Scattering of X-Rays by Matter

The distribution of electrons in matter in the crystalline, amorphous, gaseous or liquid states is described by the electron density function p(r) whose value is given in units of electrons per unit volume [eA ] or [enm ]. The number of electrons contained in a volume element d is p(r)d T. This function has pronounced maxima at the centers of atoms and broad minima between them. The function also represents the X-ray scattering power per unit volume, the amplitude of the radiation scattered by the volume dh being proportional to the number of classical electrons that it contains. 

As shown in Fig. 3.11, the path difference between the wave A scattered by the volume element at the origin and the wave B scattered by the volume element at 

The total wave scattered by the sample in the direction s (Fig. 3.12) is thus given by 

The sum of all the waves given by G(S) rigorously represent the density p(r). The limit of resolution of a microscope (Section 3.1.1) is due to the fact that certain vectors S are experimentally inaccessible because the maximum value of S is 2/A (3.21) (Fig. 3.12). G(S) is a complex quantity, 

This relationship indicates the origin of the phase problem (Section 3.1.1) whose solution is one of the important tasks in X-ray crystallography there exist an infinite number of functions p(r) which give rise to the same function /(S). If p(r) is given, we can always calculate the corresponding function G(S). The passage from I G(S) to p(r), i.e. the solution of the phase problem, is only possible on the basis of models the most important will be developed in Sections 3.3.3 and 3.4.1. 

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The scattering of X rays by matter consists of two processes. The first process is the classical incoherent scattering, with no change in wavelength, called Thomson scattering, where the average intensity (S) of plane-polarized radiation, due to an electric field E, incident on a single electron is... 

Silicate— A mineral containing the elements silicon and oxygen, and usually other elements as well. X-ray diffraction— A method using the scattering of X rays by matter to study the structure of crystals. 

The physics of X-ray refraction are analogous to the well known refraction of light by optical lenses and prisms, governed by Snell s law. The special feature is the deflection at very small angles of few minutes of arc, as the refractive index of X-rays in matter is nearly one. Due to the density differences at inner surfaces most of the incident X-rays are deflected [1]. As the scattered intensity of refraction is proportional to the specific surface of a sample, a reference standard gives a quantitative measure for analytical determinations. 

BraceweU R (1999) The Fourier Transform and Its Applications, Me Graw-Hill, New York, 3rd edn. Guinier A and Fbumet G (1955) Small-Angle Scattering of X-Rays, Chapman and Hall, London, UK. Hosemann R and Bagchi S N (1962) Direct Analysis of Diffraction by Matter, North-Holland, Amsterdam, Netherland. 

There was no experimental evidence for the wave nature of matter until 1927, when evidence was provided by two independent experiments. Davisson found that a diffraction pattern was obtained if electrons were scattered from a nickel surface, and Thomson found that when a beam of electrons is passed through a thin gold foil, the diffraction pattern obtained is very similar to that produced by a beam of X-rays when it passes through a metal foil. 

The application of X-ray scattering for the study of soft matter has a long tradition. By shining X-rays on a piece of material, representative structure information is collected in a scattering pattern. Moreover, during the last three decades X-ray scattering has gained new attractivity, for it developed from a static to a dynamic method. 

The first convincing description of stationary quantum states was provided by the assumed wave nature of matter proposed by Louis de Broglie. The proposal had its roots in Einstein s explanation of the photoelectric effect and Compton s analysis of X-ray scattering. 

In the diffraction pattern from a crystalline solid, the positions of the diffraction maxima depend on the periodicity of the structure (i.e. the dimensions of the unit cell), whereas the relative intensities of the diffraction maxima depend on the distribution of scattering matter (i.e. the atoms, ions or molecules) within the repeating unit. Each diffraction maximum is characterized by a unique set of integers h, k and l (called the Miller indices) and is defined by a scattering vector h in three-dimensional space, given by h=ha +A b +Zc. The three-dimensional space in which the diffraction pattern is measured is called reciprocal space , whereas the three-dimensional space defining the crystal structure is called direct space . The basis vectors a, b and c are called the reciprocal lattice vectors, and they depend on the crystal structure. A given diffraction maximum h is completely defined by the structure factor F(h), which has amplitude F(h) and phase a(h). In the case of X-ray diffraction, F(h) is related to the electron density p(r) within the unit cell by the equation... 

Since the interaction of hard X-rays with matter is small, the kinematical approximation of single scattering is valid in most cases, except for perfect crystals near Bragg scattering. The intensity scattered by a block-shaped crystal with N, q and N, unit cells along the three crystal axes defined by the vectors Uj, a and a, takes the form ... 

X-rays interact with electrons in matter. When a beam of X-rays impinges on a material it is scattered in various directions by the electron clouds of the atoms. If the wavelength of the X-rays is comparable to the separation between the atoms, then interference can occur. For an ordered array of scattering centres (such as atoms or ions in a crystalline solid), this can give rise to interference maxima and minima. The wavelengths of X-rays used in X-ray diffraction experiments therefore typically lie between 0.6 and 1.9 A. 

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