The Structure of Solids

The same bonding mechanisms that hold a diatomic molecule together — ionic, covalent, metallic, and van der Waals — bind atoms into solids, now on a scale of Avogadro's number of atoms. When a liquid is cooled slowly its molecules arrange into a regular array that maximizes the number of bonds and minimizes the potential energy: a crystal. Cooled too fast, they freeze in a disordered snapshot: an amorphous solid such as glass.

The smallest repeating structure is the unit cell. Its geometry depends on the bonding and, when more than one kind of atom is present, on their relative sizes.

Cubic lattices

The three cubic unit cells differ only in where atoms sit relative to the corners of a cube.

The three cubic unit cells: simple cubic (atoms at corners only), body-centered cubic with one atom at the cube center, and face-centered cubic with an atom at the center of each face.

Sodium chloride crystallizes in the face-centered-cubic (fcc) structure: each $Na^{+}$ has six $Cl^{-}$ nearest neighbors and vice versa, and the ions are not paired into molecules. Cesium chloride adopts a different cubic arrangement with eight nearest neighbors. The structure is whatever minimizes the total potential energy given the ion sizes.

A face of the NaCl crystal: sodium and chlorine ions alternate on a cubic grid, each ion surrounded by nearest neighbors of the opposite charge.

The Madelung constant

The attractive Coulomb energy of one ion in the crystal is not simply $- k e^{2} / r$ , because that ion feels every other ion. Writing the net attractive energy as

U_{att} = - α \frac{k e ^{2}}{r},

the dimensionless Madelung constant $α$ collects the geometry. Each ion has 6 opposite-charge nearest neighbors at distance $r$ , then 12 like-charge ions at $2 r$ , then 8 opposite at $3 r$ , and so on, so the naive sum is

α = ? 6 - \frac{12}{2} + \frac{8}{3} - \frac{6}{4} + \dots

An ion in the NaCl lattice sees successive shells of neighbors: 6 opposite charges at r, 12 like charges at √2 r, 8 opposite at √3 r; the alternating pull and push is what the Madelung constant sums.

This alternating series does not converge. The resolution is physical: real crystals are electrically neutral in compact regions, so the sum must be organized into neutral cubic shells rather than spherical ones. Done that way it converges to $α = 1.7476$ for NaCl (and the fcc lattices LiBr, KCl, RbF, all share it), while CsCl gives $α = 1.7627$ .

Adding the exclusion-principle repulsion $A / r^{n}$ and minimizing, the total potential energy per ion pair at equilibrium is

U (r_{0}) = - α \frac{k e ^{2}}{r _{0}} (1 - \frac{1}{n}) .

The exponent $n$ is found from the measured dissociation energy. For NaCl the lattice dissociation energy is $770 kJ/mol = 7.98 eV$ per ion pair; with $r_{0} = 0.282 nm$ and $α = 1.75$ , solving the equation gives $n \approx 9$ .

Worked example — spacing of NaCl from its density. Treating each ion as occupying a cube of side $r_{0}$ , the mass of one mole ( $58.4 g$ ) fills a volume $2 N_{A} r_{0}^{3}$ . From the density $ρ = 2.16 g/cm^{3}$ ,

r_{0}^{3} = \frac{m}{2 N _{A} ρ} = \frac{58.4 g}{2 ( 6.02 \times 1 0 ^{23} ) ( 2.16 g/cm ^{3} )} = 2.24 \times 1 0^{- 23} cm^{3},

so $r_{0} = 2.82 \times 1 0^{- 8} cm = 0.282 nm$ , matching X-ray diffraction.

Cohesive energy

The dissociation energy is measured per ion pair; the cohesive energy is the same well depth expressed per atom pair, which is the quantity comparable across all bonding types. For NaCl, removing an $Na^{+} + Cl^{-}$ pair costs $7.98 eV$ , but forming $Cl^{-}$ from Cl requires $+ 3.62 eV$ while forming $Na^{+}$ from Na releases $5.14 eV$ , so removing the neutral pair costs

7.98 - 3.62 + 5.14 = 6.46 eV per pair .

Cohesive-energy well of an ionic crystal: the Madelung Coulomb attraction and the exclusion-principle repulsion give a deep minimum at r_0, whose depth is the crystal's cohesive energy and sets the melting point.

A large cohesive energy means strongly bound atoms, a high melting point, and a hard crystal — ionic and covalent crystals (NaCl, diamond) are hard with high melting points, while van der Waals crystals of noble gases are soft and melt at very low temperatures. Metallic bonding, the fourth mechanism, holds the atoms together by valence electrons that detach entirely and roam the whole lattice. That electron sea is the subject of a later module, where it also carries the electric current. The lattice geometry itself is the next concern: the Bravais lattices that classify every crystalline arrangement, and the reciprocal space in which their diffraction is read.