TheInfoList

X-ray crystallography is the experimental science determining the atomic and molecular structure of a
crystal A crystal or crystalline solid is a solid Solid is one of the four fundamental states of matter (the others being liquid A liquid is a nearly incompressible fluid In physics, a fluid is a substance that continually Deformatio ...

, in which the crystalline structure causes a beam of incident
X-rays An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Moti ...

to into many specific directions. By measuring the angles and intensities of these diffracted beams, a can produce a three-dimensional picture of the density of
electron The electron is a subatomic particle (denoted by the symbol or ) whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are general ...

s within the crystal. From this
electron density In quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of chemistry Chemistry is the scientific discipline involved with Chemical element, elements and chemical compound, compounds composed of atoms, ...
, the mean positions of the atoms in the crystal can be determined, as well as their
chemical bond A chemical bond is a lasting attraction between atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyda ...
s, their
crystallographic disorderIn X-ray crystallography, crystallographic disorder describes the cocrystallization of more than one rotamer, Conformational isomerism, conformer, or isomer where the center of mass of each form is identical or unresolvable. As a consequence of di ...
, and various other information. Since many materials can form crystals—such as
salts In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during ...
,
metal A metal (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

s,
mineral In geology and mineralogy, a mineral or mineral species is, broadly speaking, a solid chemical compound with a fairly well-defined chemical composition and a specific crystal structure that occurs naturally in pure form.John P. Rafferty, ed. (2 ...

s,
semiconductor A semiconductor material has an electrical conductivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric curre ...
s, as well as various inorganic, organic, and biological molecules—X-ray crystallography has been fundamental in the development of many scientific fields. In its first decades of use, this method determined the size of atoms, the lengths and types of chemical bonds, and the atomic-scale differences among various materials, especially minerals and
alloy An alloy is an admixture of metal A metal (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in ...
s. The method also revealed the structure and function of many biological molecules, including
vitamin A vitamin is an organic molecule , CH4; is among the simplest organic compounds. In chemistry, organic compounds are generally any chemical compounds that contain carbon-hydrogen chemical bond, bonds. Due to carbon's ability to Catenation, ...
s, drugs,
protein Proteins are large biomolecule , showing alpha helices, represented by ribbons. This poten was the first to have its suckture solved by X-ray crystallography by Max Perutz and Sir John Cowdery Kendrew in 1958, for which they received a No ...

s and
nucleic acid Nucleic acids are biopolymer Biopolymers are natural polymer A polymer (; Greek ''wikt:poly-, poly-'', "many" + ''wikt:-mer, -mer'', "part") is a Chemical substance, substance or material consisting of very large molecules, or macromolecule ...

s such as
DNA Deoxyribonucleic acid (; DNA) is a molecule File:Pentacene on Ni(111) STM.jpg, A scanning tunneling microscopy image of pentacene molecules, which consist of linear chains of five carbon rings. A molecule is an electrically neutral gro ...

. X-ray crystallography is still the primary method for characterizing the atomic structure of new materials and in discerning materials that appear similar by other
experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome oc ...

s. X-ray
crystal structure In crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek language, Greek words ''crystallon'' "co ...

s can also account for unusual
electronic Electronic may refer to: *Electronics Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter. It uses active devices to control electron flow b ...
or elastic properties of a material, shed light on chemical interactions and processes, or serve as the basis for designing pharmaceuticals against diseases. In a single-crystal X-ray diffraction measurement, a crystal is mounted on a
goniometer A goniometer is an instrument that either measures an angle or allows an object to be rotated to a precise angular position. The term goniometry derives from two Greek words, γωνία (''gōnía'') 'angle In Euclidean geometry Euclidea ...

. The goniometer is used to position the crystal at selected orientations. The crystal is illuminated with a finely focused
monochromatic during the 1889 Exposition Universelle File:Parrot EGA monochrome palette.png, A photograph of a μονόχρωμος.html" ;"title="lightness">values of one color). The term ''monochrome'' comes from the grc">μονόχρωμος">monochrom ...

beam of X-rays, producing a
diffraction pattern Diffraction refers to various phenomena that occur when a wave encounters an obstacle or opening. It is defined as the bending of waves around the corners of an obstacle or through an aperture into the region of Umbra, penumbra and antumbra, ge ...
of regularly spaced spots known as ''reflections''. The two-dimensional images taken at different orientations are converted into a three-dimensional model of the density of electrons within the crystal using the mathematical method of
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
s, combined with chemical data known for the sample. Poor resolution (fuzziness) or even errors may result if the crystals are too small, or not uniform enough in their internal makeup. X-ray crystallography is related to several other methods for determining atomic structures. Similar diffraction patterns can be produced by scattering electrons or
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behav ...

s, which are likewise interpreted by
Fourier transformation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. 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 Fiber diffraction is a subarea of scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, is forced to deviate from a straight traje ...
,
powder diffraction upright=2, X-ray powder diffraction of Y2Cu2O5 and yttrium_oxide.html"_;"title="Rietveld_refinement_with_two_phases,_showing_1%_of_yttrium_oxide">Rietveld_refinement_with_two_phases,_showing_1%_of_yttrium_oxide_impurity_(red_tickers). Powder_diff ...
and (if the sample is not crystallized)
small-angle X-ray scattering Small-angle X-ray scattering (SAXS) is a small-angle scatteringSmall-angle scattering (SAS) is a scattering technique based on deflection of collimated radiation away from the straight trajectory after it interacts with structures that are much lar ...
(SAXS). If the material under investigation is only available in the form of
nanocrystalline A nanocrystalline (NC) material is a polycrystalline material with a crystallite size of only a few nanometers. These materials fill the gap between amorphous materials without any long range order and conventional coarse-grained materials. Definit ...
powders or suffers from poor crystallinity, the methods of
electron crystallography The electron is a subatomic particle, symbol or , whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are generally thought to be e ...
can be applied for determining the atomic structure. For all above mentioned X-ray diffraction methods, the scattering is elastic; the scattered X-rays have the same
wavelength In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...

as the incoming X-ray. By contrast, ''inelastic'' X-ray scattering methods are useful in studying excitations of the sample such as
plasmon In physics, a plasmon is a quantum In physics, a quantum (plural quanta) is the minimum amount of any physical entity ( physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is refe ...
s, crystal-field and orbital excitations,
magnon A magnon is a quasiparticle, a collective excitation of the electrons' spin (physics), spin structure in a crystal lattice. In the equivalent wave picture of quantum mechanics, a magnon can be viewed as a quantized spin wave. Magnons carry a fix ...

s, and
phonon In , a phonon is a in a periodic, arrangement of s or s in , specifically in s and some s. Often referred to as a , it is an in the of the for elastic structures of interacting particles. Phonons can be thought of as quantized , similar to ...
s, rather than the distribution of its atoms.

# History

## Early scientific history of crystals and X-rays

Crystals, though long admired for their regularity and symmetry, were not investigated scientifically until the 17th century.
Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer An astronomer is a in the field of who focuses their studies on a specific question or field outside the scope of . They observe s such as s, s, , s and ...

hypothesized in his work ''Strena seu de Nive Sexangula'' (A New Year's Gift of Hexagonal Snow) (1611) that the hexagonal symmetry of was due to a regular packing of spherical water particles. The Danish scientist
Nicolas Steno Nicolas Steno ( da, Niels Steensen; Latinized to ''Nicolaus Steno'' or ''Nicolaus Stenonius''; 1 January 1638 – 25 November 1686
(1669) pioneered experimental investigations of crystal symmetry. Steno showed that the angles between the faces are the same in every exemplar of a particular type of crystal, and
René Just Haüy René Just Haüy () FRS MWS FRSE (28 February 1743 – 3 June 1822) was a French priest and mineralogist Mineralogy is a subject of geology specializing in the scientific study of the chemistry, crystal structure, and physical (including optical ...

(1784) discovered that every face of a crystal can be described by simple stacking patterns of blocks of the same shape and size. Hence, in 1839 was able to give each face a unique label of three small integers, the
Miller indices Miller indices form a notation system in crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek w ...
which remain in use today for identifying crystal faces. Haüy's study led to the correct idea that crystals are a regular three-dimensional array (a
Bravais lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of Translation operator (quantum mechanics)#Discrete Translational Symmetry, discrete translation operations described in t ...
) of atoms and
molecule A molecule is an electrically Electricity is the set of physical phenomena associated with the presence and motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon ...

s; a single
unit cell In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

is repeated indefinitely along three principal directions that are not necessarily perpendicular. In the 19th century, a complete catalog of the possible symmetries of a crystal was worked out by Johan Hessel,
Auguste Bravais Auguste Bravais (; 23 August 1811, Annonay, Ardèche – 30 March 1863, Le Chesnay, France) was a French physicist known for his work in crystallography, the conception of Bravais lattices, and the formulation of Bravais law. Bravais also studied ...
,
Evgraf Fedorov Evgraf Stepanovich Fedorov (russian: Евгра́ф Степа́нович Фёдоров, – 21 May 1919) was a Russian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Gr ...
, Arthur Schönflies and (belatedly) William Barlow (1894). From the available data and physical reasoning, Barlow proposed several crystal structures in the 1880s that were validated later by X-ray crystallography; however, the available data were too scarce in the 1880s to accept his models as conclusive.
Wilhelm Röntgen Wilhelm Conrad Röntgen (; ; 27 March 184510 February 1923) was a German mechanical engineer Mechanical may refer to: Machine * Mechanical system, a system that manages the power of forces and movements to accomplish a task * Machine (mechanica ...

discovered X-rays in 1895, just as the studies of crystal symmetry were being completed. Physicists were uncertain of the nature of X-rays, but soon suspected that they were waves of
electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

, a form of
light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that is visual perception, perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nan ...

