research papers
Determining the radial pair distribution function from Xray absorption spectra by use of the Landweber iteration method
^{a}Forschungszentrum DresdenRossendorf, Institute of Radiochemistry, Germany, and ^{b}The Rossendorf Beamline at ESRF, France
^{*}Correspondence email: rossberg@esrf.fr
The Landweber iteration approach is used to construct the radial pair distribution function (RPDF) from an Xray absorption (EXAFS) spectrum. The physical motivation for the presented investigation is the possibility to also reconstruct asymmetric RPDFs from the i.e. by a recursive application of the kernel.
spectra. From the methodical point of view the shell fit analysis in the case of complicated spectra would be much more eased if the RPDF for the first shell(s) are computed precisely and independently. The RPDF, as a solution of the fundamental integral equation, is examined for theoretical examples, and a detailed noise analysis is performed. As a real example the spectrum of curium(III) hydrate is evaluated in a stable way without supplementary conditions by the proposed iteration,1. Introduction
The atomic pair distribution function describes the density of interatomic distances in matter. The radial pair distribution function (RPDF), which is independent of orientation, is of special practical importance for the analysis of
spectra. It is a major descriptor for the atomic structure of amorphous materials and liquids.A RPDF has by definition the following properties (Babanov et al., 1981):
The first condition is the positivity constraint, which is valid for each probability distribution. The second condition states that, for large distances where the particles are uncorrelated, g(r) tends to 1.^{1} Thirdly, for distances smaller than the sum of ionic radii of the two atoms (r_{min}), where the impenetrability of the particles becomes apparent, g(r) is equal to zero. Fourthly, g(r) is normalized to 1.
The conditions in (1) must be incorporated in the construction procedure of the function g(r), thus ensuring that the sought solution is a normalized RPDF.
The general i.e. for a system with only one type of backscattering atoms, has the form
integral equation for a onecomponent system,where the kernel
results from the matrix element of the electron scattering according to Fermi's Golden Rule (see Stern, 1974; Lee & Pendry, 1975; Teo, 1986). Here χ(k) is the normalized oscillating part of the measured spectrum, k is the electron wavevector, and r is the distance from the central to the backscattering atoms. The function's backscattering amplitude F(k,r), the central atom phase shift δ(k), the backscattering atom phase shift φ(k,r) and the of the photoelectron λ(k) are obtained from the curved wave approximation using the FEFF code (Ankudinov et al., 1998). The reduction factor is set to S_{0}^{2} = 1 in this article.
Equation (2) may be simplified in spherical coordinates due to the fact that the kernel depends only on the radius r. In this case the volume element is dV = 4πr^{2}dr. For data analysis it is more demonstrative and more adapted to introduce a radial particle distribution n(r) instead of g(r),
with
so that N is the number of backscattering atoms in the spherical shell between r_{1} and r_{2}, i.e. the analogue to the coordination number.
equation (2)The function n(r), depending on the radius, is unknown and contains all the information about the spectrum.
The `shell fitting' procedure is the most frequently applied method to evaluate σ^{2},
spectra in practice. In this procedure it is usually assumed that the is represented by separated Gaussian peaks where the half width of the Gauss curve is the Debye–Waller factorThus, the integral in the can be resolved analytically and, assuming some well justified approximations, will be reduced to the equation for a number of coordination shells,
equation (6)Here, N_{m} is the and is the Debye–Waller factor of the backscattering atoms in the mth shell.
While the shell fitting procedure approximates n(r) by symmetrical Gaussians, knowledge of the exact RPDF, or equivalent n(r), is of greatest importance, e.g. for the investigation of anharmonic corrections caused by thermal vibrations (Yang & Joo, 1998; Dalba & Fornasini, 1997; Crozier et al., 1988). This is the main physical reason to determine the complete RPDF independently.
