Theoretical Background

This document summarizes the conventions and equations implemented by Glassure for isotropic X-ray total-scattering data. Glassure uses the Faber–Ziman definition of the total structure factor. This convention matters for multicomponent samples, because other definitions (for example, Ashcroft–Langreth) use different partial structure factors and normalizations.

Processing overview and notation

Glassure performs the following main steps:

1. subtract a scaled background pattern;

2. normalize the corrected intensity and subtract incoherent scattering;

3. calculate the Faber–Ziman total structure factor \(S(Q)\);

4. extrapolate \(S(Q)\) to \(Q=0\) and optionally optimize it;

5. Fourier transform \(S(Q)\) to the reduced real-space function \(F(r)\); and

6. calculate \(g(r)\) and related real-space functions.

The magnitude of the scattering vector is

\[ Q = \frac{4\pi\sin\theta}{\lambda}, \]

where \(2\theta\) is the scattering angle and \(\lambda\) is the incident X-ray wavelength. Glassure expects \(Q\) in \(\mathrm{\AA}^{-1}\) and distances in \(\mathrm{\AA}\).

The intensity supplied to the normalization routines should already contain any experiment-specific corrections that are not explicitly provided by Glassure, such as detector, polarization, absorption, and general sample-geometry corrections. The optional corrections implemented by Glassure are described below.

Atomic form factors

For a sample containing species \(i\) with atomic fractions \(c_i\), where \(\sum_i c_i=1\), Glassure calculates

\[ \langle f(Q)\rangle = \sum_i c_i f_i(Q) \]

and

\[ \langle f^2(Q)\rangle = \sum_i c_i f_i^2(Q). \]

Here \(f_i(Q)\) is the coherent atomic X-ray form factor in electron units. Notice the distinction between \(\langle f^2\rangle\) and \(\langle f\rangle^2\). The available sources provide real, neutral-atom form factors; Glassure does not currently add energy-dependent anomalous-dispersion terms \(f'\) and \(f''\).

Faber–Ziman structure factor

After normalization and incoherent-scattering subtraction, the coherent intensity per atom is related to the Faber–Ziman total structure factor by

\[ I_{\mathrm{coh}}(Q) = \langle f^2(Q)\rangle + \langle f(Q)\rangle^2[S(Q)-1]. \]

Glassure therefore calculates

\[ S(Q) = \frac{I_{\mathrm{coh}}(Q)-\langle f^2(Q)\rangle} {\langle f(Q)\rangle^2}+1. \]

This is the normalization implemented by calculate_sq().

In the Faber–Ziman formalism, the total structure factor can be written as a weighted sum of partial structure factors:

\[ S(Q) = \sum_i\sum_j w_{ij}(Q)S_{ij}^{\mathrm{FZ}}(Q), \]

with the ordered-pair weights

\[ w_{ij}(Q) = \frac{c_i c_j f_i(Q)f_j(Q)}{\langle f(Q)\rangle^2}, \qquad \sum_i\sum_j w_{ij}(Q)=1, \]

and

\[ S_{ij}^{\mathrm{FZ}}(Q) = 1 + 4\pi\rho_0\int_0^\infty r^2[g_{ij}(r)-1]\frac{\sin(Qr)}{Qr}\,dr. \]

Equivalently, when summing only unique pairs \(i\leq j\), the weight numerator contains the factor \((2-\delta_{ij})\). For X-rays, the weights generally depend on \(Q\) because the atomic form factors depend on \(Q\).

Intensity normalization

Let \(I_{\mathrm{corr}}(Q)\) denote the background-subtracted and otherwise corrected experimental intensity in arbitrary units. Glassure provides two normalization methods.

Krogh–Moe–Norman integral normalization

The integral method uses the sum rule

\[ \int_0^\infty Q^2[S(Q)-1]\,dQ = -2\pi^2\rho_0. \]

For finite experimental data, Glassure applies \(A(Q)=\exp(-aQ^2)\), where \(a\) is the configured attenuation factor. The implemented normalization factor is

\[ \alpha = \frac{ -2\pi^2\rho_0 + \int A(Q)Q^2 \dfrac{\langle f^2(Q)\rangle+I_{\mathrm{inc}}(Q)} {\langle f(Q)\rangle^2}\,dQ }{ \int A(Q)Q^2 \dfrac{I_{\mathrm{corr}}(Q)}{\langle f(Q)\rangle^2}\,dQ }. \]

The numerical integrals run over the selected experimental \(Q\) range. The coherent intensity passed to the structure-factor calculation is

\[ I_{\mathrm{coh}}(Q)=\alpha I_{\mathrm{corr}}(Q)-I_{\mathrm{inc}}(Q). \]

High-\(Q\) fit normalization

At sufficiently high \(Q\), structural oscillations become small and \(S(Q)\) tends to one. Glassure can therefore determine \(\alpha\) by fitting the measured intensity to the self-scattering plus incoherent scattering above a configured cutoff. In the most general implemented case, it minimizes residuals proportional to

\[ Q^p\left[ \alpha I_{\mathrm{corr}}(Q)-m-\beta I_{\mathrm{container,inc}}(Q) -\left(\langle f^2(Q)\rangle+I_{\mathrm{inc}}(Q)\right) \right], \]

where \(p=1\) or \(2\), \(m\) is an optional constant approximation to multiple scattering, and \(\beta\) scales an optional incoherent container contribution. Terms disabled in the configuration are omitted. This method depends on having a high-\(Q\) interval in which the slowly varying non-structural contributions can be separated reliably from the structural signal.

