# X-ray crystallography **X-ray crystallography** is the experimental science of working out where the [[Atom|atoms]] sit inside a crystal from the way the crystal scatters X-rays. Because an X-ray wavelength is comparable to the spacing between atomic planes, about a tenth of a nanometre, a [[Crystal_structure|crystal]] acts on the beam as a [[Diffraction_grating|diffraction grating]] acts on light: the scattered waves interfere, and strong beams leave the crystal only in a few sharp directions.[^up3-bragg] Those directions obey Bragg's law, `n·λ = 2·d·sin(θ)`, where `d` is the spacing of a family of atomic planes, `θ` the glancing angle of the beam against the planes, `λ` the wavelength and `n` a small integer.[^up3-bragg] The angles give the size and shape of the [[Unit_cell|unit cell]]; the intensities give the positions of the atoms within it. In the microsim below the reader chooses a cubic lattice type, its edge `a` and the wavelength `λ` (the copper Kα line at 0.154 nm is the preset), and the sim answers Bragg's law at every plane spacing `d_hkl` the cell contains, drawing the peaks as a stick pattern from which the structure-factor selection rules have removed the forbidden lines. On the [[Materials_science]] flagship this page serves Part II, Fundamentals › Structure, in the section *Diffraction from a crystal*, between the description of the lattice and the study of its defects. The Bragg condition that identifies a powdered metal in a factory laboratory has also placed every atom of a [[Hemoglobin|hemoglobin]] molecule.[^perutz-kendrew] ## History X-rays were discovered by Wilhelm Röntgen in 1895.[^nobel1901] In 1912 Walter Friedrich and Paul Knipping, following a suggestion of Max von Laue in Munich, passed a narrow X-ray beam through a crystal of copper sulfate and then of zinc blende and recorded a regular array of [[Diffraction|diffraction]] spots around the direct beam.[^laue1912] The spots proved that X-rays are waves, because they interfere, and that a crystal is a periodic arrangement of atoms fine enough to act as a three-dimensional grating for them.[^nobel1914] Within months the younger Bragg, a research student in Cambridge, showed that the same spots could be read as reflections from families of parallel atomic planes, and wrote the condition that has carried his name since.[^bragg1913] His father built an X-ray spectrometer with which the two measured reflections one plane family at a time, and by 1913 they had solved the alkali halides, [[Diamond|diamond]] and several other simple crystals.[^wlbragg1913][^diamond1913] Father and son shared the 1915 Nobel Prize in Physics; Lawrence Bragg, born in 1890, was 25.[^nobel1915][^wlbragg-bio] The powder method of Peter Debye and Paul Scherrer (1916) and Albert Hull (1917) then let a mass of tiny [[Crystallite|crystallites]] stand in for a single crystal, with the [[Reciprocal_lattice|reciprocal lattice]] as the natural geometry of the problem.[^debye-scherrer][^hull1917] ### Impact on chemistry The first structures overturned a chemical assumption. [[Sodium_chloride|Sodium chloride]] contains no NaCl [[Molecule|molecules]] at all, only an alternating lattice of sodium and chlorine ions in which every ion has six neighbours of the other kind, so that the formula names a ratio rather than a unit.[^wlbragg1913] Diamond showed each carbon atom bonded tetrahedrally to four others, a direct picture of the [[Covalent_bond|covalent bond]] that chemists had inferred but never seen.[^diamond1913] Bond lengths and angles measured from crystals became the calibration set for every later theory of the [[Chemical_bond|chemical bond]], and in 1929 Kathleen Lonsdale showed that the benzene ring in hexamethylbenzene is a flat, regular hexagon.