5. Is there any more information on PDF analysis available?

Pair Distribution Function Analysis is a method of extracting structure-related information from powder diffraction data. Unlike other better known methods, PDF analysis can provide information not only about the long-range (>100Å) atomic ordering but also for the short-range ordering in materials. This is because the technique takes into account both the Bragg as well as diffuse scattering (which is known to be related to short-range order effects). This sensitivity to the local structure has made the PDF analysis the tool of choice for structural studies of amorphous materials. Only recently (though very succesfuly) the PDF analysis has been applied to (poly)crystalline materials with significant intrinsic disorder, including  nanocrystals. Important knowledge about the short-range atomic arrangement in semiconductors, superconductors and other materials of technological importance has been obtained. Despite the increasing interest in the study of the local structure of a great variety of materials, PDF analysis is still not widely used. In our understanding, the reason for this is that the existing software for retriving structural information from experimental Pair Distribution Functions (PDF) is not as user-friendly as it should be. The aim of this manual is to show the PDF method is applied to study the structure of a typical nanocrystalline material and to outline the most important steps in the PDF study. We hope that even researchers who do not have any experience in PDF analysis can follow the steps, run the programs, compare their results with ours, and (why not?) find better solution.

1. How reliable are the results from a PDF analysis?

The best way to answer this question is to compare the results obtained from PDF analysis with those obtaned by other well established techniques. Such a comparison has been done for a number of  crystalline materials, such as Ni, GaAs, GdAl₂, etc. The results have shown excellent agreement between the structural data obtained by PDF analysis and those obtained with other methods.

Here, we compare structure data for crystalline LiMn₂O₄ (cubic, spinel-like structure) obtained by PDF analysis and the Rietveld method. This material is with cubic, spinel-like structure that may be viewed as a three-dimensional network of edge-sharing octahedra (Fig.1). The experimental powder diffraction data were collected at the 11-ID-C beamline at the Advanced Photon Source of Argonne National Laboratory. The wavelength used was 0.1078Å (short wavelength is important for obtaining diffraction data at high wave vectors). The diffraction geometry was Debye-Scherrer with a glass capillary (d=0.5mm). A file with the experimental diifraction data (corrected for flux decay, background, Compton scattering and sample absorption) can be found here.  In order to extract structural information from the diffraction data we used the Rietveld method, as implemented in the program FullProf. The fitted diffraction profile is shown in Fig.2, As can be seen, the structural model (cubic, spinel-like) accounts very well for all Bragg peaks in the diffraction pattern. The unit cell constants and fractional atomic coordinates are shown in Table 1 together with some reliability factors and goodness-of-fit indicators. The .pcr file (a macro file with the structure and the necessary commands) for the Rietveld refinement with FullProf can be found here (limn2o4_rietveld.pcr). 
The same set of diffraction data was used for the PDF analysis. At first, the diffraction pattern was corrected for background (using a separate diffraction measurement of an empty glass capillary to account for the contributions of the capillary walls and air scattering), Compton scattering, detector dead-time, absorption, diffraction geometry, pollarization. All these corrections were done with the program RAD. Then, the corrected X-ray diffraction data were scaled into electron units and the interference function was calculated (all these calculations are again done with RAD). Finally, The structure function was Fourrier transformed to a PDF. The experimental PDF can be found here (spinel.rdf).


It is worth noting that the agreement factors R_G used in the PDF-based refinement appears higher when compared to the corresponding R_wp factor used in the Rietveld refinement. This does not indicate that the PDF-based fit is of inferior quality since it yelds structural parameters that agree quite well with those obtained by the Rietveld method. Rather, it reflects the fact that the experimental quantities being fit in real and reciprocal space are not the same.

Table 1. Structural parameters for LiMn2O4 obtained by the Rietveld refinement method and PDF analysis. Atomic positions are refined in the space group Fd-3m.

