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Display of crystallographic symmetry in IT A

A crystallographic symmetry operation may be visualized geometrically by its `geometric element', mostly called symmetry element. The symmetry element is a point, line, or plane related to the symmetry: depending on the symmetry operation, it is the center of inversion or (for rotoinversions) the inversion point; the rotation, screw rotation, or rotoinversion axis; the mirror or glide plane. Only the identity operation I and the translations T do not define a symmetry element. Whereas the symmetry element of a symmetry operation is uniquely defined, more than one symmetry operation may belong to a symmetry element. For example, to a 4-fold rotation axis belong the symmetry operations $4^1=4, 4^2=2$, and $4^3=4^{-1}$ around this axis.

[There is some confusion concerning the terms symmetry element and symmetry operation. It is caused by the fact that symmetry operations are the group elements of the symmetry groups (space groups, site-symmetry groups, or point groups). Symmetry operations can be combined resulting in other symmetry operations and forming a symmetry group. Symmetry elements can not be combined such that the combination results in a uniquely determined other symmetry element. As a consequence, symmetry elements do not form groups, and group theory can not be applied to them. Nevertheless, the description of symmetry by symmetry elements is very useful, as will be seen now.]

In IT A, crystallographic symmetry is described in 4 ways:

  1. The analytical description of symmetry operations by matrix-column pairs will be considered in Chapter 4. The listing of these matrix-column pairs as the `General position' of IT A in a kind of short-hand notation will be discussed in Section 4.6.
  2. The geometric meaning of the matrix-column pair can be determined, see Section 5.2. In order to save the user this determination, in IT A the geometric meaning is listed for each matrix-column pair, see Section 4.6.
  3. A visual geometric description of a space group (more exactly: a space-group type) is possible by displaying the framework of symmetry elements in a diagram, see this section.
  4. In another diagram the space-group symmetry is represented by a set of points which are symmetrically equivalent under the operations of the space group, see this section.

In IT A, for each space group there are at least 2 diagrams displaying the symmetry (there are more diagrams for space groups of low symmetry). In this section only one example for each kind of diagrams can be discussed in order to explain the principles of this way of symmetry description. A full explanation of the details is found in IT A, Section 2.6 `Space-group diagrams', dto. in the Brief Teaching Edition of IT A.

The Figs. 3.5.1 and 3.5.2 are taken from IT A, space-group table No. 86, $P4_2/n$ ($HM$ symbol for this space-group type), $C_{4h}^4$ (SCHOENFLIES symbol for this space-group type). In both diagrams, displayed is an orthogonal projection of a unit cell of the crystal onto the paper plane. The direction of projection is the c axis, the paper plane is the projection of the a-b plane (if c is perpendicular to a and b, then the paper plane is the a-b plane). The thin lines outlining the projection are the traces of the side planes of the unit cell. Because opposite lines represent translationally equivalent side planes of the unit cell, the line pairs can be considered as representing the basic translations a and b. The origin (projection of all points with coordinates 00$z$) is placed in the upper left corner; the other vertices represent the edges 10$z$ (lower left), 01$z$ (upper right), and 11$z$ (lower right).

The following diagram is always placed on the left side of the page in IT A.

Fig. 3.5.1 Symmetry elements. A small circle represents a center of inversion $\bar{1}$, the attached number $\frac{1}{4}$ is its $z$ coordinate (height above paper in units of the lattice constant $c$). There are black squares with 2 small tails: 4-fold screw-rotation axes $N=4,\ n=2$, see Section 3.3, $HM$ symbol $4_2$. A partly filled empty square represents a 4-fold rotoinversion axis, HM symbol $\bar{4}$. The $\bar{4}$ axes are parallel to c, they are projected onto points. The right angle drawn outside the top left of the unit cell indicates a horizontal glide plane with the direction of its arrow as the glide vector. Missing $z$ coordinates mean either `$z = 0$', e.g. for the centers of $\bar{4}$, or `$z$ meaningless', as for the screw axes.

In the unit cell or on its borders are (only 1 representative of each set of translationally equivalent elements is listed):

centers of inversion in 1/4,1/4,1/4; 1/4,1/4,3/4; 3/4,1/4,1/4;
3/4,1/4,3/4; 1/4,3/4,1/4; 1/4,3/4,3/4; 3/4,3/4,1/4; 3/4,3/4,3/4;
$4_2$ axes in 1/2,0,$z$; 0,1/2,$z$;
$\bar{4}$ axes in 0,0,$z$; 1/2,1/2,$z$ with inversion points in 0,0,0; 0,0,1/2; 1/2,1/2,0; 1/2,1/2,1/2;
glide planes $x,\,y,\,1/4;\ x,\,y,\,3/4$ with glide vector 1/2,1/2,0.
The following diagram is always placed on the right side of the page in IT A.

Fig. 3.5.2 Starting with the point in the upper left corner of the unit cell, marked by an open circle and with the sign `+', all points in and near the unit cell are drawn which are images of the starting point under some symmetry operation of the space group. The starting point is a point with site symmetry 1, i.e. identity only. (Note that the high symmetry of a circle does not reflect the site symmetry 1 of its center properly. The circle is chosen for historical reasons.) Then all image points have site symmetry 1 too. The $x$ and $y$ coordinates of all points can be taken from the projection; the $z$ coordinate of the starting point is indicated by `+' (= +$z$), the other points have either `+', or `$-$' (= $-z$), `1/2+' (= 1/2+$z$); or `$1/2-$' (= 1/2$-z$); in other diagrams $1/4+,\ 1/4-,\ 3/4+$, $3/4-$, etc.

A '-sign (comma) in the circle means that this point is an image of the starting point by a symmetry operation of the second kind, see Section 3.1. If the empty circles are assumed to represent right gloves, then the circles with a comma represent left gloves, and vice versa.

The correspondence between the 2 diagrams is obvious: With some practice each of the diagrams can be produced from the other. Therefore, they are completely equivalent descriptions of the same space-group symmetry. Nevertheless, both diagrams are displayed in IT A in order to provide different aspects of the same symmetry. Because of the periodicity of the arrangement, the presentation of the contents of one unit cell is sufficient.

Answer to the question in Section 3.2.

$\bar{3}^3=\bar{1}$, ($\bar{3}^6=1$); $\bar{4}^2=2$; $\bar{6}^2=3$; and $\bar{6}^3=m$, where the normal of the mirror plane is parallel to the rotoinversion axis of $\bar{6}$ (the mirror plane itself is perpendicular to the rotoinversion axis).

The following statements hold always:

  1. The even powers of rotoinversions are rotations.
  2. The order of an $N$-fold rotoinversion is 2$N$ for odd $N$.

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