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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Fiber diffraction</span></span>

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</p><p><b>Fiber diffraction</b> is a subarea of <a href="Scattering" title="Scattering">scattering</a>, an area in which molecular structure is determined from scattering data (usually of X-rays, electrons or neutrons). In fiber diffraction, the scattering pattern does not change, as the sample is rotated about a unique axis (the fiber axis). Such uniaxial symmetry is frequent with filaments or fibers consisting of biological or man-made <a href="Macromolecule" title="Macromolecule">macromolecules</a>. In <a href="Crystallography" title="Crystallography">crystallography</a>, fiber symmetry is an aggravation regarding the determination of crystal structure, because reflections are smeared and may overlap in the fiber diffraction pattern. <a href="Materials_science" title="Materials science">Materials science</a> considers fiber symmetry a simplification, because almost the complete obtainable structure information is in a single <b>two-dimensional (2D)</b> diffraction pattern exposed on photographic film or on a 2D detector. 2 instead of 3 co-ordinate directions suffice to describe fiber diffraction.

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<p>The ideal fiber pattern exhibits <b>4-quadrant symmetry</b>. In the ideal pattern, the fiber axis is called the <b>meridian</b>, the perpendicular direction is called <b>equator</b>. In case of fiber symmetry, many more reflections than in single-crystal diffraction show up in the 2D pattern. In fiber patterns these reflections clearly appear arranged along lines (<b>layer lines</b>) running almost parallel to the equator. Thus, in fiber diffraction the layer line concept of <a href="Crystallography" title="Crystallography">crystallography</a> becomes palpable. Bent layer lines indicate that the pattern must be straightened. Reflections are labelled by the <a href="Miller_index" title="Miller index">Miller index</a> hkl, i.e. 3 digits. Reflections on the <i>i</i>-th layer line share l=<i>i</i>. Reflections on the meridian are 00l-reflections. In <a href="Crystallography" title="Crystallography">crystallography</a> artificial fiber diffraction patterns are generated by rotating a single crystal about an axis (<b>rotating crystal method</b>).

</p><p>Non-ideal fiber patterns are obtained in experiments. They only show mirror symmetry about the meridian. The reason is that the fiber axis and the incident beam (X-rays, electrons, neutrons) cannot be perfectly oriented perpendicular to each other. The corresponding geometric distortion has been extensively studied by <a href="Michael_Polanyi" title="Michael Polanyi">Michael Polanyi</a> introducing the concept of <i>Polanyi's sphere</i> (German: "Lagenkugel") intersecting <a href="Ewald's_sphere" title="Ewald's sphere">Ewald's sphere</a>. Later <a href="Rosalind_Franklin" title="Rosalind Franklin">Rosalind Franklin</a> and <a href="Raymond_Gosling" title="Raymond Gosling">Raymond Gosling</a> have carried out their own geometrical reasoning and presented an approximative equation for the fiber tilt angle β. Analysis starts by mapping the distorted 2D pattern on the representative plane of the fiber. This is the plane that contains the cylinder axis in <a href="Reciprocal_space" class="mw-redirect" title="Reciprocal space">reciprocal space</a>. In <a href="Crystallography" title="Crystallography">crystallography</a> first an approximation of the mapping into <a href="Reciprocal_space" class="mw-redirect" title="Reciprocal space">reciprocal space</a> is computed that is refined iteratively. The digital method frequently called <i>Fraser correction</i> starts from the Franklin approximation for the tilt angle β. It eliminates fiber tilt, unwarps the detector image, and corrects the scattering intensity. The correct equation for the determination of β has been presented by Norbert Stribeck.

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<div class="mw-heading mw-heading2"><h2 id="Historical_role">Historical role</h2></div>

<p>Fibrous materials such as wool or cotton easily form aligned bundles, and were among the first biological macromolecules studied by X-ray diffraction, notably by <a href="William_Astbury" title="William Astbury">William Astbury</a> in the early 1930s. Fiber diffraction data led to several important advances in the development of <a href="Structural_biology" title="Structural biology">structural biology</a>, e.g., the original models of the <a href="Alpha_helix" title="Alpha helix">α-helix</a> and the Watson-Crick model of double-stranded <a href="DNA" title="DNA">DNA</a>.

