When considering crystal structures, idealized as crystal patterns, frequently the values of distances between the atoms (bond lengths) and of the angles between atomic bonds (bonding angles) are wanted. These quantities can not be calculated from the coordinates of the points (centers of the atoms) directly. Distances and angles are independent of the choice of the origin but the point coordinates depend on the origin choice, see Section 1.4. Therefore, bond distances and angles can only be calculated using the vectors (distance vectors) between the points participating in the bonding. In this section the necessary formulae for such calculations will be derived.
We assume the crystal structure to be given by the coordinates of the atoms (better: of their centers) in a conventional coordinate system. Then the vectors between the points can be calculated by the differences of the point coordinates.
Let be the vector from point to point , , see equation 1.4.1. The scalar product of r with itself is the square of the length of r. Thus
Using this equation, bond distances can be calculated if the coefficients of the bond vector and the lattice constants of the crystal are known.
The general formula (1.6.1) becomes much simpler for the higher symmetric crystal systems. For example, referred to an orthonormal basis, equation (1.6.1) is reduced to
Using the sign and abbreviating ( is defined for : then ), this formula can be written
|Fig. 1.6.1 The bonding angle between the bond vectors and .|
The (bonding) angle between the (bond) vectors
is calculated using the equation
Again one can use the coefficients to obtain, see also Subsection 2.6.2,
For orthonormal bases, equation (1.6.4) is reduced to
and equation 1.6.5 is replaced by
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