X-ray crystallography 34151 220644951 2008-06-20T21:32:18Z Reciproc08 7287033 /* See also */ [[Image:X ray diffraction.png|thumb|200px|right|upright|Workflow for solving the structure of a molecule by X-ray crystallography.]] '''X-ray crystallography''' is the science of determining the arrangement of [[atom]]s within a [[crystal]] from the manner in which a beam of [[X-ray]]s is scattered from the [[electron]]s within the crystal. The method produces a three-dimensional picture of the density of [[electron]]s within the crystal, from which the mean [[atomic]] positions, their [[chemical bond]]s, their [[disorder]] and sundry other information can be derived. The key step in X-ray crystallography is the [[X-ray diffraction|diffraction of X-rays]] from a crystalline material. A crystal is a solid in which a particular arrangement of atoms (its [[unit cell]]) is repeated indefinitely along [[Bravais lattice| three principal directions]] known as the ''basis'' (or ''lattice'') ''vectors''. A wide variety of materials can form crystals — such as [[salt (chemistry)|salts]], [[metal]]s, [[mineral]]s, [[semiconductor]]s, as well as various inorganic, organic and biological molecules — which has made X-ray crystallography fundamental to many scientific fields. Although it is most informative to diffract X-rays from a single, large crystal with few [[crystal defect|defects]], such crystals may be difficult to obtain; for simple materials, it may be possible to reconstruct the atomic structure from the X-ray diffraction of [[polycrystalline]] samples, a technique known as X-ray [[powder diffraction]]. After a [[crystal]] has been obtained or grown in the laboratory, it is mounted on a [[goniometer]] and bombarded with X-rays, producing a diffraction pattern of regularly spaced spots known as ''reflections''. The crystal is gradually rotated and a diffraction pattern is collected for each distinct orientation of the crystal. These two-dimensional images are converted into a three-dimensional model of the density of electrons within the crystal using the mathematical method of [[Fourier transform]]s and chemical data on the sample. The positions of the atomic nuclei are deduced from this electron density and chemical data, producing a model of the atoms within the crystal. X-ray crystallography is useful in identifying known materials, characterizing new materials and in discerning materials that appear similar by other [[experiment]]s. X-ray [[crystal structure]]s can also account for unusual [[electronic]] or [[elastic]] properties of a material, shed light on chemical interactions and processes, or serve as the basis for understanding [[enzyme|enzymatic mechanisms]] and [[drug design|designing pharmaceuticals against diseases]]. == 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 are reflected from evenly spaced planes of a single crystal, producing a ''diffraction pattern'' of spots called ''reflections''.<ref>An analogous diffraction pattern may be observed by shining a laser pointer on a [[compact disc]] or [[DVD]]; the periodic spacing of the CD tracks corresponds to the periodic arrangement of atoms in a crystal.</ref> Each reflection corresponds to one set of evenly spaced planes within the crystal. The density of electrons within the crystal is determined from the position and brightness of the various reflections observed as the crystal is gradually rotated in the X-ray beam; this density, together with supplementary data, allows the atomic positions to be inferred. 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 [[Ångström]] and to within a few tenths of a [[degree (angle)|degree]], respectively. The data also allow the static and dynamic disorder in the atomic positions to be estimated, which is usually less than a few tenths of an Ångström. === 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 100 [[micrometre]]s in all dimensions), pure in composition and regular in structure, with no significant internal [[crystal defect|imperfections]] such as cracks or [[crystal twinning|twinning]]. A small or irregular crystal will give fewer and less reliable data, from which it may be impossible to determine the atomic arrangement. 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. 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 reflection intensities. 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]]'' — is usually stored in a public database. === Limitations === {{see also|Resolution (electron density)}} 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 are often discerned, "small-molecule" and "macromolecular" crystallography. ''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 its 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 [[virus]]es with hundreds of thousands of atoms. ==History== ===Scientific pre-history of crystals and X-rays=== [[Image:Kepler conjecture 1.jpg|thumb|right|Drawing of square (Figure A, above) and hexagonal (Figure B, below) packing from [[Johannes Kepler|Kepler's]] work, ''Strena seu de Nive Sexangula''.]] Crystals have long been admired for their regularity and symmetry, but they were not investigated scientifically until the 17th century. [[Johannes Kepler]] hypothesized in his work ''Strena seu de Nive Sexangula'' (1611) that the hexagonal symmetry of [[snow|snowflake crystals]] was due to a regular packing of spherical water particles.<ref>{{cite book | last = Kepler | first = J | authorlink = Johannes Kepler | year = 1611 | title = Strena seu de Nive Sexangula | publisher = G. Tampach | location = Frankfurt}}</ref> [[Image:Snowflake8.png|thumb|left|As shown by X-ray crystallography, the hexagonal symmetry of snowflakes results from the [[tetrahedron|tetrahedral]] arrangement of [[hydrogen bond]]s about each water molecule. The water molecules form a [[diamond cubic|diamond lattice]], which has hexagonal symmetry when viewed along a principal axis.]] Crystal symmetry was first investigated experimentally by [[Nicolas Steno]] (1669), who showed that the angles between the faces are the same in every exemplar of a particular type of crystal,<ref>{{cite book | last = Steno | first = N | authorlink = Nicolas Steno | year = 1669 | title = De solido intra solidum naturaliter contento dissertationis prodromus | publisher = Florentiae}}</ref> and by [[René Just Haüy]] (1784), who discovered that every face of a crystal can be described by three small integers, the so-called [[Miller index|Miller indices]]. These studies led Haüy to the correct idea that crystals are a regular three-dimensional array (a [[Bravais lattice]]) of [[atom]]s and [[molecule]]s; a single [[unit cell]] 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 [[Johann Hessel]],<ref>{{cite book | last = Hessel | first = JFC | year = 1831 | title = Kristallometrie oder Kristallonomie und Kristallographie | publisher = Leipzig}}</ref> [[Auguste Bravais]],<ref>{{cite journal | last = Bravais | first = Auguste | authorlink = Auguste Bravais | year = 1850 | title = Mémoire sur les systèmes formés par des points distribués regulièrement sur un plan ou dans l'espace | journal = J. l'Ecole Polytech. | volume = 19 | pages = 1&ndash;?}}</ref> [[Yevgraf Fyodorov]],<ref>{{Cite journal|author=I. I. Shafranovskii and N. V. Belov|title=E. S. Fedorov|journal=50 Years of X-Ray Diffraction, ed. Paul Ewald (Springer) |date=1962| id=ISBN 9027790299| url=http://www.iucr.org/iucr-top/publ/50YearsOfXrayDiffraction/fedorov.pdf | pages=pp 351–353 }}</ref>, [[Arthur Moritz Schönflies|Arthur Schönflies]]<ref>{{cite book | last = Schönflies | first = A | authorlink = Arthur Moritz Schönflies | year = 1891 | title = Kristallsysteme und Kristallstruktur | publisher = Leipzig}}</ref> and (belatedly) [[William Barlow]]. On the basis of the available data and physical reasoning, Barlow proposed several crystal structures in the 1880s that were ultimately proven correct by X-ray crystallography;<ref>{{cite journal | author = Barlow W | date = 1883 | title = Probable nature of the internal symmetry of crystals | journal = Nature | volume = 29 | pages = 186&ndash;? | doi = 10.1038/029186a0}} See also Barlow W, ''Nature'', '''29''', 205, 383, 404 (1883-1884).</ref> however, the available data were too few in the 1880s to accept his models as conclusive. [[Image:3D model hydrogen bonds in water.jpg|thumb|right|X-ray crystallography shows the arrangement of water molecules in ice, revealing the hydrogen bonds that confer its hexagonal symmetry. Few other methods can determine the structure of matter with such sub-atomic precision (''resolution'').]] X-rays were discovered by [[Wilhelm Conrad Röntgen]] in 1895, just as the studies of crystal symmetry were being concluded. Physicists were initially uncertain of the nature of X-rays, although it was soon suspected (correctly) that they were waves of [[electromagnetic radiation]], in other words, another form of [[light]]. At that time, the wave model of [[light]] — specifically, the [[James Clerk Maxwell|Maxwell]] theory of [[electromagnetic radiation]] — was well accepted among scientists, and experiments by [[Charles Glover Barkla]] showed that X-rays exhibited phenomena associated with electromagnetic waves, including transverse [[polarization]] and spectral lines akin to those observed in the visible wavelengths. Single-slit experiments in the laboratory of [[Arnold Sommerfeld]] suggested the [[wavelength]] of X-rays was roughly 1 [[Angström]], one ten millionth of a millimetre. Being composed of [[photon]]s, X-rays also exhibit particle-like properties, e.g., in the ionization of gases; these properties led [[William Henry Bragg]] to suggest in 1907 that X-rays were ''not'' electromagnetic radiation,<ref>{{cite journal | author = Bragg WH | authorlink = William Henry Bragg | date = 1907 | title = The nature of Röntgen rays | journal = Transactions of the Royal Society of Science of Australia | volume = 31 | pages = 94&ndash;98}}<br />{{cite journal | author = Bragg WH | authorlink = William Henry Bragg | date = 1908 | title = The nature of γ- and X-rays | journal = Nature | volume = 77 | pages = 270&ndash;271 | doi = 10.1038/077270a0}} See also ''Nature'', '''78''', 271, 293&ndash;294, 665 (1908).