Characterization of Materials, edited by Elton N. Kaufmann.Copyright � 2012 John Wiley & Sons, Inc.

PAIRDISTRIBUTIONFUNCTIONSANALYSIS

VALERI PETKOV

Department of Physics, Central Michigan University,Mt. Pleasant, MI, USA

INTRODUCTION

The atomic-scale structure, that is, how atoms arearranged in space, is a fundamental material’s prop-erty. Pair distribution functions (PDF)s analysis is awidely used technique for characterizing the atomic-scale structure of materials of limited structural coher-ence. It was first applied on liquids and glasses (War-ren, 1934). Recently, it was extended to crystals withintrinsic disorder (Egami andBillinge, 2003) and nano-sized particles (Petkov, 2008). The technique is basedon the fact that any condensed material acts as adiffraction grating when irradiated with x-rays produc-ing a diffraction pattern that is a Fourier transform ofthe distribution of the distinct atomic pair distances inthat grating (Klug and Alexander 1974). Therefore, bycollecting an x-ray diffraction (XRD) pattern and Four-ier transforming it the distribution of the atomic pairdistances in any condensed material can be obtained.In an atomic PDF that distribution appears as asequence of peaks starting at the shortest and continu-ing up to the longest distinct atomic pair distance amaterial shows. The areas under the PDF peaks areproportional to the number of atomic pairs occurring atthe respective distances and the widths of the peaks—to the root-mean-square (rms) scatter, Uij, about thosedistances. In particular, the full width at halfmaximumof a PDFpeak equals 2Uij

ffiffiffiffiffiffiffiffiffiffiffiffiffi2 ln 2

pwhere iand jdenote the

particular atomic pair type. The atomic rms scatteramplitudes Uij (see articles X-RAY POWDER DIFFRACTION

and SINGLE-CRYSTAL X-RAY STRUCTURE DETERMINATION) maybe dynamic (e.g., thermal) or static (e.g., due to strain)in nature reflecting correlated or uncorrelated atomicmotion (Jeong et al., 1999).

Anexampleof experimental atomicPDFs foroneof themost abundant material on Earth—water, in its solidcrystalline and liquid forms, is shown in Figure 1. ThePDFpeaks below2A

�reflect interatomic distanceswithin

the water molecules and those at longer distances—between atoms from different water molecules. Watermolecules are arranged into a long-range ordered, peri-odic structure with an average hexagonal symmetry incrystalline ice (Kuhs and Lehmann, 1983). The degree ofstructural coherence in crystalline ice is high and so therespectivePDFshowsaseries ofwell-definedpeaksup tovery high interatomic distances. Molecules in liquidwater are only short-range ordered reflecting its lowdegree of structural coherence (Malenkov, 2009).Accordingly, the respective PDF peaks to distances ofabout 1nmonly. Depending on the degree of their struc-tural coherence,materialsmay showPDFsbehaving likethat for solid crystalline iceor that for liquidwater, or likesomething in between. Thus by examining the profile of

an experimental atomic PDF, the degree of structuralcoherence in a condensed material can be easily recog-nized. Moreover, as exemplified below, by analyzing thepositions andareas of the PDF’s peaks the characteristicfor a particular material distribution of distinct inter-atomic pair distances and numbers, also known asatomic coordination sphere radii (Ri) andnumbers (CNij),can be obtained over the whole length of structuralcoherence a material shows. For a given condensedmaterial, the {Ri, CNij} distribution is unique and so itcan be used as its “structural fingerprint.” Also, a PDFcan be easily computed for any model configuration ofatoms and then compared with an experimental PDF.This allows convenient testing and refining of three-dimensional structure models for materials of anydegree of structural coherence. Themodels may be peri-odic or not periodic in nature allowing crystals andnoncrystals to be considered on the same footing. Fromthe model, atomic configurations important material’sproperties, for example, the electronic band structureand conductivity type (Petkov, 2002), may be computedand so better understood.

COMPETITIVE AND RELATED TECHNIQUES

ThefirstNobel prizewasawarded toWilhelmR€ontgen forthe discovery of x-rays back in 1901. The Nobel prize in1914 went to Max von Laue for the discovery of x-raysdiffraction in crystals. Since then x-ray diffractionhas been a major scientific tool for the determinationof the structure of single crystals. The techniques ben-efits from the fact that several independent diffractionpatterns can be collected for different orientations of thesingle crystal specimen with respect to the x-ray beam(Giacovazzo, 1998). The recorded several tens to manyhundreds or even thousands sharp diffraction spots,also known as Bragg peaks, provide a firm basis for thedetermination of the atomic-scale structure of singlecrystals from simple solids to proteins (see alsoSINGLE-CRYSTAL X-RAY STRUCTURE DETERMINATION). Theatomic PDF resembles the so-called Patterson functionthat is widely used in traditional crystallography(Giacovazzo, 1998). However, while the Patterson func-tion peaks at interatomic distanceswithin theunit cell ofa crystal, the atomic PDF peaks (see Fig. 1) at all distinctinteratomic distances occurring in a material. Inter-atomic distances of different spatial orientations butsame magnitude will come as a single PDF peak sincethe atomic PDF is one dimensional, spherically, that is,all atomic-scale structure orientations averaged repre-sentation of the respective diffraction grating/material(see Fig. 2). For this reason, atomic PDFs analysis haslittle advantage to offer in single crystal structure stud-ies. Its advantages are much more obvious in studyingthe atomic-scale structure of materials that are notperfect single crystals in nature.

Typical polycrystalline materials consist of a largenumber of randomly oriented crystallites. As a result,polycrystalline materials exhibit one-dimensional dif-fraction patterns where all atomic-scale structure

PAIR DISTRIBUTION FUNCTIONS ANALYSIS 1361

orientations are randomly, that is, spherically averaged(see article X-RAY POWDER DIFFRACTION). Although thestructure orientations are averaged out in x-ray powderdiffraction patterns they still show several tens up to afew hundred sharp Bragg peaks allowing a successfulcrystal structure refinement and even determinationto be carried out in an almost routine way (Davidet al., 2002; Samy et al., 2010). Depending on theirdegree of structural coherence materials will show pow-der diffraction patterns with only sharp Bragg peaks(e.g., crystalline ice) or only broad, diffuse-type diffrac-tion features (e.g., water), or both. Powder diffractionanalysis mostly concentrates on the sharp Bragg peaksand considers them in terms of structure models basedon infinite periodic lattices (Hahn, 2002). On the otherhand, atomic PDFs analysis uses both the Bragg peaksand the diffuse-type scattering components in the dif-fraction pattern. In this way both the existing atomicorder, manifested in the Bragg-like peaks, and all struc-tural “imperfections” (e.g., strain, defects, very small/nanosize particle’s dimension) that are responsible forits limited extent, manifested in the diffuse component ofthe diffraction pattern, are reflected in the experimentalPDFs. This renders the atomic PDFs analysis much bet-ter suited to study materials where the periodicity of theatomicarrangement ispartiallybrokendue to localstruc-tural distortions (Petkov, 1999) or the atomic arrange-ment is not periodic at all (Roux et al., 2011). In thisrespect, atomic PDFs analysis goes beyond traditionalpowder x-ray diffraction analysis that yields only theperiodic features of the atomic-scale structure.

