Chapter 6.19. Real-Space X-ray Absorption Package (RSXAP)

The Real-Space X-ray Absorption Package (RSXAP; Booth, 2016) is a collection of data-reduction and fitting routines for analyzing both EXAFS and XANES data that grew out of the analysis codes of Hayes & Boyce (1982). The programs currently only run on Unix- and Linux-based systems. The main routines for EXAFS are reduce and rsfit. Most programs can be run on batches of files for rapid processing. The guiding principle in these codes is to avoid certain approximations that are typically used in EXAFS analysis. For instance, a specific k-dependent background subtraction is performed that properly normalizes the data and does not require corrections to the Debye–Waller factors. Another example is that the usual k-space approximation of the effect of the pair-distribution variance, exp(−2k²σ²), is not used in favour of directly applying a pair-distribution function g(r), typically a Gaussian, in r-space. These and other important aspects of the methodology are discussed below.

The descriptions below focus on the reduce and rsfit programs, but many other codes are included in RSXAP for performing tasks such as converting data to RSXAP format, performing dead-time corrections (see, for example, Kappen et al., 2024), performing energy calibrations etc. Table 1 lists some of the most-used auxiliary programs. Some highlights include codes that also allow self-absorption corrections (Booth & Bridges, 2005), either within reduce or from the standalone code sabcor (Bridges & Booth, 2024), an F-test (Downward et al., 2007) with the standalone code hamilton or partially automated for individual scattering shells from within rsfit (Booth, 2024a), and an iterative technique for the determination of μ₀(E) (Bridges et al., 1995), among many other features.

Table 1 Some of the most-used RSXAP auxiliary programs Program Purpose convgerm Converts as-collected data to the RSXAP binary format glitch Used for removing glitches in batches of files dead Fits dead-time curves to extract the dead time for both windowed and integrated count-rate (ICR) data from multi-channel fluorescence detectors sug For averaging multi-channel detector data, performing dead-time corrections (including to ICR data) and removing data from individual channels over restricted energy ranges (such as from a Bragg diffraction) sabcor Given information about the sample such as chemical makeup, perform a self-absorption correction to k-space data (Booth & Bridges, 2005; Bridges & Booth, 2024) fites Allows the fitting of a XANES spectrum to a shifted sum of reference XANES spectra fitcD, fitD, fitE For fitting σ2 versus T data to a correlated Debye, Debye or Einstein model fitedge For fitting edge data to sums of pseudo-Voigts and integrated pseudo-Voigt (arctan-like) functions gete0 Determines the energy of the first inflection point in a batch of files shiftes Using the output of gete0, shifts a batch of files to an agreed reference energy hamilton Evaluates the level of confidence that one fit is statistically significantly better that the other based on (%) and the degrees of freedom of the two fits (Downward, 2007; see also Booth, 2024a)

The program reduce allows data reduction from e-space to k-space and r-space by pre- and post-edge subtractions. There are several methods for determining various options, and only some are discussed here. It is important to note that the pre-edge subtraction method used by reduce to remove all unwanted background absorption processes from transmission data gives a proper energy-dependent normalization such that no so-called `McMaster corrections' (see Rehr et al., 1991) are required to obtain accurate Debye–Waller factors (Li et al., 1995). The importance of this normalization and the accuracy of the Debye–Waller factors obtained by this method are not generally appreciated by all EXAFS practitioners, and so we describe it more fully here. Pre-edge subtraction is accomplished for transmission data by constructing a function μ_pre(E) such that the absorption due to the atomic edge of interest, μ_e(E) = μ(E)t − μ_pre(E)t, above the threshold energy E₀ generally follows the energy dependence expected from a Victoreen formula (Victoreen, 1949). This is accomplished by automatically determining the step height at E₀, Δμ_e(E₀), by extrapolating linear fits in the pre- and post-edge regions to the energy at the half-height of the edge step, defining E₀. The Victoreen values μ_vic(E) for the absorption of the given edge are determined above the edge from standard tables (Teo, 1986). A function f_pre(E) is then defined differently above and below the edge. For E < E₁,and for E > E₂,where E₁ and E₂ define a range around the edge that is not used in the fit. The final μ_pre(E)t function is then typically determined by a fourth-order polynomial fit through f_pre(E). This procedure is portrayed in Fig. 1, where μ_e is seen to be about 75% of Δμ_e(E₀) at 1 keV above the edge.

Figure 1 Example of Cu K-edge transmission data and pre-edge absorption based on Victoreen absorption coefficients. The raw data, fpre (equation 1) and μpre, are shown in (a) and the normalized isolated absorption from the Cu K edge, μe, is shown in (b). The size of the gap in the function fpre(E) in the vicinity of E0 is determined by the user, but is typically chosen between the initial upturn at the edge and about 150–200 eV above E0. The dashed lines in (b) are meant to help to visually demonstrate the proper fall-off of the data with E, where μe is about 75% of Δμ(E0) at 1 keV above the edge for the Cu K edge.

