X Ray Scattering

12 Key excerpts on "X Ray Scattering"

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  Structural and Resonance Techniques in Biological Research

  • Denis Rousseau(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  Significant advances in experimental technology have opened a new vista of small-angle scattering studies. For instance, dynamic structural studies can now be performed with nanosecond time resolution. Structural deter-mination of the location of trace elements at a concentration of one atom in ten thousand is also practical. Of the possibilities, only a few have been performed. Based on these positive results, it is clear that these new methods will find wide application in the coming decade. The scope of this chapter is to introduce the technology and methods made available since the early 1970s. A very brief review of elementary theory will first be given, to provide a necessary background, followed by a description of the technology. Application of the new experimental tech-niques will then be presented to illustrate the methods which have been spawned. II. X-Ray Scattering and Diffraction The distinction between scattering and diffraction is not well defined (partic-ularly in biological systems). Both areas are really one, from a theoretical point of view. If a sample is placed in a collimated beam of χ rays, a pattern will be formed on an x-ray-sensitive film placed at some distance behind the specimen. The central beam is prevented from striking the film and obliter- 6. SMALL-ANGLE X-RAY SCATTERING AND DIFFRACTION 439 ating the pattern by a beam stop. If the pattern consists of sharp spots or rings, then the phenomena is known as diffraction. If, on the other hand, the pattern consists of very diffuse features, then the term scattering applies. Diffraction occurs if the atomic structure is arranged in a periodic fashion with long-range order. Scattering occurs if the structure lacks such order (e.g., a gas or a liquid). Biological samples display a range of structural order extending from highly crystalline (e.g., arrays of lipid bilayers) to nearly total disorder (e.g., a protein solution).

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  Non-destructive Micro Analysis of Cultural Heritage Materials

  • K. Janssens, R. Van Grieken(Authors)
  • 2004(Publication Date)
  • Elsevier Science

    (Publisher)

  Janssens 4.2.4 Scattering and diffraction Scattering is the interaction between radiation and matter which causes a photon to change direction. If the energy of the photon is the same before and after scattering, the process is called elastic or Rayleigh scattering. Elastic scattering takes place between photons and bound electrons and forms the basis of XRD. If the photon loses some of its energy, the process is called inelastic or Compton scattering. Accordingly, the total cross-section for scattering s i can be written as the sum of two components: s i ¼ s R ; i þ s C ; i ð 4 : 7 Þ Fig. 4.5. Variation of characteristic line wavelengths with atomic number. Fig. 4.6. Characteristic X-ray lines observed in the K series. 137 X-ray based methods of analysis where s R, i and s C, i , respectively, denote the cross-sections for Rayleigh and Compton scatter of element i . Compton scattering occurs when X-ray photons interact with weakly bound electrons. After inelastic scattering over an angle f , a photon (see Fig. 4.7), with initial energy E , will have a lower energy E 0 given by the Compton equation: E 0 ¼ E 1 þ E m 0 c 2 ð 1 2 cos f Þ ð 4 : 8 Þ where m 0 denotes the electron rest mass. 4.2.4.1 Interference and diffraction When a wave front of X-rays strikes an atom, the electrons in this atom will scatter the X-rays (see above: Rayleigh scattering). The elastically scattered wave is immediately re-emitted in all directions and can be imagined as a spherical wave front (see Fig. 4.8). When a line of identical atoms, a distance a apart, is considered, and a wave (with wavelength l and wave vector ~ k ; where l ~ k l ¼ 1 = l ) approaches this line under an angle u , the wave front arriving at the second atom will have a path different d 1 ¼ OP ¼ a cos( u ) relative to that arriving at the first atom Fig. 4.7. Geometry for Compton scattering of X-ray photons. 138 K. Janssens

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  Synchrotron Radiation

  Sources and Applications to the Structural and Electronic Properties of Materials

