Rules unit cell choice

The four unit cells shown in Figure 1.21 have the same symmetry (a twofold rotation axis, which is perpendicular to the plane of the projection and passes through the center of each unit cell), but they have different shapes and areas (volumes in three dimensions). Furthermore, the two unit cells located on top of Figure 1.21 do not contain lattice points inside the unit cell, while each of the remaining two has an additional lattice point in the middle. We note that all unit cells depicted in Figure 1.21 satisfy the rule for the monoclinic crystal system established in Table /.//. It is quite obvious, that more unit cells can be selected in Figure 1.21, and an infinite number of choices is possible in the infinite lattice, all in agreement with Table 1.11. 

Without adopting certain conventions, different unit cell dimensions might and most definitely would be assigned to the same material based on preferences of different researchers. Therefore, long ago the following rules Table 1.12) were established to designate a standard choice of the unit cell, dependent on the crystal system. This set of rules explains both the unit cell shape and relationships between the unit cell parameters listed in Table 1.11 (i.e. rule number one), and can be considered as rule number two in the proper selection of the unit cell. 

Applying the rules established in Table 1.12 to two of the four unit cells shown on top of Figure 1.21, the cell based on vectors aj and bi is the standard choice. The unit cell based on vectors aj and b2 has the angle between the vectors much farther from 90 than the first one. The remaining... 

The first two reduction rules are normally employed only during the indexing. They usually do not produce a standard choice of the unit cell since at this stage the space group symmetry, and often even the lattice type, are not involved. For example, in the orthorhombic space group symmetry Pnma (a standard setting) the condition a< b< c is not necessarily obeyed. 

Often one has several options for choosing the net, or reciprocal lattice axes. The rule is that an axial system is always chosen that preserves the highest symmetry of the diffraction pattern. That is, the axes are chosen to be consistent with the real unit cell of highest symmetry. Sometimes, before the diffraction symmetry is fully clear, incorrect axes may be chosen. The axial system, or reciprocal lattice net, can, however, always be reassigned and the hkl indexes of the reflections reindexed at a later time. The choice of axes determines the crystal class. 

Figure 93 Crystal unit cell of Cl there is a" arbitrary rule in the choice on the unit cell.

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