Bond Angle Distribution¶

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

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.

Usage¶

`compute_angles(structure, center_type, neighbor_type, cutoff, bins)`¶

from amorphouspy import compute_angles

# O-Si-O angles (intra-tetrahedral): center=Si (14), neighbor=O (8)
bin_centers, angle_hist = compute_angles(
    structure=glass_structure,
    center_type=14,    # Atomic number of central atom (Si)
    neighbor_type=8,   # Atomic number of ligand atoms (O)
    cutoff=2.0,        # Bond cutoff (Å)
    bins=180,          # Number of histogram bins
)

Parameters:

Parameter	Type	Default	Description
`structure`	`Atoms`	—	ASE Atoms object
`center_type`	`int`	—	Atomic number of the central atom
`neighbor_type`	`int`	—	Atomic number of the ligand atoms
`cutoff`	`float`	—	Bond cutoff for center-neighbor pairs (Å)
`bins`	`int`	`180`	Number of bins for the angle histogram (1°–180°)

Returns: A tuple (bin_centers, angle_hist):

Variable	Type	Description
`bin_centers`	`np.ndarray`	Bin centers in degrees
`angle_hist`	`np.ndarray`	Normalized angle histogram

Example: Complete bond angle analysis¶

from amorphouspy import compute_angles
import plotly.graph_objects as go

# Intra-tetrahedral: O-Si-O (center=Si=14, neighbor=O=8)
o_si_o_bins, o_si_o_hist = compute_angles(glass_structure, center_type=14, neighbor_type=8, cutoff=2.0)

# Inter-tetrahedral: Si-O-Si (center=O=8, neighbor=Si=14)
si_o_si_bins, si_o_si_hist = compute_angles(glass_structure, center_type=8, neighbor_type=14, cutoff=2.0)

fig = go.Figure()
fig.add_trace(go.Scatter(x=o_si_o_bins, y=o_si_o_hist, name="O-Si-O"))
fig.add_trace(go.Scatter(x=si_o_si_bins, y=si_o_si_hist, name="Si-O-Si"))
fig.update_layout(xaxis_title="Angle (°)", yaxis_title="P(θ)")
fig.show()

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.