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<div class="mw-heading mw-heading2"><h2 id="Fiber_diffraction_geometry">Fiber diffraction geometry</h2></div>

<p>The animation shows the geometry of fiber diffraction. It is based on the notions proposed by <a href="Michael_Polanyi" title="Michael Polanyi">Michael Polanyi</a>. Reference direction is the primary beam (label: X-ray). If the fiber is tilted away from the perpendicular direction by an angle β, as well the information about its molecular structure in reciprocal space (trihedron labelled <i>s-space</i>) is tilted. In reciprocal space the <a href="Ewald_sphere" class="mw-redirect" title="Ewald sphere">Ewald sphere</a> has its center in the sample. Its radius is 1/λ, with λ the wavelength of the incident radiation. On the surface of the <a href="Ewald_sphere" class="mw-redirect" title="Ewald sphere">Ewald sphere</a> all the points of reciprocal space are found that are seen by the detector. These points are mapped on the pixels of the detector by central projection.

</p><p>In s-space each reflection is found on its Polanyi-sphere. Intrinsically the ideal reflection is a point in s-space, but fiber symmetry turns it into a ring smeared out by rotation about the fiber direction. <i>Two</i> rings represent each reflection on the Polanyi sphere, because scattering is <a href="Point_symmetry" class="mw-redirect" title="Point symmetry">point symmetric</a> with respect to the origin of s-space. Mapped onto the detector are only those points of the reflection in s-space that are both on the <a href="Ewald_sphere" class="mw-redirect" title="Ewald sphere">Ewald sphere</a> and on the Polanyi sphere. These points form the <b>reflection circle</b> (blue ring). It does not change as the fiber is tilted. As with a slide projector the reflection circle is projected (red moving rays) on the detector (<b>detector circle</b>, blue ring). There up to 4 images (red spots) of the monitored reflection can show up. The position of the reflection images is a function of the orientation of the fiber in the primary beam (<b>Polanyi equation</b>). Inverted, from the positions of the reflection images the orientation of the fiber can be determined, if for the <a href="Miller_index" title="Miller index">Miller index</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle hkl}">

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<annotation encoding="application/x-tex">{\displaystyle hkl}</annotation>

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</math></span><img src="./b11c00a7a2bb17351170f0ecbdcd61cd1581daf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.243ex; height:2.176ex;" alt="{\displaystyle hkl}" loading="lazy"></span> both <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |h|+|k|\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle |h|+|k|\neq 0}</annotation>

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</math></span><img src="./19a3b5f4ae1f4abe501d4953bc3b9f4734605de3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.239ex; height:2.843ex;" alt="{\displaystyle |h|+|k|\neq 0}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle l\neq 0}</annotation>

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</math></span><img src="./5a10d3efece67b4c745a2e283d3fbbda9cad0515.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.954ex; height:2.676ex;" alt="{\displaystyle l\neq 0}" loading="lazy"></span> is valid. From the Polanyi representation of fiber diffraction geometry the relations of the fiber mapping are established by <a href="Geometry" title="Geometry">elementary</a> and spherical geometry.

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<div class="mw-heading mw-heading2"><h2 id="Pattern_correction">Pattern correction</h2></div>

<p>The figure on the left shows a typical fiber pattern of <a href="Polypropylene" title="Polypropylene">polypropylene</a> before mapping it into reciprocal space. The mirror axis in the pattern is rotated by the angle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\phi }">

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</math></span><img src="./113645d4ee5b40587d6364c530eb01be60890d14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.966ex; height:2.509ex;" alt="{\displaystyle ~\phi }" loading="lazy"></span> with respect to the vertical direction. This shortcoming is compensated by simple rotation of the picture. 4 straight arrows point at 4 reflection images of a chosen reference reflection. Their positions are used to determine the fiber tilt angle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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</p><p>After determination of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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</math></span><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span> the distance between sample and detector is computed using known crystallographic data of the reference reflection, a uniformly gridded map for the representative fiber plane in reciprocal space is constructed and the diffraction data are fed into this map. The figure on the right shows the result. Change of scattering intensity has been considered in the unwarping process. Because of the curvature of the surface of the <a href="Ewald_sphere" class="mw-redirect" title="Ewald sphere">Ewald sphere</a> there remain white spots at the meridian, in which structure information is missing. Only in the center of the image and at an s-value related to the scattering angle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\beta }">

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<annotation encoding="application/x-tex">{\displaystyle 2\beta }</annotation>

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</math></span><img src="./fbd74eba3e136b73b2770f298e7708fae1a3b0a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.494ex; height:2.509ex;" alt="{\displaystyle 2\beta }" loading="lazy"></span> there is structure information on the meridian. Of course, there is now 4-quadrant symmetry. This means that in the example pattern part of the missing information may be copied "from the lower half to the upper half" into the white areas. Thus, it frequently makes sense to tilt the fiber intentionally.