Pre-edge subtractions for fluorescence data do not use tabulated absorption data, since the fluorescence efficiency is also a function of E. In addition, the progression of the measured fluorescence flux, I_f(E), is itself a strong function of the angle of the sample with respect to the beam and the detector [see Bridges & Booth (2024) on self-absorption corrections]. Fortunately, modern fluorescence detectors can often discriminate against absorption processes outside the edge of interest, and in these cases one need only fit the pre-edge data to a constant and subtract it. Other empirical methods are used when background fluorescence processes affect the measured data, and are discussed in the documentation.

Available methods to determine the post-edge atomic background function μ₀(E) each use polynomial or spline fits, but there are many options. Spline knots can be at fixed intervals in some power of E, or the knots can be allowed to vary in energy. In both spline and polynomial fits the data can be weighted in different ways, such as increasing the weight by a power of E or forcing the data through the first data point in the fit range (E_min) or another data point of the investigator's choosing (E_fix). Other function options exists. Once a function is determined, typically E_min (and sometimes E_fix, if used) is allowed to vary such that the low-r part of the modulus of the Fourier transform (FT) is either minimized, fits best to a line or fits best to a quadratic function. Minimizing the modulus works best for systems with relatively far nearest neighbours, such as a metal, and a quadratic function works best for systems with close nearest neighbours, such as an oxide.

Using the low-r part of a transform as an indicator of the quality of the determination of μ₀ is effective, but some atomic backgrounds have features that can limit the effectiveness of such a method, such as those due to so-called `atomic EXAFS' (AXAFS; Rehr et al., 1994) and multielectron excitations (Filipponi et al., 1988; Li et al., 1992; D'Angelo et al., 1996; Gomilšek et al., 2009). In this case, a theoretical model of the EXAFS can be employed to obtain better estimates of μ₀. The method employed in RSXAP (Bridges et al., 1995) is to perform an iterative procedure where a first-guess estimate of μ₀ is made using the techniques just described, followed by a shell fit to the data. The fit residual is then generated in e-space and a new estimate of μ₀ is made by fitting to this residual. The process is iterated until a satisfactory fit to the Fourier transform of the data is obtained.

All EXAFS fitting using rsfit is performed in r-space. Furthermore, unlike many other EXAFS fitting routines, the pair-distribution function g(r) is applied and numerically convolved to obtain the final fitting function, allowing the actual Gaussian or any other form to be used rather than using the k-space exp(−2σ²k²) approximation, typically asThe convolution can be applied either in k-space or in r-space. In k-space, one may write χ(k) as a sum over shells n_s aswhere a standard EXAFS curve for a given scattering (or multiple scattering) path i at a distance R′ is given byC_3,i and C_4,i are the third and fourth cumulants as described in Bunker (1983), Tranquada & Ingalls (1983) and Yang et al. (1977). Note that the second cumulant term that is normally present in the argument of the sine term of a k-space formulation is missing in equation (5), since it is already included by explicitly integrating the Gaussian with the 1/R² term (equation 4) in this r-space formulation (Hayes & Boyce, 1982). Threshold energy shifts ΔE_0,i are included by defining the wavevector k = [k′² − (2m_e/ℏ²)E_0,i]^1/2, where k′ is the original wavevector of the standard curve. The standard curve can either be from an experimental standard or from a code such as FEFF (Rehr et al., 2010; Kas et al., 2024), in which case a single value of ΔE₀ for all paths is generally used. χ(k) can then be Fourier-transformed to r-space, where W(k) is a window function that accounts for the transform range (RSXAP uses a Gaussian-rounded window function).

The methodology employed in RSXAP reverses the order of the integration over R′ in equation (4) and the FT in equation (6). In this case, the core fit function is written aswhere is the FT of the standard ,Note that the factor |F_i(k, R′)| implicitly includes any polarization dependence. A nice feature of this methodology is the easy and explicit R′ dependence of , which is trivially shifted in r when calculating a sample fit function in equation (7). It is important to note that the Fourier transform is a complex function with a real and imaginary part. The quality-of-fit (residual) parameter is defined aswhere and are the real and imaginary parts of the FT of the data using the same W(k) as the similar fit function parts and .

The factor is minimized using the fitting routine STEPIT and uncertainties determined by FIDO (Chandler, 1965). As such, no derivatives are necessary or calculated. To estimate the parameter uncertainties, the parameter in question p_i is varied while holding all other parameters fixed until the statistical χ² is increased by unity, as described in Booth (2024a) and Booth & Hu (2009). This method not only allows better uncertainty estimates in complicated fitting landscapes where the assumption that the statistical χ² is quadratic near its minimum is poor, but also naturally allows asymmetric uncertainties, such as are commonly observed for σ² parameters.

