An fcc gold icosahedral-twin butterfly, one fifth (not counting the icosahedral core) of a twenty-face tetrahedral-crystal cluster that shares one or more sets of crystallographic planes. This part of the icosahedral twin therefore "lights up as one" in optical and digital darkfield images of the cluster taken under the right conditions. It reminds me of Darth Vader's twin ion engine (TIE) fighter in the original episode of StarWars, even though nature has been manufacturing these things since well before George Lucas was born. In any case, what planes do all four of the outlying tetrahedra share in common? Is the interatom spacing correct for gold atoms, and if not how should it be changed? Also, can you tell if the lattice between adjacent tetrahedra is feeling compressive or tensile strain, and if either how much?
Below to the left see how bowties (top) and butterflies (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?
Here 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 how the non-intersection area of two crossed visibility bands reduces to a group of four spherical caps as the band half-width exceeds Pi/4 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 2nm (narrow bands), through 1.5nm, down to 1nm (widest bands).
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.