Spectroscopy techniques such as expended x-rayabsorption fine structure (EXAFS) andnuclearmagneticresonance (NMR) are also frequently used to study thelocal atomic arrangement in materials, including deter-mining of atom–atom separations and coordinationnumbers (Czichos et al., 2006; see articles NMR SPEC-

TROSCOPY IN THE SOLID STATE andXAFSSPECTROSCOPY). Thesetechniques have a better atomic species sensitivity (lessthan 1at%) than x-ray diffraction-based atomic PDFsanalysis (down to few atomic percent). Spectroscopytechniques, however, yield structural informationrelatednot to all presentbut only to theparticular atomicspecies probed. Besides, this information is limited tointeratomic distances up to 5–6A

�only.

Imaging techniques such as high-resolution trans-mission electron microscopy (TEM) can provide infor-mation about material’s structure with atomicresolution (Czichos et al., 2006; see articles SCANNING

ELECTRON MICROSCOPY and TRANSMISSION ELECTRON MICROS-

COPY). However, as any other image, TEM images are justa projection down an axis and so are not easy to beinterpreted in terms of a unique three-dimensionalatomic arrangement. The situation may change forbetter with the recent advances in TEM tomography(Midgley and Dunin-Borkowski, 2009). Nevertheless,data from present day EXAFS and TEM experiments arevery useful, in particular in providing independentconstraints for structure modeling guided by atomicPDFs (Roux et al., 2011).

PRINCIPLES OF THE METHOD

Atoms in condensed matter interact via chemical bondsthat impose preferred atom–atom separations and coor-dination numbers that do not change a lot even when amaterial appears in phase states of very different struc-tural coherence.For example,bothquartz crystals (SiO2)and silica glass (SiO2) aremadeof rigidSi–O4 tetrahedralunits that share all of their corners. The difference is thatin quartz crystals, the tetrahedra are assembled into aperiodic network of hexagonal symmetry (Page andDonnay, 1976), whereas in the glass that network iscompletely random (Zachariasen, 1932). The situationwith crystalline ice andwater is similar (Malenkov, 2009)regardless of the fact that the hydrogen–oxygen bonds init are much weaker than the Si-O covalent bonds insilicates. Given the presence of well-defined chemicalbonds and the imposed by them distinct atom–atomseparations and coordination numbers, a quantitycalled atomic pair distribution function can be definedfor any condensedmaterial. In particular, the frequentlyused reduced atomic PDF, G(r), gives the number ofatoms in a spherical shell of unit thickness at a distancer from a reference atom as follows:

GðrÞ ¼ 4pr rðrÞ�ro½ � ð1Þ

where r(r) and ro are the local and average atomic num-ber densities, respectively. As illustrated in Figure 2, theatomic PDF is a one-dimensional function that oscillates

50403020100–1.0

–0.5

0.0

0.5

1.0

1.5

2.0

50454035302520151050–1.50

–0.75

0.00

0.75

1.50

2.25

(b)

Ato

mic

PD

F G

(r)

Radial distance r (Å)

(a)

Figure 1. Experimental atomicPDFs for solid crystalline (a) andliquid water (b).

1362 X-RAY TECHNIQUES

around zero and shows positive peaks at distances sep-arating distinct pairs of atoms, that is, where the localatomicdensityexceeds theaverageone.Asdemonstratedin Figure 1, the oscillations are very pronounced andcharacteristic to the particular phase state of the studiedmaterial. Since thewavelength of x-rays is comparable tothedistancesbetweenatoms incondensedmatter, x-rayscan scatter constructively from the grating of uniformlyseparated and coordinated atoms in a condensed mate-rial.Moreover, the resulteddiffractionpatternreflects thespatial characteristics of that grating (Giacovazzo, 1998;Klug and Alexander 1974). Indeed, the PDF G(r) is theexact Fourier transform of the atomic-scale structuresensitive part of the scattered x-ray intensities, alsoknown as structure function, S(Q), that is,

GðrÞ ¼ ð2=pÞðQmax

Q¼Qmin

Q SðQÞ�1½ �sinðQrÞdQ ð2Þ

where Q is the magnitude of the wave vector (Q¼4psiny/l), 2y is the angle between the incoming andoutgoing x-rays and l is the wavelength of the x-raysused (Wagner, 1969).Note the structure function,S(Q), isrelated to only the coherent/elastic part of the scatteredx-ray intensities, Icoh.(Q), as follows:

SðQÞ ¼ 1þ Icoh:ðQÞ�X

ci fiðQÞj j2h i X

ci fiðQÞ��� ���2;

�ð3Þ

where ci and fi(Q) are the atomic concentration and x-rayscattering factor, respectively, for the atomic species of

type i. Note, other definitions of S(Q) and the atomic PDFare also known but less frequently used (Keen, 2001).Also, the integral inEquation2 is taken fromQmin toQmax

where the former is the lowest (typically of the order of0.3–0.5A

� �1) and the latter the highest wave vectors,respectively, reached in the diffraction experiment. Inother words, in this common derivation of G(r) thesmall-angle scattering intensities that appear at wavevectors approaching zero are not included. If thosewere included a modified atomic distribution function,G0(r)¼4pr[r(r)� go(r)ro] would be obtained, where go(r) isfunction reflecting large-scale density fluctuations in thematerial studied, including the shape of its constituentcrystallites/particles (Farrow and Billinge, 2009). Noexamples of atomic PDF studies that incorporatesmall-angle scattering intensities are known yet, thoughthis may change in future. In this review article, we stickto the common derivation (given in Equations 1, 2 and 3)andusageof atomicPDFs thatdonot includesmall-anglescattering information. Within this common derivation,the integral inEquation2canbe represented inadiscreteform to take into account the discrete nature of XRDexperiment, that is,

GðrkÞ ¼ ð2=pÞXNi¼1

Qi SðQiÞ�1½ �sinðQirkÞDQ ð4Þ

where N is the number of actual experimental datapoints, Qi, collected in equidistant DQ steps as usuallydone in XRD studies. Here rk is a real-space distance atwhichG(r) has been chosen to be evaluated. For practicalpurposesG(r) isalsoevaluatedatanumberof equidistantreal space distances rk¼kDr points, where k is an integernumber. Typically Dr is of the order of 0.01 to 0.05A

�.

However, following the Nyquist-Shannon samplingtheorem the step Dr is best to be set close to p/Qmax

(Thijsse, 1984; Farrow et al., 2011).Therefore, to obtain an experimental atomic PDF an

XRD data set should be collected, only the coherentpart, Icoh.(Q), of the collected intensities extracted,reduced to a structure factor S(Q) (Equation 3) and thenFourier transformed (Equations 2 and 4). For a materialcomprising n atomic species a single diffractionexperiment would yield a total atomic distribution func-tion, G(r), which is a weighted sum of n(n þ 1)/2 partialPDFs, G(rij), that is,

GðrÞ ¼Xi;j

wijGijðrÞ; ð5Þ

where wij are weighting factors (Wagner, 1969) depend-ing on the concentration and scattering power of theatomic species as follows:

wij ¼ cicjfiðQÞfjðQÞX

ci fiðQÞ��� ���2:

�ð6Þ

For practical purposeswij’s are often evaluated forQ¼0.A total PDF for a multielement material, however,

will comprise quite a few partial atomic correlationswhich may render its interpretation ambiguous.