[^lonsdale1929] The Cambridge Structural Database, begun in 1965, now holds more than a million small-molecule structures.[^ccdc-about] ### Impact on mineralogy Minerals were the natural first subjects because they grow as good crystals. Through the 1920s Lawrence Bragg's Manchester group solved the silicates and found that almost all of them are built from one unit, the SiO₄ tetrahedron, joined at corners into isolated groups, chains, sheets and frameworks; the classification of the [[Silicate_mineral|silicate minerals]] used by [[Mineralogy|mineralogy]] since then follows that topology rather than the chemical formula.[^bragg-silicates] [[Linus_Pauling|Linus Pauling]] rationalised the observed structures in 1929 as rules about coordination polyhedra and the balance of electrostatic bond strength around each ion.[^pauling1929] ### Impact on Biology Biological molecules are large, and their crystals diffract weakly and hold thousands of atoms per cell, so the method reached them only in the 1950s. In 1953 the diffraction photographs of DNA fibres taken by Rosalind Franklin and Raymond Gosling, with those of Maurice Wilkins, gave James Watson and Francis Crick the dimensions they needed to propose the double helix.[^franklin-gosling][^watson-crick] In 1958 John Kendrew's group reported the first three-dimensional model of a protein, myoglobin, at 6 Å resolution, and Max Perutz followed with hemoglobin; the two shared the 1962 Nobel Prize in Chemistry.[^kendrew1958][^perutz-kendrew] The Protein Data Bank was founded in 1971 with seven structures; it now holds more than two hundred thousand, and its models of [[Enzyme|enzymes]], receptors and viruses guide drug design in [[Medicine|medicine]].[^pdb2000] ## Methods A structure determination runs through the same stages whatever the sample: grow a crystal, collect the pattern, find the cell and symmetry, solve the phase problem, build and refine a model, and deposit it. The microsim covers the geometry that fixes where the beams go, the part used most in [[Characterization_(materials_science)|materials characterization]]. ### Overview Every diffracted beam is indexed by three integers `h, k, l`, the [[Miller_index|Miller indices]] of the plane family that reflects it. For a cubic cell of edge `a` the plane spacing is `d_hkl = a / sqrt(h² + k² + l²)`, and Bragg's law with `n = 1` gives the glancing angle of each reflection as `sin(θ_hkl) = λ / (2·d_hkl)`.[^up3-bragg] A reflection exists only if `λ / (2·d_hkl) ≤ 1`, so a short wavelength reaches more planes than a long one; and the angle is measured from the plane, not its normal, which is why a diffractometer reports `2θ`, the angle between the direct and the diffracted beam.[^up3-bragg] The [[Bragg's_law|Bragg's law]] child page treats one plane family; this sim runs the law over every `d_hkl` of a cell. Geometry alone predicts too many beams. Whether a reflection carries intensity is decided by the [[Structure_factor|structure factor]] `F_hkl = Σ_j f_j·exp(2πi·(h·x_j + k·y_j + l·z_j))`, the sum over the atoms in the cell of each atom's scattering power `f_j` times a phase set by its fractional position; the intensity is proportional to `|F_hkl|²`.[^kittel-sf] In a body-centred cubic cell the atom at `(½, ½, ½)` scatters exactly out of phase with the corner atom whenever `h + k + l` is odd, so `F_hkl = 0` and the reflection is absent; in a face-centred cubic cell the three face atoms cancel the corner unless `h, k, l` are all even or all odd. A simple cubic cell shows every line. In the sim the reader sets the lattice type (simple, body-centred or face-centred cubic), the cell edge `a` and the wavelength `λ`, with the copper Kα line preset at 0.154 nm.