Equation (6) may be rewritten in a condensed form as
This is a Fredholm integral equation of the first type connecting the known spectrum χ(k) with the unknown function n(r). The inversion of equation (9), i.e. the determination of n(r), is a socalled illposed problem. From the three conditions for a well posed problem, suggested by Hadamard (1932), i.e. (i) the existence, (ii) the uniqueness and (iii) the stability of the solution depending continuously on the data, condition (iii) is most often violated owing to the uncertainties in the data, i.e. experimental error. The two most important well established methods for the solution of this type of integral equations, like equation (9), are Tikhonov's regularization (Tikhonov & Arsenin, 1981) and Landweber's iteration (Landweber, 1951) procedures. The principal difference between the methods is the following. The integral equation, like equation (9) or (10), is solved using the Tikhonov regularization by multiplication with the inverse matrix (i.e. the inverse integral operator), whose irregularities are avoided by addition of a regularization matrix. The Landweber iteration (see details below) does not use the inverse matrix and therefore does not contain irregularities. Both methods are independent and different.
In the mathematical literature some comparisons of the two methods exist (Kirsch, 1996; Latham, 1998). They stated that for nonnoisy model spectra both methods lead to the same result. A general comparison of both methods for the case of realistic noisy data is not possible because of the different character of data errors for each individual case.
It has been confirmed that for the inversion of equation (9), and for data with experimental error, more stable results are supplied by the Landweber iteration (Kirsch, 1996), which on the other hand needs more iteration steps. The latter statement agrees with our initial experiences of the analysis of spectra using the Landweber iteration. However, no direct comparison of both methods, applied to real noisy spectra, exists. In §3 a qualitative noise analysis for the Landweber iteration using model spectra is presented.
Babanov and coworkers (Babanov et al., 1981, 2007; Ershov et al., 1981; Kunicke et al., 2005; see also Yang & Bunker, 1996; Khelashvili & Bunker, 1999; Bunker et al., 2005; Ageev et al., 2007) have applied the Tikhonov regularization to the analysis of spectra. Using special assumptions, adequate RPDF of various systems were reconstructed. In this context no systematic noise analysis exists.
Lee & Yang (2006) presented a new analytical method, the Matched RPDF Projection (MEPP), to obtain the RPDF of an spectrum via Fourier filtering.
An interesting article by Yamaguchi et al. (1999) deals with the application of the Landweber iteration to the inversion of the equation. Their work focuses on the application of the wavelet Galerkin method for the reconstruction of the integral kernel A(k,r) in equation (9). The Landweber iteration method contains two parameters (see below): a convergence parameter α and a stopping number ν_{opt}, which indicates the optimal number of iterations. Both parameters were set by Yamaguchi et al. (1999) by `trial and error' to 5000.
In the present work we present a methodical introduction and some examples for the analysis of α, which depends on the norm A^{*}A^{2}, and the stopping parameter ν_{opt}, which depends on the error in the data. The calculation of the matrix A is performed by the use of the FEFF8.2 program (Ankudinov et al., 1998). We also give a physical interpretation of the iteration steps of the Landweber method for the inversion of the equation. In favor of simplicity and clarity, the article does not contain analysis of multicomponent spectra and of multiple scattering effects.
spectra using the Landweber iteration method, including the convergence parameter2. Landweber's iteration for a onecomponent backscattering system without multiple scattering effects
2.1. Landweber's iteration and constraints
2.1.1. Definition
equation (6)when multiplied by αA^{*}, is identically rewritten as
Based on this equation, with initial condition n_{0} = 0 and a parameter α > 0, Landweber (1951) suggested the iteration scheme
where ν is the iteration number. For mathematical details, see for example the textbooks by Kirsch (1996) and Louis (1989). This iteration procedure can be interpreted as the steepest descent algorithm to minimize the functional An − χ with α as convergence parameter. We remark that the acceleration term (χ − An_{ν}) is the difference spectrum between the original spectrum χ and the last iteration spectrum An_{ν}.
