Bond Angle Distribution

Bond angle distributions provide insight into the local polyhedral geometry of network-forming cations and the inter-polyhedral connectivity structure.

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Theory

Intra-polyhedral angles: O–X–O

The O–X–O angle distribution measures the angles between oxygen atoms bonded to the same network-forming cation X. For perfect tetrahedral coordination (e.g., Si in SiO₂), the ideal angle is:

\[ \theta_{\text{tet}} = \arccos\left(-\frac{1}{3}\right) \approx 109.47° \]

Deviations from this value indicate distorted tetrahedra. The width of the distribution reflects the degree of structural disorder.

Common ideal angles:

Geometry	Coordination	Ideal angle
Tetrahedral	4	109.47°
Trigonal planar	3	120.00°
Octahedral	6	90.00°

Inter-polyhedral angles: X–O–X

The X–O–X angle distribution measures the angles at bridging oxygens connecting two network-forming polyhedra. This angle controls the medium-range order and ring topology.

For SiO₂ glass, the Si–O–Si angle distribution is centered around ~144° with a broad range of ~120–180°. This is a key structural parameter that distinguishes glass polymorphs and relates to the density anomaly.

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Usage

compute_bond_angle_distribution(structure, cutoff, center_types, ligand_type, nbins)

from amorphouspy import compute_bond_angle_distribution

# O-Si-O angles (intra-tetrahedral)
osi_o = compute_bond_angle_distribution(
    structure=glass_structure,
    cutoff=2.0,
    center_types=["Si"],      # Central atom
    ligand_type="O",           # Ligand atoms
    nbins=180,                 # Number of histogram bins
)

Parameters:

Parameter	Type	Default	Description
structure	Atoms	—	ASE Atoms object
cutoff	float	—	Bond cutoff for center-ligand pairs (Å)
center_types	list[str]	—	Element symbols of central atoms
ligand_type	str	—	Element symbol of ligand atoms
nbins	int	180	Number of bins for the angle histogram (1°–180°)

Returns: A dictionary with:

Key	Type	Description
"angles"	np.ndarray	Bin centers in degrees
"counts"	np.ndarray	Histogram counts (or normalized probability)
"mean"	float	Mean angle in degrees
"std"	float	Standard deviation in degrees

Example: Complete bond angle analysis

from amorphouspy import compute_bond_angle_distribution
import plotly.graph_objects as go

# Intra-tetrahedral: O-Si-O
o_si_o = compute_bond_angle_distribution(
    glass_structure, cutoff=2.0,
    center_types=["Si"], ligand_type="O",
)

# Inter-tetrahedral: Si-O-Si
si_o_si = compute_bond_angle_distribution(
    glass_structure, cutoff=2.0,
    center_types=["O"], ligand_type="Si",
)

fig = go.Figure()
fig.add_trace(go.Scatter(
    x=o_si_o["angles"], y=o_si_o["counts"],
    name=f"O-Si-O (mean={o_si_o['mean']:.1f}°)",
))
fig.add_trace(go.Scatter(
    x=si_o_si["angles"], y=si_o_si["counts"],
    name=f"Si-O-Si (mean={si_o_si['mean']:.1f}°)",
))
fig.update_layout(xaxis_title="Angle (°)", yaxis_title="P(θ)")
fig.show()

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Typical Results for Oxide Glasses

Angle	Expected peak	Width	Significance
O-Si-O	~109°	Narrow (~5°)	Nearly perfect tetrahedra
O-Al-O	~109°	Broader (~10°)	More distorted tetrahedra
O-B-O (III)	~120°	Narrow	Trigonal planar BO₃
O-B-O (IV)	~109°	Moderate	Tetrahedral BO₄
Si-O-Si	~144°	Very broad (120–180°)	Inter-tetrahedral flexibility
Si-O-Al	~130–140°	Broad	Mixed linkage

Tip: A bimodal O-B-O distribution (peaks near 109° and 120°) indicates a mix of 3-fold and 4-fold coordinated boron — the ratio relates to the glass composition and modifier content.