Ring Statistics¶

Ring analysis determines the distribution of closed loops in the atomic network, revealing medium-range order that is invisible to pair correlation functions like the RDF.

Theory¶

Guttman Rings¶

A Guttman ring is a closed path through the T-O-T network that satisfies the primitiveness (shortest-path) criterion: no shortcut exists through the rest of the network between any two non-adjacent ring nodes. Specifically, for every pair of non-adjacent ring atoms, the shortest path in the full network graph equals their arc distance along the ring.

The ring size is counted in terms of the number of network-forming cation nodes (T atoms, e.g. Si, Al) — not total atoms — in the loop.

For example, in SiO₂: - A 3-membered ring contains 3 Si atoms connected by bridging oxygens: Si-O-Si-O-Si-O-Si - A 6-membered ring (most common in vitreous silica) contains 6 Si atoms

Algorithm¶

The implementation is a pure-Python networkx-based BFS approach:

Build a T-T connectivity graph where two network formers share an edge if they are both bonded to the same bridging oxygen (coordination ≥ 2).
For every edge (u, v): temporarily remove it, find all shortest paths from u back to v, restore the edge.
Each candidate ring is tested against the Guttman primitiveness criterion using shortest-path lengths in the full graph.
Canonical ring forms (rotation- and reflection-invariant) prevent double-counting.

Physical Significance¶

Ring statistics connect structure to properties:

Ring size	Structural feature
3-membered	Associated with the D₂ Raman band (~606 cm⁻¹) in SiO₂
4-membered	Associated with the D₁ Raman band (~492 cm⁻¹) in SiO₂
5–7	Dominant in vitreous silica; peak at 6
Large (>8)	Less strained; common in open network structures

Small rings (3, 4) are energetically strained but kinetically trapped during the quench. Their population is sensitive to: - Cooling rate (faster quench → more small rings) - Composition (modifiers break rings) - Temperature (high T → more small rings)

Usage¶

`compute_guttmann_rings(structure, bond_lengths, max_size)`¶

from amorphouspy.analysis.rings import compute_guttmann_rings, generate_bond_length_dict

# Generate bond length cutoffs for all element pairs
bond_lengths = generate_bond_length_dict(
    glass_structure,
    specific_cutoffs={('Si', 'O'): 1.8, ('Al', 'O'): 1.95},
    default_cutoff=2.0,
)

# Compute ring statistics
histogram, mean_size = compute_guttmann_rings(
    structure=glass_structure,
    bond_lengths=bond_lengths,
    max_size=12,
)

print(f"Mean ring size: {mean_size:.2f}")
print(histogram)
# Example: {3: 12, 4: 45, 5: 120, 6: 210, 7: 98, 8: 30}

Parameters:

Parameter	Type	Default	Description
`structure`	`Atoms`	—	ASE Atoms object
`bond_lengths`	`dict[tuple[str, str], float]`	—	Cutoff distances per element pair in Å
`max_size`	`int`	`24`	Maximum ring size (number of T atoms) to search for

Returns: (histogram, mean_ring_size) where:

Value	Type	Description
`histogram`	`dict[int, int]`	Mapping from ring size to ring count
`mean_ring_size`	`float`	Mean ring size weighted by count

`generate_bond_length_dict(atoms, specific_cutoffs, default_cutoff)`¶

Generates all symmetric element-pair combinations from the structure and assigns cutoff values.

from amorphouspy.analysis.rings import generate_bond_length_dict

bond_lengths = generate_bond_length_dict(
    glass_structure,
    specific_cutoffs={('Si', 'O'): 1.8},
    default_cutoff=-1.0,   # -1.0 marks pairs to ignore (e.g. T-T, O-O)
)

Parameters:

Parameter	Type	Default	Description
`atoms`	`Atoms`	—	ASE Atoms object (determines element set)
`specific_cutoffs`	`dict` or `None`	`None`	Per-pair cutoff overrides
`default_cutoff`	`float`	`-1.0`	Fallback for unspecified pairs; negative values are ignored by the ring finder

Typical Results¶

Vitreous SiO₂ (MD simulation)¶

Ring size	Count (fraction)
3	~1–3%
4	~5–10%
5	~20–25%
6	~30–35% (peak)
7	~15–20%
8	~5–10%
9+	~2–5%

Effect of modifiers¶

Adding network modifiers (Na₂O, CaO) to SiO₂: - Reduces the average ring size - Broadens the distribution - Decreases the 6-membered ring population - Can increase the fraction of small (3, 4) rings in some compositions

References¶

Guttman, L. Ring structure of the crystalline and amorphous forms of silicon dioxide. J. Non-Cryst. Solids 116, 145–147 (1990). https://doi.org/10.1016/0022-3093(90)90686-G