# Neumann's principle

(Difference between revisions)
 Revision as of 07:48, 11 November 2010 (view source)← Older edit Latest revision as of 13:11, 16 November 2017 (view source) (Tidied translations and corrected German (U. Mueller)) (3 intermediate revisions not shown) Line 1: Line 1: - Principe de Neumann (''Fr''). Neumannsche Prinzip (''Ge''). Principio de Neumann (''Sp''). Principio di Neumann (''It'') + Principe de Neumann (''Fr''). Neumannsches Prinzip (''Ge''). Principio di Neumann (''It''). ノイマンの法則 (''Ja''). Principio de Neumann (''Sp''). == Definition == == 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, 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]]. + Neumann's principle, or principle of symmetry, states that, if a crystal is invariant with respect to certain [[symmetry operation]]s, 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 == == 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. + 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 == == 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. + 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]. == See also == == See also == - [[Curie laws]]
+ *[[Curie laws]] - [http://www.iucr.org/iucr-top/comm/cteach/pamphlets/18/ An introduction to crystal physics]  (Teaching Pamphlet of the ''International Union of Crystallography'')
+ *[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/18/ ''An introduction to crystal physics'']  (Teaching Pamphlet No. 18 of the International Union of Crystallography) - Section 1.1.4 of ''[http://it.iucr.org/D/ International Tables of Crystallography, Volume D]'' + *Chapter 1.1.4 of ''[http://it.iucr.org/D/ International Tables for Crystallography, Volume D]'' - + - ---- + [[Category:Physical properties of crystals]]
[[Category:Physical properties of crystals]]

## Latest revision as of 13:11, 16 November 2017

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].