International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.1, p. 11

## Section 1.1.4.1. Introduction – Neumann's principle

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France

#### 1.1.4.1. Introduction – Neumann's principle

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We saw in Section 1.1.1 that physical properties express in general the response of a medium to an impetus. It has been known for a long time that symmetry considerations play an important role in the study of physical phenomena. These considerations are often very fruitful and have led, for instance, to the discovery of piezoelectricity by the Curie brothers in 1880 (Curie & Curie, 1880, 1881). It is not unusual for physical properties to be related to asymmetries. This is the case in electrical polarization, optical activity etc. The first to codify this role was the German physicist and crystallographer F. E. Neumann (1795–1898). In a series of papers (Neumann, 1832, 1833, 1834) he had studied the relations between the orientations of the mechanical, thermal and optical axes on the one hand and that of the crystalline axes on the other. His principle of symmetry 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, 1885). It is now called Neumann's principle: 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.

This principle may be illustrated by considering the optical properties of a crystal. In an anisotropic medium, the index of refraction depends on direction. For a given wave normal, two waves may propagate, with different velocities; this is the double refraction effect. The indices of refraction of the two waves vary with direction and can be found by using the index ellipsoid known as the optical indicatrix (see Section 1.6.3.2 ). Consider the central section of the ellipsoid perpendicular to the direction of propagation of the wave. It is an ellipse. The indices of the two waves that may propagate along this direction are equal to the semi-axes of that ellipse. There are two directions for which the central section is circular, and therefore two wave directions for which there is no double refraction. These directions are called optic axes, and the medium is said to be biaxial. If the medium is invariant with respect to a threefold, a fourfold or a sixfold axis (as in a trigonal, tetragonal or hexagonal crystal, for instance), its ellipsoid 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 that has only one circular central section and one optic axis. These crystals are said to be uniaxial. In a cubic crystal that has four threefold 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.

### References

Curie, J. & Curie, P. (1880). Développement par pression de l'électricité polaire dans les cristaux hémièdres à faces inclinées. C. R. Acad. Sci. 91, 294–295.
Curie, J. & Curie, P. (1881). Contractions et dilatations produites par des tensions électriques dans les cristaux hémièdres à faces inclinées. C. R. Acad. Sci. 93, 1137–1140.
Neumann, F. (1832). Theorie der doppelten Strahlenbrechung, abgeleitet den Gleichungen der Mechanik. Poggendorf Ann. Phys. Chem. 25, 418–454.
Neumann, F. (1834). Über das Elastizitätsmass kirstallinscher Substanzen der homoëdrischen Abteilung. Poggendorf Ann. Phys. Chem. 31, 177–192.
Neumann, F. (1885). Vorlesungen über die Theorie der Elastizität der festen Körper und des Lichtäthers, edited by O. E. Meyer. Leipzig: B. G. Teubner-Verlag.