bluuebunny/crossref_metadata_2025 huggingface dataset:
Belated Professionalization of Parliamentary Elites: Hungary 1848—1999
Authors: Ilonszki Gabriella
Published: November 2023

Hungarian parliamentary tradition begins in 1848 and in the following 150 years its parliament has existed virtually uninterrupted: that is, throughout the long liberal period up to 1918; in the interwar authoritarian system, with its façade of democratic institutions; in the diverse Communist periods; not to mention the intermezzos of 1918 and 1945 when attempts were made to establish democratic parliamentary government. Nevertheless, despite its long existence, the Hungarian parliament could only rarely act as an agent of modernization and democratization and parliamentary elites show clear trends of professionalization only in the last two legislative terms, that is, after the collapse of Communism in 1990.
Monitoring and evaluating status
Authors: New T R
Published: November 2023

Single species studies have been a major impetus for invertebrate conserva¬tion, and will continue to capture the imagination and to test the ingenuity of biologists and managers alike. Studies on assemblages or particular taxo-nomic groups commonly purport to assess change in diversity or relative abundance over space or time. Much decision-making in both these broad contexts depends on the outcome of status evaluation, and monitoring the changes and trends in abundance and distribution of the species (singular or plural) under study. The problems of monitoring invertebrates, and methods for estimating population size and structure are outlined in this chapter, as they are essential components of many surveys which emphasize conservation studies. Unlike much of the work discussed earlier in this book, this may need to be achieved without killing, harming, or, even, capturing the delicate animals involved.
Alternative approaches to species-focused conservation
Authors: New T R
Published: November 2023

The expense and complexity of single species invertebrate conservation renders it impracticable as the sole-or, even, major-strategy to pursue, because only a minute proportion of the taxa needing conservation could possibly be treated, even though the wide taxonomic spectrum of taxa considered provides a solid framework for development. Funding and exper¬tise for invertebrate conservation is limited, and concentrating on species alone for conservation would rapidly deploy this support with many taxa remaining neglected, and lingering doubts over its most effective use. Empha¬sis has thus also been placed on the prime need for conservation of any taxa, the protection of natural habitats. Surveys, sometimes of particular target groups, play important roles in selecting optimal areas for protection-or in justifying need for protection of remnant habitats where such a choice is unavailable.
Involving people in invertebrate conservation
Authors: New T R
Published: November 2023

The major practical barriers to increasing wider interest in invertebrate conservation and conducting effective surveys have been stressed repeatedly in earlier chapters, but devolve into three major categories. 1. The adverse images and public profile or, at least, general apathy with which invertebrates are regarded. 2. The widespread lack of appreciation, even within the community of zoologists and ecologists, of their diversity and fundamental importance in natural ecosystems. 3. The lack of resources, including expertise, for studying invertebrate diversity and biology, and for incorporating this information effectively into broader conservation programmes by standard sampling and analyti¬cal protocols. These categories, and their many subdivisions, emphasize the need to involve people in many walks of life, to improve and foster public sympathy for invertebrates in pragmatic ways, and to enlist whatever help can be mustered to undertake practical aspects of sampling, recording, distribution mapping, helping in well co-ordinated surveys and in protecting and manag¬ing sites, as a major collective contribution to sustaining the earth’s biota.
Banach and topological algebras
Authors: Dales H G
Published: November 2023

We now turn to the definitions and fundamental properties of (in general, non commutative) Banach algebras and some more general topological algebras. Thus we are moving from the foundations given in Chapter 1, and shall now examine the implications of the additional assumption that the algebras that we are considering also have a topology; in each case, the topology is connected to the algebraic structure by the simple requirement that the algebraic operations be continuous.
Commutative Banach Algebras
Authors: Dales H G
Published: November 2023

In this chapter, we shall give a rather detailed description of many specific commutative Banach algebras; we shall study the examples for their own interest and for the sake of applications to automatic continuity theory in the final Chapter 5. Throughout, we shall be concerned with the basic structural properties of these examples-their character spaces, their polynomial generators, and their ideals-and we shall build up the machinery which will be required in Chapter 5.
Recent Developments in Dense Numerical Linear Algebra
Authors: Higham Nicholas J
Published: November 2023

Numerical linear algebra with dense matrices is still today, as it was at the time of the first IMA meeting on the State of the Art in Numerical Analysis in 1965, an extremely active area of research. There are several reasons. First, we still do not fully understand some of the classical algorithms. For example, the behaviour of the growth factors for Gaussian elimination and its variants with partial and complete pivoting is not yet completely understood. Second, novel computer architectures force us to design new algorithms, modify old ones, and reassess algorithms that were once discarded. In particular, parallel computers have made the flop counts traditionally used to measure the cost of an algorithm of dubious relevance, so that some algorithms hitherto considered inefficient are now of interest again. Third, applications stimulate the development of new theory and techniques. For example, real-time signal processing applications have stimulated much work on rank-revealing factorizations and the updating of factorizations after low rank changes.
Hilbert space methods
Authors: Partington Jonathan R
Published: November 2023