[^deslattes2003] The sim lists every `(hkl)` with `d_hkl` and `2θ_hkl`, drops the reflections the selection rules forbid, and draws the survivors as a stick pattern on a `2θ` axis. Shortening `λ` slides the pattern to smaller angles and adds lines at the high-angle end; enlarging `a` does the same; switching from body-centred to face-centred changes which lines exist without moving the ones they share. The stick heights are ILLUSTRATIVE: they show the multiplicity of each plane family, not the full intensity, which also depends on atomic form factors, thermal motion and instrument geometry. ### Crystallization Nothing can be measured without a crystal, and for most samples this is the slowest and least predictable step. A crystal grows when a solution is driven gently past its [[Solubility|solubility]] limit, so that [[Nucleation|nucleation]] happens at a few points and [[Crystal_growth|crystal growth]] then proceeds slowly enough for each new layer to find its lattice site.[^mcpherson2014] Proteins, which carry a shell of water and are easily denatured, are usually crystallised by vapour diffusion: a microlitre drop of protein and precipitant equilibrates against a reservoir of stronger precipitant, and the drop's [[Supersaturation|supersaturation]] rises over days.[^mcpherson2014] ### Data collection A single crystal is mounted on a goniometer and rotated in a monochromatic beam while a two-dimensional detector records the spots; the rotation brings successive plane families through the Bragg condition.[^af-lattice] Laboratory instruments use the characteristic radiation of a metal anode, usually [[Copper|copper]] (Kα₁ at 0.154056 nm) or molybdenum (0.0709 nm).[^deslattes2003] For weak samples the source is a synchrotron, and protein crystals are held near 100 K in cold nitrogen so that the beam does not destroy them before the data are complete.[^garman2010] The powder method replaces the rotation with statistics. Many randomly oriented crystallites present every plane family at every orientation at once, so each reflection becomes a cone of half-angle `2θ` around the incident beam, and a detector scanning through `2θ` records the one-dimensional pattern of lines that the sim draws.[^debye-scherrer] The pattern of a phase is a fingerprint, and [[Powder_diffraction|powder diffraction]] is the everyday tool for identifying phases, measuring lattice parameters and, through the Rietveld method of 1969, refining whole structures against the pattern.[^rietveld1969] [[Neutron_diffraction|Neutron diffraction]], which sees nuclei rather than [[Electron|electrons]], and [[Electron_diffraction|electron diffraction]], which works on nanometre-sized volumes, follow the same geometry. ### Crystal symmetry, unit cell, and image scaling The first thing done with a pattern is to index it: to assign `(hkl)` to every reflection and extract the cell. For a cubic powder pattern the recipe is arithmetic. Since `sin²(θ) = λ²·(h² + k² + l²)/(4·a²)`, the `sin²θ` values of successive lines stand in the ratios of the integers `h² + k² + l²`, and the sequence identifies the lattice: simple cubic gives 1, 2, 3, 4, 5, 6, 8, …, body-centred 2, 4, 6, 8, 10, 12, 14, … and face-centred 3, 4, 8, 11, 12, 16, ….[^cullity-index] The sim's presets make the point concrete. [[Nickel|Nickel]] is face-centred cubic with `a = 0.3524 nm`; with copper Kα₁ radiation its first three lines fall at `2θ` of 44.5° (111), 51.9° (200) and 76.4° (220), the (100) and (110) lines that a simple cubic lattice would show at 25.3° and 36.0° being absent.[^af-lattice][^deslattes2003] [[Tungsten|Tungsten]] is body-centred cubic with `a = 0.3165 nm`; its lines appear at 40.3° (110), 58.3° (200) and 73.2° (211), and the (100) line is missing.