Four-fold rotation axis

We have recently demonstrated the ability of six resorcin[4]arenes and eight water molecules to assemble in apolar media to form a spherical molecular assembly which conforms to a snub cube (Fig. 9.3). [10] The shell consists of 24 asymmetric units - each resorcin[4]arene lies on a four-fold rotation axis and each H2O molecule on a three-fold axis - in which the vertices of the square faces of the polyhedron correspond to the corners of the resorcin[4]arenes and the centroids of the eight triangles that adjoin three squares correspond to the water molecules. The assembly, which exhibits an external diameter of 2.4 nm, possesses an internal volume of about 1.4 A3 and is held together by 60 O-H O hydrogen bonds. 

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.

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

The four-fold rotation axis Figure 1.14, left) results in four symmetrically equivalent objects by rotating the original object around the axis by 90°, 180°, 270° and 360°. 

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 2.52. The illustration of forbidden overlaps as a result of an atom being too close to a finite symmetry element mirror plane (left), three-fold rotation axis (middle), and four-fold rotation axis (right). Assuming that there are no defects in a crystal lattice, these distributions require g" = 1/2,1/3 and 1/4, respectively.

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

Four-fold rotation axis parallel to c (perpendicular to the plane of the diagram)... 

Some molecules have several rotation axes. The axis ofhighest order is called the principal axis. The complex [PtCU] ", in which the platinum atom is located in the centre of the square defined by the four chlorine atoms (6-10), possesses a C4-axis, perpendicular to the plane of the square, and four C2-axes in the plane of the square two of these are co-linear with the bonds Cli—Pt—CI3 and CI2—Pt—CI4 (C0, and the two others bisect the angles Cl—Pt—Cl (C O- The principal axis is therefore four-fold (of order four). The existence of a four-fold rotation axis implies the presence of a co-linear two-fold axis, as the operation (a rotation of 2 x 27t/4) is identical to the operation C (a rotation of 1 X in/2). 

Taylor et al. [156] suggested that the crystals belong to the two-sided plane group C12, in which there are four ATPase molecules per unit cell of 9 113 A, with ATPase dimers related by a two-fold rotational axis within the membrane plane parallel to the b cell axis. While the arrangement of ATPase molecules was highly ordered within... 

An example of a molecule with a three-fold rotation axis is the conformation of. vym-1,3,5-triethylcyclohexane shown in Figure B.l. Note that all molecules possess a trivial Ci axis (indeed, an infinite number of them). Note also that if we choose a Cartesian coordinate system where the proper rotation axis is the z axis, and if the rotation axis is two-fold, then for every atom found at position (x,y,z) where x and y are not simultaneously equal to 0 (i.e., not on the z axis itself) there will be an identical atom at position (—x,—y,z). If the rotation axis is four-fold, there will be an identical atom at the three positions (—x,y,z), (x,—y,z), and (—x,—y,z). Note finally that for linear molecules the axis of the molecule is a proper symmetry axis of infinite order, i.e., Cao-... 

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. 

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. 

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. 

In the previous examples Figure 1.16 and Table 1.5), the two-fold rotation axis and the mirror plane are perpendicular to one another. However, in general, symmetry elements may intersect at various angles (( )). When crystallographic symmetry elements are of concern and since only one-, two-, three-, four- and six-fold rotation axes are allowed, only a few specific angles ( ) are possible. In most cases they are 0° (e.g. when an axis belongs to a plane), 30°, 45°, 60° and 90°. The latter means that symmetry elements are mutually perpendicular. Furthermore, all symmetry elements should intersect along the same line or in one point, otherwise a translation and, therefore, an infinite symmetry results. 

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).

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... 

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