Figure 2. Spherically averaged distribution of interatomicdistancesandnumbers inahypothetical square latticeof atoms(points). The distribution peaks at distances separating pairs ofatoms; peak areas are proportional to the number of atoms atthose distances.

PAIR DISTRIBUTION FUNCTIONS ANALYSIS 1363

Element specificity may be added by employing the so-called resonant XRD, which involves measuring twodiffraction data sets close to but below the absorptionedge of an atomic species, taking the difference betweenthese two data sets, and Fourier transforming it into aquantity called a differential atomic PDF. Similarly toEXAFS spectroscopy, the differential atomic PDF willreflect only correlations relative to the element whoseabsorption edge is probed. However, unlike EXAFS, itwill show these correlations to the longest interatomicdistances to which they extend (Petkov and Shastri,2010). As demonstrated recently, differentialatomic PDFs can be very useful in revealing very finestructural features of complex materials (Petkovet al., 2010a).

PRACTICAL ASPECTS OF THE METHOD

To ensure good quality results XRD experiments aimedat atomic PDFs analysis should be conducted payingspecial attention to the following details:

Source of X-ray Radiation: XRD data up to high wavevectors should be collected so that the respective atomicPDF is of good enough real-space resolution to reveal allimportant structural features of the material studied.High-wave vectors canbe reachedbyemployingx-raysofa shorter wavelength, that is, of higher energy. Forexample, by using synchrotron radiation x-rays ofenergy 60keV (Petkov et al., 1999) structure functionsfor (In–Ga)As semiconductors extending to Qmax¼45A� �1 were possible to be obtained (see Fig. 3). The respec-tive atomic PDFs have an excellent real-space resolu-tion, dr¼2p/Qmax¼0.14A

�, allowing to reveal the

presence of distinct Ga–As (2.44A�) and In–As (2.61A

�)

bonds in this semiconductor alloy (see Fig. 4). If x-rays ofenergy 8keV (CuKa radiation) or 22keV (AgKa) radiationwere used XRD data would have been possible to becollected toQmax values of approximately only 8A

� �1 and20A

� �1, respectively (see the broken lines inFig. 3). UsingXRD data with Qmax¼20A

� �1 in the Fourier transforma-tion of Equation 2 and 4 would not have allowed toresolve the distinct Ga-As and In-As bonds but yetallowed to reveal well the characteristic sequence ofcoordination spheres in (In–Ga)As semiconductor alloys(see Fig. 5). Using XRD data with Qmax¼8A

� �1 wouldhave produced a very low-resolution PDF where theindividual peaks are merged (see the 5–8A

�region in

Fig. 5) beyond recognition. Such low real-space resolu-tion PDF data may be very misleading, leading to erro-neous interpretation. The example emphasizes theimportance of collecting and using XRD data up to ashigh wave vectors as possible in atomic PDF studies.This can be achieved on in house equipment usingsealed x-ray tubes with a Mo or better Ag and definitelynot Cu anode, or by employing higher energy synchro-tron radiation sources.

XRDDataStatistics andCollectionTime: Regardless ofthe source of high-energy x-rays used the diffractiondata should be collected with a very good statisticalaccuracy, usually 2–3 orders of magnitude better than

that required for traditional powder XRD (e.g., Rietveldanalysis) studies. An example illustrating the impor-tance of collecting XRD of good statistical accuracy isgiven in Figure 6. An XRD pattern for 5nm CdTe quan-tum dots (QDs) collected with 104 counts per Qi datapoint yieldsastructure function that is of goodstatisticalaccuracy at low-Q values only. Due to the presence of amultiplicativeQ factor in the kernel of the Fourier trans-formation (see Equation 2) the data noise, that is, hardlyvisible at low-Q values, appears greatly amplified athigher wave vectors. Following the Fourier transforma-tion this data noise leads to pronounced high-frequencyripples throughout the respective atomic PDF. The rip-ples distort its profile and even may be falsely taken for“real” interatomic distances. An XRD pattern collectedwith 106 counts per Qi data point yields an almost noisefree structure factor and an atomic PDF revealing theatomic-scale structure of CdTe QDS in very accuratedetail (Pradhan et al., 2007). To achieve good statisticalaccuracy obviously longer than usual XRD data collec-tion time is necessary to be used. Data collection timeestimates relevant to atomic PDF studies are discussedin (Toby and Egami, 1992; Peterson et al. 2003; MullenandLevin, 2011). Also, the stepDQ (or the equivalent to itstep in the Bragg angles) with which the XRD data are

454035302520151050

–5

0

5

10

15

20

25

30

35

40

45

50

55

60

65Ag KαCu Kα

In0.83

Ga0.17

As

In0.17

Ga0.83

As

In0.33

Ga0.67

As

In0.50

Ga0.50

As

GaAs

InAs

Q[S

(Q)-

1]

Wavevector Q (Å–1)

Figure 3. Experimental structure factors for (In–Ga)As semi-conductor alloys obtained with synchrotron x-rays of energy60keV (l¼0.205A

�). Broken lines show the maximum wave

vector Q that could have been reached if CuKa (l¼1.54A�) or

AgKa (l¼0.509A�) radiation were used instead.

1364 X-RAY TECHNIQUES

collected should be small enough (e.g., ensuring at least5–7 Qi data points under a Bragg-like peak) so that nofine diffraction features are missed. In total, all this mayresult in many tens of hours of data collection timeper sample if a sealed x-ray tube source and a singlepoint (e.g., scintillation) detector are employed.Synchrotron x-rays and large area detectorsmay reducethe data collection time to seconds (Chupas et al., 2003;Lee et al., 2008a).

Experimental Setup (Q-space) Resolution: In general,structure studies on materials of limited structuralcoherence do not require experimental setups with veryhigh reciprocal (Q)-space resolution because of theinherently diffuse nature of the XRD patterns suchmaterials show. However, care should be taken that thereciprocal space resolution of the experimental set up,including the detector, is not too low either. As an exam-ple, atomic PDFs for BaZr0.1Ti0.9O3 ceramics obtainedfrom XRD data sets collected with two different types ofdetectors, an image plate (IP) detector, and a detector setof 12 single crystals (Lee et al., 2008b) are shown inFigure 7. The quite low Q-space resolution of the XRDdata collected with an area IP detector, that is, the quitelarge detector introduced broadening of the XRD peaks,leads to a very fast, unphysical decay (Qiu, 2004a) in the

respective atomic PDF. As a result, it appears completelyflat for distances above 50–60A

�that are much shorter

than the actual length of structural coherence in theseceramics. By contrast, the much higher Q-space reso-lution of the XRD data collected with the detector set of12 single crystals results in an atomic PDF showingphysical oscillations, that is, the presence of distinctatomic coordination spheres, to very high interatomicdistances allowing studying of long-range atomic order-ing effects. Therefore, to avoid unwanted loss of infor-mation in the higher region of atomic PDFs, the Q-spaceresolution of the experimental set up, in particular thatof the detector, should be adjusted accordingly.

