Absence systematic

As noted at the beginning of this section, crystallographic symmetry has an effect on the structure amplitude, and therefore, it affects the intensities of Bragg peaks. The presence of translational symmetry causes certain combinations of Miller indices to become extinct because symmetrical contributions into Eq. 2.110 result in the cancellation of relevant trigonometric factors in Eq. 2.107. It is also said that some combinations of indices are forbidden due to the occurrence of translational symmetry. 

Consider a body-centered lattice, in which every atom has a symmetrically equivalent atom shifted by (1/2, 1/2, 1/2). The two matrices A (Eq. 2.109) for every pair of the symmetrically identical atoms are  

When these arguments are substituted into the most general equation (2.107) and the summation is carried over every pair of the symmetrically equivalent atoms, the resulting sums of the corresponding trigonometric factors are 

All prefactors in Eq. 2.107 (i.e. g, t,f, Af, and Af ) are identical for the pairs of the symmetrically equivalent atoms. Hence, Bragg reflections in which the sums of all Miller indices are odd should have zero structure amplitude and zero intensity in any body-centered crystal structure. In other words, they become extinct or absent, and therefore, could not be observed and are forbidden. 

This property, which is introduced by the presence of a translational symmetry, is called the systematic absence (or the systematic extinction). Therefore, in a body-centered lattice only Bragg reflections in which the sums of all Miller indices are even (i.e. h + k + l = 2n and = 1, 2, 3,. ..) may have non-zero intensity and be observed. It is worth noting that some (but not all) of the Bragg reflections with h + k + I = 2n may become extinct because their intensities are too low to be detected due to other reasons, e.g. a specific distribution of atoms in the unit cell, which is not predetermined by symmetry. 

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All tenus in the sum vanish if / is odd, so (00/) reflections will be observed only if / is even. Similar restrictions apply to classes of reflections with two indices equal to zero for other types of screw axis and to classes with one index equal to zero for glide planes. These systematic absences, which are tabulated m the International Tables for Crystallography vol A, may be used to identify the space group, or at least limit die... 

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. 

No systematic extinctions were found in addition to those characteristic of body-centering. The only space groups with Laue symmetry Th allowed by this observation are T, T3, and T6. No non-systematic absences were recorded. 

These values were used to correct the intensities for nine representative data sets, organic crystals, organometallics and minerals, and the data compared with respect to systematic absences and space group assignment (i) with no corrections to the data, (ii) with an absorption correction (SADABS [9]), (iii) with only 1/2 correction, and (iv) with both absorption and X/2 correction. Analogously four different refinements per sample were carried out based on F2. 

The magnitudes of the corrections varied quite widely over the nine data sets, the average correction being less than one esd, however the maximum correction was 47.4s. In all cases there was an improvement in the number of observed systematic absences and, hence, space group assignment. 

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. 

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. 

The pattern of observed lines for the two other cubic crystal systems, body-centred and face-centred is rather different from that of the primitive system. The differences arise because the centring leads to destructive interference for some reflections and these extra missing reflections are known as systematic absences. 

It is possible to characterize the type of Bravais lattice present by the pattern of systematic absences. Although our discussion has centred on cubic crystals, these absences apply to all crystal systems, not just to cubic, and are summarized in Table 2.3 at the end of the next section. The allowed values of are listed in Table 2.2 for... 

TABLE 2.3 Systematic absences due to translational symmetry elements... 

SYSTEMATIC ABSENCES DUE TO SCREW AXES AND GLIDE PLANES... 

The reflections are indexed, and from the systematic absences the Bravais lattice and... 

Lamellar, single crystals of ivory-nut mannan were studied by electron diffraction. The base-plane dimensions of the unit cell are a = 0.722 nm and b = 0.892 nm. The systematic absences confirmed the space group P212121. The diffraction pattern did not change with the crystallization temperature. Oriented crystallization ofD-mannan with its chain axis parallel to the microfibril substrates, Valonia ventricosa and bacterial cellulose, was discovered ( hetero-shish-kebabs ). 

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. 

Fig. 107. The systematic absences in this reciprocal lattice indicate that a larger reciprocal cell (that is, a smaller real cell) can be chosen. The new reciprocal cell is heavily outlined.

For all other crystal planes there are no simple phase relations between waves from M and those from N, and therefore no further systematic absences. Thus, the distance x of the atoms from the screw axis in the direction of the a axis is not, except by accident, a submultiple of a0i and therefore there are no systematic absences of kOO reflections. One or two of these may not appear on the photograph because the structure amplitudes happen to be very small but the point is that there are no systematic absences. The same is true for all other planes—101 for instance (Fig. 140 6), since the distance s between such a plane of atoms as NQ and the plane through P is not, except by accident, a simple submultiple of the spacing d1Q1. 

Thus the only systematic absences caused by a twofold screw axis are the odd orders of reflection from the plane perpendicular to the screw axis. 

Fig. 141. Ordinary twofold axes. No systematic absences of reflections.

Ordinary reflection planes m cannot be detected in this way because they cause no systematic absences thus when two molecules related by a reflection plane are seen from a direction normal to the reflection plane, one molecule is exactly eclipsed by the other there is no apparent halving of an axis or a diagonal, and therefore there are no systematic absences due to a plane of symmetry. 

Thus, while screw axes and glide planes can be detected and distinguished from each other by observing which types of reflections are absent, ordinary rotation axes and reflection planes cannot be detected in this way, since neither type leads to any systematic absences of reflections. 

In examining a list of X-ray reflections for this purpose, it is best to look first for evidence of the lattice type—whether it is simple (P) or compound systematic absences throughout the whole range of reflections indicate a compound lattice, and the types of absences show whether the cell is body-centred (/), side-centred ( 4, P, Or C), or face-centred (F). When this is settled, look for further absences systematic absences throughout a zone of reflections indicate a glide plane normal to the zone axis, while systematic absences of reflections from a single principal plane indicate a screw axis normal to the plane. The result... 

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