. The theory of
electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

was well accepted among scientists, and experiments by
Charles Glover Barkla Charles Glover Barkla FRS FRSE Fellowship of the Royal Society of Edinburgh (FRSE) is an award granted to individuals that the Royal Society of Edinburgh, Scotland's national academy of science and Literature, letters, judged to be "emine ...

showed that X-rays exhibited phenomena associated with electromagnetic waves, including transverse
polarization Polarization or polarisation may refer to: In the physical sciences *Polarization (waves), the ability of waves to oscillate in more than one direction, in particular polarization of light, responsible for example for the glare-reducing effect of ...
and
spectral line A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spect ...
s akin to those observed in the visible wavelengths. Barkla created the x-ray notation, as well, noting in 1909 two separate types of diffraction beams, at first, naming them “A” and “B” and then 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 Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German people, German theoretical physicist who pioneered developments in atomic physics, atomic and quantum physics, and also educated and mentored many students f ...
wavelength In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...

angstrom The angstromEntry "angstrom" in the Oxford online dictionary. Retrieved on 2019-03-02 from https://en.oxforddictionaries.com/definition/angstrom.Entry "angstrom" in the Merriam-Webster online dictionary. Retrieved on 2019-03-02 from https://www.m ...

. X-rays are not only waves but are also
photon The photon ( el, φῶς, phōs, light) is a type of elementary particle In , an elementary particle or fundamental particle is a that is not composed of other particles. Particles currently thought to be elementary include the fundamental s ...

s, and have particle properties causing Sommerfeld to coin the name, Bremsstrahlung, for this wavelike type of diffraction.
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ...

introduced the photon concept in 1905, but it was not broadly accepted until 1922, when
Arthur Compton Arthur Holly Compton (September 10, 1892 – March 15, 1962) was an American physicist who won the Nobel Prize in Physics in 1927 for his 1923 discovery of the Compton effect, which demonstrated the particle physics, particle nature of ele ...

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 Sir William Henry Bragg (2 July 1862 – 12 March 1942) was an English physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical method ...

to argue in 1907 that X-rays were ''not'' electromagnetic radiation. Bragg's view proved unpopular and the observation of
X-ray diffraction X-ray crystallography (XRC) is the experimental science determining the atomic and molecular structure of a crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a ...
by
Max von Laue Max Theodor Felix von Laue (; 9 October 1879 – 24 April 1960) was a German physicist who received the Nobel Prize in Physics in 1914 for his discovery of the diffraction of X-rays by crystals. In addition to his scientific endeavors with co ...

in 1912 confirmed for most scientists that X-rays are a form of electromagnetic radiation.

## X-ray diffraction

Crystals are regular arrays of atoms, and X-rays can be considered waves of electromagnetic radiation. 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 Elastic scattering is a form of particle scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, is forced to deviate from a straight t ...
, 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 In physics, interference is a phenomenon in which two waves Superposition principle, superpose to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive interference result from the interaction of waves tha ...

, they add constructively in a few specific directions, determined by
Bragg's law In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spa ...

: :$n \lambda=2d \sin \theta$ Here ''d'' is the spacing between diffracting planes, $\theta$ is the incident angle, ''n'' is any integer, and λ is the wavelength of the beam. These specific directions appear as spots on the
diffraction pattern Diffraction refers to various phenomena that occur when a wave encounters an obstacle or opening. It is defined as the bending of waves around the corners of an obstacle or through an aperture into the region of Umbra, penumbra and antumbra, ge ...
called ''reflections''. Thus, X-ray diffraction results from an electromagnetic wave (the X-ray) impinging on a regular array of scatterers (the repeating arrangement of atoms within the crystal). X-rays are used to produce the diffraction pattern because their wavelength λ is typically the same order of magnitude (1–100 angstroms) as the spacing ''d'' between planes in the crystal. In principle, any wave impinging on a regular array of scatterers produces
diffraction Diffraction refers to various phenomena that occur when a wave In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which r ...

, as predicted first by
Francesco Maria Grimaldi Francesco Maria Grimaldi (2 April 1618 – 28 December 1663) was an Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ** Itali ...

in 1665. To produce significant diffraction, the spacing between the scatterers and the wavelength of the impinging wave should be similar in size. For illustration, the diffraction of sunlight through a bird's feather was first reported by James Gregory in the later 17th century. The first artificial
diffraction grating In optics, a diffraction grating is an optical component with a periodic structure that diffraction, diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form ...

s for visible light were constructed by
David Rittenhouse David Rittenhouse (April 8, 1732 – June 26, 1796) was an American astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomica ...

in 1787, and
Joseph von Fraunhofer Joseph Ritter von Fraunhofer (; ; 6 March 1787 – 7 June 1826) was a Bavarian physicist and optical lens manufacturer. He made optical glass and achromatic telescope objective lenses, invented the spectroscope, and developed diffraction gra ...

in 1821. However, visible light has too long a wavelength (typically, 5500 angstroms) to observe diffraction from crystals. Prior to the first X-ray diffraction experiments, the spacings between lattice planes in a crystal were not known with certainty. The idea that crystals could be used as a
diffraction grating In optics, a diffraction grating is an optical component with a periodic structure that diffraction, diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form ...

for
X-ray An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Moti ...

s arose in 1912 in a conversation between
Paul Peter Ewald Paul Peter Ewald, FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * Family Resourc ...
and
Max von Laue Max Theodor Felix von Laue (; 9 October 1879 – 24 April 1960) was a German physicist who received the Nobel Prize in Physics in 1914 for his discovery of the diffraction of X-rays by crystals. In addition to his scientific endeavors with co ...

in the
English Garden The English landscape garden, also called English landscape park or simply the English garden (french: Jardin à l'anglaise, it, Giardino all'inglese, german: Englischer Landschaftsgarten, pt, Jardim inglês, es, Jardín inglés), is a sty ...

in
Munich Munich ( ; german: München ; bar, Minga ) is the capital and most populous city of Bavaria. With a population of 1,558,395 inhabitants as of 31 July 2020, it is the List of cities in Germany by population, third-largest city in Germany, ...

. Ewald had proposed a resonator model of crystals for his thesis, but this model could not be validated using
visible light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that is visual perception, perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nano ...
, since the wavelength was much larger than the spacing between the resonators. Von Laue realized that electromagnetic radiation of a shorter wavelength was needed to observe such small spacings, and suggested that X-rays might have a wavelength comparable to the unit-cell 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 Copper sulfate may refer to: * Copper(II) sulfate Copper(II) sulfate, also known as copper sulphate, are the inorganic compound In chemistry, an inorganic compound is typically a chemical compound that lacks carbon–hydrogen bonds, that is, a com ...
crystal and record its diffraction on a
photographic plate photographic plates, 1880 Image:Femme-au-chien neg.jpg, Negative plate Photographic plates preceded photographic film Photographic film is a strip or sheet of transparent film base coated on one side with a gelatin photographic emulsion, emu ...
. After being developed, the plate showed a large number of well-defined spots arranged in a pattern of intersecting circles around the spot produced by the central beam. Von 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 The Nobel Prize in Physics is a yearly award given by the Royal Swedish Academy of Sciences for those who have made the most outstanding contributions for mankind in the field of physics. It is one of the five Nobel Prizes established by the will ...
in 1914.

## Scattering

As described in the mathematical derivation below, the 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 Thomson scattering is the elastic scattering of electromagnetic radiation In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural sc ...
, the interaction of an electromagnetic ray with a free electron. This model is generally adopted to describe the polarization of the scattered radiation. The intensity of
Thomson scattering Thomson scattering is the elastic scattering of electromagnetic radiation In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural sc ...
for one particle with mass ''m'' and elementary charge ''q'' is: : $I_o = I_e \left\left(\frac\right\right)\frac = I_e7.94.10^\frac = I_ef$ Hence the atomic nuclei, which are much heavier than an electron, contribute negligibly to the scattered X-rays.

## Development from 1912 to 1920

After Von Laue's pioneering research, the field developed rapidly, most notably by physicists
William Lawrence Bragg Sir William Lawrence Bragg, (31 March 1890 – 1 July 1971) was an Australian-born British physicist and X-ray crystallography, X-ray crystallographer, discoverer (1912) of Bragg's law, Bragg's law of X-ray diffraction, which is basic for t ...
and his father
William Henry Bragg Sir William Henry Bragg (2 July 1862 – 12 March 1942) was an English physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical method ...

. In 1912–1913, the younger Bragg developed
Bragg's law In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spa ...

, which connects the observed scattering with reflections from evenly spaced planes within the crystal. The Braggs, father and son, shared the 1915 Nobel Prize in Physics for their work in crystallography. The earliest structures were generally simple and marked by one-dimensional symmetry. However, as computational and experimental methods improved over the next decades, it became feasible to deduce reliable atomic positions for more complicated two- and three-dimensional arrangements of atoms in the unit-cell. The potential of X-ray crystallography for determining the structure of molecules and minerals—then only known vaguely from chemical and hydrodynamic experiments—was realized immediately. The earliest structures were simple inorganic crystals and minerals, but even these revealed fundamental laws of physics and chemistry. The first atomic-resolution structure to be "solved" (i.e., determined) in 1914 was that of
table salt Salt is a mineral composed primarily of sodium chloride (NaCl), a chemical compound belonging to the larger class of Salt (chemistry), salts; salt in the form of a natural crystallinity, crystalline mineral is known as rock salt or halite. ...
. The distribution of electrons in the table-salt structure showed that crystals are not necessarily composed of
covalently bonded A covalent bond is a chemical bond A chemical bond is a lasting attraction between atoms, ions or molecules that enables the formation of chemical compounds. The bond may result from the Coulomb's law, electrostatic force of attraction bet ...
molecules, and proved the existence of
ionic compound In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during ...
s. The structure of
diamond Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. At Standard conditions for temperature and pressure, room temperature and pressure, another solid form of ...

was solved in the same year, proving the tetrahedral arrangement of its chemical bonds and showing that the length of C–C single bond was 1.52 angstroms. Other early structures included
copper Copper is a chemical element In chemistry, an element is a pure Chemical substance, substance consisting only of atoms that all have the same numbers of protons in their atomic nucleus, nuclei. Unlike chemical compounds, chemical elem ...

