Rotation-inversion axis

Figure 2.33. Illustration of crystallographic point group operations. Shown are (a) rotation axis, (b) rotation-inversion axis, and (c) mirror plane.

FIGURE 4.6. A rotatory-inversion axis involves a rotation and then an inversion across a center of symmetry. Since, by the definition of a point group, one point remains unmoved, this must be the point through which the rotatory-inversion axis passes and it must lie on the inversion center (center of symmetry). The effect of a fourfold rotation-inversion axis is shown in two steps. By this symmetry operation a right hand is converted to a left hand, and an atom at x,y,z is moved to y,—x,—z. (a) The fourfold rotation, and (b) the inversion through a center of symmetry. 

Tetragonal Z-axis is always parallel to the unique four-fold rotation (inversion) axis. X-and T-axes form a 90 angle with the Z-axis and with each other None... 

Fig. 2-6 Some symmetry elements of a cube, (a) Reflection plane. Ai becomes A2. (b) Rotation axes. 4-fold axis Ai becomes A2 3-fold axis A becomes A 2-fold axis Ai becomes A4,. (c) Inversion center. A becomes A2. (d) Rotation-inversion axis. 4-fold axis A becomes A inversion center A becomes A2.

A body has an inversion center if corresponding points of the body are located at equal distances from the center on a line drawn through the center. A body having an inversion center will come into coincidence with itself if every point in the body is inverted, or reflected, in the inversion center. A cube has such a center at the intersection of its body diagonals [Fig, 2-6(c)]. Finally, a body may have a rotation-inversion axis, either 1-, 2-, 3-, 4-, or 6-fold. If it has an -fold rotation-inversion axis, it can be brought into coincidence with itself by a rotation of 360°/n about the axis followed by inversion in a center lying on the axis. Figure 2-6(d) illustrates the operation of a 4-fold rotation-inversion axis on a cube. 

Cubic Tetragonal Orthorhombic Rhombohedral Hexagoral Monoclinic Triclinic Four 3 - fold rotation axes One 4 - fold rotation (or rotation - inversion) axis Three perpendicular 2-fold rotation (or rotation - inversion) axes One 3-fold rotation (or rotation - inversion) axis One 6-fold rotation (or rotation - inversion) axis One 2-fold rotation (or rotation-inversion) axis None... 

Fig. 5.7. The C o molecule viewed down a threefold rotation-inversion axis. The upper half of the molecule is identified with filled, the lower half with unfilled bond lines. Molecular twofold axes are also shown. The unfilled bond lines also represent the upper half of the second orientation of Cgo found in the crystal...

Tetragonal One fourfold rotation (or rotation-inversion) axis... 

Now look at the illustration above more closely. Perform a four-fold rotation about an axis through the center of the face and then an inversion through the center of the unit cell (it may be helpful to watch the zinc ions). You should observe that there is a four-fold rotation-inversion axis. 

It is a four-fold rotation-inversion axis. If you have trouble seeing this, look down the c-axis, rotate 90°, look at the new position of the atoms, and then invert them through the center. The new configuration of atoms will be indistinguishable from the original one. 

It is common to use rotation-inversion axes (rather than rotation-reflection axes) to classify the symmetry of crystals. Any S axis is equivalent to a rotation-inversion axis (symbolized by p) whose order p may differ from n. A rotation-inversion operation consists of rotation by 2ir/p radians followed by inversion. Show that... 

The irreducible tensor representations (McClain, 1971) allowed by symmetry are collected in Table 1 together with the point groups for which they occur and their depolarization ratios piijn) for freely rotating molecules. In addition to the representations discussed above, this table also list degenerate representations that occur in point groups with a three- or higher fold rotation or rotation-inversion axis. The tensor elements in the plane... 

Monoclinic a b c a — y = 90° 7 fi One 2-fold rotation or rotation-inversion axis... 

Illustrative examples of crystallographic symmetry operations are shown in Figure 2.35. An integer label, n, indicates the regeneration of an equivalent lattice point when an object in the crystal lattice is rotated 360°/n about an axis. A rotation-inversion axis is designated by n, featuring rotation about an axis (360°/n), followed... 

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