Equivalent reflections

An extreme example of the reduction in radiation damage is that of data collection at the SRS on purine nucleoside phosphorylase. On a conventional source usually a crystal can give only one 3 A resolution still photograph before the crystal suffers serious damage. At the synchrotron three crystals will give a comptete set of 4 equivalent reflections to a resolution limit of 3 A (see case study below)... 

The Laue symmetry of the diffraction pattern is now reduced to the symmetry of the non-centrosymmetric point group to which the crystal belongs (see Table 2 a listing of symmetry-equivalent reflections in the non-centrosymmetric point groups is available26). Under these circumstances, it is possible to determine different manifestations of non-centrosymmetry in a crystal, such as ... 

In Table III, the numbers of equivalent reflections in crystals of all possible symmetries are set down. These numbers are applicable to powder photographs, in which all planes having the same spacing give... 

Figure 9 Aerosol attenuation factor I) (A)= 1- F er (A)/ Fckjr (A) at 325 nm for nonabsorbing (triangles) and absorbing (diamonds) aerosols as a function of the TOMS Lambert-equivallent reflectivity (R-0.05)/0.9 (the true surface reflectivity Rj= 0.05). Solar zenith angle is 50°, satellite zenith angle 32° and azimuth 90°...

If the unit-cell contents are symmetric, then the reciprocal lattice is also symmetric and certain sets of reflections are equivalent. In theory, only one member of each set of equivalent reflections need be measured, so awareness of unit-cell symmetry can greatly reduce the magnitude of data collection. In practice, modest redundancy of measurements improves accuracy, so when more than one equivalent reflection is observed (measured), or when the same reflection is observed more than once, the average of these multiple observations is considered more accurate than any single observation. 

Reflectance data can be more deceiving with respect to the quantitative composition than transmittance data, as illustrated in Figure 6. Equivalent reflectance and transmittance spectra (Figure 6A) are converted into absorbance Ae and F(p) (Figure 6B and C). The F(p) representation... 

Discrepancy factor from merging equivalent reflections expect <5% for good and <10% for weak data if higher, absorption correction or crystal symmetry may be... 

The ( ) refers to the direction of the spin (up or down). In a powder experiment q must be averaged over equivalent reflections (55). 

The previously determined unit-cell and space-group data were confirmed from precession photographs, the 6/wtnwt diffraction symmetry being further confirmed by careful intensity measurements with scintillation counters of several sets of symmetry-equivalent reflections. 

A crystal was mounted on an Enraf-Nonius CAD-4 four-circle diffractometer, and accurate cell dimensions were obtained by a least-squares fit to three sets of eight symmetry-equivalent reflections with 26 between 25 and 29°. The cell dimensions and data collection parameters are summarized in Table I. 

Equivalent reflections For a complete set of intensity data there are eight measurements for each h,kj, corresponding to combinations of positive and negative values of each. Some of these, equivalent by the symmetry of the crystal, have (within experimental error) identical intensities. For high-symmetry crystals, other Bragg reflections may also be equivalent (e.g., hkl, klh, and Ihk for cubic crystals). 

Earlier, we considered the effect of lattice symmetry on the multiplicity factors in powder diffraction only in terms of the number of completely overlapping equivalent Bragg peaks (section 2.10.3). Because Friedel s law is violated in non-centrosymmetric structures in the presence of anomalously scattering chemical elements, the evaluation of multiplicity factors should also include Eqs. 2.114 and 2.115. Thus, the multiplicity factor,maybe comprehensively defined as the number of equivalent reflections ihkT), which satisfy Eq. 2.114 in addition to having identical lengths of the... 

Similar kinds of relationships would arise among equivalent reflections for threefold or sixfold axes in the crystal as well. This tells us that symmetry elements in real space, the crystal, may be identified by searching the appropriate zones, or planes of reciprocal space for symmetrical patterns of diffraction intensities. If we see fourfold or threefold or sixfold distributions of reflections in the diffraction patterns, then they must imply corresponding symmetry relationships in the crystal. 

Now, one might think that this should not be a serious problem, the scaling of reflections, but it is, and furthermore it often serves as one of the most serious sources of error in the creation of a data set when it is done improperly or carelessly. If we consider, for example, the hkl and —hk — l symmetry equivalent reflections from a monoclinic crystal, or the hkl and —h — k — l reflections from any crystal (again ignoring anomalous dispersion) we would expect any measured intensity hu to be exactly equal to its symmetry or Friedel mate. They seldom are, however, and we may consider the reasons why. 

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