Radial Distribution Function & Coordination¶

The radial distribution function (RDF) is the most fundamental structural descriptor for amorphous materials. It describes how atomic density varies as a function of distance from a reference atom.

Theory¶

Pair RDF¶

The partial radial distribution function \(g_{\alpha\beta}(r)\) for species \(\alpha\) and \(\beta\) is defined as:

\[ g_{\alpha\beta}(r) = \frac{V}{4\pi r^2 \Delta r \, N_\alpha N_\beta} \sum_{i \in \alpha} \sum_{j \in \beta} \delta(r - r_{ij}) \]

where \(V\) is the volume, \(N_\alpha\) and \(N_\beta\) are the number of atoms of each species, and \(\Delta r\) is the bin width.

In practice, we histogram pair distances into bins and normalize by the ideal gas density:

\[ g_{\alpha\beta}(r) = \frac{n_{\alpha\beta}(r)}{4\pi r^2 \Delta r \, \rho_\beta} \]

where \(n_{\alpha\beta}(r)\) is the number of \(\beta\) atoms in the shell \([r, r + \Delta r)\) around each \(\alpha\) atom, and \(\rho_\beta = N_\beta / V\) is the number density of species \(\beta\).

Interpretation¶

\(g(r) = 1\) → atoms are at the same density as the average
\(g(r) > 1\) → atoms are more likely to be found at this distance (peaks = coordination shells)
\(g(r) < 1\) → atoms are less likely to be found (troughs = exclusion zones)
\(g(r) \to 1\) as \(r \to \infty\) → long-range disorder (amorphous structure confirmed)

Running Coordination Number¶

The running coordination number \(n_{\alpha\beta}(R)\) counts the average number of \(\beta\) atoms within distance \(R\) of each \(\alpha\) atom:

\[ n_{\alpha\beta}(R) = 4\pi \rho_\beta \int_0^R g_{\alpha\beta}(r) \, r^2 \, dr \]

The coordination number is usually read at the first minimum of \(g(r)\), which corresponds to the boundary between the first and second coordination shells.

Computing RDFs¶

`compute_rdf(structure, r_max, dr, pairs)`¶

from amorphouspy import compute_rdf

# Compute all partial RDFs up to 8 Å with 0.02 Å bin width
rdfs = compute_rdf(
    structure=glass_structure,
    r_max=8.0,       # Maximum distance (Å)
    dr=0.02,         # Bin width (Å)
    pairs=None,      # None → all unique pairs
)

Parameters:

Parameter	Type	Default	Description
`structure`	`Atoms`	—	ASE Atoms object with periodic boundaries
`r_max`	`float`	`8.0`	Maximum distance for RDF calculation (Å)
`dr`	`float`	`0.02`	Bin width (Å) — smaller = smoother RDF
`pairs`	`list[tuple]` or `None`	`None`	Specific element pairs, e.g. `[("Si", "O"), ("Na", "O")]`. `None` = all pairs

Returns: A dictionary with:

Key	Type	Description
`"r"`	`np.ndarray`	Bin center positions (Å)
`"g_r"`	`dict[str, np.ndarray]`	Partial RDFs keyed by pair label (e.g. `"Si-O"`)
`"n_r"`	`dict[str, np.ndarray]`	Running coordination numbers

Example: Typical oxide glass RDF analysis¶

from amorphouspy import compute_rdf
import plotly.graph_objects as go

rdfs = compute_rdf(glass_structure, r_max=8.0, dr=0.02)

fig = go.Figure()
for pair, g_r in rdfs["g_r"].items():
    fig.add_trace(go.Scatter(x=rdfs["r"], y=g_r, name=pair))

fig.update_layout(
    xaxis_title="r (Å)",
    yaxis_title="g(r)",
    title="Partial Radial Distribution Functions",
)
fig.show()

Typical peak positions for oxide glasses¶

Pair	First peak (Å)	Coordination
Si-O	~1.62	4 (tetrahedral)
Al-O	~1.75	4–5
B-O	~1.37 (III) / ~1.47 (IV)	3 or 4
Ca-O	~2.35	6–7
Na-O	~2.30	5–6
O-O	~2.63	—

Computing Coordination Numbers¶

`compute_coordination(structure, cutoff_dict)`¶

Extracts integer coordination numbers from neighbor lists using element-pair-specific cutoff distances.

from amorphouspy import compute_coordination

# Define cutoffs for each pair (first minimum of g(r))
cutoffs = {
    ("Si", "O"): 2.0,   # Å
    ("Al", "O"): 2.2,
    ("Na", "O"): 3.0,
    ("Ca", "O"): 3.0,
}

coord = compute_coordination(glass_structure, cutoff_dict=cutoffs)

# Returns average coordination numbers
print(f"Si-O coordination: {coord['Si-O']:.2f}")  # Should be ~4.0
print(f"Al-O coordination: {coord['Al-O']:.2f}")  # Should be ~4.0–5.0