\(Q^n\) Distribution & Network Connectivity¶

The \(Q^n\) distribution characterizes the connectivity of network-forming polyhedra (typically tetrahedra) in oxide glasses by counting bridging oxygens (BOs).

Theory¶

\(Q^n\) Species¶

Each network-forming cation (e.g., Si, Al, B in 4-fold coordination) is bonded to a certain number of oxygen atoms. Some of these oxygens are bridging (shared with another network former) and some are non-bridging (NBOs, bonded to only one former).

The \(Q^n\) label indicates the number of bridging oxygens:

Species	Bridging O	Non-bridging O	Network role
\(Q^0\)	0	4	Isolated tetrahedron
\(Q^1\)	1	3	Chain end
\(Q^2\)	2	2	Chain middle
\(Q^3\)	3	1	Sheet/branching
\(Q^4\)	4	0	Fully cross-linked (vitreous silica)

Bridging Oxygen Definition¶

An oxygen is classified as bridging if it is bonded to \(\geq 2\) network-forming cations (within the specified cutoff distance).

Network Connectivity¶

The network connectivity (NC) is the average number of bridging oxygens per network former:

\[ \text{NC} = \frac{\sum_{n=0}^{4} n \cdot f(Q^n)}{\sum_{n=0}^{4} f(Q^n)} = \sum_{n=0}^{4} n \cdot x(Q^n) \]

where \(f(Q^n)\) is the fraction of formers with \(n\) bridging oxygens and \(x(Q^n)\) is the normalized \(Q^n\) distribution.

Typical values: - Pure SiO₂ glass: NC ≈ 4.0 (all \(Q^4\)) - 75SiO₂-25Na₂O: NC ≈ 3.0 (mainly \(Q^3\) and \(Q^4\)) - 50SiO₂-50Na₂O: NC ≈ 2.0 (mainly \(Q^2\) and \(Q^3\))

The NC provides a single number that correlates with many glass properties including viscosity, fragility, glass transition temperature, and elastic moduli.

Usage¶

`compute_qn(structure, cutoff, former_types, o_type)`¶

from amorphouspy import compute_qn, compute_network_connectivity

# Compute Qn distribution (atomic numbers: Si=14, O=8)
total_qn, partial_qn = compute_qn(
    structure=glass_structure,
    cutoff=2.0,          # Si-O bond cutoff (Å)
    former_types=[14],   # Atomic numbers of network formers (Si)
    o_type=8,            # Atomic number of oxygen
)

# total_qn: {n: count} e.g. {4: 180, 3: 18, 2: 2}
# Compute network connectivity from Qn distribution
nc = compute_network_connectivity(total_qn)
print(f"Network connectivity: {nc:.2f}")

Parameters:

Parameter	Type	Default	Description
`structure`	`Atoms`	—	ASE Atoms object
`cutoff`	`float`	—	Bond cutoff distance for former-O bonds (Å)
`former_types`	`list[int]`	—	Atomic numbers of network formers (e.g., `[14, 13]` for Si, Al)
`o_type`	`int`	—	Atomic number of oxygen (typically `8`)

Returns: A tuple (total_qn, partial_qn):

Variable	Type	Description
`total_qn`	`dict[int, int]`	Total \(Q^n\) distribution: mapping from \(n\) to count
`partial_qn`	`dict[int, dict[int, int]]`	Per-former-type \(Q^n\): former atomic number →

Multi-former analysis¶

When multiple network formers are present, the \(Q^n\) distribution is computed separately for each former species:

total_qn, partial_qn = compute_qn(
    structure=glass_structure,
    cutoff=2.2,
    former_types=[14, 13],  # Si=14, Al=13
    o_type=8,
)

# partial_qn keyed by atomic number:
# {14: {4: 120, 3: 30, ...}, 13: {4: 80, 3: 40, ...}}

Note: The cutoff should be chosen to match the first minimum in the former-O RDF. For Si-O this is typically ~2.0 Å; for Al-O it is ~2.2 Å. Using a single cutoff for multiple formers is an approximation — choose a value that works reasonably for all.

Intermediates as formers: Elements like Al, Ti, and Zr are classified as network intermediates but are passed in former_types and treated identically to true formers for \(Q^n\) analysis. This is consistent with their role in the glass network (e.g. AlO₄ tetrahedra in charge-balanced alumino-silicates).

Relationship to Composition¶

For binary alkali silicate glasses \(x\text{M}_2\text{O} \cdot (1-x)\text{SiO}_2\), the theoretical \(Q^n\) distribution can be predicted from composition using the binary model:

\[ \text{NC} = 4 - \frac{2x}{1-x} \]

This assumes each M₂O adds one non-bridging oxygen, converting one \(Q^n\) → \(Q^{n-1}\).

Deviations from this model (measured by the actual \(Q^n\) distribution) reveal important structural features: - Disproportionation: \(2Q^3 \to Q^2 + Q^4\) indicates local unmixing - Aluminum avoidance: Al prefers \(Q^4\) to satisfy Loewenstein's rule