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fraction, as described in Fig. 4.24, should be a maximum. These principles are analyzed on a series of examples of macromolecular crystals in Sects. 5.1.9 and 5.1.10. Crystals with less-well-defined motifs are described in Sect. 5.1.11.

5.1.2 Lattice Description

The space-lattice of Fig. 5.3 helps in the description of crystals. One must, however, observe that the lattice is only a mathematical abstraction and not the actual crystal. The lattice points may or may not coincide with actual motif positions. All motifs, however, have a fixed relationship to the lattice. The lattice descriptor is the unit cell, drawn in heavy lines. The three non-coplanar axes of the unit cell have the lengths: a, b, c, and the angles: , , . One lattice point occupies the corners of the unit cell. Since for any given unit cell only 1/8 of its corners lies inside the unit cell, each primitive unit cell shown, contains only one lattice point.

Fig. 5.3

The vectors identifying any lattice point, P, relative to the origin 0,0,0 can be written with three integers, as indicated in the figure. The example chosen corresponds to 2,1,3. Note also that the chosen axis system is a right-handed system,

meaning that if one looks along the vector c , the rotation from a to b is clockwise.

5.1.3 Unit Cells

Crystals are also described by unit cells, similar to lattices. All unit cells can be grouped into seven crystal systems, as listed in Fig. 5.4. The cubic system has the highest symmetry, the triclinic system has the lowest. The seven unit cells of the crystals can be linked to a total of 14 different lattices, called the Bravais lattices. The

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Fig. 5.4

unit cells of the Bravais lattices are also listed in Fig. 5.4. To identify the lattices by their external crystal symmetry, it is customary to sometimes give a somewhat larger unit cell than the primitive unit cell (P). The unit cell I (for German “innenzentriert,” body-centered) has two lattice points; the unit cell F, (face-centered) has four; and C, has two (base-centered, face ab). The orthorhombic crystal system with its four possible Bravais lattices is drawn as an example in Fig. 5.4. The primitive unit cells that belong to the larger unit cells are indicated by dashed lines. It is quite obvious that from the primitive cells, it is not possible to identify the crystal systems.

5.1.4 Miller Indices

To navigate throughout a crystal, a set of indices has been devised, the Miller indices. Figure 5.5 illustrates the construction necessary to find the Miller index of a plane. The plane to be described may be a macroscopic, naturally occurring crystal surface, or it may be a specific lattice plane needed to describe a microscopic, atomic-scale problem. Once the axes and the plane are identified, the plane is moved parallel to itself towards the origin of the axis system until the least set of whole-numbered multiples of the reciprocals of the intercepts with the axes in units of a, b, and c is reached. All planes parallel to the original plane are given the same Miller index (h k l). In the microscopic descriptions of two planes that are parallel, but refer to different occupancies with motifs, it is permitted to use larger than the least multiples to distinguish differences in packing of motifs inside a crystal, i.e., the (111) and (222) planes may be different and are then given these different indices.

Fortunately, indices higher than 3 are rare, so that it is easy to visualize the various planes without calculation. Figure 5.5 shows the difference between a rare (232) plane, and the more frequent (111) planes. Planes parallel to a unit cell axis intersect

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Similar to the planes, directions can be named by specifying in brackets [ ] the smallest integer of multiple lattice-spacings u, v, w which is met by a line from the

is [213]. The X-ray origin in the chosen direction. In Fig. 5.3, the direction OP

diffraction peak due to a plane (h k l) is given by specifying h k l (no parentheses), and a lattice point is given by its coordinates. Point P in Fig. 5.3 is described as 2,1,3. A summary of these definitions is given in Fig. 5.5. They allow a precise description of surfaces, directions, and points within crystals and lattices.

5.1.5 Symmetry Operations

Basic to the representation of crystals, lattices, and motifs is their symmetry. Group theory, summarized in Appendix 14 with Fig. A.14.1, is the branch of mathematics dealing with interrelationships between the symmetry elements. Some elementary group theory is needed for the operations, described next.

Point groups are made up of an internally consistent set of symmetry operations that leave at least one point unchanged when operating on an object to give a closed operation. Figure 5.7 illustrates the simplest, closed symmetry operations. The unit operation “1” is represented by the Statue of Liberty. Only a full rotation about the axis of symmetry will reproduce the original statue. There is no other symmetry in

Fig. 5.7

this figure, as is also observed in most other things one sees. Each body will have this symmetry 1. Additional symmetry is exceptional, and always worth mentioning in the description of an object.

