Bravais monoclinic

According to Hahn (2002), mS and oS are the standard setting independent symbols for the centred monoclinic and the one-face-centred orthorhombic Bravais lattices. Symbols between parentheses oC, oA, etc. represent alternative settings of these Bravais lattices. 

Only fourteen space lattices, called Bravais lattices, are possible for the seven crystal systems (Fig. 328). Designations are P (primitive), / (body-centered), F (face-centered),34 C pace-centered in one set of laces), and R (rhombohedral) Thus our monoclinic structure P2Jc belongs to the monoclinic crystal system and has a primitive Bravais lattice. 

We can now complete our answer to the question, What information is conveyed when we read that the crystal structure of a substance is monodime P2JC7" The structure belongs to the monoclinic crystal system and has a primitive Bravais lattice. It also possesses a two-fold screw axis and a glide plane perpendicular to it. The existence of these two elements of symmetry requires that there also be a center of inversion. The latter is not specifically included in the space group notation as it would be redundant. 

In direct analogy with two dimensions, we can define a primitive unit cell that when repeated by translations in space, generates a 3D space lattice. There are only 14 unique ways of connecting lattice points in three dimensions, which define unit cells (Bravais, 1850). These are the 14 three-dimensional Bravais lattices. The unit cells of the Bravais lattices may be described by six parameters three translation vectors (a, b, c) and three interaxial angle (a, (3, y). These six parameters differentiate the seven crystal systems triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. 

The only possible cells in two dimensions are oblique (p only), rectangular (p and c) and hexagonal (p). For each of the seven three-dimensional crystal systems primitive and centred cells can be chosen, but centring is not advantageous in all cases. In the case of triclinic cells no centred cell can have higher symmetry than the primitive and is therefore avoided. In all there are 14 different lattice types, known as the Bravais lattices Triclinic (P), Monoclinic (P,C), Orthorhombic (P,C,I,F), Trigonal (R), Tetragonal (P,I), and Cubic (P,I,F). 

Altogether there are 14 possible types of unit cell and we call these the Bravais lattices. For dmgs there are three common types of unit cell triclinic, monoclinic and orthorhombic. 

For example, think about the monoclinic point group m in the standard setting, where m is perpendicular to b (Table 1.8). According to Table 1.14, the following Bravais lattices are allowed in the monoclinic crystal system P and C. There is only one finite symmetry element (mirror plane m) to be considered for replacement with glide planes (a, b, c, n and d) ... 

The relationships a = h c,o. = = 90°, and y 90" point to a monoclinic crystal system, except that a = h. What is the reduced (standard) Bravais lattice in this case Provide equations that reduce this lattice to one of the 14 standard Bravais types. 

Figure 2.8 The 14 Bravais lattices. Note that the lattice points are exaggerated in size and are not atoms. The monoclinic lattices have been drawn with the b-axis vertical, to emphasise that it is normal to the plane containing the a- and c-axes...

The two monoclinic Bravais lattices, mP and mC, are generated in a similar way. Each layer of the lattice is similar to the plane mp lattice. In this case, the second and subsequent layers of the lattice are placed directly over the first layer, to generate the mP lattice, or displaced so that each lattice point in the second layer is vertically over the cell centre in the first layer to give the mC lattice. Because of this geometry, the diad axis of the plane lattice is preserved, and runs along the unique b- or c-axis of the lattice. Moreover, the fact that the layers are stacked vertically means that a mirror plane runs perpendicular to the diad, through... 

Bravais lattices - The 14 distinct crystal lattices that can exist in three dimensions. They include three in the cubic crystal system, two in the tetragonal, four in the orthorhombic, two in the monoclinic, and one each in the triclinic, hexagonal, and trigonal systems. 

Standard ASTM E157-82a has the Bravais lattices designations as following C - primitive cubic B - body-centered cubic F - face-centered cubic T - primitive tetragonal U - body-centered tetragonal R - rhombohedral H - hexagonal O - primitive orthorhombic P - body-centered orthorhombic Q - base-centered orthorhombic S - face-centered orthorhombic M - primitive monoclinic N - centered monoclinic A - triclinic. 

The primitive orthorhombic lattice can be thought of as arising from a primitive monoclinic lattice with the added restriction that the third angle is also 90°. In that case, all the unit cell translation vectors are 90° to one another but have different lengths. In the orthorhombic system, one can construct a C-centered cell, which can also be described as an A- or B-lattice by an interchange of the orthogonal axes. In addition, there can also be an all-face-centered F-lattice structure and a body-centered I-lattice. Thus for the orthorhombic crystal system there are four unique Bravais lattices, P, I, F, and C. 

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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