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The matrix-column pairs of crystallographic symmetry operations

For general matrices, multiplication and inversion may be rather tedious manipulations. These are unavoidable if the geometry of the object is complicated and if there is no way to simplify it. In crystallography one is in a better situation. By definition crystals are periodic, and their periodicity is not that of the continuum but that of the lattice. Therefore, primitive bases for the lattices can always be found, see definition (D 1.5.2). As a consequence, the matrix-column pairs for the crystallographic symmetry operations are simple if an appropriate coordinate system has been chosen. The conventional coordinate systems as used in the space- and plane-group tables of IT A are chosen under this aspect.

The matrix parts shall be considered first.

Suppose, a primitive lattice basis has been chosen as the coordinate basis. We take from the last section that the mapping of vectors by a crystallographic symmetry operation W is described only by the matrix part W of the $(4\times 4)$ matrix $\mos{W}$. The image of a lattice vector under a symmetry operation must be a lattice vector, otherwise the lattice would not be mapped onto itself as a whole. Being referred to a primitive basis, all lattice vectors have integer coefficients. Therefore, the matrix parts W of the crystallographic symmetry operations must have integer coefficients, they are integer matrices.

On the other hand, a crystallographic symmetry operation W is an isometry. Therefore, referred to an orthonormal basis, the matrix is an orthogonal matrix, see Subsection 2.6.1, p. [*]. When leaving all distances invariant, also the volume is invariant. Analytically, this means $\det({\mbox{\textit{\textbf{W}}}})=\pm1$.

If a matrix is integer and orthogonal, then in each row and column there are exactly one entry $\pm1$ and 2 zeroes. The matrix has thus 3 coefficients $\pm1$ and 6 zeroes. How many $(3\times 3)$ matrices of this kind do exist ? There are 6 arrangements to distribute the non-zero coefficients among the positions of the matrix. In addition, there are 3 signs with $2^3$ possibilities of distributing + and $-$. Altogether there are $6\times8=48$ different orthogonal integer matrices.

The matrix parts of crystallographic symmetry operations form groups which describe the point groups, see Sections 3.4 and 4.4. The highest order of a crystallographic point group is 48, and referred to the conventional basis this point group is described by the group of the 48 orthogonal integer matrices. It is the point group of copper, gold, rocksalt, fluorite, galena, garnet, spinel, and many other crystalline compounds. The symmetries of 24 other point groups are contained as subgroups in this highest symmetry, so that 25 of the 32 types of point groups (crystal classes) can be described by orthogonal integer matrices. The advantages of these matrices are:

  1. The product of 2 matrices is easily calculated because of the many zeroes; it is again an orthogonal integer matrix.
  2. Due to the orthogonality, the inverse of a matrix is the transpose matrix and does not need calculation.
  3. The determinant of a matrix, see equation 2.5.2, is the product of 3 coefficients $\pm1$, adjusted for the sign.

What about the necessary bases ? The matrix part of an isometry is orthogonal if referred to an orthonormal basis; it may also be orthogonal if referred to another basis. The restrictions to the basis depend on the point group. For example, the matrices describing the identity mapping and the inversion are orthogonal in any basis, viz the unit matrix and the negative unit matrix. The conventional bases in crystallography are lattice bases (not orthonormal bases). They are mostly chosen such that the matrices are integer orthogonal matrices. As already mentioned, this is possible for 25 of the 32 crystal classes of point groups.

The matrix is even simpler, if it is a diagonal orthogonal matrix, i.e. a diagonal matrix with coefficients $\pm1$. There are $2^3=8$ such matrices, among them the unit matrix I and the inversion $\bar{\mbox{\textit{\textbf{I}}}}$. If the symmetry of the crystal is low enough, all matrices are diagonal. There are 8 crystal classes (of the 25) permitting such a description. Crystals with this symmetry are also called optically biaxial crystals because of their optical properties (birefringence).

The point groups of the remaining 7 crystal classes can not be described by orthogonal integer matrices. Referred to a primitive basis, their matrices are integer, of course. However, this representation is not orthogonal. One can choose an orthonormal basis instead but then the matrices are no longer integer matrices. These point groups are hexagonal and belong to the hexagonal crystal family.

Only in crystal physics the non-integer orthogonal representation is used for hexagonal point groups, in crystallography the representation by integer matrices is preferred. One introduces the so-called hexagonal basis, referred to which the matrices consist of at least 5 zeroes and 4 coefficients $\pm1$. In the conventional settings of IT A, there occur up to 16 such matrices, the other up to 8 matrices are orthogonal integer matrices. Although not necessary, also trigonal point groups are mostly referred to the hexagonal basis, because this description is for many crystals more natural than the decription by integer orthogonal matrices.

The column parts will be discussed now.

Provided a conventional coordinate system is chosen, also the coefficients of the columns are simple. They are determined

  1. by the choice of the basis which, however, is more or less fixed already by the matrix considerations.
  2. by the origin choice. If the origin is chosen in a fixed point of a symmetry operation W, then w is the o column. Clearly, one choice of the origin is such that as many of the symmetry operations as possible have the origin as fixed point.
Doing this it turns out, that the remaining non-zero coefficients of the columns w are fractions with denominators at most 6.


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Next: The `General Position' in IT A Up: The description of mappings by ... Previous: Transformation of vector coefficients

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