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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.

Powder diffraction is complementary to the book's main technique of single-crystal diffraction; it can provide a check on purity and homogeneity of samples, and may be the only possible approach for some materials. The two methods are compared. The main limitation in powder diffraction is the one-dimensional nature of the data. Many variants are possible for the basic powder diffraction experiment: data may be measured in reflection or transmission mode, with monochromatic radiation and angle scanning, or with an energy-dispersive arrangement, and there are different kinds of detector available. Careful corrections for various effects are necessary for high accuracy. A powder diffraction pattern may be used for phase identification, qualitative and quantitative analysis of mixtures, information on grain size, strain, and preferred orientation. Reflection indexing and intensity profile measurement can lead to structure solution and refinement by Rietveld methods. Powder diffraction is well suited to non-ambient studies (temperature and pressure).

Major increases in brightness are anticipated with the upcoming coherent X‐ray lasers. Extrapolation suggests that a single macromolecule ‘sample’ may be sufficient to generate a measurable X‐ray diffraction pattern, a continuous Fourier transform of the molecule giving easier phase determination. The practical difficulties are enormous in the recording of such diffraction patterns, not least the so‐called ‘molecule blow‐up problem’ due to the thermal and radiation blast that a single molecule must take. By taking the exposure at a sufficiently small time‐flash, a few femtoseconds, this may be practical. For 3D structure determination multiple sample orientations are needed. A risk is too few photons in the femtosecond pulse that must be used to take data before sample damage occurs. A nanocluster of molecules would be a way of compensating for that and a ‘jet stream’ of these would lead to powder diffraction patterns rather than single‐molecule patterns.

There is an old adage in powder diffraction that ‘neutrons’ are for structure refinement only, whilst X-rays are for structure determination. This chapter reviews the technique of neutron powder diffraction and shows that there are some cases where neutrons are in fact the preferred probe for structure determination. The fundamental properties of the neutron that make it such a penetrating and powerful probe are discussed and in particular, it is shown that the lack of form-factor fall-off allows diffraction data to be collected to very high (sometime sub-Ångstrom) spatial resolution, which obviously benefits structure solution. Several examples of neutron-only structure solutions are given, as are various combined X-ray/neutron solutions. The chapter concludes with an examination of possible future uses of neutron powder diffraction, considering both its strengths and weaknesses.

This chapter, which assumes a familiarity with basic crystallography, introduces the concept of structure determination from powder diffraction data. It does so by examining the ‘structure determination maze’, which describes the various possible routes that one can take from the polycrystalline sample to a refined crystal structure. In so doing, it introduces the various stages of the process that are discussed in much greater detail in subsequent chapters. Touching briefly on the relationship between single-crystal and powder structure solution methods, it concludes with a forward look to the range of applications of powder methods to structure determination problems in the particular area of molecular organic compounds.

Structure refinement is a way to get more accurate atomic coordinates using ED intensities. A preliminary structure obtained from HRTEM or ED data can be refined with ED or X-ray powder diffraction data. R-values are used to follow how the refinement progresses towards a better structure. The importance of high-quality data and data completeness are stressed. After refinement, the atomic coordinates can reach an accuracy of 0.02 Å.

This chapter recalls the early developments of X-ray crystallography and how it spread throughout the world, and the first theoretical and experimental investigations that led to the determination of crystal structures, from the simpler trial-and-error methods to the systematic use of space groups and the introduction of Fourier syntheses. The Lorentz and polarization factors and the atomic scattering factor were analysed. W. H. Bragg introduced the concept of integrated intensities, and W. L. Bragg that of absolute intensities. Expressions for the diffracted intensity by small and large perfect crystals were obtained by C. G. Darwin, who also introduced the notion of mosaic crystal to account for the observed diffracted intensities. It is shown how W. L. Bragg determined the structure of the trigonal carbonates from the simple observation of diffracted intensities by various reflecting planes. The discovery of powder diffraction by Debye and Sherrer in Germany and by Hull in the United States is recounted. The first applications of the rotating crystal method to crystal structure determinations are described. Finally, the determination of three landmark crystal structures is explained: hexamethylene tetramine, graphite, and the benzene ring.