This chapter collects together a variety of interpolation techniques which exploit the additional structure available in a Hilbert space. To begin with, we shall introduce the notion of a reproducing kernel Hilbert space, where the values of any function are given by its Inner products with a set of kernel functions. Next we shall analyse the recently introduced matching pursuit algorithm, which has a wide variety of applications in data analysis. We shall then move on to consider minimal-norm interpolation, both in a Hilbert space and a general Banach space setting.
Sampling
Authors: Partington Jonathan R
Published: November 2023

In this chapter we shall be concerned with another situation in which we wish to recover an analytic function from noisy measurements. In this case the function will be defined on R, and we have in mind applications to signal processing and speech processing. We shall begin by introducing the Paley–Wiener spaces, as the appropriate reproducing kernel Hilbert spaces for the problem. It turns out that questions of sampling are much simpler when we are able to sample at regular time intervals, and we shall begin by giving a treatment of the Shannon sampling theorem and the modification required for robust sampling in the presence of noise. The question of aliasing will also be treated.
Applications
Authors: Partington Jonathan R
Published: November 2023

This final chapter is written mainly for the benefit of the mathematician who wants to see how some of the theoretical results of the previous chapters can be applied to engineering problems. Being written by a mathematician, it will inevitably contain a personal and occasionally unrealistic view of what are some of the important problems in systems and control. We shall begin with an introduction to linear systems, showing how functions spaces such as the Hardy space H∞ arise naturally from physical situations, in particular from systems modelled by ordinary and partial differential equations. This will bring us naturally onto the subject of H∞ control, where mathematical problems arise which can be solved using techniques from interpolation and approximation theory. We shall then look at identification itself, considering some recent examples where the ideas presented in this book are used effectively.
Kirkman triple systems
Authors: Anderson Ian
Published: November 2023

Query 6 on page 48 of the Lady’s and Gentleman’s Diary for 1850 was posed by Kirkman as follows. Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily, so that no two shall walk twice abreast. Writing later that year, Kirkman said of the problem that it was ‘hardly worthy of mention if it were not that it has excited, as I hear, some attention among a far higher class of readers than those for whom the first 48 pages of the Diary are intended. I hit upon this pretty problem four years ago...’. It had certainly aroused interest.
Balanced tournament designs
Authors: Anderson Ian
Published: November 2023

A balanced tournament design is factored if, for each of the n grounds, there are n games played on that ground which together involve each of the 2n teams. The n pairs of teams are said to form a factor. Certainly no column contains any team more than twice. As far as the rows are concerned, row 0 and rows 2k + 1 to 4k each contain each team once, but rows 1 to 2k are unsatisfactory and require alteration. We show how to move entries around so as to obtain the required properties.
Whist tournaments
Authors: Anderson Ian
Published: November 2023

The concept of a whist (or doubles tennis) tournament was introduced in Section 2.3. Recall that a Wh(4n) consists of 4n -1 rounds of n games; in each game two players play against two others, and altogether (i) each player plays in one game per round; (ii) each player partners every other player exactly once; (iii) each player opposes every other player exactly twice.
Approach Properties
Authors: Lowen R
Published: November 2023

In this chapter we will study some natural invariants or properties of approach spaces. In view of the fact that approach spaces generalize at the same time topological spaces and metric spaces, the properties which we study will be generalizations of similar topological or metric properties. As a general rule we will adopt the following convention. If (P) is a topological property then we will say that an approach space has (P) if and only if its topological coreflection has (P). This rule will not be followed for properties of metric spaces, however.
DNA mismatches in solution
Authors: Chou Shan-Ho , Reid Brian R
Published: November 2023

The DNA double helix, with its complementary G:C and A:T Watson—Crick base pairing, is a remarkably efficient device for the storage and expression of information and the stable transmission of this information through successive generations. Although normal, or Watson-Crick, base pairing is mediated through hydrogen bonding between A and T residues and between G and C residues, the double helix is also stabilized by a variety of other ‘stacking’ interactions which obviously differ from one sequence to the next. In the process of copying each of the two strands of DNA to produce two identical double helices, i.e. daughter cells with the same genetic composition, incorrect or mismatch pairings (G or C with T or A, or with themselves) inevitably occur.
Basic error processing
Authors: Pretzel Oliver
Published: November 2023

The theoretical properties of a class of codes only bear fruit if there exists a practical error processing scheme that exploits them. So there have been continuing efforts to produce practical error processors for Goppa codes. The first processing algorithm was published by Justesen et al. (1989). It applied only to a somewhat restricted class of codes based on smooth plane curves and could not correct errors up to the full designed capability of the code.
Full error processing
Authors: Pretzel Oliver
Published: November 2023