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<p>The three-dimensional sketch demonstrates that in the example experiment the collected information on the molecular structure of the polypropylene fiber is almost complete. By rotation of the plane pattern about the meridian the scattering data collected in 4 s fill an almost spherical volume of s-space. In the example the 4-quadrant symmetry has not yet been considered to fill part of the white spots. For clarity a quarter of the sphere has been cut out, but keeping the equatorial plane itself.

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li>Arnott S & Wonacott A J, The Refinement of the Molecular & Crystal Structures of Polymers Using X-Ray Data and Stereochemical Constraints, Polymer 1966 <b>7</b> 157 – 166</li>

<li>Bian W, Wang H, McCullogh I, Stubbs G (2006). "WCEN: a computer program for initial processing of fiber diffraction patterns". <i>J. Appl. Crystallogr.</i>, <b>39</b>, 752–756.</li>

<li>Bunn C W, Chemical Crystallography, University of Oxford, 2nd Ed, 1967</li>

<li>Campbell Smith P J & Arnott S, LALS (etc.) Acta Crystallogr 1978 <b>A34</b> 3 - 11</li>

<li>Cochran W, Crick FHC, and Vand V (1952). "The Structure of Synthetic Polypeptides. I. The Transform of Atoms on a Helix". <i>Acta Crystallogr.</i>, <b>5</b>, 581–586.</li>

<li>Donohue J, and Trueblood, K N, On the unreliability of the reliability index, Acta Crystallogr, 1956, <b>9</b>, 615</li>

<li>Franklin RE, Gosling RG (1953) "The Structure of Sodium Thymonucleate Fibres. II. The Cylindrically Symmetrical Patterson Function". <i>Acta Crystallogr.</i>, <b>6</b>, 678–685</li>

<li>Fraser RDB, Macrae TP, Miller A, Rowlands RJ (1976). "Digital Processing of Fibre Diffraction Patterns". <i>J. Appl. Crystallogr.</i>, <b>9</b>, 81–94.</li>

<li>Hamilton W C, R-Factors, Statistics and Truth, Paper H5, Amer Cryst Ass Program & Abstracts, Boulder, Colorado, 1961</li>

<li>Hamilton W C, Significance Tests on the Crystallographic R Factor, Acta Crystallogr 1965 <b>18</b> 502 – 510</li>

<li>James T W & Mazia D, Surface Films of Desoxyribonucleic Acid, Biochim Biophys Acta 1953 <b>10</b> 367 - 370</li>

<li>Marvin DA (2017) "Fibre diffraction studies of biological macromolecules". <i>Prog. Biophys. Mol. Biol.</i> <b>127</b>, 43–87.</li>

<li>Millane RP, Arnott S (1985) "Digital Processing of X-Ray Diffraction Patterns from Oriented Fibers". <i>J. Macromol. Sci. Phys.</i>, <b>B24</b>, 193-227</li>

<li>Polanyi M (1921) "Das Röntgen-Faserdiagramm (Erste Mitteilung)". <i>Z. Physik</i>, <b>7</b>, 149-180</li>

<li>Polanyi M, Weissenberg K (1923) "Das Röntgen-Faserdiagramm (Zweite Mitteilung)". <i>Z. Physik</i>, <b>9</b>, 123-130</li>

<li>Rajkumar G, AL-Khayat H, Eakins F, He A, Knupp C, Squire J (2005) "FibreFix — A New Integrated CCP13 Software Package", <i>Fibre Diffraction Rev.</i>, <b>13</b>, 11-18</li>

<li>Stribeck N (2009). "On the determination of fiber tilt angles in fiber diffraction" <i>Acta Crystallogr.</i>, <b>A65</b>, 46-47</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Text_books">Text books</h3></div>

