Errors diffraction

Also, the result of any diffraction-based trial-and-error fitting is not necessarily unique it is always possible that there exists another untried structure that would give a better fit to experiment. Hence, a multi-teclmique approach that provides independent clues to the structure is very fniithil and common in surface science such clues include chemical composition, vibrational analysis and position restrictions implied by other structural methods. This can greatly restrict the number of trial structures which must be investigated. 

In practical appHcations, diffraction instmments may exhibit certain problems. Eor example, there may be poor resolution for the larger droplets. Also, it is not possible to obtain an absolute measure of droplet number density or concentration. Furthermore, the Fraunhofer diffraction theory cannot be appHed when the droplet number density or optical path length is too large. Errors may also be introduced by vignetting, presence of nonspherical... 

The simplest analytical procedure is to oxidize a sample in air below the fusion point of the ash. The loss on ignition is reported as graphitic carbon. Refinements are deterrninations of the presence of amorphous carbon by gravity separation with ethylene bromide, or preferably by x-ray diffraction, and carbonates by loss of weight on treating with nitric acid. Corrections for amorphous carbon and carbonates are appHed to the ignition data, but loss of volatile materials and oxidation may introduce errors. 

The amplitudes and the phases of the diffraction data from the protein crystals are used to calculate an electron-density map of the repeating unit of the crystal. This map then has to be interpreted as a polypeptide chain with a particular amino acid sequence. The interpretation of the electron-density map is complicated by several limitations of the data. First of all, the map itself contains errors, mainly due to errors in the phase angles. In addition, the quality of the map depends on the resolution of the diffraction data, which in turn depends on how well-ordered the crystals are. This directly influences the image that can be produced. The resolution is measured in A... 

Figure 14.17 Structures of (a) the tetraperoxochromate(V) ion [Cr (02)4] , (b) the pyridine oxodiperoxo-chromium(VI) complex [Cr 0(02)2py], and (c) the triamminodiperoxochromium(IV) complex [Cr" (NH3)3(02)2] showing important interatomic distances and angles. (This last compound was originally described as a chromium(II) superoxo complex [Ci (NH3)3(02)2] on the basis of an apparent 0-0 distance of 131 pm/ and is a salutary example of the factual and interpretative errors that can arise even in X-ray diffraction studies. " ...

Because of the difficulty of obtaining satisfactory photometer records of electron diffraction photographs of gas molecules, we have adapted and extended the visual method to the calculation of radial distribution curves, by making use of the values of (4t sin d/2)/X obtained by the measurement of ring diameters (as in the usual visual method) in conjunction with visually estimated intensities of the rings, as described below. Various tests of the method indicate that the important interatomic distances can be determined in this way to within 1 or 2% (probable error). 

Car-C.i = 1.52 A. and C -C = 1.37 A., these distances being 0.02 A. smaller than those given by our photographs. The rings given by mesitylene and hexamethylbenzene are unusually sharp and well defined, and we estimate, by comparison with other substances for which the electron-diffraction method has been tested,81 that the error in our values of sa is not greater than about 0.5%. 

Values of interatomic distances and bond angles found in this investigation are collected in Table XIV. It should be mentioned that Wierl s early electron-diffraction work seems to be more reliable than he considered it to be the mean difference of his C-C values and ours for the six hydrocarbons common to the two investigations is 0.03 A., much less than the mean of the errors assigned by him, 0.07 A. 

Imaging is considered to be diffraction-limited for S > 0.8— the Marechal criterion. (This corresponds to a wavefront error of A/14.) The Strehl ratio decreases rapidly with increasing wavelength since ro oc A /. ... 

R,1S isomer. However, this proposal is tentative, because X-ray diffraction has shown that another specimen of asperlin, possessing a different crystalline form, has structure 49b. It should be noted that 1 tumbles somewhat anisotropically, with D /D = 1.3, as deduced from C relaxation measurements. If, however, the anisotropic motion of 1 were not properly corrected for, the largest error in the measurement of its interproton distances would not exceed 4%. 

Each atom in the lattice acts as a scattering centre, which means that the total intensity of the diffracted beam in a given direction depends on the extent to which contributions from individual atoms are in phase. Relating the underlying structure to the observed diffraction pattern is not straightforward, but is essentially a trial-and-error search involving extensive computer-based calculations. 

The main sources of error in charge density studies based on high-resolution X-ray diffraction data are of an experimental nature when special care is taken to minimise them, charge density studies can achieve an accuracy better than 1% in the values of the structure factor amplitudes of the simplest structures [1, 2]. The errors for small molecular crystals, although more difficult to assess, are reckoned to be of the same order of magnitude. 

Iversen, B.B., Jensen, J.L. and Danielsen, J. (1997) Errors in maximum-entropy charge-density distributions obtained from diffraction data, Acta Cryst., A53, 376-387. 

The MEM is a powerful new method which is especially useful in cases with limited data sets (powder diffraction). Monte Carlo simulations have shown that the MEM introduces systematic features into the reconstructed density and caution should be exercised when interpreting fine details of an MEM density. It must be emphasized that because the present MEM algorithms do not contain any models, they cannot filter out inconsistencies in the data stemming from systematic errors. The MEM densities may therefore contain non-physical features not only because of systematic bias in the calculation but also because of systematic errors in the data. 

The advent of CCD detectors for X-ray diffraction experiments has raised the possibility of obtaining charge density data sets in a much reduced time compared to that required with traditional point detectors. This opens the door to many more studies and, in particular, comparative studies. In addition, the length of data collection no longer scales with the size of the problem, thus the size of tractable studies has certainly increased but the limit remains unknown. Before embracing this new technology, it is necessary to evaluate the quality of the data obtained and the possible new sources of error. The details of the work summarized below has either been published or submitted for publication elsewhere [1-3]. 

Blessing, R.H. (1987) Data reduction and error analysis for accurate single crystal diffraction intensities, Cryst. Rev., 1, 3-58. 

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