[^af-lattice] Once a line is indexed the cell follows from `a = d_hkl·sqrt(h² + k² + l²)`, which is how lattice parameters are measured to five significant figures. For single crystals software fits a reciprocal lattice to the spot positions, deduces the [[Crystal_system|crystal system]] and [[Bravais_lattice|Bravais lattice]] from the cell's symmetry and the systematic absences, scales thousands of images to a common intensity scale, and picks the space group, one of 230 possibilities classified by [[Group_theory|group theory]]. ### Initial phasing The data are intensities, `|F_hkl|²`, but reconstructing the electron density by [[Fourier_analysis|Fourier synthesis]] requires the complex structure factors, magnitude and phase, and the phase is lost in measurement. This is the phase problem. For small molecules the solution since the 1950s has been direct methods, the statistical relations among the phases of strong reflections worked out by Herbert Hauptman and Jerome Karle, who received the 1985 Nobel Prize in Chemistry for them.[^hauptman-karle] For proteins the classic route is isomorphous replacement: a heavy atom such as mercury is soaked into the crystal, the small change in intensities locates it, and its known contribution fixes the phases of the rest.[^kendrew1958] Anomalous scattering near an absorption edge does the same with selenium substituted for sulfur, and molecular replacement borrows phases from a homologous structure already in the database, now the most common route of all.[^pdb2000] ### Model building and phase refinement With initial phases, an electron-density map is computed and an atomic model is built into it. The model is then refined by [[Least_squares|least squares]] or maximum likelihood, adjusting coordinates and displacement parameters until the calculated structure factors match the observed ones; the agreement is reported as the R factor, `R = Σ ||F_obs| − |F_calc|| / Σ |F_obs|`, a few per cent for a good small-molecule structure.[^shelx] Because a flexible model can be fitted to noise, a randomly chosen 5–10 % of reflections are set aside and never used in refinement; their agreement, the free R value introduced in 1992, is the honest measure of the model.[^rfree] ### Disorder Atoms are never still and never all in the same place. Thermal vibration, coupled through the [[Phonon|phonons]] of the lattice, smears each atom's electron density and weakens the high-angle reflections by the Debye–Waller factor `exp(−B·sin²θ/λ²)`, so the displacement parameter `B`, in Ų, is refined for every atom and drawn as a thermal ellipsoid when made anisotropic.[^shelx] Static disorder is described by occupancy: a side chain seen in two conformations is modelled as two half-atoms. ### Applied computational data analysis From indexing to deposition the work is done in software. The SHELX programs, begun by George Sheldrick around 1970 and still the standard for small-molecule solution and refinement, are among the most cited pieces of scientific code ever published.[^shelx] Macromolecular pipelines integrate data reduction, phasing, automated model building and validation.[^rfree] ### Deposition of the structure A finished structure is deposited in a public database in the Crystallographic Information File format: cell, space group, coordinates, displacement parameters and the reflection data themselves. Small molecules go to the Cambridge Structural Database, inorganic structures to the Inorganic Crystal Structure Database or the open Crystallography Open Database, and biological macromolecules to the Protein Data Bank, which has required the experimental structure factors alongside the model since 2008.[^pdb2000][^ccdc-about] [[Materials_science|Materials science]] increasingly mines these archives directly for materials with a target structure. ## Contribution of women to X-ray crystallography Crystallography was unusual among the physical sciences of the twentieth century in the number of women who led it. Kathleen Lonsdale proved the planarity of the benzene ring in 1929 and in 1945 became one of the first two women elected Fellows of the Royal Society.