmainlywhenthestructuresofmoleculesareconcerned.
Structuralchemistrydealswiththeelucidationanddescriptionofthespatialorderofatoms
in a compound,with the explanationof the reasons that lead to this order,and with the
propertiesresultingtherefrom.Italsoincludesthesystematicorderingoftherecognized
structuretypesandthedisclosureofrelationshipsamongthem.
Structuralchemistryisanessentialpartofmodernchemistryintheoryandpractice.To
understandtheprocessestakingplaceduringachemicalreactionandtorenderitpossible
todesignexperimentsforthesynthesisofnewcompounds,aknowledgeofthestructures
ofthecompoundsinvolvedisessential. Chemicalandphysicalpropertiesofasubstance
can only be understood when its structure is known. The enormous influence that the
structure of a material has on its properties can be seen by the comparison of graphite
anddiamond:bothconsistonlyofcarbon,andyettheydifferwidelyintheirphysicaland
chemicalproperties.
Themostimportantexperimentaltaskinstructuralchemistryisthestructuredetermi-
nation.ItismainlyperformedbyX-raydiffractionfromsinglecrystals;furthermethods
include X-ray diffraction from crystalline powders and neutron diffraction from single
crystalsandpowders.Structuredeterminationistheanalyticalaspectofstructuralchem-
istry; the usual result is a static model. The elucidation of the spatial rearrangementsof
atomsduringachemicalreactionismuchlessaccessibleexperimentally.Reactionmecha-
nismsdealwiththisaspectofstructuralchemistryinthechemistryofmolecules.Topotaxy
isconcernedwithchemicalprocessesinsolids,inwhichstructuralrelationsexistbetween
theorientationofeductsandproducts.Neitherdynamicaspectsofthiskindaresubjects
ofthisbook,northeexperimentalmethodsforthepreparationofsolids,togrowcrystals
ortodeterminestructures.
Crystals are distinguished by the regular, periodic order of their components. In the
following we will focus much attention on this order. However, this should not lead to
theimpressionofaperfectorder.Realcrystalscontainnumerousfaults,theirnumberin-
creasingwithtemperature.Atomscanbemissingormisplaced,anddislocationsandother
imperfectionscanoccur.Thesefaultscanhaveanenormousinfluenceonthepropertiesof
amaterial.
InorganicStructuralChemistry,SecondEdition UlrichMu¨ller
c 2006JohnWiley&Sons,Ltd.
(cid:0)
2
2 Description of Chemical Structures
Inordertospecifythestructureofachemicalcompound,wehavetodescribethespatial
distributionoftheatomsinanadequatemanner.Thiscanbedonewiththeaidofchem-
ical nomenclature, which is well developed, at least for small molecules. However, for
solid-statestructures,thereexistsnosystematicnomenclaturewhichallowsustospecify
structural facts. One manages with the specification of structure types in the following
manner: ‘magnesium fluoride crystallizes in the rutile type’, which expresses for MgF
2
a distributionof Mg and F atomscorrespondingto that ofTi and O atomsin rutile. Ev-
ery structure type is designated by an arbitrarily chosen representative. How structural
informationcanbeexpressedinformulasistreatedinSection2.1.
Graphic representations are useful. One of these is the much used valence-bondfor-
mula,whichallowsasuccinctrepresentationofessentialstructuralaspectsofamolecule.
Moreexactandmoreillustrativeareperspective,true-to-scalefigures,inwhichtheatoms
aredrawnasballsor—ifthealwayspresentthermalvibrationsaretobeexpressed—as
ellipsoids. To achievea better view,the balls orellipsoidsare plotted ona smaller scale
than that correspondingto the effectiveatomic sizes. Covalentbondsare represented as
sticks.Thesizeofathermalellipsoidischosentorepresenttheprobabilityoffindingthe
atomaveragedovertime(usually50%probabilityoffindingthecenteroftheatomwithin
theellipsoid;cf.Fig.2.1b).Formorecomplicatedstructurestheperspectiveimagecanbe
made clearer with the aid of a stereoscopic view (cf. Fig. 7.5, p. 56). Differenttypes of
drawingscanbeusedtostressdifferentaspectsofastructure(Fig.2.1).
Quantitativespecificationsaremadewithnumericvaluesforinteratomicdistancesand
angles.Theinteratomicdistanceisdefinedasthedistancebetweenthenucleioftwoatoms
Fig.2.1
Graphic Cl Cl
representationsfor
Cl Cl Cl
amoleculeof
U U
(UCl ,alldrawn
5 2
tothe(cid:0)samescale. Cl Cl Cl
(a)Valence-bond (a) Cl Cl (b)
formula.
