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