N-fold rotational axes

Proper rotational operations are represented by the n-fold rotation axes n 1000 (n = 2, 3,4, 6). Rotation-inversion axes such as the 2 axis are improper rotation operations, while screw axes and glide planes are combined rotation-translation operations. 

The limitation on the types of n-fold rotational axes can be easily visualized by considering the analogous task of completely tiling a two-dimensional plane with polygon tiles, a process called tessellation. Congment regular polygons (equilateral and... 

Is there, then, an improper axis S(Note that if n > 2, the n-fold rotation axis C is by convention taken to be the vertical (z) axis). You have replied that there is indeed an axis Sjn. However, are there other binary axes perpendicular to the If not, the symmetry of your molecule is described by one of the groups Ja, (Note that if n is odd, there is a center of inversion). However, this result is subject to doubt, as there are very few molecules of symmetry J ... 

Consider a crystal having a n-fold rotation axis, often simply referred to as an n-axis. In Fig. 9.1.5, OE, OE, OE" represent the horizontal projections of three vertical planes generated by successive rotations of 2n/ about the symmetry axis, which passes through point O. As reference axes, we adopt the n-axis, OE, and OE", and the latter two are assigned unit length. Next, we consider a plane E F which is drawn parallel to OE. The intercepts of E F on the three axes are oo, 1, and (sec 2n/ri)/2. The Miller indices of plane E F are therefore (0,1,2 cos 2n/n) which must be integers. Since cos 2n/n < 1, the only possible values of 2 cos 2n/n are 2, 1, and 0. Hence... 

If a molecule has several n-fold symmetry axes, these must intersect at a point and their spatial arrangement must be such that a rotation Cn about one of the axes results in an interchange of the other axes. This condition severely limits the number of possible groups containing more than one n-fold axis. We shall discuss the various values of n successively below. 

Z)5, Z)6, D-j,. .., D This series can be continued by analogy. It is characterized by one n-fold rotation axis and n twofold rotation axes perpendicular to the n-fold axis. 

An n-fold rotation axis is a line, rotation about which through 360°/n leaves the object indistinguishable from the original. Only axes with n = 2, 3, 4, and 6 can be present in crystals, all others being incompatible with translational symmetry. 

Rotation C through an angle Ink/n about the n-fold rotation axis C (C with maximal n denotes a principal axis which should be coincident with a cartesian z axis, the rotational axes perpendicular to the principal axis are denoted by... 

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. 

N and N are major N-fold rotation and inversion axes, respectively. 

We now come to the second point concerning plane patterns. An isolated object (for example, a polygon) can possess any kind of rotational symmetry but there is an important limitation on the types of rotational symmetry that a plane repeating pattern as a whole may possess. The possession of n-fold rotational symmetry would imply a pattern of -fold rotation axes normal to the plane (or strictly a pattern of -fold rotation points in the plane) since the pattern is a repeating one. In Fig. 2.4 let there be an axis of -fold rotation normal to the plane of the paper at /, and at Q one of the nearest other axes of -fold rotation. The rotation through Ivjn about Q transforms P into F and the same kind of rotation about P transforms Q into Q. It may happen that P and Q coincide, in which case n = 6. n all other cases PQ must be equal to, or an integral multiple of, PQ (since Q was chosen as one of the nearest axes), i.e. 4. The permissible values of n are therefore 1, 2, 3, 4, and 6. Since a 3-dimensional lattice may be regarded as built of plane nets the same restriction on kinds of symmetry applies to the 3-dimensional lattices, and hence to the symmetry of crystals. 

The symmetry of the atomic arrangement within the crystal can be described by space group theory, that is, the theory of the various ways of arranging objects in three dimensions such that a continuation of the symmetry operations gives the next unit cell and so forth (24). For protein molecules, which are by nature asymmetric, the important symmetry operations are rotation axes and screw axes. An object is said to have an n-fold rotation axis if, when an object is rotated (360/n)0, it appears like the original. For isolated objects, by point group theory, n may have any value. On the other hand, if the object is in a crystal (with its regular... 

The various symmetry elements that can be present in a crystal with three-dimensional lattices are (1) rc-fold rotation axes, (2) ra-fold rotation-inversion axes, (3) mirror planes, (4) rc-fold screw axes, and (5) glide planes. The n-fold rotation... 

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). 

As in Chapter 1 the symbol Cn means an n-fold rotation axis. In the Cn groups the n-fold axis is the only symmetry operation and there are no reflection planes, rotation-reflection axes, or inversions. 

D indicates the dihedral group containing an n-fold rotation axis plus n twofold axes perpendicular to that axis. 

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