Neumann’s principle

From this general law it is possible to infer probable properties, since according to the principle of Neumann the properties cannot be less symmetrical than the structure. Neumann s principle states that "The symmetry elements of any physical property of a crystal must include the symmetry elements of the point group of the crystal". Thus, a centro-symmetric crystal cannot by pyroelectric, since it would require that the two symmetrically related ends behave differently towards a change of temperature. 

The diffusivity tensor has special forms for particular choices of coordinate axes if the diffusing body itself has special symmetry (e.g., if it is crystalline). Neumann s principle states ... 

A consequence of Neumann s symmetry principle is that direct tensor Onsager coefficients (such as in the diffusivity tensor) must be symmetric. This is equivalent to the addition of a center of symmetry (an inversion center) to a material s point group. Thus, the direct tensor properties of crystalline materials must have one of the point symmetries of the 11 Laue groups. Neumann s principle can impose additional relationships between the diffusivity tensor coefficients Dij in Eq. 4.57. For a hexagonal crystal, the diffusivity tensor in the principal coordinate system has the form... 

Additional symmetries arise when the tensors Xj and/or Y, are symmetric, and from crystal symmetry in accordance with Neumann s principle, as seen in Section 15.2. These symmetries are properties of the tensor and the crystal point group, and, if different physical properties may be represented by the same kind of tensor, it will exhibit the same structure, irrespective of the actual physical property under consideration. 

There are two important points to remember regarding the applicability of Neumann s principle. First, forces imposed on a crystal, including mechanical stresses and electric fields, can have any arbitrary direction or orientation. These types... 

Unlike stress and strain, which are field tensors, elasticity is a matter tensor. It is subject to Neumann s principle. Hence, the number of independent elastic coefficients is further reduced by the crystal symmetry. The proof is beyond the scope of this book (the interested reader is referred to Nye, 1957), here the results will merely be presented. For example, even with triclinic crystals, the lowest symmetry class, there are only 21 independent elastic-stiffness coefficients ... 

Spontaneous strain is a symmetrical second rank tensor property and must conform to Neumann s principle in relation to symmetry. A general spontaneous strain with all six of the independent strain components having non-zero values can be referred to an alternative set of axes by diagonalisation to give... 

Neumann s principle states that under any symmetry operation on the system, the sign and the amplitude of the physical property should remain unchanged. This has a severe consequence for second-order effects only non-centrosymmetric systems are allowed. A system is centrosymmetric when its physical properties remind unchanged under the inversion symmetry transformation (x -x, v -> —y, z —z). 

Thus physical properties can be used as a probe of symmetry, and can reveal the crystallographic point group of the phase. Note that Neumann s principle states that the symmetry elements of a physical property must include those present in the point group, and not that the symmetry elements are identical with those of the point group. This means that a physical property may show more symmetry elements than the point group, and so not all properties are equally useful for revealing tme point group symmetry. For example, the density of a crystal is controlled by the unit cell size and contents, but the symmetry of the material is irrelevant, (see Chapter 1). Properties similar to density, which do not reveal symmetry are called non-directional. Directional properties, on the other hand, may reveal symmetry. 

Theorem 9 Neumann s principle states that the symmetry elements of any physical property of a crystal must include all the symmetry elements of the point group of the crystal. 

Once the spatial group determined, from the structure determination, this structure can be further characterized following all the correlated properties with the existing symmetry elements and operations (in accordance with the so-called Neumann s principle, se also the Curie Principle of Section 2.5.8.2). Consequently, there is again emphasized the importance of the accuracy with which the structure is determined or the method is refined. 

Piezo is derived from the Greek piezein, which means to squeeze or press. The piezoelectric effect is the phenomena where some materials generate an electric potential in response to an applied mechanical stress. Piezoelectricity was discovered by Pierre and Jacques Curie in 1880. Based on crystal symmetry and Neumann s principle, it is known that the piezoelectric effect only exists in crystals that do not have a center of symmetry. There are 20 classes of crystal point symmetry groups that exhibit the piezoelectric effect. The piezoelectric effect can be described by [2] ... 

