Search Results

4 Simulating experimental data

Reinhard B. Neder and Thomas Proffen

in Diffuse Scattering and Defect Structure Simulations: A cook book using the program DISCUS

In most cases, simulations of disordered materials are performed to understand experimental observations, in this case diffraction data. This chapter discusses the calculation of several experimental ... More

In most cases, simulations of disordered materials are performed to understand experimental observations, in this case diffraction data. This chapter discusses the calculation of several experimental quantities: single crystal diffuse scattering, powder diffraction, and the atomic pair distribution function (PDF). Since diffraction data are obtained via a Fourier transform, the finite size of the model crystal as well as issues concerning coherence are discussed in detail. The PDF is basically calculated from the atomic structure directly. Different ways to incorporate thermal motion are illustrated.Less

1 Introduction

Reinhard B. Neder and Thomas Proffen

in Diffuse Scattering and Defect Structure Simulations: A cook book using the program DISCUS

This chapter gives a brief introduction into diffuse scattering and provides a discussion of disordered materials within the scope of this book.

Analysis of the molecular short‐range order

Naomi E. Chayen, John R. Helliwell, and Edward H. Snell

in Macromolecular Crystallization and Crystal Perfection

Whilst the transition from short‐range order to long‐range order is not a sharp transition, i.e. there is a steady progression, there are metrics to judge the shorter‐range effects. These are ... More

Whilst the transition from short‐range order to long‐range order is not a sharp transition, i.e. there is a steady progression, there are metrics to judge the shorter‐range effects. These are described. Diffuse scattering occurs, from effects associated with both short‐range order and long‐range order. For convenience, and the special nature of the topic, we have kept these two aspects in this chapter. Diffuse scattering as a source of measurement error is described.Less

STRUCTURE

Ted Janssen, Gervais Chapuis, and Marc de Boissieu

in Aperiodic Crystals: From Modulated Phases to Quasicrystals

This chapter discusses the methods for determining the structure of crystals. Topics covered include diffraction techniques, determination of modulated phases and composites, structure determination ... More

This chapter discusses the methods for determining the structure of crystals. Topics covered include diffraction techniques, determination of modulated phases and composites, structure determination of quasi-crystals, and diffraction by an imperfect crystal.Less

Diffuse scattering – thermal, short-range order, gaseous, liquid, and amorphous scattering

Erich H. Kisi and Christopher J. Howard

in Applications of Neutron Powder Diffraction

This chapter deals with diffuse scattering attributable to departures from perfect crystallinity and from gaseous, liquid, or amorphous states. First, the way in which thermal vibration of atoms ... More

This chapter deals with diffuse scattering attributable to departures from perfect crystallinity and from gaseous, liquid, or amorphous states. First, the way in which thermal vibration of atoms leads not only to reduced peak intensities but also to thermal diffuse scattering (TDS) is discussed. The detail depends on whether the probing neutrons are slower or faster than the speed of sound. For alloys, it is shown that short range order (SRO) gives rise to features in the diffuse scattering that may become superlattice peaks when long range order is established. Next the scattering from gases, liquids, and amorphous materials is developed from the Debye scattering equation. A structure factor is given which is related, by Fourier transform, to the radial and (when more than one element is present) pair distribution functions of interest. Studies of diffuse scattering utilising isotopic substitution, model-based computer simulation, and Reverse Monte Carlo analysis are reviewed.Less

Physical properties

Ted Janssen, Gervais Chapuis, and Marc de Boissieu

in Aperiodic Crystals: From Modulated Phases to Quasicrystals: Structure and Properties

Physical properties of aperiodic crystals present some theoretical challenges due to the lack of three-dimensional periodicity. For the description of the structure there is a periodic representation ... More

Physical properties of aperiodic crystals present some theoretical challenges due to the lack of three-dimensional periodicity. For the description of the structure there is a periodic representation in higher-dimensional space. For physical properties, however, this scheme cannot be used because the mapping between interatomic forces and the high-dimensional representation is not straightforward. In this chapter methods are described to deal with these problems. First, the hydrodynamic theory of aperiodic crystals and then the phonons and phasons theory are developed and illustrated with some examples. The properties of electrons in aperiodic crystals are also presented. Finally, the experimental findings of phonon and phason modes for modulated and quasicrystals are presented. The chapter also discusses diffuse scattering, the Debye–Waller factor, and electrical conductivity.Less

