# Bravais lattices and irreducible Brillouin zones {#app:BS}

  Bravais lattice              BZ variation   Criteria
  ---------------------------- -------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------
  Simple cubic                 CUB            $a_1=a_2=a_3$ **and** $\theta_{12}=\theta_{23}=\theta_{31}=\pi/2$
  Face-centered cubic          FCC            $a_1=a_2=a_3$ **and** $\theta_{12}=\theta_{23}=\theta_{31}=\pi/3$
  Body-centered cubic          BCC            $a_1=a_2=a_3$ **and** $\theta_{12}=\theta_{23}=\theta_{31}=\mathrm{acos}(-1/3)$
  Rhombohedral                 RHL1           $a_1=a_2=a_3$ **and** $\theta_{12}=\theta_{23}=\theta_{31} < \pi/2$
  Rhombohedral                 RHL2           $a_1=a_2=a_3$ **and** $\theta_{12}=\theta_{23}=\theta_{31} > \pi/2$
  Body-centered tetragonal     BCT1           $a_1=a_2=a_3$ **and** $\theta_{23}=\theta_{31}$ **and** $\|\mathbf{a}_2+\mathbf{a}_3\| > \|\mathbf{a}_1+\mathbf{a}_2\|$
  Body-centered tetragonal     BCT2           $a_1=a_2=a_3$ **and** $\theta_{23}=\theta_{31}$ **and** $\|\mathbf{a}_2+\mathbf{a}_3\| < \|\mathbf{a}_1+\mathbf{a}_2\|$
  Hexagonal                    HEX            $(a_1=a_2  \lor a_2=a_3 \lor a_3=a_1)$ **and** $[\theta_{12},\theta_{23},\theta_{31}] = \pi[2/3,1/2,1/2]$
  Tetragonal                   TET            $a_1 = a_2$ **and** $\theta_{12}=\theta_{23}=\theta_{31}=\pi/2$
  Orthorhombic                 ORC            $a_1 < a_2 < a_3$ **and** $\theta_{12}=\theta_{23}=\theta_{31}=\pi/2$
  Face-centered orthorhombic   ORCF1          $\Theta_{12}=\Theta_{23}=\Theta_{31}=\pi/2$ **and** $A_1 < A_2 < A_3$ **and** $1/A_1^2 > 1/A_2^2+1/A_3^2$
  Face-centered orthorhombic   ORCF2          $\Theta_{12}=\Theta_{23}=\Theta_{31}=\pi/2$ **and** $A_1 < A_2 < A_3$ **and** $1/A_1^2 < 1/A_2^2+1/A_3^2$
  Face-centered orthorhombic   ORCF3          $\Theta_{12}=\Theta_{23}=\Theta_{31}=\pi/2$ **and** $A_1 < A_2 < A_3$ **and** $1/A_1^2 = 1/A_2^2+1/A_3^2$
  Body-centered orthorhombic   ORCI           $\Theta_{12}=\Theta_{23}=\Theta_{31}=\pi/2$ **and** $A_1 < A_2 < A_3$
  Base-centered orthorhombic   ORCC           $\Theta_{12}=\Theta_{23}=\Theta_{31}=\pi/2$ **and** $A_1 < A_2$
  Monoclinic                   MCL            $a_1 < a_3$ **and** $a_2 < a_3$ **and** $\theta_{12}=\theta_{31}=\pi/2$ **and** $\theta_{23} < \pi/2$
  Base-centered monoclinic     MCLC1          $\Theta_{12}=\Theta_{31}=\pi/2$ **and** $\Theta_{23} < \pi/2$ **and** $\gamma_{12} > \pi/2$
  Base-centered monoclinic     MCLC2          $\Theta_{12}=\Theta_{31}=\pi/2$ **and** $\Theta_{23} < \pi/2$ **and** $\gamma_{12} = \pi/2$
  Base-centered monoclinic     MCLC3          $\Theta_{12}=\Theta_{31}=\pi/2$ **and** $\Theta_{23} < \pi/2$ **and** $\gamma_{12} < \pi/2$ **and** $A_2/A_3\cos(\Theta_{23})$ $+ A_2^2/A_1^2\sin(\Theta_{23})^2 < 1$
  Base-centered monoclinic     MCLC4          $\Theta_{12}=\Theta_{31}=\pi/2$ **and** $\Theta_{23} < \pi/2$ **and** $\gamma_{12} < \pi/2$ **and** $A_2/A_3\cos(\Theta_{23})$ $+ A_2^2/A_1^2\sin(\Theta_{23})^2 = 1$
  Base-centered monoclinic     MCLC5          $\Theta_{12}=\Theta_{31}=\pi/2$ **and** $\Theta_{23} < \pi/2$ **and** $\gamma_{12} < \pi/2$ **and** $A_2/A_3\cos(\Theta_{23})$ $+ A_2^2/A_1^2\sin(\Theta_{23})^2 > 1$

  : Type of lattices recognized by RESCU. The lattice vectors are
  designated by $\mathbf{a}_1$, $\mathbf{a}_2$ and $\mathbf{a}_3$ and
  the reciprocal lattice vectors by $\mathbf{b}_1$, $\mathbf{b}_2$ and
  $\mathbf{b}_3$. Lower case letters refer to primitive cell vectors
  while capital letters refer to conventional cell vectors.

