1.1 Bravais lattices

Types of Bravais lattices

A Bravais lattice is an infinite array of discrete points generated by a set of translation vectors such that the arrangement looks exactly the same from every point. Every crystalline solid can be described by one of 14 Bravais lattices, grouped into seven crystal systems based on the symmetry of the unit cell.

The seven crystal systems differ in their lattice parameters (edge lengths $a$, $b$, $c$) and the angles between axes ($\alpha$, $\beta$, $\gamma$). Within each system, you can have different centering types (primitive, body-centered, face-centered, or base-centered), which is how 7 systems produce 14 distinct lattices.

Cubic lattice systems

All three lattice parameters are equal ($a = b = c$) and all angles are 90°. This is the most symmetric crystal system, and it supports three Bravais lattices:

• Simple cubic (SC): Lattice points only at the cube corners. Each corner is shared among 8 unit cells, so there's effectively 1 lattice point per cell. Rare in nature; polonium is the classic example.
• Body-centered cubic (BCC): Adds one lattice point at the center of the cube (2 points per cell). Found in iron (at room temperature), chromium, tungsten.
• Face-centered cubic (FCC): Adds a lattice point at the center of each face (4 points per cell). Found in copper, aluminum, gold, silver.

Tetragonal lattice systems

Two equal lattice parameters with the third different ($a = b \neq c$), all angles 90°. Think of it as a cubic cell stretched or compressed along one axis.

• Simple tetragonal: Lattice points at the corners of a rectangular prism with a square base.
• Body-centered tetragonal: Adds a point at the center of the prism.

Note that face-centered and base-centered tetragonal lattices aren't listed among the 14 because they can always be redescribed as one of the two above with a smaller unit cell.

Orthorhombic lattice systems

All three lattice parameters are unequal ($a \neq b \neq c$), but all angles remain 90°. This system has the most centering variants of any crystal system:

• Simple (P): Points at corners only.
• Base-centered (C): Adds points to the centers of one pair of opposite faces.
• Body-centered (I): Adds a point at the body center.
• Face-centered (F): Adds points to the centers of all faces.

That gives four Bravais lattices, all with mutually perpendicular axes but three distinct edge lengths.

Hexagonal lattice system

Two equal lattice parameters ($a = b \neq c$) with $\alpha = \beta = 90°$ and $\gamma = 120°$. Only one Bravais lattice exists here: the simple hexagonal lattice.

A common source of confusion: the hexagonal close-packed (HCP) structure (found in magnesium, titanium, zinc) is not itself a Bravais lattice. HCP is a crystal structure built on the simple hexagonal Bravais lattice with a two-atom basis.

Trigonal (rhombohedral) lattice system

Three equal lattice parameters ($a = b = c$) with all angles equal but not 90° ($\alpha = \beta = \gamma \neq 90°$). There is only one Bravais lattice: the rhombohedral lattice.

• The unit cell is a rhombohedron, like a cube squashed or stretched along a body diagonal.
• Found in materials like calcite ($CaCO_3$), bismuth, and the rhombohedral form of quartz.
• Some trigonal crystals can alternatively be described using hexagonal axes, which can cause confusion. The key distinction is the underlying point group symmetry.

Monoclinic lattice systems

Three unequal lattice parameters ($a \neq b \neq c$) with $\alpha = \gamma = 90°$ and $\beta \neq 90°$. One angle tilts away from 90°, reducing symmetry.

• Simple monoclinic (P): Points at corners of the tilted parallelepiped.
• Base-centered monoclinic (C): Adds points to the centers of one pair of faces.

Common in many minerals (gypsum, orthoclase) and organic molecular crystals.

Triclinic lattice system

Three unequal lattice parameters ($a \neq b \neq c$) with no right angles ($\alpha \neq \beta \neq \gamma \neq 90°$). This is the least symmetric crystal system, with only one Bravais lattice (primitive).

• The unit cell is a general parallelepiped with no special angular relationships.
• Found in some complex minerals (e.g., plagioclase feldspar) and certain organic compounds.

Lattice parameters

Lattice parameters are the set of lengths and angles that fully specify the geometry of a unit cell. Together, the three edge lengths ($a$, $b$, $c$) and three angles ($\alpha$, $\beta$, $\gamma$) give you six numbers that define the size and shape of any unit cell.

