Pair Distribution Function

5 Key excerpts on "Pair Distribution Function"

• 

  eBook - PDF

  Neutron Scattering in Earth Sciences

  • Hans Rudolf Wenk(Author)
  • 2018(Publication Date)
  • De Gruyter

    (Publisher)

  In general diffuse scattering contains information about two-body correlations in the structure including chemical SRO, distortions, correlated motion or orientation SRO of molecules to name a few examples. In the next sections, we will discuss the details of the PDF approach from measurements to modeling as well as present a few examples selected from our own work. An in depth discussion of the PDF method, its formalism and many more applications can be found in a recent book (Egami and Billinge 2003). THE ATOMIC Pair Distribution Function (PDF) What is a PDF? The PDF gives the probability of finding an atom at a given distance r from another atom. In other words it can be understood as a weighted bond length distribution. Looking back at Figure If, it is immediately apparent that the nearest neighbor distance in our example structure is 2.5 A. In fact the PDF method originated from the study of liquids and glasses (see Wilding and Benmore 2006, this volume; and Cole et al. 2006, this volume) which have no long range order at all. The strength of the PDF approach is the fact that it can be applied to a wide range of materials from amorphous materials to crystals. This is illustrated in Figure 2 showing the experimental PDFs for three samples: (a) silica glass rod, (b) crystalline quartz powder and (c) gold nano-particles with an average diameter of ~3.5 nm. In all cases the PDF is shown out to atom-atom distances of 40 A. One can see immediately that the glass has no long range order, since there are no PDF peaks at larger atom-atom distances. However, the PDF shows two sharp clearly defined nearest neighbor peaks corresponding to Si-O and O-O nearest neighbors. In addition some broader medium range order peaks can be observed before the PDF goes to zero.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Disordered Pharmaceutical Materials

  • Marc Descamps(Author)
  • 2016(Publication Date)
  • Wiley-VCH

    (Publisher)

  /λ, to limit the effects of cut-off in the Fourier transform. This requires working with short wavelengths less accessible to laboratory instruments, and has long limited the use of the PDF to the field of amorphous materials and liquids. More recently, the PDF analysis took a new development thanks to its usefulness in the study of local structural effects and nanomaterials to the emergence of instruments well suited for PDF measurements at synchrotron and neutron sources and to the development of user-friendly software to process and analyze the data.

  Depending on the authors and the communities, there are different definitions of the PDF. We will not go into the details here, and the interested reader can refer to the article by Keen [6], where the relationships between different definitions are explained. We will here follow the formalism developed by Egami and Billinge [7].

  The PDF G(r) yields the probability of finding a pair of atoms separated by a distance r. It can be calculated from a structural model describing the distribution of atoms in a sample according to the formula:

  10.1

  where the summation is throughout the atoms contained in the structural model, separated by the distance rij . bi is the scattering power of atom i (its Fermi length for neutrons, its form factor for X-rays or electrons), ⟨b⟩ is the average scattering power of the sample, and ρ0 is the numerical density of compound, that is, the number of atoms per unit volume. This function will have peaks for values of r corresponding to interatomic distance in the model, the intensity of these peaks being proportional to the product of the scattering factors of the atoms forming the pair. All the contributions of all pairs of atoms add up in the PDF. However, it is possible to calculate partial PDFs by taking into account selected atoms only (Figure 10.1 ). It is worth noting that Eq. (10.1 ) is strictly valid only for neutron scattering for which the Fermi length b is independent of Q in reciprocal space so that ⟨b⟩ can be calculated unambiguously. For X-rays and electrons, the form factors are Q-dependent and the shape of F(Q) depends on the element considered so that Eq. (10.1

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  In-situ Characterization of Heterogeneous Catalysts

  • José A. Rodriguez, Jonathan C. Hanson, Peter J. Chupas, José A. Rodriguez, Jonathan C. Hanson, Peter J. Chupas(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  5 Pair Distribution Function Analysis of High-Energy X-ray Scattering Data Karena W. Chapman and Peter J. Chupas

  5.1 Introduction

  Pair Distribution Function (PDF ) analysis is an emerging structure characterization tool with the potential to provide valuable new insights into a broad range of problems in heterogeneous catalysis. The method yields atomic scale structural information with crystallographic resolution, without the need for crystalline order implicit in conventional Bragg diffraction analysis. Accordingly, it can be used to study the nanoscale and/or defect-rich materials systems often involved in heterogeneous catalysis.

