Crystal surfaces

Tasks 9.4.1-9.4.4. References: pymatgen core.surface, analysis.wulff, analysis.adsorption; Wulff (1901); Zur and McGill (1984).

Symmetry-unique Miller indices

The point group comes from a symmetry search of the structure. Every reduced (h k l) inside the index bound is transformed by each rotation and by its negative (the Laue class always contains inversion); the family is the set of distinct transformed triples. The representative is the family member with the largest positive component count and the largest absolute components, which picks (1 0 0), (1 1 0) and (1 1 1) for a cubic crystal. Multiplicity is the size of the family and the d-spacing follows the reciprocal metric tensor.

Slab terminations

The atomic layers are the fractional heights in the surface basis (surface_basis, Task 1.8), taken modulo one. Every gap between consecutive layers defines a cut shift at the midpoint; make_slab builds the slab for each shift. A signature over (element, z, x, y) removes cuts that produce the same decoration. A symmetric slab mirrors every atom across the middle of the film. The net dipole is sum q_i z_i along the surface normal, with the oxidation states of Task 9.3.3 as charges when available, and a dipole above 1e-4 e Angstrom marks a polar surface. The minimal slab for a thickness uses ceil(thickness / d_hkl) layers.

Wulff construction

For a surface energy gamma(h k l), the facet plane lies at distance gamma / |g| from the origin, where g is the reciprocal-lattice vector of (h k l). The Wulff polyhedron is the intersection of the half-spaces n . x <= gamma / |g| for every direction; a finite set with positive energies is bounded and contains the origin. The half-space clipper of the Voronoi module reports the vertices, faces, volume and area. Each facet is matched to the direction with the largest normal cosine, and its fraction is the facet area over the total surface area. Lower energies therefore produce larger facets, as the construction requires.

Adsorption sites

The top layer is the set of atoms within a fractional tolerance of the maximum z. Top sites sit on each surface atom; bridge sites at the midpoint of two surface atoms within 1.25 * (r_i + r_j) covalent radii; hollow sites at the centroid of three pairwise-bonded surface atoms. Duplicate sites are removed by position. place_molecule translates the molecule centroid to the site plus an offset along the surface normal and appends its atoms in the slab cell.