Indirect Fourier transform technique

Use an indirect Fourier transform technique which involves a series of Fourier transforms between the real and the reciprocal space to fit the scattering data. ° It is quite possible, from a mathematical perspective, to get negative size distributions which are difficult to interpret. 

All measured SAXS data were analyzed by the generalized indirect Fourier transformation (GIFT) technique with the Boltzmann simplex simulated annealing (BSSA) algorithm [34, 35]. The GIFT calculation is based on the analytical or numerical solution of the Ornstein-Zernike (OZ) equation that describes the interplay between the total (h(r)) and direct (c(r)) correlation functions ... 

A typical system composed of EAN/l-hexadecyl-3-methyllmidazolium chloride [Cj mim][Cl]/dodecane at ambient temperature with high thermal stability (stable between 30 and 150°C) was reported by Zech et al. [100]. Inherent properties were characterized by SANS, DLS, generalized indirect Fourier transformation, and Teubner and Strey model. A phase behavioral study with respect to temperature and IL mass fraction of a system comprising water/TX-100/[bmim][PFJ was reported by Anjum et al. [101]. SANS and polarized microscopic techniques revealed the existence of discontinnons-type microemulsion droplets in the system. 

Several analytical techniques have been used to characterize the polymer/ silane coupling agent interphase. Culler et aL [2] used Fourier transform infrared (FT-IR) spectroscopy to characterize the chemical reactions at the matrix/silane interphase of composite materials. They correlated the extent of reaction of the resin with the coupling agent (as determined by FT-IR) with the extent of interpenetration. Culler et al. [2] have also used observations of improved resistance of the interphase region to solvent attack as indirect evidence to support the interpenetrating network theory. 

A considerable number of different techniques has been employed in the past to characterize the porosity and surface chemistry of porous carbon materials. These include gas adsorption (mostly N2 and CO2) [9-14], immersion calorimetry [9], small-angle X-ray [11,15] and neutron [14] scattering, inverse gas chromatography [12,13], differential thermal analysis [12], Fourier transform infrared [12], Raman [16] and X-ray photoelectron [17] spectroscopies and electron spin resonance [16]. It is worth mentioning that the information about the porous structure of the material provided by this array of techniques is only indirect... 

In the early days there was a sensitivity problem when using CW techniques which has been overcome - at least in part - by indirect multiple-resonance experiments, but with the advent of pulse Fourier Transform spectrometers (ca. 1970) sensitivity is no longer an obstacle. 

Spin-coupled wavefunctions have proved to be very useful in studies of momentum-space properties " . Except for very simple systems, it is rather difficult to solve the Schrodinger equation directly in the momentum representation fortunately, the momentum-space wavefunction is also given by the Fourier transform of that in position space and this indirect approach proves to be much more tractable. The momentum-space formalism is particularly convenient for the interpretation of various scattering techniques such as Compton scattering and binary (e, 2e) spectroscopy. 

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