S(q) and XRD: methodology notes

This page archives the investigation behind AmorphGen’s choice of S(q) implementation. The user-facing recommendation in Analysis is the direct q-vector method (structure_factor_direct()); this note keeps the underlying comparison and physical reasoning for posterity.

Two methods that were on the table

AmorphGen ships both implementations because they have genuinely different properties:

Method	API	Speed	Peak intensity
FT-of-g(r)	`structure_factor()`	Fast (seconds)	Damped ~2× by finite `rmax`
Direct q-vector (Debye sum)	`structure_factor_direct()`	Slower (~20s × 20 structs)	Quantitatively correct

Both implement well-established physics; the difference is only how the Fourier integral is handled in a finite simulation cell.

Why peak intensities differ between the two

FT-of-g(r). Starts from the ensemble-averaged radial distribution function:

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

In a finite cell with side \(L\), the integral has to be truncated at \(r_{\max} = L/2\) because beyond this the minimum-image convention becomes ambiguous. For a typical 400-atom amorphous-oxide cell \(L \approx 16\) Å so \(r_{\max} = 8\) Å — exactly where the medium-range correlations responsible for the FSDP live. Cutting them off damps the FSDP intensity by about 50 %.

Direct q-vector. Evaluates the Debye scattering equation at the reciprocal-lattice vectors of the periodic cell:

\[S(\vec G) = \frac{1}{N\langle f\rangle^{2}}\,\Bigl|\sum_i f_i\, \mathrm{e}^{\mathrm{i}\vec G\cdot\vec r_i}\Bigr|^{2}, \qquad \vec G = 2\pi(n_1 \vec b_1 + n_2 \vec b_2 + n_3 \vec b_3)\]

No truncation, no minimum-image issues. Spherical averaging then gives a clean S(q) curve.

Validation that drove the decision

We benchmarked both methods on the published a-Ga₂O₃ DFT-PBE0 ensemble (Kaewmeechai, Strand & Shluger, Phys. Rev. B 111 (2025) 035203). Comparing against the experimental X-ray S(Q) and the GAP_500 simulation from the same reference (Fig. S2b):

Method	FSDP intensity at q = 2.4 Å⁻¹	Match to experiment (~1.8-2.0)?
FT-of-g(r), unweighted	0.84	❌ ~2× low
FT-of-g(r), X-ray weighted	0.84	❌ ~2× low
Direct q-vector, X-ray weighted	2.00	✅
GAP_500 (Csányi group)	~1.8	✅
Experiment (Fig. S2b)	~1.8-2.0	✅ reference

The direct method matches both the experimental S(Q) and the GAP simulation from the same reference. The FT method positions peaks correctly but consistently under-shoots their height.

Why the FT method is still in the package

Even though the direct method is preferred for paper figures and experimental comparison, structure_factor() (FT-of-g(r)) remains useful for:

Ensemble-vs-ensemble comparisons — the systematic damping cancels when comparing two ensembles computed the same way.
Quick sanity checks — seconds rather than tens of seconds.
Backwards compatibility — existing scripts and the JOSS paper validation figures use the FT method; preserving the API avoids silent behaviour changes.

The default weighting="unweighted" of structure_factor() is fine for ensemble comparison and matches the historical AmorphGen behaviour. The Faber–Ziman weighted total (weighting="xray") is available for users who want X-ray-like intensities from the fast method, with the caveat that the FSDP height will still be damped.

A worked example of the FSDP cancellation in unweighted sums

For a-Ga₂O₃ at q = 2.5 Å⁻¹ (from the PRB ensemble):

\(S_{\rm Ga-Ga}(2.5) \approx 1.23\) (the FSDP itself)
\(S_{\rm Ga-O}(2.5) \approx 0.25\) (an anti-peak — Ga–O correlations are anti-phase at this q)
\(S_{\rm O-O}(2.5) \approx 1.25\)

Unweighted sum (\(f_\alpha = 1\)):

\[S^{(\rm unwt)}(2.5) = c_{\rm Ga}^{2}(1.23) + 2 c_{\rm Ga} c_{\rm O}(0.25) + c_{\rm O}^{2}(1.25) \approx 0.84\]

The Ga–O dip cancels the like-pair peaks — the FSDP disappears.

X-ray weighted (\(Z_{\rm Ga} = 31, Z_{\rm O} = 8\)):

\[S^{(\rm xray)}(2.5) = \frac{0.4^2 \cdot 961 \cdot 1.23 + 2\cdot 0.4\cdot 0.6\cdot 248\cdot 0.25 + 0.6^2\cdot 64\cdot 1.25}{17.2^{2}} \approx 2.0\]

Heavy Ga–Ga dominates (\(Z^2 = 961 \gg 64\)). The FSDP survives — and matches experiment.

This is why X-ray diffraction sees the FSDP that an unweighted total or a chemistry-blind “first-shell-only” analysis would miss.

References

The methodology decision and the FT-vs-direct comparison are documented for the JCTC methods paper (in preparation). Primary references for the underlying physics are listed in the Analysis “Physical validity” tab.