peaksdirectspace

PartIConcepts2PartIConceptsInthisrstpartofthebook,thebasicconceptsandtoolsarepresentedforthedescriptionofquasicrystalsandtheirstructurallycloselyrelatedperiodicapproximants.
Wewillusebothd-dimensional(dD)andn-dimensional(nD)approaches,wheredisthedimensionofthephysicalspaceandnthatofthehigher-dimensionalembeddingspace(n>d).
IndDphysicalspace,quasiperiodicstructurescanbedescribedbasedontilingsorcoverings.
Bytilingwemeanagaplesspackingofnon-overlappingcopiesofanitenumberofunittiles.
Inanalogytoacrystallographiclattice,suchatilingmaybeseenasaquasilatticewithmorethanoneunitcellofgen-eralshape.
Inacovering,oneormoretypesofpartiallyoverlappingcoveringclustersfullycoveratilingorquasiperiodicpattern.
InthenDdescription,dDquasiperiodicstructuresresultfromirrationalphysical-spacecutsofap-propriateperiodicnDhypercrystalstructures.
Rationalapproximantscanbeobtainedinthesamewayaftershearinghypercrystalstructuresintothere-spectiverationalcutorientations.
InthenDapproach,otherwisehiddenstructuralcorrelationsarerevealed.
Forinstance,theformationofdiractionpatternswithBraggreectionsand5-foldsymmetry,causingsomuchcontroversyinthersttimeafterthediscov-eryofquasicrystals,1canbeeasilyexplainedinthisway.
ThenDapproachalsoclearlyidentiesaparticularkindofcorrelatedatomicjumps(phasonips)asoriginatingfromphasonmodes,whichareexcitationsalreadyknownfromthestudyofincommensuratelymodulatedstructures.
DespitethepowerandeleganceofthenDapproach,onehastokeepinmind,however,thatrealquasicrystalsare3Dobjectsandthattheirphysicalinteractionstakeplaceinthreedimensions,indeed.
WhatisaCrystalBeforewedenethetermquasicrystalweshouldclarifywhatwemeanbycrystalandnD(hyper)crystal,ingeneral.
IntheInternationalTablesforCrys-tallography,VolA,chapter8.
1Basicconcepts,2onewillndthefollowing:Crystalsareniterealobjectsinphysicalspacewhichmaybeidealizedbyinnitethree-dimensionalperiodiccrystalstructuresinpointspace.
Three-dimensionalperiodicitymeansthattherearetranslationsamongthesymmetryoperationsoftheobjectwiththetranslationvectorsspanningathree-dimensionalspace.
Extendingthisconceptofcrystalstructuretomoregeneralperiodicobjectsandton-dimensionalspace,oneobtainsthefollowingdenition:1see,e.
g.
,W.
Steurer,S.
Deloudi(2008):FascinatingQuasicrystals.
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A64,1–11,andreferencestherein.
2H.
Wondratschek:BasicConcepts.
In:InternationalTablesforCrystallography,vol.
A,KluwerAcademicPublisher,Dordrecht/Boston/London,pp.
720–740(2002)PartIConcepts3Denition:Anobjectinn-dimensionalpointspaceEniscalledann-dimensionalcrystallographicpatternor,forshort,crystalpatternifamongitssymmetryoperations(i)therearentranslations,thetranslationvectorst1,tnofwhicharelinearlyindependent,(ii)alltranslationvectors,exceptthezerovectoro,havealengthofatleastd>0.
Condition(i)guaranteesthen-dimensionalperiodicityandthusexcludessubperiodicsymmetrieslikelayergroups,rodgroupsandfriezegroups.
Condition(ii)takesintoaccountthenitesizeofatomsinactualcrystals.
Acrucialpropertyofideal,fullyorderedcrystalsofanydimensionisthattheypossesspurepointFourierspectra.
ThismeansthattheirdiractionpatternsshowBraggreections(Diracδ-peaks)only,andnostructuraldiusescattering.
Arealcrystalcanbedescribedbycomparingitwiththemodelofanidealcrystalandbyclassifyingthedeviationsfromit.
Inthefollowing,sometermsarelistedwhichareusedforthedescriptionofrealcrystalsortheiridealizedmodels:IdealcrystalThecounterparttoarealcrystal.
Innitemathematicalobjectwithanidealizedcrystalstructure;anidealcrystalcanbeorderedordisordered(disorderedidealcrystal);ifitisdisordered,itisnotperiodicanymore,however,ithasaperiodicaveragestructure.
RealcrystalThecounterparttoanidealcrystal.
Reallyexistingcrystalwhichcanbeperfectorimperfect.
PerfectcrystalCrystalinthermodynamicequilibrium,whichcanbeor-deredordisordered;theonlydefectspossiblearepointdefectssuchasthermalvacancies,impurities.
