Space Groups

It has already been discussed in section 5.10 that theoretically there are 32 different ways of arranging the elements of symmetry about a point. These different arrangements are called point groups. However in a complete space lattice, because of the regular repetition of the pattern in the three dimensional space, two additional symmetry elements called glide planes and screw axes become possible. The detailed discussion of these new elements of symmetry is beyond the scope of the present book. However it is interesting to mention that because of this additional element of symmetry, the number of arrangements possible for the elements of symmetry about a point of a space lattice becomes manifold, viz., and equal to 230. These 230 possible arrangements of the elements of symmetry about a lattice point are called space groups. The split-up of these space groups corresponding to the different crystal systems is given in Table below:

Table 5.5. Distribution of space groups among crystal systems

Systems                                No. of space groups

1. Cubic                                              36 2. Tetragonal                                        68 3. Orthorhombic                                    59 4 Monoclinic                                         13 5. Triclinic                                              2 6. Trigonal or Rhombohedral                  30 7. Hexagonal                                        22

To Sum up:         Number of Crystallographic Systems = 7
                            Number of point Groups                  = 32
                            Number of Bravais Lattices              = 14
                            Number of Space Groups                = 230

The location f any point within a unit cell is described by giving its co-ordinates (x/b, y/b, /z/c) as fractions of the lengths of the vetoes that define the unit cell. For example he face-centered positions in fcc unit cell

are   and   . Similarly, the body-centered position in a bcc unit cell is.

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