X-ray diffraction

X-ray diffraction is a generic term for phenomena associated with changes in the direction of X-ray beams due to interactions with the electrons around atoms. It occurs due to elastic scattering, when there is no change in the energy of the waves. The resulting map of the directions of the X-rays far from the sample is called a diffraction pattern. It is different from X-ray crystallography which exploits X-ray diffraction to determine the arrangement of atoms in materials, and also has other components such as ways to map from experimental diffraction measurements to the positions of atoms. This article provides an overview of X-ray diffraction, starting with the early history of x-rays and the discovery that they have the right spacings to be diffracted by crystals. In many cases these diffraction patterns can be Interpreted using a single scattering or kinematical theory with conservation of energy (wave vector). Many different types of X-ray sources exist, ranging from ones used in laboratories to higher brightness synchrotron light sources. Similar diffraction patterns can be produced by related scattering techniques such as electron diffraction or neutron diffraction.…

X-ray diffraction is a generic term for phenomena associated with changes in the direction of X-ray beams due to interactions with the electrons around atoms. It occurs due to elastic scattering, when there is no change in the energy of the waves. The resulting map of the directions of the X-rays far from the sample is called a diffraction pattern. It is different from X-ray crystallography which exploits X-ray diffraction to determine the arrangement of atoms in materials, and also has other components such as ways to map from experimental diffraction measurements to the positions of atoms. This article provides an overview of X-ray diffraction, starting with the early history of x-rays and the discovery that they have the right spacings to be diffracted by crystals. In many cases these diffraction patterns can be Interpreted using a single scattering or kinematical theory with conservation of energy (wave vector). Many different types of X-ray sources exist, ranging from ones used in laboratories to higher brightness synchrotron light sources. Similar diffraction patterns can be produced by related scattering techniques such as electron diffraction or neutron diffraction. If single crystals of sufficient size cannot be obtained, various other X-ray methods can be applied to obtain less detailed information; such methods include fiber diffraction, powder diffraction and (if the sample is not crystallized) small-angle X-ray scattering (SAXS). ==History== When Wilhelm Röntgen discovered X-rays in 1895 physicists were uncertain of the nature of X-rays, but suspected that they were waves of electromagnetic radiation. The Maxwell theory of electromagnetic radiation was well accepted, and experiments by Charles Glover Barkla showed that X-rays exhibited phenomena associated with electromagnetic waves, including transverse polarization and spectral lines akin to those observed in the visible wavelengths. Barkla created the x-ray notation for sharp spectral lines, noting in 1909 two separate energies, at first, naming them "A" and "B" and, supposing that there may be lines prior to "A", he started an alphabet numbering beginning with "K." Single-slit experiments in the laboratory of Arnold Sommerfeld suggested that X-rays had a wavelength of about 1 angstrom. X-rays are not only waves but also have particle properties causing Sommerfeld to coin the name Bremsstrahlung for the continuous spectra when they were formed when electrons bombarded a material. Albert Einstein introduced the photon concept in 1905, but it was not broadly accepted until 1922, when Arthur Compton confirmed it by the scattering of X-rays from electrons. The particle-like properties of X-rays, such as their ionization of gases, had prompted William Henry Bragg to argue in 1907 that X-rays were not electromagnetic radiation. Bragg's view proved unpopular and the observation of X-ray diffraction by Max von Laue in 1912¹ confirmed that X-rays are a form of electromagnetic radiation. X-ray interference patterns published in Von Laue's 1912 paper ]] The idea that crystals could be used as a diffraction grating for X-rays arose in 1912 in a conversation between Paul Peter Ewald and Max von Laue in the English Garden in Munich. Ewald had proposed a resonator model of crystals for his thesis, but this model could not be validated using visible light, since the wavelength was much larger than the spacing between the resonators. Von Laue realized that electromagnetic radiation of a shorter wavelength was needed, and suggested that X-rays might have a wavelength comparable to the spacing in crystals. Von Laue worked with two technicians, Walter Friedrich and his assistant Paul Knipping, to shine a beam of X-rays through a copper sulfate crystal and record its diffraction pattern on a photographic plate. After being developed, the plate showed rings of fuzzy spots of roughly elliptical shape. Despite the crude and unclear image, the image confirmed the diffraction concept. The results were presented to the Bavarian Academy of Sciences and Humanities in June 1912 as "Interferenz-Erscheinungen bei Röntgenstrahlen" (Interference phenomena in X-rays). After seeing the initial results, Laue was walking home and suddenly conceived of the physical laws describing the effect. Laue developed a law that connects the scattering angles and the size and orientation of the unit-cell spacings in the crystal, for which he was awarded the Nobel Prize in Physics in 1914. After Von Laue's pioneering research the field developed rapidly, most notably by physicists William Lawrence Bragg and his father William Henry Bragg. In 1912–1913, the younger Bragg developed Bragg's law, which connects the scattering with evenly spaced planes within a crystal. The Braggs, father and son, shared the 1915 Nobel Prize in Physics for their work in crystallography. The earliest structures were generally simple; as computational and experimental