Four-fold inversion axis

Furthermore, as we will see in sections 1.5.3 and 1.5.5, below, transformations performed by the three-fold inversion and the six-fold inversion axes can be represented by two independent simple symmetry elements. In the case of the three-fold inversion axis, 3, these are the threefold rotation axis and the center of inversion acting independently, and in the case of the six-fold inversion axis, 6, the two independent symmetry elements are the mirror plane and the three-fold rotation axis perpendicular to the plane, as denoted in Table 1.4. The remaining four-fold inversion axis, 4, is a unique symmetry element (section 1.5.4), which cannot be represented by any pair of independently acting symmetry elements. 

Four-fold rotation axis and four-fold inversion axis... 

The four-fold inversion axis Figure 1.14, right) also produces four symmetrically equivalent objects. The original object, e.g. any of the two clear p)ramids with apex up, is rotated by 90° in any direction and then it is immediately inverted from this intermediate position through the center of inversion. This transformation results in a shaded pyramid with its apex down in the position next to the original pyramid but in the direction opposite to the direction of rotation. By applying the same transformation to this shaded pyramid, the third symmetrically equivalent object would be a clear pyramid next to the shaded pyramid in the direction opposite to the direction of rotation. The fourth object is obtained in the same fashion. Unlike in the case of the three-fold inversion axis (see above), this combination of four objects cannot be produced by appl)dng the four-fold rotation axis and the center of inversion separately, and therefore, this is a unique symmetry element. As can be seen from Figure 1.14, both four-fold axes also contain a two-fold rotation axis (180° rotations) as a sub-element. 

Figure 1.17. Mirror plane (m) and two-fold rotation axis (2) intersecting at 45" (left) result in additional symmetry elements mirror plane, two-fold rotation axis and four-fold inversion axis (right).

The tetrahedron illustrates the operation of a four-fold inversion axis, 4, (pronounced four bar ). A tetrahedron inscribed in a cube allows the three Cartesian axes to be defined... 

Four-fold inversion axis 4 Four-fold rotation axis with centre of symmetry... 

Figure 1.8. From left to right horizontal two-fold rotation axis (top) and its alternative symbol (bottom), diagonal three-fold inversion axis inclined to the plane of the projection, horizontal four-fold rotation axis, horizontal, and diagonal mirror planes. Horizontal or vertical lines are commonly used to indicate axes located in the plane of the projection, and diagonal lines are used to indicate axes, which form an angle other than the right or zero angles with the plane of the projection.

Earlier (see Figure 1.7) we established that there are four simple symmetry operations, namely rotation, reflection, inversion and translation. Among the four, reflection in a mirror plane may be represented as a complex symmetry element - two-fold inversion axis - which includes simultaneous two-fold rotation and inversion. Therefore, in order to minimize the number of simple symmetry operations, we will begin with rotation, inversion and translation, noting that complex operations can be described as simultaneous applications of these three simple transformations. 

Figure 4.6a). With respect to these axes, the four-fold inversion axes lie parallel to the x-, y-and z-axes. One such axis is drawn in Figure 4.6b. The operation of this element is to move vertex A (Figure 4.6b) by a rotation of 90° in a counter-clockwise direction and then inversion through the centre of symmetry to generate vertex D. In subsequent application of the 4 operator, vertex D is transformed to vertex B, B to C and C back to A. 

Both are body-centered Bravais lattices and for both the site symmetry of the origin is identical with the short space group symbol. The body-center position is of the lowest multiplicity (two-fold) and highest symmetry, and thus is considered as the origin in the lA/mmm space group. However, in the tetragonal lattice, a = b c. Hence, the body center position is not an inversion center. It possesses four-fold rotational symmetry (the axis is parallel to c) with a perpendicular mirror plane and two additional perpendicular mirror planes that contain the rotation axis. 

Symbols of finite crystallographic symmetry elements and their graphical representations are listed in Table 1.4. The fiill name of a symmetry element is formed by adding "N-fold" to the words "rotation axis" or "inversion axis". The numeral N generally corresponds to the total number of objects generated by the element, and it is also known as the order or the multiplicity of the symmetry element. Orders of axes are found in columns two and four in Table 1.4, for example, a three-fold rotation axis or a fourfold inversion axis. 

The mirror plane is, therefore, a derivative of the two-fold rotation axis and the center of inversion located on the axis. The derivative mirror plane is perpendicular to the axis and intersects the axis in a way that the center of inversion also belongs to the plane. If we start from the same pyramid A and apply the center of inversion first (this results in pyramid D) and the twofold axis second (i.e. A -> B and D C), the resulting combination of four symmetrically equivalent objects and the derivative mirror plane remain the same. 

Tetragonal Unique four-fold axis, either rotation or inversion... 

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... 

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... 

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. 

The numbers 2, 3, 4 and 6 are used as symbols of the corresponding axes of symmetry while the symbols 3, 4 and 6 (3 bar, 4 bar, etc.) are used for the three-four- and six-fold (roto) inversion axes, corresponding to a counter-clockwise rotation of 360% around an axis followed by an inversion through a point on the axis. 

We consider four kinds of symmetry elements. For an n fold proper rotation axis of symmetry Cn, rotation by 2n f n radians about the axis is a symmetry operation. For a plane of symmetry a, reflection through the plane is a symmetry operation. For a center of symmetry /, inversion through this center point is a symmetry operation. For an n-fold improper rotation axis Sn, rotation by lir/n radians about the axis followed by reflection in a plane perpendicular to the axis is a symmetry operation. To denote symmetry operations, we add a circumflex to the symbol for the corresponding symmetry element. Thus Cn is a rotation by lit/n radians. Note that since = o, a plane of symmetry is equivalent to an S, axis. It is easy to see that a 180° rotation about an axis followed by reflection in a plane perpendicular to the axis is equivalent to inversion hence S2 = i, and a center of symmetry is equivalent to an S2 axis. 

Consequently, there are four symmetry elements the n-fold axis of rotation, labeled C the plane of symmetry, labeled o the center of inversion, /, and the n-fold rotation-reflection axis, labeled S . Because of mathematical reasons, it is necessary to include the identity symmetry element, /. 

Screw axes perform a rotation simultaneously with a translation along the rotation axis. In other words, the rotation occurs around the axis, while the translation occurs parallel to the axis. Crystallographic screw axes include only two-, three-, four- and six-fold rotations due to the three-dimensional periodicity of the crystal lattice, which prohibits five-, seven- and higher-order rotations. Hence, the allowed rotation angles are the same as for both rotation and inversion axes (see Eq. 1.2). 

A symmetry operation transforms an object into a position that is physically indistinguishable from the original position and preserves the distances between all pairs of points in the object. A symmetry element is a geometrical entity with respect to which a symmetry operation is performed. For molecules, the four kinds of symmetry elements are an n-fold axis of symmetry (C ), a plane of symmetry (cr), a center of symmetry (i), and an n-fold rotation-reflection axis of symmetry (5 ). The product of symmetry operations means successive performance of them. We have " = , where E is the identity operation also, 5, = o-, and Si = i, where the inversion operation moves a point at x,y, zto -X, -y, -z.Two symmetry operations may or may not commute. 

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