<br />{{cite journal | author = Bragg WH | authorlink = William Henry Bragg | date = 1910 | title = The consequences of the corpuscular hypothesis of the γ- and X-rays, and the range of β-rays | journal = Philosophical Magazine | volume = 20 | pages = 385&ndash;416}}<br />{{cite journal | author = Bragg WH | authorlink = William Henry Bragg | date = 1912 | title = On the direct or indirect nature of the ionization by X-rays | journal = Philosophical Magazine | volume = 23 | pages = 647&ndash;650}}</ref> since the concept of the photon was relatively new (1905)<ref name="Einstein1905">{{cite journal | last = Einstein | first = A | authorlink = Albert Einstein | year = 1905 | title = Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt (trans. A Heuristic Model of the Creation and Transformation of Light) | journal = [[Annalen der Physik]] | volume = 17 | pages = 132&ndash;148}} {{de icon}}. An [[s:A Heuristic Model of the Creation and Transformation of Light|English translation]] is available from [[Wikisource]].</ref> and not generally accepted.<ref name="Einstein1909">{{cite journal | last = Einstein | first = A | authorlink = Albert Einstein | year = 1909 | title = Über die Entwicklung unserer Anschauungen über das Wesen und die Konstitution der Strahlung (trans. The Development of Our Views on the Composition and Essence of Radiation) | journal = Physikalische Zeitschrift | volume = 10|pages = 817&ndash;825}} {{de icon}}. An [[s:The Development of Our Views on the Composition and Essence of Radiation|English translation]] is available from [[Wikisource]].</ref><ref>{{cite book | last = Pais | first = A. | authorlink = Abraham Pais | year = 1982 | title = Subtle is the Lord: The Science and the Life of Albert Einstein | publisher = Oxford University Press }}</ref> However, Bragg's view was itself not broadly accepted and the observation of [[X-ray diffraction]] in 1912<ref name="Laue1912" /> confirmed for most scientists that X-rays were a form of electromagnetic radiation. Any remaining doubt was resolved in 1922, when [[Arthur Compton]] confirmed the photon model by studying the scattering of X-rays from [[electron]]s.<ref name="Compton1923">{{cite journal | last = Compton | first = A | authorlink = Arthur Compton | year = 1923 | title = [http://www.aip.org/history/gap/Compton/01_Compton.html A Quantum Theory of the Scattering of X-rays by Light Elements] | journal = [[Physical Review]] | volume = 21 | pages = 483&ndash;502 | doi = 10.1103/PhysRev.21.483}}</ref> ===X-rays analysis of crystals === [[Image:Bragg diffraction.png|thumb|left|The incoming beam (coming from upper left) causes each scatterer to re-radiate a small portion of its energy as a spherical wave. If scatterers are arranged symmetrically with a separation ''d'', these spherical waves will be in synch (add constructively) only in directions where their path-length difference 2 ''d'' sin θ equals an integer multiple of the [[wavelength]] λ. In that case, part of the incoming beam is deflected by an angle 2θ, producing a ''reflection'' spot in the [[diffraction pattern]].]] Crystals are regular arrays of atoms and X-rays can be considered waves of electromagnetic radiation. Atoms scatter X-rays, primarily through their [[electron]]s; 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 ''scattering'', and the electron (or lighthouse) is known as the ''scatterer''. A regular array of scatterers produces a regular array of spherical waves. Although these waves cancel one another out in most directions ([[destructive interference]]), they add constructively in a few choice directions, determined by [[Bragg's law]] :<math> 2d \sin \theta = n \lambda </math> where ''n'' is any integer. These "choice directions" appear as spots on the [[diffraction pattern]], often 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 [[Ångström]]s) as the spacing ''d'' between planes in the crystal. In principle, any wave impinging on a regular array of scatterers produces [[diffraction]], as predicted first by [[Francesco Maria Grimaldi]] in 1665. To produce significant diffraction, the spacing between the scatterers and the wavelength of the impinging wave should be roughly similar in size. For illustration, the diffraction of sunlight through a bird's feather was first reported by [[James Gregory (astronomer and mathematician)|James Gregory]] in the later 17th century. The first man-made [[diffraction grating]]s for visible light were constructed by [[David Rittenhouse]] in 1787, and [[Joseph von Fraunhofer]] in 1821. However, visible light has too long a wavelength (typically, 5500 [[Ångström]]s) to observe diffraction from crystals. However, prior to the first X-ray diffraction experiments, the spacings between unit cells in a crystal were not known with certainty. The idea that crystals could be used as a [[diffraction grating]] for [[X-ray]]s arose in 1912 in a conversation between [[Paul Peter Ewald]] and [[Max von Laue]] in the [[Englischer Garten (Munich)|English Garden]] in [[Munich]]. Ewald had proposed a resonator model of crystals for his thesis, but this model could not be validated using [[visible light]], since the [[wavelength]] was much larger than the spacing between the resonators. Von Laue realized that electromagetic 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 [[sphalerite]] crystal and record its diffraction on a [[photographic plate]]. 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.<ref name="Laue1912" >{{cite journal | author = Friedrich W, Knipping P, von Laue M | year = 1912 | title = Interferenz-Erscheinungen bei Röntgenstrahlen | journal = Sitzungsberichte der Mathematisch-Physikalischen Classe der Königlich-Bayerischen Akademie der Wissenschaften zu München | volume = 1912 | pages = 303&ndash;322}}</ref> 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]] in 1914.<ref>Dana ES, Ford WE (1932) ''A Textbook of Mineralogy'' fourth edition New York: John Wiley & Sons p 28</ref> 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 an atomic electron, the scattering may be modeled as [[Thomson scattering]], 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 declines as 1/''m''² with the [[mass]] ''m'' of the charged particle that is scattering the radiation; hence, the atomic nuclei, which are thousands of times heavier than an electron, contribute negligibly to the scattered X-rays. ===Development from 1912 to 1920=== [[Image:Diamond and graphite.jpg|thumb|right|Although [[diamond]]s (top left) and [[graphite]] (top right) are identical in chemical composition — being both pure [[carbon]] — X-ray crystallography revealed the arrangement of their atoms (bottom), which accounts for their different properties. In diamond, the carbon atoms are arranged [[diamond cubic|tetrahedrally]] and held together by single [[covalent bond]]s, making it strong in all directions. By contrast, graphite is composed of stacked sheets, in which the carbon atoms are bonded hexagonally by [[delocalized electron|delocalized]] single and double bonds; there are no covalent bonds between sheets, making graphite easy to flake.]] After von Laue's pioneering research, the field developed rapidly, most notably by physicists [[William Lawrence Bragg]] and his father [[William Henry Bragg]]. In 1912-1913, the younger Bragg developed [[Bragg's law]], which connects the observed scattering with reflections from evenly spaced planes within the crystal.<ref>{{cite journal | author = Bragg WL | authorlink = William Lawrence Bragg | date = 1912 | title = The Specular Reflexion of X-rays | journal = Nature | volume = 90 | pages = 410 | doi = 10.1038/090410b0 <!--Retrieved from CrossRef by DOI bot-->}}<br />{{cite journal | author = Bragg WL | authorlink = William Lawrence Bragg | date = 1913 | title = The Diffraction of Short Electromanetic Waves by a Crystal | journal = Proceedings of the Cambridge Philosophical Society | volume = 17 | pages = 43&ndash;57}}<br />{{cite journal | author = Bragg WL | authorlink = William Lawrence Bragg | date = 1914 | title = Die Reflexion der Röntgenstrahlen | journal = Jahrbuch der Radioaktivität und Elektronik | volume = 11 | pages = 350}}</ref> 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 generally simple inorganic crystals and minerals. The first atomic-resolution structure to be solved (in 1914) was that of [[sodium chloride|table salt]],<ref>{{cite journal | last= Bragg | first = WL | authorlink = William Lawrence Bragg | year = 1914 | title = The Structure of Some Crystals as Indicated by their Diffraction of X-rays | journal = Proceedings of the Royal Society (London) | volume = A89 | pages = 248&ndash;277}}<br />{{cite journal | author = [[William Lawrence Bragg|Bragg WL]], James RW, Bosanquet CH | year = 1921 | title = The Intensity of Reflexion of X-rays by Rock-Salt | journal = Philosophical Magazine | volume = 41 | pages = 309&ndash;337}}<br />{{cite journal | author = [[William Lawrence Bragg|Bragg WL]], James RW, Bosanquet CH | year = 1921 | title = The Intensity of Reflexion of X-rays by Rock-Salt. Part II | journal = Philosophical Magazine | volume = 42 | pages = 1&ndash;17}}<br />{{cite journal | author = [[William Lawrence Bragg|Bragg WL]], James RW, Bosanquet CH | year = 1922 | title = The Distribution of Electrons around the Nucleus in the Sodium and Chlorine Atoms | journal = Philosophical Magazine | volume = 44 | pages = 433&ndash;449}}</ref> which proved the existence of [[ionic compound]]s and that crystals are not necessarily comprised of [[molecule]]s. The structure of [[diamond]] was solved in the same year,<ref name="diamond" >{{cite journal | author = Bragg WH, Bragg WL | year = 1913 | title = The structure of the diamond | journal = Nature | volume = 91 | pages = 557 | doi = 10.1038/091557a0 <!