Related reciprocal-space functions

Glassure uses

\[ i(Q)=S(Q)-1 \]

for the interference function and

\[ F(Q)=Q[S(Q)-1] \]

for the reduced total-scattering function in reciprocal space. The symbol \(F(r)\) below denotes its sine transform; naming conventions differ across the total- scattering literature, where \(F(r)\) is also often called \(G(r)\).

Limiting behavior

At high \(Q\),

\[ S(Q)\rightarrow 1. \]

By default, Glassure extrapolates to

\[ S_{\mathrm{default}}(0) =1-\frac{\langle f^2(0)\rangle}{\langle f(0)\rangle^2} =1-\frac{\langle Z^2\rangle}{\langle Z\rangle^2}, \]

because \(f_i(0)=Z_i\). This boundary value follows from setting the coherent forward-scattering intensity in the equation above to zero; it is zero for a single-component sample and can be negative for a multicomponent sample.

This is a default extrapolation assumption, not a universal thermodynamic value of \(S(0)\). Compressibility, concentration fluctuations, small-angle scattering, and the treatment of the forward beam can change the physical low-\(Q\) limit. The target can therefore be overridden with ExtrapolationConfig.s0 when an independently justified value is available.

Fourier transform to real space

The reduced real-space pair-distribution function is

\[ F(r)=\frac{2}{\pi}\int_0^\infty Q[S(Q)-1]\sin(Qr)\,dQ. \]

For measured data, Glassure evaluates the finite-range transform

\[ F(r)=\frac{2}{\pi}\int_0^{Q_{\max}} Q[S(Q)-1]M(Q)\sin(Qr)\,dQ, \]

after extrapolating the measured structure factor from \(Q_{\min}\) to zero.

Lorch modification function

The optional Lorch modification function is

\[ M(Q)=\frac{\sin(\pi Q/Q_{\max})}{\pi Q/Q_{\max}}, \]

with \(M(0)=1\) by continuity. It suppresses the discontinuity at \(Q_{\max}\) and therefore reduces termination ripples, at the cost of broader real-space peaks and reduced real-space resolution.

Glassure supports direct numerical integration and an FFT implementation. The FFT method requires uniformly spaced \(Q\) data; rebin irregularly spaced data before using it.

Real-space functions and their interpretation

Glassure defines

\[ F(r)=4\pi r\rho_0[g(r)-1] \]

and hence

\[ g(r)=1+\frac{F(r)}{4\pi r\rho_0}, \]

where \(\rho_0\) is the atomic number density in atoms/\(\mathrm{\AA}^3\).

For a single-component material, \(g(r)\) is the usual pair correlation function: it approaches zero in an excluded-volume region and approaches one at long range. For a multicomponent X-ray measurement, the total \(S(Q)\) and the derived real-space functions are scattering-weighted combinations of the partial correlations. Since the X-ray weights are \(Q\) dependent, the resulting total \(g(r)\) is not generally a simple number-weighted pair probability. Peak areas should therefore not be read as element-specific coordination numbers without partial-structure information or a structural model.

Radial distribution function

Glassure calculates

\[ \operatorname{RDF}(r)=4\pi r^2\rho_0g(r). \]

For a single-component system, integrating this function over a coordination shell gives the corresponding coordination number. For multicomponent X-ray totals it is an effective, scattering-weighted shell distribution with the caveat above.

Total correlation function

Glassure also calculates

\[ T(r)=4\pi r\rho_0g(r) =F(r)+4\pi r\rho_0. \]

Atomic number density

For atomic fractions \(c_i\), define the mean molar mass per atom as

\[ \overline{M}=\sum_i c_iM_i. \]

The conversion from mass density \(\rho_m\) in g/cm\(^3\) to atomic number density is

\[ \rho_0 =\frac{\rho_mN_A}{\overline{M}}10^{-24}, \]

where \(N_A\) is Avogadro’s constant and \(10^{-24}\) converts cm\(^{-3}\) to \(\mathrm{\AA}^{-3}\). Equivalently, for stoichiometric coefficients \(n_i\),

\[ \rho_0 =\frac{\rho_mN_A\sum_i n_i}{\sum_i n_iM_i}10^{-24}. \]

Structure-factor optimization

Optimization reduces slowly varying systematic errors by using the physically empty real-space region below the shortest interatomic distance. It should not be used to replace missing experimental corrections, and the cutoff must remain below the first physical pair-correlation peak.

Iterative Kaplow–Eggert method

The iterative method uses the condition \(g(r)=0\) for \(r<r_{\mathrm{cutoff}}\), or equivalently

\[ F(r)=-4\pi r\rho_0. \]

Glassure repeatedly transforms \(S(Q)\) to \(F(r)\), back-transforms the discrepancy in the low-\(r\) interval, and updates \(S(Q)\). This reduces unphysical low-\(r\) oscillations but can also modify the structure factor if the cutoff, density, background, or correction model is inappropriate.

