page_title: Kikuchi lines (physics)
section_title: Bend contours and rocking curves
hierarchical_section_title: Kikuchi lines (physics) / Real space analogs / Bend contours and rocking curves
caption_attribution_description: Thickness-normalized plot of bend contour amplitude, and lattice fringe visibility, as a function of tilt from edge-on. This particular plot is a purely kinematic simulation for contrast from lattice planes of spacing d=0.25 nanometres in spherical particles of diameter t with electrons of energy 300 keV and wavelength λ near 1.9 picometres.
context_page_description: 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 not certain 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 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.
context_section_description: Rocking curves (left) are plots of scattered electron intensity, as a function of the angle between an incident electron beam and the normal to a set of lattice planes in the specimen. As this angle changes in either direction from edge-on (at which orientation the electron beam runs parallel to the lattice planes and perpendicular to their normal), the beam moves into Bragg diffracting condition and more electrons are diffracted outside the microscope's back focal plane aperture, giving rise to the dark-line pairs (bands) seen in the image of the bent silicon foil shown in the image at right.
  The [100] bend contour "spider" of this image, trapped in a region of silicon that was shaped like an oval watchglass less than a micrometre in size, was imaged with 300 keV electrons. If you tilt the crystal, the spider moves toward the edges of the oval as though it is trying to get out. For example, in this image the spider's [100] intersection has moved to the right side of the ellipse as the specimen was tilted to the left.
  The spider's legs, and their intersections, can be indexed as shown in precisely the same way as the Kikuchi pattern near [100] in the section on experimental Kikuchi patterns above. In principle, one could therefore use this bend contour to model the foil's vector tilt (with milliradian accuracy) at all points across the oval.
caption: Bend contour and lattice fringe visibility as a function of specimen thickness and beam tilt
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