Neumann's principle

From Online Dictionary of Crystallography

Principe de Neumann (Fr). Neumann Prinzip (Ge). Principio de Neumann (Sp).

Definition

Neumann's principle, or principle of symmetry, states that, if a crystal is invariant with respect to certain symmetry elements, any of its physical properties must also be invariant with respect to the same symmetry elements.

Example

This principle may be illustrated by considering the optical indicatrix of a crystal, which is an ellipsoid. If the medium is invariant with respect to a three--fold, a four--fold or a six--fold axis (as in a trigonal, tetragonal or hexagonal crystal, for instance), its optical indicatrix must also be invariant with respect to the same axis, according to Neumann's principle. As an ellipsoid can only be ordinary or of revolution, the indicatrix of a trigonal, tetragonal or hexagonal crystal is necessarily an ellipsoid of revolution. These crystals are said to be uniaxial. In a cubic crystal which has four three--fold axes, the indicatrix must have several axes of revolution, it is therefore a sphere and cubic media behave as isotropic media for properties represented by a tensor of rank 2.

History

Neumann F.E.(1833). Poggendorf Ann. Phys., 27, 240-274, Die thermischen, optischen und kristallographischen Axen des Kristallsystems des Gypses.