<ul><li>Alexander LE (1979) "X-Ray Diffraction Methods in Polymer Science", Wiley, New York</li>

<li>Klug HP, Alexander LE (1974) "X-Ray Diffraction Procedures For Polycrystalline and Amorphous Materials", 2nd ed, Wiley, New York</li>

<li>Warren BE (1990) "X-Ray Diffraction". Dover, New York</li>

<li><a rel="nofollow" class="external text" href="https://drive.google.com/open?id=0B3L_EN9hIuFTTkhuN2lrWEU4RDQ&authuser=0">Saad Mohamed (1994) "Low resolution structure and packing investigations of collagen crystalline domains in tendon using Synchrotron Radiation X-rays, Structure factors determination, evaluation of Isomorphous Replacement methods and other modeling." PhD Thesis, Université Joseph Fourier Grenoble 1</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://fibernet.vanderbilt.edu/software/wcen/">WCEN</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120723043425/http://fibernet.vanderbilt.edu/software/wcen/">Archived</a> July 23, 2012, at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a> — Software (Linux, Mac, Windows) for the analysis of fiber patterns</li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070908163254/http://www.mpimf-heidelberg.mpg.de/~holmes/fibre/branden.html">Fiber Diffraction</a> — an introduction provided by Prof. K.C. Holmes, Max Planck Institute for Medical Research, Heidelberg.</li></ul>

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</style><div id="Crystallography625" style="font-size:114%;margin:0 4em"><a href="Crystallography" title="Crystallography">Crystallography</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Timeline_of_crystallography" title="Timeline of crystallography">Timeline of crystallography</a>

<ul><li>Crystallographers</li></ul></li>

<li><a href="Metallurgy" title="Metallurgy">Metallurgy</a></li>

<li>Biocrystallography</li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Crystal_structure" title="Crystal structure">Structure</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bravais_lattice" title="Bravais lattice">Bravais lattice</a></li>

<li><a href="Miller_index" title="Miller index">Miller index</a></li>

<li><a href="Crystallographic_point_group" title="Crystallographic point group">Point group</a></li>

<li><a href="Reciprocal_lattice" title="Reciprocal lattice">Reciprocal lattice</a></li>

<li><a href="Crystallographic_restriction_theorem" title="Crystallographic restriction theorem">Restriction theorem</a></li></ul></li>

<li><a href="Periodic_table_(crystal_structure)" title="Periodic table (crystal structure)">Periodic table</a></li>

<li><a href="Crystal_structure_prediction" title="Crystal structure prediction">Structure prediction</a></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Systems26" scope="row" class="navbox-group" style="width:1%"><a href="Crystal_system" title="Crystal system">Systems</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cubic_crystal_system" title="Cubic crystal system">Cubic</a></li>

<li><a href="Hexagonal_crystal_family" title="Hexagonal crystal family">Hexagonal</a></li>

<li><a href="Monoclinic_crystal_system" title="Monoclinic crystal system">Monoclinic</a></li>

<li><a href="Orthorhombic_crystal_system" title="Orthorhombic crystal system">Orthorhombic</a></li>

<li><a href="Tetragonal_crystal_system" title="Tetragonal crystal system">Tetragonal</a></li>

<li><a href="Triclinic_crystal_system" title="Triclinic crystal system">Triclinic</a></li></ul>

</div></td></tr></tbody></table><div>

<ul><li><a href="Crystal_growth" title="Crystal growth">Growth</a>

<ul><li><a href="Crystallite" title="Crystallite">Crystallite</a></li>

<li><a href="Equiaxed_crystal" title="Equiaxed crystal">Equiaxed</a></li></ul></li>

<li><a href="Crystal_twinning" title="Crystal twinning">Twinning</a>

<ul><li><a href="Fiveling" title="Fiveling">Fiveling</a></li></ul></li>

<li><a href="Aperiodic_crystal" title="Aperiodic crystal">Aperiodic crystal</a>

<ul><li><a href="Quasicrystal" title="Quasicrystal">Quasicrystal</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Phase_transition" title="Phase transition">Phase<br>transition</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Phase_diagram" title="Phase diagram">Phase diagram</a>

