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X?rayAbsorptionFineStructureDebye?WallerFactors?byAnnaV? PoiarkovaA dissertationsubmitted in partial ful?llmentofthe requirements forthe degree ofDo ctorofPhilosophyUniversity of Washington

1999Approved by

?Chairp ersonof Sup ervisory Committee?

ProgramAuthorizedtoO?er Degree

Date

In presenting this dissertation in partial ful?llment of the requirements for the Do c?toraldegreeattheUniversityofWashington?IagreethattheLibraryshallmakeitscopiesfreelyavailableforinsp ection?Ifurtheragreethatextensivecopyingofthisdissertationisallowableonlyforscholarlypurp oses?consistentwith?fairuse?asprescrib edintheU?S?CopyrightLaw?Requestsforcopyingorrepro ductionofthisdissertationmayb ereferredtoUniversityMicro?lms?1490EisenhowerPlace?P?O?Box975?AnnArb or?MI48106?towhomtheauthorhasgranted?therighttorepro duceandsell?a?copiesofthemanuscriptinmicroform and?or?b?printedcopies ofthe manuscript made from microform??Signature

Date

University of WashingtonAbstractX?rayAbsorptionFineStructureDebye?WallerFactors?by AnnaV? PoiarkovaChairp ersonofSup ervisory Committee:ProfessorJohnJ? RehrDepartment of PhysicsForaccuratex?ray absorption?nestructure ?XAFS? sp ectracalculations? esp eciallyincomplexanddisorderedsystems?itiscrucialtohaveane?cientandreliablemetho dforobtainingmultiple?scatteringXAFSDebye?Wallerfactors?Traditionalphenomenologicalmo delssuchasthecorrelatedDebyeandEinsteinmo dels?oftenfailtoprovidesu?cientaccuracyinthemeansquarehalf?pathlength?uctuation??

2?Toovercomelimitationsofsuchisotropicmo delsweintro ducetwoalternativemetho dsfortheDebye?Wallerfactorcalculations:theequation?of?motionmetho dandthe recursion metho d?Theseare generalized foramultiple?scattering casefromtheir original single?scattering formulation?The equation?of?motion metho d is an e?cient lo cal metho d for calculation of themean?uctuations?

2j

inXAFS Debye?Waller factorsforageneralscatteringpathj?Givenafewlo calforceconstants?themetho dyields?

2j

viathepro jecteddensitiesof mo des or via the displacement?displacement correlation function in real time? overafewvibrationcycles?Sampleapplicationsofthemetho darepresentedforcrys?tallineCuandGe?andforseveralorganometallicmolecules?XAFSDebye?Wallerfactorsin anionoftetrachloroferrate ?I I?were calculated viathe equation?of?motionmetho dusingdynamicalmatrixobtainedfromabinitiocomputationviadensity

functionaltheorybymeansoftheDGaussprogram?TheseabinitioDebye?Wallerfactors were then used in XAFS calculations in tetramethylammonium tetrachlorofer?rate ?I I?? Debye?Waller factors were also calculated for single? and multiple scatteringpathsin a molecule of oxidizedPyrococcus furiousrubredoxin anda molecule of zinctetraimidazole basedonforce constants?tted to exp erimental vibrational sp ectra?Also?e?cientlo calrecursionmetho dispresentedfor?

2j

calculations?Insteadofcomputingentirepro jecteddensitiesofmo des?thecalculationsarebasedonadouble??function representation?Sample application ofthe metho d ispresented forCu crystal?Both metho ds have b een implemented as FORTRAN 77feffcompatiblecomputer programssigemandsigrm?Discussion on calculation of anharmonic andspherical wave corrections is presented?

2j ???forthe?rstshellofCucalculatedusingtheEMmetho d withN? 459andk1

? 27?9N?m ?solid??incomparisonwith the CD?long dashes?andCE?shortdashes?mo dels???????112?1Exampleofthedisplacement?displacementauto correlationfunctionwithacuto?term?F????hQ1

???jQ1 ?0?ie

?Fouriertransform of this function de?nes pro jected VDOS ?see Eq? 2?6????142?2Mean square amplitudes?