,
calcium fluoride Calcium fluoride is the inorganic compound In chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chem ...

(CaF2, also known as ''fluorite''),
calcite Calcite is a carbonate mineral Carbonate minerals are those mineral In geology Geology (from the Ancient Greek γῆ, ''gē'' ("earth") and -λoγία, ''-logia'', ("study of", "discourse")) is an Earth science concerned with the solid E ...

(CaCO3) and
pyrite The mineral In geology and mineralogy, a mineral or mineral species is, broadly speaking, a solid chemical compound with a fairly well-defined chemical composition and a specific crystal structure that occurs naturally in pure form.John P. R ...

(FeS2) in 1914;
spinel Spinel () is the magnesium Magnesium is a chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only o ...

(MgAl2O4) in 1915; the
rutile Rutile is an oxide mineral The oxide mineral class includes those mineral In geology Geology (from the Ancient Greek γῆ, ''gē'' ("earth") and -λoγία, ''-logia'', ("study of", "discourse")) is an Earth science concerned with the so ...

and
anatase Anatase is a metastable In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), mot ...
forms of
titanium dioxide Titanium dioxide, also known as titanium(IV) oxide or titania , is the inorganic compound In chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical ...
(TiO2) in 1916; pyrochroite Mn(OH)2 and, by extension,
brucite Brucite is the mineral In geology Geology (from the Ancient Greek γῆ, ''gē'' ("earth") and -λoγία, ''-logia'', ("study of", "discourse")) is an Earth science concerned with the solid Earth, the rock (geology), rocks of which it i ...

Mg(OH)2 in 1919. Also in 1919,
sodium nitrate Sodium nitrate is the chemical compound with the chemical formula, formula . This alkali metal nitrate salt (chemistry), salt is also known as Chile saltpeter (large deposits of which were historically mined in Chile) to distinguish it from ord ...
(NaNO3) and caesium dichloroiodide (CsICl2) were determined by
Ralph Walter Graystone WyckoffRalph Walter Graystone Wyckoff, Sr. (born August 9, 1897 in Geneva, New York; died November 3, 1994 in Tucson, Arizona) was an American scientist and pioneer of X-ray crystallography. He was elected member of the National Academy of Sciences in 1949 ...
, and the
wurtzite Wurtzite is a zinc and iron sulfide mineral with the chemical formula , a less frequently encountered Polymorphism (materials science), structural polymorph form of sphalerite. The iron content is variable up to eight percent.Palache, Charles, Harr ...

(hexagonal ZnS) structure became known in 1920. The structure of
graphite Graphite (), archaically referred to as plumbago, is a Crystallinity, crystalline form of the element carbon with its atoms arranged in a Hexagonal crystal system, hexagonal structure. It occurs naturally in this form and is the most stable for ...

was solved in 1916 by the related method of
powder diffraction upright=2, X-ray powder diffraction of Y2Cu2O5 and yttrium_oxide.html"_;"title="Rietveld_refinement_with_two_phases,_showing_1%_of_yttrium_oxide">Rietveld_refinement_with_two_phases,_showing_1%_of_yttrium_oxide_impurity_(red_tickers). Powder_diff ...
, which was developed by
Peter Debye Peter Joseph William Debye (; ; March 24, 1884 – November 2, 1966) was a Dutch-American physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches ...

and
Paul Scherrer Paul Hermann Scherrer (3 February 1890 – 25 September 1969) was a Swiss physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area ...
and, independently, by
Albert Hull Albert Wallace Hull (19 April 1880 – 22 January 1966) was an American physicist and electrical engineer who made contributions to the development of vacuum tube A vacuum tube, an electron tube, valve (British usage) or tube (North America ...
in 1917. The structure of graphite was determined from single-crystal diffraction in 1924 by two groups independently. Hull also used the powder method to determine the structures of various metals, such as iron and magnesium.

## Cultural and aesthetic importance

In 1951, the Festival Pattern Group at the
Festival of Britain#REDIRECT Festival of Britain The Festival of Britain was a national exhibition and fair that reached millions of visitors throughout the United Kingdom The United Kingdom of Great Britain and Northern Ireland, commonly known as the Unite ...

hosted a collaborative group of textile manufacturers and experienced crystallographers to design lace and prints based on the X-ray crystallography of
insulin Insulin (, from Latin ''insula'', 'island') is a peptide hormone produced by beta cells of the pancreatic islets; it is considered to be the main Anabolism, anabolic hormone of the body. It regulates the metabolism of carbohydrates, fats and p ...

,
china clay Kaolinite ( ) is a clay mineral Clay minerals are , sometimes with variable amounts of , , s, s, and other s found on or near some s. Clay minerals form in the presence of water and have been important to life, and many theories of in ...
, and
hemoglobin Hemoglobin or haemoglobin (spelling differences Despite the various English dialects Dialect The term dialect (from Latin , , from the Ancient Greek word , , "discourse", from , , "through" and , , "I speak") is used in two distinct wa ...

. One of the leading scientists of the project was Dr. Helen Megaw (1907–2002), the Assistant Director of Research at the Cavendish Laboratory in Cambridge at the time. Megaw is credited as one of the central figures who took inspiration from crystal diagrams and saw their potential in design. In 2008, the Wellcome Collection in London curated an exhibition on the Festival Pattern Group called "From Atom to Patterns".

# Contributions to chemistry and material science

X-ray crystallography has led to a better understanding of
chemical bond A chemical bond is a lasting attraction between atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyda ...
s and
non-covalent interactions A non-covalent interaction differs from a covalent bond A covalent bond is a chemical bond that involves the sharing of electron pairs between atoms. These electron pairs are known as shared pairs or bonding pairs, and the stable balance of at ...
. The initial studies revealed the typical radii of atoms, and confirmed many theoretical models of chemical bonding, such as the tetrahedral bonding of carbon in the diamond structure, the octahedral bonding of metals observed in ammonium hexachloroplatinate (IV), and the resonance observed in the planar carbonate group and in aromatic molecules.
Kathleen Lonsdale Dame Kathleen Lonsdale (née Yardley; 28 January 1903 – 1 April 1971) was an Irish pacifist Pacifism is opposition to war, militarism (including conscription Conscription, sometimes called the draft in the United States, is the man ...
's 1928 structure of
hexamethylbenzene Hexamethylbenzene, also known as mellitene, is a hydrocarbon with the molecular formula C12H18 and the condensed structural formula C6(CH3)6. It is an aromatic compound and a derivative (chemistry), derivative of benzene, where benzene's six ...

established the hexagonal symmetry of
benzene Benzene is an organic Organic may refer to: * Organic, of or relating to an organism, a living entity * Organic, of or relating to an anatomical organ (anatomy), organ Chemistry * Organic matter, matter that has come from a once-living organ ...

and showed a clear difference in bond length between the aliphatic C–C bonds and aromatic C–C bonds; this finding led to the idea of
resonance Resonance describes the phenomenon of increased amplitude The amplitude of a Periodic function, periodic Variable (mathematics), variable is a measure of its change in a single Period (mathematics), period (such as frequency, time or Wavelen ...
between chemical bonds, which had profound consequences for the development of chemistry. Her conclusions were anticipated by
William Henry Bragg Sir William Henry Bragg (2 July 1862 – 12 March 1942) was an English physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical method ...

, who published models of
naphthalene Naphthalene is an organic compound In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavio ...

and
anthracene Anthracene is a solid polycyclic aromatic hydrocarbon (PAH) of formula C14H10, consisting of three fused benzene Benzene is an organic chemical compound A chemical compound is a chemical substance composed of many identical molecules (or ...

in 1921 based on other molecules, an early form of molecular replacement. Also in the 1920s, Victor Moritz Goldschmidt and later
Linus Pauling Linus Carl Pauling (; February 28, 1901 – August 19, 1994) was an American chemist, biochemist, chemical engineer, peace activist, author, and educator. He published more than 1,200 papers and books, of which about 850 dealt with scientific t ...

developed rules for eliminating chemically unlikely structures and for determining the relative sizes of atoms. These rules led to the structure of brookite (1928) and an understanding of the relative stability of the
rutile Rutile is an oxide mineral The oxide mineral class includes those mineral In geology Geology (from the Ancient Greek γῆ, ''gē'' ("earth") and -λoγία, ''-logia'', ("study of", "discourse")) is an Earth science concerned with the so ...