--Retrieved from CrossRef by DOI bot-->}}<br />{{cite journal | author = Bragg WH, Bragg WL | year = 1913 | title = The structure of the diamond | journal = Proceedings of the Royal Society (London) | volume = A89 | pages = 277&ndash;291 | doi = 10.1098/rspa.1913.0084}}</ref> proving the tetrahedral arrangement of its chemical bonds and showing that the C-C single bond was 1.52 [[Ångström]]s. Other early structures included [[copper]],<ref>{{cite journal | author = Bragg WL | authorlink = William Lawrence Bragg | date = 1914 | title = The Crystalline Structure of Copper | journal = Philosophical Magazine | volume = 28 | pages = 355&ndash;360}}</ref> [[calcium fluoride]] (CaF<sub>2</sub>, also known as ''fluorite''), [[calcite]] (CaCO<sub>3</sub>) and [[pyrite]] (FeS<sub>2</sub>)<ref name="carbonate" >{{cite journal | author = Bragg WL | authorlink = William Lawrence Bragg | date = 1914 | title = The analysis of crystals by the X-ray spectrometer | journal = Proceedings of the Royal Society (London) | volume = A89 | pages = 468&ndash;489}}</ref> in 1914; [[spinel]] (MgAl<sub>2</sub>O<sub>4</sub>) in 1915;<ref>{{cite journal | author = Bragg WH | authorlink = William Henry Bragg | date = 1915 | title = The structure of the spinel group of crystals | journal = Philosophical Magazine | volume = 30 | pages = 305&ndash;315}}<br />{{cite journal | author = Nishikawa S | date = 1915 | title = Structure of some crystals of spinel group | journal = Proc. Tokyo Math. Phys. Soc. | volume = 8 | pages = 199&ndash;209}}</ref> the [[rutile]] and [[anatase]] forms of [[titanium dioxide]] (TiO<sub>2</sub>) in 1916;<ref>{{cite journal | author = Vegard L | date = 1916 | title = Results of Crystal Analysis | journal = Philosophical Magazine | volume = 32 | pages = 65&ndash;96}}</ref> [[pyrochroite]] and, by extension, [[brucite]] [Mn(OH)<sub>2</sub> and Mg(OH)<sub>2</sub>, respectively] in 1919;<ref>{{cite journal | author = Aminoff G | authorlink = Gregori Aminoff | date = 1919 | title = Crystal Structure of Pyrochroite | journal = Stockholm Geol. Fören. Förh. | volume = 41 | pages = 407&ndash;433}}<br />{{cite journal | author = Aminoff G | authorlink = Gregori Aminoff | date = 1921 | title = Über die Struktur des Magnesiumhydroxids | journal = Z. Kristallogr. | volume = 56 | pages = 505&ndash;509}}</ref> and [[wurtzite]] (hexagonal ZnS) in 1920.<ref>{{cite journal | author = Bragg WL | authorlink = William Lawrence Bragg | date = 1920 | title = The crystalline structure of zinc oxide | journal = Philosophical Magazine | volume = 39 | pages = 647&ndash;651}}</ref> The structure of [[graphite]] was solved in 1916<ref>{{cite journal | author = [[Peter Debye|Debije P]], [[Paul Scherrer|Scherrer P]] | date = 1916 | title = Interferenz an regellos orientierten Teilchen im Röntgenlicht I | journal = Physikalische Zeitschrift | volume = 17 | pages = 277&ndash;283}}</ref> by the related method of [[powder diffraction]],<ref>{{cite journal | last = Friedrich | first = W | date = 1913 | title = Eine neue Interferenzerscheinung bei Röntgenstrahlen | journal = Physikalische Zeitschrift | volume = 14 | pages = 317&ndash;319}}</ref> which was developed by [[Peter Debye]] and [[Paul Scherrer]] and, independently, by [[Albert Hull]] in 1917.<ref>{{cite journal | last = Hull | first = AW | authorlink = Albert Hull | date = 1917 | title = A New Method of X-ray Crystal Analysis | journal = Physical Review | volume = 10 | pages = 661&ndash;696 | doi = 10.1103/PhysRev.10.661}}</ref> The structure of graphite was determined from single-crystal diffraction in 1924 by two groups independently.<ref>{{cite journal | author = Bernal JD | authorlink = John Desmond Bernal | date = 1924 | title = The Structure of Graphite | journal = Proceedings of the Royal Society (London) | volume = A106 | pages = 749&ndash;773}}</ref><ref>{{cite journal | author = Hassel O, Mack H | date = 1924 | title = Über die Kristallstruktur des Graphits | journal = Zeitschrift für Physik | volume = 25 | pages = 317&ndash;337 | doi = 10.1007/BF01327534 <!--Retrieved from CrossRef by DOI bot-->}}</ref> Hull also used the powder method to determine the structures of various metals, such as iron<ref>{{cite journal | last = Hull | first = AW | authorlink = Albert Hull | date = 1917 | title = The Crystal Structure of Iron | journal = Physical Review | volume = 9 | pages = 84&ndash;87}}</ref> and magnesium.<ref>{{cite journal | last = Hull | first = AW | authorlink = Albert Hull | date = 1917 | title = The Crystal Structure of Magnesium | journal = Proceedings of the National Academy of Science USA | volume = 3 | pages = 470&ndash;473 | doi = 10.1073/pnas.3.7.470}}</ref> ==Contributions to chemistry and material science== X-ray crystallography has led to a better understanding of [[chemical bond]]s and non-covalent interactions. The initial studies revealed the typical radii of [[atom]]s, and confirmed many theoretical models of chemical bonding, such as the tetrahedral bonding of carbon in the diamond structure,<ref name="diamond" /> the octahedral bonding of metals observed in ammonium hexachloroplatinate (IV),<ref>{{cite journal | author = Wyckoff RWG, Posnjak E | date = 1921 | title = The Crystal Structure of Ammonium Chloroplatinate | journal = Journal of the American Chemical Society | volume = 43 | pages = 2292&ndash;2309 | doi = 10.1021/ja01444a002 <!--Retrieved from CrossRef by DOI bot-->}}</ref> and the resonance observed in the planar carbonate group<ref name="carbonate" /> and in aromatic molecules.<ref name="bragg_anthracene" /><ref name="lonsdale_1928" /> [[Kathleen Lonsdale]]'s 1928 structure of [[hexamethylbenzene]]<ref name="lonsdale_1928" >{{cite journal | last = Lonsdale | first = K | authorlink = Kathleen Lonsdale | year = 1928 | title = The structure of the benzene ring | journal = Nature | volume = 122 | pages = 810 | doi = 10.1038/122810c0}}</ref> established the hexagonal symmetry of [[benzene]] 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 (chemistry)|resonance]] between chemical bonds, which had profound consequences for the development of [[chemistry]].<ref>{{cite book | last = Pauling | first = L | authorlink = Linus Pauling | title = The Nature of the Chemical Bond | edition = 3rd edition | publisher = Cornell University Press | location = Ithaca, NY}}</ref> Her conclusions were anticipated by [[William Henry Bragg]], who published models of naphthalene and anthracene in 1921 based on other molecules, an early form of [[molecular replacement]].<ref name="bragg_anthracene" >{{cite journal | author = Bragg WH | authorlink = William Henry Bragg | date = 1921 | title = The structure of organic crystals | journal = Proceedings of the Royal Society (London) | volume = 34 | pages = 33&ndash;50}}<br />{{cite journal | author = Bragg WH | authorlink = William Henry Bragg | date = 1922 | title = The crystalline structure of anthracene | journal = Proceedings of the Royal Society (London) | volume = 35 | pages = 167&ndash;169}}</ref> Also in the 1920s, [[Victor Moritz Goldschmidt]] and later [[Linus Pauling]] 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]], [[brookite]] and [[anatase]] forms of [[titanium oxide]]. The distance between two covalently 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,<ref>{{cite journal | last = Brosset | first = Cyrill | year = 1935 | title = Unknown title | journal = Arkiv för Kemi, Mineralogi och Geologi | volume = 12A | pages = No. 4}}<br />{{cite journal | author = Powell HM, Ewens RVG | year = 1939 | title = The crystal structure of iron enneacarbonyl | journal = J. Chem. Soc. | pages = 286&ndash;292 | doi = 10.1039/jr9390000286 <!--Retrieved from CrossRef by DOI bot-->}}<br />{{cite journal | author = Bertrand JA, Cotton FA, Dollase WA | year = 1963 | title = The Metal-Metal Bonded, Polynuclear Complex Anion in CsReCl<sub>4</sub> | journal = Journal of the American Chemical Society | volume = 85 | pages = 1349&ndash;1350 | doi = 10.1021/ja00892a029}}<br />{{cite journal | author = Robinson WT, Fergusson JE, Penfold BR | year = 1963 | title = Configuration of Anion in CsReCl<sub>4</sub> | journal = Proceedings of the Chemical Society of London | pages = 116&ndash;?}}</ref> metal-metal quadruple bonds,<ref>{{cite journal | author = Cotton FA, Curtis NF, Harris CB, Johnson BFG, Lippard SJ, Mague JT, Robinson WR, Wood JS | year = 1964 | title = Mononuclear and Polynuclear Chemistry of Rhenium (III): Its Pronounced Homophilicity | journal = Science | volume = 145 | pages = 1305&ndash;1307 | doi = 10.1126/science.145.3638.1305 <!--Retrieved from CrossRef by DOI bot-->}}<br />{{cite journal | author = Cotton FA, Harris CB | year = 1965 | title = The Crystal and Molecular Structure of Dipotassium Octachlorodirhenate(III) Dihydrate, K<sub>2[Re<sub>2</sub>Cl<sub>8</sub>]2H<sub>2</sub>O | journal = Inorganic Chemistry | volume = 4 | pages = 330&ndash;333 | doi = 10.1021/ic50025a015}}<br />{{cite journal | last = Cotton | first = FA | title = Metal-Metal Bonding in [Re<sub>2</sub>X<sub>8</sub>]<sup>2-</sup> Ions and Other Metal Atom Clusters | journal = Inorganic Chemistry | volume = 4 | pages = 334&ndash;336 | doi = 10.1021/ic50025a016 | year = 1965}}</ref> and three-center, two-electron bonds.<ref>{{cite journal | author = Eberhardt WH, Crawford W, Jr., Lipscomb WN | year = 1954 | title = The valence structure of the boron hydrides | journal = Journal of Chemical Physics | volume = 22 | pages = 989&ndash;1001 | doi = 10.1063/1.1740320}}</ref> X-ray crystallography — or, strictly speaking, an inelastic Compton scattering experiment — has also provided evidence for the partially covalent character of [[hydrogen bond]]s.<ref>{{cite journal | author = Martin TW, Derewenda ZS | date = 1999 | title = The name is Bond — H bond | journal = Nature Structural Biology | volume = 6 | pages = 403&ndash;406 | doi = 10.1038/8195 <!