  • D. Phil Woodruff(Author)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  5 Local Structural Techniques 5.1 Introduction The structural techniques described in Chapter 4 exploit the special way in which crystalline materials – materials in which the constituent atoms are arranged on a three-dimensionally periodic lattice – scatter (‘diffract’) X-rays that have a wavelength comparable to typical interatomic distances. Not all materials have this property, and some that do cannot be grown as crystals that are sufficiently large to enable a complete structure determination, even by powder diffraction, a technique designed for the study of small crystals. Nevertheless, non-crystalline materials may possess local atomic-scale ordering; for example a particular elemental species in the material may arrange to have a specific number of atoms of another element at a preferred separation. Not only are there X-ray scattering techniques that cast light on this ordering, but there are also electron scattering techniques that provide this information but rely on the supply of X-rays of tuneable energy to deliver these electrons; specifically, they exploit the photoelectrons emitted within the material by these incident X-rays. X-ray scattering may also be used to determine structural and morphological aspects of materials on a significantly longer distance scale. This chapter outlines the principles of some of these methods that have been found to benefit from, or are totally dependent on, synchrotron radiation. 5.2 X-Ray Scattering from Non-Crystalline Samples: General Considerations In Chapter 4 the constraints imposed on the scattering vector Δk by X-ray scattering from a three-dimensionally periodic (crystalline) solid were discussed in terms of the reciprocal lattice. Fig. 5.1 shows the basic definition of the scattering vector with no reference to whether or not the scatterers are in a crystalline material. Notice that by convention the angle between k in and k out is defined as 2θ.

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  X-Ray Diffraction

  Modern Experimental Techniques

  • Oliver H. Seeck, Bridget Murphy, Oliver H. Seeck, Bridget Murphy(Authors)
  • 2015(Publication Date)
  • Jenny Stanford Publishing

    (Publisher)

  This means that electromagnetic waves can be used to investigate the atomic structure of matter without affecting the atomic potentials. X-rays are such electromagnetic waves with wave lengths in the same order as the typical atomic distances in matter, namely, 0.1…10 Å. Therefore, X-rays are especially sensitive to the atomic potential. X-ray scattering and diffraction means that the incoming electromagnetic wave field is coherently deflected at the electric or magnetic potential of sample [see 3]. The full dynamical X-ray scattering theory, which includes multiple scattering effects, is rather complex [see 4–6]. Fortunately, in many cases approximations can be applied, in particular the kinematical approximation or Born approximation for elastic scattering. The Born approximation neglects any kind of multiple scattering effects or energy gain or loss and gives a very descriptive insight into scattering theory. It can be used to explain a large number of different scattering experiments [7]. In this chapter, the fundamental terms of X-ray diffraction and scattering are introduced, and the Born approximation is explained. Specific examples are presented for which the Born approximation yields good results. The conditions where the Born approximation fails are also shown. For these cases, methods from the dynamical scattering theory are presented. Special techniques such as coherent scattering and tomography are discussed in following chapters. 1.1 Scattering at Single Electrons The scattering process of X-rays at single electrons has been described in text books [see 3,7,8] and is not explained here in detail. Instead, the scattering will be introduced in simple terms so that in the following the Born approximation of X-ray scattering can be introduced. 3 For this, it is assumed that the incident electromagnetic wave is a plane wave with wave length l . The wave length and the photon energy of electromagnetic waves are connected by E =  w .

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  High-Intensity X-rays - Interaction with Matter

  Processes in Plasmas, Clusters, Molecules and Solids

  • Stefan P. Hau-Riege(Author)
  • 2012(Publication Date)
  • Wiley-VCH

    (Publisher)

  Chapter 3 Scattering of X-Ray Radiation

  In this chapter we review the principles of x-ray scattering. When an electromagnetic wave irradiates an atom or ion, a photon may be absorbed or scattered. If a photon is scattered and its energy is conserved, the photon is said to undergo elastic or Thomson or Rayleigh scattering. Elastically scattered light is useful for structural investigations using x rays, such as for x-ray crystallography or for coherent diffractive imaging. If the photon transfers energy to the atom and changes its wavelength, it is said that the photon undergoes inelastic, modified, Compton or Raman scattering. In the first part of this chapter we will describe scattering of photons by free charges like electrons and nuclei, followed by a description of scattering by bound electrons, crystals, and plasmas.

  3.1 Scattering by Free Charges

  We will first review the classical electromagnetic theory for scattering which is based on the periodic motion of charges in a time-varying electric field that leads to the emission of scattered radiation [1, 2]. The phenomenon of Compton scattering demonstrates that actual x-ray scattering does not follow classical electromagnetic theory and needs to be treated in a quantum theoretical framework. Nevertheless, we will still discuss the classical theory since scattering is often described and tabulated in units of classically derived expressions.