, brookite and
anatase Anatase is a metastable In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), mot ...
forms of
titanium dioxide Titanium dioxide, also known as titanium(IV) oxide or titania , is the inorganic compound In chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical ...
. The distance between two bonded atoms is a sensitive measure of the bond strength and its bond order; thus, X-ray crystallographic studies have led to the discovery of even more exotic types of bonding in inorganic chemistry, such as metal-metal double bonds, metal-metal quadruple bonds, and three-center, two-electron bonds. X-ray crystallography—or, strictly speaking, an inelastic Compton scattering experiment—has also provided evidence for the partly covalent character of hydrogen bonds. In the field of organometallic chemistry, the X-ray structure of ferrocene initiated scientific studies of sandwich compounds, while that of Zeise's salt stimulated research into "back bonding" and metal-pi complexes. Finally, X-ray crystallography had a pioneering role in the development of supramolecular chemistry, particularly in clarifying the structures of the crown ethers and the principles of host–guest chemistry. X-ray diffraction is a very powerful tool in catalyst development. Ex-situ measurements are carried out routinely for checking the crystal structure of materials or to unravel new structures. In-situ experiments give comprehensive understanding about the structural stability of catalysts under reaction conditions. In material sciences, many complicated inorganic and organometallic systems have been analyzed using single-crystal methods, such as fullerenes, porphyrin, metalloporphyrins, and other complicated compounds. Single-crystal diffraction is also used in the pharmaceutical industry, due to recent problems with Polymorphism (materials science), polymorphs. The major factors affecting the quality of single-crystal structures are the crystal's size and regularity; Recrystallization (chemistry), recrystallization is a commonly used technique to improve these factors in small-molecule crystals. The Cambridge Structural Database contains over 1,000,000 structures as of June 2019; over 99% of these structures were determined by X-ray diffraction.

## Mineralogy and metallurgy

Since the 1920s, X-ray diffraction has been the principal method for determining the arrangement of atoms in minerals and
metal A metal (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

s. The application of X-ray crystallography to mineralogy began with the structure of garnet, which was determined in 1924 by Menzer. A systematic X-ray crystallographic study of the silicates was undertaken in the 1920s. This study showed that, as the silicon, Si/oxygen, O ratio is altered, the silicate crystals exhibit significant changes in their atomic arrangements. Machatschki extended these insights to minerals in which aluminium substitutes for the silicon atoms of the silicates. The first application of X-ray crystallography to metallurgy likewise occurred in the mid-1920s. Most notably,
Linus Pauling Linus Carl Pauling (; February 28, 1901 – August 19, 1994) was an American chemist, biochemist, chemical engineer, peace activist, author, and educator. He published more than 1,200 papers and books, of which about 850 dealt with scientific t ...

's structure of the alloy Mg2Sn led to his theory of the stability and structure of complex ionic crystals. On October 17, 2012, the Curiosity rover on the Mars, planet Mars at "Rocknest (Mars), Rocknest" performed the first X-ray diffraction analysis of Martian soil. The results from the rover's CheMin, CheMin analyzer revealed the presence of several minerals, including feldspar, pyroxenes and olivine, and suggested that the Martian soil in the sample was similar to the "weathered Basalt, basaltic soils" of Hawaii Volcanoes, Hawaiian volcanoes.

## Early organic and small biological molecules

The first structure of an organic compound, hexamethylenetetramine, was solved in 1923. This was followed by several studies of long-chain fatty acids, which are an important component of biological membranes. In the 1930s, the structures of much larger molecules with two-dimensional complexity began to be solved. A significant advance was the structure of phthalocyanine, a large planar molecule that is closely related to porphyrin, porphyrin molecules important in biology, such as heme, corrin and chlorophyll. X-ray crystallography of biological molecules took off with Dorothy Crowfoot Hodgkin, who solved the structures of cholesterol (1937), penicillin (1946) and vitamin B12, vitamin B12 (1956), for which she was awarded the Nobel Prize in Chemistry in 1964. In 1969, she succeeded in solving the structure of
insulin Insulin (, from Latin ''insula'', 'island') is a peptide hormone produced by beta cells of the pancreatic islets; it is considered to be the main Anabolism, anabolic hormone of the body. It regulates the metabolism of carbohydrates, fats and p ...

, on which she worked for over thirty years.

## Biological macromolecular crystallography

Crystal structures of proteins (which are irregular and hundreds of times larger than cholesterol) began to be solved in the late 1950s, beginning with the structure of sperm whale myoglobin by John Kendrew, Sir John Cowdery Kendrew, for which he shared the Nobel Prize in Chemistry with Max Perutz in 1962. Since that success, over 130,000 X-ray crystal structures of proteins, nucleic acids and other biological molecules have been determined. The nearest competing method in number of structures analyzed is Protein nuclear magnetic resonance spectroscopy, nuclear magnetic resonance (NMR) spectroscopy, which has resolved less than one tenth as many. Crystallography can solve structures of arbitrarily large molecules, whereas solution-state NMR is restricted to relatively small ones (less than 70 katomic mass unit, Da). X-ray crystallography is used routinely to determine how a pharmaceutical drug interacts with its protein target and what changes might improve it. However, intrinsic membrane proteins remain challenging to crystallize because they require detergents or other Denaturation (biochemistry), denaturants to solubilize them in isolation, and such detergents often interfere with crystallization. Membrane proteins are a large component of the genome, and include many proteins of great physiological importance, such as ion channels and receptor (biochemistry), receptors. Helium cryogenics are used to prevent radiation damage in protein crystals. On the other end of the size scale, even relatively small molecules may pose challenges for the resolving power of X-ray crystallography. The structure assigned in 1991 to the antibiotic isolated from a marine organism, diazonamide A (C40H34Cl2N6O6, molar mass 765.65 g/mol), proved to be incorrect by the classical proof of structure: a synthetic sample was not identical to the natural product. The mistake was attributed to the inability of X-ray crystallography to distinguish between the correct -OH / -NH and the interchanged -NH2 / -O- groups in the incorrect structure. With advances in instrumentation, however, similar groups can be distinguished using modern single-crystal X-ray diffractometers. Despite being an invaluable tool in structural biology, protein crystallography carries some inherent problems in its methodology that hinder data interpretation. The crystal lattice, which is formed during the crystallization process, contains numerous units of the purified protein, which are densely and symmetrically packed in the crystal. When looking for a previously unknown protein, figuring out its shape and boundaries within the crystal lattice can be challenging. Proteins are usually composed of smaller subunits, and the task of distinguishing between the subunits and identifying the actual protein, can be challenging even for the experienced crystallographers. The non-biological interfaces that occur during crystallization are known as crystal-packing contacts (or simply, crystal contacts) and cannot be distinguished by crystallographic means. When a new protein structure is solved by X-ray crystallography and deposited in the Protein Data Bank, its authors are requested to specify the "biological assembly" which would constitute the functional, biologically-relevant protein. However, errors, missing data and inaccurate annotations during the submission of the data, give rise to obscure structures and compromise the reliability of the database. The error rate in the case of faulty annotations alone has been reported to be upwards of 6.6% or approximately 15%, arguably a non-trivial size considering the number of deposited structures. This "interface classification problem" is typically tackled by computational approaches and has become a recognized subject in structural bioinformatics.

# Scattering techniques

## Elastic vs. inelastic scattering

X-ray crystallography is a form of
elastic scattering Elastic scattering is a form of particle scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, is forced to deviate from a straight t ...
; the outgoing X-rays have the same energy, and thus same wavelength, as the incoming X-rays, only with altered direction. By contrast, ''inelastic scattering'' occurs when energy is transferred from the incoming X-ray to the crystal, e.g., by exciting an inner-shell electron to a higher energy level. Such inelastic scattering reduces the energy (or increases the wavelength) of the outgoing beam. Inelastic scattering is useful for probing such excitations of matter, but not in determining the distribution of scatterers within the matter, which is the goal of X-ray crystallography.
X-ray An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Moti ...

s range in wavelength from 10 to 0.01 nanometers; a typical wavelength used for crystallography is 1 angstrom, Å (0.1 nm), which is on the scale of covalent
chemical bond A chemical bond is a lasting attraction between atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyda ...
s and the radius of a single atom. Longer-wavelength photons (such as ultraviolet electromagnetic radiation, radiation) would not have sufficient resolution to determine the atomic positions. At the other extreme, shorter-wavelength photons such as gamma rays are difficult to produce in large numbers, difficult to focus, and interact too strongly with matter, producing pair production, particle-antiparticle pairs. Therefore, X-rays are the "sweetspot" for wavelength when determining atomic-resolution structures from the scattering of
electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

.

## Other X-ray techniques

Other forms of elastic X-ray scattering besides single-crystal diffraction include
powder diffraction upright=2, X-ray powder diffraction of Y2Cu2O5 and yttrium_oxide.html"_;"title="Rietveld_refinement_with_two_phases,_showing_1%_of_yttrium_oxide">Rietveld_refinement_with_two_phases,_showing_1%_of_yttrium_oxide_impurity_(red_tickers). Powder_diff ...
, Small-Angle X-ray Scattering (SAXS) and several types of X-ray
fiber diffraction Fiber diffraction is a subarea of scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, is forced to deviate from a straight traje ...
, which was used by Rosalind Franklin in determining the double helix, double-helix structure of
DNA Deoxyribonucleic acid (; DNA) is a molecule File:Pentacene on Ni(111) STM.jpg, A scanning tunneling microscopy image of pentacene molecules, which consist of linear chains of five carbon rings. A molecule is an electrically neutral gro ...

. 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 to interpret the back reflection Laue photograph.

