On the surface of the Ewald sphere all the points of reciprocal space are found that are seen by the detector. Its radius is 1/λ, with λ the wavelength of the incident radiation. In reciprocal space the Ewald sphere has its center in the sample. 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 s-space) is tilted. Reference direction is the primary beam (label: X-ray). It is based on the notions proposed by Polanyi. The animation shows the geometry of fiber diffraction. In the animation 1 Polanyi sphere with 1 reflection on it is monitored Structure information is in reciprocal space (black axes), expanded on surfaces of Polanyi spheres. Fiber diffraction data led to several important advances in the development of structural biology, e.g., the original models of the α-helix and the Watson-Crick model of double-stranded DNA.įiber diffraction geometry Fiber diffraction geometry changes as the fiber is tilted (tilt-angle β is between the blue rigid axis and the axis labelled s-space). The correct equation for the determination of β has been presented by Norbert Stribeck.įibrous materials such as wool or cotton easily form aligned bundles, and were among the first biological macromolecules studied by X-ray diffraction, notably by William Astbury in the early 1930s. It eliminates fiber tilt, unwarps the detector image, and corrects the scattering intensity. The digital method frequently called Fraser correction starts from the Franklin approximation for the tilt angle β. In crystallography first an approximation of the mapping into reciprocal space is computed that is refined iteratively. This is the plane that contains the cylinder axis in reciprocal space. Analysis starts by mapping the distorted 2D pattern on the representative plane of the fiber. Later Rosalind Franklin and Raymond Gosling have carried out their own geometrical reasoning and presented an approximative equation for the fiber tilt angle β. The corresponding geometric distortion has been extensively studied by Michael Polanyi introducing the concept of Polanyi's sphere (German: "Lagenkugel") intersecting Ewald's sphere. The reason is that the fiber axis and the incident beam (X-rays, electrons, neutrons) cannot be perfectly oriented perpendicular to each other. They only show mirror symmetry about the meridian. Non-ideal fiber patterns are obtained in experiments. In crystallography artificial fiber diffraction patterns are generated by rotating a single crystal about an axis ( rotating crystal method). Reflexions on the meridian are 00l-reflexions. Reflexions on the i-th layer line share l= i. Reflexions are labelled by the Miller index hkl, i.e. Bent layer lines indicate that the pattern must be straightened. Thus, in fiber diffraction the layer line concept of crystallography becomes palpable. In fiber patterns these reflexions clearly appear arranged along lines ( layer lines) running almost parallel to the equator. In case of fiber symmetry, many more reflexions than in single-crystal diffraction show up in the 2D pattern. In the ideal pattern the fiber axis is called the meridian, the perpendicular direction is called equator. The ideal fiber pattern exhibits 4-quadrant symmetry. High intensity is represented by dark color. Ideal fiber diffraction pattern of a semi-crystalline material with amorphous halo and reflexions on layer lines. 2 instead of 3 co-ordinate directions suffice to describe fiber diffraction. Materials science considers fiber symmetry a simplification, because almost the complete obtainable structure information is in a single two-dimensional (2D) diffraction pattern exposed on photographic film or on a 2D detector. In crystallography fiber symmetry is an aggravation regarding the determination of crystal structure, because reflexions are smeared and may overlap in the fiber diffraction pattern. Such uniaxial symmetry is frequent with filaments or fibers consisting of biological or man-made macromolecules. In fiber diffraction the scattering pattern does not change, as the sample is rotated about a unique axis (the fiber axis). Subarea of scattering, an area in which molecular structure is determined from scattering dataįiber diffraction is a subarea of scattering, an area in which molecular structure is determined from scattering data (usually of X-rays, electrons or neutrons).
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