Symmetry of Crystals and Molecules PDF
Preview Symmetry of Crystals and Molecules
SYMMETRY OF CRYSTALS AND MOLECULES (cid:2) (cid:3) TheorthorhombicspacegroupFddd Fd2d2d2 ;originon222,at−1/8,−1/8,−1/8from1¯. Symmetry of Crystals and Molecules MARK LADD FormerlyHeadofChemicalPhysics,UniversityofSurrey “Athingissymmetricalifthereissomethingyoucandotoitsothatafteryou havefinisheddoingit,itlooksthesameasbefore.” HermannWeyl(1983):Symmetry 3 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries (cid:2)c MarkLadd2014 Themoralrightsoftheauthorhavebeenasserted FirstEditionpublishedin2014 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2013944301 ISBN978–0–19–967088–8 Printedandboundby ClaysLtd,StIvesplc LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. Foreword The observation that stone-age people produced objects with a measure of symmetry is not as unexpected as often made out. The opposite would actu- ally be much more surprising. Totally asymmetric objects in the real world are extremely rare, if not altogether non-existent. Some objects are admit- tedly more symmetrical than others, and this rule applies without exception. Perfectlysymmetricalobjectsarethereforeasrareasthetotallyasymmetrical. The most perfectly fashioned sphere or the most exquisite crystal contains some blemish when examined in sufficient detail. The distinction drawn by chemistsbetweenchiralandachiralmoleculesismeaningless.Iconsideritall a matter of resolution; any molecule has a degree of chirality between zero andone. Thepowerfulconceptofsymmetryappliedwithsuchgoodeffectinscience isanabstractidealizationwithprecisedefinitiononlyinthesenseoflogic.Like the absolute zero of temperature or the relativistic limit to material velocity it can be described mathematically, but remains physically unreachable. The mathematicaltoolthatdescribessymmetryisknownasgrouptheory.Interms ofKlein’sfamousErlangenprogramanygeometryischaracterizedbyagroup oftransformationsunderwhichitspropositionsremainvalid.Inthissensethe Euclideangroupismorerestrictivethantheprojectivegroup,andbothofthese are less general than the continuous topological group. Topology allows any amountofstretchingorcompression,withouttearing;inthismostgeneralcase thepopularnotionofsymmetrydisappears,butremainsembeddedinthegroup structure. Toproduceamonographonatopicaswideassymmetryneedssomecour- age. In doing so Mark Ladd wisely stipulated at the outset that the book is restrictedtocrystalandmolecularstructure,forwhichheisperhapsuniquely qualified. Molecular structure in this context strictly refers to the nuclear framework as revealed in crystallographic analyses. The appropriate sym- metry group in this case is the three-dimensional discrete Euclidean group, commonlyreferredtoasspace-groupsymmetry. As a seasoned crystallographer, albeit of forgotten vintage, I do not find much new in this presentation, but I marvel at the comprehensive detail and rigour.Inthedaysofvisuallyestimatedx-rayintensitiesandBeevers–Lipson strips,crystalgeometryandspace-groupsymmetrywereexperiencedasprac- ticalnecessities.Itisnotclearwhereandhowthenewgeneration,whotwiddle a few knobs to produce a crystal structure, acquires this know-how. Working through the numerous examples could well provide the necessary hands-on experience. This is where I see this book becoming essential reading, if not prescribed for intensive study. Too many papers are being published around dubioussymmetryarguments. vi Foreword Up to a point the analysis is based on Newton’s approach that relies only onconclusionswhichcanbeformulatedmathematically—withouttheneedof hypotheses.Inoticeinparticularhowtheauthorwiselyrefrainsfromoffering physicalinterpretationsforobservedaspectssuchaschirality,opticalactivity andothersymmetryrelatedphenomena,identifiedandmanipulatedpurelyas point group variables. Unfortunately, the temptation to reinterpret molecular symmetryintermsofhybridorbitals,aconceptpopularinquantumchemistry, wasclearlyirresistible. ItiscommonchemicalpracticetovisualizeLewistypeelectronpaircova- lent interaction in terms of real orthogonal classical functions, called atomic orbitals.Thesefunctions,suchastheso-calledp ,p andp orbitalsareoften x y z confusedwithquantummechanicalwavefunctions,whichtheyresembleafter a fashion. However, these functions cannot be described by quantum num- bers,conflictwiththeexclusionprinciple,donotrepresentsurfaceharmonics, which are complex and hence cannot be used to describe electronic orbital angularmomentumnorspin.Atomicorbitalsareroutinelyusedinlinearcom- bination, in the same way as Euclidean unit vectors, to visualize directed covalent interaction, inferred from crystallographic analyses, to derive point groupsymmetryofmolecularspecies.Theseconceptsare,admittedly,noteas- ily avoided, being so thoroughly entrenched in chemical group theory, but a wordofwarningisperhapsnotmisplaced.Fromthesymmetrypointofview, real p and p functions, which represent linear vibrations, can never be con- x y sideredequivalent totherotationdefined byacomplex function. Itisnotthe pointgroupdescriptionofmoleculesthatisatissue,butthewayitisusedto representhybridorbitals.Beingoffundamental importance, itcouldserveas anencouragementforthereadertomakeanindependentassessment,oneway ortheother. ProfessorJanBoeyens UniversityofPretoria Preface The study of crystal and molecular symmetry is often considered difficult by thosemeetingitforthefirsttime.Ibelievethattherearetwoparticularreasons forthisproblem.First,crystalsandmolecules,unlikethemorefamiliarplane figures,arethree-dimensionalbodiesanditcanbedifficulttotakeinsimultan- eously all parts of such a body and see them in relation to the whole. In