Lattice constants

The lattice constants $a$, $b$, and $c$ are the lengths of the three edges of the unit cell.

• Typically measured in angstroms (1 Å = $10^{-10}$ m) or nanometers.
• For copper (FCC), $a = 3.61$ Å. For silicon (diamond cubic), $a = 5.43$ Å.
• They change with temperature (thermal expansion) and pressure (compression), which is why materials expand when heated.
• Experimentally determined using X-ray diffraction, where peak positions relate directly to lattice spacings.

Angles between lattice vectors

The angles $\alpha$, $\beta$, and $\gamma$ describe the shape of the unit cell:

• $\alpha$ is the angle between vectors $\mathbf{b}$ and $\mathbf{c}$
• $\beta$ is the angle between vectors $\mathbf{a}$ and $\mathbf{c}$
• $\gamma$ is the angle between vectors $\mathbf{a}$ and $\mathbf{b}$

In cubic, tetragonal, and orthorhombic systems, all three angles are 90°. As you move to lower-symmetry systems (monoclinic, triclinic), one or more angles deviate from 90°, making the geometry progressively more complex.

Symmetry operations

Symmetry operations are transformations that map the crystal structure onto itself. They constrain which lattice parameters are independent and determine the crystal system.

• Rotations: $n$-fold rotation axes (2-fold, 3-fold, 4-fold, 6-fold are the only ones compatible with translational periodicity).
• Reflections: Mirror planes that map one half of the structure onto the other.
• Inversions: Mapping every point $(x, y, z)$ to $(-x, -y, -z)$ through a center of symmetry.
• Translations: The defining symmetry of any Bravais lattice.

The combination of all symmetry operations at a lattice point defines the point group, and combining point group symmetry with translational symmetry gives the space group. There are 32 crystallographic point groups and 230 space groups total.

Primitive vs conventional cells

The same lattice can be described by different choices of unit cell. The two most common choices are the primitive cell and the conventional cell, and understanding when to use each is important.

Definition of primitive cell

A primitive cell is the smallest unit cell that tiles all of space when translated by the lattice vectors. It contains exactly one lattice point (corners are shared, so 8 corners × 1/8 = 1 point).

• Its volume equals the total crystal volume divided by the number of lattice points.
• For an FCC lattice, the primitive cell is a rhombohedron, not a cube. It's harder to visualize but contains only 1 lattice point instead of 4.
• Primitive cells are preferred in theoretical work (band structure calculations, phonon dispersion) because they give the smallest possible basis for Bloch's theorem.
• The downside: a primitive cell may not visually reflect the full symmetry of the lattice.

Wigner-Seitz primitive cell

The Wigner-Seitz cell is a special primitive cell that does preserve the full point symmetry of the lattice. You construct it as follows:

1. Pick a lattice point as the origin.
2. Draw lines from that point to all neighboring lattice points.
3. Construct the perpendicular bisector plane of each line.
4. The smallest enclosed volume around the origin is the Wigner-Seitz cell.

This cell contains exactly one lattice point and has the same symmetry as the lattice itself. Its reciprocal-space counterpart is the first Brillouin zone, which plays a central role in electronic band theory.

Conventional unit cells

A conventional cell is chosen to make the symmetry of the lattice visually obvious, even if it contains more than one lattice point.

• The conventional FCC cell is a cube with 4 lattice points (8 corners × 1/8 + 6 faces × 1/2 = 4).
• The conventional BCC cell is a cube with 2 lattice points.
• These cells are much easier to work with for describing Miller indices, crystal planes, and directions.
• In crystallography and materials science, conventional cells are the standard choice for communication and visualization.

Reciprocal lattice

The reciprocal lattice is a mathematical construction in wavevector ($\mathbf{k}$) space that encodes the periodicity of the real-space (direct) lattice. It's indispensable for understanding diffraction and electronic structure.

Definition and properties

The reciprocal lattice is the set of all wavevectors $\mathbf{G}$ such that $e^{i\mathbf{G} \cdot \mathbf{R}} = 1$ for every direct lattice vector $\mathbf{R}$. In other words, these are the wavevectors of plane waves that have the same periodicity as the crystal.