2.1.2. Convergence parameter
It has been proven, e.g. by Louis (1989), that for α in the range 0 < α < 2/A^{*}A, the Landweber algorithm is equivalent to a linear regularization procedure. This means that the iteration converges to an optimal solution n_{νopt} at an optimal number of iterations ν_{opt}. In the following examples, the convergence parameter α^{EXAFS} = 1/A^{*}A will be chosen.
2.1.3. Stopping parameter
While α defines only the `speed' of the iteration, the optimal stopping parameter ν_{opt} must be defined for each problem separately. On one hand, iterating too long causes the experimental error in the data to increasingly disturb the solution. On the other hand, too few iterations will lead to a loss of resolution. This convergence behavior is called semiconvergence (see also Fig. 1a).
Apparently, the stopping parameter ν_{opt} depends on the error in the data (δ), e.g. caused by noise, spline errors etc. All other sources of uncertainty, e.g. the choice of α and the model matrix A, are described in the literature (Elfving & Nikazad, 2007; Kirsch, 1996) by an additional parameter τ. There, it is proved that the iterative method is convergent for 0 < τ < 2. Our initial model calculations (see below) show best results for τ ≃ 1.
If the total data error τδ were known, the stopping rule defining ν_{opt} would simply be
This means that the solution is such that the norm of the residual An − χ is of the same order as the total data error τδ (Fig. 1b). We have used this condition several times with simulated noise, i.e. with known δ and τ ≃ 1, to validate the following procedure developed for realistic noisy data.
2.1.4. Lcurve criterion
In the case of real τδ is not known. To determine the stopping parameter in the Landweber iteration we adapted the concept of the Lcurve criterion, derived for the determination of an optimized regularization parameter for the Tikhonov regularization (Hansen, 1992, 2001; Hansen & Oleary, 1993; Kunicke et al., 2005). In general, the Lcurve is a log–log plot of the norm of a regularized solution versus the norm of the corresponding residual. Both norms depend parametrically on ν_{opt}. The resulting curve shows a shape like the capital letter L (Fig. 2). The optimal regularization parameter is chosen as the maximum of the Lcurve curvature, corresponding to the corner of the L. As ν increases up to the value ν_{opt}, the norm of the residual decreases much faster than the norm of the solution increases. A region of ν values follows where the norm of the solution starts to increase dramatically. This means that the value ν_{opt}, corresponding to the corner of the Lcurve, gives an optimum in the effort to minimize simultaneously the norm of the residual and the norm of the solution.
measurements, the total errorIn the ideal case the Lcurve is defined by a smooth computable formula. Then the maximum of the curvature of the graph logn_{ν} over logAn_{ν} − χ gives us the value of ν_{opt}.
The curvature c_{ν} of the Lcurve is obtained using equation (14) where x = logAn_{ν} − χ and y = logn_{ν}. The prime denotes the derivative with respect to ν if both functions depend parametrically and continuously on the parameter ν,
Unfortunately, in the case of realistic data the Lcurve does not have the clear form shown in Fig. 2, and is limited by the knowledge of only a finite set of points. As a first approximation for the numerical differentiations in equation (14) we used the integral values of the parameter ν as nodes. The corresponding stopping parameter is further denoted by ν_{curv}. The use of ν_{curv} resulted in satisfying curvature maxima for all of our examples.
A complete discussion of the numerical problems in locating the corner of the Lcurve, e.g. by use of cubic spline curves, is given by Hansen & Oleary (1993).
2.1.5. Positivity constraint
Generally, different constraints are used for the construction of solutions of illposed problems in order to improve their stability and convergence. A common method is the utilization of positivity constraints (Kirsch, 1996). The positivity constraint [see equation (1)] is included in the Landweber iteration steps because it is an inherent part of the definition of the RPDF. It should be noted that the consideration of the positivity constraint leads to a slight degradation of the smoothness of the Lcurve. Instead of the terminus `positivity constraint' the terminus `projected inversion' is equivalently used in the relevant literature (Bunker, 2009).
