Key Concepts of Bravais Lattices

Why This Matters

Understanding Bravais lattices is fundamental to everything you'll encounter in crystallography and solid-state physics. These 14 unique three-dimensional arrangements describe every possible way atoms can periodically repeat in space—and that periodicity directly determines a material's symmetry, electronic behavior, mechanical properties, and optical characteristics. When you're analyzing X-ray diffraction patterns, predicting material properties, or understanding phase transitions, you're working with Bravais lattice concepts.

You're being tested on more than just memorizing lattice names and parameters. Exams expect you to connect lattice geometry to physical properties, explain why certain materials adopt specific structures, and predict how symmetry affects behavior like isotropy versus anisotropy. Don't just memorize the seven crystal systems—know what constraints define each one and how those constraints manifest in real materials.

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High-Symmetry Systems: Cubic and Hexagonal

These lattices possess the highest symmetry in their respective families, making them the most common structures for metals and simple compounds. High symmetry means equivalent properties in multiple directions, which simplifies both analysis and prediction.

Cubic (Simple, Body-Centered, Face-Centered)

• Equal edge lengths ($a = b = c$) and all 90° angles—this maximizes symmetry and creates isotropic properties in polycrystalline samples
• Atoms per unit cell vary by type: simple cubic (1), body-centered/BCC (2), face-centered/FCC (4)—directly affects density and packing efficiency
• Dominates metallic structures because close-packing (FCC) and efficient space-filling (BCC) minimize energy; copper and gold are FCC, iron is BCC

Hexagonal

• Two equal basal edges with $\gamma = 120°$ and a distinct c-axis perpendicular to the basal plane—creates six-fold rotational symmetry
• Two atoms per unit cell in the primitive hexagonal cell; the related HCP structure has even higher packing efficiency
• Strong anisotropy makes properties direction-dependent; graphite's layered structure and magnesium's deformation behavior both stem from hexagonal geometry

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Rectangular Systems: Tetragonal and Orthorhombic

These systems maintain 90° angles but relax the constraint of equal edge lengths. The resulting lower symmetry creates anisotropic properties while preserving relatively simple mathematical descriptions.

Tetragonal (Simple, Body-Centered)

• Two equal edges ($a = b \neq c$) with all 90° angles—essentially a "stretched" or "compressed" cubic lattice along one axis
• Simple tetragonal has 1 atom per unit cell, body-centered has 2—no face-centered variant exists because it would reduce to body-centered
• Common in phase transitions from cubic structures; tin's allotropic forms and many ceramic materials (like $BaTiO_3$) adopt tetragonal symmetry

Orthorhombic (Simple, Body-Centered, Face-Centered, Base-Centered)

• All edges unequal ($a \neq b \neq c$) but all angles remain 90°—the lowest symmetry system that still maintains perpendicular axes
• Four centering options (P, I, F, C) provide the most variants of any crystal system—reflects the geometric flexibility
• Found in sulfur, olivine, and many minerals—the unequal axes create distinct properties along each crystallographic direction

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Oblique-Angle Systems: Monoclinic and Triclinic

When angles deviate from 90°, symmetry drops significantly. These lower-symmetry systems accommodate complex molecular structures and are common in minerals and organic crystals.

Monoclinic (Simple, Base-Centered)

• One oblique angle ($\beta \neq 90°$) while $\alpha = \gamma = 90°$—the single tilted axis creates a "leaning" unit cell
• Simple has 1 atom per unit cell, base-centered has 2—only two centering types are unique; others reduce to these
• Gypsum, many feldspars, and organic compounds crystallize here—the flexibility accommodates molecules that don't pack efficiently in higher-symmetry systems

Triclinic

• All edges unequal AND all angles oblique ($\alpha \neq \beta \neq \gamma \neq 90°$)—the most general case with no special constraints
• Only one lattice type exists (primitive)—any centering would create a smaller primitive cell
• Lowest symmetry of all Bravais lattices means the most complex property tensors; kyanite and many organic molecules crystallize here

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The Rhombohedral Exception

This system occupies a unique position—high edge-length symmetry combined with non-orthogonal angles creates distinctive properties.

Rhombohedral (Trigonal)

• All edges equal ($a = b = c$) but angles equal and non-90° (typically ~60° or ~109°)—can be visualized as a cube stretched along its body diagonal
• Often described using hexagonal axes for convenience; the relationship between rhombohedral and hexagonal settings is a common exam topic
• Quartz, calcite, and bismuth adopt this structure—the three-fold rotational symmetry creates optical properties like birefringence in calcite

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Quick Reference Table

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Self-Check Questions

1. Which two crystal systems share the constraint of equal edge lengths but differ in their angular relationships? How does this affect their symmetry operations?

2. A material exhibits identical mechanical properties regardless of the direction of applied stress. Which Bravais lattice type(s) would you expect, and why?

3. Compare and contrast tetragonal and orthorhombic systems: what geometric constraint distinguishes them, and why does tetragonal have fewer centering options?

4. If you were indexing a diffraction pattern and found that no two angles equal 90°, which crystal system must you be working with? What additional challenge does this create?

5. Explain why face-centered tetragonal is not listed as a distinct Bravais lattice, even though face-centered cubic and face-centered orthorhombic both exist.