The reciprocal lattice vectors are defined by:

$\mathbf{b}_1 = 2\pi \frac{\mathbf{a}_2 \times \mathbf{a}_3}{\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)}$

with $\mathbf{b}_2$ and $\mathbf{b}_3$ obtained by cyclic permutation. The denominator $\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)$ is the volume of the direct-space unit cell.

A key property: large real-space lattice constants correspond to small reciprocal-space lattice constants, and vice versa. This inverse relationship is why reciprocal space is so useful for diffraction, where short wavelengths probe large real-space structures.

Brillouin zones

Brillouin zones are the Wigner-Seitz cells of the reciprocal lattice.

• The first Brillouin zone is the region of reciprocal space closer to the origin than to any other reciprocal lattice point. It contains all the unique $\mathbf{k}$-points needed to describe electronic states (by Bloch's theorem).
• Higher-order Brillouin zones are defined by successive perpendicular bisector planes at increasing distance from the origin. Each zone has the same volume as the first.
• The shape of the first Brillouin zone reflects the symmetry of the crystal. For FCC, it's a truncated octahedron. For BCC, it's a rhombic dodecahedron.
• High-symmetry points in the Brillouin zone (labeled $\Gamma$, $X$, $L$, $K$, etc.) are where band structure diagrams are typically plotted.

Reciprocal lattice vectors

The three basis vectors $\mathbf{b}_1$, $\mathbf{b}_2$, $\mathbf{b}_3$ satisfy the orthogonality condition:

$\mathbf{a}_i \cdot \mathbf{b}_j = 2\pi \delta_{ij}$

where $\delta_{ij}$ is the Kronecker delta (1 if $i = j$, 0 otherwise). This means each reciprocal vector is perpendicular to two of the three direct lattice vectors.

These vectors are used to:

• Express any reciprocal lattice point as $\mathbf{G} = h\mathbf{b}_1 + k\mathbf{b}_2 + l\mathbf{b}_3$, where $h, k, l$ are integers (directly related to Miller indices).
• Calculate structure factors in diffraction: the amplitude of a diffracted beam depends on the atomic positions evaluated at each $\mathbf{G}$.
• Describe crystal planes, since the reciprocal lattice vector $\mathbf{G}_{hkl}$ is perpendicular to the $(hkl)$ plane in real space.

Crystal structures

A crystal structure = Bravais lattice + basis. The lattice tells you the periodicity; the basis tells you what sits at each lattice point (one atom, a group of atoms, a molecule, etc.). Many important structures share the same Bravais lattice but differ in their basis.

Close-packed structures

Close-packed structures maximize the packing fraction for identical spheres, achieving 74% packing efficiency. There are two ways to stack close-packed layers:

• FCC (cubic close-packed): Stacking sequence ABCABC... Three distinct layer positions before the pattern repeats. Found in copper ($a = 3.61$ Å), aluminum, gold, silver.
• HCP (hexagonal close-packed): Stacking sequence ABAB... Two distinct layer positions. Found in magnesium, titanium, zinc.

Both have a coordination number of 12 (each atom touches 12 neighbors). The difference in stacking sequence affects slip systems and therefore mechanical behavior: FCC metals tend to be more ductile than HCP metals because FCC has more available slip planes.

Body-centered vs face-centered

• BCC: 2 atoms per conventional cell, coordination number 8, packing fraction 68%. Examples: iron ($\alpha$-Fe), chromium, tungsten, molybdenum. BCC metals tend to be strong but can be brittle at low temperatures.
• FCC: 4 atoms per conventional cell, coordination number 12, packing fraction 74%. Examples: copper, aluminum, gold, platinum. FCC metals are generally more ductile with more slip systems (12 vs. 48 for BCC, though BCC slip is more complex).

The difference in coordination number and packing directly affects density, thermal properties, and how these metals deform under stress.

Diamond and zinc blende structures

The diamond structure consists of two interpenetrating FCC lattices, with the second offset by $(a/4, a/4, a/4)$ from the first. Each atom is tetrahedrally bonded to 4 neighbors, giving a coordination number of 4 and a packing fraction of only 34%.

• Found in carbon (diamond), silicon, germanium.
• The low packing fraction reflects the directional covalent bonding.

The zinc blende structure is identical in geometry but the two FCC sublattices are occupied by different atom types (e.g., Ga and As in GaAs, or Zn and S in ZnS). This breaks inversion symmetry, which has important consequences for piezoelectric and optical properties. Zinc blende semiconductors like GaAs and InP are the basis of much of modern optoelectronics.