2j for a 459?atom cluster of Cu vs temp eratureascalculatedfrom asingleforceconstant?k1

? 27?9N?m? mo del forthe ?rst shell ?EM SS?and forthe 111triangular MS path?EM 111??The CD mo del ??D

? 315 K? calculations for the ?rst shell ?CD SS? andthe111triangularMSpath?CD111?andtheCEmo del forthe?rstshell ?CE SS? are given for comparison?Points represent exp erimentalvalues of?

???forthe?rstshell?solid?andforthe111triangular MS path ?dashes? for Cu calculated via the EM metho d?b?TotalVDOS????andpro jectedVDOS?R

2?4Radialdep endenceofthecorrelationfunctionCR

calculatedforCuatdi?erenttemp eraturesusingfeffco deandCDmo delwith?D

2?11Lowfrequencypartofthe?rstshellpro jectedVDOS?R

???forthe23?parameter mo del ofZn?tetraimidazole ?N? 21??Thelines at296?277? 274?228? 206?205? 204?184and 174cm

2j

for Cu at 295 K as calculated with a single force constant?k?27?9N?m?withRM?RMcorrectedwiththe9?8factor?RMc??EM andCEmetho ds vsMS pathindex?Twoexp erimental values?2?corresp onding to the ?rst and second shell SS are given for comparison?524?1Pro jected VDOS for three distinct triangular MS ?a? and SS ?b? pathsinFeCl

?24 ?Thelinesat45?68?78?23?100?64?109?89?116?57?245?34?248?39? 281?77? and 284?02 cm ?Cl4 andFe1 ?Cl5 ?andtwoMSpaths?Fe1 ?Cl2 ?Cl3 andFe1 ?Cl2 ?Cl4 3 ?4 ?2 ?FeCl4

4?6s?DOSforthecentralFe

?2atom ?solid line? and??k? ?long dashes? fortetramethylammonium TCF?I I?ascalculatedwithfeff8xincom?parisonwith p?DOScalculated by UniChem ?shortdashes???????664?8d?DOSforthecentralFe

4?15Magnitude of the phase corrected Fourier transform ???R? ? FT?k

rm1?dat? b?s2rm2?dat?andc?s2 rm1?dat? b?s2rm2?dat?andc?s2 LISTOFTABLES2?1ValuesoftheparametersinBadger?smo delfordiatomicmolecules?HereCij is in such unitsthatkR 21
andMS? 23
for111triangularMSpathascalculatedusingEMmetho dincomparisonwithCDmo del??D 2j calculation in zinctetraimidazole?Here ?Nand

?Carepseudo?atoms?seetext??All angleb endsare scaled by corresp ondingnear?neighb ordistances??????392?4ForceconstantsusedintheVFF 3for?

2j 2j at20KcalculatedforfourcentralMSpathsofthetyp eZn k ?1??N ?1? ??N ?2? k

?2??Zninzinctetraimidazoledep endingonthenumb eroftheforceconstants???usedintheVFFmo del?HerejistheMSpathindex?Rj

itse?ectivelength??j thescatteringan?gleN ?1??Zn?N ?2?indegrees?k ?i?theforceconstantfortheb ondZn?N ?i?and?is the b ending force constant for the corresp onding?j 21

at 20 K for the weak Zn?N b ond in zinc tetraimidazoledep endingonthenumb er oftheforceconstants???usedintheVFFmo del?Here?exp

? 100?? 21
2exp 2exp and?? 100?? 21
2LP T

2LP Twith?

2exp ? ?2?5?0?2??10 ?3?A 2and? 2LP T ? 2?62?10 ?3?A

2?????43ix

2?7Valuesofselectedb ondsandanglesinzinctetraimidazoleobtainedfromAM1geometryoptimizationincomparisonwithexp erimentalvaluesgivenbyLo e?enetal??4??Allb ondsareinunitsof

2j ?10 ?3?A 2j ?10 ?3?A

2at295Kfora225?atomclusterofCuascalculatedwithasingleforceconstant?k?27?9N?m?mo delusingRM??