  3.1.1 Classical Description (Thomson Formula)

  We consider the case of a monochromatic plane wave with amplitude irradiating a free particle of charge q and mass m located at the origin. E 0 is the electric field vector, and k 0 is the wave vector of magnitude 2π /λ . In vacuum, c = ωλ /2π , where c is the velocity of light, λ is the wavelength, and ω is the frequency. The electric field accelerates the particle, and therefore radiation is emitted. In classical nonrelativistic electrodynamics, the incident and emitted radiation have the same frequency. From Maxwell equations (1.5) to (1.8) it can be derived [3] that the amplitude of the scattered light at a position r far from the particle (r >> λ

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  High Resolution X-Ray Diffractometry And Topography

  • D.K. Bowen, Brian K. Tanner(Authors)
  • 1998(Publication Date)
  • CRC Press

    (Publisher)

  4 X-ray Scattering Theory In this chapter we first extend the kinematical theory, discussing the strengths of X-ray reflections, forbidden reflections, the thin crystal solution, anomalous dispersion and reciprocal space geometry. We then review the results of the dynamical theory: deviation parameters, dispersion space and geometry, boundary conditions, the range of Bragg reflections, polarisation, intensity formulae and penetration depth. We briefly discuss spherical wave theory, and Penning Polder theory for distorted crystals. Finally we give a summary of useful formulae. 4.1 Introduction The theory of X Ray Scattering is highly practical! It is an accurate theory, based on a few sound assumptions, and with implementations on personal computers it may be used to interpret the structures of advanced industrial materials and thereby to assist in process development and quality control. The characterisation scientist or engineer who has a good grasp of the theory will therefore be able to design better measurements and to interpret them more accurately. It is not our intention to provide full derivations of X-ray scattering theory, since this is primarily of interest to the specialist and may be found in many excellent books and reviews. The characterisation scientist needs a qualitative understanding in order to appreciate general features of experiment design and the interpretation of high resolution rocking curves and images. He or she also needs specific numbers such as the ideal rocking curve width or the penetration depth for a particular specimen. Our aim is therefore simply to explain the aspects of the theory that are relevant to later chapters and to summarise the important formulae. It is conventional and useful to approach X-ray scattering theory on two levels, the so-called kinematical and dynamical theories.

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  The Materials Physics Companion

  • Anthony C. Fischer-Cripps(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  Atomic number Z 0 10 20 30 40 When x-rays pass through a material, the intensity of the beam is reduced by absorption and scattering. 1.4.5 X-Ray Absorption The Materials Physics Companion 66 Consider a beam of x-rays incident on a unit area of material containing N atoms. Each atom within the material presents a cross- sectional area  to the rays through which the x-rays cannot pass. If there are n atoms per unit volume, then the fractional loss of intensity for all the atoms is equal to:   I o I o I x I x n I I x       1 V N n  where I/I x x x n x x I e I e I I x n I I dx n dI I o               o o o 0 ln 1  is called the linear attenuation coefficient. V A N N V m n       Now, A i density When the material contains a 1 x I/I o A A A A A N A N A N n n N A V N n V N V          Atomic weight thus, mixture of atoms, n 1 , n 2 , …, the mass attenuation coefficient is given by the weighted sum: 2 2 1 1 2 2 1 1 2 2 2 1 1 1 m m m m m w w A N A N N A N A              weight fraction Avogadro’s number m A A      mass attenuation coefficient.  m is independent of the state (liquid, gas, etc.) of matter. Attenuation of x-rays by a target material occurs due to scattering and photoelectric effects. 1.4.6 Attenuation of X-Rays 67 1.4 X-Ray Diffraction In coherent scattering, an x-ray sets an electron in a target atom into oscillation and x-rays at the same wavelength are re-radiated in all directions. This is called Thomson scattering. In incoherent scattering, energy from the x-ray is transferred to an electron (but not enough to cause a transition as in photoelectric attenuation), and a longer wavelength x-ray is re-radiated.

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  X-ray Scattering From Semiconductors And Other Materials (3rd Edition)

  • Paul F Fewster(Author)
  • 2015(Publication Date)
  • WSPC

    (Publisher)

  Chapter 2

  THE THEORY OF X-RAY SCATTERING

  This chapter presents the scattering theories applicable to analyzing thin film and polycrystalline materials. The 2-beam dynamical theory is derived, because of its usefulness in simulating scattering from epitaxial semiconductors. The limitations of this model are discussed along with extensions and alternatives that are more exact, including an approach that generates all the appropriate reflections making it a truly multiple-beam dynamical theory. The kinematic model is discussed and its applicability to thin films and imperfect materials, and how the optical theory is adequate for specular reflectivity. An alternative theory for X-ray diffraction is also presented that considers the scattering throughout space, which is particularly relevant for studying polycrystalline materials. It will also become obvious how the whole process is intricately linked to the instrument. The distorted-wave Born Approximation is discussed along with applications in modeling diffuse scattering.