Lattice planes and directions

Crystallographers need a compact notation to describe planes and directions within a crystal. Miller indices provide this, and they connect directly to the reciprocal lattice.

Miller indices

Miller indices $(hkl)$ label a set of parallel planes in the crystal. To determine them:

1. Find where the plane intercepts the three crystallographic axes (in units of $a$, $b$, $c$).
2. Take the reciprocals of those intercepts.
3. Clear fractions to get the smallest set of integers.

For example, a plane that intercepts at $a$, $\infty$, $\infty$ (parallel to $b$ and $c$ axes) has reciprocals $(1, 0, 0)$, giving Miller indices $(100)$.

• Negative intercepts are denoted with a bar: $(1\bar{1}0)$.
• Curly braces $\{hkl\}$ denote the family of all symmetry-equivalent planes.
• In hexagonal systems, four-index Miller-Bravais notation $(hkil)$ is often used, where $i = -(h+k)$.

Interplanar spacing

The distance between adjacent $(hkl)$ planes is the interplanar spacing $d_{hkl}$. For a cubic system:

$d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$

This formula gets more complex for lower-symmetry systems (additional terms involving $b$, $c$, and the angles). Interplanar spacing is directly measurable via Bragg's law: when $n\lambda = 2d\sin\theta$, the peak position $\theta$ tells you $d$.

Planes with small Miller indices have large spacings and tend to be the most physically significant (cleavage planes, dominant growth faces, primary slip planes).

Crystallographic directions

Directions are written in square brackets $[uvw]$, where $u$, $v$, $w$ are the smallest integers proportional to the direction vector components.

• $[100]$ points along the $a$-axis; $[111]$ points along the body diagonal of a cube.
• Angle brackets $\langle uvw \rangle$ denote the family of all symmetry-equivalent directions.
• In cubic systems, the direction $[hkl]$ is perpendicular to the plane $(hkl)$. This is not generally true in non-cubic systems.
• Crystallographic directions matter for understanding anisotropic properties: Young's modulus, thermal conductivity, and wave propagation speeds all depend on direction in a crystal.

Lattice defects

Real crystals are never perfect. Defects break the ideal periodicity and have an outsized influence on material properties. Even defect concentrations of parts per million can dramatically change electrical or mechanical behavior.

Point defects

Point defects are localized disruptions at or near a single lattice site.

• Vacancies: Missing atoms. Their equilibrium concentration increases exponentially with temperature: $n_v \propto e^{-E_v / k_B T}$, where $E_v$ is the vacancy formation energy.
• Interstitials: Extra atoms squeezed into spaces between regular lattice sites. They cause local lattice distortion and are important for diffusion.
• Substitutional impurities: Foreign atoms replacing host atoms. This is the basis of semiconductor doping (e.g., adding phosphorus to silicon to create n-type material).

Point defects control diffusion rates, enable doping in semiconductors, and create color centers in ionic crystals.

Line defects

Line defects, or dislocations, are one-dimensional defects that run through the crystal. They are the primary mechanism of plastic deformation in metals.

• Edge dislocation: An extra half-plane of atoms inserted into the lattice. The dislocation line runs along the edge of this half-plane, and the Burgers vector is perpendicular to the dislocation line.
• Screw dislocation: Atoms are displaced in a helical pattern around the dislocation line. The Burgers vector is parallel to the dislocation line.
• Most real dislocations are mixed, with both edge and screw character.

Dislocation density (typically $10^6$ to $10^{12}$ per cm² in metals) directly affects yield strength. Work hardening occurs because dislocations multiply and tangle, making further motion more difficult.

Planar defects

Planar defects are two-dimensional disruptions in the crystal.

• Grain boundaries: Interfaces between crystallites (grains) of different orientation in a polycrystalline material. They impede dislocation motion, which is why fine-grained materials are stronger (Hall-Petch relationship).
• Twin boundaries: The crystal on one side is a mirror image of the other. Twinning can occur during growth or deformation.
• Stacking faults: Errors in the stacking sequence of close-packed layers (e.g., ABCABABC instead of ABCABC in FCC). The stacking fault energy determines how easily partial dislocations can separate, which affects deformation mechanisms.