The arrow at the top in Fig. 5.7 has a twofold axis of symmetry. It is indistinguishable after rotation about this axis by 180°. The operation 2 reproduces the same arrow after half a turn. Naturally this is only true if the surface which is not seen in the

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drawing is identical to the dotted surface. In addition, we will see below that there is additional symmetry in the arrow, namely two mirror planes.

The other possible rotation axes 2 /n in crystals are analogously labeled by “n” and their symbols are indicated in the center in form of polygons. Axes with n > 6 cannot describe a space-filling lattice, and even a fivefold axis of symmetry is not possible. Motifs, on the other hand, may have any symmetry. The rotation axis is, for example, exhibited by the cylinder shown on the right of Fig. 5.7. It is also shown that the stacking of cylinders as motifs in a crystal cannot make use of this high symmetry, their crystal symmetry is only 3, 4, or 6. The pencil of Fig. 5.7 has a sixfold rotation axis which is not disturbed by the higher symmetry of the rounded sections. The group of arrows in the center of Fig. 5.7 has fourfold symmetry, assuming that the two sides of each arrow are different in the same way as indicated.

A final, sixth, symmetry-element of the point groups is the center of inversion, i, marked by an open circle ( ). Its operation involves a projection of each point of the object through its center to a point equally distant on the other side. The projection is illustrated by the letter A in Fig. 5.8.

Fig. 5.8

Note that the letter is different in front and back. The other symmetry elements of the letter A are also indicated. The six basic point-symmetry elements (1, 2, 3, 4, 6, and i) can describe the crystal symmetry as it is macroscopically recognizable by inspection, if needed, helped by optical microscopy.

The basic symmetry elements can also be carried out in sequence. This combination is called the product of the elements (see Appendix 14). Two fourfold rotations about the axis of symmetry are thus equal to one twofold rotation, two sixfold rotations, a threefold one, and three sixfold rotations, a twofold one. When

the element i is included in the products, one can generate the inversion axes 1, 2, 3,

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Figure 5.9 shows on the upper left the operation of a screw axis 31 on a lattice point. After a rotation of 2 /3, a translation by 1/3 in the direction c is carried out. After two more of such 31 operations, the identical point of the lattice in the next unit cell is reached. Three 31 operations, thus, are identical to one unit cell translation, c. Again, the symbol indicating a screw axis 31 is drawn next to the axis indication. The also shown action of a screw axis 32 has a rotation of 2 /3 followed by a translation of 2c/3. Figure 5.9 shows that identity of the filled circle is reached two unit-cell lengths away from the initial point. Filling-in the missing points, by translation back by c for each point to reach the open circles, gives a screw axis that is identical to 31, except that it rotates counterclockwise instead of clockwise as can be seen by following the dotted path with six counterclockwise rotations by 2 /3 and a translation 1c/3. The other possible screw axes are listed together with their symbols. As in threefold screw axes, 41 and 43, 61 and 65, as well as 62 and 65 are of opposite handedness, they are enantiomorphs (Gk. ( and "#, of opposite form, as in stereoisomers, see Appendix 14).

Glide planes are also shown in Fig. 5.9. Shown is the glide plane a, the product of a mirror operation followed by a glide of a/2. The glide planes with glides along b and c are analogous to a, with glides of b/2 and c/2. The glide planes n have a diagonal glide (a + b)/2, (b + c)/2, or (c + a)/2. The glide plane d is the so-called diamond glide, found in centered lattices with a 1/4 glide along the diagonal. The total of 230 space groups represents the possible repetition schemes that act on the motif(s) placed in the unit cell and generate the crystal.

5.1.6 Helices

Crystallization of flexible linear macromolecules is only possible if the molecule assumes an ordered conformation of low energy. In the treatment of the size and shape of random coils of macromolecules in Sects. 1.3.3 5 it is found that the backbone bonds of the molecules have only few conformations with low energy, the rotational isomers, illustrated in Figs. 1.36 and 1.37. A conformation is defined in Fig. A.14.3 and distinguished from a configuration. Repeating low-energy conformations along the chain commonly leads to helices. Based on lowest-energy, the helices may result in any rotational symmetry and are not necessarily rational in their repetition along their axis. For the description of rational and irrational numbers, see Fig. A.14.4. The screw-axes in crystals, in contrast, as described in Sect. 5.1.5, are limited to 1-, 2-, 3-, 4-, or 6-fold rotations with rational repeats in the direction of the axes. In addition, screw axes do not necessarily recognize the continuity of the molecule, as illustrated by the 32 screw axis, described in Fig. 5.9. To rectify this problem, polymer scientists have developed the helix-lattice concept [2,3].

