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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