The SY error processor is quite simple, but it does not exploit the full correction capacity t of the code it is designed for. It only corrects up to t –r 1 /2 errors for a code based on a curve of genus T This weakness is quite serious because the best geometric Coppa codes necessarily come from curves with a large genus.
Introduction: the algebraic approach
Authors: Pretzel Oliver
Published: November 2023

In Part I we saw how the algebraic structures associated with a curve can be used to construct good codes and error processors for them. The main aim of this part is to prove the theorems stated in Chapter 4. The simplest proofs by far are those that look only at the algebra of the function field of the curve.
Roger Penrose—a Personal Appreciation
Authors: Atiyah Michael
Published: November 2023

Roger Penrose and I were research students together in Cambridge from 1952 to 1955. Moreover, we were both algebraic geometers at the time. I did my research under Hodge, and Roger, after starting with Hodge, moved over to work with Todd. In view of later events it is perhaps interesting to review briefly what areas our research was concerned with. Under Hodge I was steered towards differential geometry and topology. From Hodge’s book I learnt about harmonic forms and their origin in Maxwell’s equations, while from the work of the French school (Leray, Cartan, Serre) I learnt about sheaf cohomology. Although I had attended Dirac’s course on quantum mechanics it was not until some years later that I became acquainted with spin and the Dirac operator.
Quantum Geometric Origin of All Forces in String Theory
Authors: Veneziano Gabriele
Published: November 2023

I am a great admirer of Roger, but I cannot pretend to know him as well as most of you here. Let me, however, start my talk with a small episode of my life indirectly related to him. A couple of years ago I bought The Emperor’s New Mind just before leaving for summer vacations in southern France. I took the book with me, hoping, of course, to enjoy my holiday so much as to “forget” the book in my suitcase. Things went differently: at my first tennis lesson I was struck by lumbago and finished my vacation reading the book over and over again by the hotel’s swimming pool ... This morning I went to a local bookstore and bought Shadows of the Mind. I am planning to take holidays in Italy in a couple of weeks and start to feel a strange ache at my left shoulder: I am not superstitious, but⃛
Interaction-Free Measurements
Authors: Vaidman Lev
Published: November 2023

I am greatly indebted to Roger Penrose. I have learned very much from his papers, from his exciting books, and from our (too short) conversations. I am most grateful to Roger for developing the idea of Avshalom Elitzur and myself on interaction-free measurements (IFM). The version of IFM Penrose described in his book (1994) is conceptually different from our original proposal, and although it is much more difficult for practical applications it has the advantage of demonstrating even more striking quantum phenomena. So I will start with presenting Penrose’s version of IFM.
Geometrically Finite Kleinian Groups
Authors: Matsuzaki Katsuhiko , Taniguchi Masahiko
Published: November 2023

Kleinian groups which are cofinite (namely, which correspond to complete hyperbolic 3-manifolds with finite volume) have several desirable properties. The cofiniteness can be generalized to the concept of geometric finiteness, which we discuss in this chapter. In Section 3.1, we give various definitions of geometrically finite Kleinian groups and prove that they are equivalent. Next in Section 3.2 we explain some of the fundamental properties of such groups. In Section 3.3, we consider the variety and rigidity of geometrically finite hyperbolic structures admitted in a given topological manifold or represented by isomorphic Kleinian groups.
Finitely generated Kleinian groups
Authors: Matsuzaki Katsuhiko , Taniguchi Masahiko
Published: November 2023

In this chapter, we consider hyperbolic 3-manifolds whose fundamental groups are finitely generated. Although this algebraic finiteness is equivalent to geometric finiteness in the 2-dimensional case, the concepts are not coincident in the 3-dimensional case.
Structure invariants, seminvariants and the origin problem
Authors: Giacovazzo Carmelo
Published: November 2023

Any ordered arrangement of atoms, that is any crystal structure, can be mathematically described provided the origin of the reference system is fixed. Vice versa, the phases of the structure factors define (through a Fourier transform) the atomic positions with respect to a given origin. Thus a mechanism must exist according to which phasing reflections simultaneously involves the fixing of the origin in direct space. This intriguing correspondence will be elucidated throughout this chapter by making use of the concepts of structure invariants and semivariants, which will prove to be central to direct methods theory.
Triplet invariant estimation and classical direct phasing procedures
Authors: Giacovazzo Carmelo
Published: November 2023

In the first part of this chapter we describe the use of the central limit theorem for the probabilistic estimation of the most important s.i., the triplet invariant. We limit our treatment to the case in which X-ray diffraction data are used without anamalous dispersion effects (see Chapter 16 for such an extension).
The method of joint probability distributions
Authors: Giacovazzo Carmelo
Published: November 2023

