Below find several symmetric orientations for this icosahedral twin, with the bonds removed to make atom periodicities more visible. Can you find these orientations using the model above? Which ones have 5-fold symmetry? Three-fold?
Below to the left see how bowtie crystals (top) and butterfly bicrystals (bottom) look in digital darkfield images of model icosahedral clusters. How many butterflies can you find in the 1534 tiny experimental darkfield images shown at this URL? The images on the left are phase gradient images used to track isotropic and shear components of lattice strain, detectable experimentally in electron microscope images at the picometer level.
Below are some drawings put together for assembling the fringe visibility map of an icosahedral twin. Note that in this five-fold orientation, these crystals are being viewed from the face-centered cubic (fcc) perspective of a [110] zone. Ten bowties will be needed to form the complete structure.
The figure below illustrates Jon Bailey's observation that the non-intersection area of two crossed visibility bands reduces to a group of four spherical caps as the band half-width exceeds π/4 radians or 45 degrees.
This figure shows how equal-width visibility bands with the symmetry of a face-centered cubic crystal begin to cover the sphere, in this case as band half-angle moves beyond 1.6 radians.
The image below does something similar, except that in this case band widths are linked to particle diameter, which ranges from 3nm (narrow bands) down to 1nm (widest bands). Non-band areas (orientations at which no fringes show up in the image) are colored grey. Note that all orientations show fringes when the crystal is only 1 nm in diameter.
Below find the visibility band map for icosahedral twins with crystallite thickness in the 5 nm (wide band) to 20 nm (narrow band) range. All ten fcc crystals are included, with (111) and (200) spacings (i.e. spacings in metals typically larger than 2 Angstroms) in bright white and (220) spacings in grey. Bands from one of the ten fcc crystals are colored yellow instead of white. The figure at right shows the interesting pattern of gaps between bands if one's microscope can't image the smaller (220) spacings. In either case, it's easy to see that the fraction of "non-band orientations" (i.e black orientations which won't show a bowtie or butterfly in digital darkfield) is very small even for particles larger than 10 nm in size, i.e. on the large end of the "visibly active quantum dot" size range.
If your microscope can reliably detect (220) periodicities as well, as shown in the visibility map for 10nm icotwins below at right, the fraction of particles not recognizable from digital darkfield analysis (i.e. the fractional area of non-band patches) will go lower still. A quick estimate of the areas of the 6*20 six-fold spot arrays, and the 10*12 ten-fold spot arrays, suggests that in such a microscope more than 98 percent of all randomly-oriented 10nm icosahedral twins will show butterfly or bowtie signatures on first encounter. Smaller particles are even more likely to reveal their icosahedral nature.
Oops. There are possible band thickness errors in the red-band figures above. Below find the visibility map for a 20 nm icotwin with corrected bandwidths. The (220) bands are now in dark grey. It's clear here that even 200 Angstrom clusters will for the most part be recognizable under digital darkfield analysis, with a microscope capable of delivering 1.4A metal periodicities reliably.
The fringe visibility map above is available in interactive form here, although it will likely slow down even the fastest of computers. By zooming in on that model, you can see that some of the (111) bands almost, but not quite, overlap in pairs (as with the bands running vertically through the center of the image below). Some of the (220) bands (dark grey) similarly run together in groups of five (as with the curved grey bands running across the top of the image below). In an unrelaxed (non-fcc) MacKay cluster, the overlap would be exact. Non-band regions in the figure below are highlighted in magenta.
The area of magenta non-band regions for 10nm thick crystals, below left, is significantly smaller than it is in the plot for 20nm thick crystals above. Even the largest non-band area there is almost gone in the 6nm crystal-thickness map section at right. Thus fcc icotwin particles with crystallites thinner than 5nm will show lattice fringes in any orientation. Digital darkfield amplitude maps of these lattice fringe footprints can in most cases be used to unambiguously confirm or deny the particle's icosahedral nature.
The figure below homes in on the triangular asymmetric unit in the orientation space of an fcc icosahedral twin, and labels some of the key structures. Ten asymmetric units per 5-fold zone, times twelve 5-fold zones, means that the orientation sphere is completely tiled with 120 of the triangles outlined in the lower right half of the figure below. What do you estimate is the fraction of that triangle's area covered by one of its fourteen green patches? Similarly, what fraction of the orientation area is crossed by two or more bands (thus offering more than one "digital darkfield fingerprints" of the particle's icosahedral form)?
