Diffraction patterns systematic absences

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... 

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. 

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. 

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. 

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. ... 

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. 

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. 

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

Systematic absences (or extinctions) in the X-ray diffraction pattern of a single crystal are caused by the presence of lattice centering and translational symmetry elements, namely screw axes and glide planes. Such extinctions are extremely useful in deducing the space group of an unknown crystal. 

The diffraction pattern of a crystal has its own syimnetry (known as Lane syrmnetry), related to the symmetry of the stmcture, thns in the absence of systematic errors (particnlarly absorption), reflections with different, bnt related, indices should have equal intensities. According to Friedel s law, the diffraction pattern of any crystal has a center of inversion, whether the crystal itself is centrosymmetric or not, that is, reflections with indices hkl and hkl ( Friedel equivalents ) are equal. Therefore Lane symmetry is equal to the point-group symmetry of a crystal plus the inversion center (if it is not already present). There are 11 Lane symmetry classes. For example, if a crystal is monoclinic (fi 90°), then I hkl) = I hkl) = I hkl) = I hkl) I hkl). For an orthorhombic crystal, reflections hkl, hkl, hkl, hid and their Friedel equivalents are equal. If by chance a monochnic crystal has f 90°, it can be mistaken for an orthorhombic, but Lane symmetry will show the error. 

Refers to the Miller indices (Shkl) values) that are absent from the diffraction pattern. For instance, a body-centered cubic lattice with no other screw axes and glide planes will have a nonzero intensity for all reflections where the sum oi Qi + k + 1) yields an odd number, such as (100), (111), etc. other reflections from planes in which the sum of their Miller indices are even, such as (110), (200), (211), etc. will be present in the diffraction pattern. As these values indicate, there are three types of systematic absences three-dimensional absences (true for all hkl) resulting from pure translations (cell centering), two-dimensional absences from glide planes, and one-dimensional absences from screw axes.[261... 

There have been a number of reports of polymorphic systems in which the reader might be led to expect some similarity in the powder pattern of two polymorphs because they crystallize in the same space group. There is no physical basis for this expectation. Except for the systematic absences of certain reflections due to space group symmetry polymorphic structures in the same space group and different cell constants will have different powder diffraction patterns. On the other hand, polymorphs with similar cell constants but different space groups may exhibit some similarity in X-ray powder diffraction patterns, but these cases are very rare vide infra). See also Section 2.4.3. 

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. 

The internal symmetry of the crystal is revealed in the symmetry of the Bragg reflection intensities, as discussed in Chapter 4. The crystal system is derived by examining the symmetry among various classes of reflections. Key patterns in the diffraction intensities indicate the presence of specific symmetry operations and lead to a determination of the space group. The translational component of the symmetry elements, as in glide planes or screw axes, causes selective and predictable destructive interference to occur. These are the systematic absences that characterize these symmetry elements, described in Chapter 4. 

Powder diffraction pattern of a compound with unknown crystal structure was indexed with the following unit cell parameters (shown approximately) a = 10.34 A, h = 6.02 A, c = 4.70 A, a = 90°, P = 90° and y = 90°. The list of all Bragg peaks observed from 15 to 60° 20 is shown in Table 2.19. Analyze systematic absences (if any) present in this powder diffraction pattern and suggest possible space groups symmetry for the material. 

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