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

12

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

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