[^lonsdale1929] Dorothy Crowfoot Hodgkin, in Oxford, determined the structures of penicillin during the Second World War, of vitamin B₁₂ in 1955 and of insulin in 1969, and received the 1964 Nobel Prize in Chemistry.[^hodgkin] Rosalind Franklin's fibre diffraction of DNA at King's College London in 1952–53, with Raymond Gosling, produced the photographs whose helical cross fixed the pitch and diameter of the double helix.[^franklin-gosling] Isabella Karle developed the symbolic addition procedure that made direct methods usable on real crystals, the work for which her husband Jerome Karle and Herbert Hauptman received the 1985 prize.[^hauptman-karle] Olga Kennard founded the Cambridge Structural Database in 1965, and Ada Yonath shared the 2009 Nobel Prize in Chemistry for the ribosome's structure.[^ccdc-about][^yonath] ## Nobel Prizes involving X-ray crystallography The technique's Nobel record begins with the first physics prize ever awarded, to Röntgen in 1901 for the discovery of X-rays, and continues with von Laue in 1914 and the two Braggs in 1915.[^nobel1901][^nobel1914][^nobel1915] In chemistry the prizes followed the growth of the method, from the first protein structures in 1962 to the ribosome in 2009, while the 1962 prize in Physiology or Medicine to Crick, Watson and Wilkins rested on the DNA diffraction work.[^perutz-kendrew][^dna-nobel] Dan Shechtman's 2011 chemistry prize for the discovery of [[Quasicrystal|quasicrystals]] came from an electron diffraction pattern with a symmetry the 230 space groups forbid.[^shechtman] | Year | Prize | Laureates | Work | |---|---|---|---| | 1901 | Physics | Röntgen | discovery of X-rays[^nobel1901] | | 1914 | Physics | von Laue | diffraction of X-rays by crystals[^nobel1914] | | 1915 | Physics | W. H. and W. L. Bragg | crystal structure analysis by X-rays[^nobel1915] | | 1962 | Chemistry | Perutz, Kendrew | structures of globular proteins[^perutz-kendrew] | | 1964 | Chemistry | Hodgkin | structures of important biochemical substances[^hodgkin] | | 2009 | Chemistry | Ramakrishnan, Steitz, Yonath | structure and function of the ribosome[^yonath] | ## See also - [[Bragg's_law]] — one plane family, with θ as the control - [[Powder_diffraction]] - [[Structure_factor]] - [[Neutron_diffraction]] - [[Electron_diffraction]] - [[Reciprocal_lattice]] - [[Crystal_structure]] - [[Miller_index]] - [[Unit_cell]] - [[Crystallography]] ## Notes The glancing angle `θ` in Bragg's law is measured between the beam and the atomic plane, not the plane's normal; diffractometers report the scattering angle `2θ`. The integer `n` (written `m` in some textbooks) is usually absorbed into the indices by treating the `n`-th order reflection from `(hkl)` as the first-order reflection from `(nh nk nl)`. Footnotes are collected under References. ## References [^up3-bragg]: OpenStax (Sanny, J.; Ling, S. J.). *University Physics Volume 3* (2016), Ch. 4 Diffraction, §4.6 X-Ray Diffraction, pp. 168–170 (Bragg's law with θ measured from the planes; Example 4.7, NaCl planes at d = 0.252 nm). https://openstax.org/books/university-physics-volume-3/pages/4-6-x-ray-diffraction [^af-lattice]: OpenStax (Flowers, P.; Neth, E. J.; Robinson, W. R.; et al.). *Chemistry: Atoms First 2e* (2019), Ch. 10 Liquids and Solids, §10.6 Lattice Structures in Crystalline Solids, pp. 475–544 (page to pin: the cubic unit cells, the nickel 0.3524 nm and tungsten 3.165 Å cell edges in the worked examples and exercises, and the X-ray diffraction subsection with the diffractometer figure). https://openstax.org/books/chemistry-atoms-first-2e/pages/10-6-lattice-structures-in-crystalline-solids [^bragg1913]: Bragg, W. H.; Bragg, W. L. (1913). "The Reflection of