Glossary

Compact reference for terms used across the documentation, with pointers to where each one is unpacked in depth. The Enumlib-specific terms come first; the Materials-science background section at the bottom collects the field acronyms (FCC, DFT, POSCAR, …) that the tutorials and how-to pages use as shorthand.

Bitmap

The dense bit array the concentration-restricted algorithms use to mark which colorings have already been visited: one bit per candidate coloring, indexed by its mixed-radix hash. Fast and cache-friendly, but its size is fixed by the number of candidate colorings, which is why :auto refuses it when that exceeds the memory budget and falls back to the recursive-stabilizer algorithm. See multinomial-2012.

Burnside's lemma

The counting identity underneath every orbit count here: the number of orbits equals the average number of colorings left unchanged by a group element, (1/|G|) Σ_g |fix(g)|. Also called the orbit-counting theorem; the Pólya count is this average evaluated in closed form rather than by enumeration. See polya-counting and Burnside's lemma.

Bravais lattice

A periodic point lattice with one site per primitive cell. ParentLattice(A) with no dset argument (or a single-element dset) is Bravais. Contrast with multilattice.

Coloring

The atom-species vector for one supercell — a Vector{Int8} of indices in {0, 1, ..., k-1}. The accessor function is to_labeling; "coloring" and "labeling" are interchangeable in the literature, but we use "coloring" in body prose to free up "labeling" for API contexts. Contrast with configuration (the supercell-plus-coloring pair).

Concentration

The composition of a coloring: the fraction (or count) of each species. See Concentration and the concentration-and-multiplicity explanation.

Configuration

A supercell paired with a coloring — what an EnumeratedStructure value represents. The output of enumerate(...) is a list of symmetry-inequivalent configurations. We avoid the word structure in body prose because of the collision with Julia's struct and the type name EnumeratedStructure; the original HF 2008 paper calls these derivative structures.

Cycle index

The polynomial that records the cycle structure of every element of a permutation group. Evaluating it at the right arguments produces a coloring count — with all k species free it collapses to k raised to the number of cycles; at fixed composition it becomes a coefficient-extraction problem. See polya-counting.

Degeneracy

Synonym for orbit_size — how many distinct colorings collapse into one orbit. Also called multiplicity. Reported per structure in struct_enum.out and carried on EnumeratedStructure.

Derivative structure

The literature name (HF 2008 paper title) for what we call a configuration. Same thing.

dset (basis sites)

For a multilattice, the list of basis-site fractional coordinates in the primitive cell. For FCC, dset = [(0,0,0)] (single site). For HCP, dset = [(0,0,0), (1/3, 2/3, 1/2)] (two sites).

Enumeration

The full set of symmetry-inequivalent derivative structures for a given (parent, sites, supercell selection) tuple. Returned by enumerate(...); see Enumeration.

Exhaustive algorithm

Iterates k^n colorings on each supercell, hashes each into the orbit-canonical representative, emits one structure per orbit. The simplest and most direct algorithm; not :auto's default. See exhaustive-2008.

HF 2008

Hart & Forcade, Algorithm for generating derivative structures, PRB 77, 224115 (2008). The original derivative-structure enumeration algorithm — single-lattice case.

HF 2009

Hart & Forcade, Generating derivative structures from multilattices, PRB 80, 014120 (2009). The multilattice extension that Enumlib implements for uniform-sublattice (Regime B) inputs.

HF 2012

Hart, Nelson, & Forcade, Generating derivative structures at a fixed concentration, Comp. Mat. Sci. 59, 101 (2012). Adds the multinomial mixed-radix hash for concentration-restricted enumeration.

HNF (Hermite Normal Form)

An integer-matrix canonical form for a supercell relative to its parent. Each HNF uniquely identifies a supercell shape up to parent symmetry. See HNF.

Labeling

Synonym for coloring — the API uses this name (the accessor function is to_labeling). Body prose uses coloring; both refer to the same Vector{Int8}.

Mixed-radix hash

The indexing scheme that turns a fixed-composition coloring into a dense integer with no gaps, so it can address a bitmap directly. "Mixed radix" because each position's place value depends on how many of each species remain, rather than being a constant base. From HF 2012. See multinomial-2012.

Möbius inversion

The bookkeeping that turns "at least" counts into "exactly" counts. The two-set case is the familiar one: to count the things in A or B you add |A| and |B|, then subtract |A ∩ B| because you counted the overlap twice. With more overlapping cases the corrections keep alternating — subtract, add back, subtract again — and that alternating pattern is what combinatorics calls inclusion–exclusion.

Möbius inversion is the same idea on a partially ordered set rather than a handful of sets. Here the set is the lattice of subgroups of a supercell's translation group, and the Möbius function supplies the ± weight for each subgroup so the alternation comes out right. Enumlib needs it because "labelings fixed by at least these translations" is easy to count, while what the aperiodic (non-super-periodic) count actually wants is "labelings fixed by none". For the simplest case — a cyclic translation group of prime order — it collapses to "all labelings minus the constant ones". See polya-counting and Möbius inversion.

Multilattice

A parent lattice with two or more dset sites per primitive cell (HCP, diamond, perovskite). Contrast with Bravais lattice (single dset site).

Multinomial algorithm

The concentration-restricted bitmap algorithm. Iterates only colorings matching the target concentration via mixed-radix indexing. Best when the multinomial coefficient is much smaller than k^n. See multinomial-2012.

Orbit

An equivalence class of colorings under the supercell's permutation group. Each orbit corresponds to one symmetry-inequivalent derivative structure. orbit_size is the number of colorings in an orbit (also called multiplicity or degeneracy).

