Smoothing, desmearing, and Fourier transformation

Glatter, O.

doi:10.1107/97809553602060000581

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International Tables for Crystallography (2006). Vol. C. ch. 2.6, pp. 101-103

Section 2.6.1.6.3. Smoothing, desmearing, and Fourier transformation

O. Glatter^a

2.6.1.6.3. Smoothing, desmearing, and Fourier transformation

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There are many methods published that offer a solution for this problem. Most are referenced and some are reviewed in the textbooks (Glatter, 1982a; Feigin & Svergun, 1987). The indirect transformation method in its original version (Glatter, 1977a,b, 1980a,b) or in modifications for special applications (Moore, 1980; Feigin & Svergun, 1987) is a well established method used in the majority of laboratories for different applications. This procedure solves the problems of smoothing, desmearing, and Fourier transformation [inversion of equations (2.6.1.9) or (2.6.1.54), (2.6.1.55)] in one step. A short description of this technique is given in the following.

Indirect transformation methods . The indirect transformation method combines the following demands: single-step procedure, optimized general-function system, weighted least-squares approximation, minimization of termination effect, error propagation, and consideration of the physical smoothing condition given by the maximum intraparticle distance. This smoothing condition requires an estimate D_max as an upper limit for the largest particle dimension: $[D_{\rm max}\ge D.\eqno (2.6.1.57)]$ For the following, it is not necessary for $[D_{\rm max}]$ to be a perfect estimate, but it must not be smaller than D.

As [p(r)=0] for $[r\ge D_{\rm max}]$ , we can use a function system for the representation of p(r) that is defined only in the subspace $[0\leq r\leq D_{\rm max}]$ . A linear combination $[p_A(r)=\textstyle\sum\limits^N_{v=1} c_v\varphi_v(r)\eqno (2.6.1.58)]$ is used as an approximation to the PDDF. Let N be the number of functions and [c_v] be the unknowns. The functions $[\varphi_v(r)]$ are chosen as cubic B splines (Greville, 1969; Schelten & Hossfeld, 1971) as they represent smooth curves with a minimum second derivative.

Now we take advantage of two facts. The first is that we know precisely how to calculate a smeared scattering function $[\bar I (h)]$ from I(h) [equation (2.6.1.56)] and how p(r) or D(R) is transformed into I(h) [equations (2.6.1.9) or (2.6.1.54), (2.6.1.55)], but we do not know the inverse transformations. The second fact is that all these transformations are linear, i.e. they can be applied to all terms in a sum like that in equation (2.6.1.58) separately. So it is easy to start with our approximation in real space [equation (2.6.1.58)] taking into account the a priori information D_max. The approximation [I_A(h)] to the ideal (unsmeared) scattering function can be written as $[I_A(h)=\textstyle\sum\limits^N_{v=1}c_v\Psi_v(h),\eqno (2.6.1.59)]$ where the functions $[\Psi_v(h)]$ are calculated from $[\varphi_v(r)]$ by the transformations (2.6.1.9) or (2.6.1.54), (2.6.1.55), the coefficients [c_v] remain unknown. The final fit in the smeared, experimental space is given by a similar series $[{\bar I}_A(h)=\textstyle\sum\limits ^N_{v=1}c_v\chi_v(h),\eqno (2.6.1.60)]$ where the $[\chi_v(h)]$ are functions calculated from $[\psi_v(h)]$ by the transform (2.6.1.56). Equations (2.6.1.58), (2.6.1.59), and (2.6.1.60) are similar because of the linearity of the transforms. We see that the functions $[\chi_v(h)]$ are calculated from $[\varphi_v(r)]$ in the same way as the data $[\bar I_{\rm exp}(h)]$ were produced by the experiment from p(r). Now we can minimize the expression $[L=\textstyle \sum\limits ^M_{k=1} [\bar I_{\rm exp} (h_k)-\bar I_A(h_k)]^2/\sigma ^2(h_k),\eqno (2.6.1.61)]$ where M is the number of experimental points. Such least-squares problems are in most cases ill conditioned, i.e. additional stabilization routines are necessary to find the best solution. This problem is far from being trivial, but it can be solved with standard routines (Glatter, 1977a,b; Tikhonov & Arsenin, 1977).

