Automatic physical phasing X-ray crystallography

Phase invariants are important pieces of information about the atomic structures of crystals. There are several mathematical methods in X-ray crystallography to estimate phase invariants. The multi-wave diffraction phenomenon offers a unique opportunity of

AutomaticphysicalphasingX-raycrystallography

S´ergioL.Morelh ao,1, LuisH.Avanci,1andStefanKycia2

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arXiv:cond-mat/0409487v1 [cond-mat.mtrl-sci] 19 Sep 2004

InstitutodeF´ sica,UniversidadedeS aoPaulo,CP66318,05315-970S aoPaulo,SP,Brazil

Laborat´orioNacionaldeLuzS´ ncrotron/LNLS,CP6192,13084-971Campinas,SP,Brazil

(Dated:February2,2008)

Phaseinvariantsareimportantpiecesofinformationabouttheatomicstructuresofcrystals.ThereareseveralmathematicalmethodsinX-raycrystallographytoestimatephaseinvariants.Themulti-wavedi ractionphenomenono ersauniqueopportunityofphysicallymeasuringphaseinvariants.Inthiswork,theunderneathprincipalsfordevelopinganautomaticproceduretoextractaccuratephase-invariantvaluesaredescribed.Ageneralsystematicprocedureisdemonstrated,inpractice,byanalyzingintensitydatafromaKDPcrystal.

PACSnumbers:61.10.Nz;61.10.Dp

Keywords:X-raydi raction,semiconductors,nanomaterials

I.INTRODUCTION

InX-raycrystallography,thephasesofthedi ractedwavesareroughlyestimatedbymathematicalmethods,knowasDirectMethods1,2,foranalyzingintensitydatasetscomposedofalargenumberofre ections.Thesemethodsexploitalgebraicorprobabilisticrelationshipsamongthephasevalues.Someofsuchrelationshipsaretripletphaseinvariants;theyareinvariantfromthechoiceoforigininthecrystallattice.Experimentalpro-ceduresallowingphysicalmeasurementsofphaseinvari-antsareofgreatinterestsince,inprinciple,theycouldextendthee ciencyoftheDirectMethodstocomplexstructuressuchasproteins.Itwouldhavetobecom-paredtootherproceduresthatareactuallyusedtothesamepurposes,suchasmultipleanomalousdispersionandmultipleisomorphousreplacement3.

Physicalmeasurementsoftripletphaseinvariantsarepossiblebymeansofthree-beamdi raction(3BD)ex-periments4,5wheretheinterferenceofsimultaneouslydi ractedwavesprovideinformationonphasevalues.However,besidesallexperimentalandanalyticaldi -cultiesinvolvedinphasedeterminationfrom3BDex-periments,themostseriousandpracticallimitationofphysical-phasingcrystallography(PPC)isthereducenumberof3BDcasessuitableforphasing.Thereciprocalspaceofcomplexmoleculecrystalsarefullofre ectionswhereisolated3BDcaseshavebecomeevenmorerare;phasinggeneraln-beamcases(n>3)isnotfeasibleatthemomentduetotheoreticalde ciencies.Therefore,itisimportanttomentionthat,regardingcomplexcrystals,theusefulnessofPPCisquitelimitedwhencomparedtotheavailablephasingprocedures.Nevertheless,thereareresearchesfocusedondevelopingandoptimizingexper-imentaldatacollectionproceduresforPPC6.Ontheotherhand,the3BDexperimentso eranuniqueop-portunityforaccuratedeterminationoftripletphasein-variants,andconsequently,forstudyingcrystallinestruc-turesviameasurementsoftheseinvariants.Forexample,dependingontheachievedexperimentalaccuracy,elec-trondensityofchemicalbondingcharges7orevendis-tortionofmoleculesunderappliedelectric eldcanbe

