💎 Crystal Structures — BCC, FCC, HCP, Diamond Lattices

The atomic packing factor (APF) is the fraction of the unit cell volume occupied by atoms, calculated as APF = (N x V_atom) / V_cell, where N is the number of atoms per unit cell and V_atom is the volume of one atom treated as a sphere. Simple Cubic gives APF = 0.52, BCC = 0.68, FCC and HCP = 0.74 (the theoretical maximum for equal hard spheres, known as close-packing), and Diamond Cubic only 0.34 due to its open tetrahedral bonding.

Simple Cubic is extremely rare — only polonium adopts it. BCC is common in the transition metals iron (below 912 degrees C), tungsten, chromium, and molybdenum. FCC appears in aluminium, copper, gold, silver, and nickel. HCP is found in magnesium, titanium, zinc, and cobalt. The Diamond Cubic structure describes carbon (diamond), silicon, and germanium. The NaCl (rock-salt) structure is adopted by common salt, magnesium oxide, and iron(II) oxide, while CsCl applies to caesium chloride, caesium bromide, and ammonium chloride.

Hardness in diamond does not come from dense packing but from the nature of the bonds. Each carbon atom forms four strong, directional covalent bonds (sp3 hybridisation) arranged in a perfect tetrahedron. Deforming the crystal requires breaking many of these bonds simultaneously throughout the three-dimensional network, which demands enormous energy. BCC iron has a higher packing factor but much weaker metallic bonds, making it far softer than diamond.

Max von Laue demonstrated in 1912 that X-rays diffract through crystals, proving both the wave nature of X-rays and the periodic structure of crystals. William Henry Bragg and his son William Lawrence Bragg then derived Bragg's Law (n*lambda = 2d*sin(theta)) in 1913, enabling scientists to calculate atomic spacings from diffraction patterns. Both Braggs shared the 1915 Nobel Prize in Physics. X-ray crystallography later revealed the double helix structure of DNA in 1953.

Crystal structures are closely linked to topics such as phonons and lattice vibrations (heat conduction), electron band theory (why metals conduct and semiconductors do not), and materials failure (dislocation motion through BCC or FCC lattices governs ductility). Fluid simulations such as SPH Fluid are the complementary case where atoms lack long-range order. Phase transitions between crystal types, like iron switching from BCC to FCC at 912 degrees C, connect to thermodynamics simulations.

Silicon and germanium both adopt the Diamond Cubic structure, and their semiconductor properties arise directly from the sp3 tetrahedral bonding. Integrated circuits are fabricated on single-crystal silicon wafers cut along specific crystallographic planes (usually the (100) or (111) plane) because surface reactivity and electron mobility depend on orientation. Compound semiconductors like gallium arsenide (GaAs) use the closely related Zinc Blende structure, a variant of Diamond Cubic with two different atom types.

Computational crystal structure prediction — using density functional theory and machine learning force fields to predict which structure a new compound will adopt before it is ever synthesised — is a rapidly growing field. The Cambridge Structural Database holds over one million experimentally determined crystal structures, and AI models trained on this data can now propose novel stable crystal phases for battery materials, pharmaceuticals, and high-entropy alloys. Quasicrystals, discovered by Dan Shechtman in 1982 (Nobel Prize 2011), also remain an active area; they show diffraction patterns with five-fold symmetry that is impossible in any conventional Bravais lattice.