X-rays by Crystals." *Proceedings of the Royal Society A* 88 (605): 428–438. https://doi.org/10.1098/rspa.1913.0040 [^wlbragg1913]: Bragg, W. L. (1913). "The Structure of Some Crystals as Indicated by their Diffraction of X-rays." *Proceedings of the Royal Society A* 89 (610): 248–277. [^diamond1913]: Bragg, W. H.; Bragg, W. L. (1913). "The Structure of the Diamond." *Proceedings of the Royal Society A* 89 (610): 277–291. [^kittel-sf]: Kittel, C. *Introduction to Solid State Physics*, 8th ed. (2005), Ch. 2 Wave Diffraction and the Reciprocal Lattice, "Structure factor of the bcc lattice" and "fcc lattice" (page to pin). [^cullity-index]: Cullity, B. D.; Stock, S. R. *Elements of X-Ray Diffraction*, 3rd ed. (2001), Ch. 10 Determination of Crystal Structure, the indexing of cubic powder patterns (page to pin). [^mcpherson2014]: McPherson, A.; Gavira, J. A. (2014). "Introduction to protein crystallization." *Acta Crystallographica Section F* 70 (1): 2–20 (DOI to pin). [^garman2010]: Garman, E. F. (2010). "Radiation damage in macromolecular crystallography: what is it and why should we care?" *Acta Crystallographica Section D* 66 (4): 339–351 (DOI to pin). [^laue1912]: Friedrich, W.; Knipping, P.; Laue, M. (1912). "Interferenz-Erscheinungen bei Röntgenstrahlen." *Sitzungsberichte der Mathematisch-Physikalischen Classe der Königlich-Bayerischen Akademie der Wissenschaften zu München*: 303–322. [^nobel1901]: The Nobel Prize in Physics 1901 (Wilhelm Conrad Röntgen). NobelPrize.org. https://www.nobelprize.org/prizes/physics/1901/summary/ [^nobel1914]: The Nobel Prize in Physics 1914 (Max von Laue). NobelPrize.org. https://www.nobelprize.org/prizes/physics/1914/summary/ [^nobel1915]: The Nobel Prize in Physics 1915 (Sir William Henry Bragg and William Lawrence Bragg). NobelPrize.org. https://www.nobelprize.org/prizes/physics/1915/summary/ [^wlbragg-bio]: William Lawrence Bragg — Biographical. NobelPrize.org. https://www.nobelprize.org/prizes/physics/1915/wl-bragg/biographical/ [^debye-scherrer]: Debye, P.; Scherrer, P. (1916). "Interferenzen an regellos orientierten Teilchen im Röntgenlicht. I." *Physikalische Zeitschrift* 17: 277–283. [^hull1917]: Hull, A. W. (1917). "A New Method of X-Ray Crystal Analysis." *Physical Review* 10 (6): 661–696. https://doi.org/10.1103/PhysRev.10.661 [^rietveld1969]: Rietveld, H. M. (1969). "A profile refinement method for nuclear and magnetic structures." *Journal of Applied Crystallography* 2 (2): 65–71. https://doi.org/10.1107/S0021889869006558 [^deslattes2003]: Deslattes, R. D.; Kessler, E. G.; Indelicato, P.; de Billy, L.; Lindroth, E.; Anton, J. (2003). "X-ray transition energies: new approach to a comprehensive evaluation." *Reviews of Modern Physics* 75 (1): 35–99 (Cu Kα₁ = 8047.8 eV, λ = 0.154056 nm; Mo Kα₁ = 0.070932 nm). https://doi.org/10.1103/RevModPhys.75.35 [^lonsdale1929]: Lonsdale, K. (1929). "The structure of the benzene ring in C₆(CH₃)₆." *Proceedings of the Royal Society A* 123 (792): 494–515. [^bragg-silicates]: Bragg, W. L. (1930). "The structure of silicates." *Zeitschrift für Kristallographie* 74: 237–305. [^pauling1929]: Pauling, L. (1929). "The principles determining the structure of complex ionic crystals." *Journal of the American Chemical Society* 51 (4): 1010–1026. [^franklin-gosling]: Franklin, R. E.; Gosling, R. G. (1953). "Molecular Configuration in Sodium Thymonucleate." *Nature* 171 (4356): 740–741. https://doi.org/10.1038/171740a0 [^watson-crick]: Watson, J. D.; Crick, F. H. C. (1953). "Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid." *Nature* 171 (4356): 737–738. https://doi.org/10.1038/171737a0 [^kendrew1958]: Kendrew, J. C.; Bodo, G.; Dintzis, H. M.; Parrish, R. G.; Wyckoff, H.; Phillips, D. C. (1958). "A Three-Dimensional Model of the Myoglobin Molecule Obtained by X-Ray