  2.1. The interaction of X-ray photons with the sample

  The X-ray photon interacts with the sample in many different ways and the form of interaction depends on the photon energy and the nature of the sample. X-ray photons are electromagnetic and it is the electric field vector that interacts most strongly with the sample. The magnetic interaction is small and is only observable under special conditions with very intense X-ray sources. There are several forms of interaction depending on the photon energy and the nature of the electron state. Electrons loosely bound to atoms, for example the valence electrons, may absorb part of the energy of a photon and the emitted photon will have a lower energy and longer wavelength. If it is assumed that the electron is stationary and totally unbound then this wavelength change is given by;

  This basically reflects the kinetic energy taken up by the electron. This interaction is termed Compton scattering, Compton (1923). The wavelength change is therefore independent of the wavelength of the incident photon but varies with scattering angle, 2θ, and is small (~0.024Å at most). An electron is not stationary or totally unbound in a solid and this will influence the energy (and wavelength) spread of the scattered photon. This makes the Compton scattering process a very useful tool for studying electron momenta in solids, etc. Because the wavelength change is so small, typical X-ray detectors used in diffraction experiments cannot discriminate this contribution from elastic scattering processes, therefore Compton scattering appears as a background signal. Each photon involved in this process will scatter independently. The scattering probability of coherent and Compton scattered photons for any given atom are of the same magnitude. However waves scattered in phase redistribute this intensity into sharp maxima that give intensities approximately related to N2 (where N is the number of contributing atoms) compared to N for Compton scattering. Since N is generally very large the proportional contribution of Compton scattering is negligible, unless we are dealing with samples of very poor crystallinity. Equation 2.1

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  Practical Radiotherapy

  Physics and Equipment

  • Pam Cherry, Angela M. Duxbury, Pam Cherry, Angela M. Duxbury(Authors)
  • 2019(Publication Date)
  • Wiley-Blackwell

    (Publisher)

  The electrons may interact with other atoms in the medium causing ionisation and excitation (see Chapters 2 and 3). This results in the chemical and biological changes important to radiotherapy. Scattering occurs following a collision interaction between the primary X‐ray beam and the atoms of the medium through which the X‐ray beam is passing. The incident photon is deflected out of the path of the primary beam and travels onward in a new direction. This collision may or may not involve transference of energy from the incident photon to the medium. The size of the nucleus of an atom is extremely small compared with the overall size of the atom and so most of the atom is considered as space. Therefore, there is a high probability that some X‐rays will pass straight through a medium without undergoing absorption or scattering (Figure 5.1). The X‐rays are said to be transmitted and it is these X‐rays that play a part in the production of a radiographic image and contribute to the exit dose of a beam of X‐rays used for radiation treatment. FIGURE 5.1 Process of attenuation. (A) transmission; (B) scattering; (C) absorption. 5.1.2 Exponential Relationship Although it is impossible to predict which photons will interact with the medium, it is possible to predict the fraction of total photons that will undergo interaction. Experimentally it can be shown that for a narrow, homogeneous beam of X‐rays, i.e. a beam of photons of similar energy, the intensity of radiation transmitted is reduced in an ‘exponential’ manner (Figure 5.2). Equal thicknesses of uniform attenuating material placed in the path of the beam produce equal fractional reductions in the intensity of radiation transmitted, e.g. a thickness of attenuator (t) reduces the intensity of the beam initially by 50% (100% reduced to 50%), then again by 50% (50% reduced to 25%)

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  Modern Diagnostic X-Ray Sources