Method	Rietveld	PDF
a, Å	8.265(2)	8.252(1)
x(O)a	0.3873(2)	0.3873(4)
B(Mn), Å2	0.57(1)	0.85(1)
B(Li), Å2	1.4(3)	4.1(1)
B(O), Å2	1.28(7)	2.84(8)
Rwp(RG), %	10.2	16.6

^a The three atoms in the asymmetric unit occupy the following
Wyckoff positions: Mn 16d (5/8,5/8,5/8); Li 8a(0,0,0); O 32e(x,x,x).

Files used in this example:

1. limn2o4.dat                 Experimental diffraction data collected with λ = 0.1078Å
2. limn2o4_rietveld.pcr    File used with FullProf  for Rietveld refinement
3. spinel.rdf                    Experimental Atomic Pair Distribution Function
   
4. spinel.stru                    File with structure data used by PDFFIT
5. spinel.mac                   Macro file with PDFFIT commands 
   

2. How PDF can be applied to nano-crystalline materials?
Many materials that lack long range order (show very few, if any, sharp Bragg peaks) still possess well defined local struture on the nanometer length scale. The structure can be well described with a small number of parameters, such as unit cell and symmetry. The PDF-guided structural determination of nanocrystalline materials starts with the choice of a plausible structural model. Such can be found among the crystalline counterparts of the material.  If there are several crystalline compounds with similar chemical compositions, or if the material exists in a different polymorphic form, all crystalline counterparts should be tested. Since no sharp Bragg peaks exist in the diffraction pattern of nanocrystalline materials, the usual crystallographic procedures that can be used to distinguish between competing structural models are of no help.  Neither the extinction conditions formalism, nor the automated routines for unit-cell search can be applied. So, apart from some crystal chemistry considerations, the only way to tell which is the correct structure is to compare the PDFs calculated for each of the models with the experimental PDF. Then, the model that accounts best for the features of the experimental PDF is used to refine the structural parameters of the material under study. 
In this practical example we show results from a structural study of a nanocrystalline material with the approximate composition Li_0.54K_0.3MnO_3.0-δI_0.10, prepared  by reacting aqueous solution of KMnO₄ with 1.5 equiv LiI at room temperature[1]. From the thermogravimetric  analysis, it was obtained that the water content is about 0.2-0.4 mole water per unit formula. X-ray diffraction data from the sample were collected at the beamline 11-ID-C (Advanced Photon Source, Argonne National Laboratory) using X-rays of energy 115.013 keV (λ=0.1078Å). The higher energy x-ray were used to extend the region of reciprocal space  (i.e. to obtain data at higher wave vectors, Q) which is important for the success of PDF analysis. The experimental diffraction data are shown on Fig. 4. The file hyd.dat contains the diffraction data in ASCII x-y format.



