Crystal Structures
Bravais lattices, unit cell, Miller indices and Bragg diffraction
A crystal forms when the forces between atoms — chemical bonds, electrostatic interactions, van der Waals forces — drive the system toward an ordered, periodic structure. This structure corresponds to the lowest energy state. Atoms arrange themselves so that their interaction energy is minimized — this is the "logic" of a crystal lattice. Different atoms, different bonds — different structures: FCC copper, BCC iron, diamond carbon — all are results of this "logic".
1784 — René Just Haüy first introduced the crystal lattice concept. 1848 — Auguste Bravais showed only 14 periodic lattices are possible in 3D space (Bravais lattices). 1912 — Max von Laue discovered X-ray diffraction by crystals — first direct evidence of crystal structure (Nobel Prize 1914). 1913 — William Bragg formulated Bragg's law: 2d·sin(θ) = nλ — still used in crystallography today.
A crystal lattice is defined by three primitive vectors a₁, a₂, a₃. Any lattice node is obtained by:
where n₁, n₂, n₃ are integers. These three vectors define the unit cell — the smallest periodic unit of the crystal. The reciprocal lattice is defined by vectors b₁, b₂, b₃:
The reciprocal lattice is used in physics to describe diffraction, band structure, and k-space.
| Symbol | Name | Unit | Typical Value |
|---|---|---|---|
| a, b, c | Lattice parameters | nm, Å | 0.1–1 nm |
| α, β, γ | Lattice angles | ° | 60°–120° |
| Z | Atoms per unit cell | — | 1–8 |
| CN | Coordination number — number of nearest neighbors | — | 4, 6, 8, 12 |
| APF | Atomic Packing Factor — fraction of volume occupied by atoms | % | 34–74% |
| dhkl | Interplanar spacing | nm | 0.05–0.5 nm |
| θB | Bragg angle | ° | 5°–85° |
There are only 14 Bravais lattices in 3D space, in 7 crystal systems. Most common cubic structures:
| Structure | Abbreviation | CN | APF (%) | Examples |
|---|---|---|---|---|
| SC — Simple Cubic | SC | 6 | 52 | Po |
| BCC — Body-Centered Cubic | BCC | 8 | 68 | Fe, W, Mo |
| FCC — Face-Centered Cubic | FCC | 12 | 74 | Al, Cu, Au |
| Diamond Structure | DC | 4 | 34 | C, Si, Ge |
| HCP — Hexagonal Close-Packed | HCP | 12 | 74 | Mg, Ti, Zn |
| NaCl — Rock Salt Structure | NaCl | 6 | 79 | NaCl, MgO |
Miller indices (h,k,l) are integers that define the orientation of a plane in a crystal. Plane (hkl) intercepts the lattice axes at a/h, b/k, c/l. Interplanar spacing for cubic system:
h, k, l — Miller indices. a — lattice parameter. d_hkl encompasses the full set of parallel planes.
Bragg's law (1913): 2d_hkl·sin(θ_B) = nλ — where θ_B is the Bragg angle, λ is the X-ray wavelength, n is an integer (diffraction order). This law is the foundation of crystallography.
Crystal Structure → Band Structure: lattice parameter a directly determines the Brillouin zone size (π/a) and the tight-binding E(k) dispersion. Crystal Structure → DFT: DFT calculations always begin with defining the crystal structure — atomic coordinates in the unit cell. Crystal Structure → Nanophysics: Si diamond structure determines its semiconductor properties; GaAs zinc-blende structure — its optoelectronic properties.
X-ray Crystallography (XRD): Using Bragg's law to determine crystal structure — this method revealed the DNA double helix (1953, Watson & Crick). Semiconductor devices: Si, GaAs crystal structures are the foundation of all electronic devices. Nanotechnology: Quantum dots, nanowires — nanoscale control of crystal structure. Materials science: FCC/BCC structure determines a metal's strength, plasticity, and conductivity.
APF (Atomic Packing Factor) — higher means denser structure. FCC and HCP are densest (74%). CN (Coordination Number) — number of nearest neighbors; higher CN → more bonds → stronger but less ductile material. Diamond structure is least dense (APF=34%) — sp³ hybrid bonds in 4 directions.
Q1. How many atoms does an FCC unit cell contain?
Q2. What process does Bragg's law describe?
Q3. What crystal structure does Si belong to?
Q4. FCC and HCP have the same APF. What is this value?