Systematic absences in the diffraction pattern

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... 

Similarly we might have considered a 3i, a 43, a 61, or any other kind of screw axis as well as the 2i. Note, however, that the translational components for these screw axes are not, but 5, j and, respectively. Thus we might expect that not every other reflection will be present, but only every third (00/ = 3n), fourth (00/ = 4n), or sixth (00/ = 6n), respectively. A pure rotation axis, having no translational component, gives no systematic absences in the diffraction pattern. Absences produced by higher symmetry screw axes are shown in Figure 6.7. 

D20.2 We can use the Debye-Scherrer powder diffraction method, follow the procedure of Example 20.3, and in particular look for systematic absences in the diffraction patterns. We can proceed through the following sequence... 

A crystal can have additional symmetry based on rotation plus translation elements. The combination of a rotation axis and a translation is a screw axis (e.g. a 2X screw axis involves a 180° rotation and translation by half a unit cell dimension). The combination of a mirror plane and a translation is a glide plane. These symmetry elements lead to systematic absences in a diffraction pattern e.g. for a 2i axis in the c... 

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. 

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... 

In Table 7.1, where real and reciprocal cell dimensions, or other distances are related, an orthogonal system is assumed for the sake of simplicity. For nonorthogonal systems, the relationships are somewhat more complicated and contain trigonometric terms (as we saw in Chapter 3), since the unit cell angles must be taken into account. Rotational symmetry is preserved in going from real to reciprocal space, and translation operations create systematic absences of certain reflections in the diffraction pattern that makes them easily recognized. As already noted, because of Friedel s law a center of symmetry is always present in diffraction space even if it is absent in the crystal. This along with the absence of... 

A careful tabulation of systematic absences in the X-ray diffraction pattern allows the space group of the crystal to be assigned. This, after the determination of unit cell dimensions, is the next piece of information that the X-ray crys-tallographer obtains. Space group determination is done systematically considering each class of Bragg reflection (/i00, OkO, 001, hk0, AO/, 0kl, hkl) and checking that examples of reflections in these classes with appreciable intensities for even and odd values of each index either are observed or are consistently... 

Though the intensity of the double reflected beam is usually very weak, sometimes both hikili and /i2 2 2 are very strong giving rise to spurious reflections. However, as this case is not very frequent in the diffraction pattern, the presence of one weak reflection violating the norm of systematic absences may be considered as spurious and can be safely ignored. 

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. 

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. 

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

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. 

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. 

A white crystalline powder, prepared by hydrothermal treatment at 200°C of a mixture of molybdic acid, H2M0O4, and methylammonium ma) chloride, CH3NH3CI, taken in a 1 2 molar ratio and acidified with hydrochloric acid, HCl, to pH = 3.5, resulted in a complex powder diffraction pattern shown in Figure 6.29. It was indexed in the monoclinic crystal system as was discussed in section 5.12.2. The space group C2/c (or its acentric subgroup Cc) was established from the analysis of the systematic absences, and the unit cell dimensions were refined using 120 resolved reflections below 20 = 60° a = 23.0648(6) A, b = 5.5134(2) A, c = 19.5609(5) A, p = 122.931(1)°, and the sample displacement 8 = -0.098(3) mm for a 250 mm goniometer radius. The unit cell volume is 2087.8 A. ... 

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