The joint probability distribution of a set of n normalized structure factors was first introduced by Hauptman and Karie (1953b) and was developed further by several authors (e.g. Bertaut 1955a, b, 1960a, b; Klug 1958; Naya et al. 1964, 1965; Vaughan 1959). The importance of their work lies in the introduction of the idea that a certain phase or combination of phases may be estimated when other related s.f.s have their observed values.
Representations of structure invariants and seminvariants
Authors: Giacovazzo Carmelo
Published: November 2023

In Chapter 5 we described the method of the joint probability distribution functions of structure factors. The aim was to describe the mathematical techniques for the estimation of a given s.i. or an s.s.
The probabilistic estimation of one- and two-phase structure seminvariants
Authors: Giacovazzo Carmelo
Published: November 2023

We briefly describe in this chapter the probabilistic formulae obtained via the method of joint probability distribution functions. Estimates of s.s.s can also be obtained via direct space methods, i.e. from Patterson properties. For this topic the reader is referred to Chapter 4.
Direct methods versus powder data
Authors: Giacovazzo Carmelo
Published: November 2023

Powder diffractometry plays (and will probably continue to play in the near future) a central role in research and technology because it allows one to investigate materials which are not available as a single crystal of adequate size or quality. Therefore much effort has recently been devoted to the development of powder diffraction.
Maximum entropy techniques for crystal structure solution
Authors: Giacovazzo Carmelo
Published: November 2023

Maximum entropy techniques have found application in many scientific fields, like statistical mechanics, spectroscopy, nuclear physics, radio astronomy, image enhancement, economics, and last, but not least, in crystallography. The method has its root in probability theory and can be considered a modern extension of this: it aims to derive the most convincing conclusions from given sets of data and prior information eventually available.
Names
Authors: Freer Stephen
Published: November 2023

Nomenclature , the second (151) foundation of botany, after the arrangement has been done (152), should first of all apply the names. If you do not know the names of things, the knowledge of them is lost too. Where there is a single genus, there will be a single name. Section 215. The names given by the ancients are mostly excellent; those given by more recent [writers] have been inferior. [My] Critica Botanica proposed nomenclature for genera, species, and varieties, with reasons and examples; hence there is not much about that subject in the present work.
Recognizing integrable functions II
Authors: Priestley H A
Published: November 2023

We may amalgamate earlier results to obtain the following integrability criterion. It should be treated as guidance on how to proceed in examples rather than a result to quote verbatim.
Measurable functions
Authors: Priestley H A
Published: November 2023

Elementary real analysis concentrates on functions which are, at a minimum, continuous. Integrability and continuity do not mesh well: there are continuous functions which are ‘too big’ to be integrable and integrable functions which are not continuous at any point (examples?). The appropriate concept of minimal good behaviour in integration theory is measurability.
Transformations of ℝk
Authors: Priestley H A
Published: November 2023

The preceding chapter emphasized the connections between integrals on JR and on JR2. There are aspects of the integral on ℝ2 (and likewise on ℝkfor k > 2) which are interesting and important in their own right, notably the relationship between Lebesgue measure and the rich geometry of ℝ We have defined the measure of a bounded measurable subset S of ℝkto be ∫ 𝒳S. This agrees, by definition, with the expected area when S is a bounded interval. Is the same true for other familiar figures, such as parallelograms? What happens to measure under transformations of the plane? Substitution is a valuable technique for evaluating integrals on JR. In the same way, it is useful to be able to make changes of variables in higher dimensional integrals. In full generality these topics involve multi-dimensional calculus beyond the scope of this book. Our aim here is to validate in the Lebesgue framework the use of the transformations which are most useful in practice. Beginners may wish to omit the proof of the main theorem, 27.4.
L2-spaces as Hilbert spaces
Authors: Priestley H A
Published: November 2023

In analysing an L2-space we can profitably exploit the interplay of its inner product structure with the convergence notions associated with its norm. An inner product space X is called a Hilbert space if the associated norm makes X a Banach space (that is, Cauchy sequences in X converge). Theorem 28.16 tells us that L2 is a Hilbert space, as is L2 (I) for any interval I.
Introduction
Authors: Cong Nguyen Dinh
Published: November 2023

Originating in the work of the French mathematician Henri Poincare in the last century, the theory of dynamical systems has become a major branch of the mathematical sciences. Its importance is explained by the fact that everything in the world evolves with time, so that a dynamical approach is required. Even those processes which seem to be unchanged with time can be viewed as stationary solutions (fixed points) of more general models. The theory of dynamical systems deals with long-term behavior of systems. It has three sub-areas: measurable dynamics (or ergodic theory), topological dynamics and differentiable dynamics. The set-up of the system under investigation determines whether we should deal with measurable, topological or differentiable properties. Of course, to make a mathematical investigation of possible long-term behavior, the evolution law is assumed to preserve the structure of the system.
Random Dynamical Systems: Foundations
Authors: Cong Nguyen Dinh
Published: November 2023

