Here's a model of two interleaved face centered cubic (fcc) carbon lattices, each with a respectable metallic inter-atom distance of 2.52 Å and a kissing (nearest-neighbor) number of 12. This kissing number and bond distance is probably also typical of carbon atoms in the melt, although in that case the symmetries may be more icosahedral than face centered cubic.
However, because carbon atoms in tetrahedral coordination (with a kissing number of only 4) will tolerate the much shorter interatom distance of about 1.54 Å (still larger than the 1.42 Å carbon-carbon distance in a graphene sheet), careful placement of interstitial carbon atoms in an fcc lattice allows one to put twice as many atoms into almost exactly the same space! Of course sneaking twice as many carbon atoms into that original fcc lattice is not so easy, which might be one reason why large diamonds are not easy to make.
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