The Solid State

A solid has a fixed shape and volume because its particles are locked in place, vibrating about fixed points. But not all solids are built the same. A crystalline solid has long-range order: its particles repeat in a regular, three-dimensional pattern that extends across the whole crystal. An amorphous solid (from Greek a-morphē, "no form") has only short-range order — orderly in small patches but disordered overall, like a frozen snapshot of a liquid.

Crystalline solids are sorted by the kind of particle at the lattice points and the force holding them together. Four classes cover almost everything, and the binding force predicts every bulk property — hardness, melting point and conductivity.

A crystal lattice is the regular three-dimensional array of points that represents the repeating pattern of a crystal. The smallest repeating block that, stacked in three dimensions, reproduces the whole lattice is the unit cell. A unit cell is defined by three edge lengths \(a,\,b,\,c\) and three angles \(\alpha,\,\beta,\,\gamma\) between them.

By choosing different combinations of edges and angles, only seven distinct shapes are possible — the seven crystal systems. Adding face-, body- and end-centring gives the 14 Bravais lattices.

The three cubic cells dominate the syllabus. To count how many atoms a cell "owns", remember that atoms are shared with neighbouring cells. The share depends on position:

To pack spheres as tightly as possible, lay down a hexagonal first layer (A), nest the second layer (B) into its hollows, then choose where the third goes. Stacking ABAB… gives hexagonal close packing (hcp); stacking ABCABC… gives cubic close packing (ccp), which is identical to the face-centred cubic lattice. Both reach a coordination number of 12 and the same packing efficiency.

Wherever spheres meet, empty pockets — voids — remain. A pocket touched by four spheres is a tetrahedral void; one touched by six spheres is an octahedral void. For \(N\) close-packed spheres there are exactly \(N\) octahedral voids and \(2N\) tetrahedral voids. Which voids the smaller ions occupy fixes the structure of an ionic compound.

Packing efficiency is the fraction of a unit cell's volume actually filled by spheres. It rises as the coordination number rises — close packing wastes the least space.

Because a unit cell's mass and volume are both known once we know the structure, a crystal's density links directly to atomic mass and edge length. This single formula lets you find Avogadro's number, the type of cell, or an unknown atomic mass from X-ray data.

For an ionic crystal, the ratio of the cation radius to the anion radius decides which void the cation fits into, and therefore its coordination number. A cation must be just large enough to touch its surrounding anions without rattling — that lower limit is the radius ratio rule.

No real crystal is perfect. Above absolute zero, thermodynamics demands some disorder. Point defects are irregularities around a single lattice point. Stoichiometric defects keep the formula intact; non-stoichiometric defects change the ratio of ions.

The same lattice that fixes a solid's shape also fixes how it carries current and responds to a magnet. Band theory sorts solids by the gap between the filled valence band and the empty conduction band: no gap → conductor, small gap → semiconductor, large gap → insulator.

A pure intrinsic semiconductor like silicon conducts feebly. Doping transforms it: adding a group-15 element (P, As) donates extra electrons to give an n-type semiconductor; adding a group-13 element (B, Al) creates electron "holes" to give a p-type semiconductor. Joining the two builds the p–n junction at the heart of every diode and transistor.

For each item, first decide which idea it tests — structure, packing, density, or defects — then apply the relevant rule. Difficulty rises down the list.