Finite Particle’s Size Effect on Atomic PDFs: AtomicPDFs analysis is very well suited to study the atomicarrangement in nanosized particles (Gilbert et al., 2004;Petkov, 2008). The finite size and highly anisotropicshape of nanosized, in particular 1–10nm in size, par-ticles may affect the shape and intensity of the peaks inexperimental atomic PDFs substantially (Petkovet al., 2009). The effect of particle’s finite size on theatomic PDFs is somewhat similar to that of the low-Qspace resolution discussed above so care should betaken that those are not confused with each other. Thefinite particle’s size effect can be taken into account byusing appropriate particle’s shape functions (Kodama

10987654321

–5

0

5

10

15

20

25

30

35

40

45

50

55

60

65In-As

Ga-As

In0.83

Ga0.17

As

In0.17

Ga0.83

As

InAs

In0.5

Ga0.5

As

In0.33

Ga0.67

As

GaAs

Ato

mic

PD

F G

(r)

Radal distance r (Å)

Figure 4. Experimental PDFs for (In–Ga)As semiconductorsobtained by Fourier transforming the structure factors ofFigure 3. Broken lines with arrows mark the shortest Ga–As(2.44A

�) and In–As (2.61A

�) bond lengths in the alloy samples.

8765432

0

2

4

6

In-As Ga-As

(c)

(b)

(a)

In0.33

Ga0.67

As

Ato

mic

PD

F G

(r)

Radial distance r (Å)

Figure 5. Experimental PDFs for In0.33Ga0.67As alloy obtainedby Fourier transforming the respective structure factor ofFigure 3 with Qmax set to 45A

� �1 (a), 20A� �1 (b), and 8A

� �1 (c).Arrow marks the shortest Ga–As (2.44A

�) and In–As (2.61A

�)

bond lengths in this material. Those are clearly resolved onlywhen XRD data collected up to 45A

� �1 are included in theFourier transformation.

PAIR DISTRIBUTION FUNCTIONS ANALYSIS 1365

et al. 2006; Farrow et al., 2007) or by building finite size,real particle’s shape structure models (Korsunskyet al., 2007; Petkov et al., 2010b).

Background Scattering Treatment: Air, sample holderetc. background-type scattering should be kept to aminimum since atomic PDFs are related to only thecoherent/elastic part, Icoh.(Q), of the x-ray intensitiesscattered from the sample alone (see Equation 3). Aspractice has repeatedly shown it is always easier tocorrect for a weak background signal than for a strongone. Therefore, as high as possible sample to back-ground scattering ratio is recommended, especially athigh wave vectors where that ratio should at least be ofthe order of 4–5 to 1. Once reduced to a minimum, thebackground scattering should be measured with thesame statistical accuracy as the sample scattering andso used in the process of reducing the experimental XRDdata into an atomic PDF.

Sample Related “Unwanted” Scattering: X-rays areboth scattered from and absorbed inside materials viavarious processes (Klug and Alexander, 1974). Theabsorption of high-energy x-rays is relatively low and

usually does not posemuch of a problem in atomic PDFsstudies. The same is true for multiple scattering of high-energyx-rays.As illustrated inFigure8, inelastic (Comp-ton) scattering, however, may be very strong and evenexceed the elastic scattering, especially at high wavevectors (Petkov, 2002). Inelastic, that is, scattered withmodified energy x-ray photons should be eliminatedfrom the experimental XRD pattern since only its coher-ent/elastic part is related to the atomic PDF (seeEquation 3). The elimination is best to be done duringdata collection by using x-ray energy sensitive detectors(Ruland, 1964; Petkov, 1999, 2000). Alternatively, it can

14121086420

–0.2

0.0

0.2

(a)

(b)

Ato

mic

PD

F G

(r)

Radial distance r(Å)

5 nm CdTe QDs

Str

uct

ure

fact

or

Q[S

(Q)-

1]

Wave vector Q(Å–1)

302520151050

–0.05

0.00

0.05

0.10

Figure 6. Experimental structure factors for 5nm CdTe quan-tum dots (QD)s collected with 104 (full line in gray) and 106

(symbols) counts per data point (a). The respective atomic PDFsare shown in (b). That obtained from the data set of lowerstatistical accuracy suffers pronounced high-frequency,unphysical ripples.

50403020100

–2

–1

0

1

2

3

4

980960940920900

–0.1

0.0

0.1

10008006004002000–2

0

2

4

(b)

Radial distance r(Å)

Radial distance r(Å)

Ato

mic

PD

F G

(r)

Radial distance r(Å)

BaZr0.1

Ti0.9

O3

(c)

(a)

12-crystals detector data set

Image plate detector data set

Figure 7. (a) Experimental atomic PDFs for BZT ceramicsobtained with an image plate detector (line in black) and adetector set of 12 single crystals (line in dark gray). The lowQ-resolution image plate detector yields an atomic PDF thatdecays to zero at distances of 50–60A

�that are much shorter

than the average domain size in the ceramics material studied.The higher Q-resolution detector set of 12 crystals yields anatomic PDF that shows physical oscillations up to distances of1000A

�, and more, that are comparable to the average domain

size in this material. (b, c) Fragments of the experimental PDFdata as fit with a model featuring the well-known perovskitestructure of BZT (line in light gray). Experimental and modelPDF’s peaks match well to very high interatomic distancesconfirming the physical origin of the PDF’s oscillations.

1366 X-RAY TECHNIQUES

be done analytically during the reduction of the exper-imental XRD data into an atomic PDF. The analyticalelimination of Compton scattering is, however, inevita-bly based on some theoretical approximations (Hajduand Palinkas, 1972) which may lead to difficulties (Qiu,2004a) in obtaining good quality atomic PDF data.

METHOD AUTOMATION

Atomic PDFs analysis is fully automated in all of itsaspects, includingXRDdata collection,XRDdata reduc-tion into atomic PDFs, the interpretation of experimentalPDFs in terms interatomic distances and numbers, andatomic PDFs guided structure modeling rendering thetechnique a rather convenient and useful scientific tool.

XRD data of quality good enough for a successfulatomic PDFs study can be collected on in-house equip-ment for powder XRD. However, instead of the standardCu Ka radiation source, x-ray tubes with a Mo or Aganode or rotating (Mo or Ag) anode sources should beemployed so that the XRD data are collected to at least15–20A

� �1. Also, extra care shouldbe exercised to reducethe inherently high background scattering inside the

enclosures of the standard powder XRD instruments.XRD data of quality good enough for a successful PDFstudy aremuch easily collected at synchrotron radiationsourcesbecauseof themuchhigher intensity andenergyof the x-rays produced by them allowing Qmax values of30–45A

� �1 to be achieved. Several beam lines at numer-ous synchrotron radiation facilities are available all overthe world, including instruments entirely dedicated toatomic PDFs studies (Chupas et al., 2003; Leeet al. 2008a; Kohara et al., 2001).

Software for correcting experimental XRDpatterns forbackground and other “unwanted” scattering as well asfor x-rayabsorption,polarization, detectordead timeandx-ray energy resolution, normalizing the corrected inten-sities into absolute units, reducing them into structurefunctions S(Q), and finally performing a Fourier trans-formation to obtain an atomic PDF is readily available(Petkov, 1989; Qiu, 2004b; Soper and Barney, 2011).