Polynomial fit method

The method inspired by PDFgetX3 fits a polynomial \(P_n(Q)\) through the origin to

\[ F(Q)=Q[S(Q)-1] \]

and applies

\[ S_{\mathrm{corrected}}(Q)=S(Q)-\frac{P_n(Q)}{Q}. \]

Glassure chooses an effective degree near

\[ n\simeq\frac{Q_{\max}r_{\mathrm{cutoff}}}{\pi} \]

and blends the fits for adjacent integer degrees when needed. The intent is to remove slowly varying reciprocal-space components whose Fourier signal is concentrated below \(r_{\mathrm{cutoff}}\), while retaining correlations at larger \(r\). This is an ad hoc correction and its stability should be checked by varying \(Q_{\max}\) and \(r_{\mathrm{cutoff}}\).

Incoherent scattering and optional corrections

Incoherent (Compton) scattering

For a mixture, Glassure uses

\[ I_{\mathrm{inc}}(Q)=\sum_i c_i I_{\mathrm{inc},i}(Q). \]

The default brown_hubbell source interpolates tabulated incoherent scattering functions from Hubbell et al. (1975), together with Brown coherent form factors. The hajdu source uses the analytic approximation

\[ I_{\mathrm{inc},i}(Q) =\left[Z_i-\frac{f_i^2(Q)}{Z_i}\right] \left[1-M_i\left(e^{-K_i s}-e^{-L_i s}\right)\right], \qquad s=\frac{Q}{4\pi}, \]

with element-specific parameters. An xraylib source is also available. The simple expression \(Z-f^2/Z\) is only an independent-atom approximation; the incorrect \(Z^2-f^2\) expression is not an incoherent intensity per atom.

Klein–Nishina correction

For high-energy X-rays, the optional Klein–Nishina correction accounts for photon recoil in the angular dependence of Compton scattering. Let

\[ p=\frac{E'}{E} =\left[1+\frac{h}{\lambda m_ec}(1-\cos 2\theta)\right]^{-1}. \]

Glassure multiplies the incoherent scattering function by the Klein–Nishina to Thomson ratio

\[ K(Q)= \frac{p^3-p^2\sin^2(2\theta)+p}{1+\cos^2(2\theta)}, \]

using \(Q=4\pi\sin\theta/\lambda\). This is distinct from the unpolarized Thomson polarization factor \((1+\cos^2 2\theta)/2\).

Soller-slit DAC correction

For the configured diamond-anvil-cell workflow, Glassure can apply separate angle-dependent transfer functions for scattering from the sample region and the additional diamond incoherent contribution. This correction is specific to the implemented DAC/Soller-slit geometry and is not a general absorption or geometry correction.

Extrapolation to \(Q=0\)

Because measurements normally begin at finite \(Q_{\min}\), Glassure supports:

1. a step extension at the target \(S(0)\);

2. a linear connection to \(S(0)\);

3. a polynomial extrapolation over a configurable overlap; and

4. a spline extrapolation over a configurable overlap.

Unless ExtrapolationConfig.s0 is set explicitly, these methods use the default boundary value described above. Polynomial and spline extrapolations can optionally replace measured points in the overlap region.

References

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2. Krogh-Moe, J. (1956). A method for converting experimental X-ray intensities to an absolute scale. Acta Crystallographica, 9, 951–953. https://doi.org/10.1107/S0365110X56002655

3. Norman, N. (1957). The Fourier transform method for normalizing intensities. Acta Crystallographica, 10, 370–373. https://doi.org/10.1107/S0365110X57001085

4. Lorch, E. (1969). Neutron diffraction by germania, silica and radiation-damaged silica glasses. Journal of Physics C: Solid State Physics, 2, 229–237. https://doi.org/10.1088/0022-3719/2/2/305

5. Kaplow, R., Strong, S. L., & Averbach, B. L. (1965). Atomic arrangement in vitreous selenium. Physical Review, 138, A1336–A1345. https://doi.org/10.1103/PhysRev.138.A1336

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8. Hubbell, J. H., Veigele, W. J., Briggs, E. A., Brown, R. T., Cromer, D. T., & Howerton, R. J. (1975). Atomic form factors, incoherent scattering functions, and photon scattering cross sections. Journal of Physical and Chemical Reference Data, 4, 471–538. https://doi.org/10.1063/1.555523

9. Hajdu, F. (1972). Revised parameters of the analytic fits for coherent and incoherent scattered X-ray intensities of the first 36 atoms. Acta Crystallographica Section A, 28, 250–252. https://doi.org/10.1107/S0567739472000671

10. Keen, D. A. (2001). A comparison of various commonly used correlation functions for describing total scattering. Journal of Applied Crystallography, 34, 172–177. https://doi.org/10.1107/S0021889800019993

11. Egami, T., & Billinge, S. J. L. (2012). Underneath the Bragg Peaks: Structural Analysis of Complex Materials (2nd ed.). Elsevier.

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