2.2. Physical interpretation of the transformation of the spectrum by the transposed matrix A^{*}
In the first step of the Landweber iteration (n_{0} = 0) and with α = 1, equation (12) has the form n_{1} = A^{*}χ, where χ is the spectrum. In each further iteration step the operator A^{*} acts on a difference of two spectra (χ − An_{ν}). We consider the effect of the transposed matrix A^{*} acting on the spectrum χ(k),^{3}
and rewrite (15) according to (6) in its continuous form,
from which the following interpretation of transformation (16) is obvious. The oscillating part of the integration kernel, sin(2kr), is the imaginary part of the Fourier transform, which results usually in a very precise determination of the distances of the backscattering atoms. The argument of the sine function is phase corrected by the phases of the central and backscattering atoms, and an amplitude window function is formed by the atomic amplitudes and the mean free paths. Therefore the transform (16) may be interpreted as a windowed and phasecorrected Fourier transform, whereas the window and phase functions originate from the underlying model.
2.3. Construction of the matrix A
For the discretization of equation (6) by an equalspaced step function, n(r_{m}) is approximated by a linear function in each Δr interval.
Consequently, the righthand side of (8) represents the columns of the matrix A_{im}, which correspond to the kdependence of the kernel for each distance r_{m}. If the in each interval is set to N_{m} = 1, we obtain the columns of A_{im},
Therefore, for the distances r_{m}, the columns of A_{im} are calculated directly from the FEFF program (the chi.dat files without Debye–Waller factors) in the form of the theoretical spectra. We used step sizes of Δr = 0.02 Å or Δr = 0.002 Å in our calculations.
Most algorithms use for the description of an RPDF peak the theoretical scattering functions calculated for one radial distance supplied by the FEFF structural model. It is well known that the theoretical scattering functions depend on the radial distance. Therefore the theoretical scattering functions must be calculated for a narrow grid of radial distances in the rinterval of the RPDF. If the theoretical scattering functions are calculated only for one radial distance then, especially for strong disordered systems with a broad RPDF, the calculated RPDF might be incorrect.
2.4. Determination of the shift ΔE_{0}
The kaxis, which is the basis for the computation of the matrix A, is defined by
where is Plank's constant, m is the electron mass and E is the photon energy. E_{0} is the i.e. the minimum energy to free the electron. To define the kaxis of the real experiment, usually a correction term ΔE_{0} is introduced. ΔE_{0} is a priori unknown and usually it is determined by a shell fit of the spectrum before an inversion method is applied. If the shell fit model is incorrect then the determined ΔE_{0} is erroneous and an inverse method would give a wrong RPDF. This leads in turn to the conclusion that the recent inversion procedures are only applicable if the structural model for the investigated system is already known.
In our approach, ΔE_{0} is defined by the iteration procedure itself, without recourse to fit calculations or models. The only assumption is that for each ν an optimal ΔE_{0} exists, and that over a broad range around ν_{opt} the best values of ΔE_{0} are similar. By using this assumption for a given ν, the optimal ΔE_{0} is determined simply by a oneparameter search, using the minimization of the standard deviation SD_{ν}, defined as
SD_{ν} is the norm of the residual of the equation (10), and depends on the iteration number ν, normalized to the maximum number of data points in rspace (i_{max}).
We start with arbitrarily chosen values ν_{start} and ΔE_{0}. With this start shift (e.g. ΔE_{0} = 0) the columns in the matrix A_{im} are recalculated on the new kaxis, based on equation (18). Then the Landweber iteration is performed up to ν_{start} and the new SD_{ν} is determined. In the next cycle, ΔE_{0} is modified so that SD_{ν} decreases. If ΔE_{0} converges to a constant value, the approximation procedure based on ν_{start} is stopped. This first optimal ΔE_{0} is already near the true value, and we calculate with this ΔE_{0} the new Lcurve and the new ν_{curv}. Then the algorithm to find ΔE_{0} is repeated by using ν_{curv} as the new ν_{start}. If the actual ν_{start} is equal to the next ν_{curv}, then ΔE_{0} and SD_{ν} have reached their final values.