Structure

Ted Janssen, Gervais Chapuis, and Marc de Boissieu

in Aperiodic Crystals: From Modulated Phases to Quasicrystals: Structure and Properties

This chapter discusses the X-ray and neutron diffraction methods used to study the atomic structures of aperiodic crystals, addressing indexing diffraction patterns, superspace, ab initio methods, ... More

This chapter discusses the X-ray and neutron diffraction methods used to study the atomic structures of aperiodic crystals, addressing indexing diffraction patterns, superspace, ab initio methods, the structure factor of incommensurate structures; and diffuse scattering. The structure solution methods based on the dual space refinements are described, as they are very often applied for the resolution of aperiodic crystal structures. Modulation functions which are used for the refinement of modulated structures and composite structures are presented and illustrated with examples of structure models covering a large spectrum of structures from organic to inorganic compounds, including metals, alloys, and minerals. For a better understanding of the concept of quasicrystalline structures, one-dimensional structure examples are presented first. Further examples of quasicrystals, including decagonal quasicrystals and icosahedral quasicrystals, are analysed in terms of increasing shells of a selected number of polyhedra. The notion of the approximant is compared with classical forms of structures.Less

Magnetic Diffraction

Andrew T. Boothroyd

in Principles of Neutron Scattering from Condensed Matter

The basic concepts of magnetic order in crystals are reviewed, including magnetic unit cells, propagation vectors and magnetic domains. Some commonly-occuring magnetic structures are discussed, such ... More

The basic concepts of magnetic order in crystals are reviewed, including magnetic unit cells, propagation vectors and magnetic domains. Some commonly-occuring magnetic structures are discussed, such as ferromagnets, antiferromagnets, ferrimagnets, and noncollinear and incommensurate magnetic structures. The differential cross-section for neutron diffraction from a magnetic structure is derived, and the magnetic structure factor is defined. The use of neutron polarization analysis, including spherical neutron polarimetry, in the determination of magnetic structures and of the spatial distribution of magnetization is described in detail. Diffuse magnetic scattering due to magnetic frustration and magnetic phase transitions is discussed, and the relevance of the static approximation is explained. Neutron diffraction studies of nuclear spin order are described.Less

Analysis of extended inorganic structures

John Evans

in Crystal Structure Analysis: Principles and Practice

Extended (polymeric) inorganic structures present different problems to crystallographers than molecular structures. The samples are often tiny crystals, atomic scattering powers vary widely, and ... More

Extended (polymeric) inorganic structures present different problems to crystallographers than molecular structures. The samples are often tiny crystals, atomic scattering powers vary widely, and structures are frequently disordered or twinned, strongly absorbing, or prone to temperature-induced phase transitions. Pseudo-symmetry is also a frequent problem. Disorder includes atomic substitution on a single site and partial occupancy by atoms of variable valency; some forms of disorder lead to diffuse scattering. Phase transitions may cause incommensurate structures to form. Extended structures require different approaches to testing and validation; these include bond valence calculations. This chapter illustrates the issues with two detailed case histories of mixed oxide materials.Less

Diffraction in the Static Approximation

Andrew T. Boothroyd

in Principles of Neutron Scattering from Condensed Matter

The basic principles of crystallography are reviewed, including the lattice, basis and reciprocal lattice. The Bragg diffraction law and Laue equation, which describe coherent scattering from a ... More

The basic principles of crystallography are reviewed, including the lattice, basis and reciprocal lattice. The Bragg diffraction law and Laue equation, which describe coherent scattering from a crystalline material, are derived, and the structure factor and differential cross-section are obtained in the static approximation. It is explained how the presence of defects, short-range order, and reduced dimensionality causes diffuse scattering. For non-crystalline materials, such as liquids and glasses, the pair distribution function and density-density correlation function are introduced, and their relation to the static structure factor established. For molecular fluids, the form factor is defined and calculated for a diatomic molecule, and the separation of intra- and inter-molecular scattering is discussed. The principles of small-angle neutron scattering are described.Less