  Technology, Manufacturing, Reliability

  • Rolf Behling(Author)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  As in the case of Thomson scattering of X-rays in the energy range for medical imaging, the electrons again oscillate in antiphase with the incoming X-ray wave. Therefore, Rayleigh scattering of high-energy photons also delivers an index of refraction below unity. However, the description of this process is more complex than for Thomson scattering. Although its total cross section turns out to be smaller than that of other processes that drive photons off the rectilinear direction of the primary X-ray beam, its peculiar angular distribution, which features a pronounced forward enhancement, is of practical importance. Low- Z materials, such as carbon, glass, beryllium, and aluminum, are often in the beam to serve as a vacuum container or for filtration purposes. The X-ray window of many X-ray tube types comprises a sheet of oil for cooling. These materials scatter the primary radiation. If the scattering angle is small, this radiation may end up in the used beam and appear as off-focal radiation, emitted from the window area of the source. Without further countermeasures, it may deteriorate the image quality. Dyson (1990, Chapter 5) provides a practical treatment of Rayleigh scattering, which shall be the basis of the following. An elaborate treatment can also be found in Podgoršak (2010, Section 7.4). The angular distribution of Rayleigh scattered photons differs markedly from that of Thomson scattering, which occurs at loosely bound electrons. As entire atoms are affected, the notion of electron scattering cross section is not adequate. Instead, the cross section should be stated for X-rays scattered at atoms

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  Nanobeam X-Ray Scattering

  Probing Matter at the Nanoscale

  • Julian Stangl, Cristian Mocuta, Virginie Chamard, Dina Carbone(Authors)
  • 2013(Publication Date)
  • Wiley-VCH

    (Publisher)

  2

  X-ray Diffraction Principles

  In this chapter some basic concepts and nomenclature of diffraction are introduced to make the book self-consistent and to provide a formal framework for the understanding of the experimental issues discussed later in this book. There exist many well-written books on diffraction [3, 4], and the reader who is interested in a deeper understanding of the subject is referred to them. Our choice is to keep this introduction very informative to benefit the reader who has little familiarity with this topic.

  2.1 A Brief Introduction to Diffraction Theory

  2.1.1 Interference of X-ray Waves

  The basic concept behind X-ray diffraction is the elastic scattering of an electromagnetic wave by the electrons present in the sample. An X-ray wave is scattered elastically if the scattering process does not cause a change of its wavelength, but only of its direction. The elastic scattering, therefore, is a conserving process, that is it does not involve any energy loss, neither by transfer of energy to the electrons, nor by generation of electronic or phonon excitation of the sample.

  X-rays, as electromagnetic waves, are described by a wave vector k (cf. Figure 2.1 ), related to the wavelength λ by the following relation:

  (2.1)

  where λ depends on the energy E (for X-rays typically in the range of 102 –105 eV) in the following way:

  (2.2)

  with h and c being the universal constant and the speed of light, respectively.

  Figure 2.1 The electric and magnetic fields are mutually perpendicular, and perpendicular to the propagation direction k . The wavelength λ is highlighted as a distance between two consecutive maxima of the electric (magnetic) field.

  The phenomenon of diffraction consists of the interference of waves elastically scattered by all the electrons present in the material illuminated by the radiation. This interference produces an angular distribution of intensity of radiation that holds information about density and distribution of electrons inside the specimen.

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  Characterization of Biomaterials

  • M Jaffe, W. Hammond, P Tolias, T Arinzeh(Authors)
  • 2012(Publication Date)
  • Woodhead Publishing

    (Publisher)

  Fig. 2.4a . X-ray sources are typically sealed tubes or rotating anodes. The wavelength of the radiation is  ~1 Å. Synchrotron radiation is now widely used for specialized studies, as well as for studying small sample, weakly scattering systems and for time- or temperature-resolved experiments.

  2.4 Instrumentation for (a) WAXS and (b) SAXS. (Source: Images courtesy of Bruker AXS Inc.)

  A few basic examples of the use of X-ray scattering will be described here. The availability of synchrotron sources has provided an opportunity to explore other aspects of phase behavior not discussed here. These include the use of microbeam X-ray diffraction in which a beam less that 5 μm can be used to map the phase separation on micrometer length scales. Glancing angle techniques are used to study the changes in the phase behavior as a function of depth near surfaces. Combining techniques, such as SAXS/WAXS with thermal and mechanical tests, provides further opportunities to study phase behavior with heat, and the effect of phase morphology on mechanical properties.

  2.3.1 Crystal structure analysis

  Atoms in metals and ceramics are almost always arranged into regular lattices. Polymer molecules too can be arranged into similar lattices. At the smallest length scales, established crystallographic techniques are used to determine the crystal structures from which the atomic coordinates can be derived. Ideally, single crystals are a prerequisite to determine the crystal structure. However, numerous structures have been solved using data from fibers and polycrystalline powders. These techniques are very relevant to biomaterials that cannot always be obtained as single crystals. An example of the arrangement of atoms that can be derived from such crystal structure analysis from fiber-diffraction data is shown in Fig. 2.5 .4

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