This procedure will be illustrated below, where ΔE_{0} is determined for the noise contaminated spectrum of example #6.
3. Application of Landweber's iteration to model spectra
In this section, we examine theoretically how the Landweber iteration method responds to noisy `closetoreality' spectra. Noise is added in ten steps with a noise generator to a noisefree theoretical spectrum (Table 1, example #1). Subsequently, the ten noisecontaminated spectra are analyzed. The change of the structural parameter (N, r, σ^{2}) and the stopping parameter are inspected as a function of the amplitude of the added noise.

Additionally, the difference between the theoretical stopping parameter ν_{theor} (equivalent to ν_{opt} in Fig. 1a) and the stopping parameter ν_{curv} (equivalent to ν_{opt} in Fig. 2), determined from the curvature of the Lcurve, will be discussed.
By using equation (7) we calculated for one oxygen shell a Gaussianshaped model RPDF (n_{theor}) with N = 1, r = 2.50 Å and σ^{2} = 0.005 Å^{2}. The terms in equation (8) [F(k,r), λ(k), 2δ(k) and φ(k,r)] are calculated by FEFF8.20 (see §2.3), where the amplitude reduction factor is set to S_{0} = 1. The FEFF structural model consists of a curium hydrate cluster based on the results of Skanthakumar et al. (2007). The matrix A_{im} is then calculated using (17) with an energy shift of ΔE_{0} = 5 eV [see (18)] and the model spectrum χ is calculated using (10). For the calculation of the kaxis we used (18). We added to the unweighted spectrum χ a noise equal to C_{noise}·rnd, where the random number generator (rnd) gives values in the range −0.5 < rnd < 0.5 (Table 1). Fig. 3 shows for examples #2, #6 and #10 (Table 1) the theoretical RPDF and the noisecontaminated spectra.
For examples #2–#10 the Lcurves are shown in Fig. 4 with an inset showing example #10 on a more convenient scale. For the presented theoretical examples, n_{theor} is known, hence for each example the theoretical optimum number of iterations ν_{theor} can be determined as the minimum in n_{ν} − n_{theor} normalized to the maximum number of data points in rspace (m_{max}). Figs. 5(a) and 5(b) illustrate the determination of ν_{theor} and ν_{curv} for the examples #2–#10.
The inspection of the shape of the Lcurves (Fig. 4) leads to the conclusion that for examples with a high noise level the norm of the residual An_{ν} − χ decreases and the norm of the solution n_{ν} increases faster than for the examples with a lower noise level; hence the higher the experimental error in the data the lower is the number of iterations needed for the solution. This trend is in accordance with the shift of the minimum in n_{ν} − n_{theor} and c(ν) to smaller ν_{theor} and ν_{curv}, in case the noise level increases [Figs. 5(a) and 5(b)]. These observations are in line with the semiconvergent behavior of the Landweber iteration. The observed good accordance between ν_{curv} and ν_{theor} (Table 1) shows the reliability of the Lcurve concept to estimate the number of required iterations for noisecontaminated spectra.
For all Lcurves, the position of the maximum curvatures determined by equation (14) shows a very similar norm n_{ν} (open circles connected by line in Fig. 4). This is a first indication that for all calculated RPDFs the will be very similar as discussed later.
For all examples, the determined n_{υcurv} and the theoretical n_{theor} are in good accordance, as can be seen for examples #2, #6 and #10 in Fig. 3. Also, the spectra calculated using n_{υcurv} are in good agreement with the noisecontaminated spectra within the noise level (Fig. 3).
The sought structural data (N, r, σ^{2}) in Table 1 are derived from a comparison of the RPDFs with a Gaussianshaped RPDF [see equation (7)]. For all noise levels the structural parameters (N, r) are in good agreement with those of the initial model (Table 1, #1). The deviation of σ^{2} from its true value increases systematically and significantly with the noise level. For example #2, no significant deviation is observed, whereas, for example #10, σ^{2} is 14% below the true value (Table 1). It is interesting to note that the precision in the determination of the structural parameters follows the order r > N > σ^{2}. This trend is also visible in the deviations between n_{υcurv} and n_{theor} (Fig. 3). The position of the maxima in n_{υcurv} does not change with the noise level, while the left and righthand sides of n_{υcurv} show deviations from n_{theor} with increasing noise (Fig. 3). It is important to realise that these deviations influence the shape of the RPDF and that therefore σ^{2} is the parameter which is most affected by noise, i.e. experimental error, of the spectrum.
