determination

BrukerAXS

ThePairDistributionFunction(PDF)canbedetermined

fromlaboratoryorsynchrotronX-rayorneutron

isthekeytoinvestigateshort,

hod

canbeappliedtocrystallineandamorphoussolid

stallinematerials,

PDFisapowerfultooltodeterminethedeviationofthe

shortrangeorderfromtheaveragecrystalstructure.

Theexamplesofnano-anataandC

60

measured

withAg-radiationshowthegreatpotentialofthe

D8ADVANCEforsuchinvestigations.

WhatisaPDF?

ThePDFanalysisisaprincipallywellunderstoodmethod

thatwasalreadydescribedintheclassical1969textbook

ofWarrenonX-raydiffraction(AddisonWesley;New-York,

Dover).Itdescribesaprobabilityoffindinganytwoatoms

atgiveninter-atomicdistancesr(Fig.1).Theprimary

applicationofPDFwasthestudyofmaterialsthatdonot

showalong-rangelatticeperiodicity,sorglass.

TheirPDFsshowbroadfeaturesanddonotextendoverthe

short-rangeofthefirstfewcoordinationspheres.

APDFfromcrystallinematerialismuchmorestructured

andgivesinformationonamedium-rangeofinter-atomic

complimentarytotheRietveldmethod

(TOPAS),whichmodelstheaveragecrystalstructurefrom

theintensitiesoftheBraggreflectionsinadiffraction

cthermalvibrationsthatarealwaysprent

areconsideredintheRietveldmethodbythethermal

DeterminationofPairDistributionFunctions(PDF)

-4

-3

-2

-1

0

1

2

3

4

5

123456

G(

r

)

r[A]

012345678

r(Å)

-5

0

5

10

15

20

G

(

Å-

2

)

C60_5d_VCT

C60_1d_VCT

r

2

r

1

r

3

displacementparametersandtreatedasrandomdeviations.

Otherdeviationsfromtheidealsymmetry,-range

orderdonotcontributetotheBraggreflectionsbuttothe

diffuscattering.

G(r)

r(Å)

r

1

r

2

r

3

Figure1:-atomicdistancesr

i

caumaximainthePDFG(r).Theareabelowthepeaks

correspondtothenumberofneighbors,scaledbythe

scatteringpoweroftherespectiveatoms

r

2

r

1

r

3

HowtocalculateaPDF?

TheexperimentalPDFG(r)ortheradialdistributionfunction

4pr²G(r)aredirectlyobtainedfromthediffractiondataby

Fouriertransformingthenormalizedtotalstructurefactor

S(Q)withQ=4p(sinq)/l.S(Q)isthemeasuredintensity

correctedforbackground,Comptonandmultiplescattering,

absorption,calculation

ofS(Q)andG(r)academicsoftware(X2,RAD)is

availablefromtheinternet,.

Note:Thesoftwarelicensonlyallowusingthesoftware

cialurs

needtoobtainpermissiontouthissoftwarefromthe

,thesoftwarecomes

withoutsupportfromBrukerAXS.

TheinterpretationofexperimentalPDFsisverysimilar

turalmodeloftheatomic

agreementwiththeexperimentalPDFisobtainedby

,

aprogramforthefullprofilestructuralleastsquares

samesupportandlicenissuesasdescribedabovealso

applytoPDFfit.

Experimentalaspects

Threefactorsmainlyinfluencethequalityoftheexperimen-

talPDF:Fourierterminationerrors,resolutionandcounting

ationerrorsareminimizedbymeasuringto

imumvalueofQislimitedby

theinstrumentttingforthehighestdiffractionangle(typi-

callyupto160°2q)andthewavelengthoftheX-raysource.

Foranassumedlargestangleof160°2qthemaximumQ

valueforCu-radiationis8Å-ofMo-radiationmore

thandoublesthisvalueto17.4Å-onalextensionis

rgestheaccessible

Q-rangeto24Å-1(compareFig.2).Conquently,the

lowestterminationeffectscanbeexpectedforAg-radiation.

BrukerAXSoffersacomplete

anduniquesolutionforhigh

energyX-raypowderdiffrac-

lutionconsistsof

eitheraMoorAgaledtube,

aprimaryfocusingmonochro-

matorforMoorafocusing

GoebelmirrorforMoorAg

radiation,andtheLYNXEYE

detector.

Anexampletupofthenew

D8ADVANCEdiffractometer,

whichisperfectlysuitedfor

structurerearchisshown

temiquipped

withaprimaryfocusing

Johanssonmonochromatorfor

Mo-radiationandaLYNXEYE

detectoroptimizedforhard

X-rays,andairscatterscreens

instrumentdeliversveryhigh-

resolutiondataatoptimumflux

tosufficientlyhighQvalues.

ThisnewandinnovativeinstrumentisuniquetoBruker

inesahighfluxofhardX-raysandfastdata

collectionwiththesuperiorresolutionofthefocusing

soutstandingcapabilities

forcrystalstructuredeterminationandrefinementwith

TOPASsuchaninstrumentisalsooptimallyequippedfor

GoniometerwithprimaryfocusingJohanssonmonochromatorandLYNXEYEdetector

,itistheperfectchoiceforstructure

rearchingeneral.

Ifthescatteringpropertiesofthesampleshouldjustify

theuofAg-radiationandanapplicationshouldrequire

theuofveryhigh-QdatathecombinationofAgaled

tubeandfocusingGoebelmirrorprovidesthehighest

conceivabledataquality.

