Analysis in Reciprocal Space

Long-range order does not exist [12], This means that correlation between two widely separated points does not exist. In accordance with the above, at large values of r, the electron densities would be independent, and they could be replaced by the average value p. In this way, the stmcture is represented by the finite region where p (r) deviates from the mean value, and therefore values of r widely separated do not provide information. 

An experimental intensity curve as a function of the scattering vector is produced in SAXS. From the isotropic patterns of a melt-crystallized polymer, a slice is taken and, when projected in the plane I q) against q, it shows a scattering maximum with a wide statistical distribution. The evaluation of structural parameters from the intensity curve requires the whole scattering curve. Nonetheless, the lower and upper ends of the curve cannot be determined because of the nature of the scattering process. Therefore, mathematical approximations are used to access both ends of the curve. According to the characteristics of the dispersion function measured experimentally, the intensity curve is divided into three parts, which are described in the following sections. 

Debye et al. (13b) developed a model for random scatterers, which is given by the expression 

1 Bragg s Law In the analysis of semicrystalline polymers, in a first approach, it can be assumed that L is associated with the distance between the lamellar planes. These distances can be related through the Bragg s law with the diffracted waves that are in phase and are reinforced in certain directions (angles) [15]. This analysis give values that qualitatively coincide with the structure observed by means of electron microscopy and with sizes of crystals indicated by means of WAXS [16]. The periodicity L is related to the vector q in the maximum of dispersion according to [17] 

2 The Lorentz Factor Lq The factors that rule the diffraction intensity I hkl) are the stmcture factor for atomic dimensions F(hkl), the polarization factor P, the Lorentz factor Lq, and the multiplicity factor j. However, in studies of SAXS, all factors except the Lorentz factor can be omitted [9]. 

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