<ul><li><a href="Eutectic_system" title="Eutectic system">Eutectic</a></li>

<li><a href="Miscibility_gap" title="Miscibility gap">Miscibility gap</a></li>

<li><a href="Crystal_polymorphism" title="Crystal polymorphism">Polymorphism</a></li>

<li><a href="Liquid_crystal" title="Liquid crystal">Liquid crystal</a></li></ul></li>

<li><a href="Phase_transformation_crystallography" title="Phase transformation crystallography">Phase transformation crystallography</a></li>

<li><a href="Precipitation_hardening" title="Precipitation hardening">Precipitation</a></li>

<li><a href="Segregation_(materials_science)" title="Segregation (materials science)">Segregation</a></li>

<li><a href="Spinodal_decomposition" title="Spinodal decomposition">Spinodal decomposition</a></li>

<li><a href="Supersaturation" title="Supersaturation">Supersaturation</a></li>

<li><a href="Ostwald_ripening" title="Ostwald ripening">Ostwald ripening</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Crystallographic_defect" title="Crystallographic defect">Defects</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Grain_boundary" title="Grain boundary">Grain boundary</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Disclination" title="Disclination">Disclination</a></li>

<li><a href="Grain_growth" title="Grain growth">Growth</a></li>

<li><a href="Abnormal_grain_growth" title="Abnormal grain growth">Abnormal growth</a></li></ul>

</div></td></tr></tbody></table><div>

<ul><li><a href="Perfect_crystal" title="Perfect crystal">Perfect crystal</a></li>

<li><a href="Stacking_fault" title="Stacking fault">Stacking fault</a></li>

<li><a href="Dislocation" title="Dislocation">Dislocation</a>

<ul><li><a href="Burgers_vector" title="Burgers vector">Burgers vector</a></li>

<li><a href="Partial_dislocation" title="Partial dislocation">Partial dislocation</a></li>

<li><a href="Cross_slip" title="Cross slip">Cross slip</a></li>

<li><a href="Frank%E2%80%93Read_source" title="Frank–Read source">Frank–Read source</a></li>

<li><a href="Cottrell_atmosphere" title="Cottrell atmosphere">Cottrell atmosphere</a></li>

<li><a href="Peierls_stress" title="Peierls stress">Peierls stress</a></li>

<li><a href="Lomer%E2%80%93Cottrell_junction" title="Lomer–Cottrell junction">Lomer–Cottrell junction</a></li></ul></li>

<ul><li><a href="Slip_bands_in_metals" title="Slip bands in metals">Slip bands</a></li></ul></li>

<li><a href="Interstitial_defect" title="Interstitial defect">Interstitials</a>

<ul><li><a href="Bjerrum_defect" title="Bjerrum defect">Bjerrum defect</a></li>

<li><a href="Frenkel_defect" title="Frenkel defect">Frenkel defect</a></li>

<li><a href="Wigner_effect" title="Wigner effect">Wigner effect</a></li></ul></li>

<li><a href="Vacancy_defect" title="Vacancy defect">Vacancy</a>

<ul><li><a href="Schottky_defect" title="Schottky defect">Schottky defect</a></li>

<li><a href="F-center" title="F-center">F-center</a></li></ul></li>

<li><a href="Stone%E2%80%93Wales_defect" title="Stone–Wales defect">Stone–Wales defect</a></li>

<li><a href="Crystallographic_defects_in_diamond" title="Crystallographic defects in diamond">Defects in diamond</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Laws</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Bragg's_law" title="Bragg's law">Bragg's law</a></li>

<li><a href="Friedel's_law" title="Friedel's law">Friedel's law</a></li>

<li><a href="Law_of_constancy_of_interfacial_angles" title="Law of constancy of interfacial angles">Steno's law (constancy of interfacial angles)</a></li>

<li><a href="Law_of_rational_indices" title="Law of rational indices">Law of rational indices</a></li>

<li><a href="Law_of_symmetry_(crystallography)" title="Law of symmetry (crystallography)">Law of symmetry</a></li>

</div></td></tr></tbody></table><div>

<ul><li><a href="Bragg_plane" title="Bragg plane">Bragg plane</a></li>

<li><a href="Ewald's_sphere" title="Ewald's sphere">Ewald's sphere</a></li>

<li><a href="Hermann%E2%80%93Mauguin_notation" title="Hermann–Mauguin notation">Hermann–Mauguin notation</a></li>

