Complete Guide to Crystal Structure: Bravais Lattices & Diffraction
To understand the physics of solids, we must define the precise geometry of their internal structure. This post covers the journey from atomic bonds to the 14 fundamental Bravais lattices that describe all crystals. We also explore the band structure of Graphene and derive the diffraction equations that allow us to "see" atomic arrangements using waves.
1. Fundamentals of Solids
Types of Bonds
The stability of a crystal is determined by the interactions between atoms:
- Ionic: Electrostatic attraction between ions (e.g., \(Na^+\) and \(Cl^-\)).
- Covalent: Electron sharing between atoms (e.g., Silicon, Diamond).
- Metallic: Delocalized "sea" of electrons shared across the lattice.
- Van der Waals: Weak intermolecular forces holding molecules together.
The Definition of a Crystal
A solid is defined as a collection of atoms in a periodic manner, meaning the arrangement looks identical from every lattice point. This leads to the fundamental equation of crystallography:
- Lattice: An infinite array of abstract mathematical points in space.
- Basis: The atom or group of atoms attached to each lattice point.
2. Case Study: Graphene (Dimensionality)
Graphene illustrates how dimensionality affects properties.
- Monolayer: A single 2D sheet of carbon atoms.
- Bilayer: Placing a second layer on top changes the electronic properties due to electron hopping between the layers.
Band Structure Approximation
The energy dispersion relation for electrons in a lattice involves cosine terms (from the tight-binding model). For example: \[ E = -2t \cos(k_x a) \]
Near the bottom of the band (small \(k\)), we can expand this using a Taylor series \(\cos(x) \approx 1 - x^2/2!\): \[ E \approx -2t + 2t \frac{(k_x a)^2}{2!} - \dots \]
Ignoring the constant offset, this simplifies to a parabolic relationship:
This shows that even in a lattice, low-energy electrons behave similarly to free electrons (\(E \propto k^2\)).
3. Lattice Geometry & Symmetries
Vectors and Unit Cells
Translation Vectors: Any point in the lattice is reachable via \(\vec{R} = n_1\vec{a}_1 + n_2\vec{a}_2 + n_3\vec{a}_3\).
Unit Cell: The volume that generates the full crystal when translated.
Primitive Unit Cell: The smallest possible cell containing effectively only one lattice point.
Wigner-Seitz Construction
A geometric method to find the primitive cell:
1. Connect a lattice point to all its neighbors.
2. Draw perpendicular bisectors (normals) to these lines.
3. The smallest volume enclosed is the Wigner-Seitz cell.
Symmetry Operations
Lattices are invariant under:
• Translation
• Rotation: (e.g., Square = 4-fold, Hexagon = 6-fold).
• Inversion: \(I(x,y,z) \to (-x,-y,-z)\).
4. The 14 Bravais Lattices
In 3 dimensions, there are exactly 14 unique ways to arrange points in space such that every point has identical surroundings. These are grouped into 7 Crystal Systems based on the lengths of lattice vectors (\(a,b,c\)) and angles (\(\alpha, \beta, \gamma\)).
| System | Axial & Angular Relations | Bravais Lattices |
|---|---|---|
| 1. Cubic | \(a=b=c\) \(\alpha=\beta=\gamma=90^\circ\) |
Simple (P), Body-Centered (I), Face-Centered (F) |
| 2. Tetragonal | \(a=b \neq c\) \(\alpha=\beta=\gamma=90^\circ\) |
Simple (P), Body-Centered (I) |
| 3. Orthorhombic | \(a \neq b \neq c\) \(\alpha=\beta=\gamma=90^\circ\) |
Simple (P), Body-Centered (I), Face-Centered (F), Base-Centered (C) |
| 4. Hexagonal | \(a=b \neq c\) \(\alpha=\beta=90^\circ, \gamma=120^\circ\) |
Simple (P) |
| 5. Trigonal (Rhombohedral) | \(a=b=c\) \(\alpha=\beta=\gamma \neq 90^\circ\) |
Simple (R) |
| 6. Monoclinic | \(a \neq b \neq c\) \(\alpha=\gamma=90^\circ \neq \beta\) |
Simple (P), Base-Centered (C) |
| 7. Triclinic | \(a \neq b \neq c\) \(\alpha \neq \beta \neq \gamma\) |
Simple (P) |
5. Detailed Cubic Structures
The notes specifically focus on the Cubic system structures:
A. Simple Cubic (SC)
Atoms are located only at the 8 corners.
Total Atoms: \(8 \times \frac{1}{8} = 1\) atom per cell.
B. Body-Centered Cubic (BCC)
Atoms at 8 corners + 1 atom in the exact center.
Total Atoms: \((8 \times \frac{1}{8}) + 1 = 2\) atoms per cell.
C. Face-Centered Cubic (FCC)
Atoms at 8 corners + atoms at the center of all 6 faces.
Total Atoms: \((8 \times \frac{1}{8}) + (6 \times \frac{1}{2}) = 1 + 3 = 4\) atoms per cell.
Note: Since a primitive cell contains 1 lattice point, the FCC conventional unit cell contains 4 primitive cells.
6. Diffraction Theory
Since the wavelength of visible light is too large to resolve atoms, we use X-ray, Neutron, or Electron scattering. The crystal acts as a 3D diffraction grating.
Scattering Amplitude Derivation
Consider a wave source \(Q\) and detector \(P\). We analyze the scattering from an atom at position \(\vec{R} + \vec{r}\) (relative to origin \(O\)).
The wave amplitude \(A_P\) at the detector depends on the phase difference caused by the path length. The phase factor is:
Where \(\vec{k}\) is the scattering wavevector. The total intensity observed is proportional to the square of the summed amplitudes: \(I \propto | \sum A_P |^2\). This summation leads to the Laue Equations or Bragg's Law, where constructive interference (bright spots) only occurs for specific angles.