Edging towards an understanding of CH/CH$_2$ on nano-diamonds: Regular
  and semi-regular polyhedra and diamond network models
  Nano-diamonds have been observed in only a handful of circumstellar regions
$10-100$\,\tiny{ A.U.} from moderately bright stars ($T_{\rm eff} \sim
8,000-10,000$\,K). They have also been extracted from primitive meteorites;
some of these are clearly pre-solar, that is to say that they formed far from
the solar system and therefore traversed the interstellar medium, where they
must exist but, because we see no evidence of them, must be extremely well
hidden. Our goal is to understand if it is possible to constrain the sizes and
shapes of nano-diamonds in circumstellar media using the observed ratio,
[CH]/[CH$_2$], of their surface CH$_2$ and CH infrared bands at $\simeq
3.43\,\mu$m and $\simeq 3.53\,\mu$m, respectively. We calculated the CH and
CH$_2$ abundances on nano-diamonds using two approaches. The first assumes
regular and semi-regular polyhedra (tetrahedra, octahedra, and cubes and their
truncated forms). The second uses a diamond bonding network to derive the
structures of tetrahedral and octahedral particles, and their truncated
variants, and also of spherical nano-diamonds. As a function of the particle
size and shape, and for the two different calculation methods, we derived the
relative abundance ratio [CH]/[CH$_2$], which can then be weighted by their
laboratory-measured infrared band intensities. The two methods give good
agreement and indicate that the spread in values, over the different particle
forms, is more that an order of magnitude for any size. We conclude that the
ratio [CH]/[CH$_2$], and their infrared band ratio, strongly depend upon
particle size and shape. For a given shape or size, the ratio can vary by more
than an order of magnitude. It may therefore be difficult to constrain
nano-diamond sizes using the observed $3-4\,\mu$m spectra alone. James Webb
Space Telescope (JWST) mid-infrared spectra may help, but only if bands are
size-specific.
