Crystallography & Symmetry
patterns derived from their groups
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Three shapes, and nothing else

Three shapes, and nothing else
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.

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.

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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. What symmetry decides

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.

 15 classes may rotate light, 11 of them chiral. A crystal is chiral when its point group contains no improper operation, and there are 11 such classes. A crystal may rotate the plane of polarisation when its class permits a non-zero gyration tensor, and there are 15. The four in the difference — 4̅, m, 4̅2m, mm2 — are achiral and may still rotate light, which is why the two words are not synonyms. In each of the four, symmetry forces the tensor to be traceless, so the rotation changes sign with direction and cancels in any average over directions. What symmetry decides

Fifteen may rotate light, and eleven are chiral

Optical rotation and handedness are treated as the same thing and are not. Eleven crystal classes are chiral; fifteen permit a crystal to rotate the plane of polarisation; and the four in between are the reason quartz and sodium chlorate are the examples everybody uses.

 Where the axes are free, they move. Five different invariant tensors of each of three classes, with the trace of each tensor's principal axes on the page. In the orthorhombic class every sample gives the same three directions: the axes are the two-fold axes and symmetry has fixed them. In the monoclinic class one direction is common to every sample and the other two rotate freely in the plane across it. In the triclinic class nothing is common at all. Each sample stands for a different material, or the same material at a different wavelength — which is what makes the middle picture the dispersion of the optic axes. What symmetry decides

The axes a class pins down

A property tensor has a shape and an orientation, and symmetry treats them differently. Three principal directions fixed for ever in an orthorhombic crystal; one in a monoclinic one, with the other two turning as the wavelength changes.

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