Crystals with various point group symmetries belong to different crystal classes, which are synonymous terms. Despite being in the same class, crystals may have distinct shapes, like cubes and octahedra. There are 32 three-dimensional point groups, all of which are systematically divided into seven crystal systems.
The basic cubic crystal system, exemplified by NaCl, features orthogonal vectors (α = β = �� = 90°) of equal lengths (a = b = c). When specific requirements are not imposed on the defining vectors, indicated by a ≠ b ≠ c, and α ≠ β ≠ ��, termed a triclinic lattice. The triclinic lattice lacks the selection of a unique vector set, but conventionally, the three shortest vectors are chosen. Its only symmetry elements are inversion centers, and it is necessarily primitive.
A monoclinic lattice necessitates one vector perpendicular to the plane of the other two, demonstrating two-fold rotational symmetry and perpendicular planes of symmetry. Monoclinic lattices can be centered or primitive.
Further constraints on the defining vector set lead to an orthorhombic lattice, where three vectors of different lengths (a ≠ b ≠ c) are orthogonal (α =β = �� = 90°). This lattice showcases three mutually perpendicular sets of twofold axes and reflection planes. Trigonal, also called Rhombohedral system, has three equal-length vectors (a =b = c), all intersecting at 120°.
Introducing the condition of two vectors being of equal length (a = b) yields a tetragonal lattice. This lattice shares mirror planes and twofold axes with an orthorhombic lattice but introduces fourfold axes parallel to the c direction. If two vectors are orthogonal and equal in length, the third vector at a 120° angle forms a hexagonal lattice system.
In conclusion, the 32 point groups are categorized into seven crystal systems based on the unit-cell lengths and angles. As geometric constraints increase—from triclinic through monoclinic and orthorhombic to tetragonal, trigonal/rhombohedral, hexagonal, and finally cubic—the symmetry elements become more defined. Even within the same symmetry class, crystals can adopt different external shapes.
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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