Reciprocal lattice and Brillouin zone

The real-space picture — points, bonds, sublattices — is half the story. The other half lives in momentum space: the reciprocal lattice, the Brillouin zone, and the structure factor. Everything LatticeCore exposes under reciprocal_lattice, monkhorst_pack, BraggPeakSet, and structure_factor ultimately comes from the content of this column.

The reciprocal lattice

Given a Bravais lattice with real-space basis $\mathbf{a}_1, \dots, \mathbf{a}_d$, the reciprocal basis $\mathbf{b}_1, \dots, \mathbf{b}_d$ is defined by the orthogonality relation

\[\mathbf{a}_i \cdot \mathbf{b}_j = 2\pi \, \delta_{ij}.\]

Writing $A$ for the real-space basis matrix and $B$ for the reciprocal one,

\[B = 2\pi \, (A^{-\top}).\]

The reciprocal lattice is then the set

\[\mathbf{G} = m_1 \mathbf{b}_1 + \dots + m_d \mathbf{b}_d, \qquad m_i \in \mathbb{Z}.\]

Reciprocal lattice vectors are distinguished by a single property: for every real-space lattice point $\mathbf{R}$,

\[e^{-i \mathbf{G} \cdot \mathbf{R}} = 1.\]

In other words, $\mathbf{G}$ and $\mathbf{G} + \mathbf{b}_i$ produce identical plane waves sampled at the crystal points. Up to reciprocal lattice translations, only $\mathbf{k}$ values in a single primitive cell of reciprocal space are physically distinct. That primitive cell is the Brillouin zone.

In LatticeCore, reciprocal_basis(ml) returns the $B$ matrix for a PeriodicMomentumLattice, and the orthogonality relation is one of the package's unit tests: a_i · b_j ≈ 2π δ_ij is checked directly against the reference square lattice.

The Brillouin zone

The first Brillouin zone is the Wigner–Seitz cell of the reciprocal lattice: the set of k-points that are closer to the origin $\Gamma$ than to any other reciprocal lattice vector. For the square lattice it is the square $(-\pi, \pi] \times (-\pi, \pi]$; for the triangular lattice it is a hexagon; for the honeycomb, a hexagon rotated by 30°.

The Brillouin zone matters because every physical quantity that depends on momentum is periodic in reciprocal space. Band structures, structure factors, form factors — they all repeat with period $\mathbf{G}$, so all the independent information fits inside a single BZ.

LatticeCore's reference implementations do not enumerate Wigner–Seitz cells (that is topology-specific work for Lattice2D.jl). What they do provide is a uniform sampler of the BZ interior — the monkhorst_pack mesh.

Sampling: Monkhorst–Pack and Γ-centred grids

Monte Carlo simulations on a finite $L_1 \times L_2 \times \dots \times L_d$ lattice can only resolve a finite set of k-points, namely those compatible with the sample's periodicity:

\[\mathbf{k} \in \left\{ \frac{n_i}{L_i} \mathbf{b}_i \right\}_{n_i = 0}^{L_i - 1}.\]

LatticeCore ships two equivalent ways to enumerate these points:

gamma_centered(basis, mesh) uses $n_i / L_i$, including the $\Gamma$ point exactly.
monkhorst_pack(basis, mesh) uses $(n_i + 1/2)/L_i - 1/2$, which centres the mesh on $\Gamma$ but avoids it. This is the convention solid-state codes use for integrating smooth functions over the BZ.

Both return the same number of k-points, $\prod_i L_i$, matching the number of sites in the corresponding real-space sample.

The structure factor

Given a scalar field $s_i$ defined on the lattice points, the structure factor is the modulus squared of its spatial Fourier transform:

\[S(\mathbf{k}) \;=\; \frac{1}{N}\, \left| \sum_{i = 1}^{N} s_i \, e^{-i \mathbf{k} \cdot \mathbf{r}_i} \right|^{2}.\]

It tells you, at a single configuration, how much Fourier weight sits at each momentum. Three sanity checks encode the physics:

Ferromagnet, $s_i \equiv 1$. The sum is simply $N$, so $S(0) = N$, and — crucially — $S(\mathbf{G}) = N$ for every reciprocal lattice vector, because every such $\mathbf{G}$ is equivalent to $\Gamma$. Away from $\mathbf{G}$ the sum over sites vanishes by destructive interference. This is verified in LatticeCore's test suite.
Néel antiferromagnet on a bipartite lattice, $s_i = (-1)^{\mathbf{x} + \mathbf{y}}$ on a square. The sum peaks at $\mathbf{k} = (\pi, \pi)$, giving $S(\pi, \pi) = N$ and $S(0) = 0$. Bipartiteness is exactly the condition that this momentum lies on the reciprocal lattice.
Disorder gives $S(\mathbf{k}) \approx 1$ on average — a useful null baseline when looking for order.

Those checks are what LatticeCore's test_structure_factor.jl actually runs against structure_factor.

Implementation notes

LatticeCore uses the naive O(N) per k-point formula above because it is transparent, correct on any lattice — including quasicrystals, which have no FFT-friendly mesh — and fast enough for the reference implementations. Production-grade codes on regular meshes can specialise structure_factor for FFT-based evaluation without changing the API.

For quasicrystals, $\mathbf{k}$ is no longer drawn from a discrete reciprocal lattice but from a dense Fourier module projected from a higher-dimensional periodic lattice. LatticeCore ships the type skeletons HyperReciprocalLattice and BraggPeakSet; the enumeration algorithm itself lives in QuasiCrystal.jl. See Quasiperiodic order for the column on this.