Diffraction space symmetries

Fig. 8. Diffraction space according to the "disordered stacking model" (a) achiral (zigzag) tube (b) chiral tube. The parallel circles represent the inner rims of diffuse coronae, generated by streaked reflexions. The oo.l nodes generate sharp circles. In (a) two symmetry related 10.0 type nodes generate one circle. In the chiral case (b) each node generates a separate corona [9].

Fig. 11. Simulated diffraction space of a chiral (40, 5) SWCNT. (a) Normal incidence diffraction pattern with 2mm symmetry (b),(c),(d) and (e) four sections of diffraction space at the levels indicated by arrows. Note the absence of azimuthal dependence of the intensity. The radii of the dark circles are given by the zeros of the sums of Bessel functions [17].

Several sections of the diffraction space of a chiral SWCNT (40, 5) are reproduced in Fig. 11. In Fig. 11(a) the normal incidence pattern is shown note the 2mm symmetry. The sections = constant exhibit bright circles having radii corresponding to the maxima of the Bessel functions in Eq.(7). The absence of azimuthal dependence of the intensity is consistent with the point group symmetry of diffraction space, which reflects the symmetry of direct space i.e. the infinite chiral tube as well as the corresponding diffraction space exhibit a rotation axis of infinite multiplicity parallel to the tube axis. 

Fig. 12. S imulated diffraction space for a (10, 10) armchair tube, (a) Normal incidence pattern, note the absence of oo.l spots, (b) Equatorial section. The pattern has 20-fold symmetry, (c) The section The pattern contains 20 radial...

Fig. 13. Simulated diffraction space of a 10-layer monochiral MWCNT with Hamada indices (40+8/ , 5+k) with / =0,...,9. In (a), (a ) and (02) the initial stacking at ( q was ABAB. whereas in (b), (b[) and (b2) the initial stacking was random, (a) The normal incidence pattern has a centre of symmetry only. (3 )(a2) The cusps are of two different types. The arc length separating the cusps is c (b) The normal incidence pattern now exhibits 2mm symmetry. (b )(b2) The cusps are distributed at random along the generating circles of the evolutes. These sections represent the diffuse coronae referred to in the "disordered stacking model" [17].

Diffraction Pattern Symmetry and Crystal Space Group... 

FIGURE 1.14 Seen here is the hk0 zone diffraction pattern from a crystal of M4 dogfish lactate dehydrogenase obtained using a precession camera. It is based on a tetragonal crystal system and, therefore, exhibits a fourfold axis of symmetry. The hole at center represents the point where the primary X-ray beam would strike the film (but is blocked by a circular beamstop). Note the very predictable positions of the diffraction intensities. All the intensities, or reflections, fall at regular intervals on an orthogonal net, or lattice. This lattice in diffraction space is called the reciprocal lattice. 

What has been said here is true but obscures another fundamental property of the Fourier transform, one that complicates matters a bit but not hopelessly so. The Fourier transform fails to directly carry translational relationships from one space to another, in particular, from real space into reciprocal space. This means that the transform does not discriminate between asymmetric units based on the distances between them. The immediate relevance of this is that a set of asymmetric units related by a screw axis symmetry operator (which has translational components) in real space is transformed into diffraction space as though it simply contained a pure rotation axis. The translational components are lost. If our crystal has a 6i axis, we will see sixfold symmetry in the diffraction pattern. If we have 2i2j2i symmetry in real space, the diffraction pattern will exhibit 222 (or more properly, mmm) symmetry. 

What is the symmetry of the reciprocal lattice That is, what are the symmetry operators that relate sets of identical intensities The symmetry group that we observe for a crystal in reciprocal space, namely the diffraction pattern symmetry, is called the Laue symmetry, or Laue group. 

There are many insights, tricks, and unexpected relationships in the symmetries of diffraction patterns, and they can only be appreciated by practice and experience. Navigation in diffraction space is the high art of X-ray crystallography, and learning it takes time and patience. For the novice as well as the expert, however, there is always the International Tables, Volume I, to serve as guide and reference, and it should be consulted freely. 

Virtually everything that exists or happens in real space has a corresponding property or effect in diffraction space, and vice versa. The correspondences are established through the Fourier transform, which, as we have seen, operates symmetrically in both directions, getting us from real space into reciprocal space and back again. It may occasionally appear that this rule is violated, but in fact it is not. For example, the chirality of molecules and the handedness of their arrangement in real space would seem to be lost in reciprocal space as a consequence of Friedel s law and the addition of a center of symmetry to reciprocal space. If, however, we could record phases of reflections in reciprocal space, we would see that in fact chirality is preserved in phase differences between otherwise equivalent reflections. The phases of Fhu, for example, are 0, but the phase of F-h-k-i are —0. Fortunately the apparent loss of chiral information is usually not a serious problem in the X-ray analysis of proteins, as it can usually be recovered at some point by consideration of real space stereochemistry. 

In Table 7.1, where real and reciprocal cell dimensions, or other distances are related, an orthogonal system is assumed for the sake of simplicity. For nonorthogonal systems, the relationships are somewhat more complicated and contain trigonometric terms (as we saw in Chapter 3), since the unit cell angles must be taken into account. Rotational symmetry is preserved in going from real to reciprocal space, and translation operations create systematic absences of certain reflections in the diffraction pattern that makes them easily recognized. As already noted, because of Friedel s law a center of symmetry is always present in diffraction space even if it is absent in the crystal. This along with the absence of... 

The otiier type of noncrystalline solid was discovered in the 1980s in certain rapidly cooled alloy systems. D Shechtman and coworkers [15] observed electron diffraction patterns with sharp spots with fivefold rotational synnnetry, a syimnetry that had been, until that time, assumed to be impossible. It is easy to show that it is impossible to fill two- or tliree-dimensional space with identical objects that have rotational symmetries of orders other than two, tliree, four or six, and it had been assumed that the long-range periodicity necessary to produce a diffraction pattern with sharp spots could only exist in materials made by the stacking of identical unit cells. The materials that produced these diffraction patterns, but clearly could not be crystals, became known as quasicrystals. 

In a difiraction experiment one observes the location and shapes of the diffracted beams (the diffraction pattern), which can be related to the real-space structure using kinematic diffraction theory. Here, the theory is summarized as a set of rules relating the symmetry and the separation of diffracted beams to the symmetry and separation of the scatterers. 

The parallelization of crystallites, occurring as a result of fiber drawing, which consists in assuming by crystallite axes-positions more or less mutually parallel, leads to the development of texture within the fiber. In the case of PET fibers, this is a specific texture, different from that of other kinds of chemical fibers. It is called axial-tilted texture. The occurrence of such a texture is proved by the displacement of x-ray reflexes of paratropic lattice planes in relation to the equator of the texture dif-fractogram and by the deviation from the rectilinear arrangement of oblique diffraction planes. With the preservation of the principle of rotational symmetry, the inclination of all the crystallites axes in relation to the fiber axis is a characteristic of such a type of texture. The angle formed by the axes of particular crystallites (the translation direction of space lattice [c]) and the... 

The structure factor for the 104-atom complex with almost perfect icosahedral symmetry determines the intensities of the diffraction maxima, in correspondence with the inverse relationship between intensity in reciprocal space and the atom-pair vectors in real space that was discovered fifty years ago by Patterson.27 The icosahedral nature of the clusters in the cubic crystal explains the appearance of the Fibonacci numbers and the golden ratio. 

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