Symmetry element multiplicity

R = (i/ r) require translations t in addition to rotations j/. The irreducible representations for all Abelian groups have a phase factor c, consistent with the requirement that all h symmetry elements of the symmetry group commute. These symmetry elements of the Abelian group are obtained by multiplication of the symmetry element./ = (i/ lr) by itself an appropriate number of times, since R = E, where E is the identity element, and h is the number of elements in the Abelian group. We note that N, the number of hexagons in the ID unit cell of the nanotube, is not always equal h, particularly when d 1 and dfi d. 

The orientation of symmetry elements is referred to a coordinate system xyz. If one symmetry axis is distinguished from the others by a higher multiplicity ( principal axis ) or when there is only one symmetry axis, it is set as thez axis. 

An inversion center is mentioned only if it is the only symmetry element present. The symbol then is 1. In other cases the presence or absence of an inversion center can be recognized as follows it is present and only present if there is either an inversion axis with odd multiplicity (N, with N odd) or a rotation axis with even multiplicity and a reflection plane perpendicular to it (N/m, with N even). 

The mutual orientation of different symmetry elements is expressed by the sequence in which they are listed. The orientation refers to the coordinate system. If the symmetry axis of highest multiplicity is twofold, the sequence is x-y-z, i.e. the symmetry element in the x direction is mentioned first etc. the direction of reference for a reflection plane is nomal to the plane. If there is an axis with a higher multiplicity, it is mentioned first since it coincides by convention with the z axis, the sequence is different, namely z-x-d. The symmetry element oriented in the x direction occurs repeatedly because it is being multiplied by the higher multiplicity of the z axis the bisecting direction between x and its next symmetry-equivalent direction is the direction indicated by d. See the examples in Fig. 3.7. 

Looked upon purely as an arithmetic multiplication table, the products are all correct. This condition remains valid whatever row is selected initially4. Because of the close association between tables 1 and 2 each row in Table 1 may indeed be regarded as a representation of the symmetry elements. 

A special position in the crystal is repeated in itself by at least one symmetry element, that is, r = r. According to Eq. (B.2), this means that s must be zero if a symmetry element is to give rise to a special position. It follows that translations, screw operations, and glide planes do not generate special positions. On the other hand, positions located on proper rotation axes or centers of symmetry have lower multiplicity than general positions in the unit cell. 

First, it is necessary to define the structure. The structure of a planar zig-zag polyethylene chain is shown in Fig. 2, together with its symmetry elements. These are C2 — a two-fold rotation axis, C — a two-fold screw axis, i — a center of inversion, a — a mirror plane, and og — a glide plane. Not shown are the indentity operation, E, and the infinite number of translations by multiples of the repeat (or unit cell) distance along the chain axis. All of these symmetry operations, but no others, leave the configuration of the molecule unchanged. 

Second, a multiplication table for the factor group is written down. The space group formed by the above symmetry elements is infinite, because of the translations. If we define the translations, which carry a point in one unit cell into the corresponding point in another unit cell, as equivalent to the identity operation, then the remaining symmetry elements form a group known as the factor, or unit cell, group. The factor... 

The number of equivalent general positions (the multiplicity) is the same as the number of symmetry elements. 

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. 

---