How can one find the geometric meaning of a matrix-column pair ? Large parts of the following recipe apply not only to crystallographic symmetry operations but also to general isometries.
Example. The matrix (in IT A shorthand notation) describes a 6-fold anti-clockwise rotation if referred to a hexagonal basis. If referred to an orthonormal basis it does not describe an isometry at all but contains a shearing component.
In general the coefficients of the matrix depend on the choice of the basis; a change of basis changes the coefficients, see Section 5.3.2. However, there are geometric quantities which are independent of the basis. Correspondingly, there exist characteristic numbers of a matrix from which the geometric features may be derived and vice versa.
. The rotation angle of the rotation or of the rotation part of a rotoinversion can be calculated from the trace by the formula
One can list this correlation in a table
By this table the type of operation may be found, as far as it is determined by the matrix part. For example, one takes from the table that a specific operation is a two-fold rotation but one does not know if the operation is a rotatation or a screw rotation, what the direction of the rotation axis is and where it is located in space. This characterization will be done in the following list for the crystallographic symmetry operations.
The coefficients of w are the coefficients of the translation vector.
For type , reflections or glide reflections, u is the direction of the normal of the (glide) reflection plane.
If t = o holds, then (W,w) describes a
rotation or reflection. For
(W,w) describes a screw rotation or
glide reflection. One forms the so-called
reduced operation by subtracting the intrinsic
translation part t/k from
If W is a diagonal matrix, i.e. if only the coefficients are non-zero, then either is and is a screw or glide component, or and is a location component. If W is not a diagonal matrix, then the location part has to be calculated according to equation 5.2.6.
The formulae of this section enable the user to find the geometric contents of any symmetry operation. In reality, IT A have provided the necessary information for all symmetry operations which are listed in the plane-group or space-group tables. The entries of the General position are numbered. The geometric meaning of these entries is listed under the same number in the block Symmetry operations in the tables of IT A. The explanation of the symbols for the symmetry operations is found in Sections 2.9 and 11.2 of IT A.
The section shall be closed with an exercise.
Problem 2B. Symmetry described by matrix-column pairs.
For the solution, see p. .
The matrix-column pairs (A,a), (B,b) (C,c), and (D,d) have been listed or derived in Problem 2A, p. , which dealt with their combination and reversion.
(A,a), (B,b), (C,c), and (D,d).
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