Based on the symmetry of a material and Neumann s principle, the number of independent elements for each material property can be reduced [2], For instance, for an unstretched polymer, like poly(vinylidene fluoride) (PVDF) poled along its 3-direction, its structure belongs to the point group oom and its properties are ... 

According to Neumann s principle, physical properties in an anisotropic crystal are also anisotropic and they can be described by tensors of different ranks, such as the second-rank... 

Those terms in Eq. (44) which have nonzero averages depend on the symmetry of Xap which by Neumann s principle must contain the symmetry of the phase to which it relates. If the phase is uniaxial, the principal components of x a become ... 

A well-known example of this is that cubic crystals are optically isotropic, which means that the dielectric permittivity has spherical symmetry in a cubic crystal. Another example is that the thermal expansion coefficient of a cubic crystal is independent of direction. In fact, if it were not, the crystal would lose its cubic symmetry if it were heated. Thus, as far as thermal expansion is concerned, a cubic crystal looks isotropic just as it does optically. Since, according to Neumann s principle, the physical properties of a crystal may be of higher symmetry than the crystal, we will generally find that they range from the symmetry of the crystal to the symmetry of an isotropic body. A more general example of higher symmetry in properties is that such physical properties characterized by polar second rank tensors must be centrosymmetric, whether the crystal has a center of symmetry or not, cf. Fig. 27. For, if a second rank tensor T connects the two vectors p and q according to... 

The general application of the Neumann principle in condensed matter is normally much more formalized (for instance, involving rotation matrices) than we will have use for. Few textbooks perform such demonstrations on an elementary level, but there are exceptions, for example, the excellent treatise by Nussbaum and Phillips [90]. For a liquid crystal, we illustrate the simplest way of using the symmetry operations of the medium in Fig. 13. (The same discussion, in more detail, is given in [46]). We choose the z-direction along the director as shown in Fig. 13a and assume that there is a nonzero polarization P=(Px, Py, Pz) i nematic or smectic A. Rotation by 180° around the y-axis transforms and P into and -P, and hence both of these components must be zero, because this rotation is a symmetry operation of the medium. Next, we rotate by 90 ° around the z-axis, which transforms the remaining Py into P. If therefore Py were nonzero, we would see that the symmetry operations of the medium are not synunetry operations of the property, in violation of Neumann s principle. Hence P =0 and P will vanish. 

Figure 13. Neumann s principle applied to (a) the nematic and smectic A, (b) the smectic C and smectic C phases.

If we want to investigate the conditions for an elastic stress to induce a macroscopic polarization, we have to turn to the Curie principle, which in a sense is a generalization of Neumann s principle. It cannot be proven in the same way as that principle in fact it is often the violations (or maybe seeming violations) that are the most interesting to study. Among these cases are the phase Iran-... 

We could also have reasoned formally in the following way If a property P (polarization in this case) should appear in the medium K as a result of the external action E (stress ain this case), then P must be compatible with the symmetry of the strained medium, according to Neumann s principle... 

It is often said that group 432 is too symmetric to allow piezoelectricity, in spite of the fact that it lacks a center of inversion. It is instructive to see how this comes about. In 1934 Neumann s principle was complemented by a powerful theorem proven by Hermann (1898-1961), an outstanding theoretical physicist with a passionate interest for symmetry, whose name is today mostly connected with the Hermann-Mau-guin crystallographic notation, internationally adopted since 1930. In the special issue on liquid crystals by ZeitschriftfUr Kristal-lographie in 1931 he also derived the 18 symmetrically different possible states for liquid crystals, which could exist between three-dimensional crystals and isotropic liquids [100]. His theorem from 1934 states [101] that if there is a rotation axis C (of order n), then every tensor of rank rcubic crystals, this means that second rank tensors like the thermal expansion coefficient a, the electrical conductivity Gjj, or the dielectric constant e,y, will be isotropic perpendicular to all four space diagonals that have threefold symme-... 

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