FIG.1:Experimental(opencircles)andsimulated(solidlines) -scansofthe260/11¯2/152three-beamdi ractioninaKDPcrystaltakenatdi erentpolarizationangle,χ(right-handsideofeachscan).[001]isthereferencedirection( =0,seeinset),X-rayphotonenergyis7482eV,andfurtherexper-imentaldetailscanbefoundelsewhere8.Theintensityscaleislinear,butforvisualizationpurposestheordinatesofsomescansareshiftedfromtheiractualvalues,givenattheleft(incps).The -scanatχ=16 (grayscan)wasmistakenlycollectedattheshoulderofthe260re ection( ω=0.003 ,30%oftheFWHM=0.01 ).The exibilityofthe ttingequation,Eq.toreproducethese -scansisexploitedinFig.3(b).

investigatedbymonitoringafewtripletphases.Notethateachtripletphaseisanabsolutevaluesinceital-readyisthephasedi erencesbetweentwodi ractedX-raywaves,andnotarelativequantitysuchasobtainedinpeakpositionorintensitymeasurements.

Thisworkhasbeenmotivatedbyourdesiredofdevel-opingatLNLSasystematicandpracticalprocedurefordeterminingphaseinvariantswithgoodaccuracy.Exper-imentalandanalyticalproceduresarestilltobeimprovedtopushphasemeasurementsfromthestate-of-arttorou-

Phase invariants are important pieces of information about the atomic structures of crystals. There are several mathematical methods in X-ray crystallography to estimate phase invariants. The multi-wave diffraction phenomenon offers a unique opportunity of

tinelyandautomaticphasingprocedures;otherwiseitwillbeverydi culttonon-expertuserstotakeadvan-tagesofthenewpossibilitieso eredbymeasuringthisphysicalquantity.Datacollectionproceduresarealreadyproposed4,andundergoingimprovement8,buttheac-tualchallengeristhedataanalysisprocedure5.Here,weoutlinetheunderneathprincipalsfordevelopinganau-tomaticproceduretoextractaccuratephasevaluesfrom3BDinterferencepro les.Ageneralsystematicproce-dureisdemonstrated,inpractice,byanalyzing3BDin-tensitydatafromaKDPcrystal,andthemajorsources

oferrorsarepointedout.

II.THEORETICALBASIS

Ingeneral,the3BDintensitypro lesaredominatedbytheinterferenceoftwodi ractedwaves.Itleadstoarelativelysimpleparametricequationthatcanbeusedto tmostoftheexperimentalintensitypro lesandtoextractthephasevalues.Itisgivenby5

I( )=

+u0 u0

(1 b|f(u)|2)|DA|2+|DBC(u)|2+ξDA·DBC(u)+ξDA·DBC(u)G( u)du

(1)

whereDA=D0vAandDBC( )=D0Rf( )eiδTvBC

aretheamplitudesoftheprimaryandsecondaryelec-tricdisplacementwave eldsgeneratedbytheprimaryre ection,A,andbythedetourre ection(alsoknownasUmwegre ection)formedbytwoconsecutivere ections,BandC.Rstandsformaximumamplituderatioofthesewaves.vAandvBCarepolarizationfactorsforlinearlypolarizedincidentradiation.δTisthephasedi erencebetweenthesewaves,whichisthetripletphaseinvari-ant.Agaussianconvolution,G(u)withFWHM=wGandu0=±2.5wG,isnecessarytoaccountforthein-strumentalwidthwG.f( )=wS/[2( 0) iwS]isalinepro lefunction(FWHM=w,wS=±w)describingtheintrinsic3BDpro leasafunctionoftheazimuthalrotationangle .bandξarerelatedtoenergybalancemechanismsamongthedi ractedbeamsandcrystallineimperfections,respectively5.

Essentially,theanalyticalprobleminaccuratephasedeterminationresidesonhowtoadjustthevectorofpa-rameters,p=[w,R,ξ,b, 0,wG],withoutcompromisingtheextractedvaluesforδT.Here,asimpleandfastevo-lutionaryalgorithm(DEA)9hasbeenusedfor ttingtheexperimentalpro leswheretheimprovementsofthe t-tingsareguidedbyamean-absolutedeviationfunction,E(p).Thebasicstrategyisthento ndoutthemini-mumofE(p)asafunctionofδT,i.e.E0(δT),whilepiskeptwithinreasonablerangesofallowedvalues.TheminimumoftheE0(δT)curve, E0/ δT=0,providetheexperimentalvalueforδT.

