Crystal System

Every mineral belongs to one of six crystal systems, defined by the geometry of its unit cell - specifically the lengths of the three cell edges (a, b, c) and the angles between them (α, β, γ). ^[1] The six systems capture every geometrically distinct unit cell shape that can tile three-dimensional space without gaps. Knowing a mineral’s crystal system is one of the first steps in characterising it, because the system determines which symmetry elements are possible and constrains how the crystal faces and cleavage planes are oriented.

The Six Systems

Triclinic is the system with the least symmetry. All three unit cell edges are different lengths, and none of the three angles between them equals 90° (a ≠ b ≠ c; α ≠ β ≠ γ ≠ 90°). ^[1] Triclinic minerals have no rotation axes higher than 1-fold and at most a centre of symmetry. This minimum symmetry allows the greatest freedom in unit cell geometry, which is why the cell edges and angles are all independent.

Monoclinic has three edges of different lengths. Two of the three inter-axial angles are 90°, but the angle β between the a and c axes is greater than 90° (a ≠ b ≠ c; α = γ = 90°, β > 90°). ^[1] The single axis of higher symmetry (a 2-fold rotation or mirror) is the b axis, which is why the angle departing from 90° is always the one involving a.

Orthorhombic has three edges of different lengths but all three inter-axial angles are exactly 90° (a ≠ b ≠ c; α = β = γ = 90°). ^[1] The right-angled axes reflect the presence of three mutually perpendicular 2-fold axes or mirror planes.

Tetragonal has two equal horizontal edges and one vertical edge of different length, with all angles at 90° (a = b ≠ c; α = β = γ = 90°). The c axis may be either longer or shorter than a and b. ^[1] The equality a = b reflects the presence of a 4-fold rotation axis parallel to c.

Hexagonal includes two sub-systems: the hexagonal division and the trigonal division. The unit cell has two equal axes (a = b) at 120° to each other, with a third axis (c) perpendicular to them and of different length (a = b ≠ c). ^[1] The hexagonal division is characterised by a 6-fold axis and the trigonal division by a 3-fold axis, both parallel to c.

Isometric (cubic) has three equal edges and all angles at 90° (a = b = c; α = β = γ = 90°). ^[1] The isometric system has the highest symmetry of the six, with four 3-fold rotation axes running diagonally through the unit cell. It is this combination of symmetry elements - not just the equal axes - that defines membership in this system.

Identifying Crystal Systems

Point group symmetry can be determined from a well-formed hand specimen by identifying the symmetry elements present in the crystal faces. ^[1] The Bravais lattice type within a crystal system - whether the lattice is primitive, body-centred, or face-centred - cannot be distinguished in a hand sample and requires X-ray diffraction. ^[1] The crystal system feeds directly into the mineral’s optical properties - whether it is optically isotropic or anisotropic, uniaxial or biaxial - and governs how cleavage planes and twinning are described.

Determining Crystal Class from Hand Specimens

When a well-formed crystal is available, the crystal class can be worked out by systematically examining its faces for three types of symmetry elements. The procedure follows a fixed sequence so that no element is missed.

The first step is to test for a center of symmetry (inversion center). Choose several points on one side of the crystal - corners are easiest - and check whether an imaginary line from each point, passed straight through the center of the crystal, arrives at a geometrically equivalent point on the opposite side. ^[1] If it does for all points tested, the crystal has a center of symmetry.

The second step is to identify any mirror planes. A mirror plane splits the crystal into two halves that are exact reflections of each other, so the angles between faces on one side of the mirror must be identical to the corresponding angles on the other side. ^[1]

The third step is to locate rotation axes. A rotation axis acts as an axle around which the crystal can be rotated to bring the same collection of faces back into view repeatedly within one full turn. ^[1] A 2-fold axis requires 180° of rotation to reproduce the same face arrangement; a 3-fold axis requires 120°; a 4-fold axis requires 90°; and a 6-fold axis requires 60°. ^[1] Once all symmetry elements have been identified, they are combined to match one of the 32 point groups - which places the mineral in its crystal class and therefore its crystal system.

How Minerals Are Distributed Across Crystal Systems

Minerals are not spread evenly across the six crystal systems or the 32 crystal classes. Monoclinic minerals dominate, accounting for 35.1% of all minerals whose crystal class is known, and within the monoclinic system the 2/m class alone accounts for 31.0% - the single most populated crystal class. ^[1] Orthorhombic minerals form the second-largest group at 19.3%, with the 2/m 2/m 2/m class contributing 14.0% of all minerals. The isometric system, despite the visual familiarity of its cubic and octahedral crystals, accounts for only 9.6% of minerals, with the 4/m 3̄ 2/m class representing 5.9%.

The practical implication of this distribution is that the minerals a geologist encounters most often - feldspars, amphiboles, pyroxenes, micas, carbonates - crystallise predominantly in the monoclinic and orthorhombic systems. Familiarity with those two systems gives a working understanding of the crystal geometry of the majority of rock-forming minerals.

