X-ray diffraction

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Hemisphere of Diffraction Hemisphere of Diffraction.png
Hemisphere of 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.

Contents

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 [1] 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." [2] [3] Single-slit experiments in the laboratory of Arnold Sommerfeld suggested that X-rays had a wavelength of about 1 angstrom. [4] 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. [3] Albert Einstein introduced the photon concept in 1905, [5] but it was not broadly accepted until 1922, [6] [7] when Arthur Compton confirmed it by the scattering of X-rays from electrons. [8] 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. [9] [10] [11] [12] Bragg's view proved unpopular and the observation of X-ray diffraction by Max von Laue in 1912 [13] confirmed that X-rays are a form of electromagnetic radiation.

One of the copper sulfate X-ray interference patterns published in Von Laue's 1912 paper . Interferenz-Erscheinungen bei Rontgenstrahlen Tafel II Fig. 5.jpg
One of the copper sulfate 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. [14] :43 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). [15] [16]

After seeing the initial results, Laue was walking home and suddenly conceived of the physical laws describing the effect. [14] :44 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. [17]

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. [1] [18] [19] [20] 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

The incoming beam (coming from upper left) causes each scatterer to re-radiate a small portion of its intensity as a spherical wave. If scatterers are arranged symmetrically with a separation d, these spherical waves will be in sync (add constructively) only in directions where their path-length difference 2d sin th equals an integer multiple of the wavelength l. In that case, part of the incoming beam is deflected by an angle 2th, producing a reflection spot in the diffraction pattern. Bragg diffraction 2.svg
The incoming beam (coming from upper left) causes each scatterer to re-radiate a small portion of its intensity as a spherical wave. If scatterers are arranged symmetrically with a separation d, these spherical waves will be in sync (add constructively) only in directions where their path-length difference 2d sin θ equals an integer multiple of the wavelength λ. In that case, part of the incoming beam is deflected by an angle 2θ, producing a reflection spot in the diffraction pattern.

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 (or lighthouse) 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. [21] [22] [23]

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 (h, k, l), and their spacing by d. William Lawrence Bragg proposed a model where the incoming X-rays are scattered specularly (mirror-like) from each plane; from that assumption, X-rays scattered from adjacent planes will combine constructively (constructive interference) 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 (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. [21]

Ewald's sphere

Representation of the Ewald construction for elastic diffraction of x-rays. Ewald3.png
Representation of the Ewald construction for elastic diffraction of x-rays.

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. [23]

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: [22]

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. [21]

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 kin, 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(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

where S is the proportionality constant.

Consider the fraction of scattered waves that leave with an outgoing wave-vector of kout and strike a screen (detector) at rscreen. Since no energy is lost (elastic, not inelastic scattering), the wavelengths are the same as are the magnitudes of the wave-vectors |kin| = |kout|. From the time that the photon is scattered at r until it is absorbed at rscreen, the photon undergoes a change in phase

The net radiation arriving at rscreen is the sum of all the scattered waves throughout the crystal

which may be written as a Fourier transform

where g = kout  kin 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 [21] [22]

The above assumes that the crystalline regions as 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 e more complicated. [23] [24]

X-ray sources

Rotating anode

Small scale diffraction experiments can be done with a local X-ray tube source, typically coupled with an image plate detector. These have the advantage of being relatively inexpensive and easy to maintain, and allow for quick screening and collection of samples. However, the wavelength of the X-rays produced is limited by the availability of different anode materials. Furthermore, the intensity is limited by the power applied and cooling capacity available to avoid melting the anode. In such systems, electrons are boiled off of a cathode and accelerated through a strong electric potential of ~50  kV; having reached a high speed, the electrons collide with a metal plate, emitting bremsstrahlung and some strong spectral lines corresponding to the excitation of inner-shell electrons of the metal. The most common metal used is copper, which can be kept cool easily due to its high thermal conductivity, and which produces strong Kα and Kβ lines. The Kβ line is sometimes suppressed with a thin (~10 μm) nickel foil. The simplest and cheapest variety of sealed X-ray tube has a stationary anode (the Crookes tube) and runs with ~2 kW of electron beam power. The more expensive variety has a rotating-anode type source that runs with ~14 kW of e-beam power.