--Retrieved from CrossRef by DOI bot-->}}</ref> In the field of [[organometallic chemistry]], the X-ray structure of [[ferrocene]] initiated scientific studies of [[sandwich compounds]],<ref>{{cite journal | author = Dunitz JD, Orgel LE, Rich A | year = 1956 | title = The crystal structure of ferrocene | journal = Acta Crystallographica | volume = 9 | pages = 373&ndash;375 | doi = 10.1107/S0365110X56001091}}<br />{{cite journal | author = Seiler P, Dunitz JD | year = 1979 | title = A new interpretation of the disordered crystal structure of ferrocene | journal = Acta Crystallographica | volume = B35 | pages = 1068&ndash;1074}}</ref> while that of [[Zeise's salt]] stimulated research into "back bonding" and metal-pi complexes in general.<ref>{{cite journal | author = Wunderlich JA, Mellor DP | year = 1954 | title = A note on the crystal structure of Zeise's salt | journal = Acta Crystallographica | volume = 7 | pages = 130 | doi = 10.1107/S0365110X5400028X <!--Retrieved from CrossRef by DOI bot-->}}<br />{{cite journal | author = Jarvis JAJ, Kilbourn BT, Owston PG | year = 1970 | title = A re-determination of the crystal and molecular structure of Zeise's salt, KPtCl<sub>3</sub>.C<sub>2</sub>H<sub>4</sub>.H<sub>2</sub>O. A correction | journal = Acta Crystallographica | volume = B26 | pages = 876}}<br />{{cite journal | author = Jarvis JAJ, Kilbourn BT, Owston PG | year = 1971 | title = A re-determination of the crystal and molecular structure of Zeise's salt, KPtCl<sub>3</sub>.C<sub>2</sub>H<sub>4</sub>.H<sub>2</sub>O | journal = Acta Crystallographica | volume = B27 | pages = 366&ndash;372}}<br />{{cite journal | author = Love RA, Koetzle TF, Williams GJB, Andrews LC, Bau R | year = 1975 | title = Neutron diffraction study of the structure of Zeise's salt, KPtCl<sub>3</sub>(C<sub>2</sub>H<sub>4</sub>).H<sub>2</sub>O | journal = Inorganic Chemistry | volume = 14 | pages = 2653&ndash;2657 | doi = 10.1021/ic50153a012}}</ref> Finally, X-ray crystallography had a pioneering role in the development of [[supramolecular chemistry]], particularly in clarifying the structures of the [[crown ether]]s and the principles of [[host-guest chemistry]]. In material sciences, many complicated [[inorganic]] and [[organometallic]] systems have been analyzed using single-crystal methods, such as [[fullerene]]s, [[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 is crystal's size and regularity; [[recrystallization]] is a commonly used technique to improve these factors in small-molecule crystals. The [[Cambridge Structural Database]] contains over 400,000 structures; over 99% of these structures were determined by X-ray diffraction. ===Mineralogy and metallurgy=== [[Image:Zeolite-ZSM-5-3D-vdW.png|thumb|right|X-ray crystal structure of a [[zeolite]], an [[aluminosilicate]] with many important applications, e.g., [[water purification]].]] In [[mineralogy]], a systematic X-ray crystallographic study of the [[silicate]]s was undertaken in the 1920s, beginning with the structure of garnet in 1924 by Menzer. As the [[silicon|Si]]/[[oxygen|O]] ratio is altered, the resulting crystals exhibited significant changes in their internal arrangements. Machatschki extended these insights to minerals in which [[aluminium]] substitutes for [[silicon]] in the silicates. In [[metallurgy]], the first alloy crystal structures began to be determined in the mid-1920s.<ref>{{cite journal | author = Westgren A, Phragmén G| date = 1925 | title = X-ray Analysis of the Cu-Zn, Ag-Zn and Au-Zn Alloys| journal = Philosophical Magazine | volume = 50 | pages = 311&ndash;341}}<br />{{cite journal | author = Bradley AJ, Thewlis J | date = 1926 | title = The structure of γ-Brass | journal = Proceedings of the Royal Society (London) | volume = 112 | pages = 678&ndash;692 | doi = 10.1098/rspa.1926.0134}}<br />{{cite journal | author = Hume-Rothery W | date = 1926 | title = Researches on the Nature, Properties and Conditions of Formation of Intermetallic Compounds (with special Reference to certain Compounds of Tin) | journal = Journal of the Institute of Metals | volume = 35 | pages = 295&ndash;361}}<br />{{cite journal | author = Bradley AJ, Gregory CH | date = 1927 | title = The Structure of certain Ternary Alloys | journal = Nature | volume = 120 | pages = 678}}<br />{{cite journal | author = Westgren A | date = 1932 | title = Zur Chemie der Legierungen | journal = Angewandte Chemie | volume = 45 | pages = 33&ndash;40 | doi = 10.1002/ange.19320450202 <!--Retrieved from CrossRef by DOI bot-->}}<br />{{cite journal | author = Bernal JD | authorlink = John Desmond Bernal | date = 1935 | title = The Electron Theory of Metals | journal = Annual Reports on the Progress of Chemistry | volume = 32 | pages = 181&ndash;184}}</ref> [[Linus Pauling|Linus Pauling's]] structure of the alloy Mg<sub>2</sub>Sn<ref>{{cite journal | last = Pauling | first = L | authorlink = Linus Pauling | title = The Crystal Structure of Magnesium Stannide | journal = Journal of the American Chemical Society | volume = 45 | pages = 2777&ndash;2780 | doi = 10.1021/ja01665a001 | year = 1923}}</ref> led to his theory governing the stability and structure of complex ionic crystals.<ref>{{cite journal | last = Pauling | first = L | authorlink = Linus Pauling | title = The Principles Determining the Structure of Complex IOnic Crystals | journal = Journal of the American Chemical Society | volume = 51 | pages = 1010&ndash;1026 | doi = 10.1021/ja01379a006 | year = 1929}}</ref> === Early organic and small biological molecules=== [[Image:penicillin.png|thumb|right|The three-dimensional structure of [[penicillin]], for which [[Dorothy Crowfoot Hodgkin]] was awarded the [[Nobel Prize in Chemistry]] in 1964. The green, white, red and blue spheres represent atoms of [[carbon]], [[hydrogen]], [[oxygen]] and [[nitrogen]], respectively.]] The first structure of an organic compound, [[hexamethylenetetramine]], was solved in 1923.<ref>{{cite journal | author = Dickinson RG, Raymond AL | year = 1923 | title = The Crystal Structure of Hexamethylene-Tetramine | journal = Journal of the American Chemical Society | volume = 45 | pages = 22&ndash;29 | doi = 10.1021/ja01654a003}}</ref> This was followed by several studies of long-chain [[fatty acid]]s, which are an important component of biological membranes.<ref>{{cite journal | author = Müller A | date = 1923 | title = The X-ray Investigation of Fatty Acids | journal = Journal of the Chemical Society (London) | volume = 123 | pages = 2043&ndash;2047}}<br />{{cite journal | author = Saville WB, Shearer G | date = 1925 | title = An X-ray Investigation of Saturated Aliphatic Ketones | journal = Journal of the Chemical Society (London) | volume = 127 | pages = 591&ndash;598}}<br />{{cite journal | author = Bragg WH| authorlink = William Henry Bragg | date = 1925 | title = The Investigation of thin Films by Means of X-rays | journal = Nature | volume = 115 | pages = 266&ndash;269 | doi = 10.1038/115266a0 <!--Retrieved from CrossRef by DOI bot-->}}<br />{{cite journal | author = [[Maurice de Broglie|de Broglie M]], Trillat JJ | date = 1925 | title = Sur l'interprétation physique des spectres X d'acides gras | journal = Comptes rendus hebdomadaires des séances de l'Académie des sciences | volume = 180 | pages = 1485&ndash;1487}}<br />{{cite journal | author = Trillat JJ | date = 1926 | title = Rayons X et Composeés organiques à longe chaine. Recherches spectrographiques sue leurs structures et leurs orientations | journal = Annales de physique | volume = 6 | pages = 5&ndash;101}}<br />{{cite journal | author = Caspari WA | date = 1928 | title = Crystallography of the Aliphatic Dicarboxylic Acids | journal = Journal of the Chemical Society (London) | volume = ? | pages = 3235&ndash;3241}}<br />{{cite journal | author = Müller A | date = 1928 | title = X-ray Investigation of Long Chain Compounds (n. Hydrocarbons) | journal = Proceedings of the Royal Society (London) | volume = 120 | pages = 437&ndash;459 | doi = 10.1098/rspa.1928.0158}}<br />{{cite journal | author = Piper SH | date = 1929 | title = Some Examples of Information Obtainable from the long Spacings of Fatty Acids | journal = Transactions of the Faraday Society | volume = 25 | pages = 348&ndash;351 | doi = 10.1039/tf9292500348 <!--Retrieved from CrossRef by DOI bot-->}}<br />{{cite journal | author = Müller A | date = 1929 | title = The Connection between the Zig-Zag Structure of the Hydrocarbon Chain and the Alternation in the Properties of Odd and Even Numbered Chain Compounds | journal = Proceedings of the Royal Society (London) | volume = 124 | pages = 317&ndash;321 | doi = 10.1098/rspa.1929.0117}}</ref> 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]],<ref>{{cite journal | last = Robertson | first = JM | year = 1936 | title = An X-ray Study of the Phthalocyanines, Part II | journal = Journal of the Chemical Society | pages = 1195&ndash;1209}}</ref> a large planar molecule that has an approximate four-fold symmetry and resembles [[porphyrin]]s found in nature, 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), [[vitamin B12]] (1945) and [[penicillin]] (1954), for which she was awarded the [[Nobel Prize in Chemistry]] in 1964. In 1969, she succeeded in solving the structure of [[insulin]], on which she worked for over thirty years.<ref>{{cite journal | author = Crowfoot Hodgkin D | authorlink = Dorothy Crowfoot Hodgkin | date = 1935 | title = X-ray Single Crystal Photographs of Insulin | journal = Nature | volume = 135 | pages = 591&ndash;592 | doi = 10.1038/135591a0 <!--Retrieved from CrossRef by DOI bot-->}}</ref> [[Image:Myoglobin.png|thumb|left|[[Ribbon diagram]] of the structure of [[myoglobin]], showing colored [[alpha helix|alpha helices]]. Such [[protein]]s are long, linear [[molecule]]s with thousands of atoms; yet the relative position of each atom has been determined with sub-atomic resolution by X-ray crystallography. Since it is difficult to visualize all the atoms at once, the ribbon shows the rough path of the protein [[polymer]] from its N-terminus (blue) to its C-terminus (red).]] ===Protein crystallography=== Crystal structures of [[protein]]s (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 [[Max Perutz]] and [[John Kendrew|Sir John Cowdery Kendrew]], for which they were awarded the [[Nobel Prize in Chemistry]] in 1962.