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. 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 small-angle X-ray scattering.

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 1912 confirmed that X-rays are a form of electromagnetic radiation.
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".
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

Basics

Crystals are regular arrays of atoms, and X-rays are electromagnetic waves. Atoms scatter X-ray waves, primarily through the atoms' electrons. Just as an ocean wave striking a lighthouse produces secondary circular waves emanating from the lighthouse, so an X-ray striking an electron produces secondary spherical waves emanating from the electron. This phenomenon is known as elastic scattering, and the electron is known as the scatterer. A regular array of scatterers produces a regular array of spherical waves. Although these waves cancel one another out in most directions through destructive interference, they add constructively in a few specific directions.
An intuitive understanding of X-ray diffraction can be obtained from the Bragg model of diffraction. In this model, a given reflection is associated with a set of evenly spaced sheets running through the crystal, usually passing through the centers of the atoms of the crystal lattice. The orientation of a particular set of sheets is identified by its three Miller indices, and their spacing by d. William Lawrence Bragg proposed a model where the incoming X-rays are scattered specularly from each plane; from that assumption, X-rays scattered from adjacent planes will combine constructively when the angle θ between the plane and the X-ray results in a path-length difference that is an integer multiple n of the X-ray wavelength λ.
A reflection is said to be indexed when its Miller indices 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 k₀ the only wavevectors with the same energy lie on the surface of a sphere. In the diagram, the wavevector k₁ lies on the Ewald sphere and also is at a reciprocal lattice vector g₁ so satisfies Bragg's law. In contrast the wavevector k₂ differs from the reciprocal lattice point and g₂ 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:
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
At a position r within the sample, consider a density of scatterers f; 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
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 at r_screen. Since no energy is lost, 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
The net radiation arriving at r_screen is the sum of all the scattered waves throughout the crystal
which may be written as a Fourier transform
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
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.