Crystals point group

Papalexi, N., 356,371,377,381 Paramagnetic crystals point groups for, 737 symmetry properties of eigenstates, 745... 

However, its was found possible to infer all four microscopic tensor coefficients from macroscopic crystalline values and this impossibility could be related to the molecular unit anisotropy. It can be shown that the molecular unit anisotropy imposes structural relations between coefficients of macroscopic nonlinearities, in addition to the usual relations resulting from crystal symmetry. Such additional relations appear for crystal point group 2,ra and 3. For the monoclinic point group 2, this relation has been tested in the case of MAP crystals, and excellent agreement has been found, triten taking into account crystal structure data (24), and nonlinear optical measurements on single crystal (19). This approach has been extended to the electrooptic tensor (4) and should lead to similar relations, trtten the electrooptic effect is primarily of electronic origin. 

Figure 6. Classification of the noncentrosymmetric crystal point groups by decreasing value of the maximal efficient phase-matchable nonlinear coefficient per molecule...

If we combine the 32 crystal point groups with the 14 Bravais lattices we find 230 three-dimensional space groups that crystal structures can adopt (i.e., 230... 

Polyhedral crystals bounded by flat crystal feces usually take characteristic forms controlled by the symmetry elements of the crystal (point) group to which the crystal belongs and the form and size of the unit cell (see Appendix A.5). When a unit cell is of equal or nearly equal size along the three axes, crystals usually take an isometric form, such as a tetrahedron, cube, octahedron, or dodec-... 

In order to apply group theory to the physical properties of crystals, we need to study the transformation of tensor components under the symmetry operations of the crystal point group. These tensor components form bases for the irreducible repsensentations (IRs) of the point group, for example x, x2 x3 for 7(1) and the set of infinitesimal rotations Rx Ry Rz for 7(1 )ax. (It should be remarked that although there is no unique way of decomposing a finite rotation R( o n) into the product of three rotations about the coordinate axes, infinitesimal rotations do commute and the vector o n can be resolved uniquely... 

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. 

Equation (4) holds generally at the face center but is valid over the whole face if the crystal point group contains a reflection plane through the zone center that is parallel to the face. It also holds for all k vectors that terminate on a line in the BZ face that is parallel to a binary axis. The E(k) may be described either by a singlevalued function of k (with k > 0), which is called the extended zone scheme, or by a multivalued function of k within the first BZ, the reduced zone scheme (see Figure 17.2). 

Chapter 2 Crystals, Point Groups, and Space Groups Table 2.1. Crystal systems and the 14 Bravais Lattices. 

J. A. Weil, T. Buch and J. E. Clapp, Crystal point group symmetry and microscopic tensor properties in magnetic resonance spectroscopy. Adv. Magn. Reson., 1973, 6,183-257. 

Figure 9-12. Representation of the 32 crystal point groups by actual minerals (after Buerger Dana andZorky) [20],...

Figure 9-13. Representation of the 32 crystal point groups by stereographic projections.

Although the word crystal in its everyday usage is almost synonymous with symmetry, there are severe restrictions on crystal symmetry. While there are no restrictions in principle on the number of symmetry classes of molecules, this is not so for crystals. AH crystals, as regards their form, belong to one or another of only 32 symmetry classes. They are also called the 32 crystal point groups. Figures 9-12 and 9-13 illustrate them by examples of actual minerals and by stereographic projections with symmetry elements, respectively. 

TABLE 5.3 The 7 Crystal Systems and the 32 Crystal Point Groups ... 

The relationship between crystal point group symmetry and dielectric... 

The relation between the diffraction groups and the crystal point groups can be seen in Table 3. 

Usually, for the standard method of crystal point group and space group determination four types of pattern are required whole, bright field, dark field... 

Table 3. Relation between diffraction groups and crystal point groups (from Buxton et al [17])...

There is a one-to-one correspondence between irreps of crystal point group D h and irreps of the space group at F point — l, 2l2g,u — — 2, B2g,u —... 

In principle molecules may belong to any of the possible point groups permitted by mathematical group theory. In reality, however, the actual chemical examples of specific point groups are somewhat restricted. In dealing with crystals only one-, two-, three-, four-, and sixfold axes are permissible. With such a restriction, there are only thirty-two possible combinations of symmetry elements yielding the thirty-two crystal point groups. 

---