X-rays are generally filtered (by use of X-ray filters) to a single wavelength (made monochromatic) and collimated to a single direction before they are allowed to strike the crystal. The filtering not only simplifies the data analysis, but also removes radiation that degrades the crystal without contributing useful information. Collimation is done either with a collimator (basically, a long tube) or with an arrangement of gently curved mirrors. Mirror systems are preferred for small crystals (under 0.3 mm) or with large unit cells (over 150 Å).

Microfocus tube

A more recent development is the microfocus tube, which can deliver at least as high a beam flux (after collimation) as rotating-anode sources but only require a beam power of a few tens or hundreds of watts rather than requiring several kilowatts.

Synchrotron radiation

Synchrotron radiation sources are some of the brightest light sources on earth and are some of the most powerful tools available for X-ray diffraction and crystallography. X-ray beams are generated in synchrotrons which accelerate electrically charged particles, often electrons, to nearly the speed of light and confine them in a (roughly) circular loop using magnetic fields.

Synchrotrons are generally national facilities, each with several dedicated beamlines where data is collected without interruption. Synchrotrons were originally designed for use by high-energy physicists studying subatomic particles and cosmic phenomena. The largest component of each synchrotron is its electron storage ring. This ring is not a perfect circle, but a many-sided polygon. At each corner of the polygon, or sector, precisely aligned magnets bend the electron stream. As the electrons' path is bent, they emit bursts of energy in the form of X-rays.

The intense ionizing radiation can cause radiation damage to samples, particularly macromolecular crystals. Cryo crystallography can protect the sample from radiation damage, by freezing the crystal at liquid nitrogen temperatures (~100 K). [25] Cryocrystallography methods are applied to home source rotating anode sources as well. [26] However, synchrotron radiation frequently has the advantage of user-selectable wavelengths, allowing for anomalous scattering experiments which maximizes anomalous signal. This is critical in experiments such as single wavelength anomalous dispersion (SAD) and multi-wavelength anomalous dispersion (MAD).

Free-electron laser

Free-electron lasers have been developed for use in X-ray diffraction and crystallography. [27] These are the brightest X-ray sources currently available; with the X-rays coming in femtosecond bursts. The intensity of the source is such that atomic resolution diffraction patterns can be resolved for crystals otherwise too small for collection. However, the intense light source also destroys the sample, [28] requiring multiple crystals to be shot. As each crystal is randomly oriented in the beam, hundreds of thousands of individual diffraction images must be collected in order to get a complete data set. This method, serial femtosecond crystallography, has been used in solving the structure of a number of protein crystal structures, sometimes noting differences with equivalent structures collected from synchrotron sources. [29]

Other X-ray techniques

Other forms of elastic X-ray scattering besides single-crystal diffraction include powder diffraction, small-angle X-ray scattering (SAXS) and several types of X-ray fiber diffraction, which was used by Rosalind Franklin in determining the double-helix structure of DNA. In general, single-crystal X-ray diffraction offers more structural information than these other techniques; however, it requires a sufficiently large and regular crystal, which is not always available.

These scattering methods generally use monochromatic X-rays, which are restricted to a single wavelength with minor deviations. A broad spectrum of X-rays (that is, a blend of X-rays with different wavelengths) can also be used to carry out X-ray diffraction, a technique known as the Laue method. This is the method used in the original discovery of X-ray diffraction. Laue scattering provides much structural information with only a short exposure to the X-ray beam, and is therefore used in structural studies of very rapid events (time resolved crystallography). However, it is not as well-suited as monochromatic scattering for determining the full atomic structure of a crystal and therefore works better with crystals with relatively simple atomic arrangements.

The Laue back reflection mode records X-rays scattered backwards from a broad spectrum source. This is useful if the sample is too thick for X-rays to transmit through it. The diffracting planes in the crystal are determined by knowing that the normal to the diffracting plane bisects the angle between the incident beam and the diffracted beam. A Greninger chart can be used [30] to interpret the back reflection Laue photograph.

