Symmetry in reciprocal space

The atomic stmeture of liquid metals evolves at rapid cooling and is inherited in solid state at crystallization or amorphization. Many key properties (electrical, magnetic and mechanical) of material obtained are determined by the combination of long-range translation order and local orientational order. Quasi-crystals [1], which were discovered in 1984, can be used as an example they are prepared by rapid cooling from melt and show ieosahedral symmetry in reciprocal space, which is incompatible with periodicity. 

One important point to keep in mind is that Kramers theorem always applies, which for systems with equal numbers of spin-up and spin-down electrons implies inversion symmetry in reciprocal space. Thus, even if the crystal does not have inversion symmetry as one of the point group elements, we can always use inversion symmetry, in addition to all the other point group symmetries imposed by the lattice, to reduce the size of the irreducible portion of the Brillouin Zone. In the example we have been discussing, C2 is actually equivalent to inversion symmetry, so we do not have to add it explicitly in the list of symmetry operations when deriving the size of the IBZ. 

Any perturbation from ideal space-group symmetry in a crystal will give rise to diffuse scattering. The X-ray diffuse scattering intensity at some point (hkl) in reciprocal space can be written as... 

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. 

WAXS and MAXS. Fiber symmetry means that, even in WAXS and MAXS, the scattering pattern is completely described by a slice in reciprocal space that contains the fiber axis. Nevertheless, for 20 > 9° the tangent plane approximation is no longer valid and the detector plane is mapped on a spherical surface in reciprocal space. 

Isotropization in the Case of Fiber Symmetry. If methods for the analysis of isotropic data shall be applied to scattering patterns with uniaxial orientation, the corresponding isotropic intensity must be computed. By carrying out this integration (the solid-angle average in reciprocal space) the information content of the fiber pattern is reduced. One should consider to apply an analysis of the longitudinal and the transversal structure (cf. Sect. 8.4.3). 

CDFs are computed from scattering data which are anisotropic and complete in reciprocal space. Thus the minimum requirement is a 2D SAXS pattern of a material with fiber symmetry taken in normal transmission geometry (cf. p. 37, Fig. 4.1). Required pre-evaluation of the image is described in Chap. 7. 

Oblique texture patterns have almost perfect 2mm symmetry and thus the whole set of diffraction spots is represented by the reflections in one quadrant. The arcs are exactly symmetrically placed relative to the major axis, being sections of the same spherical band in reciprocal space. The reflections on the lower half of the pattern are sections of reciprocal lattice rings, which are Friedel partners and thus equivalent to those giving the reflections of the upper half assuming a flat surface of the Ewald sphere. Actually, if the curvature of the Ewald sphere is taken into account, the upper and lower parts of a texture pattern will differ slightly. 

In this chapter, I will discuss the geometric principles of diffraction, revealing, in both the real space of the crystal s interior and in reciprocal space, the conditions that produce reflections. I will show how these conditions allow the crystallographer to determine the dimensions of the unit cell and the symmetry of its contents and how these factors determine the strategy of data collection. Finally, I will look at the devices used to produce and detect X rays and to measure precisely the intensities and positions of reflections. 

Mathematical approximations to the periodic minimal surfaces can be constructed from terms which are each the result of adding symmetry-related sinusoidal density waves for the appropriate symmetry group, and then taking the nodal surface the boundary between regions of positive and of negative density. The waves that correspond to a face-centred figure in real space are the body-centred terms in reciprocal space, namely ... 

There has been no basic formula analogous to the Poisson summation formula, characteristic of translational invariance, on which to base an analysis of quasicrystal diffraction patterns. Here successive values of reciprocal space have geometric ratios instead of the arithmetic spacing of the peaked functions observed with ordinary crystalline diffraction. Fig. 2.15 illustrates a two-dimensional section in reciprocal space of a diffraction pattern. The five-fold symmetry is exact, and typically six indices instead of three are required to index each point, with the choice of origin arbitrary, and for assigiunent of indices, ambiguous. The features of interest are ... 

Regardless of the nature of the diffraction experiment, finding the unit cell in a conforming lattice is a matter of selecting the smallest parallelepiped in reciprocal space, which completely describes the array of the experimentally registered Bragg peaks. Obviously, the selection of both the lattice and the unit cell should be consistent with crystallographic conventions (see section 1.12, Chapter 1), which impose certain constraints on the relationships between unit cell symmetry and dimensions. 

Below we will examine some practical applications of the theory of kinematical diffraction to solving crystal structures from powder diffraction data. When considering several rational examples in reciprocal space, we shall implicitly assume that the crystal structure of each sample is unknown and that it must be solved based solely on the information that can be obtained directly from a powder diffraction experiment and from a few other, quite basic properties of a polycrystalline material. The solution of a number of crystal structures in direct space will be based on the previously known structural data and supported by the results of powder diffraction analysis, such as unit cell dimensions and symmetry. 

There has been one lattice dynamics study of leucite by inelastic neutron scattering (Boysen 1990). The low-energy dispersion curves were measured for the high-temperature cubic phase along a few symmetry directions in reciprocal space. The results... 

This might seem of scant use in protein crystallography, since we have no centric space groups. Crystals of biological macromolecules, as previously pointed out, cannot possess inversion symmetry. Sets of centric reflections frequently do occur in the diffraction patterns of macromolecular crystals, however, because certain projections of most unit cells contain a center of symmetry. The correlate of a centric projection, or centric plane in real space, is a plane of centric reflections in reciprocal space. A simple example is a monoclinic unit cell of space group P2. The two asymmetric units have the same hand, as they are related by pure rotation, and for every atom in one at xj, yj, Zj there is an equivalent atom in the other at —Xj, yj, —zj. If we project the contents of the unit cell on to a plane perpendicular to the y axis, namely the xz plane, by setting y = 0 for all atoms, however, then in that... 

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. 

The evidence for the existence of screw axis symmetry is manifested in certain subclasses of reflections that are systematically absent. These systematic absences, we will see, fall along axial lines (/tOO, OkO, 00/) in reciprocal space and clearly signal not only whether an axis in real space is a screw axis or a pure rotation axis, but what kind of a screw axis it is, for instance, 4i or 42, 6i or 63. Thus the inherent symmetry of the diffraction pattern (which we call the Laue group), plus the systematic absences, allow us to unambiguously identify (except for a few odd cases) the space group of any crystal. 

This idea has some useful consequences in terms of interpreting diffraction patterns. For example, consider the case of a twofold axis in real space, along b and perpendicular to the ac plane, as we would have in a monoclinic crystal of space group P2 or C2. The projection of all of the electron density in the unit cell having this dyad symmetry onto the ac plane, would of course also have twofold symmetry. Because this projection has dyad symmetry, the corresponding diffraction pattern, which is the MO plane of reflections in reciprocal space, would also have twofold symmetry, namely reflections F o = F-h-kO-... 

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. 

As noted already, symmetry elements containing translational components, such as screw axes, appear in reciprocal space as the pure rotational element. The translational component, if it exists, must be deduced from systematic absences. These, however, are all explicitly described in the International Tables, Volume I, for each space group. 

In identifying symmetry elements present in reciprocal space, we are seeking to establish symmetry relationships between intensities in various parts of the three-dimensional diffraction pattern. In doing so, it is necessary to remember that a symmetry relationship observed for a single plane of the diffraction pattern, because of Freidel s law, may not pertain to the entire pattern, and this can only be ascertained by examining additional planes through reciprocal space. 

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. 

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