Stokes deconvolution

Several peaks of interest (ideally higher order reflections of the same type hkl, 2h, 2k, 21, 3h, 3k, 31,. .., nh, nk, nl) are fitted by Fourier series the same procedure is applied to the diffraction lines of a reference sample, in which size and strain effects are negligible, in order to determine the instrumental line broadening. Such information is used in order to deconvolute instrumental broadening from sample effects (Stokes-Fourier deconvolution [36]). 

A model-free method for the analysis of lattice distortions is readily established from Eq. (8.13). It is an extension of Stokes [27] method for deconvolution and has been devised by Warren and Averbach [28,29] (textbooks Warren [97], Sect. 13.4 Guinier [6], p. 241-249 Alexander [7], Chap. 7). For the application to common soft matter it is of moderate value only, because the required accuracy of beam profile measurement is rarely achievable. On the other hand, for application to advanced polymeric materials its applicability has been demonstrated [109], although the classical graphical method suffers from extensive approximations that reduce its value for the typical polymer with small crystal sizes and stronger distortions. 

The two most common methods used to correct resolved peak profiles for the broadening imposed by the finite width of the X-ray beam in the diffractometer, are due to Jones (15) and Stokes (16). Both are essentially unfolding or deconvolution methods, but the Jones method defines specific functions for both the uncorrected and the instrumental broadening profile. If the uncorrected profile is Gaussian, then... 

Results. We have made a detailed study of the effect of using both the Jones and the Stokes corrections on crystallite size measurements obtained from the most crystalline samples of cellulose I and II, Ramie and Fortisan (18). Our conclusion was that the Jones corrections were always within 5% of the Stokes corrections for both half-width and integral breadth. Current practise is always to use a Stokes deconvolution procedure for the correction of all resolved peak profiles, evaluating an... 

The different approaches used to extract the pure profile by deconvolution were described and compared in particular by Louer and Weigel [LOU 69b], and more recently by Armstrong and Kalceft [ARM 99]. Cernansky [CER 99] gave a detailed description of the inherent mathematical aspects of this deconvolution process. Aside from Stokes historical method [STO 48], the various approaches will not be explained here, only the basic ideas will. 

Constraint free deconvolution method Stokes method... 

This approach was first applied to the study of X-ray diffraction peaks by Louer and his collaborators [LOU 69a, LOU 69b, LOU 72]. These authors showed that this method can prevent the oscillations observed after the Stokes deconvolution. 

Vibrational Raman band intensities and frequencies are also dependent on temperature, applied pressure, and the intrinsic microstructure of the material. These second-order parameters may be extracted from measured spectra. Both X-ray diffraction lines and Raman bands from polycrystalline materials show increased broadening as the microcrystallite grain sizes decrease. In fact, for the hexagonal phase of BN, bandwidths vary linearly with the reciprocal grain size (13). Inherent stress in thin films is manifested in vibrational line shifts. Based on pressure-dependent measurements of vibrational frequencies in bulk solids, inherent stress and stress inhomogeneity can be determined in thin films. Since localized stress can influence the optical and electronic properties of a thin film, it appears to be an important parameter in film characterization studies. Vibrational features also exhibit temperature-dependent frequency shifts. Therefore, an independent measurement of temperature is sometimes necessary to deconvolute these effects. Reference to Figure 1 shows that the molecular temperature of a material may be determined from the Stokes/anti-Stokes... 

To characterize the time-dependent red-shift, each spectrum was fit to a lognormal curve, and the peaks of the curves were plotted as a function of time. Fig.2 shows these data for BChl-a in 1-propanol, as well as a biexponential fit obtained by deconvolution with an excitation function. Similar data were obtained for BChl-a in methanol. Time constants were estimated to be 0.7 ps and 58 ps for BChl in 1-propanol, and 0.9 ps and 18 ps for BChl in methanol. The total magnitudes of these relaxations are about 2/3 of the Stokes shifts. 

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