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This chapter describes general approaches to the ab initio solution of crystal structures from X-ray or neutron powder diffraction data. The steps in the process, unit cell determination and indexing, intensity extraction, space group determination, structure solution, and structure refinement are described. Indexing methods such as zone indexing, exhaustive methods or recently developed whole pattern methods, and the use of a figure of merit (M₂₀) are presented. Intensity extraction is shown to be reasonably straightforward but for the problem of peak overlap that occurs in powder patterns. The phase problem makes structure solution more difficult: Fourier and Patterson methods, direct methods, or global optimization methods (simulated annealing, genetic algorithms) are brought to bear. The chapter concludes with a section on advanced refinement techniques, including the interpretation of displacement and site occupancy parameters, and the use of constraints. The discussion is illustrated by frequent reference to structure solution for the Ruddlesden-Popper compound Ca₃Ti₂O₇.

This chapter provides an overview of the methods of structure solution of incommensurately modulated crystals and composite crystals. Assuming the periodic average structure is known, it is shown that modulation functions can often be determined by trial and error, employing structure refinements starting with randomly chosen but small values for the structural parameters. The presentation of systematic methods of structure determination includes Patterson function methods, direct methods, and the method of charge flipping.

Maximum entropy methods are used in crystallography for estimating reflection phases among other applications. This chapter explains the basic concepts behind maximum entropy, including entropy itself and its relationship to probability and information. They are illustrated by a simple non-scientific example of estimating missing information based on reasonable assumptions.

The idea of a variational convergence is introduced as an equality of upper and lower bounds for families of variational problems. Examples are given that illustrate some of the main applications of Gamma-convergence, starting from the gradient theory of phase transitions, through homogenization and dimension reduction, to limits of atomistic theories. A final section shows how the definition of Gamma-convergence can be deduced from the requirement that it implies the convergence of minimizers be local and stable under continuous perturbations.

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.

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).

Global optimisation methods that involve the assessment of multiple trial crystal structures against measured diffraction data offer a powerful alternative to Direct methods of structure solution. This chapter describes the problem of searching an N-dimensional hypersurface, populated with a great many function minima, for the global minimum that corresponds to the true crystal structure. Key factors discussed include model construction, agreement factor calculation and estimating the likelihood of success. Various global optimisation methods including simulated annealing and genetic algorithms are covered, and the chapter is illustrated with numerous examples of solved molecular crystal structures.

Direct methods are ab initio structure determination techniques that handle the crystallographic phase problem starting from only the observed intensities. They are undoubtedly responsible for the enormous success of structure determination from single-crystal data, but their application to powder diffraction data is hampered by the latter's lack of spatial resolution and the reflection overlap problem. This chapter details the probabilistic basis of direct methods and explains how the intensity relationships they rely on can be used to improve the intensity estimates obtained by pattern fitting for overlapping reflections. Some examples of structures solved by Direct methods and some guidelines for data normalisation are given.

This chapter focuses on the performance of a particular software package, EXPO, against a wide range of test datasets. In EXPO, the intensity extraction and Direct methods stages are intimately linked to help overcome the reflection overlap problem. Various strategies, such as calculating Patterson information in real space to provide additional input to the extraction stage, are outlined and their beneficial effects in terms of improving the quality of the extracted intensities are listed. A short list of the main pitfalls that a Direct methods user has to avoid when using powder data is also provided.

This chapter provides a detailed mathematical definition of the Direct methods modulus sum function and its associated tangent formula (S' – TF). The latter appears to be especially well-suited to coping with the resolution and reflection overlap limitations of powder data. The concepts introduced have been encoded in a computer program XLENS and its application to heavy atom and framework-type powder is discussed.

This chapter explores the analytical method of fluxions, as stated in De Methodis. Newton’s method of fluxions can be divided into two parts: The direct and the inverse. Newton considered the techniques of the direct method to be perfected, as presented in his treatise De Methodis. After making his De Methodis treatise, he also sought to develop his inverse method algorithm, while also creating a better conceptual foundation to the direct method. The chapter notes that Newton continued in improving the two methods until he composed the De Quadratura, a work which explains the most advanced refinement of his method of fluxions.

This chapter determines stellar properties from seismic data. It discusses how the average seismic properties discussed in the previous chapters are used in determining the mass and radius of a star. The chapter provides two ways of determining stellar properties: the “direct method” and grid-based modeling. The first method produces a set of model-independent results derived from a set of equations. On the other hand, the grid-based model, which can make up for the shortcomings of the direct method, derives its results from a fairly dense grid of stellar models of different metallicities that cover a wide range of masses and evolutionary stages.

As indicated at the start of Chapter 4, after the diffraction pattern has been recorded and measured, the next stage in a crystal structure determination is solving the structure—that is, finding a suitable “trial structure” that contains approximate positions for most of the atoms in the unit cell of known dimensions and space group. The term “trial structure” implies that the structure that has been found is only an approximation to the correct or “true” structure, while “suitable” implies that the trial structure is close enough to the true structure that it can be smoothly refined to give a good fit to the experimental data. Methods for finding suitable trial structures form the subject of this chapter and the next. In the early days of structure determination, trial and error methods were, of necessity, almost the only available way of solving structures. Structure factors for the suggested “trial structure” were calculated and compared with those that had been observed. When more productive methods for obtaining trial structures—the “Patterson function” and “direct methods”—were introduced, the manner of solving a crystal structure changed dramatically for the better. We begin with a discussion of so-called “direct methods.” These are analytical techniques for deriving an approximate set of phases from which a first approximation to the electron-density map can be calculated. Interpretation of this map may then give a suitable trial structure. Previous to direct methods, all phases were calculated (as described in Chapter 5) from a proposed trial structure. The search for other methods that did not require a trial structure led to these phaseprobability methods, that is, direct methods. A direct solution to the phase problem by algebraic methods began in the 1920s (Ott, 1927; Banerjee, 1933; Avrami, 1938) and progressed with work on inequalities by David Harker and John Kasper (Harker and Kasper, 1948). The latter authors used inequality relationships put forward by Augustin Louis Cauchy and Karl Hermann Amandus Schwarz that led to relations between the magnitudes of some structure factors.

