When bonds point: the end of pure packing
The last four guides told one story over and over: when bonds do not care about direction, geometry wins. Metallic and ionic bonds pull equally every way, so atoms behaved like sticky balls and packed as tightly as balls allow — 0.74 for close packing, 0.68 for BCC — or, in the ionic structures, as spheres nestled in the holes of other spheres, refereed by the radius ratio and Pauling's rules. This guide is where that story breaks. A covalent bond is a specific pair of electrons shared along a specific line between two atoms, so it very much cares which way its neighbour lies.
That directionality has teeth. Carbon has four valence electrons and, to satisfy the octet rule, wants four shared pairs — and quantum mechanics arranges those four bonds (sp3 hybrids) to point at the corners of a tetrahedron, 109.5 degrees apart. So a carbon atom does not want as many neighbours as will fit; it wants exactly four, at exactly those angles. This is bond directionality, and it flips the whole optimisation. There is even a tidy rule of thumb for the covalent elements: an atom from group N of the periodic table forms 8 minus N bonds, so carbon and silicon (group 14) each make four. The coordination number is now set by counting electrons, not by counting how many balls fit.
Watch the reversal carefully, because it upends the intuition the last four guides built. In close-packed metals, more neighbours meant lower energy, so atoms crowded together to coordination 12. In a covalent solid the opposite holds: a fifth neighbour has no bond to offer and only gets in the way, so the crystal deliberately stays open and half-empty. The hard-sphere packing model that served us so well for metals simply does not apply here — the number of neighbours is dictated by chemistry (available electrons), not by geometry (available space).
Diamond cubic: an open FCC in disguise
Here is how carbon builds those tetrahedra into a crystal. Take the FCC arrangement you already know, then hang a second, identical carbon on every atom, offset by one quarter of the body diagonal — the vector (1/4, 1/4, 1/4). Equivalently, and more memorably from the last rung: fill exactly half of the tetrahedral holes of an FCC array with more of the same atom. Either way you get the diamond cubic structure, and every atom — those on the FCC sites and those in the holes — ends up bonded to four neighbours at the perfect tetrahedral angle. Count the contents and the conventional cube holds 8 atoms: the usual 4 from FCC plus 4 sitting in half the tetrahedral holes.
Now the payoff of all that directional fuss: the diamond structure is astonishingly empty. Its packing factor is just 0.34 — the balls fill barely a third of space, less than half of close packing's 0.74. And yet diamond is the hardest natural material known. That is the honest lesson worth carrying: hardness and stiffness come from the strength and rigidity of the bonds themselves, not from how tightly the atoms are packed. A loosely packed lattice of ferociously strong, angle-locked bonds beats a densely packed pile of soft, sliding ones. Silicon, germanium, and grey tin adopt this very same open structure — which is why the diamond cubic lattice quietly underpins the entire semiconductor industry.
THE TETRAHEDRAL (sp3) FAMILY: coordination 4, packing factor ~0.34
structure lattice + motif stacking examples
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diamond cubic FCC + 2 identical atoms ABCABC C, Si, Ge, grey Sn
zinc-blende FCC + 2 DIFFERENT atoms ABCABC GaAs, ZnS, CdTe
wurtzite HCP + 2 different atoms ABAB GaN, ZnO, w-ZnS
all three: each atom bonds to 4 neighbours at 109.5 degrees
contrast: close-packed metals reach coordination 12, packing 0.74
diamond fills only 0.34 of space -- directional bonds
beat dense packingTwo atoms in the tetrahedra: zinc-blende and wurtzite
What if the two interpenetrating sublattices are not the same element? Colour the FCC sites as one species and the tetrahedral-hole atoms as another, and diamond cubic becomes the zinc-blende (sphalerite) structure. Picture an FCC array of sulfur atoms with zinc filling half the tetrahedral holes: every zinc is tetrahedrally surrounded by four sulfurs and every sulfur by four zincs, coordination 4 and 4. This is the master structure of the compound semiconductors — gallium arsenide (GaAs), cadmium telluride, most of the III-V and II-VI families that make LEDs, laser diodes and solar cells.
Zinc-blende was diamond built on an ABCABC (cubic) stack of layers. Build the identical tetrahedra on an ABAB (hexagonal) stack instead and you get wurtzite — the tetrahedra are locally the same, but the long-range stacking is the two-layer beat rather than the three-layer one. So zinc-blende is to wurtzite exactly as FCC is to HCP: same neighbours, same bonds, different stacking period. It is the polytype idea from guide 1 wearing a chemical costume. Zinc oxide and gallium nitride (the blue-LED material) take wurtzite; zinc sulfide is so evenly balanced that it shows real polytypism, flipping between the two — and silicon carbide is the champion, with hundreds of stacking variants.
Graphite: mighty sheets, feeble seams
Carbon has a second trick. Instead of spending all four electrons on four tetrahedral bonds, it can put three into strong in-plane bonds at 120 degrees (sp2 hybrids) and leave the fourth to roam. Those three bonds tile a flat honeycomb of hexagons — a single such sheet is graphene — and stacking the sheets in an ABAB sequence gives the graphite structure. Within a sheet the carbon-carbon bonds are among the strongest in all of chemistry, shorter and stiffer even than diamond's. Between the sheets, though, there is no covalent bond at all.
What holds the sheets together is only the weak van der Waals force, the faint universal stickiness between all atoms, and it leaves the layers a full 0.335 nm apart — more than twice the 0.142 nm in-plane bond length. This lopsided architecture — iron-strong in two directions, cobweb-weak in the third — is the textbook case of structural anisotropy. It explains everything you know about graphite: it conducts electricity and heat superbly along the sheets but poorly across them, and it cleaves and smears because the sheets slide over one another almost for free. That sliding is why graphite is a solid lubricant and why a pencil leaves a trail — you are shearing off stacks of sheets.
One element, two worlds: allotropy and density from the cell
Step back and marvel: diamond and graphite are both pure carbon. Same atom, same chemistry — yet one is the hardest, most transparent, electrically insulating solid we have, and the other is soft, black, greasy and conducting. Every difference between them lives in the arrangement of the atoms, nothing else. This is polymorphism, and in a single element it is called allotropy; passing between two such forms is a polymorphic transition. A quiet surprise: at ordinary temperature and pressure it is graphite, not diamond, that is the stable form — diamond is only metastable, kinetically trapped by bonds too strong to rearrange on any human timescale. 'Diamonds are forever' is really a statement about a fantastically slow transition.
And structure alone fixes density, exactly as in guide 1. Feed the diamond cell into the same recipe — theoretical density equals the mass inside one cell divided by the cell's volume — and it lands right on reality, as the steps below show. The deeper point is the comparison: run the identical calculation for graphite and you get about 2.26 g/cm^3, roughly a third lighter than diamond's 3.51, with not a single atom changed. That whole 35 percent drop in density is bought purely by the open, wide-gapped layered stacking. Structure, and structure alone, wrote the number.
- Count atoms per cell. Diamond cubic has n = 8 (the 4 FCC atoms plus 4 in half the tetrahedral holes).
- Mass of one cell = n times atomic mass / Avogadro's number = (8 times 12.01 g/mol) / (6.022 times 10^23 per mol) = 1.60 times 10^-22 g.
- Cell volume from the edge a = 3.567 angstrom = 3.567 times 10^-8 cm, so V = a^3 = 4.54 times 10^-23 cm^3.
- Divide: 1.60 times 10^-22 g / 4.54 times 10^-23 cm^3 = 3.52 g/cm^3 — essentially exact on diamond's measured 3.51.