## Electron and neutron diffraction

Other particles, such as electrons and
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behav ...

s, may be used to produce a
diffraction pattern Diffraction refers to various phenomena that occur when a wave encounters an obstacle or opening. It is defined as the bending of waves around the corners of an obstacle or through an aperture into the region of Umbra, penumbra and antumbra, ge ...
. Although electron, neutron, and X-ray scattering are based on different physical processes, the resulting diffraction patterns are analyzed using the same coherent diffraction imaging techniques. As derived below, the electron density within the crystal and the diffraction patterns are related by a simple mathematical method, the
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
, which allows the density to be calculated relatively easily from the patterns. However, this works only if the scattering is ''weak'', i.e., if the scattered beams are much less intense than the incoming beam. Weakly scattered beams pass through the remainder of the crystal without undergoing a second scattering event. Such re-scattered waves are called "secondary scattering" and hinder the analysis. Any sufficiently thick crystal will produce secondary scattering, but since X-rays interact relatively weakly with the electrons, this is generally not a significant concern. By contrast, electron beams may produce strong secondary scattering even for relatively thin crystals (>100 nm). Since this thickness corresponds to the diameter of many viruses, a promising direction is the electron diffraction of isolated macromolecular assemblies, such as virus, viral capsids and molecular machines, which may be carried out with a cryo-electron microscope. Moreover, the strong interaction of electrons with matter (about 1000 times stronger than for X-rays) allows determination of the atomic structure of extremely small volumes. The field of applications for
electron crystallography The electron is a subatomic particle, symbol or , whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are generally thought to be e ...
ranges from bio molecules like membrane proteins over organic thin films to the complex structures of (nanocrystalline) intermetallic compounds and zeolites. Neutron diffraction is an excellent method 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 much more readily from the atomic nuclei rather than from the electrons. Therefore, neutron scattering is very useful for observing the positions of light atoms with few electrons, especially hydrogen, which is essentially invisible in the X-ray diffraction. Neutron scattering also has the remarkable property that the solvent can be made invisible by adjusting the ratio of normal water, H2O, and heavy water, D2O.

# Methods

## Overview of single-crystal X-ray diffraction

The oldest and most precise method of X-ray crystallography is ''single-crystal X-ray diffraction'', in which a beam of X-rays strikes a single crystal, producing scattered beams. When they land on a piece of film or other detector, these beams make a ''diffraction pattern'' of spots; the strengths and angles of these beams are recorded as the crystal is gradually rotated. Each spot is called a ''reflection'', since it corresponds to the reflection of the X-rays from one set of evenly spaced planes within the crystal. For single crystals of sufficient purity and regularity, X-ray diffraction data can determine the mean chemical bond lengths and angles to within a few thousandths of an angstrom and to within a few tenths of a degree (angle), degree, respectively. The atoms in a crystal are not static, but oscillate about their mean positions, usually by less than a few tenths of an angstrom. X-ray crystallography allows measuring the size of these oscillations.

### Procedure

The technique of single-crystal X-ray crystallography has three basic steps. The first—and often most difficult—step is to obtain an adequate crystal of the material under study. The crystal should be sufficiently large (typically larger than 0.1 mm in all dimensions), pure in composition and regular in structure, with no significant internal crystal defect, imperfections such as cracks or crystal twinning, twinning. In the second step, the crystal is placed in an intense beam of X-rays, usually of a single wavelength (''monochromatic X-rays''), producing the regular pattern of reflections. The angles and intensities of diffracted X-rays are measured, with each compound having a unique diffraction pattern. As the crystal is gradually rotated, previous reflections disappear and new ones appear; the intensity of every spot is recorded at every orientation of the crystal. Multiple data sets may have to be collected, with each set covering slightly more than half a full rotation of the crystal and typically containing tens of thousands of reflections. In the third step, these data are combined computationally with complementary chemical information to produce and refine a model of the arrangement of atoms within the crystal. The final, refined model of the atomic arrangement—now called a ''
crystal structure In crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek language, Greek words ''crystallon'' "co ...

''—is usually stored in a public database.

### Limitations

As the crystal's repeating unit, its unit cell, becomes larger and more complex, the atomic-level picture provided by X-ray crystallography becomes less well-resolved (more "fuzzy") for a given number of observed reflections. Two limiting cases of X-ray crystallography—"small-molecule" (which includes continuous inorganic solids) and "macromolecular" crystallography—are often discerned. ''Small-molecule crystallography'' typically involves crystals with fewer than 100 atoms in their crystal structure, asymmetric unit; such crystal structures are usually so well resolved that the atoms can be discerned as isolated "blobs" of electron density. By contrast, ''macromolecular crystallography'' often involves tens of thousands of atoms in the unit cell. Such crystal structures are generally less well-resolved (more "smeared out"); the atoms and chemical bonds appear as tubes of electron density, rather than as isolated atoms. In general, small molecules are also easier to crystallize than macromolecules; however, X-ray crystallography has proven possible even for viruses and proteins with hundreds of thousands of atoms, through improved crystallographic imaging and technology. Though normally X-ray crystallography can only be performed if the sample is in crystal form, new research has been done into sampling non-crystalline forms of samples.

## Crystallization

Although crystallography can be used to characterize the disorder in an impure or irregular crystal, crystallography generally requires a pure crystal of high regularity to solve the structure of a complicated arrangement of atoms. Pure, regular crystals can sometimes be obtained from natural or synthetic materials, such as samples of
metal A metal (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

s, minerals or other macroscopic materials. The regularity of such crystals can sometimes be improved with macromolecular crystal annealing (metallurgy), annealing and other methods. However, in many cases, obtaining a diffraction-quality crystal is the chief barrier to solving its atomic-resolution structure. Small-molecule and macromolecular crystallography differ in the range of possible techniques used to produce diffraction-quality crystals. Small molecules generally have few degrees of conformational freedom, and may be crystallized by a wide range of methods, such as chemical vapor deposition and Recrystallization (chemistry)#Single perfect crystals (for X-ray analysis), recrystallization. By contrast, macromolecules generally have many degrees of freedom and their crystallization must be carried out so as to maintain a stable structure. For example, proteins and larger RNA molecules cannot be crystallized if their tertiary structure has been Denaturation (biochemistry), unfolded; therefore, the range of crystallization conditions is restricted to solution conditions in which such molecules remain folded. Protein crystals are almost always grown in solution. The most common approach is to lower the solubility of its component molecules very gradually; if this is done too quickly, the molecules will precipitate from solution, forming a useless dust or amorphous gel on the bottom of the container. Crystal growth in solution is characterized by two steps: ''nucleation'' of a microscopic crystallite (possibly having only 100 molecules), followed by ''growth'' of that crystallite, ideally to a diffraction-quality crystal. The solution conditions that favor the first step (nucleation) are not always the same conditions that favor the second step (subsequent growth). The crystallographer's goal is to identify solution conditions that favor the development of a single, large crystal, since larger crystals offer improved resolution of the molecule. Consequently, the solution conditions should ''disfavor'' the first step (nucleation) but ''favor'' the second (growth), so that only one large crystal forms per droplet. If nucleation is favored too much, a shower of small crystallites will form in the droplet, rather than one large crystal; if favored too little, no crystal will form whatsoever. Other approaches involves, crystallizing proteins under oil, where aqueous protein solutions are dispensed under liquid oil, and water evaporates through the layer of oil. Different oils have different evaporation permeabilities, therefore yielding changes in concentration rates from different percipient/protein mixture. It is extremely difficult to predict good conditions for nucleation or growth of well-ordered crystals. In practice, favorable conditions are identified by ''screening''; a very large batch of the molecules is prepared, and a wide variety of crystallization solutions are tested. Hundreds, even thousands, of solution conditions are generally tried before finding the successful one. The various conditions can use one or more physical mechanisms to lower the solubility of the molecule; for example, some may change the pH, some contain salts of the Hofmeister series or chemicals that lower the dielectric constant of the solution, and still others contain large polymers such as polyethylene glycol that drive the molecule out of solution by entropic effects. It is also common to try several temperatures for encouraging crystallization, or to gradually lower the temperature so that the solution becomes supersaturated. These methods require large amounts of the target molecule, as they use high concentration of the molecule(s) to be crystallized. Due to the difficulty in obtaining such large quantities (milligrams) of crystallization-grade protein, robots have been developed that are capable of accurately dispensing crystallization trial drops that are in the order of 100 nanoliters in volume. This means that 10-fold less protein is used per experiment when compared to crystallization trials set up by hand (in the order of 1 microliter). Several factors are known to inhibit or mar crystallization. The growing crystals are generally held at a constant temperature and protected from shocks or vibrations that might disturb their crystallization. Impurities in the molecules or in the crystallization solutions are often inimical to crystallization. Conformational flexibility in the molecule also tends to make crystallization less likely, due to entropy. Molecules that tend to self-assemble into regular helices are often unwilling to assemble into crystals. Crystals can be marred by Crystal twinning, twinning, which can occur when a unit cell can pack equally favorably in multiple orientations; although recent advances in computational methods may allow solving the structure of some twinned crystals. Having failed to crystallize a target molecule, a crystallographer may try again with a slightly modified version of the molecule; even small changes in molecular properties can lead to large differences in crystallization behavior.

## Data collection

### Mounting the crystal

The crystal is mounted for measurements so that it may be held in the X-ray beam and rotated. There are several methods of mounting. In the past, crystals were loaded into glass capillaries with the crystallization solution (the mother liquor). Nowadays, crystals of small molecules are typically attached with oil or glue to a glass fiber or a loop, which is made of nylon or plastic and attached to a solid rod. Protein crystals are scooped up by a loop, then flash-frozen with liquid nitrogen. This freezing reduces the radiation damage of the X-rays, as well as the noise in the Bragg peaks due to thermal motion (the Debye-Waller effect). However, untreated protein crystals often crack if flash-frozen; therefore, they are generally pre-soaked in a cryoprotectant solution before freezing. Unfortunately, this pre-soak may itself cause the crystal to crack, ruining it for crystallography. Generally, successful cryo-conditions are identified by trial and error. The capillary or loop is mounted on a
goniometer A goniometer is an instrument that either measures an angle or allows an object to be rotated to a precise angular position. The term goniometry derives from two Greek words, γωνία (''gōnía'') 'angle In Euclidean geometry Euclidea ...

