Space group determination from diffraction patterns

4 Space group determination from diffraction patterns 

The symmetry of the crystal is indicated in its diffraction pattern. Systematic absences in the diffraction pattern show that there are translational symmetry elements relating components in the unit cell. The translational component of the symmetry elements causes selective and predictable destructive interference to occur when the specific translation in the arrangement of atoms are simple fractions of the normal lattice 

FIGURE 4.15. The symmetry of intensities implied by FriedeFs Law. Shown are the hkO Bragg reflections. Note the symmetry of the intensities for hkO and hkO and for hkO and hkO. This gives a twodimensional view of Friedel s Law which states that intensities of Bragg reflections hkl and hkl should be the same. 

FIGURE 4.16. Laue symmetry. A diffraction photograph of tetragonal lysozyme, viewed down its unique axis (c). Note the fourfold symmetry of the diffraction pattern. 

FIGURE 4.17. Laue symmetry in monoclinic and orthorhombic ceises in addition to Friedel symmetry [I(hkl) = I hkl)]. (a) Orthorhombic, [I(hkl) = I hkl) = 

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FIGURE 4.18. Space group determinations from systematic absences in Bragg reflections. hkO and hkl diffraction patterns are shown. In each case the unit-cell dimensions are the same, but different Bragg reflections are systematically absent, (a) P2i2i2i, (b) Pnma, and (c) 1222. 

As the preceding discussion implies, the determination of cell constants from powder patterns involves large uncertainties, even more so for the determination of the crystal system and space group, because the diffraction pattern does not directly lead to the symmetries, which are instead deduced from measured angles which include experimental errors, e.g., 90° for one of the cell... 

The stmcture factors used in the determination of a model structure and the unit cell dimensions and space group are derived from diffraction patterns... 

When a powder is examined, many diffracted beams overlap, (see Section 6.11), so that the procedure of structure determination is more difficult. In particular this makes space group determination less straightforward. Nevertheless, powder diffraction data is now used routinely to determine the structures of new materials. An important technique used to solve structures from powder diffraction data is that of Rietveld refinement. In this method, the exact shape of each diffraction line, called the profile, is calculated and matched with the experimental data. Difficulties arise not only because of overlapping reflections, but also because instrumental factors add significantly to the profile of a diffracted beam. Nevertheless, Rietveld refinement of powder diffraction patterns is routinely used to determine the structures of materials that cannot readily be prepared in a form suitable for single crystal X-ray study. 

Whenever = 2n- -l, with n integer, exp[7riA ] = —1 and the structure factor vanishes. Thus, the above list of observed structure factors is indeed a direct experimental proof that in the crystal of succinic anhydride any scatterer atx,y,z has an equivalent scatterer at -x, 1/2 -b y, 1/2 - z. The same applies to (hOO) and (00/) reflections because of the other two equivalent positions of the space group. Internal symmetry is revealed by destructive interference of scattered waves from symmetry-related objects. In fact, the analysis of systematic absences is the method normally used for determining the space group from diffraction patterns. 

The main purpose of extracting the diffraction information from any kind of diffraction pattern is to continue with stmcture solution using the extracted quantitative data. This data includes the calculated unit cell parameters obtained during the indexing procedure, s mimetry determination such as a space group or a set of possible space groups and integrated intensities for indexed reflections. 

Electron-diffraction patterns were recorded for the dry and frozen-hydrate fonns of pustulan from Pustulan papullosa. The frozen-hydrate form crystallizes in a rectangular unit-cell, with a = 2.44 nm and b = 1.77 nm. The chain-axis repeat was not determined. Systematic absences led to the two-dimensional space-group Pgg. Dehydration results in a reversible, partial collapse of the crystals. 

Distinguishing Space Groups by Systematic Absences. From the symmetry and metric properties of an X-ray diffraction pattern we can determine which of the 6 crystal systems and, further, which of the 11 Laue symmetries we are dealing with. Since we need to know the specific space group in order to solve and refine a crystal structure, we would still be in a highly unsatisfactory situation were it not for the fact that the X-ray data can tell us still more. 

While it is very easy, when one knows the structure of the crystal and the wavelength of the rays, to predict the diffraction pattern, it is quite another matter to deduce the crystal structure in all Its details from the observed pattern and the known wavelength. The first step is lo determine the spacing of the atomic planes from the Bragg equation, and hence the dimensions of the unit cell. Any special symmetry of the space group of the structure will be apparent from space group extinction. A Irial analysis may (hen solve the structure, or it may be necessary to measure the structure factors and try to find the phases or a Fourier synthesis. Various techniques can be used, such as the F2 series, the heavy atom, the isomorphous series, anomalous atomic scattering, expansion of the crystal and other methods. 

As mentioned earlier, the unit-cell space group can be determined from systematic absences in the the diffraction pattern. With the space group in hand, the crystallographer can determine the space group of the reciprocal lattice, and thus know which orientations of the crystal will give identical data. All reciprocal lattices possess a symmetry element called a center cf symmetry or point of inversion at the origin. That is, the intensity of each reflection hkl is identical to the intensity of reflection -h k -1. To see why, recall from our discussion of lattice indices (Section II.B) that the the index of the (230) planes can also be expressed as (-2 -3 0). In fact, the 230 and the —2 -3 0 reflections come from opposite sides of the same set of planes, and the reflection intensities are identical. (The equivalence of Ihkl and l h k l is called Friedel s law,but there are exceptions. See Chapter 6, Section IV.) This means that half of the reflections in the reciprocal lattice are redundant, and data collection that covers 180° about any reciprocal-lattice axis will capture all unique reflections. 

During the last five years, a powerful new method of getting crystal structural information from powder diffraction patterns has become widely used. Known variously as the Rietveld method, profile refinement1, or, more descriptively, whole-pattern-fitting structure refinement, the method was first introduced by Rietveld (X, 2) for use with neutron powder diffraction patterns. It has now been successfully used with neutron data to determine crystal structural details of more than 200 different materials in polycrystalline powder form. Later modified to work with x-ray powder patterns (3, X) the method has now been used for the refinement of more than 30 crystal structures, in 15 space groups, from x-ray powder data. Neutron applications have been reviewed by Cheetham and Taylor (5) and those for x-ray by Young (6). 

An interesting side benefit is relatively precise determination of lattice parameters, even from forward reflection (20<9O°) diffraction patterns with broadened reflections. Fig. 1 shows an example. Even though LaPO, has four lattice parameters (space group P2/n) and the pattern is not well resolved, the precision in the 3 cell-edge parameters was 6 parts in 10 and that in the angle was 0.04° (6). 

Space groups (or enantiomorphous pairs) that are uniquely determined from the symmetry of the diffraction pattern and systematic absences are shown in boldface type. 

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