Experimental techniques

Crystal structures are determined experimentally by probing materials with radiation whose wavelength is comparable to interatomic spacings (on the order of angstroms).

X-ray diffraction

X-ray diffraction (XRD) is the most widely used technique for determining crystal structures. It relies on Bragg's law:

$n\lambda = 2d\sin\theta$

where $n$ is the diffraction order, $\lambda$ is the X-ray wavelength, $d$ is the interplanar spacing, and $\theta$ is the angle of incidence.

• Powder XRD uses a polycrystalline sample and produces a pattern of rings (or peaks in a 1D scan). Peak positions give lattice parameters; peak intensities give information about the basis.
• Single-crystal XRD gives a full 3D map of diffraction spots, allowing complete structure determination.
• Common X-ray sources use copper K-alpha radiation ($\lambda = 1.54$ Å), well-matched to typical interatomic spacings.

Neutron scattering

Neutron scattering complements XRD in several ways:

• Neutrons scatter off nuclei rather than electron clouds, making them sensitive to light elements (especially hydrogen) that X-rays miss.
• Neutrons have a magnetic moment, so they can probe magnetic structures and spin ordering.
• Inelastic neutron scattering measures phonon and magnon dispersion relations, giving dynamic information that XRD cannot.
• The main drawback: neutron sources (nuclear reactors or spallation sources) are large and expensive, so access is limited.

Electron microscopy

Electron microscopy provides real-space images of crystal structure, complementing the reciprocal-space information from diffraction.

• Transmission Electron Microscopy (TEM): Electrons pass through a thin sample. High-resolution TEM can directly image atomic columns. Selected-area electron diffraction (SAED) gives local crystal structure information.
• Scanning Electron Microscopy (SEM): Electrons scan the surface, producing images of topography and composition. Lower resolution than TEM but easier sample preparation.
• Electron microscopy is particularly valuable for studying defects, interfaces, grain boundaries, and nanostructures that are difficult to characterize by diffraction alone.

Applications in materials science

The concepts covered above aren't just abstract theory. They directly inform how materials are designed, selected, and engineered for real applications.

Structure-property relationships

The crystal structure of a material constrains its properties in predictable ways:

• Mechanical: FCC metals (Cu, Al) are ductile because of their many slip systems. BCC metals (Fe, W) are strong but can undergo a ductile-to-brittle transition at low temperatures. HCP metals (Ti, Mg) have limited slip systems and are often less ductile.
• Electronic: Band structure depends on lattice periodicity. Silicon's diamond structure gives it an indirect band gap of 1.12 eV, making it ideal for transistors. GaAs (zinc blende) has a direct band gap of 1.42 eV, making it better for LEDs and lasers.
• Thermal: Lattice parameters and bonding determine thermal expansion coefficients and thermal conductivity. Diamond's stiff tetrahedral bonds give it the highest thermal conductivity of any bulk material.

Crystal engineering

Crystal engineering is the deliberate design of crystalline materials to achieve target properties.

• In pharmaceuticals, controlling which polymorph (crystal form) of a drug is produced affects solubility, stability, and bioavailability. Different polymorphs of the same molecule can have very different dissolution rates.
• In catalysis, exposing specific crystal faces with favorable surface energies can dramatically improve reaction rates.
• Metal-organic frameworks (MOFs) are engineered porous crystals used for gas storage, separation, and catalysis, designed by choosing specific metal nodes and organic linkers.

Nanostructures and superlattices

When you control crystal structure at the nanoscale, quantum mechanics becomes directly relevant.

• Quantum wells: Thin layers (a few nm) of one semiconductor sandwiched between layers of a wider-gap semiconductor. Electrons are confined in one dimension, modifying the density of states and enabling laser diodes.
• Superlattices: Periodic stacks of alternating materials (e.g., GaAs/AlAs) with layer thicknesses of a few nanometers. The artificial periodicity creates new electronic band structures not found in either bulk material.
• Quantum dots: Nanoscale crystals (2-10 nm) where confinement in all three dimensions produces discrete energy levels. Used in displays, biological imaging, and next-generation solar cells.

These applications all depend on precise control of lattice matching between layers. Even a small mismatch in lattice constants between adjacent materials introduces strain and defects that degrade device performance.