3.1 Seven crystal systems and their characteristics

Crystal systems classify every possible 3D lattice into seven categories based on symmetry and unit cell geometry. They're foundational to crystallography because the symmetry of a crystal's unit cell directly controls its physical properties, from how it interacts with light to how it conducts heat. The seven systems range from cubic (highest symmetry, most constrained) to triclinic (lowest symmetry, least constrained).

Crystal Systems: Symmetry and Shapes

Seven Crystal Systems Overview

Seven crystal systems account for all possible three-dimensional lattice structures: cubic, tetragonal, orthorhombic, hexagonal, trigonal, monoclinic, and triclinic. Each system is defined by specific relationships between its lattice parameters ($a, b, c$) and interaxial angles ($\alpha, \beta, \gamma$), along with a minimum set of symmetry elements.

Symmetry decreases as you move from cubic to triclinic. Here's a quick summary of each:

• Cubic has three equal axes at right angles. Its defining symmetry is four threefold rotation axes along the body diagonals.
• Tetragonal has two equal axes and one unique axis, all at right angles. It requires one fourfold rotation axis along the unique axis.
• Orthorhombic has three unequal axes, all at right angles. It contains three mutually perpendicular twofold rotation axes or mirror planes.
• Hexagonal uses four axes: three equal coplanar axes at 120° to each other, plus a fourth axis perpendicular to that plane. Its defining element is one sixfold rotation axis.
• Trigonal can be described using either a rhombohedral or hexagonal unit cell. Its defining element is one threefold rotation axis.
• Monoclinic has three unequal axes with one oblique angle (conventionally $\beta$). It contains one twofold rotation axis or one mirror plane.
• Triclinic has three unequal axes and three unequal angles, none of which are 90°. It possesses no rotational symmetry beyond the identity operation (onefold).

Symmetry Elements and Unit Cell Shapes

Each crystal system carries a characteristic set of symmetry elements. These elements constrain which point groups and space groups are possible within that system.

• Cubic (e.g., NaCl, diamond): Four threefold axes along body diagonals, three fourfold axes along cube edges, and nine mirror planes. This is the highest symmetry achievable in 3D.
• Tetragonal (e.g., rutile, zircon): One fourfold rotation axis, four mirror planes parallel to the unique axis, and one mirror plane perpendicular to it.
• Orthorhombic (e.g., topaz, aragonite): Three twofold rotation axes with three mirror planes perpendicular to them.
• Hexagonal (e.g., beryl, apatite): One sixfold rotation axis, six mirror planes parallel to the main axis, and one perpendicular mirror plane.
• Trigonal (e.g., calcite, corundum): One threefold rotation axis with three mirror planes intersecting at 120°.
• Monoclinic (e.g., gypsum, orthoclase): Only one twofold rotation axis or one mirror plane.
• Triclinic (e.g., albite, microcline): No rotation axes beyond identity. This is the lowest-symmetry system.

Crystal System Characteristics

Lattice Parameters and Angle Constraints

These constraints define each system. Memorizing them is essential for identification.

Note that the trigonal system can also be described using hexagonal axes, in which case it follows hexagonal lattice parameter constraints but with only threefold (not sixfold) rotational symmetry. The tetragonal unique axis ($c$) can be either longer or shorter than the other two axes.

Symmetry and Physical Property Relationships

Crystal symmetry directly governs optical, electrical, magnetic, and mechanical behavior. The core principle: higher symmetry produces more isotropic (direction-independent) properties, while lower symmetry leads to greater anisotropy.

• Cubic crystals show uniform thermal expansion and optical isotropy because all directions are symmetrically equivalent.
• Tetragonal, hexagonal, and trigonal crystals exhibit uniaxial optical properties, meaning they have one unique symmetry axis that makes properties along that axis differ from those perpendicular to it.
• Orthorhombic, monoclinic, and triclinic crystals display increasingly complex anisotropic behavior, with properties varying along multiple independent directions.

Symmetry also determines whether certain physical phenomena can occur at all. For example, piezoelectricity (generating voltage under mechanical stress) requires the absence of a center of symmetry. Quartz is piezoelectric precisely because its trigonal point group lacks an inversion center. Similarly, pyroelectricity is restricted to polar crystal classes.

The symmetry elements of a crystal system constrain its possible point groups and space groups, which in turn dictate which physical phenomena are allowed.

A couple of concrete examples:

• Cordierite (orthorhombic) expands at different rates along its three crystallographic axes, which is why it's used in thermal-shock-resistant ceramics.
• Calcite (trigonal) shows strong birefringence because its refractive index differs significantly along vs. perpendicular to the optic axis.

Identifying Crystal Systems

Analysis of Lattice Parameters and Angles

To identify a crystal system from measured lattice data, work through the constraints systematically. A decision-tree approach starting from the most constrained system is most efficient:

1. Check if all three axes are equal ($a = b = c$).

   • If yes, and all angles are 90°, it's cubic.
   • If yes, and all angles are equal but not 90°, it's trigonal (rhombohedral setting).

