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Enumerate at fixed concentration

Restrict enumerate to labelings with a prescribed concentration (e.g., 4 A + 4 B in an 8-cell binary alloy).

Setup

julia> using Enumlib

julia> p = ParentLattice([0.0 0.5 0.5; 0.5 0.0 0.5; 0.5 0.5 0.0]);   # FCC

julia> sites = Sites([Site([0.0, 0.0, 0.0], [0, 1])]);              # binary

Steps

Pass a Concentration via the concentration kwarg. The constructor of choice depends on what you're holding in your head:

julia> c = concentration_count([4, 4]; n_total = 8);   # 4 A + 4 B in an 8-cell

julia> e = enumerate(p, sites; supercells = VolumeRange(8:8), concentration = c);

julia> length(e)
94

The canonical HF 2012 reference: 94 symmetry-inequivalent labelings of 8-site FCC supercells at 50/50 binary concentration.

Other constructors

When you care about a ratio rather than literal counts at a specific cell size, use concentration_ratio — scale-free, reduces the input to lowest terms:

julia> concentration_ratio([2, 4])     # 2:4 reduces to 1:2; total is 3 so 1/3:2/3
Concentration(1//3, 2//3)

When you already have fractions in hand, the canonical Concentration constructor accepts them directly:

julia> Concentration([1//4, 3//4])
Concentration(1//4, 3//4)

Behavior notes

  • Divisibility: the concentration's fractions must each multiply by the supercell volume to an integer. For example, Concentration([1//3, 2//3]) works at volumes that are multiples of 3 (3, 6, 9, …) but not at volume 4. Volumes that don't divide cleanly are silently skipped — enumerate returns whatever structures the compatible volumes produced.

    Caveat: if every volume in the range is incompatible, you get back an Enumeration of length 0 with no error raised. Calling multiplicities(c, n) directly does throw an EmptyEnumerationError — only the enumerate(...) path catches it per-volume.

  • :auto algorithm dispatch picks :multinomial for fixed concentration, or :recursive_stabilizer if the multinomial bitmap would exceed memory_budget × 0.8. See Pick an algorithm.

  • Super-periodic labelings are dropped by default. See Handle super-periodicity.

See also