Table Of ContentZ. Kristallogr. 216 (2001) 1–?? 1
(cid:13)c byR.OldenbourgVerlag,Mu¨nchen
Chemical short range order obtained from the atomic pair distribu-
2
tion function
0
0
2
n
aTh.ProffenI∗,V.PetkovII,S.J.L.BillingeII,T.VogtIII
J
3 I LosAlamosNationalLaboratory,LANSCE-12,MSH805,LosAlamos,NM87545,USA
2II DepartmentofPhysicsandAstronomy,MichiganStateUniversity,EastLansing,MI48824-1116,USA
III PhysicsDepartment,BrookhavenNationalLaboratory,Upton,NY11973,USA
]
i
cReceivedxxxxx,2001;acceptedxxxxx,2001
s
-
l
r
tAbstract. Manycrystallinematerialsshowchemicalshort information see Billinge (2001)). However, it has been un-
m
range order and relaxation of neighboring atoms. Local clear if the PDF would be sensitive enough to extract in-
.structural information can be obtained by analyzing the formation about chemical ordering in perfectly crystalline
t
aatomicpairdistributionfunction(PDF)obtainedfrompow- materials.Inapreviousstudy(Proffen,2000)wehaveused
m
derdiffraction data.Inthispaper,wepresentthesuccessful simulated data based on structures showing chemical SRO
-extractionofchemicalshortrangeorderparametersfromthe totesttheabilityofthePDFmethodtoextractSROparam-
d
nx-rayPDFofaquenchedCu3Ausample. eters.Thosetestsweresuccessfulandshowedthatinprinci-
o plesuchinformationcanbeextracted.Inthispaperwechose
c thesystemCu3Autoverifythosefindingsusing’real’data.
[ Thechemical orderinginCu3Auhasbeenstudiedforquite
1 Introduction sometime(e.g.Chenetal.(1970)).ThestructureofCu3Au
v is cubic. The system undergoes a chemical order-disorder
8 phasetransitionatT =394C◦.AboveT thetwoatomtypes
In many alloys the different atoms are not distributed ran- c c
2 arerandomlydistributed,belowT copperatomsoccupythe
domly,butshowchemicalshortrangeorder(SRO).Theonly c
4 corners of the unit cell and gold occupies the face centers.
1information that can be obtained from a routine crystallo-
Thisworkshouldbeseenasproofofprincipleandwecon-
0graphic structure refinement based on Bragg intensities is
centrateonthesimplestcasecomparingaquenchedsample
2the average occupancies of the different atomic sites. The
0chemical SRO shows up as diffuse scattering. This diffuse showingnearlyrandomorderingofCuandAuwiththefully
/ orderedcase.
tscattering, obtained from single crystal scattering experi-
a
ments, can be analyzed and SRO parameters can be ob-
m
tained.Asummaryofthismethodcanbefounde.g.inCow-
-ley(1995).However,insomecasesitmightbeimpossibleto
d Experiment
ngrowsinglecrystalsortoobtaincrystalsofsufficientquality
ofor single crystal diffuse scattering measurements. In these
Two samples were investigated: ordered and disordered
ccasesaviablealternative istheuseofpowderdiffraction as
v:we demonstrate in this paper. A recently emerging method Cu3Au . Both were obtained from the same master alloy
prepared by melting of 99.999 % Cu and Au under vac-
ito analyze the local structure based on the total scattering
X uum. The alloy was homogenized by annealing for 2 h at
pattern from crystalline powder samples is the atomic pair
r 1200 K.An ingot of the annealed master alloy was rapidly
distribution function (PDF) method. The PDF is obtained
a quencheddowntoroomtemperatureresultingindisordered
via Fourier transform from the scattering data containing
both Bragg and diffuse scattering contributions. The PDF Cu3Au . Another ingot was slowly cooled down and al-
lowedtoannealforaweekat723K.Thisallowedthemate-
gives the probability to find an atom at a distance r from
rial toreach anordered state. Both samples were then filed
another atom, in other words it gives the bond length dis-
intopowder.Thepowdersampleswerecarefullypackedbe-
tribution of the material. This approach is long known in
tweenKaptonfoilstoavoidtextureformationandsubjected
thefieldofstudyingshortrangeorderinliquidsandglasses
todiffractionexperimentsusingx-raysofenergy29.09keV
buthasrecently beenapplied tocrystalline materials show-
(λ=0.4257A˚). The measurements were carried out in sym-
ing positional disorder (e.g. Egami (1998); Proffen et al.
