**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.

**Curie laws**

Curie extended the notion of symmetry to include that of physical phenomena and stated that:

- the symmetry characteristic of a phenomenon is the highest compatible with the existence of the phenomenon;
- the phenomenon may exist in a medium which possesses that symmetry or that of a subgroup of that symmetry.

and concludes that some symmetry elements may coexist with the phenomenon but that their presence is not necessary. On the contrary, what is necessary is the absence of certain symmetry elements: ‘asymmetry creates the phenomenon’. Noting that physical phenomena usually express relations between a cause and an effect (an influence and a response), P. Curie restated the two above propositions in the following way, now known as Curie laws, although they are not, strictly speaking, laws (Curie himself spoke about 'the principle of symmetry'):

- the asymmetry of the effects must pre-exist in the causes;

- the effects may be more symmetric than the causes.

### Applications

Curie applied the above statements to determine the symmetry characteristic of physical quantities such as a polar vector, a force or an electrical field, *A*_{∞} ∞*M*, an axial vector or a magnetic field, (*A*_{∞} /*M*) *C*.

If one now considers a phenomenon resulting from the superposition of several causes in the same medium, one may note that the symmetry of the global cause is the intersection of the groups of symmetry of the various causes: the asymmetries add up. This remark can be applied to the determination of the point groups where physical properties such as pyroelectricity or piezoelectricity are possible.

### References:

[1] 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.

[2] Pierre Curie (1859-1906)'s principle of symmetry is stated in Curie P., 1894, *J. Physique*, **3**, 393-415, *Sur la symétrie dans les phénomènes physiques, symétrie d'un champ électrique et d'un champ magnétique*.

[3] https://Chemistry/Crystallography/PhysicalPropertiesCrystals/NeumannPrinciple

[4] https://oldwww.iucr.org/iucr-top/comm/cteach/pamphlets/18/node4.html

[5] https://reference.iucr.org/dictionary/Neumann%27s_principle