, which allows it to be positioned accurately within the X-ray beam and rotated. Since both the crystal and the beam are often very small, the crystal must be centered within the beam to within ~25 micrometers accuracy, which is aided by a camera focused on the crystal. The most common type of goniometer is the "kappa goniometer", which offers three angles of rotation: the ω angle, which rotates about an axis perpendicular to the beam; the κ angle, about an axis at ~50° to the ω axis; and, finally, the φ angle about the loop/capillary axis. When the κ angle is zero, the ω and φ axes are aligned. The κ rotation allows for convenient mounting of the crystal, since the arm in which the crystal is mounted may be swung out towards the crystallographer. The oscillations carried out during data collection (mentioned below) involve the ω axis only. An older type of goniometer is the four-circle goniometer, and its relatives such as the six-circle goniometer.

### =Rotating anode

= Small scale crystallography can be done with a local X-ray tube source, typically coupled with an Photostimulated luminescence, 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 light 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 Volt, 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 Atomic orbital, inner-shell electrons of the metal. The most common metal used is
copper Copper is a chemical element In chemistry, an element is a pure Chemical substance, substance consisting only of atoms that all have the same numbers of protons in their atomic nucleus, nuclei. Unlike chemical compounds, chemical elem ...

, which can be kept cool easily, due to its high thermal conductivity, and which produces strong K-alpha, 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 run with ~2 Kilowatt, kW of electron beam power. The more expensive variety has a X-ray tube#Rotating anode tube, rotating-anode type source that run 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 collimator, 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 a clever arrangement of gently curved mirrors. Mirror systems are preferred for small crystals (under 0.3 mm) or with large unit cells (over 150 Å). Rotating anodes were used by Joanna (Joka) Maria Vandenberg in the first experiments that demonstrated the power of X rays for quick (in real time production) screening of large InGaAsP thin film wafers for quality control of quantum well lasers.

### =Microfocus tube

= A more recent development is the X-ray tube#Microfocus X-ray tube, 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 sources are some of the brightest light sources on earth and are some of the most powerful tools available to X-ray crystallographers. X-ray beams generated in large machines called 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 Hutch (synchrotron), beamlines where data is collected without interruption. Synchrotrons were originally designed for use by high-energy physicists studying subatomic particles and cosmos, cosmic phenomena. The largest component of each synchrotron is its electron storage ring. This ring is actually 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. Using synchrotron radiation frequently has specific requirements for X-ray crystallography. The intense ionizing radiation can cause radiation damage to samples, particularly macromolecular crystals. Cryo bio-crystallography, Cryo crystallography protects the sample from radiation damage, by freezing the crystal at liquid nitrogen temperatures (~100 kelvin, K). However, synchrotron radiation frequently has the advantage of user-selectable wavelengths, allowing for Anomalous X-ray scattering, 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 crystallography. 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, 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.

### Recording the reflections

When a crystal is mounted and exposed to an intense beam of X-rays, it scatters the X-rays into a pattern of spots or ''reflections'' that can be observed on a screen behind the crystal. A similar pattern may be seen by shining a laser pointer at a compact disc. The relative intensities of these spots provide the information to determine the arrangement of molecules within the crystal in atomic detail. The intensities of these reflections may be recorded with photographic film, an area detector (such as a hybrid pixel detector, pixel detector) or with a charge-coupled device (CCD) image sensor. The peaks at small angles correspond to low-resolution data, whereas those at high angles represent high-resolution data; thus, an upper limit on the eventual resolution of the structure can be determined from the first few images. Some measures of diffraction quality can be determined at this point, such as the mosaicity of the crystal and its overall disorder, as observed in the peak widths. Some pathologies of the crystal that would render it unfit for solving the structure can also be diagnosed quickly at this point. One image of spots is insufficient to reconstruct the whole crystal; it represents only a small slice of the full Fourier transform. To collect all the necessary information, the crystal must be rotated step-by-step through 180°, with an image recorded at every step; actually, slightly more than 180° is required to cover reciprocal space, due to the curvature of the Ewald sphere. However, if the crystal has a higher symmetry, a smaller angular range such as 90° or 45° may be recorded. The rotation axis should be changed at least once, to avoid developing a "blind spot" in reciprocal space close to the rotation axis. It is customary to rock the crystal slightly (by 0.5–2°) to catch a broader region of reciprocal space. Multiple data sets may be necessary for certain Phase problem, phasing methods. For example, multi-wavelength anomalous dispersion phasing requires that the scattering be recorded at least three (and usually four, for redundancy) wavelengths of the incoming X-ray radiation. A single crystal may degrade too much during the collection of one data set, owing to radiation damage; in such cases, data sets on multiple crystals must be taken.

## Data analysis

### Crystal symmetry, unit cell, and image scaling

The recorded series of two-dimensional diffraction patterns, each corresponding to a different crystal orientation, is converted into a three-dimensional model of the electron density; the conversion uses the mathematical technique of Fourier transforms, which is explained #Diffraction theory, below. Each spot corresponds to a different type of variation in the electron density; the crystallographer must determine ''which'' variation corresponds to ''which'' spot (''indexing''), the relative strengths of the spots in different images (''merging and scaling'') and how the variations should be combined to yield the total electron density (''phasing''). Data processing begins with ''indexing'' the reflections. This means identifying the dimensions of the unit cell and which image peak corresponds to which position in reciprocal space. A byproduct of indexing is to determine the symmetry of the crystal, i.e., its ''space group''. Some space groups can be eliminated from the beginning. For example, reflection symmetries cannot be observed in chiral molecules; thus, only 65 space groups of 230 possible are allowed for protein molecules which are almost always chiral. Indexing is generally accomplished using an ''autoindexing'' routine. Having assigned symmetry, the data is then ''integrated''. This converts the hundreds of images containing the thousands of reflections into a single file, consisting of (at the very least) records of the Miller index of each reflection, and an intensity for each reflection (at this state the file often also includes error estimates and measures of partiality (what part of a given reflection was recorded on that image)). A full data set may consist of hundreds of separate images taken at different orientations of the crystal. The first step is to merge and scale these various images, that is, to identify which peaks appear in two or more images (''merging'') and to scale the relative images so that they have a consistent intensity scale. Optimizing the intensity scale is critical because the relative intensity of the peaks is the key information from which the structure is determined. The repetitive technique of crystallographic data collection and the often high symmetry of crystalline materials cause the diffractometer to record many symmetry-equivalent reflections multiple times. This allows calculating the symmetry-related R-factor (crystallography), R-factor, a reliability index based upon how similar are the measured intensities of symmetry-equivalent reflections, thus assessing the quality of the data.

### Initial phasing

The data collected from a diffraction experiment is a reciprocal space representation of the crystal lattice. The position of each diffraction 'spot' is governed by the size and shape of the unit cell, and the inherent Unit cell, symmetry within the crystal. The intensity of each diffraction 'spot' is recorded, and this intensity is proportional to the square of the ''structure factor'' amplitude. The structure factor is a complex number containing information relating to both the amplitude and Phase (waves), phase of a wave. In order to obtain an interpretable ''electron density map'', both amplitude and phase must be known (an electron density map allows a crystallographer to build a starting model of the molecule). The phase cannot be directly recorded during a diffraction experiment: this is known as the phase problem. Initial phase estimates can be obtained in a variety of ways: * ''Ab initio'' phasing or Direct methods (crystallography), direct methods – This is usually the method of choice for small molecules (<1000 non-hydrogen atoms), and has been used successfully to solve the phase problems for small proteins. If the resolution of the data is better than 1.4 Å (140 picometre, pm), Direct methods (crystallography), direct methods can be used to obtain phase information, by exploiting known phase relationships between certain groups of reflections. * Molecular replacement – if a related structure is known, it can be used as a search model in molecular replacement to determine the orientation and position of the molecules within the unit cell. The phases obtained this way can be used to generate electron density maps. * Anomalous X-ray scattering (''Multi-wavelength anomalous dispersion, MAD or Single wavelength anomalous dispersion, SAD phasing'') – the X-ray wavelength may be scanned past an absorption edge of an atom, which changes the scattering in a known way. By recording full sets of reflections at three different wavelengths (far below, far above and in the middle of the absorption edge) one can solve for the substructure of the anomalously diffracting atoms and hence the structure of the whole molecule. The most popular method of incorporating anomalous scattering atoms into proteins is to express the protein in a methionine auxotroph (a host incapable of synthesizing methionine) in a media rich in seleno-methionine, which contains selenium atoms. A multi-wavelength anomalous dispersion (MAD) experiment can then be conducted around the absorption edge, which should then yield the position of any methionine residues within the protein, providing initial phases. * Heavy atom methods (multiple isomorphous replacement) – If electron-dense metal atoms can be introduced into the crystal, Direct methods (crystallography), direct methods or Patterson function, Patterson-space methods can be used to determine their location and to obtain initial phases. Such heavy atoms can be introduced either by soaking the crystal in a heavy atom-containing solution, or by co-crystallization (growing the crystals in the presence of a heavy atom). As in multi-wavelength anomalous dispersion phasing, the changes in the scattering amplitudes can be interpreted to yield the phases. Although this is the original method by which protein crystal structures were solved, it has largely been superseded by multi-wavelength anomalous dispersion phasing with selenomethionine.

