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Nano-diamonds have been observed in only a handful of circumstellar regions$10-100$\,\tiny{ A.U.} from moderately bright stars ($T_{\rm eff} \sim8,000-10,000$\,K). They have also been extracted from primitive meteorites;some of these are clearly pre-solar, that is to say that they formed far fromthe solar system and therefore traversed the interstellar medium, where theymust exist but, because we see no evidence of them, must be extremely wellhidden. Our goal is to understand if it is possible to constrain the sizes andshapes of nano-diamonds in circumstellar media using the observed ratio,[CH]/[CH$_2$], of their surface CH$_2$ and CH infrared bands at $\simeq3.43\,\mu$m and $\simeq 3.53\,\mu$m, respectively. We calculated the CH andCH$_2$ abundances on nano-diamonds using two approaches. The first assumesregular and semi-regular polyhedra (tetrahedra, octahedra, and cubes and theirtruncated forms). The second uses a diamond bonding network to derive thestructures of tetrahedral and octahedral particles, and their truncatedvariants, and also of spherical nano-diamonds. As a function of the particlesize and shape, and for the two different calculation methods, we derived therelative abundance ratio [CH]/[CH$_2$], which can then be weighted by theirlaboratory-measured infrared band intensities. The two methods give goodagreement and indicate that the spread in values, over the different particleforms, is more that an order of magnitude for any size. We conclude that theratio [CH]/[CH$_2$], and their infrared band ratio, strongly depend uponparticle size and shape. For a given shape or size, the ratio can vary by morethan an order of magnitude. It may therefore be difficult to constrainnano-diamond sizes using the observed $3-4\,\mu$m spectra alone. James WebbSpace Telescope (JWST) mid-infrared spectra may help, but only if bands aresize-specific.

Edging towards an understanding of CH/CH$_2$ on nano-diamonds: Regular and semi-regular polyhedra and diamond network models