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There has been a general consensus that multigroup (sample) confirmatory factor analysis offers a strong approach to evaluate cross-cultural equivalence of measurement properties. Researchers have proposed different procedural steps in the testing of measurement equivalence hypotheses, including the equivalence of the covariance matrices of the observed indicators and the equivalence of factor means among groups. Joreskog and Sorbom (2001) recommend the testing of five general hypotheses, including equivalence of covariance matrices of observed indicators of a scale or an instrument; equivalence of factor patterns of observed indicators; equivalence of factor loadings of observed indicators on their respective factors; equivalence of measurement errors of observed indicators; and equivalence of factor variances and covariance across groups. Invariance of factor pattern and factor loadings is sufficient to determine whether a construct can be measured across different cultural, national, or racial groups.

Neutron powder diffraction finds extensive application in the solution, and more particularly, the refinement, of crystal structures. This chapter reviews the tools (space group symmetry, specification of atomic positions, crystallography, reflection conditions) for describing and solving crystal structures. The solution of the structure of the low temperature form of NaOD provides a simple example. There follows an account of structure refinement, particularly the Rietveld method. This involves consideration of the observed pattern, the calculated pattern, and the many factors that need to be taken into account (peak shapes, peak widths, background, structure factors, displacement parameters, preferred orientation, etc.), followed by the matching of the two to determine the crystal structure parameters. Le Bail extraction (from whole pattern fitting) is derived from the Rietveld method. The chapter concludes with a discussion of the practical aspects of the Rietveld method and a selection of illustrative examples.

This introductory chapter provides information on some fundamental aspects of crystal structures and their diffraction of X-rays as a basis for the rest of the book. It describes electrons, atoms, molecules, and crystals scatter X-rays, leading to the observed diffraction pattern, and introduces concepts such as the reciprocal lattice, structure factors, Fourier transforms, Bragg's law for the geometry of diffraction, the phase problem encountered in crystallography, and the meaning of resolution and how it is related to the extent of the measured diffraction pattern.

This chapter presents the quantitative analysis of the intensities of Bragg reflections of incommensurately modulated crystals and composite crystals. The observed structure factor is introduced in analogy with the observed structure factor of periodic crystals. The problems of twinning and overlap of reflections are discussed. The effects of atomic vibrations on diffracted intensities are analysed. Phasons, amplitudons, and the sliding mode do not lead to specific forms of the Debye-Waller factor. Instead, Debye-Waller factors that contain modulated anisotropic displacement parameters (ADPs) must be used. The statistical properties of diffracted intensities of incommensurately modulated crystals closely follow the statistical properties of Bragg intensities of periodic crystals.

This chapter opens with a brief description of the very wide variety of magnetically ordered structures, both commensurate (with crystal structure) and incommensurate. The concepts of magnetic Bravais lattices and magnetic space groups are introduced. For an unpolarized incident neutron beam, magnetic and nuclear scattered intensities are additive — calculation of the latter involves a magnetic form factor, a magnetic interaction vector (depending on magnetic moment relative to scattering vector), and a magnetic structure factor. Example calculations are given for anti-ferromagnetic AuMn and the incommensurate heli-magnetic Au₂Mn. Methods for solving magnetic structures, i.e., establishing the nature of the magnetic ordering, then determining the magnitude and orientation of the magnetic moments, are discussed. The solution of magnetic structures from neutron powder data is illustrated with examples taken from the recent literature.

This chapter presents the properties of thermal neutrons. Their wavelength (from the de Broglie equation) is well suited to the investigation of condensed matter, i.e., to the study of liquids, glasses (amorphous materials), and crystalline materials with varying degrees of order. That the neutrons carry magnetic moment also makes them well suited to the study of magnetic ordering. The theory of nuclear and magnetic scattering from individual atoms and from assemblies of atoms is presented, this leading to the definition of neutron scattering length and to the concepts of coherent and incoherent scattering. The focus then shifts to the direction and intensity of diffraction from crystalline materials (Bragg's law, structure factors), and to the description of this scattering when samples are presented in polycrystalline or powder form (Debye-Scherrer cones).

