From: Accurate lattice parameters from 2D-periodic images for subsequent Bravais lattice type assignments
Bravais lattice types’ names and standard abbreviation letters [25] | Parameters of the conventional unit cells | Limiting cases of lattice parameters that lead from a lower symmetric Bravais lattice type to its higher symmetric counterpart(s), also known as metric specializations in 2D | Conventional basis vectors (a′, b′) for limiting cases [29] | Number of geometric or symmetry constraints that enter the G-AIC for the assignment of Bravais lattice types to sets of lattice parameters [17] |
---|---|---|---|---|
Hexagonal, hp | a = b, γ = 120° | None as top of a hierarchy branch is reached | None as there is no such case | Four, e.g., opposite sides are parallel, diagonals are orthogonal and posses ratios of \( \sqrt 3 \) or \( \sqrt 3^{ - 1} \) (equivalent to lattice defining angle 120°) |
Square, tp | a = b, γ = 90° | None as top of a hierarchy branch is reached | None as there is no such case | Four, e.g., opposite sides are parallel, adjacent sides are orthogonal, and diagonals are both orthogonal and of equal length (equivalent to lattice defining angle 90°) |
Rectangular centered, oc | a < b ≠ a\( \sqrt 3 \), γ = 90° For primitive sub-unit cell: a♦ = b♦, γ♦ ≠ 60° or 90° | a = b → tp a\( \sqrt 3 \) = b → hp For primitive sub-unit cell: γ♦ = 60° → hp γ♦ = 90° → tp | (a + b)/2, (b − a)/2 in both cases For primitive sub-unit cell: a♦, b♦ in both cases | Three, e.g., opposite sides are parallel and adjacent sides are orthogonal while conventional unit cells encompass 2 lattice points For primitive sub-unit cell: opposite sides are parallel and diagonals are orthogonal but of different lengths so that their ratio is never 1, \( \sqrt 3 \), or \( \sqrt 3^{ - 1} \) |
Rectangular (primitive), op | a < b, γ = 90° | a = b → tp | a, b | Three, e.g., opposite sides are parallel and adjacent sides are orthogonal |
Oblique, mp | − 2b cos γ < a < b, 90° < γ | γ = 90° → op − 2b cos γ = a → oc a = b → oc | a, b a, 2b + a (a′\( \sqrt 3 \) < b′) a + b, b − a (a′\( \sqrt 3 \) > b′) | Two, e.g., (two) opposite sides are parallel |