Note that for the preceding comparison of models the energy shift for the noisefree spectrum was fixed at ΔE_{0} = 5 eV. In the following we demonstrate, using example #6, the procedure for calculating ΔE_{0} as described in §2.4. For this spectrum, ν_{theor} = 57 and ν_{curv} = 59 (Table 1, #6).
The starting values are ν_{start} = 10 and ΔE_{0} = 0 eV. At ν_{start} = 10 the resulting ΔE_{0} is 5.08 eV (Table 2), hence already near the value of a noisefree initial spectrum (Table 1, #6). The curvature shows for this first value of ΔE_{0} a maximum at ν_{curv} = 64 (Table 2). By taking this value as the next ν_{start}, a new improved ΔE_{0} is determined. After four such steps, ν_{start} is equal to ν_{curv} and ΔE_{0} has reached a final value of 5.14 eV. When starting at ν_{start} = 100, the method leads to the same result (Table 2). During the approximation cycles the value of SD_{ν} decreases and the difference between the theoretical (ΔE_{0} = 5 eV) and the calculated ΔE_{0} increases slightly. This demonstrates that a part of the noise in the spectrum influences ΔE_{0} and consequently influences the RPDF, which reflects the normal behavior of error propagation.

Table 3 shows the structural parameters for example #6, determined either by the Landweber iteration with calculated ΔE_{0} (`iteration') or by shell fit. Both results are in good agreement. Furthermore, they are in good agreement with the results obtained by keeping ΔE_{0} constant during the Landweber iteration (Table 1, #6).

4. Application of Landweber's iteration to curium(III) hydrate
The coordination environment of the hydrated Cm^{3+} ion was recently investigated by Cm L_{3} and highenergy Xray scattering (Skanthakumar et al., 2007). It was found that the Cm^{3+} ion is surrounded by nine coordinating water molecules at two different Cm—O distances. The spectrum was fitted by six oxygen atoms at 2.47 Å and three oxygen atoms at 2.63 Å, while maintaining the coordination numbers fixed to stabilize the shell fit (Skanthakumar et al., 2007).
In order to test our full Landweber approach (including the procedure for determining ΔE_{0}) for deriving the RPDF of a real world spectrum, the authors of Skanthakumar et al. (2007) kindly provided their experimental Cm^{3+} hydrate spectrum [Fig. 6(a), #1].
For the calculation of the matrix A an rinterval of 1.3–4.0 Å with a step size of Δr = 0.002 Å is used. To determine ΔE_{0}, the method described in §2.4 is applied and yields ΔE_{0} = 6.42 eV at ν_{curv} ≃ 900 (Table 4). The determined RPDF shows an asymmetry at higher rdistances [Fig. 6(c), #1]. The small features at 3.1–3.8 Å in the RPDF [Fig. 6(c), #1] may arise from reproduction of small multiplescattering contributions by the Cm—O functions or other sources. The determined asymmetric RPDF is only properly reproduced when two Gaussians are considered in equation (7) [Fig. 6(c), #1]. The resulting structural parameters for the two Cm—O shells are within the standard deviations in full agreement with those gained by a shell fit of the Cm(III) hydrate spectrum (Table 4). The coordination numbers of the two Cm—O shells were assumed to be 6 and 3 and were fixed to stabilize the shell fit. Note that in the case of the Landweber iteration the RPDF is calculated without such constraints and assumptions contrary to the shell fit.