methods improved over the next decades, it became feasible to deduce reliable atomic positions for more complicated arrangements of atoms; see X-ray crystallography for more details. ==Introduction to x-ray diffraction theory== 2 d\sin\theta = n\lambda. A reflection is said to be indexed when its Miller indices (or, more correctly, its reciprocal lattice vector components) have been identified from the known wavelength and the scattering angle 2θ. Such indexing gives the unit-cell parameters, the lengths and angles of the unit-cell, as well as its space group. ===Ewald's sphere=== Each X-ray diffraction pattern represents a spherical slice of reciprocal space, as may be seen by the Ewald sphere construction. For a given incident wavevector k 0 the only wavevectors with the same energy lie on the surface of a sphere. In the diagram, the wavevector k 1 lies on the Ewald sphere and also is at a reciprocal lattice vector g 1 so satisfies Bragg's law. In contrast the wavevector k 2 differs from the reciprocal lattice point and g 2 by the vector s which is called the excitation error. For large single crystals primarily used in crystallography only the Bragg's law case matters; for electron diffraction and some other types of x-ray diffraction non-zero values of the excitation error also matter. ===Scattering amplitudes=== X-ray scattering is determined by the density of electrons within the crystal. Since the energy of an X-ray is much greater than that of a valence electron, the scattering may be modeled as Thomson scattering, the elastic interaction of an electromagnetic ray with a charged particle. The intensity of Thomson scattering for one particle with mass m and elementary charge q is: I_o = I_e \left(\frac{q^4}{m^2c^4}\right)\frac{1+\cos^22\theta}{2} = I_e7.94\times10^{-26}\frac{1+\cos^22\theta}{2} = I_ef Hence the atomic nuclei, which are much heavier than an electron, contribute negligibly to the scattered X-rays. Consequently, the coherent scattering detected from an atom can be accurately approximated by analyzing the collective scattering from the electrons in the system. The incoming X-ray beam has a polarization and should be represented as a vector wave; however, for simplicity, it will be represented here as a scalar wave. We will ignore the time dependence of the wave and just concentrate on the wave's spatial dependence. Plane waves can be represented by a wave vector k in , and so the incoming wave at time t = 0 is given by A \mathrm{e}^{2\pi \mathrm{i}\mathbf{k}_{\mathrm{in}} \cdot \mathbf{r}}. At a position r within the sample, consider a density of scatterers f(r); these scatterers produce a scattered spherical wave of amplitude proportional to the local amplitude of the incoming wave times the number of scatterers in a small volume dV about r \text{amplitude of scattered wave} = A \mathrm{e}^{2\pi \mathrm{i}\mathbf{k} \cdot \mathbf{r}} S f(\mathbf{r}) \, \mathrm{d}V, where S is the proportionality constant. Consider the fraction of scattered waves that leave with an outgoing wave-vector of k out and strike a screen (detector) at r screen . Since no energy is lost (elastic, not inelastic scattering), the wavelengths are the same as are the magnitudes of the wave-vectors |k in | = |k out |. From the time that the photon is scattered at r until it is absorbed at r screen , the photon undergoes a change in phase e^{2\pi i \mathbf{k}_\text{out} \cdot \left( \mathbf{r}_\text{screen} - \mathbf{r} \right)}. The net radiation arriving at r screen is the sum of all the scattered waves throughout the crystal A S \int \mathrm{d}\mathbf{r} \, f(\mathbf{r}) \mathrm{e}^{2\pi \mathrm{i} \mathbf{k}_\text{in} \cdot \mathbf{r}} e^{2\pi i \mathbf{k}_\text{out} \cdot \left( \mathbf{r}_\text{screen} - \mathbf{r} \right)} = A S e^{2\pi i \mathbf{k}_\text{out} \cdot \mathbf{r}_\text{screen}} \int \mathrm{d}\mathbf{r} \, f(\mathbf{r}) \mathrm{e}^{2\pi \mathrm{i} \left( \mathbf{k}_\text{in} - \mathbf{k}_\text{out} \right) \cdot \mathbf{r}}, which may be written as a Fourier transform A S \mathrm{e}^{2\pi \mathrm{i} \mathbf{k}_\text{out} \cdot \mathbf{r}_\text{screen}} \int d\mathbf{r} f(\mathbf{r}) \mathrm{e}^{-2\pi \mathrm{i} \mathbf{g} \cdot \mathbf{r}} = A S \mathrm{e}^{2\pi \mathrm{i} \mathbf{k}_\text{out} \cdot \mathbf{r}_\text{screen}} F(\mathbf{g}), where g = k out  – k in is a reciprocal lattice vector that satisfies Bragg's law and the Ewald construction mentioned above. The measured intensity of the reflection will be square of this amplitude A^2 S^2 \left|F(\mathbf{g}) \right|^2. The above assumes that the crystalline regions are somewhat large, for instance microns across, but also not so large that the X-rays are scattered more than once. If either of these is not the case then the diffracted intensities will be more complicated. ===Electron diffraction=== Because they interact via the Coulomb forces the scattering of electrons by matter is 1000 or more times stronger than for X-rays. Hence electron beams produce strong multiple or dynamical scattering even for relatively thin crystals (>10 nm). While there are similarities between the diffraction of X-rays and electrons, as can be found in the book by John M. Cowley, the approach is different as it is based upon the original approach of Hans Bethe and solving Schrödinger equation for relativistic electrons, rather than a kinematical or Bragg's law approach. Information about very small regions, down to single atoms is possible. The range of applications for electron diffraction, transmission electron microscopy and transmission electron crystallography with high energy electrons is extensive; see the relevant links for more information and citations. In addition to transmission methods, low-energy electron diffraction is a technique where electrons are back-scattered off surfaces and has been extensively used to determine surface structures at the atomic scale, and reflection high-energy electron diffraction is another which is extensively used to monitor thin film growth.

Text derived from https://en.wikipedia.org/w/index.php?oldid=1374867018 under CC BY-SA 4.0; modifications have been made.