This chapter discusses the underlying principles X-ray scattering by a distribution of electrons. The theory is then extended to the diffraction of X-rays by an infinite lattice of molecular motifs and the concept of the structure factor, which describes the diffraction pattern, is introduced. The theory of calculating the electron density distribution within the unit cell by Fourier inversion of the structure factors is then covered. The fact that only the amplitudes and not the phases of the structure factors can be measured experimentally represents the major practical problem of diffraction analysis that is known as the phase problem. The process of calculating structure factors from a known structure can be simplified by treating each atom as a scattering centre that is assigned a scattering factor related to its atomic number. In the second half of the chapter, the rules relating the symmetry of the diffraction pattern to that of the crystal are derived, and the basis for the inherent inversion symmetry of the diffraction pattern, known as Friedel's law, is described. Situations where this important law breaks down are touched upon due to their importance in solving the phase problem. The phenomenon of systematic absences, essentially missing diffraction spots, and how they can yield key information on the symmetry of the crystal is explained.

This chapter gives a comprehensive account of the symmetry of incommensurately modulated crystals. Diffraction symmetry is shown to be given by a crystallographic point group as it is known from the crystallography of periodic crystals. A complete list of symmetry restrictions on modulation wave vectors is derived from this property. The symmetry of incommensurate crystals with an one-dimensional modulation is given by (3+1)-dimensional superspace groups. The latter are defined as a subset of the space groups in four-dimensional space. A thorough discussion is given of the notation of superspace groups, of equivalence relations between them, and of their various settings. Symmetry properties of modulation functions and other structural parameters are presented. An expression is derived for the structure factor of Bragg reflections that incorporates the full superspace symmetry of the incommensurately modulated structure.

This chapter introduces the superspace theory for the description of the crystal structures of incommensurately modulated crystals. Two alternative constructions of superspace are presented. In reciprocal space the observed scattering vectors of Bragg reflections are considered to be the projections of reciprocal lattice points in (3+1)-dimensional superspace. Lattice translations in superspace applied to the atomic positions in three-dimensional, physical space generate the structure model in superspace. Thus, the latter is a periodic structure in superspace by definition. Structure factors of Bragg reflections are shown to be the Fourier transform of the electron density on one unit cell of the superspace lattice. t-Plots are defined, and their use in structural chemistry is demonstrated by the application to the incommensurately modulated structure of Sr₂Nb₂O₇.

In this chapter the theory of the static structure factor, which is closely related to the underlying fluid structure, is described. Different methods of calculating this quantity are presented, including perturbational approaches,, integral equations (including those based on the interaction site model), and computer simulations. Wherever possible the results of these calculations are compared with experimental data obtained from X-ray, neutron, and electron scattering. The range of fluids considered is broad, including simple inorganic species, polar and nonpolar organic liquids, water, and aqueous solutions. The effect of both temperature and pressure on the fluid structure is analysed and used to distinguish between competing molecular models. Quantum effects are considered where these are significant. The utility of Reverse Monte Carlo calculations and their particular limitations for molecular fluids are also discussed.

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.

Reflection phases are essential for crystal structure solution but are not available experimentally. They can be estimated from probability and other relationships derived from known or assumed constraints on the electron density, such as its positivity and atomicity. Direct methods of estimating phases use normalized structure factors, appropriate for point atoms at rest, which are calculated from the observed amplitudes and some assumptions. Various direct methods are based on inequalities, determinants, and probability relationships for relationships among phases of reflections with related indices. This chapter introduces concepts such as triplet and quartet relationships, the tangent formula, structure invariants, and maximum entropy approaches. The main steps involved in a direct methods structure solution are outlined, including the assignment of starting phases, the use of figures of merit for recognising possible solutions, and the interpretation of electron density maps (E-maps).

