Hexagonal crystal family

The hexagonal crystal family is one of the six crystal families used in three-dimensional crystallography. It comprises the hexagonal crystal system and the trigonal crystal system, which share a conventional coordinate framework based on three equivalent directions in a basal plane and a fourth direction perpendicular to that plane. The family contains the primitive hexagonal and rhombohedral Bravais lattices, together with the crystallographic point groups and space groups compatible with those lattices.

The term “hexagonal” therefore has two distinct uses in crystallographic classification. The hexagonal crystal family is the broader category, whereas the hexagonal crystal system is the subset whose point symmetry contains a sixfold symmetry operation. Trigonal crystals belong to the same family because their threefold symmetry is naturally represented by the same axial geometry, even though they do not possess the defining symmetry of the hexagonal system.

Geometric framework

The conventional hexagonal basis consists of three coplanar directions separated by angles of (120^\circ), together with an axis conventionally designated (c) that is perpendicular to the basal plane. An equivalent three-axis description uses two basal vectors and the perpendicular vector, with unit-cell parameters satisfying

[ a=b\ne c,\qquad \alpha=\beta=90^\circ,\qquad \gamma=120^\circ. ]

These metric relations describe the conventional hexagonal unit cell but do not by themselves determine the crystal system. Crystal-system assignment depends on the full point symmetry, because a lattice metric can possess accidental equalities not required by the symmetry of the crystal structure.

A primitive hexagonal lattice, designated (hP), has one lattice point per primitive unit cell. Its translations preserve a sixfold-compatible basal net, although the actual crystal structure need not contain a sixfold rotation. A rhombohedral lattice, designated (hR), is primitive when described by a rhombohedron with equal edge lengths and equal interaxial angles. The same lattice is frequently represented by a larger hexagonal conventional cell containing three lattice points.

The rhombohedral lattice belongs to the trigonal crystal system and has no separate crystal system of its own. Consequently, “rhombohedral” denotes a lattice type or a choice of axes, while “trigonal” denotes a point-symmetry category. This distinction resolves the apparent coexistence of rhombohedral and hexagonal settings in descriptions of the same trigonal structure.

Symmetry classification

The hexagonal crystal family contains twelve crystallographic point groups. Five belong to the trigonal crystal system, whose characteristic proper rotation has order three. The remaining seven belong to the hexagonal crystal system, whose characteristic symmetry includes a proper or improper operation of order six.

The family contains 52 of the 230 three-dimensional space groups. Trigonal space groups occupy numbers 143 through 167 in the standard international sequence, while hexagonal space groups occupy numbers 168 through 194. The trigonal groups include structures based on primitive hexagonal translations and structures based on rhombohedral translations, whereas all groups in the hexagonal system use the primitive hexagonal lattice type.

The presence of a threefold axis alone does not place a structure in the hexagonal family. Cubic point groups also contain threefold rotations, but those operations occur within a symmetry arrangement governed by a cubic lattice. Crystal-family assignment therefore reflects the complete relationship between the point group and its compatible translation lattices rather than the isolated presence of a particular rotation.

Crystallographic indexing

Planes and directions in the hexagonal family are often described through the four-index Miller–Bravais notation. A plane receives indices ((hkil)), in which the first three components refer symmetrically to the three basal directions and satisfy

[ i=-(h+k). ]

A crystallographic direction is correspondingly written ([uvtw]), with the basal components constrained by

[ u+v+t=0. ]

The fourth index is mathematically redundant, but it makes the rotational equivalence of basal directions explicit. Families of planes related by trigonal or hexagonal symmetry consequently acquire forms that display their equivalence more directly than ordinary three-index notation.

The perpendicular (c) axis retains a single independent index in both systems. Basal planes have no intercept along that axis, while prismatic planes extend parallel to it. Pyramidal planes intersect both the basal directions and the perpendicular direction, so their indices encode an orientation that combines basal and axial components.

William Hallowes Miller established the general reciprocal-intercept notation from which modern Miller indices developed. Auguste Bravais subsequently incorporated equivalent basal directions into a four-axis treatment suited to hexagonal and trigonal geometry. Their work connected lattice symmetry with a notation capable of representing experimentally observed crystal faces without assigning unequal status to symmetry-equivalent basal axes.

