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 toelastic 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 fromX-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 earlyhistory of x-rays and the discovery that they have the right spacings to be diffracted by crystals. In many cases these diffraction patterns can beInterpreted using a single scattering or kinematical theory with conservation of energy (wave vector). Many different types ofX-ray sources exist, ranging from ones used in laboratories to higher brightnesssynchrotron light sources. Similar diffraction patterns can be produced byrelated scattering techniques such aselectron diffraction orneutron 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 includefiber diffraction,powder diffraction and (if the sample is not crystallized)small-angle X-ray scattering (SAXS).

WhenWilhelm Röntgen discovered X-rays in 1895^[1] physicists were uncertain of the nature of X-rays, but suspected that they were waves ofelectromagnetic radiation. TheMaxwell theory ofelectromagnetic radiation was well accepted, and experiments byCharles Glover Barkla showed that X-rays exhibited phenomena associated with electromagnetic waves, including transversepolarization andspectral 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 ofArnold Sommerfeld suggested that X-rays had awavelength of about 1angstrom.^[4] X-rays are not only waves but also have particle properties causing Sommerfeld to coin the nameBremsstrahlung 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] whenArthur 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 promptedWilliam Henry Bragg to argue in 1907 that X-rays werenot electromagnetic radiation.^{[9][10][11][12]} Bragg's view proved unpopular and the observation of X-ray diffraction byMax von Laue in 1912^[13] confirmed that X-rays are a form of electromagnetic radiation.

The idea that crystals could be used as adiffraction grating forX-rays arose in 1912 in a conversation betweenPaul Peter Ewald andMax von Laue in theEnglish Garden in Munich. Ewald had proposed a resonator model of crystals for his thesis, but this model could not be validated usingvisible 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 acopper sulfate crystal and record its diffraction pattern on aphotographic 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 theBavarian Academy of Sciences and Humanities in June 1912 as "Interferenz-Erscheinungen bei Röntgenstrahlen" (Interference phenomena in X-rays).^[13][15]

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 theNobel Prize in Physics in 1914.^[16]

After Von Laue's pioneering research the field developed rapidly, most notably by physicistsWilliam Lawrence Bragg and his fatherWilliam Henry Bragg. In 1912–1913, the younger Bragg developedBragg's law, which connects the scattering with evenly spaced planes within a crystal.^{[1][17][18][19]} 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; seeX-ray crystallography for more details.

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 aselastic scattering, and the electron (or lighthouse) is known as thescatterer. A regular array of scatterers produces a regular array of spherical waves. Although these waves cancel one another out in most directions throughdestructive interference, they add constructively in a few specific directions.^[20][21][22]

An intuitive understanding of X-ray diffraction can be obtained from theBragg 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 itsthree Miller indices (h,k,l), and their spacing byd. 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 multiplen of the X-ray wavelength λ.

${\displaystyle 2d\sin \theta =n\lambda .}$

A reflection is said to beindexed when its Miller indices (or, more correctly, itsreciprocal lattice vector components) have been identified from the known wavelength and the scattering angle 2θ. Such indexing gives theunit-cell parameters, the lengths and angles of the unit-cell, as well as itsspace group.^[20]

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 wavevectork₀ the only wavevectors with the same energy lie on the surface of a sphere. In the diagram, the wavevectork₁ lies on the Ewald sphere and also is at a reciprocal lattice vectorg₁ so satisfies Bragg's law. In contrast the wavevectork₂ differs from the reciprocal lattice point andg₂ by the vectors which is called the excitation error. For large single crystals primarily used in crystallography only the Bragg's law case matters; forelectron diffraction and some other types of x-ray diffraction non-zero values of the excitation error also matter.^[22]

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 avalence electron, the scattering may be modeled asThomson scattering, the elastic interaction of an electromagnetic ray with a charged particle.

The intensity ofThomson scattering for one particle with massm and elementary chargeq is:^[21]

${\displaystyle I_o=I_e\left(\frac{q^4}{m^2{c}^4}\right)\frac{1+{\cos }^{2}2\theta }{2}=I_{e}7.94\times {10}^{-26}\frac{1+{\cos }^{2}2\theta }{2}=I_{e}f}$

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.^[20]

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 awave vectork_in, and so the incoming wave at timet = 0 is given by

${\displaystyle A{\mathrm{e}}^{2\pi \mathrm{i}{\mathbf{k}}_{\mathrm{i}\mathrm{n}}\cdot \mathbf{r}}.}$

At a positionr within the sample, consider a density of scatterersf(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 volumedV aboutr

${\displaystyle \text{amplitude of scattered wave}=A{\mathrm{e}}^{2\pi \mathrm{i}\mathbf{k}\cdot \mathbf{r}}Sf(\mathbf{r})\mathrm{d}V,}$

whereS is the proportionality constant.

