Section 2.6.2.7.3. Real-space considerations

The scattering from a large number of randomly oriented particles at infinite dilution, and as a first approximation that of particles at sufficiently high dilution (see above), is completely determined by a function in real space, the distance-distribution function. It describes the probability p of finding a given distance r between any two volume elements within the particle, weighted with the product of the scattering-length densities of the two volume elements.

Theoretically, can be obtained by an infinite sine Fourier transform of the isolated-particle scattering curve In practice, the scattering curve can be measured neither to Q = 0 (but an extrapolation is possible to this limit), nor to . In fact, neutrons allow us to measure more easily the sample scattering in the range near Q = 0; X-rays are superior for large Q values. Indirect iterative methods have been developed that fit the finite Fourier transform of a p(r) function described by a limited number of parameters between r = 0 and a maximal chord length D_max within the particle to the experimental scattering curve. It differs from the p(r) of Section 2.6.1 by a factor of 4π.

This procedure was termed the `indirect Fourier transformation (IFT)' method by Glatter (1979), who uses equidistant B splines in real space that are correlated by a Lagrange parameter, thus reducing the number of independent parameters to be fitted. Errors in determining a residual flat background only affect the innermost spline at r = 0; the intensity at Q = 0 and the radius of gyration are not influenced by a (small) flat background.

Another IFT method was introduced by Moore (1980), who uses an orthogonal set of sine functions in real space. This procedure is more sensitive to the correct choice of D_max and to a residual background that might be present in the data.

A major advantage of IFT is the ease with which the deconvolution of the scattering intensities with respect to the wavelength distribution and to geometrical smearing due to the primary beam and sample sizes is calculated by smearing the theoretical scattering curve obtained from the real-space model. In fact, it is possible to convolute the scattering curves obtained from the single splines that are calculated only once at the beginning of the fit procedure. The convoluted constituent curves are then iteratively fitted to the experimental scattering curves.

With the exception of particle symmetry, which is better seen in the scattering curve, structural features are more easily recognized in the p(r) function (Glatter, 1982a).

Once the p(r) function is determined, the zero-angle intensity and the radius of gyration can be calculated from its integral and from its second moment, respectively.