Structure Factor S(q)¶

The structure factor \(S(q)\) is the reciprocal-space equivalent of the radial distribution function. It is the quantity directly measured in neutron and X-ray total scattering experiments, making it the primary bridge between MD simulations and experiment.

Theory¶

From real space to reciprocal space¶

Starting from the partial RDFs \(g_{\alpha\beta}(r)\), the Faber-Ziman partial structure factors are obtained via the isotropic sine transform:

\[ S_{\alpha\beta}(q) = 1 + \frac{4\pi\rho}{q} \int_0^{r_\mathrm{max}} r\,\bigl[g_{\alpha\beta}(r) - 1\bigr]\,M(r)\,\sin(qr)\,\mathrm{d}r \]

where \(\rho = N/V\) is the total number density and \(q = |\mathbf{q}|\) is the magnitude of the momentum transfer. This is the spherically-averaged (1D) form of the 3D Fourier transform, valid for isotropic systems such as glasses.

Lorch modification function¶

A finite simulation box imposes a hard cutoff at \(r_\mathrm{max}\), which causes Gibbs-like termination ripples in \(S(q)\). The Lorch modification function

\[ M(r) = \operatorname{sinc}\!\left(\frac{r}{r_\mathrm{max}}\right) = \frac{\sin(\pi r / r_\mathrm{max})}{\pi r / r_\mathrm{max}} \]

smoothly damps the integrand to zero at \(r_\mathrm{max}\), strongly suppressing these artefacts at the cost of a slight broadening of sharp peaks.

Neutron diffraction¶

For neutron scattering the total structure factor is:

\[ S(q) = 1 + \frac{1}{\langle b \rangle^2} \sum_{\alpha \le \beta} (2 - \delta_{\alpha\beta})\, c_\alpha c_\beta\, b_\alpha b_\beta\, \bigl[S_{\alpha\beta}(q) - 1\bigr] \]

where:

Symbol	Meaning
\(c_\alpha\)	atomic concentration of species \(\alpha\)
\(b_\alpha\)	coherent neutron scattering length (fm) — q-independent constant
\(\langle b \rangle = \sum_\alpha c_\alpha b_\alpha\)	composition-averaged scattering length
\(\delta_{\alpha\beta}\)	Kronecker delta (avoids double-counting like pairs)

Because \(b_\alpha\) is a nuclear property, it is independent of \(q\) and can be positive or negative (e.g. \(b_\mathrm{H} = -3.74\) fm). This makes neutron weights fixed across the entire \(q\)-range and provides unique sensitivity to light elements and isotope contrast.

Tabulated scattering lengths¶

The implementation uses the Sears (1992) / NIST coherent scattering lengths:

Element	Z	\(b\) (fm)
H	1	−3.739
O	8	5.803
Na	11	3.630
Si	14	4.149
Fe	26	9.450
Ni	28	10.300

X-ray diffraction¶

For X-ray scattering the same Faber-Ziman expression applies, but the constant scattering lengths \(b_\alpha\) are replaced by the q-dependent atomic form factors \(f_\alpha(q)\):

\[ S(q) = 1 + \frac{1}{\langle f(q) \rangle^2} \sum_{\alpha \le \beta} (2 - \delta_{\alpha\beta})\, c_\alpha c_\beta\, f_\alpha(q)\, f_\beta(q)\, \bigl[S_{\alpha\beta}(q) - 1\bigr] \]

where \(\langle f(q) \rangle = \sum_\alpha c_\alpha f_\alpha(q)\) is now itself \(q\)-dependent, so both numerator and denominator vary across the diffractogram.

International Tables parameterisation¶

The form factor is evaluated using the four-Gaussian fit from the International Tables for Crystallography (as used by pymatgen's XRD calculator):

\[ f_\alpha(q) = Z_\alpha - 41.78214\, s^2 \sum_{i=1}^{4} a_i \exp\!\left(-b_i s^2\right), \qquad s = \frac{q}{4\pi} \]

At \(q = 0\), \(f_\alpha(0) = Z_\alpha\) (total electron count). The form factor decays monotonically with \(q\), so heavy atoms dominate at low \(q\) and contributions from all species fall off at high \(q\).

Neutron vs X-ray: key differences¶

Property	Neutron	X-ray
Scatters from	Nucleus	Electron cloud
Weight \(w_\alpha\)	Constant \(b_\alpha\) (fm)	\(q\)-dependent \(f_\alpha(q)\) (electrons)
\(q\)-dependence of weights	None	Decays to ~0 at high \(q\)
Sensitivity to light elements	High (e.g. H, Li)	Low (scales with \(Z\))
Isotope contrast	Yes (H vs D)	No
Normalization denominator	\(\langle b \rangle^2\) — scalar	\(\langle f(q) \rangle^2\) — vector

For a Na₂O–SiO₂ glass the practical consequence is:

Neutron: O (\(b=5.80\) fm) and Si (\(b=4.15\) fm) contribute comparably; the spectrum is sensitive to the Si–O–Si network.
X-ray: Si (\(Z=14\)) dominates over O (\(Z=8\)) and Na (\(Z=11\)) at low \(q\), but all contributions decay by \(q \approx 15\) Å⁻¹.

Usage¶

from ase.io import read
from amorphouspy.analysis.structure_factor import compute_structure_factor

atoms = read("sodium_silicate.extxyz")

# Neutron S(q)
q, sq_n, sq_partials_n = compute_structure_factor(
    atoms,
    q_min=0.3,
    q_max=20.0,
    n_q=600,
    r_max=10.0,
    n_bins=2000,
    radiation="neutron",
    lorch_damping=True,
)

# X-ray S(q)
q, sq_x, sq_partials_x = compute_structure_factor(
    atoms,
    radiation="xray",
    lorch_damping=True,
)

# Access a partial: S_SiO(q)
sq_SiO = sq_partials_n[(8, 14)]

Parameters¶

Parameter	Type	Default	Description
`structure`	`Atoms`	—	ASE Atoms object with PBC
`q_min`	`float`	`0.5`	Minimum momentum transfer (Å⁻¹)
`q_max`	`float`	`20.0`	Maximum momentum transfer (Å⁻¹)
`n_q`	`int`	`500`	Number of \(q\) grid points
`r_max`	`float`	`10.0`	Real-space RDF cutoff (Å)
`n_bins`	`int`	`2000`	Number of radial histogram bins
`radiation`	`str`	`"neutron"`	`"neutron"` or `"xray"`
`lorch_damping`	`bool`	`True`	Apply Lorch \(M(r)\) to reduce termination ripples
`type_pairs`	`list[tuple[int,int]]` or `None`	`None`	Pairs to compute; `None` = all unique pairs

Returns¶

Name	Shape	Description
`q`	`(n_q,)`	Momentum transfer grid (Å⁻¹)
`sq_total`	`(n_q,)`	Total \(S(q)\)
`sq_partials`	`dict[(Z_a, Z_b) → (n_q,)]`	Faber-Ziman partial \(S_{\alpha\beta}(q)\), keyed by `(min(Z), max(Z))`

References¶

Faber, T. E. & Ziman, J. M. (1965). A theory of the electrical properties of liquid metals. Phil. Mag. 11, 153–173.
Lorch, E. (1969). Neutron diffraction by germania, silica and radiation-damaged silica glasses. J. Phys. C 2, 229–237.
Sears, V. F. (1992). Neutron scattering lengths and cross sections. Neutron News 3, 26–37.
International Tables for Crystallography, Vol. C (2006). Atomic scattering factors for X-rays. Kluwer Academic Publishers.