Determination of space groups

The intensity of a reflection (hki) is proportional to the square of the structural factor F(hkl), which is given by 

of positions Wyckoff notation Site symmetry Coordinates of equivalent positions 

The intensities of reflections (hkl) and (hkl) are the same, so that the diffraction pattern has an apparent center of symmetry, even if the crystal structure is not centrosymmetric. This is known as Friedel s law. Note that Friedel s law breaks down under the conditions of anomalous scattering, which happens when the wavelength is such that the X-rays are highly absorbed by the atoms in the crystal. 

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QCBED (determination of space-group, cell constants, strain mapping, charge-density measurement)... 

As menhoned above, the unit cell of a crystalline specimen can be determined by SAED. However, the accuracy of the unit cell dimensions obtained is largely dependent upon the calibrahon of camera length, which is a funchon of specimen position and the conditions of the microscope. The procedure is complicated, and in most cases the unit cell parameters determined by SAED are less accurate than those obtained from XRD or neutron diffraction. Owing to the mulhple scattering problem, determination of space group by SAED is less reliable. Consequenhy, HRTEM is not an ideal technique for final determination of crystal structure. 

Determination of Space Group from Data Obtained from Moving Film... 

The crystal structures of hematite and corundum have been determined through the use of Taue and spectral photographs, interpreted with the aid of the theory of space groups. The unit of structure is a rhombohedron with a = 55° 17 and a = 5.420 = = 0.010 A. for hematite, and with a = 55° 17 and a = 5.120 = = 0.010 A. for corundum. The space group underlying the atomic arrangement is D. ... 

If the combination of X-ray and morphological evidence does not determine the space-group uniquely, additional information may be sought by tests for piezo-electric and pyro-electric properties, and by an optical examination for any evidence of rotation of the plane of polarization. (See Chapter VIII.) The results of such tests may settle the matter, since only certain crystal classes have these properties. Only positive results are decisive the apparent absence of piezo-electric or pyro-electric effects may be due to feeble phenomena. 

The application of the theory of space-groups to crystal structure determination involves the assumption that equivalent sites in the crystal are occupied by identical atoms. Fortunately tins is true for... 

The pore arrangement in MCM-41 could be determined by XRD due to its relatively simple structure. For other mesoporous phases with much more complicated structures, such as SBA-2, determination of a complete mesopore system by XRD becomes extremely difficult. SBA-2 was first reported in 1995 [19] and was believed to consist of discrete large cages obeying the symmetry of space group P63/mmc [20,21], However, the pore system connecting these supercages had not been determined until the TEM technique was applied [10],... 

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. 

Selected examples of the use of systematic absences in the deduction of space groups are illustrated in Table 9.4.3. Note that some conditions are redundant, as they are already covered by more general conditions. Some space groups can be determined uniquely, but others can only be narrowed down to a... 

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