The theory of random dynamical systems is a relatively new field of research. It continues, extends and unites various developments in probability theory and the theory of dynamical systems. Suppose we wish to investigate a process of the real world. It often happens, one may say always happens, that this process is influenced by many factors, which are called noise. The first thing we can do is model the so-called ideal process, i.e. ignoring the noise, and assuming that the process evolves by its internal laws we model it by a dynamical system using a physical theory, certain biological or chemical laws, etc. Now, to take the influence of the external environment into account we should also model the noise.
The Linearization Of Nonlinear Random Dynamical Systems
Authors: Cong Nguyen Dinh
Published: November 2023

Linearization of a nonlinear system is the first step in the theory of topological dynamics which leads to a simplification of the system under investigation. Since Hartman-Grohman theory has successfully solved the linearization problem in the classical theory of dynamical systems one would naturally like to generalize it to the random case. It turns out that many results of the Hartman-Grohman theory permit more or less direct generalization to the random situation. In this chapter we shall present the linearization theory for smooth random dynamical systems. The last section of the chapter is devoted to the bifurcation theory of random dynamical systems.
Topological Classification of Discrete-Time Linear Hyperbolic Cocycles
Authors: Cong Nguyen Dinh
Published: November 2023

In this chapter we shall deal with the problem of the topological classification of linear hyperbolic cocyclcs. Looking back at the deterministic case, the work of Robbin [134] gives us an overview of random dynamical systems both from the ‘local’ and ‘global’ points of view. It also gives us very important tools and some initial ideas for dealing with random dynamical systems. We recall some deterministic results from Chapter 1.
Maximum cardinality matchings
Authors: Karpinski Marek , Rytter Wojciech
Published: November 2023

In this chapter we use algebraic tools (in particular parallel interpolation algorithms for large polynomials) to find maximum cardinality matchings in undirected graphs. We explore the correspondence between graphs and some black-box polynomials (Tutte’s symbolic polynomials). The crucial point is the use of determinants instead of permanents. There are NC-algorithms to compute determinants but there is a good chance (and our general belief) that there does not exist even a polynomial time sequential algorithm to compute permanents. Recall that A(n) denotes the number of processors needed to compute the determinant of an n x n matrix and the determinants of its minors in O(log2 n) time. We stated in Chapter 5 that A(n) = O(n3·5 • m), if our numbers have O(m log(m)) bits. We emphasize the importance of A since any improvement in the growth of A leads directly to an improvement in the parallel matching for general graphs. In this chapter two basic algorithms are presented: one based on partial interpolation and the second one using the smallest-degree algorithm. The first one is simpler, but deals with integers of exponential size 0(2m), so it needs O(n3·5 • m) processors, and it is O(m) times less efficient (in the sense of processor bound) than the second one. The second algorithm uses modular arithmetics in a clever way, thus reducing the size of the integers which are processed in the algorithm. Its efficiency (in the sense of the number of processors) is O(n3·5). Hence it is essentially the same as the complexity of matrix inversion: the matching problem is reduced to the computation of the determinant of the whole matrix and its minors.
Maximal independent sets
Authors: Karpinski Marek , Rytter Wojciech
Published: November 2023

The line graph of G is a graph L whose vertices are edges of G. Two vertices are neighbors in L if they have a common endpoint (as edges) in G. Then a set M of edges of G is a matching in G iff it is independent (as a set of vertices) in L. The NC-construction of a maximal independent set is also related to matchings in the following way: the basic part of the algorithm is a parallel construction of a matching with sufficiently small weight. In this chapter we show how to construct in parallel a maximal independent set. Of course construction of a maximum cardinality independent set is a much harder problem, since it is NP-complete. We present two very simple randomized algorithms due to Luby [186], one of which can be derandomized using the technique from Chapter 4. However, the number of processors of a derandomized version is too large. We then show a completely different and more efficient NC-algorithm, due to Goldberg and Spencer. We also mention some extensions to hypergraphs. The maximal independent sets problem is much harder for hypergraphs than for standard graphs and no NC-algorithm is known to solve it. Another generalisation of maximal independent sets is the notion of a k-dependent set, we cover this subject at the end of the chapter.
Pfaffians, counting the number of matchings, and planar graphs
Authors: Karpinski Marek , Rytter Wojciech
Published: November 2023

In this chapter we study ‘the notion of the Pfaffian of a graph and its connection to the problem of counting the number of matchings, and the problem of computing a Pfaffian orientation of a planar graph. In Section 6.1 we introduced the notions of Edmonds’ and Tutte’s matrices, and investigated the relation of their determinants to the problem of deciding the existence of a perfect matching. Towards the end of this section we will introduce the notion of a Pfaffian of Tutte’s matrix A corresponding to the two-cycle permutations over {1, ... , 2n}.
Some aspects of topological sequence spaces
Authors: Boos Johann , Cass Peter
Published: November 2023

