Translational repeat distance

Figure 1. The flexible quasihexagonal 2D H-bonded GS sheet as viewed normal to (upper) and along (lower) the ribbon direction (boxed). The translational repeat distances...

Why must a glide-reflection operation entail a translation of i the repeat distance and no other fraction of it ... 

X-ray diffraction analysis can be used to characterize the crystallites themselves.15,68 First, it is possible to obtain information about the chain conformation adopted by the polymer chains in the crystallites. Many polymers crystallize in helical conformations. These can be defined by the following helical parameters. The designation pn tells how to rotate and simultaneously translate along an axis to generate the specified helix. First a point of reference is rotated around the proposed axis by nip times a complete rotation of In. Simultaneously, there should be a translation of nip of the crystallographic repeat distance, a quantity that is also obtained from diffraction data. Repetition of this scheme gives a helix having p repeat units in n turns of the helix. 

The symmetry cf a helical structure is defined by an space axis, which combines a rotation of2 77/n radiusabout an axis followed by a translation cfnvh cfthe repeat distance. 

There are fourteen 3-dimensional lattices consistent with the types of rotational symmetry which a 3D repeating pattern may possess. These infinite 3D frameworks are the 14 Bravais lattices (Fig. 2.7) and Table 2.1. The repeat distances (unit translations) along the axes define the unit cell, and the full lines in Fig. 2.7 show one unit cell of each lattice. 

Screw axes are rototranslational symmetry elements. A screw axis of order n operates on an object by (a) a rotation of 2tt/h counter clockwise and then a translation by a distance t parallel to the axis, in a positive direction. The value of n is the order of the screw axis. For example, a screw axis running parallel to the c-axis in an orthorhombic crystal would entail a counter-clockwise rotation in the a-b-plane, (001), followed by a translation parallel to +c. This is a right-handed screw rotation. If the rotation component of the operator is applied n times, the total rotation is equal to 27t. The value of n is the principle symbol used for a screw axis, together with a subscript giving the fraction of the repeat distance moved in the translation. Thus, a three-fold screw axis 3i involves a counter clockwise rotation of 27t/3, followed by a translation of a distance equal to A the lattice repeat in a direction parallel to the axis. The three-fold screw axis 32 involves a counter clockwise rotation of 27t/3, followed by a translation of a distance equal to % the lattice repeat in a direction parallel to the axis. 

In general, for an atomic orbital Xj of the unit cell of a chain with repeat distance a (see Figure 15.1a) it is possible to build its associated Bloch orbitals, which are the translationally symmetry adapted orbitals, as... 

Preparation of Polydiacetylene. The preorganization for the 1,4-polymerization of diacetylenes has been discussed previously (7,14,15). Successful polymerization occurs when the diacetylenes have a translational repeat distance (d) of about 0.49 nm and an angle (tt) of about 45° with respect to the translational direction and van der Waals contact (i v) of the polydiacetylene functionalities (Fig. 1). If these structural parameters are met then the Cl and C4 carbon atoms of adjacent diacetylenes will be in a position for a topochemically controlled polymerization. Because the 0.49 nm translational repeat distance (d) of the monomer is about the same as the repeat distance in the polymer, the pol5nnerization process can occur with little disruption of the reactant packing. 

Preparation of Polytriacetylene. Soon after the early understanding of the diacetylene polymerization was reported (8,9), attempts were made to polymerize a triacetylene to produce a polytriacetylene (45). However, these early attempts as well as more recent efforts (7) were not successful. The difficulty of the topochemically controlled polymerization is the organization of the triacetylene monomer with a translational repeat distance of about 0.74 nm. 

Polytriacetylenes have recently been prepared by a topochemically controlled polymerization of a triacetylene. This was accomplished using the host-guest strategy (Fig. 8). A vinylogous amide was used to establish the required translational repeat distance and y-rays were necessary to induce the polymerization (48). 

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