ImperfectcrystalCrystalcontainingadditionallydefectsthatarenotinthermodynamicequilibriumsuchasdislocations.
NanocrystalRealcrystalwithdimensionsonthescaleofnanometers;duetothelargesurfacearea,itsstructuremayfundamentallydierfromthatoflargercrystalswiththesamecompositionandthermalhistory.
MetacrystalCrystalconsistingofbuildingunitsotherthanatoms(ions,molecules),suchasphotonicorphononiccrystals.
WhatisaQuasicrystalOneofthetermsmissingintheabovelistisaperiodiccrystalwhichisusedashypernymforincommensuratelymodulatedstructures(IMS),compositecrys-tals(CS),andquasicrystalsQC.
AlthoughtheirstructureslackdDtransla-tionalperiodicity,theirFourierspectrashowBraggpeaksonly.
ThispropertyhasbeenusedbytheIUCrAd-interimCommissiononAperiodicCrystalstoidentifyaperiodiccrystalsbytheiressentiallydiscretediractiondiagram.
33AdinterimCommissiononAperiodicCrystals.
ActaCrystallogr.
A48,928(1992)4PartIConceptsConsequently,dDtranslationalperiodicityisnomoreseenasanecessaryconditionforcrystallinity.
Thereciprocalspacedenitionofacrystalbyitsspectralpropertiescanbemuchsimplerthantheonebasedondirectspace.
Additionally,ithastheadvantageofbeingdirectlyaccessibleexperimentallybydiractionmethods.
Howeverpragmaticthisdenitionmaybe,itisalsofuzzy.
Thetermdirac-tiondiagramreferstoanexperimentallyobtainedimage,butdoesnottakeintoaccountthattheshapesofreectionsdependonthekindofradiationused,theresolutionanddynamicrangeofthedetectoraswellasthequalityandsizeofthecrystalstudied.
Astronglyabsorbing,large,andirregularlyshapedcrystalofpoorquality,forinstance,wouldnotatallgiveanessentiallydiscretediractiondiagramevenforsimpleperiodicstructures.
Consequently,theconceptofanaperiodiccrystalhastorefertoanidealaperiodiccrystalofinnitesizeandtoitsFourierspectrumratherthantoitsdiractionimage.
Adenitionofthedierenttypesofaperiodiccrystalsingeneralandofquasicrystalsinparticularwillbegiveninchapter3.
HowdoweusethetermquasicrystalinthisbookBythetermquasicrystalwedenoterealcrystalswithdiractionpatternsshowingnon-crystallographicsymmetry.
Thisexperiment-relatedreciprocalspacedenitionofquasicrystalsmakessymmetryanalysissimpleandallowstheapplicationoftoolsthatarewellestablishedinstandardcrystalstructureanalysis.
Weclearlywanttodistinguishbetweenquasicrystals(QC)inthismeaningandtheotherkindsofaperiodiccrystalswithcrystallographicsymmetrysuchasincommensuratelymodulatedstructures(IMS)andcompositestructures(CS).
Inthemathematicalmeaningofthetermquasiperiodicity,allthreeofthemarequasiperiodicstructures,whichhavesomesimilaritiesintheirhigher-dimensionaldescription.
ThemaindierencebetweenaQCandanIMSisthatanIMScanbedescribedasmodulationofaperiodiccrystalstructure.
Ifthemodulationamplitudeapproacheszero,theperiodicbasicstructureoftheIMSisobtained.
ACS,ontheotherhand,canbedescribedas,sometimesmutuallymodulated,intergrowthofperiodicstructures.
Suchadirectone-to-onerelationshiptoperiodicstructuresisnotpossibleinthecaseofQCwithnon-crystallographicsymmetry.
Furthermore,forbothIMSandCS,theorientational(rotationalpoint)symmetrydoesnotplaceanyconstraintontheirrationallengthscalesin-volved.
ThisisdierentforQC,where,forinstance,thenumberτ=2cosπ/5isrelatedto5-foldrotationalsymmetry.
Finally,wedonotusethetermsquasicrystalandquasiperiodicstruc-turesynonymously.
QCmayhavestrictlyquasiperiodicstructureswithnon-crystallographicsymmetryinanidealizeddescription.
However,theirstructuremayalsobequasiperiodiconaverageonly;or,evenonlysomehowrelatedtoquasiperiodicity.
Strictlyquasiperiodicstructuresmustobeytheclosenesscon-ditioninthenDdescription,thismaynotbethecaseforthestructureofrealQC,whichthenwouldcorrespondtoakindoflock-instate.
PartIConcepts5StructuralComplexityUnaryphasesA:If,duetoonlyisotropicinteractions,eachatomisequallydenselysurroundedbytheotheratomsintherstcoordinationshell,densespherepackingsaretheconsequence.