Positions and areas of peaks in experimental atomicPDFs can be extracted with the help of any softwarepackage used for visualization, manipulation, and plot-ting of scientific data.

Relatively small-size (up to fewhundred atoms) struc-ture models based on periodic (Bravais) lattices can beconveniently built, tested, and refined against experi-mental atomicPDFswith thehelp of theprogramPDFgui(Farrow et al., 2007).

Large-scale (many thousands of atoms) structuremodels featuringpronounced local disorder canbebuilt,tested, and refined against experimental atomic PDFswith the help of the program DISCUS (Neder andProffen, 2008).

Completely, nonperiodic model atomic configura-tions featuring glasses and liquids can be tested andrefined against experimental PDF data employingreverse Monte Carlo (Gereben et al., 2007) or moleculardynamics (Lindahl et al., 2001) type procedures.

Once constructed, structure models for materials oflimited structural coherence can be analyzed in terms ofbond length, bond angle, and partial coordination num-ber distributions, topological connectivity, local symme-try, and other structural characteristics with the help ofthe program ISAACS (Roux and Petkov, 2010).

ATOMIC PDF DATA ANALYSIS AND INTERPRETATION

An experimental atomic PDF carries a wealth of atomic-scale structure information. Some pieces of it are imme-diately evident in the PDF data, others need applying ofextra analytical procedures to be extracted and fullyexploited.

An important material’s characteristic that is evidentin the experimental atomic PDFs is the material’s phasetype (e.g., see Fig. 1). Here a phase means a part of amaterial that has a well-defined chemical compositionandatomic-scale structure, is physically and chemicallyhomogeneous within itself and so is surrounded by aboundary thatmakes itmechanically separable from therest of the material and/or the material’s environment.Bragg XRD is a major tool for qualitative phase

0

20

40

60

80

100(2%~) (~98%)Compton + Elastic

Q = 2 (Å–1)

Q = 25 (Å–1)

Q = 40 (Å–1)

Inte

nsi

ty (

a.u

.)

E (keV)

0

20

40

60

80

(~15%)

(85%~) Compton

Elastic

10090807060

0

20

40

60

80(~97%)

(~3%)

Elastic

Compton

Figure 8. X-ray scattered intensities versus x-ray energy spec-tra fromCa0.25Al0.5Si0.5O2glasscollectedat threedifferentwavevectors, Q. Data were taken with x-rays of energy 80.6keV(Petkov et al., 1999). Note the dramatic decrease of the coher-ent/elastic scattering (marked with broken line in red) with theincrease of Q.

PAIR DISTRIBUTION FUNCTIONS ANALYSIS 1367

identification of crystalline materials. XRD patterns formaterials of limited structural coherence, however, areusually quite diffuse in nature and so are difficult to beused for unambiguous phase identification. AtomicPDFs can be employed instead. As an example, experi-mental XRD patterns and atomic PDFs for a series ofnanosized graphitic materials are shown in Figure 9.The XRD patterns show only a few Bragg-like features,whereas the respective atomic PDFs show numerouspeaks coming from the sequence of well-defined atomiccoordination spheres in these graphitic materials.Since carbon atoms in the different graphitic materialsare arranged differently (e.g., forming stacks of flatsheets in graphitic carbons, folded sheets in the nano-tubes, and spheres in the C60 fullerenes) the respectiveexperimental PDFs are substantially different and socan be used to identify each of the respective graphiticphases. Reference databases of atomic PDFs can begenerated for important classes of materials of limitedstructural coherence (e.g., for silicate glasses, industrialamorphous polymers, and pharmaceuticals) and usedfor their qualitative phase identification in a mannersimilar to that implemented in the so-called PowderDiffraction File with XRD patterns of crystals (Smithand Jenkins, 1996). Examples of successfulapplication of atomic PDFs analysis for determiningthe relative fraction of phases in a mixture, that is, forquantitative phase analysis are known aswell (Gateshkiet al., 2007).

Another important material’s characteristic is thelength of structural coherence, also known as themeansize of coherent x-ray scattering domains (Klug andAlexander, 1974). Atoms from different domains arenot well lined up and so do not make consecutivesequences of well defined atomic coordination spheres.Accordingly, the peaks in the atomic PDFs suffer extraloss of sharpness for distances longer than the “domainsize” rendering the atomic PDF featureless beyondthose distances. The fact is well demonstrated in Fig-ure 10 showing experimental atomic PDFs for crystal-line Au and nanosized Au particles (Petkovet al., 2005a). The smaller the particles, the shorter thereal space distance at which the respective atomic PDFdecays to zero. It goes fromabout50A

�for 30nmdown to

10A�for 1.7 nm particles. Note, the length of structural

coherence in the particles is shorter than their actualsize since the particles (see Fig. 11A for a structuremodel of 3 nm particles) exhibit extended structuraldefects, resembling wedge disclinations, dividing theirinner part into domains that are misoriented withrespect to each other. Care should be taken that thebehavior of the longer part of the experimental PDFs isnot dominated by low Q-space resolution effects (seeFig. 7) when the length of structural coherence is thequantity of interest. Correction procedures for moder-ate Q-space resolution effects are discussed in(Gateshki et al., 2005).

ThecommonlyusedatomicPDFG(r) (seeEquations1and2) slopes as4pro�r for small r values, in particular inthe region from r¼0 to about r¼1A

�where no real

interatomic distances exist. Therefore, estimates forthe atomic number density ro (i.e., density measuredin atoms/A

� 3) of the material studied can be obtainedfrom the initial slope of experimental G(r) data as illus-trated in Figure 10. Note sinceXRDexperimental errorstend to add up close to the origin of the Fourier trans-formation (Peterson et al., 2003; Qiu, 2004a), that is,close to r¼0A

�, the ro estimate should be done with due

care.Atomic coordination numbers, CNij, are another

important structural parameter. They can be derivedfrom the area of the respective PDFs peaks. As definedthe atomic PDFG(r) oscillates about zero (see Equation 1and Fig. 1) making it inconvenient for integrating itspeaks for the purpose of obtaining an estimate for thepeak’s areas. Another atomic PDF defined as:

RDF ¼ 4pr2rðrÞ ¼ 4pr2roþ r*GðrÞ ð7Þ

where r is the radial distance, and r(r) and ro the localand average atomic number densities, respectively, isbetter to be used in this case. From the integrated RDFpeak areas, the number of atomic neighbors of type jaround atomic species of type i, that is, CNij, can beobtained as follows:

CNij ¼ cj*ðrespective RDF Peak AreaÞ=wij ð8Þ

where the atomic concentrations cj and the weightingfactorswij (see Equation 6) should strictly obey the sum

20151050

0

2000

4000

6000

8000

10000

12000

302520151050–2

0

2

4

6

8

10

12

14

Graphene sheets

Mesoporous carbon

Graphitic carbon

C nanotube/multi walled

C nanotube/single walled

C60/C70

Inte

nsi

ty, (a

rb. units

)

Bragg angle, 2 theta

Graphene sheets

Mesoporous carbon

Graphitic carbon

C nanotube/multi walled

C nanotube/single walled

C60/C70

Ato

mic

PD

F G

(r)

Radial distance r (Å)

Figure 9. Experimental XRD patterns (up) and the respectiveatomic PDFs (down) for a series of graphitic carbons.