2RM ?andCE?? 2CE

?approximationvsMSpathindexj?Twoexp erimental values?2?corresp ondingtothe?rstandsecondshell SSaregivenforcomparison?Also?givenareEinsteinfrequencies?E

?e?ectivefrequencies?1?2 ?allinTHz??andthecorresp ondingweightfactorsw1?2 used in the study:the b ond lengths ? ?24 in units of ?10

2?mdyn

?A??St 1 ???St 2 ?? where ?St i ?is in

4?5SS and MS?

2j 2j calculationinoxidized rubredoxin?Bondstretchingforceconstantsareinunitsofmdyn? ?Aandb ondb endingare in mdyn ?A?rad 2j ACKNOWLEDGMENTSThere are many p eople whom I would like to thank for their assistance in conduct?

ingthis research andfor helping me togocheerfully throughthe 4 1?2roller?coasteryearsofgraduatescho ol?Unfortunately?Iamnotabletothankindividuallyeveryoneofthem herebutIsincerely dosoinmy heart?Iamesp ecially indebtedtomythesisadvisorJohnJ?Rehrforb eingagreatmentor?forhisstrongguidanceandsupp ort?hisvaluablelessons?andforgivingmeanopp ortunitytoworkonsuchaninteresting pro ject?I am also very grateful to Edward A? Stern for his guidance? help?andencouragement? toLowell S?Brown forb eingagreatteacher andhisvery help?fulcriticism oftheearlydraftofthisdissertation?toKeith Ho dgson?Britt HedmanandGrahamGeorge?SSRL?forprovidingmuch neededexp erimental dataandtheresearchgrant?NIHRR01209??toAlexeiAnkoudinovforhisinvaluableassistanceandforsettingaremarkable example inworkandresearch?Iwouldlike toexpressmygratitudetoHannesJonsson?DaniHaskel?BruceRavel?LukeCampb ell?andMatt Newville for very valuable discussions and assistance? and to Greg Schenter andDavidA?DixonfortheirenormoushelpwiththeDGausscalculations?Andmostofall?I wouldlike tothankmy dearesthusbandRandall?my parents?andallothermemb ers of my deeply b eloved family for their endless supp ort?care? understanding?andall precious thingsthey have taughtme in life?xii

Chapter1INTRODUCTIONMake everything as simple as p ossible? but not simpler?

Alb ert Einstein? ?1879?1955?1?1OverviewoftheproblemIn the context of temp erature dep endence of XAFS sp ectra? the theory of the single?

scattering ?SS? XAFS Debye?Waller ?DW? factors and their relation to the molecularforce ?eld ?FF? was?rst intro duced by Beni andPlatzman ?8? in 1976?To day? morethan20yearslater?XAFSDWtheoryisstilllackinggeneralabinitioformulationandapplicationoftheXAFSanalysistostudyvibrationalprop ertiesofsolidshasb eenhardlyexplored?9??InrecentyearsXAFSanalysishasb ecomeanimp ortantand widely used technique for determining lo cal microscopic structure of complex anddisorderedmaterials?Thestructuralinformationitprovidesincludesaveragenear?neighb ordistancesR?their mean square?uctuations?

2R ?andco ordinationnumb ersN R ?Thequantities? 2R

whichapp earintheXAFSDWfactorarecrucialtothesuccessofthemo derntheoryofXAFSanditsapplications?TheDWfactorac?countsforthermalandstructuraldisorderandgenerallygovernsthe?melting?oftheXAFSoscillationswithresp ecttoincreasingtemp eratureandtheirdecaywithresp ecttoincreasing photo electronenergy?In practice? the DWfactorsofthe manymultiple?scattering ?MS?terms intheXAFSsignalcansigni?cantlycomplicate theanalysis?10? 12??In anattempt toovercome these di?culties we develop ed two gen?eral metho ds for calculating the DW factors in terms of a few lo cal force constants in