metric transmission geometry at the beamline X7A of the
(1999); Peterson et al. (2001); Bozˇin et al. (1999); Louca
National Synchrotron Light Source (NSLS), Brookhaven
and Egami (1999); Petkov et al. (2000), for more general
NationalLaboratory.Scattered radiationwascollected with
an intrinsic germanium detector connected to a multichan-
nel analyzer. Experimental powder diffraction patterns are
∗ Correspondenceauthor(e-mail:[email protected]) shown in Fig. 1. These data were then normalized in elec-
2 Chemicalshortrangeorderobtainedfromtheatomicpairdistributionfunction
Fig. 1. Experimental X-ray powder diffraction data for (a) ordered Fig.3. Experimental(circles)andcalculated(solidline)PDFsforthe
Cu3Au and (b) disorderedCu3Au . Peaks are labelledwith the cor- orderedCu3Ausample(a)andthedisorderedsample(b).Theresidual
respondingMillerindexes. differencesareshownbelowthedatainbothpanels.
Table1. StructuraldataoforderedanddisorderedCu3AufromPDF
refinements.IntheorderedcaseCuisat(1,0,1)andAuisat(0,0,0).
2 2
ForthedisorderedsampleCuandAuoccupybothsiteswithanoccu-
pancyof0.75and0.25,respectively.Theunitsforthelatticeparameters
areA˚ andforvaluesofhu2iA˚2.Thenumbersinparenthesesarethees-
timatedstandarddeviationonthelastdigit.TheweightedR-valueRwp
correspondstotherefinementoftheaveragestructure,Rwp(RMC)cor-
respondstotheresultoftheRMCrefinement.Formoredetailsseetext.
orderedsample disorderedsample
a 3.75162(11) 3.7585(3)
hu2i(Cu/Au) 0.01307(2)/0.00772(3) 0.00939(4)
Rwp/Rwp(RMC) 9.5/9.5 16.2/15.4
Fig.2. ExperimentalPDFsfromdisordered(circles)andordered(solid
line)Cu3Au.Thedifferenceisgivenbelowassolidline.Theestimated
experimentaluncertaintyismarkedbyhorizontalbrokenlines. (odd + even) values of (hkl). These reflections can be in-
terpreted as super-lattice peaks showing up as a result of
theordering ofcopper andgold. Thedegreeoforderinthe
tron units and converted to the structure functions S(Q)
samplecanbequantifiedbycomparingtheintensitiesofthe
(Klug and Alexander, 1974) after due correction for flux,
super-lattice and fundamental Bragg peaks (Suryanarayana
background,Comptonscattering,andabsorption.Thecorre-
andNorton,1998;Guinier,1994).However,incaseswhere
spondingatomicpairdistribution(PDF)functions,G(r),are
chemical SRO is confined to only a few nearest neighbors,
showninFig.2.Thedataforbothsampleswereterminated
at Qmax = 22.5A˚−1. All corrections and data processing oneobservesonlybroaddiffusescatteringratherthansharp
super-lattice peaks.Abroadbumpinthediffraction pattern
wasdonewiththehelpoftheprogramRAD(Petkov,1989).
at2Θ≈7omarkedbyanarrowinFig.1asuggeststhepres-
Itis clear that, as well as the diffraction patterns, the PDFs
ence of SRO diffuse scattering. We chose to use the PDF
ofthe ordered anddisordered samples aresignificantly dif-
technique to extract SRO parameters. The PDFs of the or-
ferent.
deredand disordered sample areshowninFig.2.ThePDF
peaks for both samples are in nearly the same positions,
sincecopperandgoldareoccupyingthesamecubiclattices.