(b)Perspective
viewwith
ellipsoidsof
thermalmotion.
(c)Coordination
polyhedra.
(d)Emphasisofthe
spacerequirements
ofthechlorine
atoms (c) (d)
InorganicStructuralChemistry,SecondEdition UlrichMu¨ller
c 2006JohnWiley&Sons,Ltd.
(cid:0)
2.1CoordinationNumbersandCoordinationPolyhedra 3
in their mean positions (mean positions of the thermal vibration). The most common
methodtodetermineinteratomicdistancesexperimentallyisX-raydiffractionfromsingle
crystals.Othermethodsincludeneutrondiffractionfromcrystalsand,forsmallmolecules,
electrondiffractionandmicrowavespectroscopywithgaseoussamples.X-raydiffraction
determines not the positions of the atomic nuclei but the positions of the centers of the
negativechargesoftheatomicelectronshells,becauseX-raysarediffractedbytheelec-
tronsoftheatoms.However,thenegativechargecenterscoincidealmostexactlywiththe
positions of the atomic nuclei, except for covalently bonded hydrogenatoms. To locate
hydrogenatomsexactly,neutrondiffractionisalso moreappropriatethanX-raydiffrac-
tion for anotherreason:X-raysare diffractedby the large numberof electronsof heavy
atoms to a much larger extent, so that the position of H atoms in the presence of heavy
atoms can be determined only with low reliability. This is not the case for neutrons, as
theyinteractwiththeatomicnuclei.(Becauseneutronssufferincoherentscatteringfrom
HatomnucleitoalargerextentthanfromDatomnuclei,neutronscatteringisperformed
withdeuteratedcompounds.)
2.1 CoordinationNumbers andCoordinationPolyhedra
Thecoordinationnumber(c.n.)andthecoordinationpolyhedronservetocharacterizethe
immediate surroundings of an atom. The coordination number specifies the number of
coordinatedatoms; these are the closest neighboringatoms. For manycompoundsthere
are no difficulties in stating the coordination numbers for all atoms. However, it is not
alwaysclear upto whatlimit a neighboringatomis to becountedas a closestneighbor.
Forinstance,inmetallicantimonyeverySbatomhasthreeneighboringatomsatdistances
of291pmandthreeothersatdistancesof336pm,whichisonly15%more.Inthiscaseit
helpstospecifythecoordinationnumberby3+3,thefirstnumberreferringtothenumber
ofneighboringatomsattheshorterdistance.
Statingthecoordinationofanatomasasinglenumberisnotveryinformativeinmore
complicatedcases.However,specificationsofthefollowingkindcanbemade:inwhitetin
anatomhasfourneighboringatomsatadistanceof302pm,twoat318pmandfourat377
pm.Severalpropositionshavebeenmadetocalculateameanor‘effective’coordination
number(e.c.n.orECoN)byaddingallsurroundingatomswithaweightingscheme,inthat
theatomsarenotcountedasfullatoms,butasfractionalatomswithanumberbetween0
and1;thisnumberisclosertozerowhentheatomisfurtheraway.Frequentlyagapcan
be foundin thedistributionoftheinteratomicdistancesoftheneighboringatoms:if the
shortestdistancetoaneighboringatomissetequalto1,thenoftenfurtheratomsarefound
atdistancesbetween1and1.3,andafterthemfollowsagapinwhichnoatomsarefound.
According to a proposition of G. BRUNNER and D. SCHWARZENBACH an atom at the
distanceof1obtainstheweight1,thefirstatombeyondthegapobtainszeroweight,and
allintermediateatomsareincludedwithweightsthatarecalculatedfromtheirdistances
bylinearinterpolation:
e.c.n.=∑ d d d d
i g(cid:0) i g(cid:0) 1
(cid:0) (cid:1)(cid:0)(cid:0) (cid:1)
d =distancetotheclosestatom
1
d =distancetothefirstatombeyondthegap
g
d =distancetothei-thatomintheregionbetweend andd
i 1 g
Forexampleforantimony:taking3 d 291, 3 d 336andd 391pmoneob-
(cid:0) 1(cid:0) (cid:0) i(cid:0) g(cid:0)
tainse.c.n.=4.65.Themethodishoweverofnohelpwhennocleargapcanbediscerned.