# Neumann's principle

Principe de Neumann (Fr). Neumannsches Prinzip (Ge). Principio di Neumann (It). ノイマンの法則 (Ja). Principio de Neumann (Sp).

## Definition

Neumann's principle, or principle of symmetry, states that, if a crystal is invariant with respect to certain symmetry operations, any of its physical properties must also be invariant with respect to the same symmetry operations, or otherwise stated, the symmetry operations of any physical property of a crystal must include the symmetry operations 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 rotation (as in a trigonal, tetragonal or hexagonal crystal, for instance), its optical indicatrix must also be invariant with respect to the same operation, 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's (1795-1898) 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].