The experimental PDF was obtained as follows. First, the coherently scattered intensities were extracted from the X-ray diffraction pattern by applying appropriate corrections for flux, background, Compton scattering and sample absorption. The intensities were normalized in absolute electron units, reduced to structure function and Fourier transformed to atomic PDF. All data processing was done using the program RAD. The obtained experimental PDF is shown on Fig.5. As can be seen, the oscillations of the PDF vanish at about 15Å. The nanocrystalline sample has a very well defined local atomic arrangement but lacks the extended order of usual crystals. The experimental PDF can be found in the file hyd.rdf.
    To determine the structure of  the nanocrystalline K-Li-Mn-O-I sample we adopted the following procedure: At first, plausible structural models that are consistentwith the experimental PDF data and available structural information for the materials under study were looked for. The models were matched against the experimental data to identify the most promising one. Then, the 3D structure of the nanocrystal was determined through refining that model so that it reproduces well all important details in the experimental PDF data. A careful inspection of the most prominent features in the experimental PDF reveals the following structural characteristics of the studied nanocrystal: The first peak in the experimnetal PDF data is positioned at about 1.89Å. Since this is the typical Mn-O distance in materials with octahedral oxygen coordination around the manganese atoms, it can be concluded that the material should contain MnO₆ octahedra. For comparison, the Mn-O distance within tetrahedral MnO₄ units is about 1.66Å.  Our PDF data rule out the presence of any such units in the sample. The second PDF peak appears at 2.88Å, which is the typical Mn-Mn distance for edge-sharing MnO₆ octahedra.  Thus, the analysis of the experimental PDF data suggests that the nanocrystalline sample is likely to be built of edge-sharing MnO₆ octahedra. Similar conclusions have been drawn [2] from micro-Raman and Mn K-edge XANES measurements. Thus, the search for model structures was narrowed to such showing only edge-sharing MnO6 octahedra. Several structures are known to occur with crystalline manganese oxides of chemical composition similar to that of the nanocrystals we study. We explored them in turn starting with the structure type found with crystalline LiMn₂O₄. That structure features a three-dimensional network of edge-sharing MnO₆ octahedra as shown in Fig.1. As can be seen in Fig.5a, this model reproduces well only the first two peaks in the experimental PDF data but completely fails at longer real space distances. Obviously, the structure of the nanocrystal is not a 3D network of edge-shared MnO₆ octahedra. Next a structure type found in Li₂MnO₃ was attempted. The structure features layers of edge sharing octahedral units where two thirds of the octahedral sites are occupied by Mn atoms and one third by Li atoms. Mn and Li atoms in the layers are ordered in such a way that each MnO₆ octahedron shares its edges with three other MnO₆ octahedra and three LiO₆ octahedra. A fragment of this structure is shown in Fig.6a. The model did not perform much better as the results in Fig.5b show and was not pursued further. A model based on the structure type occuring with LiCoO₂ was considered as well. This structure type features layers of MO₆ (M = Co) octahedra. In this model, each MO₆ octahedron shares edges with six other octahedra of the same type. Fragment of the structure is shown in Fig.6b. The model performed better but still could not be refined to reproduce all details in the experimental data as can be seen in  Fig.5c.


Finally, the structure of the nanocrystalline lithium manganese oxide was approached with that occuring in K_xMnO₂·yH₂O. The structure type features layers of MnO₆ octahedra encapsulating K atoms and H₂O molecules. A fragment of the structure is shown in Fig.6c. This model was an excellent starting point and could be refined to reproduce very well all important details in the experimental data as shown in Fig.5d. The refined structural parameters are summarized in Table 2. The results of the structure determination carried out show that, at the atomic scale, the nanocrystalline material may be viewed as a stack of layeres made of edge shared MnO₆ octahedra. The layers are well apart (~6.43 Å) allowing both relatively small Li and much bigger K atoms to occupy the interlayer space. This is most likely due to the aqueous preparation route employed since it is well known that water makes layered materials swell facilitating the encapsulation of bigger-size atomic species. The layers in the nanocrystal, however, does not seem to be arranged in perfect registry. As can be seen in Table 2, the temperature factors of Mn and O atoms are highly anysotropic suggesting the presence of a significant disorder in direction perpendicular to the layers. The disorder could be due to the presence of a distribution of interlayer distances centered at about 6.43 Å. The structure file and the macro file used for the fit of the experimental data  are hyd.stru and hyd.mac.

(a)	(b)	(c)

Table 2. Structural parameters for nanocrystalline K-Li-Mn-O-I obtained  through PDF analysis. Atomic positions are refined in space group R-3m. The structure is given in a hexagonal basis.

Method	PDFfit
a, Å	2.8317(2)
c, Å	19.29(2)
z(O)a	0.382(1)
U11(Mn), Å2	0.0058(1)
U33(Mn), Å2	0.148(1)
Uiso(Li/K/H2O), Å2	0.071(1)
U11(O), Å2	0.0051(1)
U33(O), Å2	0.208(1)
RG, %	20.7

^a The three atoms in the asymmetric unit occupy the following Wyckoff positions: Mn 3a (0,0,0); Li/K/H₂O 9d(1/2,0,1/2); O 6c(0,0,z).

