Constitution of Space Groups

The detailed study to know how these space groups are constituted from seven crystal systems is out of the scope of this book. Mathematically it can be derived from the group theory and geometrically, the derivation of all these 230 space groups is cumbersome and therefore, for only one crystal system, that is, for monoclinic system the procedure that is adopted is explained below. 

A particular crystal system has some definite number of point groups and for this monoclinic system it has symmetry operations like 2, m, and 2/m, that is, twofold rotation, a mirror plane, and twofold with mirror plane of symmetries. Now, for three-dimensional crystal the possible symmetry elements will include also screw axes and glide planes, and when screw axes and glide planes are added to the point group of symmetries for this system, we can say that different possibilities that may exist are 2, 2i, m, c, 2/m, 2i/m, 2/c, and 2j/c. Now each of these symmetry groups are repeated by lattice translation of the Bravais lattices of that system. As monoclinic system has only primitive P and C, all the symmetry possibilities may be associated with both P and C. Therefore, if they are worked out, they come out to be 13 in number and they are Pm, Pc, Cm, Cc, P2, P2i, C2, P2/m, P2i/m, C2/m, P2/c, P2i/c, C2/c, etc. 

Similarly the space groups of all the crystal systems can be worked out and these come out to be as mentioned earlier 230 in number. 

When two more symmetry operations move the motifs out from their original plane on which they originally exist, more lattice patterns are created. 

These symmetry operations are called microscopic symmetry operations as they can only be identified from internal structure of the crystal lattice in three dimensions and not by the geometrical shape of the crystal. 

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