Application of Energy-Resolved Measurements to Laue Diffraction: Determination of Unit-Cell Parameters, Deconvolution of Harmonics and Assignment of Systematic Absences

Abstract

The use of energy-resolved area detection of Laue diffraction patterns for the determination of unitcell parameters and systematic absences is demonstrated. Seven different crystals having previously known unit cells were re-examined using Laue diffraction methods. These crystals included four different crystal systems including cubic, orthorhombic, tetragonal and monoclinic cells. The crystals had cell sizes ranging from 179.4 to 4588.3 A~. Comparison of known and re-determined cells showed good agreement (ratio of known to measured cells = 0.987 _+ 0.18). A single procedure was suitable for all unit-cell determinations. The accuracy of the method is presently limited by the quality of the available energy measurements. Some of the crystals represent space groups containing systematic absences normally obscured by harmonic overlap when using the Laue method. These include absences due to 21 screw axes (h, k or 1 = 2n + 1) and cell centering (h + k = 2n + 1). All systematic absences were identified using a combination of multiple linear regression with either stepwise elimination or stepwise inclusion and an F test for assignment of systematic absence. The methods are discussed in detail and simulations are used to evaluate critical tolerances for future systems.

Feasibility of one-shot-per-crystal structure determination using Laue diffraction

“…We attempted to evaluate the variation in unit-cell parameters within and between drops. Determination of accurate absolute values of unit-cell parameters from Laue data requires special measures such as a filter which produces a sharp edge in the X-ray spectrum (Carr et al, 1993) or an energy-resolving detector (Hanley et al, 1997), owing to uncertainty in the exact X-ray wavelength producing each spot in 'simple' data (Ravelli et al, 1996). However, ratios between unit-cell parameters can routinely be refined.…”

Following FRET through five energy transfer steps: spectroscopic photobleaching, recovery of spectra, and a sequential mechanism of FRET

“…The mathematical model would then be: where ss 1 and ss 2 , are the sum of squared residuals for the reduced and complete models, respectively; k is the number of parameters for the complete model; g is the number of parameters for the reduced model; n is the number of data points; k-g corresponds to the number of degrees of freedom for the numerator, ν 1 ; and n-k corresponds to the number of degrees of ‡ The treatment given here is similar to one presented by Scheaffer and McClave 32 The main difference between the two treatments is that the degrees of freedom for the denominator is given here as n-k as opposed to n-(k+1). The latter assumes that there is a constant term in the model that is not included in the value of k. See also Hanley et al 34 for a description of related use of the F-test for model building. H a = The inclusion of the additional parameters is justified.…”

Advances in array detectors for X-ray diffraction techniques

“…One approach to unscrambling harmonics, determining units cells and assigning space groups is a system of calibrated absorbers allowing the energy of individual spots to be determined. One approach to this was a foil mask spectrometer system which was capable of determining small unit cells and assigning systematic absences to spots containing harmonic overlap (Hanley et al, 1997). Separation of harmonic content of Laue diffraction spots has been treated in some detail both theoretically and experimentally (Hanley et al, 1996(Hanley et al, , 1997 and will not be treated at length here.…”

“…One approach to this was a foil mask spectrometer system which was capable of determining small unit cells and assigning systematic absences to spots containing harmonic overlap (Hanley et al, 1997). Separation of harmonic content of Laue diffraction spots has been treated in some detail both theoretically and experimentally (Hanley et al, 1996(Hanley et al, , 1997 and will not be treated at length here. It is, however, likely that any improvements in foil mask spectrometers will improve the ability to separate harmonics.…”

“…The selection of 1% error and 10 000 photons has been made arbitrarily and is for comparison purposes. It should be noted that errors in unit-cell dimensions do not scale exactly with the percent error in single foil energy measurements(Hanley et al, 1997). This is due to spectrometers reporting energy based on several foils and unit-cell determinations being made based on multiple reflections.…”