Assigning hkl indices to the peaks in the powder diffraction pattern is the essential first stage of the data analysis for structure solution. This chapter begins by outlining the basic relationships linking the positions of the Bragg peaks to the underlying crystal lattice and shows why translating observed peak positions into unit cell parameters is a non-trivial operation. Three quite distinct indexing strategies, as used in the programs ITO, DICVOL, and TREOR, are discussed in detail and practical reasons for utilising more than one indexing program are given. Considerable attention is paid to the effect that measurement errors can have upon the indexing process and figures of merit for assessing potential solutions are clearly explained. The impact of impurity lines is explained and modern strategies for handling such impurity lines discussed. The importance of databases for relating determined unit cell parameters to known phases within a sample is also assessed.

This chapter introduces the problem of polydispersity for understanding properties of materials containing polymethylene chains. A model has been proposed for the assembly of molecular chains in a petroleum waxes that will remain a focal point for later discussions in the book. The energetics of molecular crystalline polymorphism is discussed as are the methods used for the construction and interpretation of binary phase diagrams. Variances to symmetry rules for chain co-solubility proposed by A. I. Kitaigorodskii are presented for further consideration in later chapters.

This chapter discusses energetic principles important for the close packing of linear polymethylene chains that lead to favoured layer packing arrays, revealing that a hard sphere repulsive model is a good first approximation. The methylene subcell concept is introduced, both from a theoretical basis as well as from observation, based on known crystal structures of representative materials. Convenient methods for identifying methylene subcells, based on powder diffraction, electron diffraction, and infrared spectroscopy, are presented.

The crystalline state is characterized by a high degree of internal order. There are two types of order that we will discuss here. One is chemical order, which consists of the connectivity (bond lengths and bond angles) and stoichiometry in organic and many inorganic molecules, or just stoichiometry in minerals, metals, and other such materials. Some degree of chemical ordering exists for any molecule consisting of more than one atom, and the molecular structure of chemically simple gas molecules can be determined by gaseous electron diffraction or by high-resolution infrared spectroscopy. The second type of order to be discussed is geometrical order, which is the regular arrangement of entities in space such as in cubes, cylinders, coiled coils, and many other arrangements. For a compound to be crystalline it is necessary for the geometrical order of the individual entities (which must each have the same overall conformation) to extend indefinitely (that is, apparently infinitely) in three dimensions such that a three-dimensional repeat unit can be defined from diffraction data. Single crystals of quartz, diamond, silicon, or potassium dihydrogen phosphate can be grown to be as large as six or more inches across. Imagine how many atoms or ions must be identically arranged to create such macroscopic perfection! Sometimes, however, this geometrical order does not extend very far, and microarrays of molecules or ions, while themselves ordered, are disordered with respect to each other on a macroscopic scale. In such a case the three-dimensional order does not extend far enough to give a sharp diffraction pattern. The crystal quality is then described as “poor” or the crystal is considered to be microcrystalline, as in the naturally occurring clay minerals. On the other hand, in certain solid materials the spatial extent of geometrical order may be less than three-dimensional, and this reduced order gives rise to interesting properties. For example, the geometrical order may exist only in two dimensions; this is the case for mica and graphite, which consist of planar structures with much weaker forces between the layers so that cleavage and slippage are readily observed.

In this chapter diffraction by crystals is considered, the heart of classical crystallography. This is a vast topic and specific choices have to be made, keeping applications to soft matter in mind. The first section summarizes basic elements of crystallography, without going into detailed symmetry considerations. The heart of the chapter is a discussion of diffraction by a crystal lattice, which includes a further discussion of the Ewald sphere and the attendant concept of reciprocal space. After treating some practical aspects of scattering by a crystal, the chapter finishes with a case study of polymer crystallization.