ThePDFiscalculatedfromtheBraggaswellasthediffu

ore,aPDFcandiscriminate

betweenshortrangeorder(reprentedbyfinitenon-

randomdisplacementsfromtheidealcrystalstructure)and

aticdeviations

fromtheidealstructurearefrequentlyobrvedinvery

ore,thedeterminationofPDFhas

enarenaissancewiththeprentinterestinnano-scaled

materials.

However,thebenefitofanextendedQ-rangeneedstobe

tradedofffortheunavoidablelossofintensitywhileusing

xofX-raysfromanAg-tubeislow

comparedtoMo,andthedetectorefficiencyislowerfor

rmore,thedecreaoftheX-ray

scatteringpoweroftheatomswithincreasingQfrequently

contributesnoadditionalinformationfromveryhighQdata.

ThepenetrationdepthofX-raysintothesampleislarger

forhigherenergies,orsmallerwavelengths,respectively.

Inreflectiongeometrythiscausalossofresolution

ore,theuof

transmissiongeometrytogetherwitheithercapillaryorflat-

samplestagesisrequiredforPDFexperiments.

Themeasurementtimeneededtocollectdataofsufficient

ry

basicinstrumentalt-upforPDFinvestigationsusing

Ag-radiation,capillarysample,andscintillationcounter

themeasurementtimeeasilyexceedsonedayperscan.

Suchlongmeasurementtimeclearlycallsforamodern

lineardetector–astheuniqueLYNXEXEwiththensor

optimizedforhigherenergies–thatdrasticallyreducesthe

countingtime.

Asuperiorcountingstatisticsisparticularlyimportantfor

erwiththeuof

aone-dimensionaldetector(LYNXEYE)thedataquality

mostlybenefitsfromusingtheVariableCountingTime

(VCT)standardBrukerAXS

solutionthatbecameavailableinDIFFRACpluswithversion

,VCTisalsocontained

.

-1

0

1

2

3

4

5

6

7

618

S(

Q)

Q[1/A]

CuMo

Sampleapplications

Powdersamplesofnano-anata(TiO

2

)andofC

60

werepre-

paredinlowbackgroundKapton®capillariesandmeasured

onaD8ADVANCEwithAgradiation,afocusingGoebel

mirror,samplespinnerandLYNXEYEdetector(optimizedfor

highenergyradiation)s-

urementrangewas2to160°2qinstepsof0.01°.Arotating

emptycapillarywasmeasuredwiththesameparameters

forbackgroundcorrection.

Figure2:TotalexperimentalstructurefactorS(Q)ofnano-

pectivecut-

offforCu-andMo-radiationfor2qmax120°isindicated.

-4

-3

-2

-1

0

1

2

3

4

5

123456

G(

r

)

r[A]

Figure3:ExperimentalPDFof

nano-anatacalculatedfrom

S(Q)tomic

distancesindicatedintheint

werecalculatedfromtheanata

plainthe

obrvedsignalverywell.

BrukerAXSGmbH

Karlsruhe,Germany

Phone+49(721)595-2888

Fax+49(721)595-4587

info@

BrukerAXSInc.

Madison,WI,USA

Phone+1(800)234-XRAY

Phone+1(608)276-3000

Fax+1(608)276-3006

info@

Allconfigurationsandspecificationsaresubjectto

-H88-EXS031.

©dinGermany.

erimentaldata

(collectedwithdifferentmeasurementtimes,bluecurves)

aremodelled(redcurves)badontheknownstructure

ofthehigh-temperatureformofC

60

.Thedifferencecurves

(green)showagoodagreementbetweenexperimentand

theory.

Authors

KarstenKnorr,BerndHinrichn,BrukerAXS

Acknowledgements

illinge(ColumbiaUniversity

andBrookhavenNationalLaboratory)andhisco-workers

fortheanalysisoftheC

60

Ehm(StonyBrookUniversityandNSLSattheBrookhaven

NationalLaboratory)withthenano-anatadataisgratefully

acknowledged.

Nano-Anata

Figure2showsthenormalizedtotalscatteringstructure

functionS(Q)respond-

ingmaximalS(Q)rangesforCu-andMo-radiationare

Q=15Å-1theS(Q)functioncontainsonly

sonforthisismainlythedecrea

y,the

sameinformationobtainedfromtheAg-datacouldhave

beengetfromMo-dataaswell.

ThepairdistributionfunctionG(r)calculatedforamaximum

Qof15Å-1showswellresolvedcorrelations(Fig.3).The

positionsofthemaximacorrespondtotheinteratomic

distancesinanata,whiletheareabelowthesignals

isdepictedintheint,whichshowsadetailfromthePDF

togetherwithstickmarkers,reprentinglengthandrelative

frequencyofinteratomicdistancesbetweenTi-Ti(red),Ti-O

(green)andO-O(blue)atomsinanata

C

60

Buckyballs

Theexampleofaquantitativeevaluationofanexperimen-

sonlynegligiblescattering

signalfromthespecimenobrvedforQ>14A-1inthe

experimentalstructurefactorcurve(Fig.4).Therefore,this

maximumQcut-offvalueistakenforthecalculationofthe

experimentalPDF,showninFig.5.

Figure4:TotalexperimentalstructurefactorS(Q)of

fulllereneCarbonC

60

measuredwithAg-radiation.

Figure5:FitoftwoexperimentalPDFswithroom

temperaturestructuredataofBuckminsterfullereneC

60

.