Crystallographic basis

R. K. Flarris, P. Flodgkinson, C. J. Pickard, J. R. Yates and V. Zorin, Chemical shift computations on a crystallographic basis some reflections and comments. Magn. Reson. Chem., 2007,45, S174-S186. 

In hexagonal and trigonal crystal systems, the fourth index is usually introduced to address the possibility of three similar choices in selecting the crystallographic basis as illustrated in Figure 7.5 7. In addition to the unit cell based on the vectors a, b and c, two other unit cells, based on the vectors a, -(a + b) and c, and -(a + b), b and c are possible due to the six-fold or the... 

Figure 1.37. Three possibilities to select the crystallographic basis in hexagonal and trigonal crystal systems and the family of (1120) crystallographic planes in the hexagonal crystal system. Indices are shown in the unit cell based on the vectors b and c. Three additional symmetrically related families of planes have indices (1120), (1210) and (2110) in the same basis and we leave their identification to the reader.

In a crystallographic basis where X- and f-axes form a 120° angle between them, and Z-axis is perpendicular to both X and Y. 

At this point, the validity of Eqs. 1.38 and 1.39 has been established when rotations were performed around an axis intersecting the origin of coordinates. We now establish their validity in the general case by considering vector X and a symmetry operation that includes both the rotational part, R, and translational part, t. Assume that the symmetry operation is applied in a crystallographic basis where the rotation axis is shifted from the origin of coordinates by a vector At. 

Crystallographic Basis ofPolytypism and Twinning in Micas REFLECTION CONDITIONS... 

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