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, [23] the approach is different as it is based upon the original approach of Hans Bethe [31] 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 [32] 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. [33]

Neutron diffraction

Neutron diffraction is used for structure determination, although it has been difficult to obtain intense, monochromatic beams of neutrons in sufficient quantities. Traditionally, nuclear reactors have been used, although sources producing neutrons by spallation are becoming increasingly available. Being uncharged, neutrons scatter more from the atomic nuclei rather than from the electrons. Therefore, neutron scattering is useful for observing the positions of light atoms with few electrons, especially hydrogen, which is essentially invisible in X-ray diffraction. Neutron scattering also has the property that the solvent can be made invisible by adjusting the ratio of normal water, H2O, and heavy water, D2O.

Related Research Articles

<span class="mw-page-title-main">Crystallography</span> Scientific study of crystal structures

Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word crystallography is derived from the Ancient Greek word κρύσταλλος, and γράφειν. In July 2012, the United Nations recognised the importance of the science of crystallography by proclaiming 2014 the International Year of Crystallography.

<span class="mw-page-title-main">Diffraction</span> Phenomenon of the motion of waves

Diffraction is the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a secondary source of the propagating wave. Italian scientist Francesco Maria Grimaldi coined the word diffraction and was the first to record accurate observations of the phenomenon in 1660.

<span class="mw-page-title-main">X-ray crystallography</span> Technique used for determining crystal structures and identifying mineral compounds

X-ray crystallography is the experimental science of determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract in specific directions. By measuring the angles and intensities of the X-ray diffraction, a crystallographer can produce a three-dimensional picture of the density of electrons within the crystal and the positions of the atoms, as well as their chemical bonds, crystallographic disorder, and other information.

Matter waves are a central part of the theory of quantum mechanics, being half of wave–particle duality. At all scales where measurements have been practical, matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave.

<span class="mw-page-title-main">Electron diffraction</span> Bending of electron beams due to electrostatic interactions with matter

Electron diffraction is a generic term for phenomena associated with changes in the direction of electron beams due to elastic interactions with atoms. It occurs due to elastic scattering, when there is no change in the energy of the electrons. The negatively charged electrons are scattered due to Coulomb forces when they interact with both the positively charged atomic core and the negatively charged electrons around the atoms. The resulting map of the directions of the electrons far from the sample is called a diffraction pattern, see for instance Figure 1. Beyond patterns showing the directions of electrons, electron diffraction also plays a major role in the contrast of images in electron microscopes.

<span class="mw-page-title-main">Neutron diffraction</span> Technique to investigate atomic structures using neutron scattering

Neutron diffraction or elastic neutron scattering is the application of neutron scattering to the determination of the atomic and/or magnetic structure of a material. A sample to be examined is placed in a beam of thermal or cold neutrons to obtain a diffraction pattern that provides information of the structure of the material. The technique is similar to X-ray diffraction but due to their different scattering properties, neutrons and X-rays provide complementary information: X-Rays are suited for superficial analysis, strong x-rays from synchrotron radiation are suited for shallow depths or thin specimens, while neutrons having high penetration depth are suited for bulk samples.

<span class="mw-page-title-main">Synchrotron light source</span> Particle accelerator designed to produce intense x-ray beams

A synchrotron light source is a source of electromagnetic radiation (EM) usually produced by a storage ring, for scientific and technical purposes. First observed in synchrotrons, synchrotron light is now produced by storage rings and other specialized particle accelerators, typically accelerating electrons. Once the high-energy electron beam has been generated, it is directed into auxiliary components such as bending magnets and insertion devices in storage rings and free electron lasers. These supply the strong magnetic fields perpendicular to the beam that are needed to stimulate the high energy electrons to emit photons.

In many areas of science, Bragg's law, Wulff–Bragg's condition, or Laue–Bragg interference are a special case of Laue diffraction, giving the angles for coherent scattering of waves from a large crystal lattice. It describes how the superposition of wave fronts scattered by lattice planes leads to a strict relation between the wavelength and scattering angle. This law was initially formulated for X-rays, but it also applies to all types of matter waves including neutron and electron waves if there are a large number of atoms, as well as visible light with artificial periodic microscale lattices.

The Ewald sphere is a geometric construction used in electron, neutron, and x-ray diffraction which shows the relationship between:

<span class="mw-page-title-main">Davisson–Germer experiment</span> Experiment contributing to the confirmation of wave-particle duality of matter

The Davisson–Germer experiment was a 1923–1927 experiment by Clinton Davisson and Lester Germer at Western Electric, in which electrons, scattered by the surface of a crystal of nickel metal, displayed a diffraction pattern. This confirmed the hypothesis, advanced by Louis de Broglie in 1924, of wave-particle duality, and also the wave mechanics approach of the Schrödinger equation. It was an experimental milestone in the creation of quantum mechanics.