<ref>{{Cite journal | doi = 10.1038/181662a0 | volume = 181 | issue = 4610 | pages = 662–666 | last = Kendrew | first = J. C. | authorlink = John Kendrew | coauthors = G. Bodo, H. M. Dintzis, R. G. Parrish, H. Wyckoff, D. C. Phillips | title = A Three-Dimensional Model of the Myoglobin Molecule Obtained by X-Ray Analysis | journal = [[Nature (journal)|Nature]]| date = 1958-03-08 }}</ref> Since that success, over 39000 X-ray crystal structures of proteins, nucleic acids and other biological molecules have been determined.<ref>[http://www.rcsb.org/pdb/statistics/holdings.do Table of entries in the PDB, arranged by experimental method.]</ref> For comparison, the nearest competing method, [[Protein nuclear magnetic resonance spectroscopy|NMR spectroscopy]] has produced roughly 6000 structures.<ref>{{cite web | url = http://pdbbeta.rcsb.org/pdb/static.do?p=general_information/pdb_statistics/index.html | title =PDB Statistics | publisher =RCSB Protein Data Bank | accessdate = 2007-05-03}}</ref> Moreover, crystallography can solve structures of arbitrarily large molecules, whereas solution-state NMR is restricted to relatively small molecules (less than 70 k[[atomic mass unit|Da]]). X-ray crystallography is now used routinely by scientists to determine how a pharmaceutical interacts with its protein target and what changes might be advisable to improve it.<ref>{{cite journal |author=Scapin G |title=Structural biology and drug discovery |journal=Curr. Pharm. Des. |volume=12 |issue=17 |pages=2087–97 |year=2006 |pmid=16796557 | doi = 10.2174/138161206777585201 <!--Retrieved from CrossRef by DOI bot-->}}</ref> However, intrinsic membrane proteins remain challenging to crystallize because they require detergents or other means to solubilize them in isolation, and such detergents often interfere with crystallization. Such membrane proteins are a large component of the genome and include many proteins of great physiological importance, such as [[ion channel]]s and [[receptor (biochemistry)|receptor]]s.<ref>{{cite journal |author=Lundstrom K |title=Structural genomics for membrane proteins |journal=Cell. Mol. Life Sci. |volume=63 |issue=22 |pages=2597–607 |year=2006 |pmid=17013556 | doi = 10.1007/s00018-006-6252-y <!--Retrieved from CrossRef by DOI bot-->}}</ref><ref>{{cite journal |author=Lundstrom K |title=Structural genomics on membrane proteins: mini review |journal=Comb. Chem. High Throughput Screen. |volume=7 |issue=5 |pages=431–9 |year=2004 |pmid=15320710}}</ref> ==Relationship to other scattering techniques== {{further |[[X-ray scattering techniques]]}} ===Elastic vs. inelastic scattering=== X-ray crystallography is a form of [[elastic scattering]]; the outgoing X-rays have the same energy as the incoming X-rays, only with altered direction. Since the energy of a [[photon]] is inversely proportional to its [[wavelength]], elastic scattering means that the outgoing photons have the same wavelength as the incoming photons. 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 changes the wavelength of the outgoing beam, making it longer and less energetic. Inelastic scattering is useful for probing such excitations of matter, but are not as useful in determining the distribution of scatterers within the matter, which is the goal of X-ray crystallography. [[X-ray]]s range in wavelength from 10 to 0.01 nanometers (one billionth of a meter); a typical wavelength used for crystallography is roughly 1&nbsp;[[Ångström|Å]] (0.1&nbsp;[[nanometre|nm]]), which is on the scale of covalent [[chemical bond]]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 ray]]s 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]]. ===Other types of X-ray scattering=== X-ray diffraction involves the scattering of X-rays from a single crystal. Other forms of elastic X-ray scattering include [[powder diffraction]], [[SAXS]] and several types of X-ray [[fiber diffraction]], which was used by [[Rosalind Franklin]] in determining the [[double helix|double-helix structure]] of [[DNA]]. In general, X-ray diffraction produces isolated spots ("reflections"), while the other methods produce smooth, continuous scattering. In general, X-ray diffraction offers more structural information than these other techniques; however, it requires a sufficiently large and regular crystal, which is not always possible to obtain. All of these scattering methods generally use ''monochromatic'' X-rays, X-rays that 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 X-ray crystallography). However, it is not as well-suited as monochromatic scattering for determining the full atomic structure of a crystal. It is better suited to crystals with relatively simple atomic arrangements, such as minerals. The Laue back reflection mode records X-rays scattered backwards also from a broad spectrum source. This is useful if the sample is too thick or bulky 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 <ref>AB Greninger Zeitschrift f. Kristallographie 91 (1935),pp. 424-432</ref> to interpret the back reflection Laue photograph. The X-calibre RTXDB and MWL 110 are commercial systems for Laue back reflection pattern recording. This technique can be used in materials analysis or non destructive inspection. ===Electron and neutron diffraction=== Other particles, such as [[electron]]s and [[neutron]]s, may be used to produce a [[diffraction pattern]]. Although electron, neutron, and X-ray scattering use very different equipment, 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]], 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 calculation of the density of scatterers. 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 very small crystals (e.g., 100&nbsp;[[micrometre|μm]]) used in X-ray crystallography. In such cases, extremely thin samples, roughly 100 [[nanometer]]s or less, must be used to avoid secondary scattering; the primary scattered electron beams leave the sample before they have a chance to undergo secondary scattering. Since this thickness corresponds roughly to the diameter of many [[virus]]es, a promising direction is the electron diffraction of isolated macromolecular assemblies, such as [[virus|viral]] [[capsid]]s and molecular machines, which may be carried out with a cryo-[[electron microscope]]. 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 reactor]]s have been used, although the new [[Spallation Neutron Source]] holds much promise in the near future. 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 of larger molecules. Neutron scattering also has the remarkable property that the solvent can be made invisible by adjusting the ratio of normal [[water]], H<sub>2</sub>O, and [[heavy water]], D<sub>2</sub>O. ===Advantages of a crystal=== A crystalline sample is by definition periodic; a crystal is composed of many [[unit cell]]s repeated indefinitely in three independent directions. Such periodic systems have a Fourier transform 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 [[Interference#Constructive and destructive interference|constructive interference]]. In a non-crystalline sample, molecules within that sample would be in random orientations and therefore would have a continuous Fourier spectrum that spreads its amplitude more uniformly and with a much reduced intensity, as is observed in [[Biological Small-Angle X-ray Scattering|SAXS]]. More importantly, the orientational information is lost. In the crystal, the molecules adopt the same orientation within the crystal, whereas in a liquid, powder or amorphous state, the observed signal is averaged over the possible orientations of the molecules. Although theoretically possible with sufficiently low-noise data, it is generally difficult to obtain atomic-resolution structures of complicated, asymmetric molecules from such rotationally averaged scattering data. An intermediate case is [[fiber diffraction]] in which the subunits are arranged periodically in at least one dimension. ==Methods== ===Crystallization=== [[Image:Protein crystal.jpg|thumb|right|A protein crystal seen under a [[microscope]]. Crystals used in X-ray crystallography are generally smaller than a millimeter across.]] {{further|[[crystallization]] and [[recrystallisation]]}} 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 for the structure of a complicated arrangement of atoms. Pure, regular crystals can sometimes be obtained from Nature or man-made materials, such as samples of [[metal]]s, [[mineral]]s or other macroscopic materials. The regularity of such crystals can sometimes be improved with [[annealing]] and other methods. However, in many cases, obtaining a diffraction-quality crystal is the chief barrier to solving its atomic-resolution structure.<ref name=Geerlof>{{cite journal |author=Geerlof A, Brown J, Coutard B, Egloff MP, Enguita FJ, Fogg MJ, Gilbert RJ, Groves MR, Haouz A, Nettleship JE, Nordlund P, Owens RJ, Ruff M, Sainsbury S, Svergun DI, Wilmanns M |title=The impact of protein characterization in structural proteomics |journal=Acta Crystallogr. D Biol. Crystallogr. |volume=62 |issue=Pt 10 |pages=1125–36 |year=2006 |pmid=17001090 | doi = 10.1107/S0907444906030307 <!--Retrieved from CrossRef by DOI bot-->}}</ref> 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#Single perfect crystals .28for X-ray analysis.29|recrystallisation]]. By contrast, macromolecules generally have many degrees of freedom and their crystallization must be carried out to maintain a stable structure. For example, [[protein]]s 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; however, 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.<ref>{{cite journal |author=Chernov AA |title=Protein crystals and their growth |journal=J. Struct. Biol. |volume=142 |issue=1 |pages=3–21 |year=2003 |pmid=12718915 | doi = 10.1016/S1047-8477(03)00034-0 <!