In this chapter we consider several aspects of sequence spaces which have their origin in summability or which are links between topological sequence spaces and summability. Some (see Sections 11.1, 11.2 and 11.4) are based on ‘inclusion theorems’ where we show the equivalence of certain functional analytic properties of a sequence space Y with a statement about summability.
Rest Stop
Authors: David Guy , Semmes Stephen
Published: November 2023

So far we have the concepts of BPI spaces and BPI equivalence and some basic properties of them. In Chapters 6 and 7 we saw how the BPI condition and BPI equivalence are related to various other conditions. The techniques were based on coverings and cutting spaces up into little pieces. In Chapters 8 and 9 we established the basic machinery of compactness.
Iterative Domain Decomposition Methods at the Discrete Level
Authors: Quarteroni Alfio , Valli Alberto
Published: November 2023

In this chapter the iterative methods introduced in Sections 1.3 and 1.5 will be reformulated at the discrete level for the solution of the finite element problem (2.1.1). Their relation with domain decomposition preconditioners will be pointed out, and several acceleration techniques will be considered for the case where many subdomains are used. The convergence analysis will be carried out in Chapter 4.
Some applications of the kinematical theory of X-ray diffraction
Authors: Bradaczek Hans
Published: November 2023

Max von Laue’s detection (1912) of the X-ray diffraction (von Laue, 1960) by crystals simultaneously proved two facts: crystals have an internal lattice structure and X-rays have a wave nature. In subsequent years, an almost closed theory of X-ray diffraction has been developed by several scientists: von Laue, the Braggs, Ewald, Guinier, Borrmann, Hosemann and, of course, many others. Using function algebra, mainly developed by Hosemann and Bagchi (1962), most of the mathematical problems of diffraction theory can be solved and described in a very elegant way. The following derivations are almost exclusively using functional algebra. Only a few samples can be presented here.
Profile fitting and analytical functions
Authors: Enzo Stefano , Schiffini Liliana
Published: November 2023

Abstract

It is common to ascribe the line-broadening effects observed in X-ray powder diffraction patterns to various microstructural features like crystallite finiteness, microstrain and stacking faults. A presentation is given on an empirical basis of the parametric functions which are most popular in the Rietveld method of structure refinement in order to account for the observed broadening. The shape parameters obtained after the best fit of one or more profiles can be successfully used to determine physical values of interest, provided that the assumptions of the theory are reasonably well respected in the sample under investigation.
Effects of instrument function, crystallite size, and strain on reflection profiles
Authors: Honkimaki V , Suortti P
Published: November 2023

Abstract

A new approach to the analysis of reflection profiles is introduced. The basic idea is to calculate the instrument function by ray-tracing and include in the whole-pattern fitting a background function of the correct dependence on the scattering vector. In this way, only the effects of crystallite size and strain remain to be modeled. The instrument function is presented in general form, which can be adapted to energy-dispersive or angle-dispersive powder diffraction. The effects of the source size and wavelength distribution, equatorial and axial divergences, beam penetration in the sample, and different slits are included. Most of these aberrations are independent and can be cast in analytical forms, and the total instrument function is found by successive convolutions. The background arising from inelastic and resonant scattering is calculated from theory, while disorder scattering is modeled by a pair-correlation function and thermal diffuse scattering (TDS) by the Debye model for acoustic phonons and by the Einstein model for optical phonons. The total TDS is calculated as the part of the Bragg reflections lost due to thermal motion. The effects of crystallite size are treated in a general formalism where the reflection profile is the Fourier transform of the common volume of the crystallite and its ‘ghost’ shifted a distance tin the direction of the scattering vector k.The effects of strain are included in a distribution function of unit-cell displacements. Long-distance strain is described by a characteristic function, and the local strain by a correction term to the structure factor. Formulae that cover transition from local strain to long-distance strain are given. Anisotropic crystallite size and strain are described in terms of spherical harmonics which obey the crystal symmetry. The diffraction pattern of PbBr2 measured using CuKa1radiation is resolved using the procedure described above. The model parameters include only the thermal motion amplitudes and coefficients of the spherical harmonics that give the anisotropic crystallite size and strain.
Classical treatment of line profiles influenced by strain, small size, and stacking faults
Authors: Houska C R , Kuiel R
Published: November 2023

