Geometry & Dielectric

Typed lattices, fractional atoms

A crystal enters the solver as a typed lattice (square, rectangular, triangular, hexagonal, or oblique, or explicit a1/a2 vectors) plus a list of basis atoms with fractional positions, radii, and per-atom permittivity. The reciprocal basis and the Brillouin-zone path presets (Γ-X-M-Γ and friends) derive from the same lattice object, so geometry and k-space can never disagree.

On the board, this is the gray control column on the left: it decides everything and moves almost no data.

Analytic subpixel smoothing

The dielectric build is where accuracy is won. Instead of sampling ε on a grid (staircase artifacts) or averaging over a sub-mesh (slow convergence), Blaze computes the exact circle-pixel intersection for every boundary pixel: filling fraction, interface normal, and interface length. Boundary pixels get the MPB-style anisotropic average

which is the physically correct mixing rule for fields parallel and perpendicular to the interface. This single kernel is why a 32×32 Blaze grid tracks MPB rather than lagging it.

Explore the result at every resolution the site ships (these are real solver outputs):

What it costs

The build is O(N · atoms) and runs once per job. The output is three real-space arrays of length N = nx·ny: ε(r), ε⁻¹(r), and a 2×2 inverse-tensor field for boundary pixels. There is no dense G-space matrix anywhere; the plane-wave basis is the FFT grid itself.

One honest data-movement note, visible on the board as a blue node: the driver currently clones the ε arrays for every k-point (board). That is an O(N) copy repeated k times where a shared reference would do. It is cheap relative to the solve, but it is exactly the kind of copy the board exists to make visible.

→ Continue with Matrix-Free Operators, which consume these arrays on every application.