Order of the axis

Axes of symmetry are denoted by the symbol n (C, where n is the order of the axis. Therefore, the rotational axis of the OF2 molecule is 2 (C2). 

The screw axis combines translation with rotation. Screw axes have the general symbol ri/where n is the rotational order of the axis (i.e., twofold, threefold, etc.), and the translation distance is given by the ratio i/n. Figure i.20 illustrates a 2] screw axis. In this example, the screw axis lies along zand so the translation must be in... 

Improper rotation axis. Rotation about an improper axis is analogous to rotation about a proper synunetry axis, except that upon completion of the rotation operation, the molecule is mirror reflected through a symmetry plane perpendicular to the improper rotation axis. These axes and their associated rotation/reflection operations are usually abbreviated X , where n is the order of the axis as defined above for proper rotational axes. Note that an axis is equivalent to a a plane of symmetry, since the initial rotation operation simply returns every atom to its original location. Note also that the presence of an X2 axis (or indeed any X axis of even order n) implies that for every atom at a position (x,y,z) that is not the origin, there will be an identical atom at position (—x,—y,—z) the origin in such a system is called a point of inversion , since one may regard every atom as having an identical... 

The values of 2% If n (where n is the order of the axis of rotation) that satisfy eq. (16) and therefore are compatible with translational symmetry, are shown in Table 16.1. It follows that the point groups compatible with translational symmetry are limited to the twenty-seven axial groups with n= 1, 2, 3, 4, or 6 and the five cubic groups, giving thirty-two... 

The symmetry operation performed by a screw axis nm is equivalent to a combination of rotation of 2n/n radians (or 360°/ri) followed by a translation of m/n in the direction of the n-fold axis, where n = 1,2,3,4, or 6 is the order of the axis and the subscript m is an integer less than n. There exist totally eleven screw axes 2i, 3i, 32, 4i, 42, 43, 6i, 62, 63, 64, and 65 in the crystal lattices. Figure 9.3.1 shows the positions of the equivalent points (also called equipoints for simplicity) around the individual 4-, 4-, 4i-, 42-, and 43-axes. 

The columns labeled "Graphical symbol" in Table 1.4 correspond to graphical representation of symmetry elements when they are perpendicular to the plane of the projection. Other orientations of rotation and inversion axes are conventionally indicated using the same symbols to designate the order of the axis with properly oriented lines, as shown in Figure 1.8. 

Translations, t, along the axis are also limited to a few fixed values, which depend on the order of the axis, and are defined as t = k/N, where N is axis order, and k is an integer number between one and N-1. For instance, for the three-fold screw axis, k = 1 and 2, and the two possible translations are 1/3 and 2/3 of the length of the basis vector parallel to this axis, whereas for the two-fold axis, k = 1, and only 1/2 translation is allowed. 

The symbol of the screw axis is constructed as Nk to identify both the order of the axis (N) and the length of the translation (k). Thus, the two three-fold screw axes have symbols 3i and 32, whereas the only possible two-fold screw axis is 2]. The International symbols, both text and graphical, and the allowed translations for all crystallographic screw axes are found in Table 1.16. 

Pairs of screw axes, in which the sums of the subscripts equal to the order of the axis are called enantiomorphous pairs since one is the mirror image of another. The latter is reflected in the graphical symbols of the corresponding pairs of the enantiomorphous axes. These are 3i and 32l 4i and 43 61 and 65 62 and 64. Two enantiomorphous axes differ only by the direction of rotation or (which is the same) by the direction of translation. 

Apart from the rotation axes that occur in both two- and three-dimensional objects, an additional type of rotation axis occurs in a solid that is not found in planar shapes, the inversion axis, n. The operation of an inversion axis consists of a rotation combined with a centre of symmetry. These axes are also called improper rotation axes, to distinguish them from the ordinary proper rotation axes. The symmetry operation of an improper rotation axis is that of rotoinversion. The initial atom position is rotated counter clockwise, by an amount specified by the order of the axis, and then inverted through the centre of symmetry. For example, the operation of a two-fold improper rotation axis 2 is thus the initial atom position is rotated 180° counter clockwise and then inverted through the centre of symmetry. 

Spq is the degree of order of the axis through p and q. Here rvq means an... 

We return to the example of dimethylether (6-1). A rotation of 180°(tt ) around the z-axis leads to a molecular configuration that is equivalent to the initial one (6-6), with the following pairs of atoms being exchanged (Cl, C2), (Hi, H4), (H2, He), and (H3, H5). The z-axis is therefore a symmetry element of the molecule. The ratio between 360° and the rotation angle associated with the symmetry operation (180° in this example) defines the order of the axis. In this case it is therefore a twofold axis (or of order two), written C2. A single symmetry operation is associated with this axis, shown in 6-6 and written C2 or simply C2. If this operation is applied twice (a rotation of 2 x 180° = 360°, written C2), we return to the initial molecular configuration (C2 = E). Note that by convention, the rotation is performed in a clockwise sense. 

This rod is called an axis of rotation. In this case it is called a two-fold axis because we must rotate the block twice before it returns to its original position. Or, to put it another way, rotation by 360°/n, where n = 2 produces an equivalent, or indistinguishable, orientation. N is also called the order of the axis. Thus, a four-fold axis, which requires rotation by 360°/4 = 90°, has an order of 4. 

We say that a body has an n-fold axis of symmetry (also called an n-fold proper axis or an n-fold rotation axis) if rotation about this axis by 360/n degrees (where n is an integer) gives a configuration physically indistinguishable from the original position n is called the order of the axis. For example, BF3 has a threefold axis of symmetry perpendicular to the molecular plane. The symbol for an n-fold rotation eixis is C .The threefold axis in BF3 is a C3 axis. To denote the operation of counterclockwise rotation by (360/ )°, we use the symbol C . The hat distinguishes synunetry operations from symmetry elements. BF3 has three more rotation axes each B—F bond is a twofold symmetry axis (Fig. 12.2). 

A screw axis involves a rotation about an axis followed by a nonprimitive translation in the direction of the rotation axis. The Schoenflies notation for an n-fold screw axis about, say, the b axis, is Cj,. The more descriptive universal notation is Up where n is the order of the axis and p/n is the fraction of a primitive translation involved in the operation. The integer p takes the values 1.2,. . . , up to (n — 1). The locations of these axes in the unit cells of the different space groups are given in the International Tables for X-ray Crystallography (8). The glide plane involves reflection in a plane followed by a nonprimitive translation along... 

The number of operations from an improper rotation axis depends on the order of the axis and on the removal of any operations that can be written more simply. The number of operations can be seen by repeatedly carrying out the improper rotation until the original configuration is obtained. For example, the result of six successive Se operations on ethane returns the molecule to its starting configuration, as shown in Figure 2.8. 

After even numbers of operations the arrow returns to the top face of the molecule, and so there is an equivalent simple rotation that can give the same result. Since the order of the axis is odd, does not return the molecule to its start point although all the atoms are in the same place as in the original configuration, the arrow is pointing down, and so 83 = CTh. Only after six applications of the operation do we find that 83 = E. In this case there are only two unique operations arising from the S3 axis S3 and 83. ... 

Cyclic groups contain only operations derived from the repeated application of a single rotational symmetry operation. The point group is C if the repeated operation is a simple rotation, and we have the point group S if it is an improper rotation axis. In both cases the subscript denotes the order of the axis. 

---