Space groups, from systematic absences

Table 9.4.3. Examples of deduction of space groups from systematic absences...

Deduction of space groups from systematic absences... 

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. 

Among the most recent advances of the powder method is the determination of crystal structures from powder diffraction data. It is an application for which the resolution of the pattern is of prime importance. A series of successive stages are involved in the analysis, including the determination of cell dimensions and identification of the space group from systematic reflection absences, the extraction of structure factor moduli I hkl y the solution to the phase problem to elaborate a structure model and, finally, the refinement of the atomic coordinates with the Rietveld method. 

One deduces the space group from the symmetry in the crystal s diffraction pattern and the systematic absence of specific reflections in that pattern. The crystal s cell dimensions are derived from the diffraction pattern for the crystal collected on X-ray film or measured with a diffractometer. An estimation of Z (the number of molecules per unit cell) can be carried out using a method called Vm proposed by Matthews. For most protein crystals the ratio of the unit cell volume and the molecular weight is a value around 2.15 AVOa. Calculation of Z by this method must yield a number of molecules per unit cell that is in agreement with the decided-upon space group. 

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. 

Laue class Symbol from systematic absences Possible space groups ... 

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. 

Unit cell dimensions are measured from the directions, 26, of diffracted beams and an application of Bragg s law, A = 2rfsin. Space groups are derived from systematic absences in the Bragg reflections. 

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. 

The process of structure solution of an unknown material from powder data can be divided into several steps. First one must record the highest-quality data possible on a (ideally) pure sample. One must then determine the unit cell parameters from observed peak positions (several software packages exist to tackle this nontrivial step). The space group symmetry must then be determined from systematic absences. At this stage, one of a number of different routes can be followed. One... 

With the final cell constants, the crystal system derived from them, and the Bravais lattice, the reflections can be indexed. The space group can be identified from systematic absences [19. pp. 21 -53]. 

It should be mentioned here that the reciprocal lattice which is three dimension cannot be recorded without any ambiguity on a two-dimensional film and so, the rotational method discussed before is not sufficient to record the diffraction spots with their three hkl indices known. Therefore, the rotational method cannot give any satisfactory information about the Space Group from the rule of systematic absences. Moreover, there is every chance of coincidence of the diffraction from one hkl and its hkl counterpart and so, they hardly can be read separately. Therefore, it becomes necessary to move the film along with the rotation of the crystal to record one reciprocal lattice net lying on a plane. 

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 two exponential tenns are complex conjugates of one another, so that all structure amplitudes must be real and their phases can therefore be only zero or n. (Nearly 40% of all known structures belong to monoclinic space group Pl c. The systematic absences of (OlcO) reflections when A is odd and of (liOl) reflections when / is odd identify this space group and show tiiat it is centrosyimnetric.) Even in the absence of a definitive set of systematic absences it is still possible to infer the (probable) presence of a centre of synnnetry. A J C Wilson [21] first observed that the probability distribution of the magnitudes of the structure amplitudes would be different if the amplitudes were constrained to be real from that if they could be complex. Wilson and co-workers established a procedure by which the frequencies of suitably scaled values of F could be compared with the tlieoretical distributions for centrosymmetric and noncentrosymmetric structures. (Note that Wilson named the statistical distributions centric and acentric. These were not intended to be synonyms for centrosyimnetric and noncentrosynnnetric, but they have come to be used that way.)... 

All of the observed reflections could be indexed on the basis of a cubic unit cell with Oo = 11.82 A the estimated probable error is 0.01 A. The only systematic absences were hhl with l odd this is characteristic of the space group 0 -PmP>n, which also was reported by von Stackelberg from his single-crystal work on sulfur dioxide hydrate. For 46 H20 and 6 Cl2 in the unit cell the calculated density is 1.26 densities reported by various observers range from 1.23 to 1.29. 

The three space groups Cmc2, C2cm and Cmcm have the same systematic absences and cannot be distinguished from diffraction data. However, their projection symmetries are different (see Table 1). Since HRTEM images maintain the phase information, it is... 

Seven crystal systems as described in Table 3.2 occur in the 32 point groups that can be assigned to protein crystals. For crystals with symmetry higher than triclinic, particles within the cell are repeated as a consequence of symmetry operations. The number of asymmetric units within the unit cell is related but not necessarily equal to the number of molecules in a unit cell, depending on how the molecules are related by symmetry operations. From the symmetry in the X-ray diffraction pattern and the systematic absence of specific reflections in the pattern, it is possible to deduce the space group to which the crystal belongs. 

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. 

There are no further systematic absences the absences of odd orders of A00, 070, and 00Z are included in the general statement that reflections having h+k- -l odd are absent. This means that, for a body-centred lattice, we cannot tell (from the systematic absences) whether twofold screw axes are present or not. The possible space-groups are... 

The type of arrangement of pattern-units is called the space-lattice . Secondly, the group of atoms forming a pattern-unit—the group of atoms associated with each lattice point—may have certain symmetries, and some of these symmetries cause further systematic absences of certain types of reflections from the diffraction pattern. The complex of symmetry elements displayed by the complete arrangement is known as the space-group. ... 

Crystal symmetries that entail centering translations and/or those symmetry operations that have translational components (screw rotations and glides) cause certain sets of X-ray reflections to be absent from the diffraction pattern. Such absences are called systematic absences. A general explanation of why this happens would take more space and require use of more diffraction theory than is possible here. Thus, after giving only one heuristic demonstration of how a systematic absence can arise, we shall go directly to a discussion of how such absences enable us to take a giant step toward specifying the space group. 

Let us now see how these conditions for systematic absences are used. Suppose we have established from the X-ray diffraction data that a crystal is monoclinic. See Table 11.7 for the monoclinic space groups. We can next see if the unit cell is primitive or centered. If we choose the unique axis to be c, we look for absences indicative of A centering (hkl, k + 2 n) or B centering... 

In all other crystal systems we encounter the same general situation, namely, that a few space groups (69, in fact) can be uniquely identified from a knowledge of diffraction symmetry and systematic absences, while the rest form mostly pairs, or small groups that are indistinguishable in this way. Table 11.9 lists for the triclinic, monoclinic, and orthorhombic crystal systems the uniquely determined space groups and the sets with identical systematic absences. 

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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