While RSXAP features accurate background-removal and fitting methodologies and is quite flexible, there are several limitations that make these codes difficult to use and employ, and some functionalities that are not included.

Regarding the methodology, multiple scattering can only be included by treating it as a separate standard curve , which is typically obtained theoretically. For instance, when using FEFF one can create a standard curve for a set of multiple-scattering paths by summing various paths together when creating a given . In this manner, the path can only be varied within the fit by varying the overall bond-length, Debye–Waller and other fitting parameters. A better method would be to recalculate the multiple scattering when bond-length changes are made in the fit. The codes do have a flexible constraint system that can be used to mimic certain types of multiple scattering, for instance constraining the effective bond length of the four-leg U–O–U–O scattering of an isolated U–O pair to be twice the U—O bond length.

As far as usability and portability are concerned, the codes only compile on Linux- and Unix-based systems, although much of the code now compiles under Windows as well. Although there is a graphical user interface, for complete functionality a text-based menu system is utilized that can be executed quite rapidly by experienced users. The routines are, however, very difficult to use for inexperienced investigators.

A missing functionality that may be added in the future is the ability to simultaneously fit multiple data files with shared fitting parameters. Another missing feature is to automate the iterative background technique (Bridges et al., 1995).

As an example of a fit to EXAFS data using RSXAP, we present the data and fit results for a copper-doped ZnS nanocrystal phosphor described in full elsewhere (Car et al., 2011). This example is for illustrative purposes only and is not meant to be a complete report of these data and fits.

The Cu K-edge data were collected at 10 K on beamline 10-2 at the Stanford Synchrotron Radiation Lightsource (SSRL) using detuned Si(111) monochromator crystals. The error bars were calculated from five scans. This simple fit model assumes only one scattering shell, with the backscattering amplitudes and phases calculated by FEFF6 (Zabinsky et al., 1995). Other fit models are considered in Car et al. (2011). The main question to resolve is how the copper substitutes into the ZnS lattice.

The fits were performed using two methods for calculating the statistical χ². Other methods are possible, and in fact parameter errors were obtained from individual fits of the five scans in Car et al. (2011). Here, the `statistical χ<sup>2</sup>' method minimizes a χ<sup>2</sup> function based on equation (1) in Booth (2024a), where the individual errors per data point in r-space, e<sub>i</sub>, are determined by the distribution per data point of the five measured scans. In the `single e' method, all e_i = e, with e chosen such that χ² = ν, where ν represents the degrees of freedom of the the fit (see Booth, 2024a,b). The data and fit using the χ² method are shown in Fig. 2 in both k-space and r-space, and the results for both methods are reported in Table 2.

Table 2 Fit results for the copper-doped ZnS data in Fig. 2 All parameters are for the Cu–S pair in this one-peak fit. The fit model is slightly altered from that in Car et al. (2011) for illustrative purposes. Results using two error-determination methods are reported as described in the text. is fixed at 0.98 for this example. The fit range is between 1.2 and 2.5 Å. The data have 7.4 independent data points (ν = 3.4). The estimated error per point from the `single e' method is 0.06, to give χ2/ν = 1.   Statistical χ2 Single e N 3.15 (9) 3.2 (2) σ2 (Å2) 0.0064 (3) 0.0065 (7) R (Å) 2.271 (2) 2.270 (9) ΔE0 (eV) −8.8 (5) −8.7 (5) (%) 4.6 4.5 χ2 16.8 3.4 (defined)

Figure 2 Data and fit results for Cu K-edge data from copper-doped ZnS in both (a) k-space and (b) r-space. This example is illustrative only; for a complete workup of these data, see Car et al. (2011). The average data and error bars are calculated from five scans, each of which was reduced separately. The data are fitted in r-space between the limits shown by the vertical dotted lines, and the transform is between 3.5 and 10.0 Å−1, narrowed using a Gaussian window with a 0.3 Å−1 width. It is important to note that while the fit quality in r-space is easily judged over the given fit range, the k-space fit (red) must be compared with the filtered data (green) backtransformed from r-space over the fit range, not the unfiltered data (black). It is somewhat difficult to see the filtered data behind the fit line in k-space.

In this case, the estimated errors are similar between the two methods, but the `single e' method gives more conservative error estimates, which is generally the case due to the enhancement of the statistical χ² from systematic uncertainties (Booth, 2024b). Note that while asymmetric errors are calculated, only the larger of the two errors is reported.

The near-neighbour coordination has four sulfur neighbours around zinc at a distance of 2.34 Å. The fit results in Table 2 are not consistent with a simple substitution of Cu for Zn atoms in this structure. In fact, copper has a reduced coordination of about 3.2 ± 0.2 neighbours at a distance of 2.27 ± 0.01 Å. A model is discussed in Car et al. (2011) that rationalizes these results with an off-centre displacement of the Cu atom within the sulfur tetrahedron in the presence of a sulfur vacancy.