### Model building and phase refinement

Having obtained initial phases, an initial model can be built. The atomic positions in the model and their respective Debye-Waller factors (or B-factors, accounting for the thermal motion of the atom) can be refined to fit the observed diffraction data, ideally yielding a better set of phases. A new model can then be fit to the new electron density map and successive rounds of refinement is carried out. This interative process continues until the correlation between the diffraction data and the model is maximized. The agreement is measured by an R-factor (crystallography), ''R''-factor defined as :$R = \frac,$ where ''F'' is the structure factor. A similar quality criterion is ''R''free, which is calculated from a subset (~10%) of reflections that were not included in the structure refinement. Both ''R'' factors depend on the resolution of the data. As a rule of thumb, ''R''free should be approximately the resolution in angstroms divided by 10; thus, a data-set with 2 Å resolution should yield a final ''R''free ~ 0.2. Chemical bonding features such as stereochemistry, hydrogen bonding and distribution of bond lengths and angles are complementary measures of the model quality. Phase bias is a serious problem in such iterative model building. ''Omit maps'' are a common technique used to check for this. It may not be possible to observe every atom in the asymmetric unit. In many cases, Crystallographic disorder smears the electron density map. Weakly scattering atoms such as hydrogen are routinely invisible. It is also possible for a single atom to appear multiple times in an electron density map, e.g., if a protein sidechain has multiple (<4) allowed conformations. In still other cases, the crystallographer may detect that the covalent structure deduced for the molecule was incorrect, or changed. For example, proteins may be cleaved or undergo post-translational modifications that were not detected prior to the crystallization.

### Disorder

A common challenge in refinement of crystal structures results from crystallographic disorder. Disorder can take many forms but in general involves the coexistence of two or more species or conformations. Failure to recognize disorder results in flawed interpretation. Pitfalls from improper modeling of disorder are illustrated by the discounted hypothesis of bond stretch isomerism. Disorder is modelled with respect to the relative population of the components, often only two, and their identity. In structures of large molecules and ions, solvent and counterions are often disordered.

### Applied computational data analysis

The use of computational methods for the powder X-ray diffraction data analysis is now generalized. It typically compares the experimental data to the simulated diffractogram of a model structure, taking into account the instrumental parameters, and refines the structural or microstructural parameters of the model using least squares based minimization algorithm. Most available tools allowing phase identification and structural refinement are based on the Rietveld refinement, Rietveld method, some of them being open and free software such as FullProf Suite, Jana2006, MAUD, Rietan, GSAS, etc. while others are available under commercials licenses such as Diffrac.Suite TOPAS, Match!, etc. Most of these tools also allow Le Bail method, Le Bail refinement (also referred to as profile matching), that is, refinement of the cell parameters based on the Bragg peaks positions and peak profiles, without taking into account the crystallographic structure by itself. More recent tools allow the refinement of both structural and microstructural data, such as the FAULTS program included in the FullProf Suite, which allows the refinement of structures with planar defects (e.g. stacking faults, twinnings, intergrowths).

## Deposition of the structure

Once the model of a molecule's structure has been finalized, it is often deposited in a crystallographic database such as the Cambridge Structural Database (for small molecules), the Inorganic Crystal Structure Database (ICSD) (for inorganic compounds) or the Protein Data Bank (for protein and sometimes nucleic acids). Many structures obtained in private commercial ventures to crystallize medicinally relevant proteins are not deposited in public crystallographic databases.

# Diffraction theory

The main goal of X-ray crystallography is to determine the density of electrons ''f''(r) throughout the crystal, where r represents the three-dimensional position vector (geometry), vector within the crystal. To do this, X-ray scattering is used to collect data about its Fourier transform ''F''(q), which is inverted mathematically to obtain the density defined in real space, using the formula : $f\left(\mathbf\right) = \frac \int F\left(\mathbf\right) e^ \mathrm\mathbf,$ where the integral is taken over all values of q. The three-dimensional real vector q represents a point in reciprocal space, that is, to a particular oscillation in the electron density as one moves in the direction in which q points. The length of q corresponds to $2\pi$ divided by the wavelength of the oscillation. The corresponding formula for a Fourier transform will be used below : $F\left(\mathbf\right) = \int f\left(\mathbf\right) \mathrm^\mathrm\mathbf,$ where the integral is summed over all possible values of the position vector r within the crystal. The Fourier transform ''F''(q) is generally a complex number, and therefore has a magnitude (mathematics), magnitude , ''F''(q), and a Phase (waves), phase ''φ''(q) related by the equation : $F\left(\mathbf\right) = \left, F\left(\mathbf\right) \\mathrm^.$ The intensities of the reflections observed in X-ray diffraction give us the magnitudes , ''F''(q), but not the phases ''φ''(q). To obtain the phases, full sets of reflections are collected with known alterations to the scattering, either by modulating the wavelength past a certain absorption edge or by adding strongly scattering (i.e., electron-dense) metal atoms such as mercury (element), mercury. Combining the magnitudes and phases yields the full Fourier transform ''F''(q), which may be inverted to obtain the electron density ''f''(r). Crystals are often idealized as being ''perfectly'' periodic. In that ideal case, the atoms are positioned on a perfect lattice, the electron density is perfectly periodic, and the Fourier transform ''F''(q) is zero except when q belongs to the reciprocal lattice (the so-called ''Bragg peaks''). In reality, however, crystals are not perfectly periodic; atoms vibrate about their mean position, and there may be disorder of various types, such as mosaicity, dislocations, various point defects, and heterogeneity in the conformation of crystallized molecules. Therefore, the Bragg peaks have a finite width and there may be significant ''diffuse scattering'', a continuum of scattered X-rays that fall between the Bragg peaks.

## Intuitive understanding by Bragg's law

An intuitive understanding of X-ray diffraction can be obtained from the Bragg diffraction, 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 Miller index, three Miller indices (''h'', ''k'', ''l''), and let their spacing be noted by ''d''. William Lawrence Bragg proposed a model in which 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. : $2 d\sin\theta = n\lambda.$ 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 lattice parameter, unit-cell parameters, the lengths and angles of the unit-cell, as well as its space group. Since
Bragg's law In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spa ...

does not interpret the relative intensities of the reflections, however, it is generally inadequate to solve for the arrangement of atoms within the unit-cell; for that, a Fourier transform method must be carried out.

## Scattering as a Fourier transform

The incoming X-ray beam has a polarization and should be represented as a vector wave; however, for simplicity, let it be represented here as a scalar wave. We also ignore the complication of 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 strength of the incoming wave at time ''t'' = 0 is given by : $A \mathrm^.$ At position r within the sample, let there be a density of scatterers ''f''(r); these scatterers should 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 : $\text = A \mathrm^ S f\left(\mathbf\right) \mathrmV,$ where ''S'' is the proportionality constant. Consider the fraction of scattered waves that leave with an outgoing wave-vector of kout and strike the screen 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 : $e^.$ The net radiation arriving at rscreen is the sum of all the scattered waves throughout the crystal : $A S \int \mathrm\mathbf f\left(\mathbf\right) \mathrm^ e^ = A S e^ \int \mathrm\mathbf f\left(\mathbf\right) \mathrm^,$ which may be written as a Fourier transform : $A S \mathrm^ \int d\mathbf f\left(\mathbf\right) \mathrm^ = A S \mathrm^ F\left(\mathbf\right),$ where q = kout – kin. The measured intensity of the reflection will be square of this amplitude : $A^ S^ \left, F\left(\mathbf\right) \^2.$

## Friedel and Bijvoet mates

For every reflection corresponding to a point q in the reciprocal space, there is another reflection of the same intensity (physics), intensity at the opposite point -q. This opposite reflection is known as the ''Friedel mate'' of the original reflection. This symmetry results from the mathematical fact that the density of electrons ''f''(r) at a position r is always a real number. As noted above, ''f''(r) is the inverse transform of its Fourier transform ''F''(q); however, such an inverse transform is a complex number in general. To ensure that ''f''(r) is real, the Fourier transform ''F''(q) must be such that the Friedel mates ''F''(−q) and ''F''(q) are complex conjugates of one another. Thus, ''F''(−q) has the same magnitude as ''F''(q) but they have the opposite phase, i.e., ''φ''(q) = −''φ''(q) : $F\left(-\mathbf\right) = \left, F\left(-\mathbf\right) \\mathrm^ = F^*\left(\mathbf\right) = \left, F\left(\mathbf\right) \\mathrm^.$ The equality of their magnitudes ensures that the Friedel mates have the same intensity , ''F'', 2. This symmetry allows one to measure the full Fourier transform from only half the reciprocal space, e.g., by rotating the crystal slightly more than 180° instead of a full 360° revolution. In crystals with significant symmetry, even more reflections may have the same intensity (Bijvoet mates); in such cases, even less of the reciprocal space may need to be measured. In favorable cases of high symmetry, sometimes only 90° or even only 45° of data are required to completely explore the reciprocal space. The Friedel-mate constraint can be derived from the definition of the inverse Fourier transform : $f\left(\mathbf\right) = \int \frac F\left(\mathbf\right) \mathrm^ = \int \frac \left, F\left(\mathbf\right) \\mathrm^ \mathrm^.$ Since Euler's formula states that ei''x'' = cos(''x'') + i sin(''x''), the inverse Fourier transform can be separated into a sum of a purely real part and a purely imaginary part : $f\left(\mathbf\right) = \int \frac \left, F\left(\mathbf\right) \\mathrm^ = \int \frac \left, F\left(\mathbf\right) \ \cos\left\left(\phi+\mathbf\cdot\mathbf\right\right) + i \int \frac \left, F\left(\mathbf\right) \ \sin\left\left(\phi+\mathbf\cdot\mathbf\right\right) = I_ + iI_.$ The function ''f''(r) is real if and only if the second integral ''I''sin is zero for all values of r. In turn, this is true if and only if the above constraint is satisfied : $I_ = \int \frac \left, F\left(\mathbf\right) \\sin\left\left(\phi+\mathbf\cdot\mathbf\right\right) = \int \frac \left, F\left(\mathbf\right) \ \sin\left\left(-\phi-\mathbf\cdot\mathbf\right\right) = -I_,$ since ''I''sin = −''I''sin implies that ''I''sin = 0.

