Three-fold inversion axis

Except for the center of inversion, which results in two objects, and three-fold inversion axis, which produces six symmetrically equivalent objects. See section 1.20.4 for an algebraic definition of the order of a symmetry element. 

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. 

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.

Three-fold rotation axis and three-fold inversion axis... 

It is easy to see that the six symmetrically equivalent objects are related to one another by both the simple three-fold rotation axis and the center of inversion. Hence, the three-fold inversion axis is not only the result of two simultaneous operations (3 and 1), Iwt it is also the result of two independent operations. In other words, 3 is identical to 3 then 1. 

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. 

The six-fold inversion axis Figure 1.15, right) also produces six symmetrically equivalent objects. Similar to the three-fold inversion axis, this symmetry element can be represented by two independent simple symmetry elements the first one is the three-fold rotation axis, which connects pyramids 1-3-5 and 2-4-6, and the second one is the mirror plane perpendicular to the three-fold rotation axis, which connects pyramids 1-4, 2-5, and 3-6. As an exercise, try to obtain all six symmetrically equivalent pyramids starting from the pyramid 1 as the original object by applying 60° rotations followed by immediate inversions. Keep in mind that objects are not retained in the intermediate positions because the six-fold rotation and inversion act simultaneously. 

The regular octahedron can be used to describe the three-fold inversion axis, 3, (pronounced three bar ). As with the regular tetrahedron, it is convenient to refer to Cartesian axes to locate the symmetry operators, (Figure 4.7a). The highest order axis, however, is not 3, but a tetrad. These axes pass through a vertex and run along the jc-, y- and z-axes. Each tetrad is accompanied by a mirror normal to it, (Figure 4.7b). This combination is written 4/m (pronounced 4 over em ). 

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 six-fold rotation axis also contains one three-fold and one two-fold rotation axes, while the six-fold inversion axis contains a three-fold rotation and a two-fold inversion (mirror plane) axes as sub-elements. Thus, any N-fold symmetry axis with N > 1 always includes either rotation or inversion axes of lower order(s), which is(are) integer divisor(s) of N. 

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. 

In general, g =l/ , where n is the multiplicity of the symmetry element which causes the overlap of the corresponding atoms. When the culprits are a mirror plane, a two fold rotation axis or a center of inversion, n = l and g = 0.5. For a three fold rotation axis = 3 and = 1/3, and so on (Figure... 

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... 

The three-fold inversion axes relate the positions of the vertices to each other. A 3 inversion axis runs through the middle of each opposite... 

Figure 4.7 Symmetry elements present in a regular octahedron (a) an octahedron in a cube, showing the three Cartesian axes (b) each tetrad rotation axis (4) lies along either x-, y- or z and is normal to a mirror plane (c) three-fold inversion axes (3) pass through the centre of each triangular face (d) a triangular face viewed from above (d) diad axes through the centre of each edge lie normal to mirror planes...

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry relationships between the molecules. Common symmetry operations are two- or three-fold screw (rotation) axes, mirror planes, inversion centers (centers of symmetry), and rotation followed by inversion. There are 230 different ways to combine allowed symmetry operations in a crystal leading to 230 space groups.12 Not all of these are allowed for protein crystals because of amino acid asymmetry (only L-amino acids are found in proteins). Only those space groups without symmetry (triclinic) or with rotation or screw axes are allowed. However, mirror lines and inversion centers may occur in protein structures along an axis. 

Orthorhombic Three mutually perpendicular two-fold axes, either rotation or inversion Trigonal Unique three-fold axis, either rotation or inversion... 

Both the rotation and inversion axes can also be the source of special positions. Consider, for example, the site 2a Table 1.18) where atoms are accommodated by the two-fold rotation axis that follows the line at which two mutually perpendicular mirror planes intersect. In this case two of the three coordinates in the triplet are fixed (x = 0 and y = 0), while the third coordinate (z) may assume any value. A similar special position is represented by the site 4c Table 1.18), where the two-fold rotation axis is parallel to Z and coincides with the line at which two mutually perpendicular... 

The skeletal structure of [Sn(OBu03l2Sr (2) shows that the polyhedron is built from two Sn03Sr trigonal bipyramids connected via a common apex with retention of the three-fold axis. The apical position, which at the same time is a center of inversion, is occupied by the strontium atom with a distorted octahedral environment (Figure 2.12.3). The Sn atoms are trigonal bipyramidally coordinated (O-Sn-O 82.3(1)°) and the average Sn-0 and Sr-0 distances are 2.078(3) and 2.523(3) A respectively. 

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... 

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. 

The cube has a three-fold axis along each body diagonal. Grinding down the vertex of a solid cube along the three-fold axis exposes a new triangular face. Grinding away from all vertices until the triangles meet, an octahedron is produced. A similar, inverse operation performed on an octahedron produces a cube. The two polyhedra are said to be reciprocal. The dodecahedron and the icosahedron are reciprocal in the same sense. 

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... 

There are five kinds of symmetry operations that one can utilize to move an object through a maximum number of indistinguishable configurations. One is the trivial identity operation E. Each of the other kinds of symmetry operation has an associated symmetry element in the object. For example, our ammonia model has three reflection operations, each of which has an associated reflection plane as its symmetry element. It also has two rotation operations and these are associated with a common rotation axis as symmetry element. The axis is said to be three-fold in this case because the associated rotations are each one-third of a complete cycle. In general, rotation by iTt/n radians is said to occur about an -fold axis. Another kind of operation—one we have encountered before is inversion, and it has a point of inversion as its symmetry element. Finally, there is an operation known as improper rotation. In this operation, we first rotate the object by some fraction of a cycle about an axis, and then reflect it through a plane perpendicular to the rotation axis. The axis is the symmetry element and is called an improper axis. 

The number of reflection planes, a, three-fold rotation axes, C3, two-fold rotation axes, C2, and inversion centers, i, that are symmetry elements for this molecule-ion axis... 

The A and B symbols attached to these representations are obtained as follows One-dimensional representations are designated by A if they are symmetric to rotation by In/n radians about the principal n-fold rotation axis (n = 2 for a 180° rotation in this case) and are designated by B if they are antisymmetric to this rotation. The subscripts 1 and 2 designate whether (in this case) they are symmetric or antisymmetric to reflection in a vertical plane. A two-dimensional representation is designated by E (not to be confused with the identity operation), and a three-dimensional representation is designated by T. Subscripts g and u are sometimes added to specify the symmetry with respect to inversion (g = gerade = even u = ungerade = odd).Arepresentation with all characters equal to 1, like the Ai representation in this case, is called the totally symmetric representation. 

We have seen in Section 4.1.4 that = n and that S2 = i, so we can immediately exclude from chirality any molecule having a plane of symmetry or a centre of inversion. The condition that a chiral molecule may not have a plane of symmetry or a centre of inversion is sufficient in nearly all cases to decide whether a molecule is chiral. We have to go to a rather unusual molecule, such as the tetrafluorospiropentane, shown in Figure 4.8, to find a case where there is no a or i element of symmetry but there is a higher-fold S element. In this molecule the two three-membered carbon rings are mutually perpendicular, and the pairs of fluorine atoms on each end of the molecule are trans to each other. There is an 54 axis, as shown in Figure 4.8, but no a or i element, and therefore the molecule is not chiral. 

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