Crystal symmetry elements

Figure 6.1 The icosahedron and some of its symmetry elements, (a) An icosahedron has 12 vertices and 20 triangular faces defined by 30 edges, (b) The preferred pentagonal pyramidal coordination polyhedron for 6-coordinate boron in icosahedral structures as it is not possible to generate an infinite three-dimensional lattice on the basis of fivefold symmetry, various distortions, translations and voids occur in the actual crystal structures, (c) The distortion angle 0, which varies from 0° to 25°, for various boron atoms in crystalline boron and metal borides.

First of all the term stress-induced crystallization includes crystallization occuring at any extensions or deformations both large and small (in the latter case, ECC are not formed and an ordinary oriented sample is obtained). In contrast, orientational crystallization is a crystallization that occurs at melt extensions corresponding to fi > when chains are considerably extended prior to crystallization and the formation of an intermediate oriented phase is followed by crystallization from the preoriented state. Hence, orientational crystallization proceeds in two steps the first step is the transition of the isotropic melt into the nematic phase (first-order transition of the order-disorder type) and the second involves crystallization with the formation of ECC from the nematic phase (second- or higher-order transition not related to the change in the symmetry elements of the system). 

The rotation of the ammonium ion in salts at ordinary temperatures provides justification for the customary treatment of the ion as spherically symmetrical in the theoretical discussion of the structure of ionic crystals. Further, the rotation of molecules such as NHj and H20 about symmetry axes accounts for the fact that these molecules occupy positions in crystals with symmetry elements not compatible with those of the non-rotating molecule. Thus in Ni(NH3)6CU the NHj molecules lie on four-fold axes, and in alum the H2O molecules on three-fold axes. The rotation of the molecules,... 

Space lattices and crystal systems provide only a partial description of the crystal structure of a crystalline material. If the structure is to be fully specified, it is also necessary to take into account the symmetry elements and ultimately determine the pertinent space group. There are in all two hundred and thirty space groups. When the space group as well as the interatomic distances are known, the crystal structure is completely determined. 

Screw rotation. The symmetry element is a screw axis. It can only occur if there is translational symmetry in the direction of the axis. The screw rotation results when a rotation of 360/1V degrees is coupled with a displacement parallel to the axis. The Hermann-Mauguin symbol is NM ( N sub M )-,N expresses the rotational component and the fraction M/N is the displacement component as a fraction of the translation vector. Some screw axes are right or left-handed. Screw axes that can occur in crystals are shown in Fig. 3.4. Single polymer molecules can also have non-crystallographic screw axes, e.g. 103 in polymeric sulfur. 

The 230 space-group types are listed in full in International Tables for Crystallography, Volume A [48], Whenever crystal symmetry is to be considered, this fundamental tabular work should be consulted. It includes figures that show the relative positions of the symmetry elements as well as details concerning all possible sites in the unit cell (cf. next section). 

The occurrence of twinned crystals is a widespread phenomenon. They may consist of individuals that can be depicted macroscopically as in the case of the dovetail twins of gypsum, where the two components are mirror-inverted (Fig. 18.8). There may also be numerous alternating components which sometimes cause a streaky appearance of the crystals (polysynthetic twin). One of the twin components is converted to the other by some symmetry operation (twinning operation), for example by a reflection in the case of the dovetail twins. Another example is the Dauphine twins of quartz which are intercon-verted by a twofold rotation axis (Fig. 18.8). Threefold or fourfold axes can also occur as symmetry elements between the components the domains then have three or four orientations. The twinning operation is not a symmetry operation of the space group of the structure, but it must be compatible with the given structural facts. 

In the case of MAP, the concept of chirality was used so as to prevent centrosymmetry a chiral molecule cannot be superimposed on its image by a mirror or center of symmetry so that a crystal made only of left or right-handed molecules can accomodate neither of these symmetry elements. This use of the chirality concept ensures exclusion of a centrosymmetric structure. However as we shall see in the following, the departure of the actual structure from centrosymmetry may be only weak, resulting in limited nonlinear efficiencies. A prerequisite to the introduction of a chiral substituent in a molecule is that its location should avoid interfering with the charge-transfer process. 

Besides the energy factors, defined by the close-packing principle, entropic factors are also involved in determining the mode of packing of molecules. A molecule in a crystal tends to maintain part of its symmetry elements, provided that this does not cause a serious loss of density. In a more symmetric position a molecule has a greater freedom of vibration, that is, the structure corresponds to a wider energy minimum.126... 

Crystal lattices can be depicted not only by the lattice translation defined in Eq. (7.2), but also by the performance of various point symmetry operations. A symmetry operation is defined as an operation that moves the system into a new configuration that is equivalent to and indistinguishable from the original one. A symmetry element is a point, line, or plane with respect to which a symmetry operation is performed. The complete ensemble of symmetry operations that define the spatial properties of a molecule or its crystal are referred to as its group. In addition to the fundamental symmetry operations associated with molecular species that define the point group of the molecule, there are additional symmetry operations necessary to define the space group of its crystal. These will only be briefly outlined here, but additional information on molecular symmetry [10] and solid-state symmetry [11] is available. 

All the possible combinations of these symmetry elements result in 32 crystallographic point-group symmetries or crystal classes their symbols are listed in Table 3.3. Notice that in putting together the symbols to denote the symmetries of any crystal classes the convention is to give the symmetry of the principal axis first for instance 4 or 4, for tetragonal classes. If there is a plane of symmetry perpendicular to the principal axis, the two symbols are associated as in 4 m or Aim (4 over m), then the symbols for the secondary axes, if any, follow, and then any other symmetry planes. In a symbol such as Almmm, the second and third m refer to planes parallel to the four-fold axis. 

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