Chapter 8 focuses on nonlinear optimal control and its applications. The chapter begins by introducing the fundamentals of optimal control and prototypical problem formulations. This is followed by the treatment of first-order necessary conditions including the Pontryagin minimum principle, dynamic programming, and the Hamilton–Jacobi–Bellman equation. Singular arcs and bang–bang controls are relevant in the solution of many minimum-time and minimum-fuel problems and so these issues are discussed with the help of examples that have been worked out in detail.This chapter then turns towards direct and indirect numericalmethods suitable for solving large-scale optimal control problems numerically.The chapter concludes with an example relating to the calculation of an optimal track curvature estimate from global positioning system (GPS) data.

This chapter details the making of The Blair Witch Project (1999). The stories that surround the production of The Blair Witch Project are as fascinating as the final product. The circumstances of the film's low-budget production are legendary with stories of the cast being filled with real fear by the ‘method’ directing techniques of the two directors. From developing the script and financing the film to the extensive post-production period and taking the finished product to the Sundance Film Festival, the entire process is an inspiring example of inventive independent filmmaking. Ultimately, The Blair Witch Project is a film that could not have been made without advancements in camera technology beyond the early days of cinema.

It is well known from nonrelativistic quantum chemistry that mean-field methods, such as the Hartree–Fock (HF) model, provide mainly qualitative insights into the electronic structure and bonding of molecules. To obtain reliable results of “chemical accuracy” usually requires models that go beyond the mean field and account for electron correlation. There is no reason to expect that the mean-field approach should perform significantly better in this respect for the relativistic case, and so we are led to develop schemes for introducing correlation into our models for relativistic quantum chemistry. There is no fundamental change in the concept of correlation between relativistic and nonrelativistic quantum chemistry: in both cases, correlation describes the difference between a mean-field description, which forms the reference state for the correlation method, and the exact description. We can also define dynamical and nondynamical correlation in both cases. There is in fact no formal difference between a nonrelativistic spin–orbital-based formalism and a relativistic spinor-based formalism. Thus we should be able to transfer most of the schemes for post-Hartree–Fock calculations to a relativistic post-Dirac–Hartree–Fock model. Several such schemes have been implemented and applied in a range of calculations. The main technical differences to consider are those arising from having to deal with integrals that are complex, and the need to replace algorithms that exploit the nonrelativistic spin symmetry by schemes that use time-reversal and double-group symmetry. In addition to these technical differences, however, there are differences of content between relativistic and nonrelativistic methods. The division between dynamical and nondynamical correlation is complicated by the presence of the spin–orbit interaction, which creates near-degeneracies that are not present in the nonrelativistic theory. The existence of the negative-energy states of relativistic theory raise the question of whether they should be included in the correlation treatment. The first two sections of this chapter are devoted to a discussion of these issues. The main challenges in the rest of this chapter are to handle the presence of complex integrals and to exploit time-reversal symmetry.

Numerical integration is an important part of the finite-element technique. As seen in Section 6.5 of Chap. 6, volume integrations as well as surface integrations should be carried out in order to represent the elemental stiffness equations in a simple matrix form. In deriving the variational principle, it is implicitly assumed that these integrations are exact. However, exact integrations of the terms included in the element matrices are not always possible. In the finite-element method, further approximations are made in the procedure for integration, which is summarized in this section. Numerical integration requires, in general, that the integrand be evaluated at a finite number of points, called Integration points, within the integration limits. The number of integration points can be reduced, while achieving the same degree of accuracy, by determining the locations of integration points selectively. In evaluating integration in the stiffness matrices, it is necessary to use an integration formula that requires the least number of integrand evaluations. Since the Gaussian quadrature is known to require the minimum number of integration points, we use the Gaussian quadrature formula almost exclusively to carry out the numerical integrations.

This chapter deals with the “direct characteristics method” approach, in which the index change between two periods is computed using separate hedonic functions estimated for each period. It compares this method, which is called the “hedonic imputation method,” to the usual time dummy approach to hedonic regressions, and derives the exact conditions under which the two approaches to hedonic regressions will give the same two main and quite distinct approaches to the measurement of hedonic price indexes: time dummy hedonic indexes and hedonic imputation indexes. The chapter considers both weighted and unweighted hedonic regressions and finds exact algebraic expressions that explain the difference between the hedonic imputation and time dummy hedonic regression models. The weighting is chosen so that we are actually in a matched model situation for the two periods being considered, then the resulting hedonic regression measures of price change resemble standard superlative index number formulae.

This chapter starts with the Fourier series and Fourier transforms, and the representation of periodic functions. It then looks at Fourier analysis in crystallography. It details electron density, the derivation of relations between Fourier coefficients and structure factors, the X-ray resolution of a crystal structure, and the structural analysis of crystals and molecules (trial and error methods, the Patterson function, interpretation of Patterson maps, heavy atom and isomorphous replacement techniques, direct methods, and charge flipping). Finally, it provides an analysis of the Fraunhofer diffraction pattern from a grating and the Abbe theory of image formation.