11.3
The Seven Crystal Systems: Overview
Crystals with various point group symmetries belong to different crystal classes, which are synonymous terms. Despite being in the same class, crystal…
Crystals are classified into seven crystal systems, namely cubic, trigonal, triclinic, monoclinic, orthorhombic, tetragonal, and hexagonal systems.
The simplest of these is the cubic system, where vectors a, b, and c are equal in length and orthogonal to each other. If only two vectors share the same length, it results in a tetragonal system.
In the trigonal crystal system, also called the rhombohedral system, the three lattice vectors are of equal length but inclined at equal, non-90-degree angles.
In a triclinic lattice, the defining vectors have no specific constraints; they are unequal in length and form unequal angles.
On the other hand, a monoclinic lattice necessitates that two of the angles are 90 degrees, while the third angle is not 90 degrees.
An orthorhombic lattice has three mutually perpendicular vectors of different lengths, whereas a tetragonal lattice also has three right angles but features two vectors of equal length.
Lastly, a hexagonal lattice system has two equal‑length vectors that subtend a 120° angle in a basal plane, while the third vector is perpendicular to this plane.
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Q1: What are the seven crystal systems and how do they differ?
The seven crystal systems are cubic, tetragonal, orthorhombic, monoclinic, triclinic, trigonal, and hexagonal. They differ based on the lengths and angles of their lattice vectors. Cubic systems have equal vectors at 90° angles, while triclinic systems have unequal vectors at unequal angles. Each system exhibits distinct symmetry properties that determine its geometric constraints.
Q2: What defines a cubic crystal system?
A cubic crystal system is the simplest system where all three lattice vectors (a, b, c) are equal in length and orthogonal to each other, with all angles at 90°. Sodium chloride (NaCl) exemplifies this system. Cubic crystals display high symmetry with multiple perpendicular axes and planes of symmetry.
Q3: How does a tetragonal lattice differ from a cubic lattice?
A tetragonal lattice has two vectors of equal length (a = b) while the third differs (c ≠ a), with all angles at 90°. Unlike cubic systems, tetragonal lattices introduce fourfold axes parallel to the c direction. This reduced symmetry compared to cubic systems creates a more elongated or compressed unit cell geometry.
Q4: What are the key characteristics of a triclinic lattice?
A triclinic lattice has no specific constraints on its defining vectors: they are unequal in length (a ≠ b ≠ c) and form unequal angles (α ≠ β ≠ γ). This system has the lowest symmetry, with only inversion centers as symmetry elements. Triclinic lattices are necessarily primitive and conventionally use the three shortest vectors.
Q5: How is a hexagonal lattice structured?
A hexagonal lattice has two equal-length vectors (a = b) that subtend a 120° angle in the basal plane, while the third vector is perpendicular to this plane. This configuration creates a six-fold rotational symmetry. The hexagonal system represents an intermediate level of symmetry between highly symmetric cubic and less symmetric triclinic systems.
Q6: What constraints define an orthorhombic lattice?
An orthorhombic lattice has three mutually perpendicular vectors of different lengths (a ≠ b ≠ c), with all angles at 90°. This system exhibits three mutually perpendicular sets of twofold axes and reflection planes. Orthorhombic lattices can be centered or primitive and represent an intermediate symmetry level between monoclinic and tetragonal systems.
Q7: Why can crystals in the same symmetry class have different external shapes?
Although crystals belong to the same point group or symmetry class, they can adopt different external shapes like cubes and octahedra because external morphology depends on growth conditions and environmental factors beyond internal symmetry. The 32 three-dimensional point groups are categorized into seven crystal systems based on unit-cell geometry, but symmetry elements alone do not determine final crystal shape.