  While the methodology itself is not new [1–3], the application of PDF methods to address problems in catalysis, and complex materials in general, is less developed than many other techniques, due to previously limited access to high-quality data. Since the early 2000s, advances in experimental approach, instrumentation, and detector technologies have increased access to PDF data, providing improved data quality in substantially reduced measurement times [4, 5]. Accordingly, it has now become possible to conduct quantitative PDF measurements with appropriate sensitivity and on timescales relevant to the structural changes in heterogeneous catalysts.

  The use of PDF methods to understand and optimize heterogeneous catalysts is rapidly expanding and the potential of this approach is only now being fully explored [6–21]. Studies are being extended beyond determining the static structure of catalytic materials, and any changes that may occur following ex-situ/offline treatment, to in-situ studies under nonambient conditions relevant to catalytic synthesis, operations, or deactivation. This includes variable temperature, pressure, and reactive chemical environments. Demonstrated in-situ

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Underneath the Bragg Peaks

  Structural Analysis of Complex Materials

  • Takeshi Egami, Simon J.L. Billinge(Authors)
  • 2012(Publication Date)
  • Pergamon

    (Publisher)

  Chapter 3

  The Method of Total Scattering and Atomic Pair Distribution Function Analysis

  Egami Takeshi Simon J.L. Billinge

  Abstract

  In this chapter the atomic Pair Distribution Function (PDF) is defined. The equations are derived from basic scattering equations. The full range of PDF related equations are laid out and a brief history of the method is presented. A number of extensions to the basic PDF method are presented such as PDFs in higher dimensions and compositionally resolved PDFs. Finally, the theory behind some practical aspects of the method are presented, such as handling and estimating errors and the application of the Nyquist-Shannon sampling theorem to the method.

  Keywords

  atomic Pair Distribution Function (PDF) equations; PDF history; multi-component PDF; compositionally resolved PDF; PDF in high dimensions; anisotropic PDF; error analysis in the PDF; termination effects; Nyquist-Shannon sampling; EXAFS

  3.1 Total Scattering and the PDF

  3.1.1 Introduction

  As we discussed in Chapter 2 , in standard crystallographic analysis, the Bragg peaks and diffuse-scattering intensities were treated separately. The structure is determined solely based upon the information provided by the position and intensity of the Bragg peaks, while additional information regarding deviations from the perfect lattice is obtained through the study of diffuse scattering. This approach makes sense when the deviations are small, but when the structure is extensively disordered it fails in practice.

  In this chapter, we present an alternative approach which treats both the Bragg and the diffuse scattering on an equal basis, the so-called total-scattering technique. Data from throughout reciprocal space, over a wide range of Q

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Structure from Diffraction Methods

  • Duncan W. Bruce, Dermot O'Hare, Richard I. Walton, Duncan W. Bruce, Dermot O'Hare, Richard I. Walton(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  etc. These localised changes are usually easier to identify.

  This process is made much less difficult by the use of the PDF, which is at the heart of this chapter.[5–8] As detailed below, the PDF can be obtained from diffraction data on crystalline powders, but it can equally well be produced from the diffraction patterns of amorphous materials such as glasses, liquids and even gases. These amorphous samples were the sole scope of the PDF from its very beginning in the 1930s until the 1990s, when its modern form was developed.

  3.2 Pair Distribution Function

  In order to obtain the full information on a crystal structure in direct space, a structure determination must be carried out. This is well established for the average crystal structure and can be carried out for both single-crystal and powder diffraction data. An analogous technique to direct methods (see Chapter 2) has not been established for disordered structures. A few special algorithms exist for limited types of defect, such as short-range order in binary alloys or stacking faults in closed packed structures. As the number of possible deviations from an average structure and of the possible types of distribution of such local defects are very large, a general theory does not seem to be available. Thus the actual structure of a disordered material cannot be obtained directly. Aside from this lack of a general theory, one can, however, directly determine information about the distribution of all interatomic distances from a diffraction pattern.

  Recall that the structure factor is essentially the Fourier transform of the scattering density:

  3.5

  where

  Sign up to read

  Learn more about book