Progress in quantitative XRD profile analysis has been closely linked with instrumental improvements, which include higher intensity fine focus diffractometers equipped with electronic detectors. Early profiles measured without automation were described by a Fourier series. Numerical transforms were obtained using Lipson-Beevers strips and non-programmable, slowly responding mechanical calculators. These forgotten tedious efforts placed limitations on further advances in analytical profile analyses. Despite the absence of time-saving instrumentation, as it is known today, some of the best data and most complete analyses of XRD profiles go back to this early period (McKeehan and Warren, 1953). With high speed desktop computers now commonly available in most laboratories, one can make full use of recent analytical advances beyond the classical approaches to attain new levels of understanding. Modern instrumentation equipped with highly monochromatic radiation, greatly improved detectors, and labor saving automation allows us to examine smaller and more subtle profile changes.
Analytic functions describing line profiles influenced by size distribution, strain and stacking faults
Authors: Houska C R , Kuiel R
Published: November 2023

This chapter, describing XRD profiles computed from basic analytic functions, is best introduced by reading Chapter 6. Here we provide a brief review of the classical Fourier series treatment of microstrain variations and particle size as described by Warren (1959, 1969). Stacking fault broadening, related asymmetries, and shifts are treated as an expansion of the original development. These are limited primarily to fee, bee, and hep metals. Statistical results of highly deformed powders are obtained from a series of Fourier transforms of the experimental data corrected for instrumental broadening. In using this procedure, results remain closely linked to experimental profiles.
The dislocation-based model of strain broadening in X-ray line profile analysis
Authors: Ungar T
Published: November 2023

Shortly after the discovery of X-ray diffraction, materials scientists were surprised to find that severely deformed metallic materials retain their crystalline structure as indicated by the almost unaffected Debye–Scherrer diffraction lines (cf. Becker, 1927; Wilkens, 1988). This ‘almost unaffectedness’ has since become the field we know now as ‘X-ray line profile analysis’. Diffraction-line broadening has many physical sources: (i) size, (ii) strain, (iii) diffuse scattering of coherent precipitates, (iv) small misfits between phases with slightly different chemical compositions, including coherent precipitates, etc. The different effects interplay with each other and very often are not easy to separate.
Diffraction-line broadening analysis of strain fields in crystalline solids
Authors: Vermeulen A C , Delhez R , Keijser Th.H de , Mittemeijer E J
Published: November 2023

In the early days of X-ray diffraction analysis, it was recognized that the diffraction-line shape contains information about imperfections in crystalline phases. This realization is of utmost practical importance since the properties of many materials in daily life largely depend on those imperfections. A striking example is mechanical strength: it is one to two orders of magnitude smaller than that of a perfectly crystalline material due to the presence of dislocations.
Paracrystallinity
Authors: Bradaczek Hans
Published: November 2023

The main idea for developing the theory of the paracrystal was to bridge the gap between the crystalline and the amorphous state of matter. Because of the large variety of structural types, it cannot be expected that the paracrystalline model represents all of them. In most cases, only an approximate or statistical correspondence can be verified. In a paracrystalline structure, the relationship between neighboring particles is the determining element of construction. Contrary to ideal crystals, the location of a nearest neighbor particle cannot be found at a fixed distance and orientation.
The model of the paracrystal and its application to polymers
Authors: Wilke W
Published: November 2023

The term ‘paracrystal’ was used by Rinne (1932) as a method of notation for liquid crystalline structures. Hosemann (1950a,b) introduced the ‘ideal paracrystal’ as a special model for the description of statistical defined lattice structures. Later, he discussed the ‘real paracrystal’ and summarized the ideas and results about para-crystalline structures in a book (Hosemann and Bagchi, 1962). Today, paracrystals can be classified as systems with quasi long-range order (QLRO). For these systems, the translational correlations decay according to a power law (Venkataraman et al.,1989).
Crystallite statistics and accuracy in powder diffraction intensity measurements
Authors: Smith Deane K
Published: November 2023

Precision and accuracy are important concepts in any science involving measurements, and powder diffraction is no exception. Precision is a measure of the reproducibility of a measurement to obtain the same value. Accuracy is a measure of the correctness of the values measured, i.e. how close the measurements are to the true value. The classic illustration of the distinctions is illustrated in Fig. 17.1. Precision can be determined experimentally by repeating the measurements many times and applying the laws of statistics to the family of values to determine the mean, mode, standard deviation, and other parameters. Determining accuracy is not so easy, especially when one has no knowledge of the true value by any other measurement. One may have to resort to understanding the theory of the measurement technique to set a limit for potential accuracy.
Reciprocal space mapping and ultra-high resolution diffraction of polycrystalline materials
Authors: Fewster Paul F , Andrew Norman L
Published: November 2023

