Monoclinic system, classes

The same type of unit cell is appropriate for class m, the a and c axes lying in the. plane of symmetry, with the 6 axis normal to this plane. It is equally appropriate for class 2jm. The three classes 2, m, and 2/m constitute the monoclinic system (Pig. 33). 

Fig. 33. Monoclinic system. (See also Fig. 26.) a. Unit aell type. 6. Left- and right-handed tartaric acid. ClasB 2. c. 2,4,6 Tribromobenzonitrile. Class m. d, p-Dinitro-benzene. Class 2/m. e. (CH3COO)8Pb.3HaO. Class 2/m.

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. 

Alpha trinitrotoluene crystallizes in long yellow needles, which belong to the monoclinic system and the prismatic class. 

The monoclinic system, shown in Figure 4.10b, is characterized, a b c and a = y = 90°, P 90°, has a twofold axis of symmefry along fhe [010] direction, and a plane of mirror symmetry (the (020) plane). The Schoenflies symbol for this class is Ctm- h indicates a horizontal mirror plane perpendicular to the axis of rofafional symmetry. The international symbol is 2/m, 2 for the axis of fwofold rofafional symmetry, and m indicates a mirror plane perpendicular to the axis of rofafional symmefry. 

This requirement excludes PI in the triclinic system as well as all those monoclinic space groups associated with crystal classes m and 2hn. Thus,... 

Merten 14> has extended the theory for the general case of a biaxial, polyatomic crystal. His theory is worked out in detail for crystals of the orthorhombic system and all crystal classes of higher symmetry, but not yet for crystals of the monoclinic and triclinic systems. 

Table 2.12. Reflection conditions for the monoclinic crystal system (Lane class 2/m, unique...

Figure 5.6. Schematic representations of the fractions of the volume of the sphere (r = 1/X) in the reciprocal space in which the list of hkl triplets should be generated in six powder Laue classes to ensure that all symmetrically independent points in the reciprocal lattice have been included in the calculation of Bragg angles using a proper form of Eq. 5.2. The monoclinic crystal system is shown in the alternative setting, i.e. with the unique two-fold axis parallel to c instead of the standard setting, where the two-fold axis is parallel to b. ...

Careful measurement of mineral specimens allowed crystals to be classified in terms of six crystal families, called anorthic, monoclinic, orthorhombic, tetragonal, hexagonal and isometric. This classification has been expanded slightly by crystallographers into seven crystal systems. The crystal systems are sets of reference axes, which have a direction as well as a magnitude, and hence are vectors1. The crystal families and classes are given in Table 1.1. 

Monoclinic crystal system the holohedric group Gift, and those sub-groups which do not belong to class 1 (one specific direction is defined) Czh Cs, Cg... 

Finally, the cube has a center of symmetry. Possession of a center of symmetry, a center of inversion, means that if any point on the cube is connected to the center by a line, that line produced an equal distance beyond the center will intersect the cube at an equivalent point. More succinctly, a center of symmetry requires that diametrically opposite points in a figure be equivalent. These elements together with rotation-inversion are the symmetry elements f or crystals. The elements of symmetry f ound in crystals are (a) center of symmetry (b) planes of symmetry (c) 2-, 3-, 4-, and 6-fold axes of symmetry and (d) 2- and 4-fold axes of rotation-inversion. Of course, every crystal does not have all these elements of symmetry. In fact, there are only 32 possible combinations of these elements of symmetry. These possible combinations divide crystals into 32 crystal classes. The class to which a crystal belongs can be determined by the external symmetry of the crystal. The number of crystal classes corresponding to each crystal system are triclinic, 2 monoclinic, 3 orthorhombic, 3 rhombohedral, 5 cubic, 5 hexagonal, 7 tetragonal, 7. 

If necessary, the symbols may be augmented by the addition of axes of order 1 to indicate a specific orientation of the symmetry elements with respect to the coordinate system. Hence the symbol 121 designates the group 2 with the b axis parallel to the twofold axis, i.e. the traditional choice of coordinate system for the monoclinic class there are no symmetry elements parallel to a or to c. The symbol 112 designates the same group 2 with c parallel to the twofold axis. The symbol mil designates the group m with a normal to the mirror plane. The symbol 3m 1... 

Each Bravais system has its corresponding minimum and maximum symmetry. Thus the Bravais lattice must be monoclinic (P or C) if the crystal has only one mirror plane or one twofold axis (crystal classes m or 2). However, the monoclinic unit cell will also allow the symmetry 2/m. Thus the symmetry of the contents of the unit cell (the motif) may be less than that of the empty cell. In this case we speak of merohedry. The formation of twins is relatively frequent for merohedral crystals. A twin (Fig. 2.28) is an interpenetration or aggregation of several crystals of the same species whose relative orientations follow well-defined laws. These orientations are related by symmetry operations which do not belong to the crystal class of the untwinned crystal, either by a rotation about a translation... 

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