Instrumentalbroadeninge ectsontheinterferencepro les,asillustrativelyshowninFig.2(a),canreduceaccuracywhencombinedwiththeuncertaintyoftheRparameter,whichisinfactthemajorsourceofinaccu-racy,asdemonstratedinFig.2(b).TheE0(δT)curvesinFig.2(b)isjustshowingthatitisnotpossibletoextractanaccuratevalueofE0(δT)fromasingle -scanwhenRisunknown.

Thebeststrategy,thatwecouldelaborate,forac-curatedeterminationoftripletphasesiscomposingpolarization-dependentsetsofazimuthalscans,astheoneinFig.1,andthen,searchforthevalueofRthatprovides E0/ δT=0ascloseaspossibleofasameδTvalue.Herethissearchstrategyhasbeenappliedintwosetsofazimuthalscans:asimulatedonethatisfreeofinstrumentale ectssuchasstatisticnoiseandsamplemisalignments,andanotherthatistheexperimentaldatainFig.1.TheE0(δT)curvesofthesimulated -scansforseveralvaluesofRareshowninFig.3(a)whileFig.3(b)showstherespectiveE0(δT)curvesfortheexperimentaldata.

IV.

CONCLUSIONS

III.RESULTSANDDISCUSSIONS

Thedataanalysespresentedherehavedemonstratedthatsystematicandreliablephasingproceduresarefea-sible.However,accuracycanbeimprovedbyoptimizingtheincidentX-raybeamopticsregardingenergyresolu-tionandangulardivergences,mainlyinthehorizontalplane.Agoodinstrumentalprecisionisalsorequiredaswellaslownoiseintheintensitydata.

Acknowledgments

Fig.1showssetof3BDdatacollectedatBrazilianSyn-chrotronLightLaboratory(LNLS)withthepolarimeter-likedi ractometerdescribedelsewhere8.Itiscomposedofseveral -scanstakenatdi erentpolarizationanglesχ,asindicatedinFig.1.

ThisworkhasbeensupportedbytheBrazilianSyn-chrotronLightSource(LNLS)underproposalNo.D12A

Phase invariants are important pieces of information about the atomic structures of crystals. There are several mathematical methods in X-ray crystallography to estimate phase invariants. The multi-wave diffraction phenomenon offers a unique opportunity of

FIG.2:(a)Simulatedinstrumentalbroadeninge ectson -scans.SimulationparametersusedintoEq.

(1):δT= 2.6 ,χ=32 ,andp=[0.0012 ,1.0,0.8,0.0,67.683 ,wG]wheretheinstrumentalwidthvalues,wG,areindicatedbyarrows. = 0.(b)TheoreticalaccuracyinphasemeasurementsasafunctionoftheinstrumentalwidthwG,andamplituderatioR.TheE0(δT)curveswereobtainedby ttingthepro lesin(a)withwG=0.001 (opencircles)andwG=0.006 (closedcircles).The ttingshavebeencarriedoutbytheDEAwithintheallowedranges:p=[0.0008 :0.0012 ,R,0.2:1.0,0.0, 0±0.012 ,0.001 :0.007 ]whereR=1.0(blacklines)orR=[0.6:1.4](graylines).De nitiononthe E0/ δT=0positiongivestheaccuracyonδT.

FIG.3:Absolute-meandeviationasafunctionofδT,E0(δT),obtainedfor(a)thesimulatedscansand(b)theexperimentalscansinFig.1.Allcurvesarenormalizedbyitsminimumvalueandaddtoanintegerforbettervisualization.Thecurveswithminimaequalto1,2,3,4,5,and6correspondtothosescanswithχ=8 ,12 ,16 ,20 ,24 ,and32 ,respectively.Allowedrangeisp=[0.0010 :0.0014 ,R,0.0:1.0,0.0:3v2, 0±0.012 ,0.001 :0.006 ]wheretheRvaluesorrangesareshowninthe gureforeachcase,andv2changestheupperlimitofthebrangewiththepolarizationangle;herev2=sin2χ.

-XRD1-1264,FAPESP(proc.No.02/10387-5),andCNPq(proc.No.301617/95-3and150144/03-2).

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Electronicaddress:morelhao@p.br.

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