--Retrieved from CrossRef by DOI bot-->}}</ref> The solution conditions that favor the first step (nucleation) are not always the same conditions that favor the second step (its 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. In some cases, the crystallographer can identify good solution conditions for growing very small crystals that do not continue to grow and which are too small for crystallography. In such cases, the tiny crystals can be transferred to new solution conditions that favor growth more strongly; the small crystals act as pre-nucleated seeds for subsequent growth. In an alternative approach, a larger but poor-quality crystal may be crushed, and the pieces used as seed crystals to obtain higher quality crystals. It is extremely difficult to predict good conditions for nucleation or growth of well-ordered crystals.<ref>{{cite journal |author=Rupp B, Wang J |title=Predictive models for protein crystallization |journal=Methods |volume=34 |issue=3 |pages=390–407 |year=2004 |pmid=15325656 | doi = 10.1016/j.ymeth.2004.03.031 <!--Retrieved from CrossRef by DOI bot-->}}</ref> 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.<ref>{{cite journal |author=Chayen NE |title=Methods for separating nucleation and growth in protein crystallization |journal=Prog. Biophys. Mol. Biol. |volume=88 |issue=3 |pages=329–37 |year=2005 |pmid=15652248 | doi = 10.1016/j.pbiomolbio.2004.07.007 <!--Retrieved from CrossRef by DOI bot-->}}</ref> Hundreds, even thousands, of solution conditions are generally tried before finding one that succeeds in crystallizing the molecules. 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 crystallisation grade protein, dispensing robots have been developed that are capable of accurately dispensing crystallisation trial drops that are of the order on 100 [[nanoliter]]s in volume. This means that roughly 10-fold less protein is used per-experiment when compared to crystallisation trials setup by hand (on the order on 1 [[microliter]]s) <ref>{{cite journal |author=Stock D, Perisic O, Lowe J |title=Robotic nanolitre protein crystallisation at the MRC Laboratory of Molecular Biology. |journal=Prog Biophys Mol Biol |volume=88 |issue=3 |pages=311–27 |year=2005 |pmid=15652247 |doi=10.1016/j.pbiomolbio.2004.07.009}}</ref>. 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. Ironically, 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 have begun to allow the structures of twinned crystals to be solved, it is still very difficult. 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==== [[Image:Kappa goniometer faster smaller.gif|frame|left|Animation showing the motions possible with a four-circle kappa goniometer. Rotation about any of the four angles φ, κ, ω and 2θ leave transparent crystal within the X-ray beam (shown in orange), but changes its orientation, allowing more of reciprocal space to be observed. Finally, the detector (shown in purple with a black screen) can be slid closer or further away from the crystal, allowing higher resolution data to be taken (if closer) or better discernment of the Bragg peaks (if further away).]] Once they are full-grown, the crystals are mounted so that they may be held in the X-ray beam and rotated. There are several methods of mounting. Although crystals were once loaded into glass capillaries with the crystallization solution (the ''mother liquor''), a more modern approach is to scoop the crystal up in a tiny loop, made of nylon or plastic and attached to a solid rod, that is then flash-frozen with [[liquid nitrogen]].<ref>{{cite journal |author=Jeruzalmi D |title=First analysis of macromolecular crystals: biochemistry and x-ray diffraction |journal=Methods Mol. Biol. |volume=364 |issue= |pages=43–62 |year=2006 |pmid=17172760}}</ref> 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 crystals often crack if flash-frozen; therefore, they are generally pre-soaked in a cryoprotectant solution before freezing.<ref>{{cite journal |author=Helliwell JR |title=Protein crystal perfection and its application |journal=Acta Crystallogr. D Biol. Crystallogr. |volume=61 |issue=Pt 6 |pages=793–8 |year=2005 |pmid=15930642 | doi = 10.1107/S0907444905001368 <!--Retrieved from CrossRef by DOI bot-->}}</ref> 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]], 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 roughly 25 micrometres 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 roughly perpendicular to the beam; the κ angle, about an axis at roughly 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. ====X-ray sources==== {{further|[[Diffractometer]] and [[Synchrotron]]}} The mounted crystal is then irradiated with a beam of [[Monochrome|monochromatic]] X-rays. The brightest and most useful X-ray sources are [[synchrotron]]s; their much higher luminosity allows for better resolution. They also make it convenient to tune the wavelength of the radiation, which is useful for [[multi-wavelength anomalous dispersion]] (MAD) phasing, described below. Synchrotrons are generally national facilities, each with several dedicated [[beamline]]s where data is collected around the clock, seven days a week. [[Image:X Ray Diffractometer.JPG|thumb|right|A diffractometer]] Smaller, weaker X-ray sources are often used in laboratories to check the quality of crystals before bringing them to a synchrotron and sometimes to solve a crystal structure. In such systems, electrons are boiled off of a cathode and accelerated through a strong electric potential of roughly 50&nbsp;[[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]], which can be kept cool easily, due to its high [[thermal conductivity]], and which produces strong K<sub>α</sub> and K<sub>β</sub> lines. The K<sub>β</sub> line is sometimes suppressed with a thin layer (0.0005&nbsp;[[inch|in.]] thick) of nickel foil. The simplest and cheapest variety of [[sealed X-ray tube]] has a stationary anode and produces ''circa'' 2&nbsp;[[Watt#Kilowatt|kW]] of X-ray radiation. The more expensive variety has a [[conventional X-ray generator|rotating-anode type source]] that produces ''circa'' 14&nbsp;[[Watt#Kilowatt|kW]] of X-ray radiation. X-rays are generally filtered to a single wavelength (made monochromatic) and collimated to a single direction before they are allowed to strike the crystal. The filtering not only simplifies the data analysis, but also removes radiation that degrades the crystal without contributing useful information. Collimation is done either with a collimator (basically, a long tube) or with a clever arrangement of gently curved mirrors. Mirror systems are preferred for small crystals (under 0.3&nbsp;mm) or with large unit cells (over 150&nbsp;[[Ångström|Å]]). ====Recording the reflections==== [[Image:X-ray diffraction pattern 3clpro.jpg|thumb|An X-ray diffraction pattern of a crystallized enzyme. The pattern of spots (called ''reflections'') can be used to determine the structure of the enzyme.]] 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 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 angle such as 90° or 45° may be recorded. The axis of the rotation should generally 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 phasing methods. For example, MAD phasing requires that the scattering be recorded at 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.<ref>{{cite journal |author=Ravelli RB, Garman EF |title=Radiation damage in macromolecular cryocrystallography |journal=Curr. Opin. Struct. Biol. |volume=16 |issue=5 |pages=624–9 |year=2006 |pmid=16938450 | doi = 10.1016/j.sbi.2006.08.001 <!--Retrieved from CrossRef by DOI bot-->}}</ref> ===Data analysis=== ====Crystal symmetry, unit cell, and image scaling==== {{further|[[Space group]]}} Having recorded a series of diffraction patterns from the crystal, each corresponding to a different crystal orientation, the crystallographer must now convert these two-dimensional images into a three-dimensional model of the density of electrons throughout the crystal using the mathematical technique of [[Fourier transform]]s. (The explanation for the relevance of this technique is given below.) Roughly speaking, 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''). In a picturesque analogy, the diffraction pattern corresponds to [[sheet music]]; the musician must determine which spot corresponds to which [[musical note]], how loudly to play each note and how the various notes should be played together to produce the final musical piece. In order to process the data, a crystallographer must first ''index'' the reflections within the multiple images recorded. 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, since they require symmetries known to be absent in the molecule itself. For example, symmetries with reflection symmetries cannot be observed in chiral molecules; thus, only 65 space groups of 243 possible are allowed for protein molecules which are almost always chiral. Indexing is generally accomplished using an ''autoindexing'' routine<ref name=Powell>{{cite journal |author=Powell HR |title=The Rossmann Fourier autoindexing algorithm in MOSFLM. |journal=Acta Crystallogr. D Biol. Crystallogr. |volume=55 |issue=Pt 10 |pages=1690–95 |year=1999 |pmid=10531518 | doi = 10.1107/S0907444999009506 <!--Retrieved from CrossRef by DOI bot-->}}</ref>. 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. This is important, since the relative intensities of the peaks is the key information from which the structure is determined. The technique of crystallographic data collection and the often high symmetry of crystalline materials, means that many symmetry-equivalent reflections are recorded multiple times - this allows a merging or symmetry related [[R-factor (crystallography)|R-factor]] to be calculated, based upon how similar the measured intensities of symmetry equivalent reflections are, thus giving a score to assess the quality of the data. ====Initial phasing==== {{further|[[Phase problem]]}} 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''' aka '''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&nbsp;[[Ångström|Å]] (140&nbsp;[[picometre|pm]]), [[Direct methods (crystallography)|direct methods]] can be used to obtain phase information, by exploiting known phase relationships between certain groups of reflections<ref>{{cite journal |author=Hauptman H |title=Phasing methods for protein crystallography |journal=Curr. Opin. Struct. Biol. |volume=7 |issue=5 |pages=672–80 |year=1997 |pmid=9345626 | doi = 10.1016/S0959-440X(97)80077-2 <!