1368 X-RAY TECHNIQUES

rulesP

cj¼1 andP

wij¼1. Note the area of a PDF/RDFpeak is not simply proportional (via the cj term in Equa-tion 8) to the number of respective atomic pairs CNij butdepends on the atomic pair’s relative scattering power(via the wij term in Equation 8) as well. Therefore, foraccurate estimates of CNij to be obtained, Equation 8should be strictly applied. The areas of individual RDF

peaks canbe obtainedby adirect integrationwhen thosepeaks are well resolved. When the RDF peaks are par-tially overlapped, which is often the case, the individualpeak areas can still be evaluated very precisely by fittingeach peak with a Gaussian function as demonstratedin Figure 12. In this example, the first coordinationnumbers of Si, Al, and Ca atoms in Cax/2AlxSi1�xO2

glasses are obtained allowing to draw importantconclusions about the type of coordination units andtheir connectivity in the glass network (Petkovet al., 2000).

The ultimate goal of structure studies is the determi-nation of the positions of the atoms constituting thematerial under study. In case of crystalline-like materi-als, PDFsanalysis canyield thepositionsof atomswithinthe unit cell of a periodic lattice by using availablesoftware (Farrow et al., 2007, Neder and Proffen, 2008).At first, a structure model featuring a unit cell of aperiodic lattice is designed using theoretical predictionsoravailable crystal structuredata formaterials of similarchemistry. The sequence of coordination radii and num-bers (ri, CNij) for that model is computed and then con-voluted with gaussians to take into account the thermaland eventually static atomic rms displacements in thematerial under study. Thus computed model PDF iscompared to the experimental one and the differencebetween the experimental and model PDF data mini-mized by adjusting the atomic positions and rms dis-placements in the initial model. Example of adetermination of the structure of a crystalline-likemate-rial via atomic PDF analysis is shown in Figure 11c.Nonperiodic type structure models that are more appro-priate for liquids and glasses can also be tested andrefined against atomic PDFs using available software(Gereben et al., 2007; Lindahl et al., 2001). In this case,the model is a large-scale atomic configuration of manythousand atoms that is a statistically representativefragment of the material under study. An example isshown in Figure 11b. Finite size models, that is, modelsnot subject to periodic boundary conditions, that arevery appropriate for nanosized metallic particles (seeFig. 11a), semiconductor quantum dots (Pradhan

50403020100

0

2

4

6

8

4προ*r

30 nm Au

1.6 nm Au

Crystalline Au

3 nm Au

15 nm Au

Ato

mic

PD

F G

(r)

Radial distance r(Å)

Figure 10. Experimental atomic PDFs for crystalline (mm-sized)and nanosized Au particles (Petkov et al., 2005a). Arrowmarksthe low-r PDF region that slopes as 4pro�r.

Figure 11. Structure models of 3nm Au particles (a), GeSe2 glass (b), and Mg ferrite (c) derivedfrom fits to the respective experimental atomic PDFs. Themodel for Au particles features atoms(red circles) arranged in an face-centered cubic type structure (Petkov et al., 2005a). ThemodelforGeSe2 glass feature a randomnetwork of corner and edge-sharingGe-Se4 tetrahedra (PetkovandMessurier, 2010). The model (Gateshki et al., 2005) for Mg ferrite features a periodic cubiclattice of Fe-O6 octahedral (gray) and Fe-O4 tetrahedral units (shaded).

PAIR DISTRIBUTION FUNCTIONS ANALYSIS 1369

et al., 2007), or large organic molecules (Petkovet al., 2005b) can also be tested and refined againstatomic PDFs. Successful attempts of using atomic PDFsfor ab initio structure determination are also known(Juhas et al., 2006). In the atomic PDF-based structuredetermination process, plausible constraints based on apriori knowledge about material’s chemistry, density,local coordination (e.g., from complimentary NMR orEXAFS experiments), and others are often needed tobe employed to discriminate between competing solu-tions. Note, atomic PDFs are one-dimensional, spheri-cally averaged representation of the atomic arrangementso the uniqueness of the three-dimensional structuresolution found by atomic PDFs analysis is not alwaysguaranteed.

SAMPLE PREPARATION

Atomic PDFs analysis can be done on samples of anysize, shape, and phase state so long those can bemounted on a typical in-house or synchrotron XRDinstrument. No special sample preparation is necessary

though optimizing the sample size and shape withrespect to maximizing the scattered intensities andreducing sample-related unwanted scattering is highlyrecommended, anytime it is possible. Also, a carefulchoice of the XRD data collection geometry, that is,reflection versus transmission versus capillary, shouldbe made according the particular sample’s phase state(Klug and Alexander, 1974; Thijsse, 1984).

SPECIMEN MODIFICATION

X-ray diffraction is anondestructive technique. Samplesmeasured remain completely unaltered which is a greatadvantage comparing with other material’s structurecharacterization techniques such as electron micros-copy and diffraction for example. Rarely organic materi-als can become damaged by the high flux of synchrotronradiation x-rays. Therefore, such samples should not beoverexposed but only measured as long as necessary toobtain good statistical accuracy.

PROBLEMS

Typical problems include distortions of the PDFs peakshape (e.g., see Fig. 5), appearance of false PDF peaks(e.g., see Fig. 6) and/or shifts in PDFs peaks positionsdue to misalignment of the XRD instrument or errors inthe x-ray wavelength calibration. Various sources oferrors and their particular effect on atomic PDFs arediscussed in (Peterson et al., 2003; Qiu et al., 2004b;Petkov and Danev, 1998). To minimize errors in atomicPDFs: (i) theXRD instrument shouldbe carefully aligned,(ii) x-ray wavelengthwell calibrated, (iii) background andsample-related unwanted scattering minimized by acareful optimization of the experimental setup, (iv) XRDdata taken to high wave vectors, (v) in appropriate DQsteps, and (vi) each with a very good statistical accuracy.Estimates for the latter are given in (Toby andEgami, 1992). The so-collectedXRDdata should be care-fully reduced to atomic PDFsusing as precise as possibledata for the chemical composition, density, and x-rayabsorption factor of the actual samplemeasured. Simplemeasures of experimental atomic PDFs quality aredescribed in (Klug and Alexander, 1974; Toby andEgami, 1992; Peterson et al., 2003). Software imple-menting these measures and correcting experimentalatomic PDFs for moderate errors is available as well(Petkov and Danev, 1998). Common problems with theinterpretation of atomic PDFs include misidentifyingan experimental artifact feature (e.g., a PDF ripple dueto the finite Qmax value) (Warren and Mozzi, 1975) as astructural feature or vice versa. To avoid such pro-blems, the particular experimental details such asinstrument resolution and Qmax should be preciselytaken into account in the PDF fitting/structure refine-ment process. Also, structure models resulted fromatomic PDFs analysis, in particular when reverse

0

2

4

Ca-O

Al-O

Si-O

O-O

Ato

mic

RD

F

Radial distance r (Å)

Ca0.125

Al0.25

Si0.75

O2

2.82.62.42.22.01.81.61.4

0

5

10Si-O

O-O

SiO2

0

2 Ca-O

Al-OSi-O

O-O

Ca0.25

Al0.5

Si0.5

O2

0

2

4Ca

0.33Al

0.67Si

0.33O

2Al-O

Si-O

O-OCa-O

Figure 12. Gaussian fit to the first peaks in the RDFs for Cax/2AlxSi1�xO2 (x¼0, 0.25, 0.5, 0.67) glasses. Experimental data:symbols; fitted data: full line; individual Gaussians: brokenline; residual difference: full line (bottom). Peaks are labeledwith the corresponding atomic pairs.

