Lattice parameters, also called cell parameters or unit-cell parameters, are the geometric quantities that define the repeating box (the unit cell) from which a crystal is built. A complete set of lattice parameters comprises three lengths and three angles. The lengths, conventionally written a, b, and c, are the edges of the unit cell along three crystallographic axes. The angles, conventionally written alpha, beta, and gamma, are the interaxial angles, where alpha is the angle between b and c, beta between a and c, and gamma between a and b.

Reduction by crystal system

Most crystal systems impose constraints that reduce the number of independent parameters. In the cubic system (diamond, garnet, spinel, fluorite), a = b = c and all angles are 90 degrees, so a single parameter a describes the cell. In the tetragonal system (zircon, scapolite, idocrase), a = b not equal to c and all angles are 90 degrees, requiring two parameters. In the hexagonal and trigonal systems (corundum, beryl, quartz, tourmaline), a = b not equal to c, alpha = beta = 90 degrees, and gamma = 120 degrees (hexagonal axes), again requiring two parameters. In the orthorhombic system (topaz, peridot, andalusite), all three lengths differ but all angles are 90 degrees, so three parameters define the cell. The monoclinic system (orthoclase, jadeite, sphene) has all lengths different, alpha = gamma = 90 degrees, and beta as a single non-orthogonal angle, giving four parameters. The triclinic system (microcline, kyanite, axinite) has no constraints and requires the full six parameters.

Measurement

Lattice parameters are determined by X-ray diffraction, in which a beam of monochromatic X-rays scatters from the periodic planes of atoms in the crystal according to Bragg's law (n lambda = 2 d sin theta), where lambda is the X-ray wavelength, d is the spacing between atomic planes, theta is the diffraction angle, and n is an integer. By measuring the angles at which diffraction peaks occur for many sets of planes, the complete set of lattice parameters can be calculated to four or five significant figures of precision.

For routine gemmology, X-ray diffraction is rarely needed. Refractive index, specific gravity, and spectroscopy give species identification with much less expensive equipment. Lattice-parameter measurement enters the trade primarily through research, treatment characterisation, and the description of new mineral species, where the International Mineralogical Association requires precise unit-cell determination as part of any approval.

Variation with composition

Lattice parameters vary with chemical composition because different atoms have different ionic radii. In a solid-solution series, parameters change linearly between end members (Vegard's law), allowing approximate composition determination from a single measurement. The pyrope-almandine garnet series provides a clear example: a ranges from approximately 11.456 angstroms (pyrope, magnesium-rich) to 11.526 angstroms (almandine, iron-rich), with intermediate compositions interpolating between.

Lattice parameters also vary slightly with temperature and pressure. Thermal expansion coefficients are tabulated in the literature and matter for high-precision X-ray work but are negligible at room temperature for trade purposes.

Distortion under stress

External stress (pressure, mechanical loading) distorts the unit cell, changing lengths and angles slightly. This is exploited in pressure calibration in diamond-anvil cell experiments, where the lattice parameters of a reference standard (typically ruby) are measured in situ to determine the applied pressure.

Reference data

Comprehensive lattice-parameter data for gemstone species are published in the GIA Gem Reference Guide, in Anthony, Bideaux, Bladh and Nichols' Handbook of Mineralogy, and in the IMA-CNMNC mineral database. The Crystallographic Open Database (COD) provides searchable structure files for thousands of mineral species at no cost. For a working gemmologist, an awareness of lattice parameters supports the deeper understanding of why a given species has a given refractive index, specific gravity, and habit, even when direct measurement is not part of routine practice.