S(q) and XRD: methodology notes

This page describes the two structure-factor implementations in amorphgen/analysis/rdf.py. See Analysis for runnable examples.

Choosing a method

Method	Python API	CLI	Finite-cell limitation
Direct reciprocal-space sum	structure_factor_direct()	--sq-method direct (default)	Discrete q vectors and sparse low-q shells
Fourier transform of g(r)	structure_factor()	--sq-method ft	Finite-r truncation and histogram resolution

Use configurations with the same composition and a valid periodic cell. The direct Python method defaults to X-ray weighting; the FT Python method defaults to unweighted scattering. The CLI defaults to X-ray weighting for both methods. Specify the weighting explicitly when comparing results.

Normalization of the direct method

For each nonzero reciprocal vector, the code calculates

\[I(\vec q)=\left|\sum_i f_i(q)e^{i\vec q\cdot\vec r_i}\right|^2,\]

then applies the Faber–Ziman convention:

\[S(\vec q)=1+\frac{I(\vec q)/N-\langle f^2(q)\rangle} {\langle f(q)\rangle^2}.\]

The composition averages are \(\langle f\rangle=\sum_\alpha c_\alpha f_\alpha\) and \(\langle f^2\rangle=\sum_\alpha c_\alpha f_\alpha^2\). For uncorrelated positions this normalization tends to 1. The uncorrected quantity \(I/(N\langle f\rangle^2)\) instead tends to \(\langle f^2\rangle/\langle f\rangle^2\), which generally differs from 1 in a mixture. These quantities should not be compared without converting them to the same convention.

With real-space cell vectors as rows of \(\mathbf A\), the q vectors are \(\vec G=2\pi\vec n\mathbf A^{-\top}\) for integer triplets \(\vec n\), excluding the origin. Values are averaged into shells by their magnitude. For a cubic cell, \(q_{\min}=2\pi/L\); increasing the number of bins does not improve that limit. n_per_bin counts the sampled vectors, and empty shells return NaN. Gaussian smoothing weights each shell by its vector count and retains the unsmoothed total as s_q_raw.

Fourier transform of g(r)

The isotropic relation used by the FT method is

\[S(q)-1=4\pi\rho\int_0^{r_{\max}}[g(r)-1] \frac{\sin(qr)}{qr}\,r^2\,\mathrm{d}r.\]

The implementation transforms each structure’s unsmoothed RDF with 500 radial bins and its own \((N-1)/V\) finite-system density prefactor before averaging the resulting curves. By default, the shared rmax is half the shortest cell-vector length across the ensemble, rounded down to 0.1 Å. Truncation can change peak heights and introduce ripples. Its size and direction depend on the structure and chosen range; there is no general factor-of-two correction, nor a guarantee that truncation errors cancel between ensembles. The direct method avoids this integral’s cutoff but retains its own cell-size and sampling limits.

For a weighted total, the FT method first computes the partials and combines them as

\[S(q)=\frac{\sum_{\alpha,\beta}c_\alpha c_\beta f_\alpha(q)f_\beta(q) S_{\alpha\beta}(q)}{\langle f(q)\rangle^2}.\]

X-ray weights use q-dependent Waasmaier–Kirfel neutral-atom form factors; neutron weights use tabulated coherent scattering lengths. Unweighted scattering sets every factor to 1. Weighting changes the relative contribution of each partial; it cannot repair finite-cell or truncation errors.

Checks and experimental comparison

TestSqNormalisation in test/test_analysis.py checks the high-q limit, the \(r^2\) FT integrand, selected scattering-table entries, q-dependent X-ray weights, smoothing and exact recombination of neutron partials. These tests verify implementation properties. They do not establish a fixed experimental accuracy or equivalence to another analysis package.

Before comparing with a measured curve, match its normalization, weights, q range, temperature and resolution. A normalized S(q) must be converted back to coherent intensity before applying any instrument-specific model; see the simulated-XRD tab in Analysis. Reference publications and software citation guidance are in Scattering methods and citations.

load_experiment() accepts measured S(q) or T(r), and compare_experiment() interpolates each calculated structure onto the measured grid before summarizing residuals and pointwise confidence intervals. Extrapolation and missing-bin gaps are excluded and counted. Measurement uncertainties, when supplied, weight Rw and chi-square; ensemble SEM is not substituted for them. No scaling, offset, resolution or background is fitted. Correlated points limit interpretation of the diagonal chi-square statistic.

xrd_pattern() restores coherent intensity per atom using each structure’s own composition, then forms an equal-weight ensemble mean. The direct method uses bin-centre form factors for this conversion, so finite shell width introduces an approximation. q maps to 2θ using the supplied wavelength; inaccessible requested q ranges are rejected. The returned profile has no Lorentz–polarization, absorption, background or instrument-response correction.