2arbitrary ap erio dic systems:the equation?of?motion ?EM? metho d ?13? 14? and the re?cursion metho d ?RM? ?15??These metho ds also provide a basis for ?tting parametersofmolecular FF mo dels directly to XAFS sp ectra?Before presenting detailed mathematical formalism? it is useful to give a qualitativedescriptionoftheoriginoftheXAFSDWfactors?rst?Absorptionofanx?rayphotonbyanatominducesexcitationofasingledeepcoreelectronwhichthenundergo esaseries ofscatteringfrom thesurroundingatomsb eforereturningtotheabsorbingcenter?The courseof scattering caninvolve either asingle scattering site?i?e?the SS pro cess? or several sites? the MS pro cess?Due to thermal vibrations??ui

? ofthescatteringandabsorbingatoms? their p ositionsb ecome smeared outaroundtheequilibrium sites?Mean square ?uctuations??

2j

? in the lengths of the photo electron?sscattering paths quantitatively account for this e?ect on the XAFS amplitude via anexp onential factor? the DW factor?The diagramin Fig? 1?1summarizes the problem ofthe XAFS DWfactorcalcu?lationpresented inthe followingsections ?fordescription ofthe symb ols usedin thediagramseeSec?1?4?1??Asmentioned ab oveXAFS???k??providesvaluablestruc?turalinformation?Because oftherelation b etween?

2j andthepro jected vibrationaldensityofstates?VDOS??j ????the?uctuations? 2j canb eusedtoobtaininforma?tiononinteratomicinteractionsintheformofthelo calforceconstants?ki

?Thisdep endence op ensa p ossibility to ?t the FF parameters directly to the exp erimentalXAFSsp ectrum?Thisisthe?socalled?inverseproblem?Ontheotherhand?itisvaluable to ?rst solve the direct problem of calculating values of?

2basedon a givendynamical mo del?Aderivationoftheformula expressing?

2j via?j ???andthepro?jectedreducedmass?j

?aswellasde?nitionsofthesenewquantities?willb egivenin the next chapter?This dissertation is aimed at solving the direct problem but alsooutlines asolution forthe inverse one?Similarly totheXAFSDWfactors?therearealsoDWfactorswhichapp earinthex?raydi?raction?XRD?andM?ossbauere?ect?Herethethermalvibrational

3 1 2 nj i i+

MS path j

rui XAFS

χ ?λ σ( ) ( , )sin( ( ))/kN S

kRf k R kR k e ej j jjeff j j jR k j j= +∑- - 02 2 2 22 2 2

σj i i ii

inu u Rj2 121

4= - ??

=∑( )Ãr r Debye-Waller factors

ωρ ωωj

jj BTd k T2

2 2( ) ( )coth=∫h h

1 1 2

12μj iin

ii iiMR R j≡+( local geometric + vibrational structures dynamical model local force constants, ki

ÃRii+

projected vibrationaldensity of statesprojected reduced mass:

NR, Rσ2

Figure 1?1:Schematic overview of the problem?

4parameteranalogousto?

2j isthemean squarevibrationofanatomiindirection ?kand is equal tou 2i?k ?h??ui ?k?

2i?It can b e calculated using the same frequency domainformula as we derived for?

2j butwith pro jected VDOS replaced by totalVDOS and? j with mass ofthe atom atthe sitei?Mi

?1?2GoalsTheprimarygoaloftheconductedresearchandthedissertationwastodevelop?implement? andtestgeneralprescriptionsforMSXAFS DWfactorcalculations?Inorderto achieve this goalthe following work hasb een done:?Generalization of the originalSS EM metho d for MS case??Computer implementation of the EMmetho d??Research andanalysis ofdi?erent FF mo dels??TestingoftheEMmetho dandanalysisofthecalculatedvibrationalsp ectraforb othcrystalline anddisordered materials usingdi?erent FF mo dels??Search for anab initioco de which would allow the calculation the FF parametersfrom ?rst principles ?DGaussis one answerfor biological molecules???Abinitiocalculation ofthe FF in biological systems??XANES and XAFS analysis in organometallic comp ounds??Generalization of the originalSS RM for aMS case??Computer implementation andtesting of the RM??Calculationof anharmonic corrections?