Consider the fraction of scattered waves that leave with an outgoing wave-vector ofk_out and strike a screen (detector) atr_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 atr until it is absorbed atr_screen, the photon undergoes a change in phase

${\displaystyle e^{2\pi i{\mathbf{k}}_{\text{out}}\cdot \left({\mathbf{r}}_{\text{screen}}-\mathbf{r}\right)}.}$

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

${\displaystyle AS\int \mathrm{d}\mathbf{r}f(\mathbf{r}){\mathrm{e}}^{2\pi \mathrm{i}{\mathbf{k}}_{\text{in}}\cdot \mathbf{r}}{e}^{2\pi i{\mathbf{k}}_{\text{out}}\cdot \left({\mathbf{r}}_{\text{screen}}-\mathbf{r}\right)}=AS{e}^{2\pi i{\mathbf{k}}_{\text{out}}\cdot {\mathbf{r}}_{\text{screen}}}\int \mathrm{d}\mathbf{r}f(\mathbf{r}){\mathrm{e}}^{2\pi \mathrm{i}\left({\mathbf{k}}_{\text{in}}-{\mathbf{k}}_{\text{out}}\right)\cdot \mathbf{r}},}$

which may be written as a Fourier transform

${\displaystyle AS{\mathrm{e}}^{2\pi \mathrm{i}{\mathbf{k}}_{\text{out}}\cdot {\mathbf{r}}_{\text{screen}}}\int d\mathbf{r}f(\mathbf{r}){\mathrm{e}}^{-2\pi \mathrm{i}\mathbf{g}\cdot \mathbf{r}}=AS{\mathrm{e}}^{2\pi \mathrm{i}{\mathbf{k}}_{\text{out}}\cdot {\mathbf{r}}_{\text{screen}}}F(\mathbf{g}),}$

whereg =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^[20][21]

${\displaystyle A^2{S}^2{\left|F(\mathbf{g})\right|}^2.}$

The above assumes that the crystalline regions are somewhat large, for instancemicrons 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.^[22][23]

Small scale diffraction experiments can be done with a localX-ray tube source, typically coupled with animage 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 differentanode 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, emittingbremsstrahlung and some strong spectral lines corresponding to the excitation ofinner-shell electrons of the metal. The most common metal used is copper, which can be kept cool easily due to its highthermal conductivity, and which produces strongK_α and K_β lines. The K_β 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 (theCrookes tube) and runs with ~2 kW of electron beam power. The more expensive variety has arotating-anode type source that runs with ~14 kW of e-beam power.

X-rays are generally filtered (by use ofX-ray filters) to a single wavelength (made monochromatic) andcollimated 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).

A more recent development is themicrofocus 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 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 insynchrotrons 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 dedicatedbeamlines where data is collected without interruption. Synchrotrons were originally designed for use by high-energy physicists studyingsubatomic particles andcosmic phenomena. The largest component of each synchrotron is its electronstorage 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 intenseionizing radiation can causeradiation damage to samples, particularly macromolecular crystals.Cryo crystallography can protect the sample from radiation damage, by freezing the crystal atliquid nitrogen temperatures (~100K).^[24] Cryocrystallography methods are applied to home source rotating anode sources as well.^[25] However, synchrotron radiation frequently has the advantage of user-selectable wavelengths, allowing foranomalous scattering experiments which maximizes anomalous signal. This is critical in experiments such assingle wavelength anomalous dispersion (SAD) andmulti-wavelength anomalous dispersion (MAD).

Free-electron lasers have been developed for use in X-ray diffraction and crystallography.^[26] These are the brightest X-ray sources currently available; with the X-rays coming infemtosecond 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,^[27] 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.^[28]

Other forms of elastic X-ray scattering besides single-crystal diffraction includepowder diffraction, small-angle X-ray scattering (SAXS) and several types of X-rayfiber diffraction, which was used byRosalind Franklin in determining thedouble-helix structure ofDNA. 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 usemonochromatic 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. AGreninger chart can be used^[29] to interpret the back reflection Laue photograph.

Because they interact via theCoulomb 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 byJohn M. Cowley,^[22] the approach is different as it is based upon the original approach ofHans Bethe^[30] and solvingSchrödinger equation forrelativistic electrons, rather than a kinematical orBragg's law approach. Information about very small regions, down to single atoms is possible. The range of applications forelectron diffraction,transmission electron microscopy and transmissionelectron 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^[31] is a technique where electrons are back-scattered off surfaces and has been extensively used to determine surface structures at the atomic scale, andreflection high-energy electron diffraction is another which is extensively used to monitor thin film growth.^[32]

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 byspallation 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, especiallyhydrogen, 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, andheavy water, D₂O.

• X-ray crystallography
• Laboratory techniques in condensed matter physics
• Crystallography
• Diffraction
• Materials science
• Synchrotron-related techniques
• X-rays

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