## Ewald's sphere

Each X-ray diffraction image represents only a slice, a spherical slice of reciprocal space, as may be seen by the Ewald sphere construction. Both kout and kin have the same length, due to the elastic scattering, since the wavelength has not changed. Therefore, they may be represented as two radial vectors in a sphere in reciprocal space, which shows the values of q that are sampled in a given diffraction image. Since there is a slight spread in the incoming wavelengths of the incoming X-ray beam, the values of, ''F''(q), can be measured only for q vectors located between the two spheres corresponding to those radii. Therefore, to obtain a full set of Fourier transform data, it is necessary to rotate the crystal through slightly more than 180°, or sometimes less if sufficient symmetry is present. A full 360° rotation is not needed because of a symmetry intrinsic to the Fourier transforms of real functions (such as the electron density), but "slightly more" than 180° is needed to cover all of reciprocal space within a given resolution because of the curvature of the Ewald sphere. In practice, the crystal is rocked by a small amount (0.25–1°) to incorporate reflections near the boundaries of the spherical Ewald's shells.

## Patterson function

A well-known result of Fourier transforms is the autocorrelation theorem, which states that the autocorrelation ''c''(r) of a function ''f''(r) : $c\left(\mathbf\right) = \int d\mathbf f\left(\mathbf\right) f\left(\mathbf + \mathbf\right) = \int \frac C\left(\mathbf\right) e^$ has a Fourier transform ''C''(q) that is the squared magnitude of ''F''(q) :$C\left(\mathbf\right) = \left, F\left(\mathbf\right) \^2.$ Therefore, the autocorrelation function ''c''(r) of the electron density (also known as the ''Patterson function'') can be computed directly from the reflection intensities, without computing the phases. In principle, this could be used to determine the crystal structure directly; however, it is difficult to realize in practice. The autocorrelation function corresponds to the distribution of vector (geometry), vectors between atoms in the crystal; thus, a crystal of ''N'' atoms in its unit cell may have ''N''(''N'' − 1) peaks in its Patterson function. Given the inevitable errors in measuring the intensities, and the mathematical difficulties of reconstructing atomic positions from the interatomic vectors, this technique is rarely used to solve structures, except for the simplest crystals.

In principle, an atomic structure could be determined from applying X-ray scattering to non-crystalline samples, even to a single molecule. However, crystals offer a much stronger signal due to their periodicity. A crystalline sample is by definition periodic; a crystal is composed of many
unit cell In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

s repeated indefinitely in three independent directions. Such periodic systems have a
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
that is concentrated at periodically repeating points in reciprocal space known as ''Bragg peaks''; the Bragg peaks correspond to the reflection spots observed in the diffraction image. Since the amplitude at these reflections grows linearly with the number ''N'' of scatterers, the observed ''intensity'' of these spots should grow quadratically, like ''N''2. In other words, using a crystal concentrates the weak scattering of the individual unit cells into a much more powerful, coherent reflection that can be observed above the noise. This is an example of constructive interference. In a liquid, powder or amorphous sample, molecules within that sample are in random orientations. Such samples have a continuous Fourier spectrum that uniformly spreads its amplitude thereby reducing the measured signal intensity, as is observed in Biological Small-Angle X-ray Scattering, SAXS. More importantly, the orientational information is lost. Although theoretically possible, it is experimentally difficult to obtain atomic-resolution structures of complicated, asymmetric molecules from such rotationally averaged data. An intermediate case is
fiber diffraction Fiber diffraction is a subarea of scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, is forced to deviate from a straight traje ...
in which the subunits are arranged periodically in at least one dimension.

# Applications

X-ray diffraction has wide and various applications in the chemical, biochemical, physical, material and mineralogical sciences. Laue claimed in 1937 that the technique "has extended the power of observing minute structure ten thousand times beyond that given us by the microscope". X-ray diffraction is analogous to a microscope with atomic-level resolution which shows the atoms and their electron distribution. X-ray diffraction, electron diffraction, and neutron diffraction give information about the structure of matter, crystalline and non-crystalline, at the atomic and molecular level. In addition, these methods may be applied in the study of properties of all materials, inorganic, organic or biological. Due to the importance and variety of applications of diffraction studies of crystals, many Nobel Prizes have been awarded for such studies.

## Drug identification

X-ray diffraction has been used for the identification of antibiotic drugs such as: eight beta-lactam antibiotic, β-lactam (ampicillin sodium, penicillin G procaine, cefalexin, ampicillin trihydrate, Benzathine benzylpenicillin, benzathine penicillin, benzylpenicillin sodium, cefotaxime sodium, Ceftriaxone sodium), three Tetracycline antibiotics, tetracycline (doxycycline , doxycycline hydrochloride, oxytetracycline, oxytetracycline dehydrate, tetracycline hydrochloride) and two macrolide (azithromycin, erythromycin estolate) antibiotic drugs. Each of these drugs has a unique X-Ray Diffraction (XRD) pattern that makes their identification possible.

## Characterization of nanomaterials, textile fibers and polymers

Forensic examination of any trace evidence is based upon Locard's exchange principle. This states that "every contact leaves a trace". In practice, even though a transfer of material has taken place, it may be impossible to detect, because the amount transferred is very small. XRD has proven its role in the advancement of nanomaterial research. It is one of the primary characterization tools and provides information about the structural properties of various nanomaterials in both powder and thin-film form. Textile fibers are a mixture of crystalline and amorphous substances. Therefore, the measurement of the degree of crystalline gives useful data in the characterization of fibers using X-ray diffractometry. It has been reported that X-ray diffraction was used to identify of a "crystalline" deposit which was found on a chair. The deposit was found to be amorphous, but the diffraction pattern present matched that of polymethylmethacrylate. Pyrolysis mass spectrometry later identified the deposit as polymethylcyanoacrylaon of Boin crystal parameters.

## Investigation of bones

Heating or burning of bones causes recognizable changes in the bone mineral that can be detected using XRD techniques. During the first 15 minutes of heating at 500 °C or above, the bone crystals began to change. At higher temperatures, thickness and shape of crystals of bones appear stabilized, but when the samples were heated at a lower temperature or for a shorter period, XRD traces showed extreme changes in crystal parameters.

## Integrated circuits

X-ray diffraction has been demonstrated as a method for investigating the complex structure of integrated circuits.

* Beevers–Lipson strip * Bragg diffraction * Crystallographic database * Crystallographic point groups * Difference density map * Electron diffraction * Energy Dispersive X-Ray Diffraction * Flack parameter * Henderson limit * International Year of Crystallography * John Desmond Bernal * Multipole density formalism * Neutron diffraction * Powder diffraction * Ptychography * Scherrer equation * Small angle X-ray scattering (SAXS) * Structure determination * Ultrafast x-ray * Wide angle X-ray scattering (WAXS)

# References

* * *

* * *

## Textbooks

* * * * * * * * * * * * * *
PDF copy of select chapters
* * *

*

## Historical

* * * * * Ewald, P. P., edito
''50 Years of X-Ray Diffraction''
(Reprinted in pdf format for the IUCr XVIII Congress, Glasgow, Scotland, International Union of Crystallography). * * *

## Tutorials

Simple, non technical introductionThe Crystallography Collection
video series from the Royal Institution
"Small Molecule Crystalization"
(PDF) at Illinois Institute of Technology website
International Union of Crystallography

demonstrating properties of the diffraction pattern of a 2D crystal.

illustrating the relationship between crystal and diffraction pattern in 2D.

* [http://nanohub.org/resources/5580 Online lecture on Modern X-ray Scattering Methods for Nanoscale Materials Analysis] by Richard J. Matyi
Interactive Crystallography Timeline
from the Royal Institution

## Primary databases

* Crystallography open database, Crystallography Open Database (COD)
Protein Data Bank
(Protein Data Bank, PDB)
Nucleic Acid Databank
(NDB)
Cambridge Structural Database
(Cambridge Structural Database, CSD)
Inorganic Crystal Structure Database
(Inorganic Crystal Structure Database, ICSD)
Biological Macromolecule Crystallization Database
(BMCD)

## Derivative databases

PDBsum

Proteopedia – the collaborative, 3D encyclopedia of proteins and other molecules

RNABase

HIC-Up database of PDB ligands
* Structural Classification of Proteins database * CATH, CATH Protein Structure Classification
List of transmembrane proteins with known 3D structure
* Orientations of Proteins in Membranes database

## Structural validation

MolProbity structural validation suite

ProSA-web

NQ-Flipper
(check for unfavorable rotamers of Asn and Gln residues)
DALI server
(identifies proteins similar to a given protein) {{DEFAULTSORT:X-Ray Crystallography X-ray crystallography, Condensed matter physics Crystallography Diffraction Materials science Protein structure Protein methods Protein imaging Synchrotron-related techniques Articles containing video clips X-rays, Crystallography