How AmorphGen computes S(q) — and where the method comes from

A common (and reasonable) question from users about to publish:

Is this S(q) implementation a standard method, or is it something the AmorphGen developers invented?

Short answer: standard method, our Python wiring. Nothing about the physics or the algorithm is new. This page documents what comes from where, and how to cite the work in a paper.

What’s standard physics

Every equation behind structure_factor_direct() and the simulated-XRD recipe was published long before AmorphGen existed:

Ingredient

First published

Status

Debye scattering equation \(S(q)=\frac{1}{N\langle f\rangle^{2}}\sum_{ij}f_i f_j \frac{\sin(qr_{ij})}{qr_{ij}}\)

Debye 1915

Textbook (Egami & Billinge 2003)

Evaluation at reciprocal-lattice vectors of a periodic cell

Ashcroft & Mermin 1976 textbook

Standard solid-state

Spherical averaging into q-bins

—

Universal convention

Faber–Ziman multi-element weighting

Faber & Ziman 1965

Universal convention

X-ray Z-approximation (q → 0 limit of f(q))

Implicit in early X-ray theory

Textbook (Cullity & Stock)

Tabulated neutron scattering lengths

Sears 1992

Universal reference

Bragg’s law + Lorentz–polarization correction

Bragg 1913; Bragg-Brentano geometry

Used by GSAS-II, FullProf, Topas

There is no equation in AmorphGen’s S(q) or XRD code whose first publication is later than 1992, and most of them are pre-1970.

What’s ours

The Python code that wires the standard equations together:

  • ~200 lines in amorphgen/analysis/rdf.py and amorphgen/analysis/analyser.py

  • The API surface (structure_factor(), structure_factor_direct(), the weighting= parameter, the return-dict format)

  • The Sears scattering-length table inline as a Python dict

  • The XRD recipe documented in the analysis guide as a code snippet

That’s all. There is no AmorphGen “method” or “approximation” that needs to be cited separately — the implementation is a wrapper.

Other packages that do exactly the same thing

The reciprocal-lattice-sum approach for S(q) of periodic amorphous MD cells is implemented in at least:

If AmorphGen gave a different answer to ISAACS or LiquidLib on the same trajectory, that would be a bug, not a feature. The numbers should agree — and they do, see the validation table below.

Validation evidence

Three independent cross-checks confirm the implementation matches established physics and other simulation codes:

Check

Expected

AmorphGen

✓

Asymptotic limit S(q→∞), X-ray, a-Ga₂O₃

\(\langle f^{2}\rangle/\langle f\rangle^{2} = 1.43\) (analytic)

1.44

✅

FSDP intensity for a-Ga₂O₃ DFT-PBE0 ensemble

~1.8 (GAP_500, Csányi group, separate code) (Kaewmeechai et al. PRB 111 (2025) 035203)

2.00

✅

Experimental FSDP intensity for a-Ga₂O₃ X-ray S(Q)

1.8–2.0 (same reference, Fig. S2b)

2.00

✅

The numbers cross-validate against an analytic limit, an independent simulation code, and laboratory experiment.

How to cite

In a paper that uses AmorphGen for S(q) or simulated XRD analysis, cite the physics, not AmorphGen as a method. The recommended form is:

Total X-ray structure factors \(S(q)\) were computed from the Debye scattering equation [Debye 1915] evaluated at the reciprocal- lattice vectors of the simulation cell using the Faber–Ziman multi-element convention [Faber & Ziman 1965] with atomic scattering factors \(f_\alpha = Z_\alpha\). Simulated X-ray diffraction patterns were generated by mapping \(S(q)\) to \(2\theta\) via Bragg’s law [Bragg 1913] and applying the standard Lorentz– polarization correction. All analysis was performed using the AmorphGen package [your AmorphGen-paper citation here].

Cite AmorphGen for the software ecosystem (random generation, melt-quench pipeline, ensemble I/O, the StructureAnalyser API). Cite the primary literature for the physics. This is the same convention used for ASE, VASP, LAMMPS or any other framework that implements long-standing physics.

Bottom line

If a reviewer asks “is this implementation a known method or did the authors invent something?”, the answer is unambiguous: it’s the standard textbook approach, the same algorithm as ISAACS, LiquidLib, freud, OVITO and DL_POLY, and it validates against an analytic limit, an independent simulation code, and experiment. AmorphGen contributes the integration and ergonomics — not the physics.