Solid State & Crystal Structures

In a simple cubic cell, 8 corner atoms each contribute 1/8, totaling 1 atom. In BCC, the 8 corners contribute 1 atom plus 1 body-centre atom = 2. In FCC, the 8 corners contribute 1 atom and 6 face-centre atoms each contribute 1/2, giving 1 + 3 = 4 atoms per unit cell. Counting shared atoms correctly is essential for density calculations.

The FCC (CCP) structure has a packing efficiency of 74.05%, the maximum possible for identical spheres. This is calculated by placing 4 atoms of radius r in a cube with edge length a = 2√2 · r, then computing the fraction of volume occupied: (4 × 4/3 πr³) / a³. The HCP structure also achieves 74% packing. BCC achieves 68% and simple cubic only 52%.

Both HCP and CCP are close-packed structures with a coordination number of 12. Each atom touches 6 neighbours in its own layer, 3 in the layer above, and 3 in the layer below. The difference lies in the stacking sequence: HCP uses ABAB… while CCP uses ABCABC… The coordination environment is identical in both cases.

In BCC, the body diagonal connects a corner atom through the body-centre atom to the opposite corner. The body diagonal length equals a√3, and it spans 4 atomic radii (corner–centre–corner contact). Therefore a√3 = 4r, giving a = 4r/√3. For comparison, in FCC atoms touch along the face diagonal, giving a = 2√2 · r, and in simple cubic a = 2r.

In the NaCl structure, both Na⁺ and Cl⁻ have a coordination number of 6 in an octahedral arrangement. Each Na⁺ is surrounded by 6 Cl⁻ ions and vice versa. The structure can be described as two interpenetrating FCC lattices offset by half a unit cell edge. This 6:6 coordination is characteristic of ions with radius ratios in the range 0.414–0.732.

Radius ratio rules predict the coordination geometry of ionic crystals. When r⁺/r⁻ > 0.732, the cation is large enough to accommodate 8 anions in a cubic arrangement. Cs⁺ (r = 181 pm) with Cl⁻ (r = 167 pm) gives a ratio of ~1.08, well above the 0.732 threshold. Na⁺ (r = 116 pm) with Cl⁻ gives ~0.69, which falls in the octahedral (6-coordinate) range of 0.414–0.732.

Both zinc blende and wurtzite have tetrahedral 4:4 coordination with the same stoichiometry (ZnS). The difference lies in how the anion sublattice is packed: zinc blende has S²⁻ in a CCP (ABCABC…) arrangement, while wurtzite has S²⁻ in an HCP (ABAB…) arrangement. In both cases, Zn²⁺ occupies half the tetrahedral holes. This parallels the CCP/HCP relationship seen in elemental close-packed metals.

In the fluorite structure, Ca²⁺ ions form an FCC lattice with 8-fold cubic coordination (surrounded by 8 F⁻ ions at the corners of a cube). Each F⁻ sits in a tetrahedral hole and is coordinated by 4 Ca²⁺ ions, giving 4-fold tetrahedral coordination. The stoichiometry requires an 8:4 ratio because there are twice as many anions as cations. The anti-fluorite structure (e.g., Na₂O) reverses the cation/anion positions.

The Born–Landé equation is: U = −(N_A |z⁺| |z⁻| e² A) / (4πε₀ r₀) × (1 − 1/n), where A is the Madelung constant (depends on crystal structure), r₀ is the equilibrium inter-ionic distance, z⁺ and z⁻ are the ion charges, and n is the Born exponent (related to compressibility). This equation captures both the Coulombic attraction and short-range repulsion between ions.

Lattice energy is inversely proportional to the inter-ionic distance r₀. Since all three have the same charges (+1/−1) and the same NaCl structure type (same Madelung constant), the key variable is ion size. F⁻ is the smallest halide, giving the shortest Na–F distance and therefore the largest lattice energy. Experimental values: NaF (930 kJ/mol), NaCl (786 kJ/mol), NaBr (747 kJ/mol).

The Born–Landé equation shows that lattice energy is proportional to |z⁺||z⁻|. For NaF: |+1|×|−1| = 1. For MgO: |+2|×|−2| = 4. So the charge product is four times larger in MgO. Combined with a slightly shorter inter-ionic distance (Mg²⁺ is smaller than Na⁺), this explains why MgO has a lattice energy roughly four times that of NaF. This charge effect also accounts for the very high melting point of MgO (2850 °C).

The Madelung constant A is a purely geometric factor that sums all pairwise Coulombic interactions (both attractive and repulsive) in an infinite ionic lattice, expressed relative to the nearest-neighbour distance. Its value depends only on the crystal structure type, not on the specific ions. For example, A = 1.7476 for the NaCl structure and A = 1.7627 for the CsCl structure. A larger Madelung constant means the ions in that arrangement achieve a more favourable electrostatic energy.

A Schottky defect consists of a matched pair of cation and anion vacancies (maintaining electrical neutrality and stoichiometry). A Frenkel defect involves an ion leaving its normal lattice site and moving to an interstitial position, creating a vacancy–interstitial pair. Schottky defects are common in NaCl (similar-sized ions), while Frenkel defects are common in AgBr (where the small Ag⁺ can fit into interstitial sites). Both defects increase in concentration with temperature.

In band theory, atomic orbitals combine to form continuous energy bands. The key parameter is the band gap (E_g) between the filled valence band and the empty conduction band. In metals, these bands overlap or the valence band is partially filled, allowing free electron movement. In semiconductors, E_g is small (typically 0.5–3 eV), so thermal excitation can promote electrons across the gap. In insulators, E_g is large (>4 eV), making thermal excitation negligible at room temperature.

In n-type semiconductors, the dopant has more valence electrons than the host (e.g., phosphorus in silicon: P has 5 valence electrons vs. Si with 4). The extra electron occupies a donor level just below the conduction band and is easily promoted, providing negative charge carriers. In p-type semiconductors, the dopant has fewer valence electrons (e.g., boron in silicon: B has 3 valence electrons). This creates an acceptor level just above the valence band, generating positive "holes" as the dominant charge carriers.