Parent Lattice

Geometric description of the parent multilattice: basis A, dset, and cached space group. See ParentLattice.

Permutation group

For a supercell, the group of position permutations induced by the parent symmetry operations that fix the supercell's lattice (rotational stabilizer) composed with the supercell's translation subgroup. The enumeration crosses out colorings under this group.

Pólya count

The Burnside-averaged count of orbits of a coloring space under the permutation group. Computed without enumerating. See polya-counting.

Recursive-stabilizer algorithm

The Morgan-Hart 2017 algorithm. Tree-walk over partial colorings; never materializes the full hash table. :auto's default for unrestricted enumeration, Regime C (see below), and any case where the bitmap doesn't fit the memory budget. See recursive-stabilizer-2017.

Regime A / B / C

A three-way split of enumeration problems by how the substitution sites are arranged, used in error messages, dispatch logic, and design notes. A — a single-site (Bravais) parent. B — a multilattice where every dset position carries the same allowed_labels ("uniform sublattices": HCP binary, diamond binary). C — a multilattice whose allowed_labels differ by dset position ("heterogeneous sublattices": zinc-blende, half- and full-Heusler, perovskite). The distinction is not cosmetic: Regime C needs the label-restricted counting formulas and a different algorithm, and rejects unrestricted (no-concentration) enumeration. See the algorithm-choice how-to.

Sites

A Sites object declares which dset positions can be substituted and which species (allowed_labels) are allowed at each. Equivalence-class ties between sites are also tracked. See Sites and the substitution-sites how-to.

SNF (Smith Normal Form)

Diagonal canonical form diag(d_1, ..., d_D) of an integer matrix with d_i | d_{i+1}. The SNF of an HNF gives the supercell's quotient-group structure Z/d_1 × ... × Z/d_D — used to index supercell sites.

Stabilizer subgroup

For a supercell, the subgroup of the parent space group whose action fixes the superlattice as a set. The supercell permutation group is built from the stabilizer's rotations composed with the supercell's translation subgroup.

Super-periodic structure

A coloring on a volume-m supercell that is also a valid coloring on some volume-m' supercell with m' | m, m' < m. By default Enumlib drops these to avoid double-counting across a volume sweep. See super-periodicity.

Supercell

A larger cell built on top of the parent lattice as A_super = A_parent · h for some integer matrix h. The HNF is the canonical representative of each shape class. See Supercell.

Union-find

The standard disjoint-set data structure, used here to merge supercell positions into joint orbits when two permutations are applied together — cheap enough that the aperiodic correction can afford to redo it for every (group element, subgroup) pair.

Supercell selection

A VolumeRange, RadiusBound, or ExplicitHNFs value passed as the supercells = ... kwarg to enumerate(...). Determines which HNFs are enumerated over. See the supercell-selection how-to.

Materials-science background

Field shorthand that the tutorials and how-to pages use without further introduction. If you arrived at Enumlib from a combinatorics / symmetry-algorithms angle rather than from solid-state chemistry, these are the ones to skim first.

BCC (body-centered cubic)

Crystallographic prototype: cubic lattice with atoms at the corners and one at the body center. In primitive form, a single-site Bravais lattice. Common in metallic alloys (α-Fe, W, Cr).

CE (cluster expansion)

A polynomial-in-occupation-variables expansion of total energy as a function of configuration. Fit to a training set of energies for symmetry-inequivalent configurations (usually from DFT) and then used downstream for Monte Carlo simulations, phase-diagram construction, ground-state searches. JuCE.jl is the cluster-expansion code that consumes Enumlib's enumerations.

DFT (density functional theory)

First-principles electronic-structure method for computing total energy, forces, stresses of a configuration. The Phase-11 POSCAR roundtrip (write_enumeration_archive → DFT calculation → read_results → attach_results) is designed for this workflow.

FCC (face-centered cubic)

Crystallographic prototype: cubic lattice with atoms at corners and face centers. In primitive form, a single-site Bravais lattice with 48-op point group. The richest-symmetry single-lattice example, and the workhorse for most reference-count comparisons in the docs. Common in metallic alloys (Cu, Al, Ag, Ni, Pt).

HCP (hexagonal close-packed)

Crystallographic prototype with a 2-atom basis on a hexagonal lattice — the simplest non-Bravais (i.e., multilattice) example. Used throughout the docs as the canonical Regime-B multilattice with uniform allowed_labels on both sublattices. Common in metallic alloys (Mg, Ti, Zn).

MLIP (machine-learning interatomic potential)

A neural-network or kernel-regression model trained on DFT data to predict energies (and often forces, stresses) much faster than DFT itself. Building the training-set of symmetry-inequivalent configurations is the Phase-11 use case alongside straight-DFT training.

POSCAR

The VASP atomic-structure input-file format. Line 1 is a free-form comment, then a scale factor, the lattice vectors, species symbols + counts, the coordinate type (Direct for fractional, Cartesian otherwise), and the per-atom coordinates. Enumlib's to_poscar / write_enumeration_archive writes POSCARs whose line 1 carries metadata (enumlib_id=, concentration=, energy_eV=, …) that read_results parses back. See write-poscars-for-dft.

PRB

Physical Review B, the journal where the Hart-Forcade 2008 and 2009 papers were published. The HF 2012 paper is in Computational Materials Science (citation in the HF 2012 glossary entry above).

VASP

The Vienna Ab initio Simulation Package — a widely-used commercial DFT code. Enumlib's POSCAR roundtrip targets VASP-5+ format (compatible with VASP 5 and VASP 6).