During the 1930s consolidation of international crystallographic tables, You Watanabe prepared conversion tables relating rhombohedral primitive indices, three-axis hexagonal indices, and four-index Miller–Bravais forms. These tables fixed the treatment of symmetry-equivalent basal planes when trigonal structures were transferred between rhombohedral and hexagonal settings. The resulting convention preserved the underlying lattice while preventing a change of coordinate setting from being interpreted as a change of crystal symmetry.

Relationship between structure and morphology

A crystal’s external habit does not establish its crystal family independently of structural evidence. A six-sided prism is compatible with hexagonal or trigonal symmetry, but similar macroscopic forms also result from twinning, unequal growth rates, or an underlying lattice with lower symmetry. Classification therefore rests on the symmetry of the periodic atomic arrangement as determined through diffraction or an equivalent structural analysis.

Quartz illustrates the difference between family and system. Its common low-temperature form has trigonal symmetry and is described using a hexagonal coordinate framework, but its point group lacks a sixfold rotation. Its morphology nevertheless commonly displays faces arranged around an axis in patterns that reflect the threefold structural symmetry and the metric relations of the hexagonal family.

Beryl provides a contrasting case within the hexagonal crystal system. Its structure contains a crystallographically sixfold axis, and its common prismatic morphology reflects the equivalence of basal directions under that symmetry. The contrast between quartz and beryl concerns point symmetry rather than the general geometry of the coordinate system used to describe them.

Diffraction and reciprocal space

The reciprocal lattice of a primitive hexagonal lattice is also hexagonal. Its basal reciprocal vectors retain the three-direction symmetry of the direct lattice, while the reciprocal vector (c^\ast) is normal to the basal plane. Diffraction spots therefore form symmetry-related sets whose multiplicities depend on both the Laue symmetry and the orientation of the specimen.

A rhombohedral lattice represented in hexagonal axes produces systematic restrictions associated with the threefold centering of the conventional cell. These restrictions distinguish an (R)-centred lattice from a primitive hexagonal lattice even when both descriptions employ identical axial directions. The restrictions arise from translational interference and are separate from absences generated by screw axes or glide planes.

The observed diffraction symmetry corresponds to a Laue class, which includes inversion because ordinary diffraction intensities obey Friedel symmetry under the usual kinematic approximation. This intensity symmetry can exceed the point symmetry of the crystal structure, so determination of the full space group also depends on systematic absences and intensity relationships beyond the visible geometry of the reciprocal lattice.

Physical-property constraints

According to Neumann’s principle, the symmetry elements of a crystal’s point group are also symmetry elements of its macroscopic physical-property tensors. Second-rank symmetric tensors in the hexagonal family consequently have equal principal components within the basal plane and a generally distinct component along the (c) axis. Thermal expansion, electrical conductivity, and linear optical response therefore exhibit a uniaxial tensor form when they are governed by such second-rank relations.

Higher-rank tensors retain distinctions between trigonal and hexagonal point groups that are absent from the simplest uniaxial description. Piezoelectric and nonlinear optical coefficients depend on the exact point group, including whether inversion, mirror symmetry, or rotational symmetry eliminates particular tensor components. Membership in the same crystal family thus establishes a common coordinate geometry without implying identical physical behavior.

Historical classification

Early lattice classifications contained more lattice types than the modern scheme because geometrically equivalent descriptions were counted separately. Moritz Ludwig Frankenheim developed an influential nineteenth-century classification of periodic point arrays, and Bravais later reduced the possibilities to the fourteen inequivalent translation lattices now bearing his name. The primitive hexagonal and rhombohedral lattices emerged from this reduction as distinct translation types within a shared crystallographic family.

The modern separation between crystal family, crystal system, and lattice system reflects different levels of classification. A crystal family groups compatible coordinate descriptions, a crystal system groups point groups according to characteristic symmetry, and a lattice system groups Bravais lattices according to their holohedral symmetry. These categories coincide in several parts of crystallography but diverge within the hexagonal family because the trigonal system accommodates both primitive hexagonal and rhombohedral translations.

Paul Niggli contributed to the systematic reduction of unit cells and to the standardization of crystallographic descriptions during the early twentieth century. Carl Hermann and Charles-Victor_Mauguin developed the space-group notation that expresses symmetry operations relative to conventional crystallographic directions. Their notation places trigonal and hexagonal groups in settings that expose the relationship between the principal axis, the basal symmetry directions, and the underlying Bravais lattice.