--Retrieved from CrossRef by DOI bot-->}}</ref> <ref>{{cite journal |author=Usón I, Sheldrick GM |title=Advances in direct methods for protein crystallography |journal=Curr. Opin. Struct. Biol. |volume=9 |issue=5 |pages=643–8 |year=1999 |pmid=10508770 | doi = 10.1016/S0959-440X(99)00020-2 <!--Retrieved from CrossRef by DOI bot-->}}</ref> {{further|[[direct methods]]}} * '''[[Molecular replacement]]''' - if a structure exists of a related crystal structure, 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''.<ref name=Taylor>{{cite journal |author=Taylor G |title=The phase problem |journal=Acta Crystallogr. D Biol. Crystallogr. |volume=59 |issue=Pt 11 |pages=1881–90 |year=2003 |pmid=14573942 | doi = 10.1107/S0907444903017815 <!--Retrieved from CrossRef by DOI bot-->}}</ref> * '''[[Anomalous X-ray scattering]]''' (''[[Multi-wavelength anomalous dispersion|MAD]] or 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 thence 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 synthesising methionine) in a media rich in Seleno-methionine, which contains [[Selenium]] atoms. A 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.<ref>{{cite journal |author=Ealick SE |title=Advances in multiple wavelength anomalous diffraction crystallography |journal=Current opinion in chemical biology |volume=4 |issue=5 |pages=495–9 |year=2000 |pmid=11006535 | doi = 10.1016/S1367-5931(00)00122-8 <!--Retrieved from CrossRef by DOI bot-->}}</ref> * '''Heavy atom methods''' (ie [[Multiple isomorphous replacement|MIR]]) - 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. Typically, a crystallographer can introduce such heavy atoms 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 MAD 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 MAD phasing with selenomethionine.<ref name=Taylor/> While all four of the above methods are used to solve the phase problem for protein crystallography, small molecule crystallography generally yields data suitable for structure solution using Direct methods/''ab initio'' phasing. ====Model building and phase refinement==== [[Image:eden.png|thumb|A [[protein]] crystal structure at 2.7&nbsp;[[Ångström|Å]] resolution. The mesh encloses the region in which the electron density exceeds a given threshold. The straight segments represent chemical bonds between the non-hydrogen atoms of an [[arginine]] (upper left), a [[tyrosine]] (lower left), a [[disulfide bond]] (upper right, in yellow), and some [[peptide|peptide groups]] (running left-right in the middle). The two curved green tubes represent [[spline]] fits to the polypeptide backbone.]] {{further|[[Molecular modelling|Molecular modeling]]}} Having obtained initial phases, an initial model can be built. This model can be used to refine the phases, leading to an improved model, and so on. Given a model of some atomic positions, these positions and their respective [[Debye-Waller factor]]s (accounting for the thermal motion of the atom - aka '''B'''-factors) 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 a further round of refinement is carried out. This 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 :<math> R = \frac{\sum_{\mathrm{all\ reflections}} \left| F_{o} - F_{c} \right|}{\sum_{\mathrm{all\ reflections}} \left| F_{o} \right|} </math> A similar quality criterion is ''R''<sub>free</sub>, 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''<sub>free</sub> should be approximately the resolution in [[Ångström]]s divided by 10; thus, a data-set with 2&nbsp;[[Ångström|Å]] resolution should yield a final ''R''<sub>free</sub> of roughly 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 of the crystallized molecule - it must be remembered that the resulting electron density is an average of all the molecules within the crystal. In some cases, there is too much residual disorder in those atoms, and the resulting electron density for atoms existing in many conformations is smeared to such an extent that it is no longer detectable in 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 posttranslational modifications that were not detected prior to the crystallization. ===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 [[Protein Data Bank]] (for protein structures) or the [[Cambridge Structural Database]] (for small molecules). Many structures obtained in private commercial ventures to crystallize medicinally relevant proteins, are not deposited in public crystallographic databases. ==Diffraction theory== {{further|[[Dynamical theory of diffraction]] and [[Bragg diffraction]]}} The main goal of X-ray crystallography is to determine the density of electrons ''f('''''r''''')'' throughout the crystal. To do this, X-ray scattering is used to collect data about its [[Fourier transform]] ''F('''''q''''')'', which is then inverted mathematically to obtain the density defined in real space, using the formula :<math> f(\mathbf{r}) = \int \frac{d\mathbf{q}}{\left(2\pi\right)^{3}} F(\mathbf{q}) e^{i\mathbf{q}\cdot\mathbf{r}} </math> The corresponding formula for a Fourier transform is :<math> F(\mathbf{q}) = \int d\mathbf{r} f(\mathbf{r}) e^{-i\mathbf{q}\cdot\mathbf{r}} </math> which will be used below. The vector '''q''' corresponds to 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 2π divided by the wavelength of the oscillation. 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 :<math> F(\mathbf{q}) = \left| F(\mathbf{q}) \right| e^{i\phi(\mathbf{q})} </math> 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. :<math>2 d\sin\theta = n\lambda\,</math> 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]] 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 focus on the wave's spatial dependence. Plane waves can be represented by a [[wave vector]] '''k'''<sub>in</sub>, and so the strength of the incoming wave at time ''t=0'' is given by :<math> A e^{i\mathbf{k}_{in} \cdot \mathbf{r}} </math> 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''' :<math> \mathrm{amplitude\ of\ scattered\ wave} = A e^{i\mathbf{k} \cdot \mathbf{r}} S f(\mathbf{r}) dV </math> where ''S'' is the proportionality constant. Let's consider the fraction of scattered waves that leave with an outgoing wave-vector of '''k'''<sub>out</sub> and strike the screen at '''r'''<sub>screen</sub>. Since no energy is lost (elastic, not inelastic scattering), the wavelengths are the same as are the magnitudes of the wave-vectors |'''k'''<sub>in</sub>| = |'''k'''<sub>out</sub>|. From the time that the photon is scattered at '''r''' until it is absorbed at '''r'''<sub>screen</sub>, the photon undergoes a change in phase :<math> e^{i \mathbf{k}_{out} \cdot \left( \mathbf{r}_{\mathrm{screen}} - \mathbf{r} \right)} </math> The net radiation arriving at '''r'''<sub>screen</sub> is the sum of all the scattered waves throughout the crystal :<math> A S \int d\mathbf{r} f(\mathbf{r}) e^{i \mathbf{k}_{in} \cdot \mathbf{r}} e^{i \mathbf{k}_{out} \cdot \left( \mathbf{r}_{\mathrm{screen}} - \mathbf{r} \right)} = A S e^{i \mathbf{k}_{out} \cdot \mathbf{r}_{\mathrm{screen}}} \int d\mathbf{r} f(\mathbf{r}) e^{i \left( \mathbf{k}_{in} - \mathbf{k}_{out} \right) \cdot \mathbf{r}} </math> which may be written as a Fourier transform :<math> A S e^{i \mathbf{k}_{out} \cdot \mathbf{r}_{\mathrm{screen}}} \int d\mathbf{r} f(\mathbf{r}) e^{-i \mathbf{q} \cdot \mathbf{r}} = A S e^{i \mathbf{k}_{out} \cdot \mathbf{r}_{\mathrm{screen}}} F(\mathbf{q}) </math> where '''q''' = '''k'''<sub>out</sub> - '''k'''<sub>in</sub>. The measured intensity of the reflection will be square of this amplitude :<math> A^{2} S^{2} \left| F(\mathbf{q}) \right|^{2} </math> The electron density ''f('''''r''''')'' is a real function, which imposes a constraint on its Fourier transform. Specifically, the Fourier transform of a negative frequency must have the same magnitude as the corresponding positive frequency, but opposite phase :<math> F(-\mathbf{q}) = \left| F(-\mathbf{q}) \right| e^{i\phi(-\mathbf{q})} = F^{*}(\mathbf{q}) = \left| F(\mathbf{q}) \right| e^{-i\phi(\mathbf{q})} </math> In other words, the Fourier transforms of the negative and positive frequency vectors are complex conjugates of one another; in X-ray crystallography, these corresponding reflections are called Friedel mates. This allows one to measure the full Fourier transform from only half the reciprocal space, e.g., by slightly more than a 180° rotation (see next section). In symmetric crystals, other reflections may have the same intensity (Bijvoet mates); in such cases, one can measure even less of the reciprocal space, e.g., slightly more than 90°. ===Ewald's sphere=== {{further|[[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 '''k'''<sub>out</sub> and '''k'''<sub>in</sub> 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]] (''add Figure to illustrate this''). In practice, the crystal is rocked by a small amount (0.25-1°) to incorporate reflections near the boundaries of the spherical Ewald shells. ===Patterson function=== {{further|[[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''''')'' :<math> c(\mathbf{r}) = \int d\mathbf{x} f(\mathbf{x}) f(\mathbf{x} + \mathbf{r}) = \int \frac{d\mathbf{q}}{\left(2\pi\right)^{3}} C(\mathbf{q}) e^{i\mathbf{q}\cdot\mathbf{r}} </math> has a Fourier transform ''C('''''q''''')''that is the squared magnitude of ''F('''''q''''')'' :<math> C(\mathbf{q}) = \left| F(\mathbf{q}) \right|^{2} </math> Therefore, the autocorrelation function ''c('''''r''''')'' of the electron density (also known as the ''Patterson function''<ref>{{cite journal | author = Patterson AL | authorlink = Arthur Lindo Patterson | date = 1935 | title = A Direct Method for the Determination of the Components of Interatomic Distances in Crystals | journal = Zeitschrift für Kristallographie | volume = 90 | pages = 517&ndash;542}}</ref>) 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 (spatial)|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. ==See also== {{wikibooks|Xray Crystallography}} * [[Structure determination]] * [[Neutron diffraction]] * [[Electron diffraction]] * [[Wide angle X-ray scattering]] (WAXS) * [[Small angle X-ray scattering (SAXS)]] * [[Bravais lattice]] * [[Crystallographic point groups]] * [[Crystallographic database]] ==References== <!