Modelling andResults However,theintensityofthePDFpeaksaredifferent,since
thePDFpeakintensitiesareweightedbytherespectivescat-
Braggpeaks inthe diffraction pattern ofdisordered Cu3Au tering powers of the atoms contributing to that interatomic
can be indexed with a cubic unit cell, space group Fm3m. distance.Fig.2alsoshowsthatthedifferenceissignificantly
TheorderedsamplecanbeindexedinspacegroupPm3m. larger than the experimental uncertainty shown as dashed
TheMiller indices (hkl) are shown in Fig. 1.One observes lines.
additional reflections for the ordered sample with mixed ThemodellingofthePDFdataiscarriedoutintwosteps:
Proffenetal. 3
First the PDFs are refined based on a model of the aver- Table2. Cu3AuRMCrefinementresults.Fordetailsseetext.
age crystal structure. In a second step Cu and Au are al-
orderedsample disorderedsample
lowed to switch sites in a Reverse Monte Carlo (RMC) re-
Neighbor expected RMC expected RMC
finement (McGreevy and Pusztai, 1988) of a large model
110 -0.33 -0.33 0.00 -0.10
crystal. Chemical SRO parameters are then extracted from 22
thefinalconfigurationoftheRMCrefinement. 100 1.00 1.00 0.00 0.19
Therefinementsoftheaveragestructurewerecarriedout 111 -0.33 -0.33 0.00 -0.05
22
using the program PDFFIT (Proffen and Billinge, 1999).
110 1.00 1.00 0.00 0.13
The refined PDFs are shown in Fig. 3. The only struc-
310 -0.33 -0.33 0.00 -0.10
turalparametersrefinedarethelatticeparameterandatomic 22
displacement parameters for Cu and Au. These parameters 111 1.00 1.00 0.00 0.05
are listed in Tab. 1. Details about the refinement procedure 311 -0.33 -0.33 0.00 -0.01
2 2
can be found in Proffen and Billinge (1999) and are not 200 1.00 1.00 0.00 -0.08
discussed here. The refinement extended over a range of
2.1 < r < 15.0 A˚. It is obvious from Fig. 3 as well as
the weighted R values listed in Tab. 1 that the refinement mentismostconvenienttodescribethechemicalSROusing
corresponding to the random sample is not as good as for correlationcoefficientsc whicharedefinedas:
ij
the ordered sample. When trying to refine site occupancies
for the random sample, their values stayed at the expected c = Pij −θ2 (2)
0.75/0.25 for Cu and Au respectively within one standard ij θ(1−θ)
deviation.However,theworseRvalueaswellasthemodu-
P isthejointprobabilitythatbothsitesiandjareoccupied
latedbackgroundofthescatteringdatafortherandomsam- ij
bythesameatomtypeandθisitsoveralloccupancy.Nega-
ple (Fig. 1),suggest somedegree of chemical SRO present
tivevaluesofc correspondtosituationswherethetwosites
inthequenchedsample,referredtoasrandominthispaper. ij
i and j tend to be occupied by different atom types while
Fortheorderedsample,weobservenosuchindication.
positivevaluesindicatethatsitesiandjtendtobeoccupied
ThenextstepoftheanalysisareRMCrefinementstoal-
bythesameatomtype.Acorrelationvalueofzerodescribes
lowCuandAutoorderlocally.Acomprehensivesummary
a random distribution. The maximum negative value of c
of the method applied to crystalline materials is given in ij
for a given concentration θ is −θ/(1 − θ) (P = 0), the
Tuckeretal.(2001).Thealgorithm worksasfollows:First, ij
maximumpositivevalueis+1(P =θ).
the PDF is calculated from the chosen crystal starting con- ij
figurationandagoodness-of-fitparameterχ2iscomputed. Table2liststhecorrelation parameters forthefirsteight
neighboring shells for the ordered and disordered sample.