In order to further confirm the validity of the results obtained for the nanocrystalline sample we calculated the powder diffraction patterns for the different trial structural models and compared them with the experimental diffraction data (Fig.7). As can be seen in the figure, although all models show some resemblance to the experimental data, the model that accounts for all observed features of the diffraction pattern is that based on  K_xMnO₂·yH₂O structure. For example, this model is the only one that produces a peak at low angles 2θ that mathes well th eone observed in the experimental pattern. It should be stressed out that this is a visual comparison only; for now there are no worked out procedures allowing to refine structure model against the broad diffraction features shown in Fig.4 .

Fig.7 Powder diffraction patterns (λ=0.1065) of nanocrystalline K-Li-Mn-O-I (a) and the following model structures: K_xMnO₂·yH₂O (b); rhombohedral LiCoO₂ (c), cubic (spinel) LiMn₂O₄ (d); monoclinic Li₂MnO₃ (e); and monoclinic (layered) LiMnO₂ (f).

Files used in this example:

1. hyd.dat           Experimental diffraction data collected with λ = 0.1078Å
2. hyd.rdf            Experimental Atomic Pair Distribution Function
3. hyd.stru          Structural file used by PDFFIT
4. hyd.mac         Macro file with PDFFIT commands 

References:

1. Hwang, S.-J., Park, D.-H., Kwon, C.-W.,Campet, G., Choy, J.-H. J. J. Power Sources 2004, 125, 119.
2. Hwang, S.-J.; Kwon, C.-W.; Portier, J.; Campet, G.; Park, H.-S.; Choy, J. H.; Huong, P. V.; Yoshimura, M.; Kakihana, M. J. Phys. Chem. B 2002, 106, 4053.

3. What do I need to start doing PDF analysis?

First of all, powder diffraction data of superb statistical quality data are necessary for both the sample and background. Also, good real-space resolution should be achieved by collecting  diffraction data to high wave vectors. This is usually achieved by employing high-energy synchrotron radiation (the wave vector is inversely proportional to the wavelength) or spallation neutron sources. One may also use a laboratory source (MoKα or Ar Ka). With diffraction data as input, the program RAD calculates the corresponding atomic Pair Distribution Function. In PDF analysis, literature search is essential. Due to the limited structural coherence and the lack of such tools as extinction conditions, or unit-cell search procedures, the structural determination ab initio can be rather difficult. The good news is that looking at the diffraction data in direct space, can help very much in identifying structural elements like polyhedral coordinations, connectivity, etc. Another useful knowledge is that nanocrystals usually have structures similar (if not the same) to crystalline materials with close chemical composition. Using the program PDFFIT, the PDFs for several trial structures can be calculated and compared to the experimental one. If the real structure (in general terms) is among the tested, the comparison should yield a structural model that represents well enough the local atomic ordering in the material. In the final step, this structural model is refined against the experimental data using the program PDFFIT.

4. Where can I find the necessary programs?

The programs used for data treatment are available as free downloads from the web sites of their authors.

• Programs for data correction and PDF extraction:

i.   RAD   A program by V. Petkov for analysis of X-ray diffraction data from amorphous materials

ii.  PdfGetX  A program by I.-K. Jeong, J. Thompson, A. M. P. Turner and S. J. L. Billinge for determining the atomic pair distribution function from x-ray powder diffraction data

iii. PdfGetN  A program by P. F. Peterson, M. Gutmann, Th. Proffen and S. J. L. Billinge for extracting the total scattering structure function and the pair distribution function from neutron powder diffraction data

• Programs for structural refinement using Pair Distribution Functions:

i.  PDFFIT  A program for full profile structural refinement of the atomic pair distribution function. Authors: Th. Proffen and S. J. L. Billinge

5. Is there more information on PDF analysis available?

There is a number of books and research publications discussing different aspects of the structural determination from Pair Distribution Functions. Here is a short list:

