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 defined per bond: the ring belonging to a given T-O bond is the shortest closed path that contains it. This is the shortest-path criterion, and it is weaker than the primitive (King/Franzblau) criterion — a Guttman ring may still have a shortcut across it, so the two definitions do not select the same set of rings.

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. Every ring alternates T and O, so an n-membered ring contains 2n atoms.

For example, in SiO₂: - A 2-membered ring is two Si sharing a polyhedron edge: Si-O-Si-O, two tetrahedra bridged by two oxygens - A 3-membered ring contains 3 Si atoms connected by bridging oxygens: Si-O-Si-O-Si-O - A 6-membered ring (most common in vitreous silica) contains 6 Si atoms

Algorithm

The search runs on the bipartite T-O atom network — formers and oxygens are both nodes, and the only edges are T-O bonds:

1. Build the T-O network from the neighbour list, keeping the minimum-image vector of every bond. Nodes with fewer than two bonds are stripped iteratively, which removes non-bridging oxygens and dangling formers before any search starts.
2. For every bond (u, v): suppress it and sweep breadth-first from u, stopping as soon as the level containing v is complete. Every shortest path found closes the bond into a candidate ring.
3. Keep only candidates that close in real space: the minimum-image bond vectors summed around the loop must vanish. A path that leaves the cell and re-enters through the opposite face returns to the same atom index but is a helix, not a ring. If every shortest closure through a bond fails this test, the search deepens by two atoms at a time until it finds the shortest one that does close.
4. Canonical ring forms (rotation- and reflection-invariant, over the full T-O cycle) prevent double-counting.

Working at the atom level rather than on a contracted T-T graph matters physically. An oxygen bonded to three or more formers — a tricluster, common in aluminosilicates and borosilicates — is a single node, so it cannot be traversed twice and does not masquerade as a small ring. Two formers bridged by two distinct oxygens form a genuine four-node cycle, so edge-sharing polyhedra are detected instead of collapsing onto one edge.

Physical Significance

Ring statistics connect structure to properties:

Ring size	Structural feature
2-membered	Edge-sharing polyhedra — two formers bridged by two oxygens. Rare in equilibrium silicates; a marker of strongly non-equilibrium or high-pressure structures
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.properties.structural.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. It bounds the breadth-first sweep itself, so raising it costs almost nothing — the sweep still stops at each ring's own radius
n_cpus	int	1	Worker processes. Only networks above ~100 000 T-O bonds benefit; below that the search stays sequential regardless, because starting workers costs more than it saves

Returns: (histogram, mean_ring_size) where:

Value	Type	Description
histogram	dict[int, float]	Mapping from ring size to ring count. The smallest reportable size is 2
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.properties.structural.rings import generate_bond_length_dict

bond_lengths = generate_bond_length_dict(
    glass_structure,
    specific_cutoffs={('Si', 'O'): 1.8},
    default_cutoff=0.0,   # 0.0 (the default) marks pairs as never bonded, e.g. T-T and O-O
)

A cutoff of 0.0 means "these two species never bond", so only the pairs named in specific_cutoffs can form an edge in the ring graph. Any non-positive value is treated the same way, so an explicit -1.0 still works, but 0.0 is preferred: it reads as a zero bonding radius rather than a negative distance.

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	0.0	Fallback for unspecified pairs. Non-positive values mark a pair as never bonded, so the default excludes every pair not named in specific_cutoffs

---

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