Conventional diffractometry of polycrystalline materials will swamp important features that characterise the microstructure of materials. The large instrumental reciprocal space ‘probe’ (or integration volume) of conventional slit-based instruments is beneficial in that the overall intensity is increased, but valuable information becomes swamped by instrumental artefacts. Ideally, if the material microstructure is poorly understood then a well-defined ‘probe’ should be used. A well-defined ‘probe’ will reduce the uncertainty in the interpretation. In reality, most investigations on ill-defined samples are performed with ill-defined probes. Here we will concentrate on how we can increase the access to more information by increasing the resolution. Ultimately, diffraction is a three-dimensional process and the micro-structure is three-dimensional and, therefore, an understanding of the nature of the full three-dimensional shape of the instrument function is required.
Texture effects in powder diffraction and their correction by simple empirical functions
Authors: Valvoda V
Published: November 2023

Non-random distribution of orientations of crystallites in powder specimens or compact polycrystalline samples can influence almost all accurate diffraction measurements. This affects mainly quantitative phase analysis and structure determination because both these methods are based on intensities of reflections from a randomly oriented assembly of particles. However, a strongly preferred orientation (texture) also causes problems in phase identification, stress measure-ments and profile analysis, especially in thin films. On the other hand, as has been pointed out by Bunge et al.(1989), texture can also help to separate completely overlapped reflections, like 330 and 411 reflections in the cubic 1-brass structure, if the sample has an appropriate preferred orientation.
Modelling of texture in whole-pattern fitting
Authors: Jiirvinen Matti
Published: November 2023

The reliability of intensity measurements in powder diffraction depends greatly on the treatment of sample effects. For instance, the texture of the sample can change the authentic intensity ratios of reflections and may, thus, introduce uncertainty in the results of an analysis. On the other hand, a powder sample, or a polycrystalline sample, with random grain orientation is difficult to prepare so that one is often obliged to use non-ideal samples in intensity measurements and attend mathematically to the texture effect. For this procedure analytical models are needed for the representation of the orientation distribution of the grains in the sample.
The role of whole-pattern databases in materials science
Authors: Smith Deane K
Published: November 2023

Now that the computer-controlled diffractometer is standard laboratory equipment, the digitized diffraction trace is the basis for most techniques of powder diffraction analysis. In addition to the geometric and structural information contained in the positions and intensities of the peaks in the diffraction pattern, the whole trace also contains the profile information that provides details on the physical state of the crystallites of the sample. Modern diffraction studies are concentrating on this profile information, and many computer programs have been prepared to carry out the calculations. The principal information which is now being exploited from diffraction patterns concerns crystallite size and size distributions, types and concentrations of defects and stacking faults, and texture.
Towards higher resolution: a mathematical approach
Authors: Reefman Derk
Published: November 2023

In the current use of X-ray diffraction, the required instrument precision is constantly increasing. With the advent of high-speed computers, more detailed models can be worked out with high qualitative precision. This in turn leads to the objective of extracting model-specific parameters from accurate experimental data. In order for this approach to be successful, a major prerequisite is that the available experimental data form a true projection of the material under study.
Use of pattern decomposition to study microstructure: practical aspects and applications
Authors: Louër Daniel
Published: November 2023

Since diffraction-line broadening is influenced by structural imperfections and the small size of crystallites (i.e. domains over which diffraction is coherent), powder diffraction is a valuable method for the determination of these properties. The theory of diffraction by imperfect solids was well established half a century after the introduction of X-ray diffraction and many applications were reported from powder diffraction data. These were essentially based on a Fourier analysis of diffraction-line broadening and, to a lesser extent, on an interpretation of the integral breadth, when an approximate characterization was sufficient (see Wagner, 1966). The detailed information, apparent crystallite size, cell-column length distribution and microdistortion, extracted from the theory based on Fourier series, has been found to be attractive and has often influenced the choice of the method for ‘precise’ line-broadening analyses, in spite of constraints (e.g. well-resolved lines are required) and propagation of errors during the treatment (Chapter 26; Young et al.,1967; Delhez et al.,1982). Experimental ‘crystallite size’ distributions are of low precision (see Bertaut, 1950; Le Bail and Louer, 1978) and their interpretation is complex, except in a few cases, such as in the direction perpendicular to uniformly-shaped disk or platelet crystallites (Le Bail and Louer 1978, 1980). As far as ‘size’ effect is concerned, Fourier apparent size ‘may appear to be of no advantage since a simple measurement of the line widths leads to another average function of the diameters’ (Guinier, 1963). Indeed, the theory of diffraction based on the integral breadth is well established (Chapter 5) and shows that average values of sizes and structural imperfections can be extracted from powder diffraction data. The general use of this approach was limited, in the past, by severe approximations (e.g. the correction of the instrumental contribution was based on simple assumptions about line shapes). The recent introduction of pattern modelling techniques, combined with the use of the properties of the Voigt function (Langford, 1978), has contributed to a renewal of interest in the use of the integral breadth for line-broadening analyses. The procedure presents the advantage of speed and convenience over the elaborate Fourier analysis, which may be a serious benefit when a number of samples for the same material have to be studied. Based on all information available in the diffraction pattern, the method is appropriate for characterizing anisotropic microstructural properties.