1370 X-RAY TECHNIQUES

Monte Carlo simulations are employed, may come outway too disordered, including predicting way tooshort/long bond lengths and/or way too distortedatomic coordination units unless suitable structureconstraints/restraints are imposed on the structuremodeling/refinement process. Simple checks foratomic bond lengths and angles feasibility and bondvalence sums consistency using available software(Roux and Petkov, 2010) can be done to certify thatsuch mishaps did not occur.

ACKNOWLEDGMENTS

Thanks are due NSF, DOE, NRL, and ICDD for providingfunds over several years for atomic PDFs studies, resultsof which are shown in this paper. Also, thanks are due toTh. Proffen for generating Fig. 2.

LITERATURE CITED

Chupas, P. J., Qiu, X., Hanson, J. C., Lee, P. L., Grey, C. P.,and Billinge, S. J. L. 2003. Rapid-acquisition pairdistribution function (RA-PDF) analysis. J. Appl. Cryst. 36:1342–47.

Czichos, H., Saito T., and Smith, L. (eds.). 2006. Handbook ofMaterials Measurements Methods. Springer. Berlin

David, W. I. F., Shakland, K., McCusker, L. B., and Baerlocher,Ch.edts. 2002. Structure Determination from Powder Dif-fraction. Oxford University Press. Oxford

Egami, T. and Billinge S. J. L. 2003. Underneath the Braggpeaks: Structural Analysis of Complex Materials. PergamonPress, Elsevier Ltd. New York

Farrow, C. L., Juh�as, P., Liu, J. W., Bryndin, D., Bo�zin, E. S.,Bloch, J., Proffen, Th., and Billinge, S. J. L. 2007.PDFfit2 and PDFgui: computer programs for studyingnanostructure in crystals. J. Phys.: Condens. Matter 19:335219.

Farrow, C. L. and Billinge, S. J. L. 2009. Relationship betweenthe atomic pair distribution function and small-angle scat-tering: Implications for modeling of nanoparticles. Acta

Cryst. A 65:232–239.

Farrow,C. L., Shaw,M., Kim,H., Juhas, P., andBillinge, S. J. L.2011.Nyquist–Shannonsampling theoremapplied to refine-ments of atomic pair distribution functions. Phys. Rev. B84:134105–7.

Gateshki, M., Petkov, V., Pradhan, S. K., and Vogt, T. 2005.Structure of nanocrystallineMgFe2O4 fromx-ray diffraction,Rietveld and atomic pair distribution function analysis J.

Appl. Cryst. 38:772–79.

Gateshki, M., Niederberger, M., Deshpande, A. S., Ren, Y., andPetkov, V. 2007. Atomic-scale structure of nanocrystallineCeO2–ZrO2 oxides by total x-ray diffraction and pair distri-bution function analysis, J. Phys.: Condens Matter.,19:156205–12.

Gereben, O., Jovari, P., Temleitner, L. and Pusztai, L. 2007. Anew version of the RMCþ þ reverse Monte Carlo pro-gramme, aimed at investigating the structure of covalentglasses. J. Optoelectron. Adv. Mater. 9:3021–7.

Giacovazzo, C. (ed.). 1998. In Fundamentals of x-ray crystal-lography. Oxford University Press.

Gilbert, B., Huang, F., Waychunas, G. A., and Banfield, J. F.2004. Nanoparticles: Strained and stiff. Science

305:651–654.

Hahn, T. (ed.). 2002. International Tables for Crystallography.Volume A. Kluwer Academic Publishers. Dordrecht, TheNetherlands

Hajdu, F. and Palinkas, G. 1972. On the determination of theabsolute intensity of X-rays scattered by a non-crystallinespecimen. J. Appl. Cryst. 5:395–401.

Jeong, I.K., Proffen,Th.,Mohiuddin-Jacobs,F.,Billinge,S. J. L.1999 Measuring correlated atomic motion using X-ray dif-fraction. J. Phys. Chem. 103:921–924.

Juhas, P., Cherba, D. M., Duxbury, P. M., Punch, W. F., andBillinge, S. J. L. 2006. Ab initio determination of solid-statenanostructure. Nature 440:655–658.

Keen, D. A. 2001. A comparison of various commonly usedcorrelation functions for describing total scattering. J. Appl.Cryst. 34:172–177.

Klug, P. K. and Alexander, L. E. 1974. X-Ray Diffraction Pro-cedures: For Polycrystalline and Amorphous Materials. 2nded., John Wiley & Sons, Inc. New York

Kodama, K., Iikibo, S., Taguchi, T., and Shamoto, S. 2006.Finite size effects of nanoparticles on the atomic pair distri-bution functions. Acta Cryst. A 62:444–453.

Kohara, S., Suzuya, K., Kashihara, Y., Matsumoto, N.,Umesaki, N., and Sakai, I. 2001. A horizontal two-axis diffractometer for high-energy x-ray diffraction usingsynchrotron radiation on bending magnet beamlineBL04B2 at SPring-8. Nucl. Instr. and Meth. A467–468:1030–33.

Korsunsky, V. I., Neder, B. R., Hofman, A., Dembski, S., Graf,Ch., and Ruhl, E. 2007. Aspects of themodeling of the radialdistribution function for small nanoparticles. J. Appl. Cryst.40:975–985.

Kuhs, W. F. and Lehmann, M. S. 1983. The structure ofthe ice Ih by neutron diffraction. J. Phys. Chem. 87:4312–13.

Lee,J.H.,Aydiner,C.C.,Almer,J.,Bernier,J.,Chapman,K.W.,Chupas, P. J. Haeffner, D., Kump, K., Lee, P. L., Lienert, U.,Miceli, A., and Vera, G. 2008a. Synchrotron applications ofan amorphous silicon flat-panel detector. J. Synchrotr. Rad.15:477–488.

Lee, P. L., Shu, D., Ramanathan, M., Preissner, C., Wang, J.,Beno, M. A., Von Dreele, R. B. Ribaud, L., Kurtz, Ch., Antao,S. M. Jiao, X., and Toby, B. H. 2008b. A twelve-analyzerdetector system for high-resolution powder diffraction.J. Synchr. Rad. 15:427–32.

Lindahl, E., Hess, B., and Spoel, D. 2001. GROMACS 3.0: Apackage for molecular simulation and trajectory analysis.J. Mol. Modeling, 7:306–17.

Malenkov, G. 2009. Liquid water and ices: Understanding thestructureandphysical properties.J. Phys.:Condens.Matter.

21:283101–35.

Midgley, P. A. and Dunin-Borkowski, R. E. 2009. Electrontomography and holography in materials science. Nat. Mat.

8:271–280.

Mullen, K. and Levin, I. 2011. Mitigation of errors in pairdistribution function analysis of nanoparticles. J. Appl.

