1. INTRODUCTION2. BACKGROUND3. MULTIPHONON CONTRIBUTIONS4. NON-GAUSSIAN LINESHAPE5. INTERMOLECULAR CORRELATIONS6. RESOLUTION FUNCTION7. DISCUSSION8. CONCLUSIONS9. ACKNOWLEDGEMENTSREFERENCESNOTES ON THE ANALYSIS OF DATA FORPAIR DISTRIBUTION FUNCTIONSM. F. Thorpe, V. A. Levashov, M. Lei and S.J.L. Billinge∗1. INTRODUCTIONIn these notes, we collect together some results that we have found useful in theanalysis of data for the Pair Distribution Function (PDF). This work is rather general, andwe illustrate the results with examples from molecules and crystalline solids. These notesare arranged into five main sections, following some definitions in the next section. Insection 3, we analyze a single PDF peak and show how it can be decomposed into zero,one, two and multiphonon contributions. These results are illustrated using the nearestneighbor PDF peak in a Ni crystal. In section 4, we examine the widely used approxima-tion that the PDF lineshape is Gaussian and find that this is indeed a very good approxi-mation under most circumstances. We give explicit expressions that can be used for thedeviations from a Gaussian lineshape in terms of the elements of the displacement–dis-placement correlation matrix. These results are illustrated for benzene and fullerenemolecules, where we show that even in these cases, which are highly anisotropic, thedeviation from a Gaussian lineshape is small. In the section 5, we show how the inter-molecular correlations can sometimes be taken into account by using a continuum ap-proximation. In section 6, we show how the experimental resolution function can betaken directly into account for a special case. In the discussion section, we show howthese results can be brought together to give both the intra– and inter– Fullerene PDF in acrystal of Fullerite.2. BACKGROUNDThe multiphonon contributions to the PDF are contained within the expression for thecoherent scattering cross-section Iqaf. This cross section1 can be written asINffe eijiijiij().( ),.( )qqr r qu u=−−∑1ij,(1) ∗ M. F. Thorpe, V. Levashov, M. Lei and S.J.L. Billinge, Physics & Astronomy Department, Michigan StateUniversity, East Lansing, MI 48824.2where the angular brackets denote that a thermal average has been taken over the dis-placements ui from the equilibrium or mean atomic positions ri, so that the instantane-ous position of the atom iis given by ruii+. The atomic form factors fqiaf are as appro-priate for x-ray scattering, where they are the Fourier transform of the atomic charge den-sity, or for neutron scattering where they are the q –independent neutron scatteringlengths. It is useful to define a reduced scattered intensity S qaf SI f f fiiiiiiqqaf af=−∑−∑FHIKLNMOQPRST|UVW|∑FHIK222(2)that has the property that after the powder average is taken, Sqaf→ 1 as q →∞.In these notes we will focus on only a single atomic species in a crystal with Nsites,as in for example a face centered crystal of Ni, and molecules such as fullerene with onlyC atoms. The generalization to many atoms is quite straightforward, and all our resultscan easily be extended to a crystal containing many atomic species. We do discuss thebenzene molecule, C6H6, in section 4. The Debye-Waller factor exp −2Waf is extractedexplicitly in the front of the expression (1) and then re-inserted, so that a multi-phononexpansion can be madeINeffe e eWijiijiWij().( ),.( )qqr r qu u=−−−∑122ij{}.(3)We expand the term inside the braces in (3) to get the Bragg peak, the one phononpeak etc, after an appropriate spherical powder average is done. To be clear on the nota-tion that is used throughout this paper, we write the spherically averaged Iq Iafaf=q, andSq Safaf=q where the bar denotes the spherical powder average, taken by averaging overthe directions of the scattering vector q . The density ρ raf is then obtained from thetransformGr r r qSq qrdqafafaf=−= −∞z42100πρ ρπsin (4)where Graf is the reduced pair distribution function (RPDF). The form (4) is appropriatefor an infinite system where the term in ρ0 is removed from the left and the term in 1from the right. For a finite system, like a molecule, this is not necessary as shown in Eq.(52). The density ρ raf can be written in terms of the individual peaks in the PDF as ρπrrPrijijaf af=≠∑142(5)where Prij() is the PDF for a pair of atoms i and j, and the density ρraf is related to thenumber Nraf of atoms in the spherical annulus between r and rdr+ viaNR dr NR r rdr+− =afaf af42πρ.(6)33. MULTIPHONON CONTRIBUTIONSIt is actually more convenient to consider this derivation of the lineshape associatedwith a single peak Prijaf in the PDF using a different starting point. The PDF peak can becalculated exactly within the harmonic approximation by the one-dimensional Fouriertransform2Pr dqe C qijiq r rijijafafdi=−−z120π,(7)where we define Cqijaf slightly differently than in our previous papers2-4 [the phase factoris taken out here] asCq e e e e e eijiqqqqWqij ijij ijij ijafafdiafdiafdi== = =−−−−−−−urur.$/.$//1212 2212 2222222 2 22 2σλσ σ λσ σ.(8)We use q as the conjugate variable to r. Here σij ij ij22=< >(.$)ur is the factor that deter-mines the width of the PDF peak and the vector uuuij i j=−, with $rij being a unit vectorfrom rrrij i j=−. The angular brackets denote a thermal average. It is useful to introducethe phonon expansion parameter λ that we use to pick out the various multiphonon con-tributions to the lineshape. This occurs as an expansion of the exponentialexp .iiiqr u+bg in (1) in powers of the displacements ui, which is equivalently an ex-pansion in q . Because two terms are required, one to create a phonon and the other todestroy it, the coefficient of the term involving qn2 or equivalently λn in (8) gives thecontribution from the n–phonon term to the PDF lineshape. The observed scattered inten-sity is obtained when λ=1. In the limit that i and j are well separated, we haveσσij222→, which leads to the usual Debye-Waller factor for the peaks at large r in thePDF. The Debye-Waller factor is defined by213222 2 2Wquqix=< >= < >= < >q.u ubg(9)for the case of cubic symmetry as in face centered cubic Ni for example. The expressionin (7) becomes12122 22222222 2222222πσπσ λσ σσσλσ σijrrijrreeij ijij ij−−−− + −=+−()/()/didiejdiej(10)where we can now make an expansion in the no phonon (elastic or Bragg), one–phonon,two–phonon processes etc. by