• Books

i. Underneath the Bragg Peaks : Structural Analysis of Complex Materials by T. Egami and S. Billinge, Elsevier Science, 2003. Buy at Amazon.com

ii. Local Structure from Diffraction by S.J.L Billinge and M.F.Thorpe (editors), Plenum, 1998. Buy at Amazon.com

iii. From Semiconductors to Proteins: Beyond the Average Structure by S.J.L. Billinge and M.F. Thorpe (editors), Plenum, 2002. Buy at Amazon.com

• Review papers

i. Th. Proffen, S. J. L. Billinge, T. Egami and D. Louca, Structural analysis of complex materials using the atomic pair distribution function - a practical guide, Z. Kristallogr. 218, 132-143 (2003).

• Journal articles

i. V. Petkov,  M. Gateshki, J Choi, E.D. Gillian and Y. Ren, " Structure of nanocrystalline GaN from X-ray diffraction, Rietveld and atomic pair distribution function analyses", J. Mater. Chem. 15 (2005), 4654 .

        ii. V. Petkov, Y. Peng, G. Williams, B. Huang, D. Tomalia and Y. Ren, “Three-dimensional structure of gold nanoparticles in water by  x-ray diffraction and computer                    simulations”,  Phys. Rev. B 72 (2005) 195402.

        iii. M. Gateshki, V. Petkov, S. K. Pradhan and T. Vogt, “Structure of Nanocrystalline MgFe2O4 From X-ray Diffraction, Rietveld and Atomic Pair Distribution                             Function  Analysis”,  J. Appl. Cryst. 38 (2005) 772.

iv. V. Petkov, P. Y. Zavalij, S. Lutta, M. S. Whittingham, V. Parvanov and S. Shastri, Structure beyond Bragg: Study of V₂O₅ nanotubes, Phys. Rev. B 69 (2004) 085410.

v. V.Petkov and T .Vogt, Positional disorder of Ba in the thermoelectric germanium clathrate Ba6Ge25, Solid State Comm. 127 (2003) 43.

vi. V. Petkov , S.J.L. Billinge, T .Vogt, A.S. Ichimura and J.L. Dye, Structure of intercalated Cs in zeolite ITQ-4: an array of metal  ions and correlated electrons confined in a pseudo-1D nanoporous host, Phys. Rev. Lett. 89 (2002) 075502.

vii. V. Petkov, E. Bozin, S.J.L. Billinge, T. Vogt, P. Trikalitis and M. Kanatzidis, Structure of nanocrystalline V2O5.nH2O xerogel determined by the atomic pair distribution function technique,  J. Am. Chem. Soc.  124 (2002) 10157.

viii. V. Petkov, S.J.L. Billinge, S.D. Shastri and B. Himmel,  High-resolution atomic distribution functions of disordered materials by high energy x-ray diffraction, J. Non-Cryst. Sol. 293/295 (2001) 726.

ix. Jeong IK, Heffner RH, Graf MJ, Billinge SJL, Lattice dynamics and correlated atomic motion from the atomic pair distribution function, Phys. Rev. B, 67 (10): Art. No. 104301.

x. Proffen T, Petkov V, Billinge SJL, Vogt T, Chemical short range order obtained from the atomic pair distribution function, Z. Kristallogr. 217 (2): 47-50 2002

• Subject related books

i. X-Ray Diffraction Procedures: For Polycrystalline and Amorphous Materials by H. P. Klug, L. E. Alexander, Wiley-Interscience, 1974, Buy at Amazon.com

ii. The Structure of Non-Crystalline Materials by Y. Waseda,  McGraw-Hill, New York, 1980.

iii. Characterization of Nanophase Materials by Zhong Lin Wang, Vch Verlagsgesellschaft Mbh, 2002 Buy at Amazon.com 

iv. X-Ray Diffraction : In Crystals, Imperfect Crystals, and Amorphous Bodies by A. Guinier, Dover, 1994, Buy at Amazon.com