Moyo conventions of standardized layer cell

[!NOTE] This document describes how the standardized layer cell MoyoLayerDataset.std_cell and the primitive standardized layer cell MoyoLayerDataset.prim_std_cell are specified in Moyo. The conventions described here apply uniformly to the core Rust implementation as well as all other language bindings.

For the bulk space-group standardization, see Moyo conventions of standardized cell. The layer-group convention deviates from the bulk one in two places: the third basis vector is the aperiodic stacking direction (not a lattice translation), and the only centerings are p (primitive) and c (rectangular-centered).

Basis vectors and the aperiodic axis

Three basis vectors \((\mathbf{a}_s, \mathbf{b}_s, \mathbf{c}_s)\) in the same column-vector convention as the input cell.

• \(\mathbf{c}_s\) is the aperiodic stacking direction. By construction it is perpendicular to \(\mathbf{a}_s\) and \(\mathbf{b}_s\).

• The magnitude of \(\mathbf{c}_s\) is preserved from the input: \(|\mathbf{c}_s| = |\mathbf{c}|\). Moyo never rescales \(\mathbf{c}\).

setting option in MoyoLayerDataset::new

A setting selects the representative of each layer-group type. Moyo supports three settings, mirroring the bulk space-group case but using layer Hall numbers (1 - 116) instead of bulk Hall numbers (1 - 530):

• LayerSetting.Standard: the BCS / ITE convention. Uses cell choice 1 for monoclinic-oblique LGs with multiple cell choices in ITE (LG 5, 7), origin choice 2 for centrosymmetric LGs with two ITE origins (LG 52, 62, 64), monoclinic-rectangular :a axis labelling (LG 8-18), and orthorhombic abc axis labelling (LG 19-48).

• LayerSetting.Spglib: the smallest layer Hall number for each LG (spglib’s first row per LG). Differs from Standard only at LG 52, 62, 64 (origin choice 1 vs 2).

• LayerSetting.HallNumber: allows users to specify a layer Hall number (an integer from 1 to 116) directly.

Moyo chooses LayerSetting.Standard as the default. This change of default behaviour from spglib’s affects the centrosymmetric LGs 52, 62, 64 (origin choice 2 vs 1).

rotate_basis option in MoyoLayerDataset::new

When rotate_basis=true (default), Moyo rotates the standardized basis so that:

• \(\mathbf{c}_s\) lies along Cartesian \(z\),

• \(\mathbf{a}_s\) lies along Cartesian \(x\),

• \(\mathbf{b}_s\) lies in the \(xy\)-plane.

When rotate_basis=false, the basis directions inherit the input orientation, modulo the LG-canonical relabelling of \((\mathbf{a}, \mathbf{b})\).

Standardized cell with setting=LayerSetting.Standard, rotate_basis=true, and right-handed input basis vectors

Let \(\mathbf{P}_{\mathrm{std}}\) be MoyoLayerDataset.std_linear and \(\mathbf{p}_{\mathrm{std}}\) be MoyoLayerDataset.std_origin_shift. The transformation \((\mathbf{P}_{\mathrm{std}}, \mathbf{p}_{\mathrm{std}})\) brings the input layer cell into the standardized layer cell (MoyoLayerDataset.std_cell).

Let \(\mathbf{A}\) be the column-wise basis vectors of the input layer cell and \(\mathbf{A}_{\mathrm{std}}\) be the column-wise basis vectors of the standardized layer cell. When rotate_basis=true, the following relation holds:

\[ \mathbf{A}_{\mathrm{std}} = \mathbf{R} \mathbf{A} \mathbf{P}_{\mathrm{std}}, \]

where \(\mathbf{R}\) is MoyoLayerDataset.std_rotation_matrix, a proper rotation matrix that brings the input basis into the orientation described above.

The third column of \(\mathbf{A}_{\mathrm{std}}\) — that is, \(\mathbf{c}_s\) — is parallel to the input \(\mathbf{c}\) and equal in length: \(|\mathbf{c}_s| = |\mathbf{c}|\).

The standardized layer-cell basis vectors \(\mathbf{A}_{\mathrm{std}}\) are defined for each LG crystal system as the following table.