two dimensional figures, such as a rectangle or a regular hexagon, it can be done readilyandweappreciatethesymmetrywithoutdifficulty.Thesecondreason, not unrelated to the first, lies in the unfamiliarity of three-dimensional rela- tionships, notwithstanding we encounter daily the three-dimensional spatial aspectsofourspace-timecontinuum. The purpose of this book is to present crystal and molecular symmetry in a straightforward and practical manner. It evolves from lecture courses given overmanyyears,atundergraduateandpostgraduatelevels,tothosepursuing crystallography, chemistry and materials science. It makes liberal use of ste- reoviewsandcomputeraidstounderstandingthetextualmaterial.Mostofthe stereoscopic illustrations were drawn with the program PLUTO, by courtesy of Dr WDS Motherwell, University of Cambridge, and aids to stereoviewing are discussed in an appendix. Each chapter concludes with a set of problems anddetailedtutorialsolutionsareprovided.Computerprogramsrelevanttothe textandtheproblemshavebeendevised,andareavailableviathepublisher’s website,<www.oup.co.uk/companion/ladd>. Itisinformativetostudyeverydayobjectsinrelationtosymmetry:aglass andabeerjug;achairandatable;atiledfloorandabrickwall;aDalmatian and a Dobermann. In this way, symmetry can become clearer—it is not only about crystals and molecules. Two notations for symmetry are in use, the Hermann–Mauguin and Schönflies notations; each has a place in crystal and molecularsymmetry,andeachisdiscussedindetail. The study of the symmetry of crystals and molecules may be begun either with lattices or with a description of the external symmetry of crystals. The formerhasadvantagesinashortlecturecourseaimedatgettingtogripswith X-ray diffraction. However, a morphological approach has been chosen for this book because, in this way, the book should have a more general appeal anditspacealittleslower.Timeisneededinordertoassimilatetheconcepts of symmetry and to consolidate them into a working knowledge of the sub- ject. It is hoped that this book will be helpful to all those meeting symmetry forthefirsttime,whatevertheirspecialization,andwillpreparethereaderfor studyofthecurrentdefinitivetextsonsymmetry,the Internationaltablesfor crystallography,VolumesAandA1. Pre-knowledgeisconfinedmainlytomathematicsandphysicalsciences,but no more than would be studied at A-level. A little more mathematics, along viii Preface lines indicated, is an advantage in certain contexts, and appropriate guidance isprovidedinappendicesinordernottointerrupttheflowofthetext. Themainnotationusedinthebookisindicatedinthefrontmatter.Inaddi- tion, equations are indicated in the form Eq. (1.2) wherever they occur, or as Eq.(A1.2)ifinAppendixA1;forcompactness,particularlyinmatrices,aneg- ativenumber,–1,forexample,isfrequentlywrittenas1;symmetryelements arewrittenas2orm,thecorrespondingsymmetryoperationsoroperatorsas 2ormandthematricesrepresentingtheseoperationsas2orm. I would like to express my thanks to those authors and publishers for permission to reproduce the figures which bear an appropriate acknowledge- ment. In particular, many illustrations have been taken from four particular sources:(a)StructuredeterminationbyX-raycrystallography.4thed.Kluwer Academic/PlenumPublishing,2003;(b)Crystalstructures:latticesandsolids instereoview,1999;(c)Structureandbondinginsolidstatechemistry,1979, both Ellis Horwood Ltd, UK; (d) International tables for X-ray crystallo- graphy,Vol.1,1965;(e)Internationaltablesforcrystallography,Vol.A;and (f)SymmetryaspectsofM.C.Escher’speriodicdrawings,thelatterthreefrom the International Union of Crystallography (IUCr). Where used, the copy- right holders will be acknowledged as ‘Reproduced by courtesy of Springer Science+BusinessMedia,NewYork(cid:2)c KluwerAcademic/PlenumPublishing’ for(a),‘ReproducedbycourtesyofWoodheadPublishing,UK’for(b)and(c), and‘ReproducedbycourtesyofIUCr’,followedby(1),(A)or(E),for(d),(e) and(f),asappropriate. It is my great pleasure to acknowledge Professor Mike Glazer of the ClarendonLaboratory,UniversityofOxford,forbothreadingthisbookinits draftstageandofferingvaluablesuggestions.MythanksalsogotoProfessor Jan Boeyens of the University of Pretoria for writing the Foreword and for timely comments. I am indebted to Sönke Adlung, Senior Commissioning Editor,PhysicalSciences,OxfordUniversityPressforencouragingmetopro- ceedwiththisbook,thepublicationofwhichcoincideswiththeInternational Year of Crystallography, celebrating 100 years of X-ray crystallography, to Jessica White, Assistant Commissioning Editor, Physical Sciences and to Erin Pearson, Production Editor, Law and Academic Science & Medicine, Oxford University Press, for advice and technical assistance at all stages of thepreparationofthebook. MarkLadd Bramshott,2014 Disclaimer Every effort has been made to ensure the correct functioning of the software associated with this book. However, the reader planning to use the software should note that, from a legal point of view, there is no warranty, expressed or implied, that the programs are free from error or will prove suitable for a particular application. By using the software the reader accepts full respons- ibility for all the results produced, and the author and publisher disclaim all liabilityfromanyconsequencesarisingfromtheuseofthesoftware.Thesoft- ware should not be relied upon for solving a problem, the incorrect solution ofwhichcouldleadtoinjurytoapersonorlossofproperty.Ifyoudousethe programs in such a manner, it is at your own risk. The author and publisher disclaim all liability for direct or consequential damages resulting from your useoftheprograms.
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