Fiber diffraction

Fiber diffraction is a method for studying the molecular structures by analyzing diffraction patterns. These images result from irradiation of the sample. For this purpose, usually X-rays, electrons or neutrons are used. The peculiarity of the optical fiber diffraction is that the spread pattern does not change when the sample to a specific axis ( fiber axis) is rotated. Such uniaxial symmetry typical for filaments or fibers of biological or synthetic macromolecules ( polymers, plastics ). For crystallography is fiber symmetry a difficulty in the determination of the crystal structure dar. Compared with the diffraction pattern of crystal reflexes are smeared in the fiber diagram may be superimposed on each other. The material science considers fiber symmetry as a simplification, because almost the entire accessible structural information is contained in a single two-dimensional (2D) diffraction pattern. Thus, an image is exposed onto photographic film or a 2D detector (such as a digital camera ). 2 instead of 3 coordinate axes are sufficient for the description of fiber diffraction.

The ideal fiber scattering image shows 4- quadrant symmetry. In such an image is, the direction of the fiber axis meridian, the direction thereof is perpendicular to the equator. Fiber symmetry prevails, then show up in the 2D image much more reflections ( lit "points" ) at the same time as in the diffraction pattern of the single crystal. These reflexes appear obviously on lines ( layer lines) arranged which are approximately parallel to the equator. Thus, the layer line concept crystallography is apparent in the fiber diffraction pattern. The bending of the layer lines is an indication that the diffraction pattern must be equalized. Reflexes are identified by Miller indices hkl. These are 3 digits. Reflections on the i- th layer line have l = i Reflections on the meridian are 00l reflections. Artificial fiber diffraction images are also produced in crystallography ( crystal rotation method). Thereby a single crystal is rotated about an axis in the X-ray beam.

Reliable fiber scattering images obtained in the experiment. They only show mirror symmetry, because the fiber axis is not oriented perfectly perpendicular to the incident beam. The corresponding geometric distortion has been studied in detail by Michael Polanyi. To describe the geometry he has the elegant concept of Polanyi sphere (originally: "Located ball " ) was introduced. Later, Rosalind Franklin and Raymond Gosling have an approximate formula for the determination of β Faserkippwinkels specified on the basis of its own geometrical considerations. In the first analysis step, the fiber scattering image is equalized and mapped to the representative fiber plane. This is the plane containing the cylinder axis of the reciprocal space. In crystallography, an approximation of the figure in the reciprocal space is computed which is iteratively refined. This is often referred to as the Fraser - correction digital method starts with the Franklin approximation. It eliminates the tilt, straighten the image and corrects the scattering intensity. The correct formula for the determination of β is given by Norbert Stribeck.

Historical role

The fiber diffraction led to several important advances in the development of structural biology, such as the first models of the α - helix and the Watson -Crick model of double-stranded DNA.

Geometry of the fiber diffraction

The geometry of the fiber diffraction shows the picture. It is based on the representation proposed by Polanyi. The reference direction is the primary beam (labeled X -ray). The fiber is tilted from the vertical by the angle β, then tilts also the information about the structure in reciprocal space (S region). S in the zone, the Ewald sphere is a sphere whose center is located in the sample. Its radius is 1 / λ, where λ is the wavelength of the incident radiation. On the surface of the Ewald sphere are all the points of the reciprocal space, which can be seen from the planar detector. They are represented by the central projection on the pixels of the detector.

In reciprocal space, each reflection is on his Polanyi sphere. Actually, the ideal reflecting a point in the space S due to the fiber symmetry dar. but it smeared to form a ring around the fiber direction. Two rings represent a reflection on the Polanyi sphere, because scattering is point symmetric about the origin of the reciprocal space. Imaged on the detector, only the points that lie on both the Ewald sphere and on the Polanyi sphere. These points form the reflection circle ( blue ring ). Also, when tilting the sample it does not change. He is like a slide projector ( red rays ) on the detector shown ( detector circle, blue ring). There appear up to 4 images of the observed reflex ( red dots). The position of the reflected images is determined by the orientation of the fiber in the beam is determined ( Polanyigleichung ). Conversely, from the position of the reflected images, the orientation of the fiber are determined when applies to the Miller indices hkl: and. From Polanyi representation is passed from the relations of the fiber image by application of elementary and spherical geometry.

Correction of the scatter image

The figure on the left shows a typical fiber Faithful Picture of polypropylene before transformation into reciprocal space. The mirror axis of the scatter image is rotated by the angle from the vertical. This deficit is compensated for by simple rotation of the image. 4 straight arrows point to four reflection images of the reference reflex. Your documents will be used to determine the Faserkippwinkels. The picture was taken with a CCD detector. It shows the logarithmic intensities in a false color representation. Bright colors denote high intensity here.

After the determination of the distance between the sample and the detector is calculated from the known crystallographic data of the reference reflection created a uniformly screened card the fiber plane of the reciprocal space and the registered data of the diffraction pattern in this card. The figure on the right shows the result. Due to the equalization also the scattering intensities change. The curvature of the Ewald sphere surface remain at the meridian white areas that lack structural information. Only in the center of the image and an s- value of the part of the scattering angle, there are structural information on the meridian. In principle, the picture shows now 4 -quadrant symmetry. In the example, could be a part of lack of information " from the bottom to the top" are copied in the white areas. So it makes sense often to tilt the fiber aware.

The three-dimensional image shows that the obtained In the example experiment, information about the molecular structure of the polypropylene fiber is almost completely. By rotation of the plane scattered image in the s- space around the Meridian fill measured in 4 s fiber scattering data a nearly spherical volume of the s- space. The 4-quadrant symmetry has not yet been called to fill white spots approach in the example. For the demonstration of the ball at the front quarter has been cut, but with the equatorial plane itself is not removed.