In the following we tested the ability of the Landweber iteration in resolving these two overlapping Cm—O shells by constructing a theoretical model. Both the model RPDF and χ were constructed using the structural parameters gained from the shell fit of the Cm(III) hydrate spectrum (Table 4) and the procedure discussed in §3. To the unweighted spectrum χ, we added noise C_{noise} (Table 4) similar to the noise amplitude of the experimental spectrum of Cm(III) hydrate [compare #1 and #2 in Fig. 6(a)]. The optimum number of iterations, ν_{curv} = 1922, is near the theoretical optimum number of iterations of ν_{theor} = 1709 (Table 4). For Cm(III) hydrate, ν_{curv} is much higher than for the theoretical examples discussed in §3, which were analyzed in the smaller rinterval of 2–3 Å. The higher ν_{curv} can be explained by the decrease of the convergence parameter α owing to the increasing norm A^{*}A (see §2.1) in the case of the larger investigated rinterval of 1.3–4.0 Å.
The resulting RPDF [Fig. 6(c), #2] reproduces the noisecontaminated spectrum [Fig. 6(a), #2] within the limits of the artificially added noise. Owing to the spatially distinct two oxygen shells, the RPDF shows an asymmetry at higher rdistances which is not visible in the corresponding Fourier transform [compare Figs. 6(b) and 6(c), #2], similar to the experimental data [compare Figs. 6(b) and 6(c), #1]. For the calculation of the structural parameters, the RPDF is reconstructed with two Gaussians according to equation (7). The resulting structural parameters (Table 4, theoretical example) are in good accordance with those taken for the construction of the theoretical RPDF (Table 4, shell fit), hence there is strong confidence in the correct calculation of the RPDF by the Landweber iteration in the case of the experimental Cm(III) hydrate spectrum. The structural parameter for Cm(III) hydrate, gained by the Landweber iteration and the shell fit (Table 4), agrees favorably with the results published by Skanthakumar et al. (2007) within the bounds of the errors in the determination of structural parameters.
5. Conclusions
The analysis of model spectra with artificial noise, as well as of an experimental
spectrum of Cm(III) hydrate, demonstrates that the Landweber iteration approach is well suited to solving the integral equation, even if the spectra contain experimental error. The robustness in relation to the error in the data represents the main advantage of this method.All variables used in the iteration approach [equation (12)] have a well defined physical meaning. The determined RPDF n(r) is the density of coordination numbers, the key transformation with the matrix A^{*} is a windowed and phasecorrected Fourier transform of the spectrum, and the acceleration term (χ − An_{ν}) is the difference spectrum between the original spectrum χ and the last iteration spectrum An_{ν}. Convergence and stopping parameters are uniquely defined by the matrix A and the Lcurve concept.
In the presented approach, the ΔE_{0} is defined by the iteration procedure itself, without recourse to fitcalculations or models.
shiftThe forthcoming investigations focus on the expansion of the stable Landweber iteration with the aim of considering more than one type of backscattering atoms and multiplescattering effects for the inversion of the
equation.Footnotes
^{1}This boundary condition is irrelevant for RPDFs, which are derived from measurements. The distance of neighboring atoms measured by spectra is smaller than the asymptotic value of r, for which g(r) → 1 is valid.
^{2}A^{*} denotes the conjugate transpose. Here and in the following the norm of a matrix is the Frobenius norm.
^{3}For the discretization of the equation (6) we use the following. The indices i,j,… run from 1 to the maximum number i_{max}, j_{max} etc. of points in kspace. The indices m,n,… run from 1 to the maximum number m_{max}, n_{max} etc. of points in rspace.
Acknowledgements
We would like to thank the authors S. Skanthakumar, M. R. Antonio, R. E. Wilson and L. Soderholm for supplying the Xray
of Cm(III) hydrate. We also thank F. Bridges (Department of Physics, University of California Santa Cruz, USA) and M. Kunicke (Forschungszentrum DresdenRossendorf, Germany) for fruitful discussions and suggestions. AR was supported through the German Science Foundation (DFG) under Contract RO 2254/31.References
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