<li><a href="Structure_factor" title="Structure factor">Structure factor</a></li>

<li><a href="Thermal_ellipsoid" title="Thermal ellipsoid">Thermal ellipsoid</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="9" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"></span><br></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Characterisation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Electron_crystallography" title="Electron crystallography">Electron</a>

<ul><li><a href="Electron_diffraction" title="Electron diffraction">Diffraction</a></li>

<li><a href="Electron_scattering" title="Electron scattering">Scattering</a></li></ul></li>

<li><a href="Neutron_crystallography" class="mw-redirect" title="Neutron crystallography">Neutron</a>

<ul><li><a href="Neutron_diffraction" title="Neutron diffraction">Diffraction</a></li>

<li><a href="Neutron_scattering" title="Neutron scattering">Scattering</a></li></ul></li>

<li><a href="Nuclear_magnetic_resonance_crystallography" title="Nuclear magnetic resonance crystallography">Nuclear magnetic resonance</a></li>

<ul><li><a href="X-ray_diffraction" title="X-ray diffraction">Diffraction</a></li>

<li><a href="X-ray_scattering" class="mw-redirect" title="X-ray scattering">Scattering</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Algorithms</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Direct_methods_(crystallography)" title="Direct methods (crystallography)">Direct methods</a></li>

<li><a href="Isomorphous_replacement" title="Isomorphous replacement">Isomorphous replacement</a></li>

<li><a href="Molecular_replacement" title="Molecular replacement">Molecular replacement</a></li>

<li><a href="Molecular_dynamics" title="Molecular dynamics">Molecular dynamics</a></li>

<li><a href="Patterson_map" class="mw-redirect" title="Patterson map">Patterson map</a></li>

<li><a href="Phase_retrieval" title="Phase retrieval">Phase retrieval</a>

<ul><li><a href="Gerchberg%E2%80%93Saxton_algorithm" title="Gerchberg–Saxton algorithm">Gerchberg–Saxton</a></li></ul></li>

<li><a href="Single_particle_analysis" title="Single particle analysis">Single particle analysis</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Software</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="CrystalExplorer" title="CrystalExplorer">CrystalExplorer</a></li>

<li><a href="OctaDist" title="OctaDist">OctaDist</a></li>

<li><a href="ShelXle" title="ShelXle">SHELX</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;"><a href="Crystallographic_database" title="Crystallographic database">Databases</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bilbao_Crystallographic_Server" title="Bilbao Crystallographic Server">Bilbao Crystallographic Server</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Journals</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Crystal_Growth_%26_Design" title="Crystal Growth & Design">Crystal Growth & Design</a></li>

<li><a href="Crystallography_Reviews" title="Crystallography Reviews">Crystallography Reviews</a></li>

<li><a href="Journal_of_Chemical_Crystallography" title="Journal of Chemical Crystallography">Journal of Chemical Crystallography</a></li>

<li><a href="Journal_of_Crystal_Growth" title="Journal of Crystal Growth">Journal of Crystal Growth</a></li>

<li><a href="Kristallografija" title="Kristallografija">Kristallografija</a></li>

<li><a href="Zeitschrift_f%C3%BCr_Kristallographie_%E2%80%93_Crystalline_Materials" title="Zeitschrift für Kristallographie – Crystalline Materials">Zeitschrift für Kristallographie – Crystalline Materials</a></li>

<li><a href="Zeitschrift_f%C3%BCr_Kristallographie_%E2%80%93_New_Crystal_Structures" title="Zeitschrift für Kristallographie – New Crystal Structures">Zeitschrift für Kristallographie – New Crystal Structures</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Awards</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Carl_Hermann_Medal" title="Carl Hermann Medal">Carl Hermann Medal</a></li>

<li><a href="Ewald_Prize" title="Ewald Prize">Ewald Prize</a></li>

<li><a href="Gregori_Aminoff_Prize" title="Gregori Aminoff Prize">Gregori Aminoff Prize</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">History</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chemical_crystallography_before_X-rays" title="Chemical crystallography before X-rays">Chemical crystallography before X-rays</a></li>

