Kikuchi lines (physics)

Kikuchi lines are patterns of electrons formed by scattering. They pair up to form bands in electron diffraction from single crystal specimens, there to serve as "roads in orientation-space" for microscopists uncertain of what they are looking at. In transmission electron microscopes, they are easily seen in diffraction from regions of the specimen thick enough for multiple scattering. Unlike diffraction spots, which blink on and off as one tilts the crystal, Kikuchi bands mark orientation space with well-defined intersections (called zones or poles) as well as paths connecting one intersection to the next. Experimental and theoretical maps of Kikuchi band geometry, as well as their direct-space analogs e.g. bend contours, electron channeling patterns, and fringe visibility maps are increasingly useful tools in electron microscopy of crystalline and nanocrystalline materials. Because each Kikuchi line is associated with Bragg diffraction from one side of a single set of lattice planes, these lines can be labeled with the same Miller or reciprocal-lattice indices that are used to identify individual diffraction spots. Kikuchi band intersections, or zones, on the other hand are indexed with direct-lattice indices i.e. indices which represent integer multiples of the lattice basis vectors a, b and c. Kikuchi lines are formed in diffraction patterns by diffusely scattered electrons, e.g. as a result of thermal atom vibrations. The main features of their geometry can be deduced from a simple elastic mechanism proposed in 1928 by Seishi Kikuchi, although the dynamical theory of diffuse inelastic scattering is needed to understand them quantitatively. In x-ray scattering, these lines are referred to as Kossel lines (named after Walther Kossel).

Recording experimental Kikuchi patterns and maps

The figure on the left shows the Kikuchi lines leading to a silicon [100] zone, taken with the beam direction approximately 7.9° away from the zone along the (004) Kikuchi band. The dynamic range in the image is so large that only portions of the film are not overexposed. Kikuchi lines are much easier to follow with dark-adapted eyes on a fluorescent screen, than they are to capture unmoving on paper or film, even though eyes and photographic media both have a roughly logarithmic response to illumination intensity. Fully quantitative work on such diffraction features is therefore assisted by the large linear dynamic range of CCD detectors. This image subtends an angular range of over 10° and required use of a shorter than usual camera length L. The Kikuchi band widths themselves (roughly λL/d where λ/d is approximately twice the Bragg angle for the corresponding plane) are well under 1°, because the wavelength λ of electrons (about 1.97 picometres in this case) is much less than the lattice plane d-spacing itself. For comparison, the d-spacing for silicon (022) is about 192 picometres while the d-spacing for silicon (004) is about 136 picometres. The image was taken from a region of the crystal which is thicker than the inelastic mean free path (about 200 nanometres), so that diffuse scattering features (the Kikuchi lines) would be strong in comparison to coherent scattering features (diffraction spots). The fact that surviving diffraction spots appear as disks intersected by bright Kikuchi lines means that the diffraction pattern was taken with a convergent electron beam. In practice, Kikuchi lines are easily seen in thick regions of either selected area or convergent beam electron diffraction patterns, but difficult to see in diffraction from crystals much less than 100 nm in size (where lattice-fringe visibility effects become important instead). This image was recorded in convergent beam, because that too reduces the range of contrasts that have to be recorded on film. Compiling Kikuchi maps which cover more than a steradian requires that one take many images at tilts changed only incrementally (e.g. by 2° in each direction). This can be tedious work, but may be useful when investigating a crystal with unknown structure as it can clearly unveil the lattice symmetry in three dimensions.

Kikuchi line maps and their stereographic projection

The figure on the left plots Kikuchi lines for a larger section of silicon's orientation space. The angle subtended between the large [011] and [001] zones at the bottom is 45° for silicon. Note that four-fold zone in the lower right (here labeled [001]) has the same symmetry and orientation as the zone labeled [100] in the experimental pattern above, although that experimental pattern only subtends about 10°. Note also that the figure at left is excerpted from a stereographic projection centered on that [001] zone. Such conformal projections allow one to map pieces of spherical surface onto a plane while preserving the local angles of intersection, and hence zone symmetries. Plotting such maps requires that one be able to draw arcs of circles with a very large radius of curvature. The figure at left, for example, was drawn before the advent of computers and hence required the use of a beam compass. Finding a beam compass today might be fairly difficult, since it is much easier to draw curves having a large radius of curvature (in two or three dimensions) with help from a computer. The angle-preserving effect of stereographic plots is even more obvious in the figure at right, which subtends a full 180° of the orientation space of a face-centered or cubic close packed crystal e.g. like that of Gold or Aluminum. The animation follows {220} fringe-visibility bands of that face-centered cubic crystal between <111> zones, at which point rotation by 60° sets up travel to the next <111> zone via a repeat of the original sequence. Fringe-visibility bands have the same global geometry as do Kikuchi bands, but for thin specimens their width is proportional (rather than inversely proportional) to d-spacing. Although the angular field width (and tilt range) obtainable experimentally with Kikuchi bands is generally much smaller, the animation offers a wide-angle view of how Kikuchi bands help informed crystallographers find their way between landmarks in the orientation space of a single crystal specimen.

Real space analogs

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