Mirror Plane — A Crystallographic Symmetry Element
Mirror Plane — A Crystallographic Symmetry Element
The plane of reflection that helps assign crystals to point groups and space groups
A mirror plane, in crystallography, is an imaginary plane that bisects a crystal such that one half is the mirror image of the other. The element is denoted by the symbol m in Hermann-Mauguin notation, or by σ (sigma) with various subscripts in Schoenflies notation. Mirror planes are one of the four fundamental symmetry operations used in classifying crystals — alongside rotation axes, inversion centres, and rotoinversion axes — and the identification of mirror planes is a key step in placing a crystal into one of the 32 crystallographic point groups and one of the 230 space groups.
How mirror planes appear in real crystals
Most well-formed crystals display mirror symmetry directly in their external faces. A cubic crystal of pyrite or galena displays multiple mirror planes — three perpendicular to the cube faces and others through the diagonals — that the trained eye can identify by inspection. A monoclinic crystal of orthoclase typically shows a single mirror plane perpendicular to the unique b axis. The internal atomic arrangement determines the external symmetry, so the visible faces are an honest record of the underlying structure.
For gemmological work, the practical relevance is that mirror planes constrain the orientation of optical, mechanical, and electrical properties within the crystal. The refractive indices of biaxial crystals, for instance, lie along three mutually perpendicular axes whose orientations are determined by the symmetry, and the same is true of cleavage directions, thermal-expansion behaviour, and any other directional property.
The 32 point groups and the role of mirrors
The 32 crystallographic point groups arise from the possible combinations of the symmetry operations consistent with a periodic atomic structure. Of these 32, eleven are centrosymmetric (containing an inversion centre), and a further subset contain mirror planes in various combinations with rotation axes. The presence or absence of mirror symmetry has important consequences for physical properties: piezoelectricity and pyroelectricity, for instance, can only occur in crystal classes lacking inversion symmetry, and many of those classes also lack mirror planes.
Quartz is the classic example of a non-centrosymmetric, mirror-deficient crystal whose lack of certain symmetry elements gives rise to its piezoelectric properties — the basis of quartz oscillators in watches and many other technologies.
In gemmology
Gemmological practice does not generally require explicit mirror-plane identification, but the consequences of crystal symmetry are everywhere. The orientation of optic axes for pleochroism and refractive index measurement, the development of cleavage in feldspar and topaz, the oriented inclusion patterns in rubies and sapphires, and the strict optical behaviour of birefringent crystals under crossed polars all derive from the underlying symmetry of which mirror planes are an element.