Cryst. 44:788–97.

PAIR DISTRIBUTION FUNCTIONS ANALYSIS 1371

Neder,R.B. andProffen, Th. 2008.Diffuse scattering anddefectstructure simulations. A Cook Book using the ProgramDISCUS. Oxford University Press. Oxford

Page, Y. Le and Donnay, G. 1976. Refinement of the crystalstructure of low-quartz. Acta Cryst. 32:2456–2459.

Peterson, P. F., Bozin, E. S., Proffen, Th., and Billinge, S. J. L.2003. Improved measures of quality of atomic pair distri-bution function quality. J. Appl. Cryst. 36:53–64.

Petkov, V. and Danev, R. 1998. IFO—a program for imagereconstruction-type calculation of atomic distributionfunctions for disordered materials. J. Appl. Cryst.

31:609–19.

Petkov, V., Jeong, I-K., Chung, J. S., Thorpe, M. F., Kycia, S.and Billinge S. J. L. 1999. High-real space resolutionmeasurement of the local structure of In xGa 1�x As semi-conductor alloys using X-ray diffraction. Phys. Rev. Lett.

83:4089–92.

Petkov, V., Billinge, S. J. L., Sashtri, S., and Himmel, B. 2000.Polyhedralunits andconnectivity in calciumaluminosilicateglasses from high-energy x-ray diffraction. Phys. Rev. Lett.85:3436–9.

Petkov, V., Billinge, S. J. L., Larson, P., Mahanti, S. D., Vogt,T., Rangan, K. K., and Kanatzidis, M. 2002. Structure ofnanocrystalline materials using atomic pair distributionfunction analysis: Study of LiMoS2 Phys. Rev. B. 65:092105.

Petkov, V., Peng, Y., Williams, G., Huang, B., Tomalia, D., andRen, Y. 2005a. Structure of gold nanoparticles suspended inwater studied by x-ray diffraction and computer simula-tions, Phys. Rev. B 72:195402–8.

Petkov, V., Parvanov, V., Tomalia, D., Swanson, D., Bergstrom,D., andVogt, T. 2005b. 3Dstructure of dendritic andhyper--branched macromolecules by x-ray diffraction”, Solid State

Commun. 134: 671–75.

Petkov, V. 2008 Nanostructure by high-energy XRD. Mater.

Today 11:28–38.

Petkov, V., Cozzoli, P. D., Buonsanti, R., Cingolani, R., andRen,Y. 2009. Size, shape and internal atomic ordering of nano-crystals by atomicpair distribution functions: a comparativestudy of gamma-Fe3O3 nanosized spheres and tetrapods. J.Am. Chem. Soc. 131:14264–6.

Petkov, V. and Shastri, S. 2010. Element-specific view of theatomic ordering in materials of limited structural coherenceby high-energy resonant x-ray diffraction and atomic pairdistribution functions analysis: A study of PtPd nanosizedcatalysts. Phys. Rev. B 81:165428.

Petkov, V., Selbach, S. M., Einarsrud, M.-A., Grande, T., andShastri, S. D. 2010a. Melting of Bi-sublattice in nanosizedBiFeO3 perovskite by resonant x-ray diffraction. Phys. Rev.Lett. 105:185501–4.

Petkov, V., Moreels, I., Hens, Z., and Ren, Y. 2010b. PbSequantum dots: Finite, off-stoichiometric, and structurallydistorted. Phys. Rev. B 81:241304–4.

Petkov, V. and Messurier, D. Le. 2010. Atomic-scale structureof GeSe2 glass revisited: a continuous or broken networkof Ge-(Se1/2)4 tetrahedra. J. Phys.: Condens. Matter 22:115402–6.

Pradhan, S. K., Deng, Z. T., Tang, F., Wang, C., Ren, Y., Moeck,P., and Petkov, V. 2007. 3D structure of CdX (X¼Se, Te)nanocrystals by total x-ray diffraction. J. Appl. Phys.

102:044304–7.

Qiu, X., Thompson, J. W., and Billinge, S. J. L. 2004a.PDFgetX2: A GUI driven program to obtain the pair

distribution function from X-ray powder diffraction data.J. Appl. Cryst. 37:678–78.

Qiu,X.,Bozin,E. S., Juhas, P., Proffen, Th., andBillinge,S. J. L.2004b. Reciprocal-space instrumental effects on the real-space neutron atomic pair distribution function. J. Appl.

Cryst. 37:110–16.

Roux, S. Le, Martin, S., Christensen, R., Ren, Y., andPetkov, V. 2011. Three dimensional structure of multicom-ponent (Na2O)0.35 [(P2O5)1�x (B2O3)x]0.65 glasses byhigh energy X-ray diffraction and constrained reverseMonte Carlo simulations. J. Phys.: Condens. Matter

23:035403.

Roux, S. Le andPetkov, V. 2010. ISAACS—Interactive structureanalysis of amorphous and crystalline systems. J. Appl.

Crystallogr. 43:181–85.

Ruland, W. 1964. The separation of coherent and incohe-rent Compton x-ray scattering. Brit. J. Appl. Phys. 15:1301–07.

Samy, A. Dinnebier, R. E. Smaalen, S. Van 2010. Maximumentropy method and charge flipping, a powerful combina-tion to visualize the true nature of structural disorder fromin situ x-ray powder diffraction data. Acta Cryst. B 66:184–195.

Smith, D. K. and Jenkins, R. 1996. The powder diffraction file:Past, present, and future. J. Res. Natl. Inst. Stand. Technol.101:259–271.

Soper, A. K. and Barney, E. R. 2011. Extracting the pair distri-bution function fromwhite-beamX-ray total scattering data.J. Appl. Cryst. 44:714–26.

Thijsse, B. J. 1984. The accuracy of experimental radial distri-bution functions for metallic glasses. J. Appl. Cryst.

17:61–76.

Toby, B. H. and Egami, T. 1992. Accuracy of pair distributionfunction analysis applied to crystalline and non-crystallinematerials. Acta Cryst. A 48:336–346.

Warren, B. E. 1934 The diffraction of X-rays in glass. Phys. Rev.B. 45:657–661.

Warren, B. E. and Mozzi, R. L. 1975. The termination effect foramorphous patterns. J. Appl. Cryst. 8:674–677.

Wagner, C. N. J. 1969. Structure of amorphous alloy films. J.Vac. Sci. Tech. 6:650–657.

Zachariasen, W. H. 1932. The atomic arrangement in glass. J.Am. Chem. Soc. 54:3841–3851.

KEY REFERENCES

Egami and Billinge, 2003. See above.This book is an excellent summary of the atomic PDFs analysis

and its application to crystals with intrinsic disorder.

Keen, 2001. See above.This paper gives an overview of the various definitions of atomic

PDFs used by the scientific community.

Klug and Alexander, 1974. See aboveThis book presents a detailed explanation of the principles of

traditional powder XRD and its extension into atomic PDFs

analysis.

Thijsse, 1984. See above.This article provides a detailed account of XRD experiments

aimed at atomic PDFs analysis.

Wagner, 1969. See above.The paper introduces the most frequently used atomic PDF

definition.

1372 X-RAY TECHNIQUES