2. Check if two axes are equal ($a = b \neq c$).

   • If all angles are 90°, it's tetragonal.
   • If $\alpha = \beta = 90°$ and $\gamma = 120°$, it's hexagonal (or trigonal in hexagonal setting; you'll need symmetry information to distinguish them).

3. Check if all three axes are unequal ($a \neq b \neq c$).

   • If all angles are 90°, it's orthorhombic.
   • If only $\beta \neq 90°$ (with $\alpha = \gamma = 90°$), it's monoclinic.
   • If all three angles differ from 90°, it's triclinic.

Keep in mind that experimental measurements always carry some error. A measured angle of 89.8° might actually be 90° within uncertainty. Always supplement lattice parameter analysis with symmetry information (rotation axes, mirror planes) to confirm your assignment.

The trigonal system deserves special attention: it can be described with either a rhombohedral or hexagonal unit cell, so you may encounter the same crystal described both ways.

Practical Considerations in Crystal System Identification

Several experimental techniques help identify crystal systems in practice:

• X-ray diffraction gives the most precise lattice parameter measurements and reveals systematic absences that point to specific symmetry elements.
• Crystal morphology provides initial clues. Cubic crystals often grow as cubes or octahedra (pyrite, fluorite). Hexagonal crystals may form hexagonal prisms (quartz, beryl).
• Optical microscopy under crossed polarizers distinguishes isotropy (cubic) from uniaxial behavior (tetragonal, hexagonal, trigonal) and biaxial behavior (orthorhombic, monoclinic, triclinic). Interference figures are particularly diagnostic.
• Cleavage patterns reflect underlying symmetry. Halite cleaves along three directions at 90° (cubic cleavage), while calcite cleaves along three directions not at 90° (rhombohedral cleavage).
• Thermal and electrical conductivity measurements can reveal anisotropy. Cubic crystals conduct isotropically; lower-symmetry crystals show directional variation.
• Twinning patterns also provide symmetry clues. Carlsbad twinning in orthoclase indicates monoclinic symmetry, while polysynthetic twinning in plagioclase is characteristic of triclinic symmetry.

Symmetry and Physical Properties

Optical Properties and Crystal Symmetry

The relationship between crystal system and optical behavior is one of the most practically useful connections in crystallography.

• Cubic crystals are optically isotropic: light travels at the same speed regardless of direction, so there's no birefringence. Diamond and garnet are examples.
• Tetragonal, hexagonal, and trigonal crystals are uniaxial: they have two principal refractive indices (ordinary and extraordinary). Birefringence is observed for light traveling perpendicular to the optic axis. Calcite's famous double refraction is a classic demonstration.
• Orthorhombic, monoclinic, and triclinic crystals are biaxial: they have three distinct principal refractive indices and produce complex interference figures. Olivine and feldspar are common examples.

Optical activity (rotation of the plane of polarized light) occurs in enantiomorphic crystal classes. Quartz is the textbook example, with left-handed and right-handed varieties rotating light in opposite directions.

Electrical and Magnetic Properties

• Piezoelectricity requires a non-centrosymmetric structure. Quartz (trigonal) and tourmaline (trigonal) are widely used piezoelectric materials.
• Pyroelectricity (generating charge from temperature change) is limited to polar crystal classes. Tourmaline and lithium niobate are examples.
• Ferroelectricity (switchable spontaneous polarization) depends on specific symmetry conditions. Barium titanate (tetragonal below its Curie temperature) and potassium dihydrogen phosphate (KDP) are well-known ferroelectrics.
• Magnetic properties are also symmetry-dependent. Hexagonal cobalt shows strong magnetocrystalline anisotropy (preferring magnetization along its $c$-axis), while cubic iron is more magnetically isotropic.

Mechanical Properties and Symmetry Relationships

The number of independent elastic constants a crystal requires is directly tied to its symmetry:

• Cubic crystals need only 3 independent elastic constants.
• Triclinic crystals require 21 independent elastic constants.

This pattern holds across all mechanical properties. Higher symmetry means fewer independent parameters needed to describe the material's behavior.

Thermal expansion follows the same trend. Cubic crystals (aluminum, copper) expand uniformly in all directions. Low-symmetry crystals can expand at very different rates along different axes. Graphite and calcite are extreme examples of anisotropic thermal expansion.

Cleavage reflects both symmetry and bonding. Halite's perfect cubic cleavage (three directions at 90°) mirrors its cubic symmetry and uniform ionic bonding. Calcite's rhombohedral cleavage (three directions not at 90°) reflects its trigonal structure.

Hardness anisotropy is notable in non-cubic crystals. Kyanite is a striking example: it has a hardness of roughly 4.5 along one crystallographic direction but about 6.5 along another, all within the same crystal.

Plastic deformation mechanisms also depend on symmetry. Face-centered cubic metals (copper, aluminum) have many slip systems and deform easily. Hexagonal close-packed metals (zinc, magnesium) have fewer slip systems and tend to deform by twinning instead.