Atomicenvironmenttypes(AET)caneitherbecuboctahedra,suchasinfcccF4-Al,ordisheptahedra,suchasinhcphP2-Mg,withthecoordinationnumberCN=12inbothcases.
Incaseofanisotropicinteractions(directionalbonding,magneticinteractions,dispro-portionationunderpressure,etc.
),morecomplexstructurescanformsuchascI58-MnoroC84-Cs-III.
4Anisotropicinteractions,however,canalsoleadtothegeometricallysimplestpossiblestructure,thatofcP1-Po.
5BinaryphasesA–B:InabinaryintermetalliccompoundAxBy,eachatomhastobesurroundedbyatleastsomeatomsoftheotherspeciesinordertomaximizethenumberofattractiveinteractions,otherwisethepureelementphaseswouldseparate.
Stoichiometry,atomicsizeratios,direction-alityofatomicinteractions,andtheelectronicbandstructuredeterminetherespectiveAET.
Thesemaycompriseseveralcoordinationshellsandareusu-allycalledclusters.
ThesizeoftheunitcellofanintermetalliccompoundisdeterminedbythemostecientpackingofitsconstitutingAET(clusters),whichisthatwiththelowestfreeenergy,ofcourse.
Consequently,themostecientpackingcanbequitedierentforhigh-andlow-temperaturephasesduetotheentropicalcontributionsofthermalvibrationsandchemicaldisorder.
ThecomplexityofbinaryintermetalliccompoundsrangesbetweencP2-NiAlandmC7,448-Yb2Cu9.
6TernaryphasesA-B-C:Ontheonehand,threedierentconstituentsgivemoreexibilityinoptimizinginteractions.
Ontheotherhand,particularlyinthecaseofrepulsiveinteractionsbetweentwoofthethreeatomtypes,itcangetmuchmorediculttorealizethemostecientpacking.
MoredierentAETorclustersmaybeneededtocreatetheoptimumenvironmentsofA,B,andC.
ThecomplexityofternaryintermetalliccompoundsrangesbetweenhP3-BaPtSb7andcF23,158-Al55.
4Cu5.
4Ta39.
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B65,308–317(2009)6PartIConceptsIncaseofquasicrystals,thenumberofdierentclustersinaparticularcompoundissmall,usuallyonlyoneortwo.
Quasiperiodiclong-rangeor-dermainlyoriginatesfromtheirnon-crystallographicsymmetrytogetherwiththeirabilitytooverlapinawell-denedwaywitheachother.
Thequestionis,howcomplexarequasicrystalscomparedtoperiodicin-termetallicsAretheymorecomplexthanthemostcomplexperiodiccom-pounds,suchascF23,158-Al55.
4Cu5.
4Ta39.
1,builtfrommuchmoredierentunitclustersthananyQCStructuralcomplexityisdiculttodene.
Itiscertainlynotsucienttojustcountthenumberofatomsperunitcell,whatwouldbeimpossibleforaquasicrystalanyway.
Forinstance,the192atomslocatedonthegeneralWyckopositioninacubicunitcellwithspacegroupsymmetryFm3m,canbedescribedjustbythecoordinatesofasingleatom,i.
e.
3parameters.
ForthesamenumberofatomsinatriclinicunitcellandspacegroupP1,576parameterswouldbeneeded.
Ontheotherhand,itisalsonotjustthenumberoffreeparameters.
AcubicstructurewithspacegroupsymmetryFm3mand4atomsperunitcellneedsthreeparameters,aswell,butitseemstobemuchsimpler.
Particularly,becauseitisjustthecubicclosestpacking.
OnepossibilityforindicatingthedegreeofcomplexitycouldbethenumberofdierentAETortheR-atlas.
TheR-atlasofastructureconsistsofalldierentatomiccongurationswithinacircleofradiusR.
Thismayworkforcomparing(quasi)periodicstructureswith(quasi)periodicones,butnotforcomparingperiodicwithquasiperiodicstructures.
Inthelattercase,onecouldcompare,forinstance,theR-atlasesuptoamaximumR,whichisgivenbythedimensionsoftheunitcell.
Anotherpossibilitywouldbetocomparetheinformationneededtofullydescribetheoneandtheotherstructureortogrowitinthecomputer.
Complexityisreectedinbroaddistributionfunctions(histogramms)ofatomicdistances,largenumberofdierentAETsforeachkindofatom,largenumberofindependentparametersforthedescriptionofastructure,lowsymmetry.
Complexityresultsfromunfavorablesizeratiosofatomshinderinggeometricallyoptimuminterac-tions,preferenceofcoordinations(AET,clusters)hinderingoptimumpackings(e.
g.
5-foldsymmetry),parametersthatareclosetooptimumbutnotoptimal(pseudosymmetry).