-- --------------------------------------------------------------- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes for a discussion of different citation methods and how to generate footnotes using the <ref> & </ref> tags and the {{Reflist}} template -------------------------------------------------------------------- --> {{Reflist|2}} ==Bibliography== === ''International Tables for Crystallography'' === * {{cite book | year = 2002 | title = International Tables for Crystallography. Volume A, Space-group Symmetry | edition = 5th edition, ed. Theo Hahn | publisher = [[Kluwer Academic Publishers]], for the [[International Union of Crystallography]] | location = Dordrecht | id = ISBN 0-7923-6590-9}} * {{cite book | year = 2001 | title = International Tables for Crystallography. Volume F, Crystallography of biological molecules | editor = eds. Michael G. Rossmann and Eddy Arnold | publisher = [[Kluwer Academic Publishers]], for the [[International Union of Crystallography]] | location = Dordrecht | id = ISBN 0-7923-6857-6}} * {{cite book | year = 1996 | title = International Tables for Crystallography. Brief Teaching Edition of Volume A, Space-group Symmetry | edition = 4th revised and enlarged edition, ed. Theo Hahn | publisher = [[Kluwer Academic Publishers]], for the [[International Union of Crystallography]] | location = Dordrecht | id = ISBN 0-7923-4252-6}} ===Bound collections of articles=== * {{cite book | year = 1997 | title = Macromolecular Crystallography, Part A (Methods in Enzymology, v. 276) | edition = edited by CW Carter, Jr. and RM Sweet | publisher = Academic Press | location = San Diego | id = ISBN 0-12-182177-3}} * {{cite book | year = 1997 | title = Macromolecular Crystallography, Part B (Methods in Enzymology, v. 277) | edition = edited by CW Carter, Jr. and RM Sweet | publisher = Academic Press | location = San Diego | id = ISBN 0-12-182178-1}} * {{cite book | year = 1999 | title = Crystallization of Nucleic Acids and Proteins: A Practical Approach | edition = 2nd edition, edited by A. Ducruix and R. Giegé | publisher = [[Oxford University Press]] | location = Oxford | id = ISBN 0-19-963678-8}} ===Textbooks=== * {{cite book | last = Blow | first = D | year = 2002 | title = Outline of Crystallography for Biologists | publisher = [[Oxford University Press]] | location = Oxford | id = ISBN 0-19-851051-9}} * {{cite book | last = Clegg | first = W | year = 1998 | title = Crystal Structure Determination (Oxford Chemistry Primer) | publisher = Oxford University Press | location = Oxford | isbn = 0-19-855-901-1}} * {{cite book | last = Drenth | first = J | year = 1999 | title = Principles of Protein X-Ray Crystallography | publisher = Springer-Verlag | location = New York | id = ISBN 0-387-98587-5}} * {{cite book | last = Giacovazzo | first = C | coauthors = Monaco HL, Viterbo D, Scordari F, Gilli G, Zanotti G, and Catti M | year = 1992 | title = Fundamentals of Crystallography | publisher = [[Oxford University Press]] | location = Oxford | id = ISBN 0-19-855578-4}} * {{cite book | last = Glusker | first = JP | coauthors = Lewis M, Rossi M | year = 1994 | title = Crystal Structure Analysis for Chemists and Biologists | publisher = VCH Publishers | location = New York | id = ISBN 0-471-18543-4}} * {{cite book | last = Massa | first = W | year = 2004 | title = Crystal Structure Determination | publisher = Springer | location = Berlin | id = ISBN 3540206442}} * {{cite book | last = McPherson | first = A | year = 1999 | title = Crystallization of Biological Macromolecules | publisher = Cold Spring Harbor Laboratory Press | location = Cold Spring Harbor, NY | id = ISBN 0-87969-617-6}} * {{cite book | last = McPherson | first = A | year = 2003 | title = Introduction to Macromolecular Crystallography | publisher = John Wiley & Sons | id = ISBN 0-471-25122-4}} * {{cite book | last = McRee | first = DE | year = 1993 | title = Practical Protein Crystallography | publisher = [[Academic Press]] | location = San Diego | id = ISBN 0-12-486050-8}} * {{cite book | last = Rhodes | first = G | year = 2000 | title = Crystallography Made Crystal Clear | publisher = [[Academic Press]] | location = San Diego | id = ISBN 0-12-587072-8}}, [http://www.chem.uwec.edu/Chem406_F06/Pages/lecture_notes/lect07/Crystallography_Rhodes.pdf PDF copy of select chapters] * {{cite book | last = Zachariasen | first = WH | year = 1945 | title = Theory of X-ray Diffraction in Crystals | publisher = Dover Publications | location = New York | id = {{LCCN|67|0|26967}}}} ===Historical=== * {{cite journal | last = Friedrich | first = W | year = 1922 | title = Die Geschichte der Auffindung der Röntgenstrahlinterferenzen | journal = Die Naturwissenschaften | volume = 10 | pages = 363&ndash;366 | doi = 10.1007/BF01565289}} * {{cite book | last = Lonsdale | first = K | authorlink = Kathleen Lonsdale | year = 1949 | title = Crystals and X-rays | publisher = D. van Nostrand | location = New York}} * {{cite book | author=[[William Lawrence Bragg|Bragg, William Lawrence]], D. C. Phillips and H. Lipson | year = 1992 | title = The Development of X-ray Analysis | publisher = Dover | location = New York | isbn = 0-486-67316-2}} * {{cite book | author = [[Paul Peter Ewald|Ewald PP]], editor, and numerous crystallographers| year = 1962 | title = Fifty Years of X-ray Diffraction | publisher = published for the [[International Union of Crystallography]] by A. Oosthoek's Uitgeversmaatschappij N.V. | location = Utrecht }} *[[Paul Peter Ewald|Ewald, P. P.]], editor [http://www.iucr.org/iucr-top/publ/50YearsOfXrayDiffraction/ ''50 Years of X-Ray Diffraction''] (Reprinted in pdf format for the IUCr XVIII Congress, Glasgow, Scotland, Copyright © 1962, 1999 International Union of Crystallography). * {{cite book | author = [[Johannes Martin Bijvoet|Bijvoet JM]], Burgers WG, Hägg G, eds. | year = 1969 | title = Early Papers on Diffraction of X-rays by Crystals (Volume I) | publisher = published for the [[International Union of Crystallography]] by A. Oosthoek's Uitgeversmaatschappij N.V. | location = Utrecht }} * {{cite book | author = [[Johannes Martin Bijvoet|Bijvoet JM]], Burgers WG, Hägg G, eds. | year = 1972 | title = Early Papers on Diffraction of X-rays by Crystals (Volume II) | publisher = published for the [[International Union of Crystallography]] by A. Oosthoek's Uitgeversmaatschappij N.V. | location = Utrecht }} ==External links== ===Tutorials=== * [http://stein.bioch.dundee.ac.uk/~charlie/index.php?section=1 Simple, non technical introduction] * [http://acaschool.iit.edu/lectures04/JLiangXtal.pdf "Small Molecule Crystalization"] ([[PDF]]) at [[Illinois Institute of Technology]] website *[http://iucr.org International Union of Crystallography] *[http://www.ruppweb.org/Xray/101index.html Crystallography 101] *[http://www.ysbl.york.ac.uk/~cowtan/sfapplet/sfintro.html Interactive structure factor tutorial] * [http://www.ysbl.york.ac.uk/~cowtan/fourier/fourier.html Book of Fourier], about a technique which is used in X-ray crystallography * [http://www.chem.uwec.edu/Chem406_F06/Pages/lectnotes.html#lecture7 Lecture notes on X-ray crystallography and structure determination] ===Primary databases=== * [http://www.rcsb.org/pdb/home/home.do Protein Data Bank] (PDB) * [http://ndbserver.rutgers.edu/ Nucleic Acid Databank] (NDB) * [http://www.ccdc.cam.ac.uk/products/csd/ Cambridge Structural Database] (CSD) * [http://www.fiz-karlsruhe.de/icsd.html Inorganic Crystal Structure Database] (ICSD) * [http://xpdb.nist.gov:8060/BMCD4/ Biological Macromolecule Crystallization Database] (BMCD) ===Derivative databases=== * [http://www.ebi.ac.uk/thornton-srv/databases/pdbsum/ PDBsum] * [http://www.proteopedia.org Proteopeida - the collaborative, 3D encyclopedia of proteins and other molecules] * [http://www.rnabase.org/ RNABase] * [http://xray.bmc.uu.se/hicup/ HIC-Up database of PDB ligands] * [[Structural Classification of Proteins]] database * [[CATH|CATH Protein Structure Classification]] * [http://blanco.biomol.uci.edu/Membrane_Proteins_xtal.html List of transmembrane proteins with known 3D structure] * [[Orientations of Proteins in Membranes database]] ===Structural validation=== * [http://swift.cmbi.kun.nl/WIWWWI// WHAT-IF structural validation suite] * [http://biotech.ebi.ac.uk/ Biotech structural validation suite] (formerly ProCheck) * [http://molprobity.biochem.duke.edu/ MolProbity structural validation suite] * [https://prosa.services.came.sbg.ac.at/prosa.php ProSA-web] * [https://flipper.services.came.sbg.ac.at/ NQ-Flipper] (check for unfavorable rotamers of Asn and Gln residues) * [http://www.ebi.ac.uk/dali/ DALI server] (identifies proteins similar to a given protein) {{Protein structure determination}} ==Gallery== <gallery> Image:X-ray crystals - slow evaporation 1 solvent.png Image:X-ray crystals - slow evaporation 2 solvent.png Image:X-ray crystals - slow gas diffusion 2 solvent.png Image:X-ray crystals - slow liquid diffusion.png Image:X-ray crystals - slow liquid diffusion - H Tube.png </gallery> [[Category:Crystallography]] [[Category:Diffraction]] [[Category:X-rays]] [[Category:Protein structure]] [[Category:Protein methods]] [[Category:Synchrotron related techniques]] [[af:X-straalkristallografie‎]] [[ca:Difracció de raigs X]] [[cs:Rentgenová strukturní analýza]] [[de:Kristallstrukturanalyse]] [[es:Cristalografía de rayos X]] [[fa:پراش اشعه ایکس]] [[fr:Diffractométrie de rayons X]] [[it:Cristallografia a raggi X]] [[he:קריסטלוגרפיה באמצעות קרני רנטגן]] [[lv:Rentgendifraktometrija]] [[nl:Röntgendiffractie]] [[ja:X線回折]] [[no:Røntgenkrystallografi]] [[pl:Rentgenografia strukturalna]] [[pt:Cristalografia de raios X]] [[ru:Рентгеноструктурный анализ]] [[scn:Diffrazzioni dî raji X]] [[sr:Рендгенска структурна анализа]] [[fi:Röntgenkristallografia]] [[sv:Röntgenkristallografi]] [[uk:Рентгеноструктурний аналіз]] [[zh:X射线晶体学]]