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We can represent every symmetry operation by a matrix A:
(8) |
(9) |
If the base system is given by the three vectors , , of a primitive lattice, the elements a_{ij} of the A matrix are necessarily integers. In fact relation (9) must hold true for every vector r of the lattice; A transforms r in another vector r: in this case the components of r and r are integers, and since relation (9) holds for every group of these integers relative to r, the elements of A must be integers.
We will now examine other restrictions on A which allow us to define the single elements a_{ij} as a function of the metric tensor. A symmetry operation obviously must not change the length of a vector or the angle between vectors. Therefore we have:
from which follows, applying relation (4):
and from (9):
r^{t}A^{t}GAr = r^{t}Gr
and finally, since the previous relation must hold for any value of r:
G = A^{t}GA | (10) |
(11) |
From relation (10), using matrix and determinant properties, we obtain:
from which, keeping in mind that |A^{t}| = |A|, follows that the determinant associated with the A matrix must be equal to 1. If the determinant is equal to +1 the symmetry operation is said to belong to the type I and it is defined as a rotation; if the determinant is equal to -1 the symmetry operation is of type II and defined as a rotoinversion.^{}
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