Scattering integral, approximation

This implies a higher cutoff angle in the scattering integral. Approximating Q according to the beam attenuation... 

The direct amplitudes involving are analogous to the distorted-wave Born approximation and are calculated by (10.31). The T-matrix element in the second amplitude of (10.51), which has the observed resonances, is calculated by solving the problem of electron scattering on He" ". The solution consists of half-on-shell T-matrix elements at the quadrature points for the scattering integral equations (6.87). The same points are used for the k integration of (10.51). 

One can see that the Born approximation reduces the forward modeling solution to simple quadrature calculation. However, this approximation holds only for a weaJi scatterer (when As (r) is relatively small). We can improve the accuracy of integral approximation significantly if, following the ideas described in Chapter 9 for an electromagnetic field, we introduce a quasi-linear approximation. 

When a distribution is anisotropic the program uses two techniques. The first technique is the discrete scatter angle approximation. POND produces 32 equiprobable angles. At a scatter, the integral part of (1 + 32z) is used to identify the appropriate angle. The angles p are obtained as solutions of Eq. (21) for... 

As in the case of the scattering integrals which appear in the Boltzmann equation, we assume that S,(w )< (r,u ) is a slowly varying function over the interval of integration and therefore may be approximated by the first two terms of a Taylor series,... 

Parker G A and Pack R T 1978 Rotationally and vibrationally inelastic scattering in the rotational lOS approximation. Ultra-simple calculation of total (differential, integral and transport) cross sections for nonspherical molecules J. Chem. Phys. 68 1585... 

It is generally accepted that the centrifugal sudden (CS) approximation is the most reliable approximate method. Its results are usually very close to those obtained by ab initio close coupling (CC) calculations. The integral and differential cross-sections of Ar inelastic scattering on nitrogen were performed for a few low-frequency rotational transitions and four different interaction potentials [205]. Much better agreement of CC with CS results was found than with IOS calculations performed in... 

Connor J. N. L., Sun H., Hutson J. M. Exact and approximate calculations for the effect of potential anisotropy on integral and differential cross-sections Ar-N2 rotationally inelastic scattering, J. 

Probable errors in assigning the integral distribution curve, as indicated by scatter of the points in Fig. 57, are magnified in the process of taking the slope for the deduction of the differential distribution. Only the approximate location of the maximum and breadth of the latter are experimentally significant. 

According to the Fraunhofer approximation of kinematic scattering theory the real space and the reciprocal space are related to each other by an integral transform known by the name Fourier transform, which shall be indicated by the operator The n-dimensional (nD) Fourier transform of h (r) is defined by... 

The cross-section of the primary X-ray beam is extended and not an ideal point. This fact results in a blurring of the recorded scattering pattern. By keeping the cross-section tiny, modern equipment is close to the point-focus collimation approximation - because, in general, the features of the scattering patterns are relatively broad. Care must be taken, if narrow peaks like equatorial streaks (cf. p. 166) are observed and discussed. The solution is either to desmear the scattering pattern or to correct the determined structure parameters for the integral breadth of the beam profile (Sect. 9.7). 

Direct calibration to absolute intensity is not a usual procedure at synchrotron beamlines. Nevertheless, the technical possibilities for realization are improving. Therefore the basic result for the total scattering intensity measured in normal transmission geometry is presented. At a synchrotron beamline point-focus can be realized in good approximation and the intensity /(s) is measured. Then integration of Eq. (7.19) results in... 

Scattering and Disorder. For structure close to random disorder the SAXS frequently exhibits a broad shoulder that is alternatively called liquid scattering ([206] [86], p. 50) or long-period peak . Let us consider disordered, concentrated systems. A poor theory like the one of Porod [18] is not consistent with respect to disorder, as it divides the volume into equal lots before starting to model the process. He concludes that statistical population (of the lots) does not lead to correlation. Better is the theory of Hosemann [158,211], His distorted structure does not pre-define any lots, and consequently it is able to describe (discrete) liquid scattering. The problems of liquid scattering have been studied since the early days of statistical physics. To-date several approximations and some analytical solutions are known. Most frequently applied [201,212-216] is the Percus-Yevick [217] approximation of the Ornstein-Zernike integral equation. The approximation offers a simple descrip-... 

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