Analysis." *Nature* 181 (4610): 662–666. https://doi.org/10.1038/181662a0 [^perutz-kendrew]: The Nobel Prize in Chemistry 1962 (Max F. Perutz and John C. Kendrew). NobelPrize.org. https://www.nobelprize.org/prizes/chemistry/1962/summary/ [^dna-nobel]: The Nobel Prize in Physiology or Medicine 1962 (Francis Crick, James Watson, Maurice Wilkins). NobelPrize.org. https://www.nobelprize.org/prizes/medicine/1962/summary/ [^hodgkin]: The Nobel Prize in Chemistry 1964 (Dorothy Crowfoot Hodgkin). NobelPrize.org. https://www.nobelprize.org/prizes/chemistry/1964/summary/ [^hauptman-karle]: The Nobel Prize in Chemistry 1985 (Herbert A. Hauptman and Jerome Karle). NobelPrize.org. https://www.nobelprize.org/prizes/chemistry/1985/summary/ [^yonath]: The Nobel Prize in Chemistry 2009 (Venkatraman Ramakrishnan, Thomas A. Steitz, Ada E. Yonath). NobelPrize.org. https://www.nobelprize.org/prizes/chemistry/2009/summary/ [^shechtman]: The Nobel Prize in Chemistry 2011 (Dan Shechtman). NobelPrize.org. https://www.nobelprize.org/prizes/chemistry/2011/summary/ [^pdb2000]: Berman, H. M.; Westbrook, J.; Feng, Z.; Gilliland, G.; Bhat, T. N.; Weissig, H.; Shindyalov, I. N.; Bourne, P. E. (2000). "The Protein Data Bank." *Nucleic Acids Research* 28 (1): 235–242 (founded 1971 at Brookhaven with seven structures; deposition practice). https://doi.org/10.1093/nar/28.1.235 — current holdings and the 2008 structure-factor requirement: RCSB Protein Data Bank, https://www.rcsb.org/ [^ccdc-about]: Cambridge Crystallographic Data Centre. "About the CCDC" and the Cambridge Structural Database (founded 1965 by Olga Kennard). https://www.ccdc.cam.ac.uk/ [^shelx]: Sheldrick, G. M. (2008). "A short history of SHELX." *Acta Crystallographica Section A* 64 (1): 112–122. https://doi.org/10.1107/S0108767307043930 [^rfree]: Brünger, A. T. (1992). "Free R value: a novel statistical quantity for assessing the accuracy of crystal structures." *Nature* 355 (6359): 472–475. https://doi.org/10.1038/355472a0 ## Further reading ### Textbooks - OpenStax, *University Physics Volume 3* (2016), Ch. 4 Diffraction — the Bragg derivation behind the Physics manual's §8.7 sim (Portal Book 079). - OpenStax, *Chemistry: Atoms First 2e* (2019), Ch. 10 §10.6 Lattice Structures in Crystalline Solids — cubic cells, coordination and the X-ray diffraction subsection (Portal Book 051). ### Historical - Bragg, W. H.; Bragg, W. L. (1913), "The Reflection of X-rays by Crystals", *Proc. R. Soc. A* 88: 428–438 — the paper in which the plane-reflection picture and the spectrometer appear together. - Friedrich, Knipping and Laue (1912), the Munich report of the first diffraction photographs. ## External links ### Tutorials - The Wikipedia pair's *External links* section lists the current tutorial sites; none is reproduced here until its URL has been checked. ### Primary databases - [RCSB Protein Data Bank](https://www.rcsb.org/) — biological macromolecules - [Cambridge Crystallographic Data Centre](https://www.ccdc.cam.ac.uk/) — the Cambridge Structural Database of small-molecule structures <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/X-ray_crystallography.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *X-ray crystallography* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/X-ray_crystallography.html" data-title="X-ray crystallography"></div> *Built from `MICROSIM_GUIDE/specs/sims/X-ray_crystallography.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/X-ray_crystallography) : [Wikitube](https://en.wikitube.io/wiki/X-ray_crystallography) · pinned revision [1372443742](https://en.wikipedia.org/w/index.php?oldid=1372443742) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Materials_science]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Materials_science row M6 · sim pending (matter/X-ray_crystallography).*