4.2 Bravais Lattices

Bravais Lattices in Crystallography

Bravais lattices describe the 14 unique ways that lattice points can be arranged in three-dimensional space while maintaining perfect translational symmetry. Every crystal structure you'll encounter in mineralogy is built on one of these 14 lattices, so understanding them gives you a direct link between atomic arrangement and the physical properties you can observe in a mineral specimen.

Fundamental Concepts

A Bravais lattice is an infinite array of discrete points where every point has an identical arrangement and orientation of surrounding points. Think of it as the repeating geometric skeleton of a crystal, onto which you place the actual atoms, molecules, or ions.

Each Bravais lattice is characterized by its unit cell, the smallest repeating unit that, when stacked in all three directions, reproduces the entire crystal structure. The unit cell is defined by three edge lengths ($a$, $b$, $c$) and three angles ($\alpha$, $\beta$, $\gamma$) between those edges. These six parameters are called the lattice parameters.

Why only 14? Mathematics constrains it. When you combine the 7 crystal systems (which define the shapes of unit cells) with the 4 possible lattice centering types (primitive, body-centered, face-centered, and base-centered), many combinations turn out to be redundant or reducible to simpler lattices. Only 14 truly unique arrangements survive.

Applications in Crystallography

• Interpreting X-ray diffraction data: the Bravais lattice determines which diffraction peaks appear and which are systematically absent, making it essential for mineral identification
• Predicting physical properties (cleavage, optical behavior, hardness) from atomic arrangements
• Classifying crystal symmetry and determining space groups, which combine lattice symmetry with additional symmetry elements
• Supporting computational modeling of crystal growth, defect behavior, and phase transitions
• Informing the design of synthetic materials like semiconductors and catalysts, where controlling crystal structure is critical

Identifying Bravais Lattices

The Seven Crystal Systems and Their Lattices

The 14 Bravais lattices distribute across the 7 crystal systems as follows:

The letters P, I, F, C, and R are standard symbols for the centering type:

• P (Primitive): lattice points only at the corners of the unit cell
• I (Body-centered): corners plus one additional point at the center of the cell
• F (Face-centered): corners plus one point at the center of each face
• C (Base-centered): corners plus points at the centers of two opposite faces
• R (Rhombohedral): a special primitive cell with equal edges and equal angles

Distinguishing Features

Several measurable properties help you tell Bravais lattices apart:

• Unit cell geometry is the first filter. Measure the edge lengths and angles, and you've narrowed it to one of the 7 crystal systems.
• Lattice point positions then distinguish which Bravais lattice within that system. A cubic mineral could be P, I, or F, and X-ray diffraction systematic absences reveal which one.
• Coordination number varies among lattices. Simple cubic has a coordination number of 6, body-centered cubic has 8, and face-centered cubic has 12.
• Packing efficiency also differs: face-centered cubic packs atoms at 74% efficiency (the theoretical maximum for equal spheres), body-centered cubic at 68%, and simple cubic at only 52%.
• Symmetry elements like rotation axes and mirror planes further differentiate lattices and connect directly to the mineral's crystal class.

Bravais Lattices and Mineral Structure

Atomic Arrangement and Lattice Selection

The Bravais lattice a mineral adopts depends on the size, shape, and bonding characteristics of its constituent atoms or ions. Once that lattice is set, it controls many of the mineral's observable properties:

• Cleavage patterns follow planes of weakest bonding in the lattice. Halite (NaCl) has perfect cubic cleavage because its face-centered cubic lattice creates equivalent weak planes along {100}.
• Optical behavior depends on lattice symmetry. Calcite's trigonal lattice produces strong birefringence because light experiences different refractive indices along different crystallographic directions.
• Mechanical properties relate to bond density along specific directions. Diamond's face-centered cubic lattice, combined with strong covalent bonding in all directions, produces extreme hardness.

The unit cell also encodes spatial relationships between atoms, including bond lengths (interatomic distances) and bond angles, both of which influence reactivity and stability.

Polymorphism and Structural Implications

Polymorphism occurs when the same chemical composition crystallizes in different Bravais lattices under different conditions. Diamond and graphite are both pure carbon, but diamond adopts a face-centered cubic lattice with tetrahedral bonding, while graphite adopts a hexagonal lattice with layered sheets. Same atoms, completely different properties.

Coordination number connects directly to the Bravais lattice's packing arrangement:

• In the NaCl structure (face-centered cubic), each $Na^+$ is surrounded by 6 $Cl^-$ ions (6-fold, or octahedral, coordination)
• In the CsCl structure (primitive cubic, not body-centered cubic despite appearances), each $Cs^+$ is surrounded by 8 $Cl^-$ ions (8-fold coordination)

The radius ratio $r_{cation}/r_{anion}$ largely determines which coordination environment is stable, which in turn determines the Bravais lattice.

Lattice defects also matter. Vacancies, interstitials, and substitutional impurities disrupt the ideal Bravais lattice and can alter properties significantly. Color centers in fluorite (caused by electron-trapping vacancies) are a classic mineralogical example.

Predicting Mineral Structures

Chemical Composition Analysis

You can often narrow down the likely Bravais lattice before running any diffraction experiments. Here's a practical approach:

1. Determine the chemical formula and stoichiometry. The ratio of atoms constrains how many of each must fit in the unit cell, which limits possible lattice types.
2. Look up atomic/ionic radii. Calculate the radius ratio for ionic compounds to predict coordination number and likely packing arrangement.
3. Assess bonding character. Ionic compounds tend toward higher-symmetry lattices (cubic, hexagonal) because non-directional electrostatic forces favor close-packing. Covalent compounds may adopt lower-symmetry structures because directional bonds constrain geometry.
4. Identify structural motifs. Certain building blocks strongly favor specific systems. Silicate tetrahedra ($SiO_4^{4-}$) in quartz point to the hexagonal system. Carbonate groups ($CO_3^{2-}$) in calcite restrict the structure to the trigonal system because of the group's 3-fold symmetry.

Predictive Methods

• Radius ratio rules provide a first approximation for ionic compounds. A ratio of 0.414–0.732 predicts octahedral coordination (as in NaCl, face-centered cubic), while a ratio above 0.732 predicts cubic coordination (as in CsCl).
• Symmetry constraints from molecular groups limit the possible lattices. If a mineral contains a building block with 3-fold symmetry, the lattice must accommodate that symmetry.
• Trends within mineral groups are powerful predictors. Spinel group minerals ($AB_2O_4$) consistently crystallize in the face-centered cubic lattice, so a newly discovered spinel-composition mineral will very likely do the same.
• Pressure and temperature effects shift lattice selection. High-pressure polymorphs generally adopt more compact, higher-coordination lattices. Knowing the formation conditions helps you predict which polymorph to expect.
• Crystal chemistry databases allow you to estimate the probability of specific Bravais lattices for hypothetical or newly synthesized minerals by comparing them to known structures with similar compositions.