# Neumann's Principle

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, or otherwise stated, the symmetry elements of any physical property of a crystal must include the symmetry elements of the point group of the crystal. It is generalized to physical phenomena by Curie laws.

### 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

Franz Neumann (1795-1898)'s principle was first stated in his course at the university of Königsberg (1873/1874) and was published in the printed version of his lecture notes (Neumann F.E., 1885, *Vorlesungen über die Theorie der Elastizität der festen Körper und des Lichtäthers*, edited by O. E. Meyer. Leipzig, B. G. Teubner-Verlag.

### See also

Section 1.1.4 of *International Tables of Crystallography, Volume D*