Crystallography & Symmetry
patterns derived from their groups
Concept

Optic axis — where it appears

A direction along which light travels through a crystal with no birefringence, both polarisations at one speed. A uniaxial crystal has one and a biaxial crystal two, and only along them can a rotation of the plane of polarisation accumulate undisturbed.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Three shapes, and nothing else. The dielectric tensor of a crystal is an ellipsoid, and averaging a generic one over a point group leaves exactly three possibilities: a sphere, where all three principal values agree and the crystal is optically isotropic; a spheroid, where two agree and there is one optic axis; and a general ellipsoid, with two. The counts are 5, 19 and 8 of the thirty-two classes, and they were found by computing the eigenvalues rather than by sorting the classes by system.

Three optical characters, and the arithmetic that assigns them

A cubic crystal cannot be birefringent, whatever it is made of. Between crossed polars it stays dark at every rotation, and the reason is that averaging any ellipsoid over a cubic point group leaves a sphere — a permission computed before anybody measures anything.

point-groups · Optical Every crystal class permits the Faraday effect. The thirty-two crystal classes, each with two counts: the number of independent components of the tensor that turns a magnetic field into an optical rotation, and the number of components of the natural gyration tensor. The first is never zero: 5 classes have 1, 11 classes have 2, 11 classes have 3, 3 classes have 5, 2 classes have 9. The second is zero in 17 classes, every one with an inversion centre or a mirror arrangement that cancels it. The Faraday count is the mean of the squared trace over the class, which is never below one because the isotropic part always survives.

A field lets every class rotate light

Seventeen of the thirty-two classes forbid a crystal to rotate the plane of polarisation on its own. Put the same crystals in a magnetic field along the ray and not one of the thirty-two forbids it, the centrosymmetric ones included. The field supplies the one thing natural rotation needs a handed structure for, and time reversal is what tells the two rotations apart.

point-groups · Optical Where the two-fold axis sits decides what the coloured optic axes keep. Interference-figure positions of the two optic axes from 400 to 700 nm for four model crystals seen down the acute bisectrix. Orthorhombic: both mirrors survive and only the angle between the axes changes. Two-fold axis along the optic normal: the axes stay on one line and slide unequally (inclined dispersion), one mirror along that line. Along the obtuse bisectrix: the axial plane stays parallel and shifts (horizontal), one mirror across it. Along the acute bisectrix: the axial plane turns about the centre (crossed), a half-turn and no mirror.

Three places for the two-fold axis

A monoclinic crystal fixes one direction of its indicatrix, the two-fold axis, and an ellipsoid has three principal directions for it to be. Which one it is decides what an interference figure keeps in white light: a single mirror along the plane of the optic axes, a single mirror across it, or a half-turn and no mirror at all. Mineralogy calls the three inclined, horizontal and crossed dispersion. They are one subgroup computed three times, and in light of a single colour all three disappear.

point-groups · Optical Down one optic axis, at three colours, for two crystals. Interference figures of two model monoclinic crystals, inclined above and crossed below, each a plate cut across the optic axis at the sodium line, at 450, 589 and 650 nm. In the inclined crystal the melatope slides along the horizontal mirror and the figure stays symmetric across it. In the crossed crystal the melatope leaves the line away from sodium, and the pattern loses the mirror.

The mirror a plate keeps in one colour

Cut a plate of a crystal across one optic axis, and in the light the plate was cut for its interference figure has a perfect mirror, whatever the crystal's class. Change the colour and the mirror survives only if the crystal really has one. Computed ring by ring, five model crystals give the prediction exactly, including the part the eye would miss: the mirror does not come back when the line is moved to follow the melatope.

point-groups · Optical

Named alongside it

The objects these essays reach for when they reach for this one.

IndicatrixBirefringenceCurie principleDispersionLaue classNeumanns principleAxial vectorBiaxialDielectric tensorGyration tensorInvariant subspaceMagnetic point group

All concepts