Crystal systems/classes

Table 3. Crystal Systems, Classes, Space Groups...

The term crystal structure in essence covers all of the descriptive information, such as the crystal system, the space lattice, the symmetry class, the space group and the lattice parameters pertaining to the crystal under reference. Most metals are found to have relatively simple crystal structures body centered cubic (bcc), face centered cubic (fee) and hexagonal close packed (eph) structures. The majority of the metals exhibit one of these three crystal structures at room temperature. However, some metals do exhibit more complex crystal structures. 

Table 3.1 The 32 crystal classes and the corresponding crystal systems...

State the crystal classes and crystal systems to which the following space groups belong ... 

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

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

Crystal System Laue Class" Non-Centrosymmetric Crystal Classesa,b Enantiomorphism Optical Activity"... 

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. 

Within each of the six crystal systems, there arc specific crystal classes. Each class displays distinctive symmetry elements. There are 32 possible classes distributed among the six crystal systems. One of the crystal classes within each system possesses all of the symmetry elements that re characteristics of its space lattice cell. These are ealled the holohedral class of that system. Other classes within each system possess somewhat fewer symmetry elements and are called tncmhedral classes. 

Two-dimensional, disc-shaped nanomaterials are the last class of nanomaterials to be discussed in more detail. Though a great variety of nanodiscs, nanosheets and nanoplatelets based on metals [256], metal oxides [257-260], graphene [261], or semiconductors [262] are frequently described in the literature, the vast majority of studies in liquid crystal systems dealt with some form of nanoclay. 

Table 2.3. The seven crystal systems and the 32 crystal classes.

There are 32 crystal classes that describe all of the possible combinations of crystal systems and symmetry elements. These are treated in other texts. 

Crystal system Crystal class Chiral (enantiomorphism) Optical activity (circular dichroism) Polar (pyroelectric) Piezoelectric, SHG... 

The thirty-two crystal classes (crystallographic point groups) described in Section 9.1.4 can also be classified into the same seven crystal systems, depending on the most convenient coordinate system used to indicate the location and orientation of their characteristic symmetry elements, as shown in Table 9.2.1. 

Table 9.3.6. The 230 three-dimensional space groups arranged by crystal systems and classes ...

As a consequence of Friedel s law, the diffraction pattern exhibits the symmetry of a centrosymmetric crystal class. For example, a crystal in class 2, on account of the 1 symmetry imposed on its diffraction pattern, will appear to be in class 2/m. The same result also holds for crystals in class m. Therefore, it is not possible to distinguish the classes 2, m, and 2/m from their diffraction patterns. The same effect occurs in other crystal systems, so that the 32 crystal classes are classified into only 11 distinct Laue groups according to the symmetry of the diffraction pattern, as shown in Table 9.4.1. 

Knowledge of the diffraction symmetry of a crystal is useful for its classification. If the Laue group is observed to be 4/mmm, the crystal system is tetragonal, the crystal class must be chosen from 422,4mm, 42m, and 4/mmm, and the space group is one of those associated with these four crystallographic point groups. 

Crystal system Crystal class Laue group... 

Crystal structure is described in terms of symmetry. In mineralogy, there are six crystal systems and within those, thirty-two crystal classes. All crystals of... 

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