LG crystal system	“Conventional” basis vectors \(\mathbf{A}_{\mathrm{std}}\)	Additional conditions
Triclinic / oblique	\(\begin{pmatrix} a_x & b_x & 0 \\ 0 & b_y & 0 \\ 0 & 0 & c \end{pmatrix}\)	2D Minkowski reduced [1]; \(a_x, b_y, c \gt 0\) [2]
Monoclinic / oblique	\(\begin{pmatrix} a_x & b_x & 0 \\ 0 & b_y & 0 \\ 0 & 0 & c \end{pmatrix}\)	2D Minkowski reduced [1]; \(a_x, b_y, c \gt 0\) [2]
Monoclinic / rectangular	\(\begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix}\)	\(a, b, c \gt 0\) [2] [3]
Orthorhombic	\(\begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix}\)	\(a, b, c \gt 0\) [2]
Tetragonal	\(\begin{pmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & c \end{pmatrix}\)	\(a, c \gt 0\) [2]
Trigonal / hexagonal	\(\begin{pmatrix} a & -a/2 & 0 \\ 0 & \sqrt{3} a / 2 & 0 \\ 0 & 0 & c \end{pmatrix}\)	\(a, c \gt 0\) [2]

Which of \(\mathbf{a}\) or \(\mathbf{b}\) carries the 2-fold or mirror normal in the monoclinic-rectangular case is a labelling choice exposed via LayerSetting (paper Fu et al. 2024 Table 5 codes :a, :b; analogously :abc / :b-ac for orthorhombic). It is not part of the reduction step.

Centering and the primitive standardized cell

Layer groups admit only two centerings: p (primitive) and c (rectangular-centered, in-plane only). There is no body, face, or rhombohedral centering. The conventional cell may double the primitive 2D cell area in the c case, but never extends along the aperiodic axis.

Let \(\mathbf{Q}\) be the transformation matrix from the primitive layer cell to the conventional layer cell:

LG Bravais class	Centering	\(\mathbf{Q}\)	\(\mathbf{Q}^{-1}\)
mp, op, tp, hp	p	\(\mathbf{Q}_p = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\)	\(\mathbf{Q}_p^{-1} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\)
oc	c	\(\mathbf{Q}_c = \begin{pmatrix} 1 & -1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\)	\(\mathbf{Q}_c^{-1} = \begin{pmatrix} 1/2 & 1/2 & 0 \\ -1/2 & 1/2 & 0 \\ 0 & 0 & 1 \end{pmatrix}\)

Note that \(\mathbf{Q}_c\) and \(\mathbf{Q}_c^{-1}\) leave the third column (the aperiodic axis) untouched.

Let \(\mathbf{P}_{\mathrm{prim}}\) be MoyoLayerDataset.prim_std_linear and \(\mathbf{p}_{\mathrm{prim}}\) be MoyoLayerDataset.prim_std_origin_shift. The transformation \((\mathbf{P}_{\mathrm{prim}}, \mathbf{p}_{\mathrm{prim}})\) brings the input layer cell into the primitive standardized layer cell (MoyoLayerDataset.prim_std_cell):

\[ (\mathbf{P}_{\mathrm{prim}}, \mathbf{p}_{\mathrm{prim}}) = (\mathbf{P}_{\mathrm{std}}, \mathbf{p}_{\mathrm{std}}) (\mathbf{Q}, \mathbf{0})^{-1}. \]

Pearson symbol

MoyoLayerDataset.pearson_symbol is built as <2D Bravais type><N>, where the Bravais type is one of mp, op, oc, tp, hp and N is the conventional-cell atom count.

Atom positions

Atom positions in MoyoLayerDataset.std_cell and MoyoLayerDataset.prim_std_cell are symmetrised to the orbits prescribed by the identified layer group, mirroring the bulk space-group standardisation.

---

[1] (1,2)

Applied regardless of rotate_basis value.

[2] (1,2,3,4,5,6)

\(c < 0\) for left-handed input basis vectors.

[3]

For monoclinic-rectangular LGs (LG 8-18) the in-plane lattice is rectangular by Bravais constraint, so 2D Minkowski reduction is trivial (length sort / sign fixing only).