5These goalshave b eensuccessfully achieved? andthe resultsare presented in thefollowing chapters?1?3DissertationoverviewIn the Intro duction ?Chap? 1? we provide a short summary of the formalism underlying

theMSXAFSDWfactortheoryaswellasabriefreviewoftwop opularisotropicmo dels? the correlated Einstein ?CE? and correlated Debye ?CD? mo dels? used for itscalculation?Chapters 2 and 3 describ e two alternative approaches to?

2calculations?the EM metho d and the RM? and their applications?The formalism in these chaptersisself?contained?althoughadditionalbackgroundondi?erentFFmo delsmightb euseful?Chapter4o?ersaprescriptionforabinitioMSDWfactorcalculationsinor?ganicsystemsonexampleoftetrachloroferrate?I I??Theresultsofthisexampleabinitiocalculation are then used in the XAFS analysis of tetramethylammonium tetra?chloroferrate?I I??Chapter5reviewssomeofthehigherordercorrectionstoXAFSDWfactors?Descriptions ofthe EM ?sigem?andRM ?sigrm?FORTRAN 77com?puter programs which were develop ed and used in the present study are given in theApp endix?And??nally?theconclusionsoftheconductedresearcharepresentedinChap?6?One might ?ndhelpful alist ofabbreviations used in the dissertation:?XAFS ? x?ray absorption?ne structure?EXAFS ?extended XAFS?XANES ? x?ray absorptionnear edge structure?MS ?multiple scattering?SS? single scattering

6?DW? Debye?Waller?CD?correlated Debye?CE?correlated Einstein?EM? equation?of?motion?RM ?recursion metho d?VDOS ?vibrational density ofstates???????FF ?force ?eld?VFF ?valence force ?eld?UFF ?Universal force ?eld?MM ?molecular mechanics1?4MultiplescatteringXAFSDebye?Wallerfactors1?4?1FormalismInthis workthe DWfactorexp ??Wj

?k??foragiven scattering pathoftotallength2rj isde?nedbythethermalandcon?gurationalaverageoftheoscillatorypartofthe XAFS signal? e i2k rj ??e i2k Rje ?Wj

?k???1?1?wheretheindexjcorresp ondstothejthscatteringpath?Curvedwavee?ectsontheDWfactorsareusuallynegligible andwill b eignoredhere?16??Wealsoneglectanharmonic corrections?In the weak disorder limit ?or harmonic approximation?? this

7DW factor is a Gaussian?Wj

?k? ? 2k 2? 2j ? where? 2j ?h?rj ?Rj

2iis the mean squarevariationinthee?ectiveorhalf?pathlengthRj

?hrj iapp earinginthestandardXAFS equation???k? ? Xj N j S 20 k R 2j jf e?j ?k ? Rj ?jsin?2k Rj ??j ?k??e ?2Rj ??e ?2? 2j k

2??1?2?Here the sum runs over all unique scattering pathsj?i?e?b oth single scattering ?SS?and MS paths? of degeneracyNj

?f e?j ?k ? Rj ? is the e?ective curved?wave backscatter?ing amplitude?S 20 is a many?b o dy amplitude reduction factor??j ?k? is the net phaseshift?k? ?2?E?EF

1?2isthewave numb er measured from thresholdEF

? and?isthe photo electronmean free path?Tob etterunderstandthenatureofMSDWfactorsitisusefultoexamine theirorigin?TheXAFSsp ectrum?isde?nedasthenormalized?oscillatorypartofthex?rayabsorptionco e?cient?? i?e???? ????0

???0 ? where?0

is the smo othatomic?backgroundabsorption?According to XAFS theory?can b e expressed as athermalaverage ?10???k? ? Im

Xj N j S 20 f e?j ?k ? rj k r 2j e i?2k rj ?2?c ??2rj ???1?3?where?c iscentralatomphaseshiftandrj

isadynamical variableequaltothein?stantaneouse?ectivelengthofascatteringpathj?Theradialdep endenceoff

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