Point groups crystal systems

The science of crystallography began, in the seventeenth century, with the stud of the shapes of crystals. It was observed that there is considerable variation in tb overall shape of crystals of a particular substance (or of crystals of one form if it polymorphic), but that however much a crystal departed from the ideal shap 42 

The characteristic symmetries of the crystal systems and also the parameters required to define the unit cells are summarized in Table 2.3. 

Attention should perhaps be drawn to the characteristic symmetry of the cubic system which is not, as might be supposed, the 4-fold (or 2-fold) axes of symmetry or planes of symmetry but four 3-fold axes parallel to the body-diagonals of the cubic unit cell. This combination of inclined 3-fold axes introduces either three 2-fold or three 4-fold axes which are mutually perpendicular and parallel to the cubic axes. Further axes and planes of symmetry may be present but are not essential to cubic symmetry and do not occur in all the cubic point groups or space groups. 

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Piezoelectric phenomena possess an essential criterion for their existence in a given crystal. Among the 21 noncentrosymmetric point groups, crystal systems possessing the 20 noncentrosynunetric point groups alone could exhibit piezoelectricity [1]. The conventional or m other words, well-known material that exhibits piezoelectric behavior is quartz. A quartz crystal when cut in a specific direction exhibits a large piezoelectric response. However, the need for high-... 

Crystal family Symbol Crystal system Crystallographic point groups (crystal classes) Number of space groups Conventional coordinate system Bravais lattices... 

Table 2.4. Space groups, crystal systems, point groups, and the Hermann-Mauguin Symbols.

There are only 32 possible eombinations of the above-mentioned elements of symmetry (ineluding the asymmetrie state), and these are ealled the 32 elasses or point groups. For eonvenienee these 32 elasses are grouped into 7 systems eharaeterized by the angles between their x, y and z axes. Crystals of eourse, ean exhibit eombination forms of the erystal systems. 

The structure factor itself is expressed as the sum of energy diffracted, over one unit-cell, of the individual scattering factors, fi, for atoms located at X, y and z. Having done this, we can then identify the exact locations of the atoms (ions) within the unit-cell, its point-group sjmimetiy, and crystal system. This then completes our picture of the structure of the material. 

Seven crystal systems as described in Table 3.2 occur in the 32 point groups that can be assigned to protein crystals. For crystals with symmetry higher than triclinic, particles within the cell are repeated as a consequence of symmetry operations. The number of asymmetric units within the unit cell is related but not necessarily equal to the number of molecules in a unit cell, depending on how the molecules are related by symmetry operations. From the symmetry in the X-ray diffraction pattern and the systematic absence of specific reflections in the pattern, it is possible to deduce the space group to which the crystal belongs. 

Crystal System Conditions Imposed on Cell Geometry Minimum Point Group Symmetry... 

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. 

Shortened version of the standard description structural types Several intermetallic phases are known which have the same (or a similar) stoichiometry and crystallize in the same crystal system and space group with the same occupied point positions. Such compounds are considered as belonging to the same structure type. About 30000 intermetallic phases have been described. These, however, may be grouped in about 2800 types. 

In the orthorhombic point group mm2 there is an ambiguity in the sense of the polar axis c. Conventional X-ray diffraction does not allow one to differentiate, with respect to a chosen coordinate system, between the mm2 structures of Schemes 15a and b (these two structures are, in fact, related by a rotation of 180° about the a or c axis) and therefore to fix the orientation and chirality of the enantiomers with respect to the crystal faces. Nevertheless, by determining which polar end of a given crystal (e.g., face hkl or hkl) is affected by an appropriate additive, it is possible to fix the absolute sense of the polar c axis and so the absolute structure with respect to this axis. Subsequently, the absolute configuration of a chiral resolved additive may be assigned depending on which faces of the enantiotopic pair [e.g., (hkl) and (hkl) or (hkl) and (hkl)] are affected. 

Microdiffraction is the pertinent method to identify the crystal system, the Bravais lattices and the presence of glide planes [4] (see the chapter on symmetry determination). For the point and space group identifications, CBED and LACBED are the best methods [5]. 

Electron Diffraction (CBED) and Large-Angle Convergent-Beam Electron Diffraction (LACBED) allow the identification of the crystal system, the Bravais lattice and the point and space groups. These crystallographic features are obtained at microscopic and nanoscopic scales from the observation of symmetry elements present on electron diffraction patterns. 

Table 2. Crystal Systems, Laue Classes, Non-Centrosymmetric Crystal Classes (Point Groups) and the Occurrence of Enantiomorphism and Optical Activity 31...

The urea molecule 0--C(NHa)a has two planes of symmetry intersecting in a single twofold axis—the symmetry found in crystals belonging to the polar class of the orthorhombic system (Kendricks, 1.928 a) the point-group symbol is mm2. 

In Section 11.4 the fourteen 3D lattices (Bravais lattices) were derived and it was shown that they could be grouped into the six crystal systems. For each crystal system the point symmetry of the lattice was determined (there being one point symmetry for each, except the hexagonal system that can have either one of two). These seven point symmetries are the highest possible symmetries for crystals of each lattice type they are not the only ones. 

For the monoclinic system it is essential to have one twofold axis, either 2(C2) or 2(m), and it is permitted, of course, to have both. When both are present the point group is that of the lattice, 2lm Cy). There are no intermediate symmetries. By proceeding in this way, we can arrive at the results shown in column 4 of Table 11.4, where each of the 32 crystallographic point groups (i.e., crystal classes) has been assigned to its appropriate crystal system. 

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