Crystal Systems

All crystals belong to one of seven crystal systems, classified by their unit cell
geometry and symmetry elements. Understanding these systems is fundamental to
crystallography and gem identification.

Crystal systems are defined by the relationship between crystallographic axes
(a, b, c) and the angles between them (α, β, γ). Each system has characteristic
symmetry elements that determine possible crystal forms.

When examining a crystal or gemstone, you can often determine its crystal system
by observing:

• Crystal habit - The external shape and face arrangement
• Optic character - Isotropic (cubic) vs. uniaxial vs. biaxial
• Interference figure - Under polarized light microscopy
• Cleavage directions - Related to internal symmetry

A point group is the set of symmetry operations (rotation axes, mirror planes, inversion
centre) that leave at least one point of a crystal unmoved. There are exactly 32 unique
combinations possible in crystals (the 32 crystal classes), expressed in Hermann–Mauguin
(H–M) notation.

Key notational elements:

• n = n-fold rotation axis (2, 3, 4, or 6)
• m = mirror plane
• n̄ = rotoinversion axis (improper rotation)
• 1̄ = inversion centre
• n/m = rotation axis perpendicular to a mirror plane

Point groups determine whether a crystal can be piezoelectric (no centre of inversion),
pyroelectric (polar point group), or optically active (enantiomorphic point group).

The gem species assignments below are drawn from standard mineralogical sources. Where
no common gem species is assigned to a class, the entry is shown as a dash; these classes
exist and are crystallographically established even without a prominent gem example.

Source: Schwarzenbach, Journal of Applied Crystallography, 2003.
DOI: 10.1107/s0021889803014778 [VERIFIED]; Read 3rd ed.
DOI: 10.4324/9780080507224 [VERIFIED]

Several point groups have gemmologically significant consequences:

• Quartz (point group 32): the enantiomorphic class explains why quartz is optically
  active (rotates plane of polarisation), exists as left-handed and right-handed crystals,
  and shows the Brazil and Dauphiné twin laws. No centre of inversion → piezoelectric.
• Tourmaline (point group 3m): polar axis along c responsible for piezoelectricity and
  pyroelectricity. The polar nature also produces pyroelectric discharge when temperature
  changes – tourmaline crystals attract dust.
• Corundum (point group 3̄m): has a centre of inversion (the 3̄ inversion axis);
  therefore neither piezoelectric nor optically active. Uniaxial negative with pleochroism
  oriented along the c-axis.
• Diamond (point group m3̄m): highest possible cubic symmetry (Oh); isotropic; single RI;
  no birefringence, no pleochroism.
• Benitoite (point group 6̄m2): hexagonal symmetry class; notable for intense SWUV
  fluorescence and dispersion equal to diamond (0.044); uniaxial positive.

Miller indices {hkl} describe the orientation of a crystal face or plane by the reciprocals
of its fractional intercepts on the crystallographic axes.

Three-index notation {hkl} – used for cubic, tetragonal, orthorhombic, monoclinic, and
triclinic systems (three axes: a, b, c):

• {100}: intercepts a at 1, parallel to b and c.
• {111}: intercepts all three axes equally – the octahedral face in cubic.
• {110}: intercepts a and b at 1, parallel to c – the dodecahedral face in cubic.

Four-index (Bravais–Miller) notation {hkil} – required for trigonal and hexagonal
systems, which have three equivalent horizontal axes (a₁, a₂, a₃) at 120° and one vertical
c-axis:

• The closure relation i = −(h+k) always holds; i is not an independent variable but
  makes the symmetry equivalence of faces explicit in the notation.
• {10-10}: the hexagonal prism face. i = -(1+0) = -1, confirming h + k + i = 0.
• {0001}: the basal pinacoid – perpendicular to the c-axis; the "table" of a hexagonal crystal.
• {10-11}: the rhombohedral face in trigonal – the twin and parting plane of corundum.

Why the redundant index? In a 6-fold-symmetric hexagonal crystal there are six
equivalent prism faces. The 4-index notation makes this equivalence explicit: {10-10},
{01-10}, {-1100}, {-1010}, {0-110}, {1-100} are all the same form, and the notation
allows this to be seen by inspection.

Source: Schwarzenbach, Journal of Applied Crystallography, 2003.
DOI: 10.1107/s0021889803014778 [VERIFIED]

Parting is a tendency for a crystal to break along planes that are not true cleavage planes.
It resembles cleavage (producing flat, reflective fracture surfaces) but is caused by
repeated twinning lamellae or by exsolution of a second phase along a crystallographic plane.

Distinction from cleavage:

• Cleavage is universal – every crystal of the species will cleave along the same planes,
  because the planes reflect inherently weak bonds in the crystal structure.
• Parting is contingent – only crystals that happen to be twinned or to have undergone
  exsolution show parting. It may be restricted to parts of the crystal and is not universal
  across all specimens of the species.

Source: Read 3rd ed. DOI: 10.4324/9780080507224 [VERIFIED];
Pignatelli 2024, DOI: 10.1007/s00710-024-00858-1 [VERIFIED]