The convergence of the van Cittert method of deconvolution

The small-angle X-ray scattering (SAXS) technique has been applied to evaluate macromolecular parameters of dewaxed cotton fibre at five different temperatures, 298, 310, 319, 327 and 334 K. For the experimental measurement of scattering intensities, a small-angle Kratky camera coupled with an automatic step-scanning device, pulse-height discriminator and a counter attachment were used. The samples were found to be non-ideal two-phase densely packed systems, deviating from Porod's law. The theories of Ruland and Vonk [Ruland (1971). J.Appl. Cryst. 4, 70-73;Vonk (1971). J. Appl. Cryst. 4, 340-342;(1973). J. Appl. Cryst. 6, 81-86] have been utilized to determine the width of the transition layer. The three-and one-dimensional correlation functions have been computed and the macromolecular parameters such as D, the transversal periodicity, S/V, the specific inner surface, lc, the length of coherence, 71, 72, the transversal lengths in matter and void respectively, lr, the range of inhomogeneity, E, the width of the transition layer and ~o~ and ~o2, the volume fractions of matter and void, have been evaluated. Details of the theories and methods of evaluating the above parameters have been thoroughly detailed in an earlier paper. The effect of temperature on the fine structural characteristics (i.e. macromolecular parameters) of cotton are reported in the present paper.

Section: Resultsmentioning

confidence: 99%

“…We applied the detailed theory and procedure for a non-ideal two-phase system to discover the fine structural characteristics as described Vonk (1973). the method of Ruland ( 1971 ).…”

Section: Resultsmentioning

confidence: 99%

The effect of temperature on the fine structural characteristics of cotton fibre – a small-angle X-ray scattering investigation using correlation functions

Misra¹,

Bisoyi²,

Khan³

et al. 1991

“…In an earlier papier (Ruland, 1971) the significance and the interpretation of systematic deviations from Porod's law have been discussed and a simple method for a quantitative evaluation of the corresponding structural parameters has been given for pin-hole collimation and infinite slit height. This method involves measurements in angular regions in which the assumption of an infinite slit-height may, in many cases of practical interest, no longer be valid (Soler & Baldrian, 1972).…”

Section: Introductionmentioning

confidence: 99%

“…It is shown that this is not a serious limitation as long as the lower limit of the validity of Porod's law is smaller than a certain dimension of the slit height and that the method of evaluation proposed in an earlier paper [Ruland (1971). J. Appl.…”

mentioning

confidence: 99%

The effect of finite slit heights on the determination of systematic deviations from Porod's law

Ruland¹

1974

Self Cite

Systematic deviations from Porod's law have often to be studied in angular regions in which the infiniteslit approximation is no longer valid. It is shown that this is not a serious limitation as long as the lower limit of the validity of Porod's law is smaller than a certain dimension of the slit height and that the method of evaluation proposed in an earlier paper [Ruland (1971). J. Appl. Cryst. 4,[70][71][72][73] can be modified to allow for the effect of finite slit heights. An example of application is given for the smallangle scattering of glassy carbons.

“…The systems we consider are spatially isotropic, so that y(r) is a function only of the magnitude r, and the scattering intensity is a function only of h= 4re(sin 0)/2; 0 = half the scattering angle. For a pointcollimated source, the intensity is (Ruland, 1971; l(h) = ~]2 Vie(h) ~ 4~zr2(sin hr)(hr)-17(r)dr,…”

Section: Introduction Basic Formulas and Experimental Detailsmentioning

confidence: 99%

Small-angle X-ray scattering analysis of catalysts: comparison and evaluation of models

Brumberger¹,

Delaglio²,

Goodisman³

et al. 1986

Small-angle X-ray scattering (SAXS) can be used to obtain interphase surface areas of a system, such as a supported-metal catalyst, composed of internally homogeneous phases with sharp interphase boundaries. Measurements of SAXS for samples of porous silica, alumina, platinum on silica, and platinum on alumina are reported. A variety of models and forms for the correlation function, the Fourier transform of which gives the X-ray scattering, are considered, and theoretical and measured intensities are compared. A criterion of fit for comparing models with different numbers of parameters is proposed. For the two-phase (unmetallized) systems the 'Debye-random' model must be rejected. Modifications of the Debye (exponential) correlation function are also not particularly good compared to an exponential-plus-Gaussian form, not derivable from a physical model, and forms based on Voronoi cell models. Since intensities can be fit to experimental error with a five-parameter correlation function, it seems incorrect to ascribe significance to the result of fitting a function with six or more parameters. It is shown that values for the single interphase surface area can be obtained independently of a model. However, fitting intensities using a modelbased correlation function gives information about the structure of the system. The two-cell-size Voronoi and the correlated Voronoi cell models are useful in this regard. For the systems containing metal, fiveparameter correlation functions again suffice to fit intensities. However, for three-phase systems a model or physical assumption is necessary to obtain values for the three surface areas from X-ray scattering intensities. The area of the surface between support and void is quite insensitive to the assumptions employed and the metal-support surface area somewhat less so, but values for the metal-void surface area $23 are consistent only to one significant figure from model to model. If the support in the three-phase catalyst is known to be unchanged from support in the absence of metal, a 'support-subtraction' model can be used to obtain reliable values for $23. In the present systems, the assumption does not seem to be borne out.