Figure 5.10 explains the helix notation for simple 1*3/1 helixes which are equal to the 31 screw axes of Fig. 5.9. The class of the helix, A, indicates the number of chain atoms that are placed in relation to a lattice point of the single helix, as will be shown below. For the description of the simple helix shown in Fig. 5.10, the class is by its definition the number 1 and as such can be omitted, i.e., one may just write 3/1 as the helix notation to show the identity to the screw axis 31 of the crystal lattice. At the bottom of the helix of Fig. 5.10, the projection of the helix is drawn.

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by translational symmetry, as shown in Fig. 5.9. The left-handed nature of a molecule when described with a 32 helix is not obvious and the helix symmetry operations do not follow the progression of the molecule. Of special interest are the equivalencies for the sixfold screw axes listed in Fig. 5.11.

Finally, one must observe that polymer helices may have any symmetry, but already of the shown helices in Fig. 5.11, the 5-, 7-, 8-, and 9-fold screw axes are not permitted as crystal symmetry, as are any not shown screw axes. Finally, the positions of the lattice points of a polymer helix can be computed from the definitions and equations in Fig. 5.11.

Selenium is an example of a class 1 helix. It is given on the left side in Fig. 5.12. Since the 1*3/1 helix is identical to the 31 screw axis, there is no difficulty to describe its crystal structure in Fig. 5.20, below. Next, planar zig-zag chains can be looked upon as degenerated helices of either no bond rotation, or a 180o rotation according to two conventions to assign the zero rotation angle, which are followed by different researchers. One must, thus, check for any helix description the zero rotation angle

Fig. 5.12

chosen! Figure 5.12 also illustrates on the top right the 1*2/1 helix of polyethylene with its CRU of CH2– (see Sect. 1.2). A class 2 helix is illustrated at the lower right with the example of poly(oxymethylene), POM. It can be visualized by rotation of each backbone bond by a 102° angle, not far from the all-gauche conformation of 120°. The helix 2*9/5 is found in a trigonal crystal, the unit cell of which is indicated in Fig. 5.12. In order to fit the helix 2*9/5 to the 31 or 32 screw axis with three lattice points per identity period, required by the trigonal crystal, the motif per lattice point is increased to three monomer units. The corresponding helix lattice is 6*3/1 with the three monomers per lattice point having 5/3 rotation about the helix axis by themselves, so that the three steps of rotation of the 31 screw axes total the five

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rotations of the helix 2*9/5. Small adjustments in the POM helix do not take a large amount of energy. Thus, closing the rotation angle to 117°, leads to an only slightly less stable orthorhombic crystal with a 2*2/1 helix and results in a second possible crystal structure (2*2/1 2*10/5). It has even been proposed that 2*29/16 or even irrational helices may be more stable than the helix 2*9/5 of POM. Experimentally such small local changes are difficult to establish by X-ray diffraction.

For the isotactic polypropylene helix of Fig. 5.13, two bond-rotation angles must be specified. The planar zig-zag at 1 = 2 = 0, the all-trans conformation, is sterically hindered by contact between two successive CH3 groups. The stable 2*3/1 helix consists of close to alternating gauche and trans conformations. When looking along one repeating unit of the chain, two stereoisomers can be recognized, marked as d and configurations. The chirality of the (LC)HC*(CH3)(RC) grouping, however, is negligible since the left chain end (LC) and the right chain end (RC) are hardly distinguishable (see also Appendix 14). On packing d and configurations in a crystal, however, largely different energies result, as discussed in Sect. 5.1.8.

Fig. 5.13

The diagram of the potential energy on the right in Fig. 5.13 illustrates the change in energy on bond-angle rotation for isotactic polypropylene. Alternate rotation directions are chosen for 1 and 2. Both the leftand right-handed helices are energetically equally stable and can interchange their handedness along a narrow path as seen in the figure.

A perspective drawing of a 2*3/1 helix observed for several vinyl polymers is given on the left of Fig. 5.14 [4]. The helix is drawn as a right-handed helix, but of the -configurations, as defined in Fig. 5.13, above. The side-groups, , are chosen as lattice points. The other two helices in Fig. 5.14 show that increasing bulkiness opens the helix from 2*3/1 to 2*7/2, and finally to 2*4/1 by decreasing 1 and 2 by