<span class="mw-page-title-main">Powder diffraction</span> Experimental method in X-ray diffraction

Powder diffraction is a scientific technique using X-ray, neutron, or electron diffraction on powder or microcrystalline samples for structural characterization of materials. An instrument dedicated to performing such powder measurements is called a powder diffractometer.

<span class="mw-page-title-main">Bragg plane</span>

In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, , at right angles. The Bragg plane is defined as part of the Von Laue condition for diffraction peaks in x-ray diffraction crystallography.

<span class="mw-page-title-main">Dynamical theory of diffraction</span>

The dynamical theory of diffraction describes the interaction of waves with a regular lattice. The wave fields traditionally described are X-rays, neutrons or electrons and the regular lattice are atomic crystal structures or nanometer-scale multi-layers or self-arranged systems. In a wider sense, similar treatment is related to the interaction of light with optical band-gap materials or related wave problems in acoustics. The sections below deal with dynamical diffraction of X-rays.

<span class="mw-page-title-main">Low-energy electron diffraction</span> Technique for determining surface structures

Low-energy electron diffraction (LEED) is a technique for the determination of the surface structure of single-crystalline materials by bombardment with a collimated beam of low-energy electrons (30–200 eV) and observation of diffracted electrons as spots on a fluorescent screen.

In condensed matter physics and crystallography, the static structure factor is a mathematical description of how a material scatters incident radiation. The structure factor is a critical tool in the interpretation of scattering patterns obtained in X-ray, electron and neutron diffraction experiments.

Diffraction topography is a imaging technique based on Bragg diffraction. Diffraction topographic images ("topographies") record the intensity profile of a beam of X-rays diffracted by a crystal. A topography thus represents a two-dimensional spatial intensity mapping (image) of the X-rays diffracted in a specific direction, so regions which diffract substantially will appear brighter than those which do not. This is equivalent to the spatial fine structure of a Laue reflection. Topographs often reveal the irregularities in a non-ideal crystal lattice. X-ray diffraction topography is one variant of X-ray imaging, making use of diffraction contrast rather than absorption contrast which is usually used in radiography and computed tomography (CT). Topography is exploited to a lesser extent with neutrons, and is the same concept as dark field imaging in an electron microscope.

<span class="mw-page-title-main">Atomic form factor</span> Measure of the scattering amplitude of a wave by an isolated atom

In physics, the atomic form factor, or atomic scattering factor, is a measure of the scattering amplitude of a wave by an isolated atom. The atomic form factor depends on the type of scattering, which in turn depends on the nature of the incident radiation, typically X-ray, electron or neutron. The common feature of all form factors is that they involve a Fourier transform of a spatial density distribution of the scattering object from real space to momentum space. For an object with spatial density distribution, , the form factor, , is defined as

Anomalous X-ray scattering is a non-destructive determination technique within X-ray diffraction that makes use of the anomalous dispersion that occurs when a wavelength is selected that is in the vicinity of an absorption edge of one of the constituent elements of the sample. It is used in materials research to study nanometer sized differences in structure.

<span class="mw-page-title-main">Laue equations</span> Equations describing diffraction in a crystal lattice

In crystallography and solid state physics, the Laue equations relate incoming waves to outgoing waves in the process of elastic scattering, where the photon energy or light temporal frequency does not change upon scattering by a crystal lattice. They are named after physicist Max von Laue (1879–1960).

<span class="mw-page-title-main">Multislice</span>

The multislice algorithm is a method for the simulation of the elastic scattering of an electron beam with matter, including all multiple scattering effects. The method is reviewed in the book by John M. Cowley, and also the work by Ishizuka. The algorithm is used in the simulation of high resolution transmission electron microscopy (HREM) micrographs, and serves as a useful tool for analyzing experimental images. This article describes some relevant background information, the theoretical basis of the technique, approximations used, and several software packages that implement this technique. Some of the advantages and limitations of the technique and important considerations that need to be taken into account are described.

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