N (G (r )−G (r ))2 Thecolumns labelled ’expected’ give the theoretical corre-
χ2 =X e i σ2 c i (1) lationparametersforthefullyorderedorfullyrandomcase.
i=1 The columns labelled ’RMC’ list the resulting correlation
coefficients oftheRMCrefinements.Incaseoftheordered
The sum is over all measured data points r , G stands for
i e sample,theRMCrefinementwasnotabletofindanyconfig-
the experimental and G forthe calculated PDF.TheRMC
c urationdifferentfromthefullyorderedcasethatwouldlead
simulation proceeds with the selection of a random site
toanimprovementofχ2.Asaresult, theresulting correla-
withinthecrystal.Thesystemvariablesassociatedwiththis
tion parameters are identical to the expected ones. In case
site, such as occupancy or displacement, are changed by a
ofthe random sample the situation is different, we observe
randomamount,andthenthemodelPDFandthegoodness-
of-fitparameterχ2arerecalculated.Thechange∆χ2ofthe values significantly different fromzerowhichdescribes the
goodness-of-fit χ2 before and after the generated move is fully random case. The weighted R value improved from
computed. Every move which improves the fit (∆χ2 < 0) 16.2%to15.4%intheRMCrefinementofthePDFfromthe
randomsample.InspectionofTab.2showsthatthechemical
isaccepted.’Bad’movesworseningtheagreementbetween
SROfoundinthesamplerefereedtoasrandomisinfactof
the observed andcalculated PDF are accepted with aprob-
ability of P = exp(−∆χ2/2). For more details about the thesamenatureasintheorderedsample,i.e.nearestCu-Cu
andAu-Auneighborsareavoidedandsecondnearestneigh-
RMCrefinement procedure used,thereader shouldrefer to
bors are preferred. However, for neighboring shells higher
Proffen (2000) and the references therein. AllRMC refine-
thanfive,theorderisindeedrandom.Inotherwords,locally
mentswerecarriedoutusingtheprogramDISCUS(Proffen
CuandAuatomsstartedorderingbelowthetransition tem-
andNeder,1997).Weusedamodelcrystalof20x20x20unit
perature,T ,beforebeingarrestedbytherapidquench.
cells insizecontaining atotalof32000atoms.Thestarting c
structure was given by the refinement result of the average
structure (seeTab.1).Theonlyallowed RMCmovewas to
haveCuandAuchangesites.Thiswaytheoverallconcen- Conclusion
tration of Cu and Au was preserved. The refinement was
aborted after15cycles whennofurther improvementinχ2 In an earlier paper (Proffen, 2000) we have shown that
was observed. A cycle is defined as the number of RMC chemical SRO parameters can successfully be extracted
movesrequiredtovisiteveryatominthemodelcrystalonce fromsimulated datausingthePDFanalysis method.Inthis
onaverage. paper we have shown that chemical SRO parameters can
To analyze the resulting structure after the RMC refine- also successfully be extracted from real data. The results
4 Chemicalshortrangeorderobtainedfromtheatomicpairdistributionfunction
Acknowledgments. We are very grateful to Marin Gospodinov for
makingtheCu3Ausamples.ThisworkwasinpartsupportedbyDOE
throughGrantNo. DEFG0297ER45651.Thiswork benefittedfrom
theuseoftheNationalSynchrotronLightSource(NSLS).Thisfacility
isfundedbytheU.S.DepartmentofEnergyundercontractDE-AC02-
98CH10886.LosAlamosNationalLaboratoryisfundedbytheUSDe-
partmentofEnergyundercontractW-7405-ENG-36.
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In summary,we have demonstrated that it is possible to
extract chemical SRO parameters of crystalline materials
from the PDF obtained from powder diffraction data. This
opens the door for rapid determination of SRO parameters
asafunction ofexternal parameters suchas temperature or
pressureusingthePDFtechnique.