The whole process of data evaluation is shown in Fig. 2.6.1.14 . Similar routines cannot be used in crystallography (periodic structures) because there exists no estimate for D_max [equation (2.6.1.57)].

Figure 2.6.1.14| top | pdf |

Function systems φ_v(r); Ψ_v(h); and χ_v(h) used for the approximation of the scattering data in the indirect transformation method.

Maximum particle dimension . The sampling theorem of Fourier transformation (Shannon & Weaver, 1949; Bracewell, 1986) gives a clear answer to the question of how the size of the particle D is related to the smallest scattering angle h₁. If the scattering curve is observed at increments $[\Delta h\leq h_1]$ starting from a scattering angle h₁, the scattering data contain, at least theoretically, the full information for all particles with maximum dimension D $[D \leq \pi /h_1.\eqno (2.6.1.62)]$ The first application of this theorem to the problem of data evaluation was given by Damaschun & Pürschel (1971a,b). In practice, one should always try to stay below this limit, i.e. $[h_1\, \lt \, \pi /D \quad\hbox{and}\quad \Delta h\ll h_1,\eqno (2.6.1.63)]$ taking into account the loss of information due to counting statistics and smearing effects. An optimum value for Δh = π/(6D) is claimed by Walter, Kranold & Becherer (1974).

Information content . The number of independent parameters contained in a small-angle scattering curve is given by $[N_{\rm max} \le h_2/h_1,\eqno (2.6.1.64)]$ with [h_1] and [h_2] being the lower and upper limits of h. In practice, this limit certainly depends on the statistical accuracy of the data. It should be noted that the number of functions N in equations (2.6.1.58) to (2.6.1.60) may be larger than $[N_{\rm max}]$ because they are not independent. They are correlated by the stabilization routine. An example of this problem can be found in Glatter (1980a).

Resolution . There is no clear answer to the question concerning the smallest structural details, i.e. details in the [p(r)] function that can be recognized from an experimental scattering function. The limiting factors are the maximum scattering angle [h_2] , the statistical error $[\sigma (h)]$ , and the weighting functions $[P(t),\ Q(x)]$ , and $[W(\lambda ')]$ (Glatter, 1982a). The resolution of standard experiments is not better than approximately 10% of the maximum dimension of the particle for a monodisperse system. In the case of polydisperse systems, resolution can be defined as the minimum relative peak distance that can be resolved in a bimodal distribution. We know from simulations that this value is of the order of 25%.

Special transforms . The PDDF [p(r)] or the size distribution function D(R) is related to I(h) by equations (2.6.1.9) or (2.6.1.54), (2.6.1.55). In the special case of particles elongated in one direction (like cylinders), we can combine equations (2.6.1.41) and (2.6.1.43) and obtain $[I(h)=2\pi^2 L \int\limits^\infty _0 p_c(r) {\textstyle J_0(hr) \over \textstyle h} {\,{\rm d}}r.\eqno (2.6.1.65)]$ This Hankel transform can be used in the indirect transformation method for the calculation of $[\psi_v(h)]$ in (2.6.1.59). Doing this, we immediately obtain the PDDF of the cross section [p_c(r)] from the smeared experimental data. It is not necessary to know the length L of the particle if the results are not needed on an absolute scale. For this application, we only need the information that the scatterers are elongated in one direction with a constant cross section. This information can be found from the overall PDDF of the particle or can be a priori information from other experiments, like electron microscopy. The estimate for the maximum dimension $[D_{\rm max}]$ (2.6.1.57) is related to the cross section in this application, i.e. the maximum dimension of the cross section must not be larger than $[D_{\rm max}]$ .

The situation is quite similar for flat particles. If we combine (2.6.1.47) and (2.6.1.49), we obtain $[I(h)=4\pi A\int\limits ^\infty _0 p_t(r) {\textstyle \cos (hr) \over \textstyle h^2} {\,{\rm d}} r,\eqno (2.6.1.66)]$ [p_t(r)] being the distance distribution function of the thickness . We have to check that the particles are flat with a constant thickness with maximum thickness $[T\le D_{\max}]$ . A is the area of the particles and would be needed only for experiments on an absolute scale.

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International Tables for Crystallography (2006). Vol. C. ch. 2.6, pp. 101-103