Alternative Nomenclature: Seven Crystal Systems

An alternative to the six-system classification is in common use, particularly in older literature. Under this scheme, minerals with a single 3-fold rotation axis are separated from the hexagonal system and placed in a distinct system called trigonal, producing a total of seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and isometric. ^[1] To accommodate this split, the scheme introduces the concept of a crystal family - a broader grouping that in this case bundles the hexagonal and trigonal crystal systems together into the hexagonal crystal family. ^[1] For the other five systems - triclinic, monoclinic, orthorhombic, tetragonal, and isometric - the crystal family is identical to the crystal system, so the family concept adds no new content for them. ^[1]

The seven-system nomenclature makes it easier to talk about trigonal minerals (calcite, quartz, tourmaline, dolomite) as a distinct group, since their 3-fold symmetry produces recognisably different crystal forms - rhombohedra, trigonal prisms - from those of true hexagonal minerals. The six-system scheme, which this text uses, assigns both divisions to a single hexagonal system and distinguishes them by talking about the hexagonal and trigonal divisions within that system. Either approach is internally consistent; the key is to know which scheme a given source is using.

Axis Orientation Conventions by Crystal System

How the crystal axes a, b, and c are assigned to a real crystal is not always uniquely determined by symmetry alone - in several systems, conventions and practical considerations guide the choice. These conventions matter because the Miller indices for any given face, cleavage, or direction depend entirely on which axis orientation has been adopted. Published sources can differ, and the same physical face may carry a different Miller index depending on which convention was used.

Triclinic. There are no symmetry constraints on axis orientation in the triclinic system, so conventions are chosen pragmatically. The c axis is generally placed parallel to the dominant zone axis of the crystal, and the a and b axes are placed parallel to crystal edges or other rational directions. ^[1] The axes are subsequently arranged to ensure that both the α and β angles exceed 90°. ^[1] Because no symmetry anchor exists, different researchers have historically adopted different axis orientations for the same mineral, and this inconsistency is common even in standard references. ^[1]

Monoclinic. The b axis is placed coincident with the 2-fold rotation axis, or perpendicular to the mirror plane - whichever applies to the mineral’s class - because the b axis is the one unique symmetry direction in the monoclinic system. ^[1] The c axis is placed parallel to a prominent zone axis, and a is tilted forward so that the β angle exceeds 90°. ^[1] This is the second setting - the standard used in all modern mineralogic literature, including this text. ^[1] The 2/m class dominates the monoclinic system and includes amphiboles, pyroxenes, micas, and many other rock-forming minerals. ^[1]

Orthorhombic. The symmetry of the orthorhombic system forces three unique directions at right angles, each corresponding to a crystal axis, but does not dictate which physical direction gets which label. ^[1] A common convention assigns the axes so that the unit cell dimensions satisfy c < a < b, but this is not universal. ^[1] Minerals with a markedly elongate habit often have their long axis assigned to c regardless of unit cell dimensions, and a mineral with a single prominent cleavage or tabular pinacoid may have c placed perpendicular to that surface. Two equivalent prismatic cleavages are oriented parallel to c when present. ^[1] The reader is advised to verify axis conventions when consulting literature on any specific orthorhombic mineral.

Tetragonal. The 4-fold rotation or rotoinversion axis, which is the defining symmetry element of the tetragonal system, always coincides with the c axis. ^[1] The mutually orthogonal a and b axes act equivalently; they are oriented to overlap with 2-fold rotation pathways or to stand perpendicular to any reflective mirrors that exist. ^[1] In the Hermann-Mauguin symbol, the first element refers to the c axis, the second to the a and b axes, and the third to the diagonal directions at 45° between them. Most tetragonal minerals crystallise with 4/m 2/m 2/m symmetry. ^[1]

Hexagonal. The c axis is always placed parallel to the 6-fold or 3-fold rotation axis, which is the unique symmetry direction in both the hexagonal and trigonal divisions. ^[1] The a and b axes are at 120° to each other, aligned parallel to 2-fold rotation axes or perpendicular to mirrors when these are present. In the Hermann-Mauguin notation, the first symbol refers to c, the second to the a and b axes, and the third to the directions perpendicular to a and b. The hexagonal prism is the most visually characteristic form of the system and is possible in all hexagonal point groups except the 6̄ and 3 classes, which makes it a useful first indicator for identifying hexagonal minerals in hand specimens. ^[1] Distinguishing between individual hexagonal point groups is often difficult in practice, because the faces that would reveal reduced symmetry relative to the 6/m 2/m 2/m ideal may be too small to recognise clearly or may not have developed at all. ^[1] On most real specimens, visual inspection can reliably distinguish only between the hexagonal division (6-fold symmetry parallel to c) and the trigonal division (3-fold symmetry parallel to c). ^[1]

Isometric. The three crystallographic axes in the isometric system are mutually perpendicular and all equal in length. They are conventionally labelled a, b, and c, though in some literature they appear as a₁, a₂, and a₃ to emphasise their equivalence. ^[1] Three symmetry axes of equivalent rank coincide with the three crystallographic axes. In three of the five isometric crystal classes, these coinciding symmetry axes are 4-fold rotation or 4-fold rotoinversion axes; in the remaining two classes they are 2-fold axes. ^[1] Hermann-Mauguin symbols for isometric crystal classes consist of up to three parts. The first refers to the symmetry of the three crystallographic axes, the second to the three body-diagonal directions [111] running corner-to-corner through the cubic unit cell, and the third to the six face-diagonal directions [110] connecting the midpoints of opposite edges. ^[1] This three-part structure is unique to the isometric system; no other crystal system requires a separate symbol for body-diagonal directions, because only the isometric system has three equivalent axes whose mutual symmetry automatically generates such diagonals as independent symmetry directions.

References

1. Nesse, W. D. (2018). Introduction to Mineralogy, 3rd ed. Oxford University Press.

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