Physics of Solids

Solids divide into two kinds by how their particles are arranged. In a crystalline solid the atoms sit in a repeating three-dimensional pattern — a lattice — that continues in every direction. Metals, salt and diamond are crystalline.

In an amorphous solid there is no long-range order; the particles are arranged much as they were in the liquid, frozen in place. Glass, rubber and most plastics are amorphous.

The difference shows up in melting. A crystalline solid has one sharp melting point, because every bond is the same and they all break at the same temperature. An amorphous solid softens gradually over a range, which is precisely what lets glass be blown and shaped.

Most metals are polycrystalline: made of many small crystals, called grains, joined at boundaries in random orientations. Those grain boundaries matter — they get in the way of deformation, which is why a fine-grained metal is harder than a coarse-grained one.

Load a wire and it stretches. To compare materials rather than particular samples, the load is divided by the cross-sectional area to give stress, and the extension divided by the original length to give strain.

For small loads the two are proportional — Hooke's law — and the constant of proportionality is the Young modulus. It measures stiffness, and it belongs to the material: every steel wire has the same Young modulus regardless of its length or thickness.

Stress has the units of pressure, pascals. Strain has no units at all, being a length divided by a length. So the Young modulus is in pascals too, and for metals it runs to hundreds of gigapascals.

σ

stressPa

ε

strain

E

Young modulusPa

e

extensionm

L

original lengthm

The gradient of the straight section is the Young modulus. Switch to glass and the plastic region vanishes entirely — it stays on the straight line until it shatters, which is what brittle means.

Below the elastic limit, deformation is elastic: remove the load and the material springs back exactly. The atoms have been pulled slightly further apart but every one has stayed with its neighbours.

Past the elastic limit, deformation becomes plastic and is permanent. Whole planes of atoms have slipped over one another, and removing the load does not slide them back. This is what lets a metal be hammered into shape, drawn into wire, or bent and left bent.

A material with a long plastic region before breaking is ductile — copper is the standard example, which is why it becomes wire. One that breaks with almost no plastic region is brittle: glass, ceramics and cast iron all snap while still obeying Hooke's law.

Brittle is not the same as weak. Glass fibre is extremely strong in tension; it simply gives no warning before it fails, which is why brittle materials are avoided where a sudden failure would be dangerous.

In an isolated atom electrons occupy sharply defined energy levels. Bring 10²³ atoms together into a solid and those levels spread into broad bands of allowed energies, separated by gaps where no electron may sit.

Two bands matter. The valence band holds the bound electrons; the conduction band above it holds electrons free to move through the material. Whether a solid conducts comes down to the gap between them.

In a conductor the bands overlap, or the upper band is only part-filled. Electrons can move into free states with almost no energy, so current flows readily.

In an insulator the gap is large — several electronvolts. At ordinary temperatures essentially no electron has enough energy to cross it, so no current flows.

A semiconductor has a small gap, around one electronvolt. At absolute zero it is an insulator, but at room temperature a few electrons have enough thermal energy to jump the gap. Heat it and more make it across — which is exactly why a semiconductor's resistance falls as it warms, while a metal's rises.