Key Crystal Classes

Why This Matters

Crystal classes form the backbone of how we classify and predict the physical properties of crystalline materials. When you're working through crystallography problems, you're being tested on your ability to connect symmetry operations to lattice geometry—understanding why a mineral cleaves a certain way, why some crystals are optically isotropic while others aren't, and how atomic arrangement dictates everything from hardness to electrical conductivity. The seven crystal systems represent a hierarchy of symmetry, and recognizing where each class falls in that hierarchy is essential for solving structure-determination problems.

Don't just memorize axis lengths and angles—know what symmetry elements define each system and how increasing symmetry constrains geometric possibilities. Exam questions often ask you to identify a crystal system from its symmetry operations, predict properties based on class membership, or explain why certain minerals belong to specific systems. Master the underlying principles, and the specific examples will make intuitive sense.

---

Low-Symmetry Systems

These crystal classes have the fewest symmetry constraints, resulting in the most general unit cell geometries. The absence of symmetry elements means all lattice parameters can vary independently.

Triclinic

• No symmetry elements beyond identity—this is the lowest possible crystal symmetry, with only a center of inversion ($\bar{1}$) at most
• Three unequal axes ($a \neq b \neq c$) at three unequal angles ($\alpha \neq \beta \neq \gamma \neq 90°$), giving maximum geometric freedom
• Minerals like plagioclase feldspar exemplify this system—their complex optical properties stem directly from this low symmetry

Monoclinic

• Single two-fold axis or mirror plane—the defining symmetry element that constrains one interaxial angle to $90°$
• Two axes perpendicular, one tilted ($\alpha = \gamma = 90°$, $\beta \neq 90°$), creating the characteristic "slanted" unit cell
• Gypsum and orthoclase belong here—note how the single symmetry direction creates distinct cleavage behavior

---

Orthogonal Systems with Unequal Axes

These systems maintain perpendicular axes while allowing different axis lengths. The $90°$ angles simplify metric tensor calculations while axis inequality preserves directional property differences.

Orthorhombic

• Three mutually perpendicular two-fold axes—this combination of symmetry elements forces all angles to $90°$
• Three unequal axis lengths ($a \neq b \neq c$, $\alpha = \beta = \gamma = 90°$) create a rectangular parallelepiped unit cell
• Olivine and barite crystallize here—the orthogonal geometry simplifies diffraction pattern indexing considerably

Tetragonal

• Four-fold rotational axis along $c$—this single constraint forces $a = b$ while $c$ remains independent
• Square cross-section ($a = b \neq c$, all angles $90°$) means properties in the $ab$-plane are equivalent
• Zircon and rutile demonstrate tetragonal symmetry—their uniaxial optical behavior follows directly from this geometry

---

Hexagonal and Trigonal Systems

These systems share a geometric relationship based on three-fold or six-fold rotational symmetry. The $120°$ angle between equivalent axes reflects the underlying rotational symmetry.

Trigonal

• Three-fold rotational axis defines this system—symmetry operations repeat every $120°$
• Three equal axes at $120°$ angles ($a = b = c$, $\alpha = \beta = \gamma \neq 90°$) in the rhombohedral setting, or can use hexagonal axes
• Quartz and calcite are classic examples—quartz's piezoelectric properties arise from its specific trigonal point group ($32$)

Hexagonal

• Six-fold rotational axis—the highest-order rotation axis possible in a periodic lattice
• Four-axis notation ($a_1 = a_2 = a_3$, $c$ perpendicular) with the Miller-Bravais system using four indices $(hkil)$
• Graphite and beryl crystallize here—graphite's layered structure and easy cleavage reflect the hexagonal symmetry of its sheets

---

Maximum Symmetry: The Cubic System

The cubic system represents the highest possible symmetry in three-dimensional lattices. The equivalence of all three axes creates isotropic or near-isotropic physical properties.

Cubic

• Four three-fold axes along body diagonals—this is the defining feature, not the $90°$ angles (tetragonal has those too)
• Complete axis equality ($a = b = c$, $\alpha = \beta = \gamma = 90°$) means only one lattice parameter needed
• Diamond, halite, and pyrite exemplify cubic symmetry—their isotropic optical properties (no birefringence) follow from the high symmetry

---

Quick Reference Table

---

Self-Check Questions

1. Which two crystal systems both have all interaxial angles equal to $90°$ but differ in axis length constraints? What symmetry element distinguishes them?

2. A mineral displays optical birefringence with a single optic axis. Which crystal systems could it belong to, and which system can you immediately rule out?

3. Compare and contrast the trigonal and hexagonal systems: What symmetry operation defines each, and why is trigonal sometimes described using hexagonal axes?

4. If you're given lattice parameters $a = 5.0$ Å, $b = 5.0$ Å, $c = 7.2$ Å, and all angles = $90°$, what crystal system is this? What single measurement change would shift it to cubic?

5. Rank the seven crystal systems from lowest to highest symmetry. For an FRQ asking you to explain how symmetry affects physical properties, which two systems at opposite ends would make the strongest contrast?