The structure factors, with their hkl indices, amplitudes and phases are dealt with in great detail. With figures, text and formulas the physical meaning of amplitudes and phases is explained. Rotation matrices and translation vectors are described mathematically as well as in figures. Plane group and space group determination from HRTEM images and ED patterns are explained. The relation between electrostatic potential and structure factors (Fourier terms, i.e. reflections) is described with formulas, structure model, EM image, ED pattern and a table of numerical values of amplitudes and phases for the structure Zr₂Se.

This chapter explores two aspects of the patterns of school victimization: firstly how the various acts of violence rank order (in terms of their frequencies), and secondly what their factor structure is. Comparisons are made in these patterns between Israel and the United States and between groups of students within these samples. Key findings include: (1) rank orders of the frequencies of violent acts, reflects the axis of the severity and seriousness of pain caused by the violent act. Overall patterns are extremely similar across gender, ethnicity/culture, school types (primary, junior-high, high school), and within and between countries; (2) victimization types fit into two major factors: those that are rare and severe, and those that are more common with less severe outcomes. These factors are very similar across all groups.

This chapter introduces the superspace description of the crystal structures of incommensurate composite crystals, and the characterization of their symmetry by superspace groups. The treatment parallels that of incommensurately modulated crystals in most aspects. The particular features of composite crystals are highlighted with respect to the diffraction and structure in superspace, superspace groups, the structure factor of Bragg reflections, and t-plots.

The principles of how to solve crystal structures from electron diffraction (ED) data are described. Recording and quantification of ED data is not trivial, considering the sharp diffraction spots that easily saturate the detector. The importance of using thin crystals is stressed. The phase problem in diffraction is presented and how it can be solved by various techniques, such as direct methods using triple relations, the Patterson function, charge flipping and the strong-reflections approach. Origin specification and semi-invariants, normalized structure factors and the Wilson plot are all described in detail.

Phase contrast, the contrast transfer function (CTF), kinematical scattering and the weak-phase object approximation are central concepts for image formation. The physics is described in great detail with one hundred formulas, leading up to the fundamental relation between electron wave phases and structure factor phases. Through-focus series are discussed as a means to collect all the information from a structure projection – some data may be lost in one HRTEM image because it falls on a node (= zero cross-over of the CTF) but this data can be obtained from another image taken at another defocus value. It is shown that singly and doubly scattered electron waves are transferred differently by the objective lens to the image. Thus, they can be separated in image processing, especially if a few images of different focus are combined.

The description of the thermodynamics of the strongly ionized plasma starts by discussing the most popular and well-studied model of the one-component plasma (OCP), which represents a system of point ions placed in a homogeneous medium of charges of opposite sign. The results of calculations by the Monte Carlo method of the binary correlation function, static structure factor, dielectric permeability, isothermal compressibility, and internal and free energies are presented. The region of existence of Wigner crystallization is determined. Pseudopotential models of multicomponent plasma are considered. The advantages and disadvantages of the quasiclassical approximation, density functional, and quantum Monte Carlo methods are discussed. A number of the proposed models in the region of increased nonideality lose thermodynamic stability, which is attributed to the possibility of a phase transition and the separation of the system into phases of different densities.

Methods for the quantitative description of intermolecular interactions, which influence the scattering pattern through the structure factor, are introduced. The relationship between the effective pair potential between solute particles, the virial coefficients in the expression of the chemical potential and the osmotic pressure provides a link between structure factor and thermodynamics. The experimental SAS studies on protein–protein interactions and the main methods used to calculate structure factors from theoretical potentials—in particular, those based on the DLVO potential—are surveyed. Ion-specific effects expressed in the Hofmeister series and those of small cosolutes including osmolytes are discussed, and the effects of large cosolutes such as polymers are described in terms of depletion interactions. The influence of temperature and pressure is reviewed, and the difficulties encountered when comparing different studies are discussed. Finally, a brief overview of studies of interactions of nucleic acids based on the Poisson–Boltzmann equation is presented.