[ Pobierz całość w formacie PDF ]angles of rotation and rotoinversion of space group operations hold as in
Section 3.2. This concerns also the rotations involved in screw rotations.
2. It is always possible to choose a primitive basis, see definition (D 1.5.2)
and the remarks to it. Referred to a primitive basis, all lattice vectors of the
crystal are integer linear combinations of the basis vectors. Each of these
lattice vectors defines a (symmetry) translation. The order of any translation
T is infinite because there is no integer number k =0 such that Tk = I.
3. Parallel to each rotation, screw rotation or rotoinversion axis as well as par-
allel to the normal of each mirror or glide plane there is a row of lattice
vectors.
4. Perpendicular to each rotation, screw rotation, or rotoinversion axis as well
as parallel to each mirror or glide plane there is a plane of lattice vectors.
5. Let ¦ = 360æ%/N be the rotation angle of a screw rotation, then the screw
rotation is called N fold. Note that the order of any screw rotation is infinite.
Let u be the shortest lattice vector in the direction of the corresponding screw
axis, and n u/N, with n =0 and integer, be the screw vector of the screw
rotation by the angle ¦. Then the HM symbol of the screw rotation is Nn.
Performing an N fold rotation N times results in the identity mapping, i. e.
the crystal has returned to its original position. After N screw rotations with
rotation angle ¦ = 360æ%/N the crystal has its original orientation but is
shifted parallel to the screw axis by the lattice vector n u.
6. Let W be a glide reflection. Then the glide vector is parallel to the glide plane
and is 1/2 of a lattice vector t. Whereas twice the application of a reflection
restores the original position of the crystal, applying a glide reflection twice
results in a translation of the crystal with the translation vector t. The order
of any glide reflection is infinite. The HM symbol of a glide reflection is
3.4 Crystallographic groups 37
g in the plane and a, b, c, d, e or n in the space. The letter indicates the
direction of the glide vector g relative to the basis of the coordinate system.
3.4 Crystallographic groups
The symmetry, i. e. the set of all symmetry operations, of any object forms a group
in the mathematical sense of the word. Therefore, the theorems and results of group
theory can be used when dealing with the symmetries of crystals. The methods of
group theory cannot be treated here but a few results of group theory for crystallo-
graphic groups will be stated and used.
We start with the definition of the terms subgroup and order of a group .
Definition (D 3.4.1) Let G and H be groups such that all elements of H are also
elements of G. ThenH is called a subgroup of G.
Remark. According to its definition, each crystallographic site symmetry group is
a subgroup of that space group from which its elements are selected.
Definition (D 3.4.2) The number g of elements of a group G is called the order of
G. In case g exists, G is called a finite group. If there is no (finite) number g, G is
called an infinite group.
Remark. The term order is an old mathematical term and has nothing to do with
order or disorder in crystals. Space groups are always infinite groups; crystallo-
graphic site symmetry groups are always finite.
The following results for crystallographic site symmetry groups S and point groups
P are known for more than 170, those for space groups R more than 100 years.
We consider site symmetry groups first.
1. The possible crystallographic site symmetry groups S are always finite groups.
The maximal number of elements of S in the plane is 12, in the space is 48.
2. Due to the periodicity of the crystal, crystallographic site symmetry groups
never occur singly. Let S be the site symmetry group of a point P , and P
be a point which is equivalent to P under a translation of R. To P belongs
a site symmetry group S which is equivalent to S. The infinite number of
translations results in an infinite number of points P and thus in an infinite
number of groups S which all are equivalent to S. In Subsection 5.3.1 is
shown, how S can be calculated from S.
Note that this assertion is correct even if not all of the groups S are different.
This is demonstrated by the following example: If the site symmetry S of
P consists of a reflection and the identity, the point P is placed on a mirror
plane. If the translation mapping P onto P is parallel to this plane, then S
of P and S of P are identical. Nevertheless, there are always translations
of R which are not parallel to the mirror plane and which carry P and S to
38 3 CRYSTALLOGRAPHIC SYMMETRY
points P with site symmetries S . These are different from but equivalent
to S. The groups S and S leave different planes invariant.
3. According to their geometric meaning the groups S may be classified into
types. A type of site symmetry groups is also called a crystal class.
4. There are altogether 10 crystal classes of the plane. Geometrically, their
groups are the symmetries of the regular hexagon, of the square and the
subgroups of these symmetries. Within the same crystal class, the site
symmetry groups consist of the same number of rotations and reflections
and have thus the same group order. The rotations have the same rotation
angles. Site symmetry groups of different crystal classes differ by the num-
ber and angles of their rotations and/or by the number of their reflections
and often by their group orders.
5. There are 32 crystal classes of groups S of the space. Their groups are the
symmetries of the cube, of the hexagonal bipyramid and the subgroups of
these symmetries. Again, the groups S of the same crystal class agree in
the numbers and kinds of their rotations, rotoinversions, reflections and thus
in the group orders. Moreover, there are strong restrictions for the possible
relative orientations of the rotation and rotoinversion axes and of the mir-
ror planes. Site symmetry groups of different crystal classes differ by the
numbers and kinds of their symmetry operations.
6. In order to get a better overview, the crystal classes are further classified into
crystal systems and crystal families.
The following exercise deals with a simple example of a possible planar crystallo-
graphic site symmetry group.
3.4 Crystallographic groups 39
Problem 1A. Symmetry of the square.
For the solution, see p. 73.
m- my m+
-1,-1 -1,1
3 2
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