Fiber Diffraction - Fiber Diffraction Geometry

Fiber Diffraction Geometry

The animation shows the geometry of fiber diffraction. It is based on the notions proposed by Polanyi. 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 s-space) is tilted. In reciprocal space the so-called Ewald sphere has its center in the sample. Its radius is 1/λ, with λ the wavelength of the incident radiation. On the surface of the Ewald sphere 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.

In s-space each reflexion is found on its Polanyi-sphere. Intrinsically the ideal reflexion is a point in s-space, but fiber symmetry turns it into a ring smeared out by rotation about the fiber direction. Two rings represent each reflexion on the Polanyi sphere, because scattering is point symmetric with respect to the origin of s-space. Mapped onto the detector are only those points of the reflexion in s-space that are both on the Ewald sphere and on the Polanyi sphere. These points form the reflexion circle (blue ring). It does not change as the fiber is tilted. As with a slide projector the reflexion circle is projected (red moving rays) on the detector (detector circle, blue ring). There up to 4 images (red spots) of the monitored reflexion can show up. The position of the reflexion images is a function of the orientation of the fiber in the primary beam (Polanyi equation). Inverted, from the positions of the reflexion images the orientation of the fiber can be determined, if for the Miller index both and is valid. From the Polanyi representation of fiber diffraction geometry the relations of the fiber mapping are established by elementary and spherical geometry.