<li><a href="Physical_crystallography_before_X-rays" title="Physical crystallography before X-rays">Physical crystallography before X-rays</a></li>

<li><a href="Timeline_of_crystallography" title="Timeline of crystallography">Timeline of crystallography</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Organisation</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="International_Organization_for_Biological_Crystallization" title="International Organization for Biological Crystallization">IOBCr</a></li>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Associations12" scope="row" class="navbox-group" style="width:1%">Associations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="European_Crystallographic_Association" title="European Crystallographic Association">Europe</a>

<ul><li><a href="French_Crystallographic_Association" title="French Crystallographic Association">France</a></li>

<li><a href="German_Crystallographic_Society" title="German Crystallographic Society">Germany</a></li>

<li><a href="Crystallographic_Society_of_Japan" title="Crystallographic Society of Japan">Japan</a></li></ul>

</div></td></tr></tbody></table><div>

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> <b>Category</b></li>

<li><span class="noviewer" typeof="mw:File"><span title="Commons page"></span></span> <b><a href="https://commons.wikimedia.org/wiki/Category:Crystallography" class="extiw external" title="commons:Category:Crystallography">Commons</a></b></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Protein_structural_analysis342" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="background:#eee8aa;"><div id="Protein_structural_analysis342" style="font-size:114%;margin:0 4em"><a href="Protein_structure" title="Protein structure">Protein structural analysis</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#eee8aa;">High resolution</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cryogenic_electron_microscopy" title="Cryogenic electron microscopy">Cryogenic electron microscopy</a></li>

<li><a href="X-ray_crystallography" title="X-ray crystallography">X-ray crystallography</a></li>

<li><a href="Electron_crystallography" title="Electron crystallography">Electron crystallography</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#eee8aa;">Medium resolution</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul>

<li><a href="Mass_spectrometry" title="Mass spectrometry">Mass spectrometry</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#eee8aa;">Spectroscopic</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Circular_dichroism" title="Circular dichroism">Circular dichroism</a></li>

<li><a href="Dual-polarization_interferometry" title="Dual-polarization interferometry">Dual-polarization interferometry</a></li>

<li><a href="Absorbance" title="Absorbance">Absorbance</a></li>

<li><a href="Fluorescence" title="Fluorescence">Fluorescence</a></li>

<li><a href="Fluorescence_anisotropy" title="Fluorescence anisotropy">Fluorescence anisotropy</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#eee8aa;">Translational diffusion</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ultracentrifuge#Analytical_ultracentrifuge" title="Ultracentrifuge">Analytical ultracentrifugation</a></li>

<li><a href="Size_exclusion_chromatography" class="mw-redirect" title="Size exclusion chromatography">Size exclusion chromatography</a></li>

<li><a href="Dynamic_light_scattering" title="Dynamic light scattering">Light scattering</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#eee8aa;">Rotational diffusion</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Fluorescence_anisotropy" title="Fluorescence anisotropy">Fluorescence anisotropy</a></li>

<li><a href="Flow_birefringence" title="Flow birefringence">Flow birefringence</a></li>

<li><a href="Dielectric_relaxation" class="mw-redirect" title="Dielectric relaxation">Dielectric relaxation</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#eee8aa;">Chemical</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hydrogen%E2%80%93deuterium_exchange" title="Hydrogen–deuterium exchange">Hydrogen–deuterium exchange</a></li>

<li><a href="Site-directed_mutagenesis" title="Site-directed mutagenesis">Site-directed mutagenesis</a></li>

<li><a href="Chemical_modification" title="Chemical modification">Chemical modification</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#eee8aa;">Thermodynamic</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Equilibrium_unfolding" title="Equilibrium unfolding">Equilibrium unfolding</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#eee8aa;">Computational</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Protein_structure_prediction" title="Protein structure prediction">Protein structure prediction</a></li>

<li><a href="Docking_(molecular)" title="Docking (molecular)">Molecular docking</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="background:#eee8aa;"><div><div style="float:left;"><a href="Protein_tertiary_structure" title="Protein tertiary structure">←Tertiary structure</a></div><div style="float:right;"><a href="Protein_quaternary_structure" title="Protein quaternary structure">Quaternary structure→</a></div></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-02" href="https://en.wikipedia.org/wiki/?title=Fiber_diffraction&oldid=1293534950">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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