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, n_bins, type_pairs)`¶

from amorphouspy import compute_rdf

# Compute all partial RDFs up to 8 Å with 500 bins
r, rdfs, cn = compute_rdf(
    structure=glass_structure,
    r_max=8.0,         # Maximum distance (Å)
    n_bins=500,        # Number of radial bins
    type_pairs=None,   # None → all unique pairs
)

Parameters:

Parameter	Type	Default	Description
`structure`	`Atoms`	—	ASE Atoms object with periodic boundaries
`r_max`	`float`	`10.0`	Maximum distance for RDF calculation (Å)
`n_bins`	`int`	`500`	Number of radial bins
`type_pairs`	`list[tuple[int, int]]` or `None`	`None`	Specific pairs as atomic numbers, e.g. `[(14, 8), (11, 8)]`. `None` = all pairs

Returns: A tuple (r, rdfs, cn_cumulative):

Variable	Type	Description
`r`	`np.ndarray`	Bin center positions (Å)
`rdfs`	`dict[tuple[int,int], np.ndarray]`	Partial RDFs keyed by atomic number pair
`cn_cumulative`	`dict[tuple[int,int], np.ndarray]`	Running coordination numbers

Example: Typical oxide glass RDF analysis¶

from amorphouspy import compute_rdf
import plotly.graph_objects as go

r, rdfs, cn = compute_rdf(glass_structure, r_max=8.0, n_bins=500)

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

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

# Access specific pairs by atomic numbers (e.g. Si=14, O=8)
g_SiO = rdfs[(14, 8)]
cn_SiO = cn[(14, 8)]

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, target_type, cutoff, neighbor_types)`¶

Computes coordination number distribution for atoms of a target type.

from amorphouspy import compute_coordination

# Si (atomic number 14) coordinated by O (atomic number 8), cutoff 2.0 Å
coord_dist = compute_coordination(
    glass_structure,
    target_type=14,       # Atomic number of central atom (Si)
    cutoff=2.0,           # Cutoff radius (Å)
    neighbor_types=[8],   # Atomic numbers of neighbors (O); None = all types
)

# coord_dist: {cn: count} e.g. {4: 195, 5: 5}