This section complements `sec:BS`{.interpreted-text role="ref"} which
introduced in passing the methodology to identify Brillouin zone
variations. We begin by listing the Brillouin zone variations recognized
by RESCU along with the criteria used in the identification in
`tab:bzTypesFull`{.interpreted-text role="numref"}. We use the following
convention for the angles

$$\begin{aligned}
\begin{aligned}
\theta_{12}=\mathbf{a}_1\angle\mathbf{a}_2\\
\theta_{23}=\mathbf{a}_2\angle\mathbf{a}_3\\
\theta_{31}=\mathbf{a}_3\angle\mathbf{a}_1\end{aligned}
\end{aligned}$$

and

$$\begin{aligned}
\begin{aligned}
\gamma_{12}=\mathbf{b}_1\angle\mathbf{b}_2\\
\gamma_{23}=\mathbf{b}_2\angle\mathbf{b}_3\\
\gamma_{31}=\mathbf{b}_3\angle\mathbf{b}_1\end{aligned}
\end{aligned}$$

Also, lower case letters refer to primitive cell vectors while capital
letters refer to conventional cell vectors.

For BCT lattices, RESCU requires that $\theta_{23}=\theta_{31}$. The
condition
$\|\mathbf{a}_2+\mathbf{a}_3\| > \|\mathbf{a}_1+\mathbf{a}_2\|$ for the
primitive lattice then correspond to $c < a$ in the conventional
lattice. If $\theta_{23}=\theta_{31}$ is not met, RESCU recognizes the
lattice and tells the user to change the domain such that
$\theta_{23}=\theta_{31}$ is satisfied. This is generally necessary for
all irreducible Brillouin zone corners to be correctly defined. In fact,
RESCU will recognize and issue a warning for any lattice which does not
meet the prescribed requirements.

For HEX lattices, RESCU requires that the lattice vector perpendicular
to the hexagonal planes be $\mathbf{a}_3$ and that the angle between the
hexagonal plane lattice vectors be 120 degrees.

For TET lattices, the equal magnitude lattice vectors must be
$\mathbf{a}_1$ and $\mathbf{a}_2$.

For ORC lattices, the lattice vectors must be ordered according to their
magnitude.

For ORCF lattices, the criteria are clearer in terms of conventional
unit cell measures. The conventional unit cell vectors are recovered
using

$$\begin{aligned}
\begin{aligned}
\mathbf{A}_1 = \mathbf{a}_2 + \mathbf{a}_3 - \mathbf{a}_1\\
\mathbf{A}_2 = \mathbf{a}_3 + \mathbf{a}_1 - \mathbf{a}_2\\
\mathbf{A}_3 = \mathbf{a}_1 + \mathbf{a}_2 - \mathbf{a}_3\end{aligned}
\end{aligned}$$

For ORCI lattices, the criteria are clearer in terms of conventional
unit cell measures. The conventional unit cell vectors are recovered
using

$$\begin{aligned}
\begin{aligned}
\mathbf{A}_1 = \mathbf{a}_2 + \mathbf{a}_3\\
\mathbf{A}_2 = \mathbf{a}_3 + \mathbf{a}_1\\
\mathbf{A}_3 = \mathbf{a}_1 + \mathbf{a}_2\end{aligned}
\end{aligned}$$

For ORCC lattices, the criteria are clearer in terms of conventional
unit cell measures. The conventional unit cell vectors are recovered
using

$$\begin{aligned}
\begin{aligned}
\mathbf{A}_1 = \mathbf{a}_1 + \mathbf{a}_2\\
\mathbf{A}_2 = \mathbf{a}_1 - \mathbf{a}_2\\
\mathbf{A}_3 = \mathbf{a}_3\end{aligned}
\end{aligned}$$

For MCLC lattices, the criteria are clearer in terms of the reciprocal
vectors and conventional unit cell measures. The conventional unit cell
vectors are recovered using

$$\begin{aligned}
\begin{aligned}
\mathbf{A}_1 = \mathbf{a}_1 - \mathbf{a}_2\\
\mathbf{A}_2 = \mathbf{a}_1 + \mathbf{a}_2\\
\mathbf{A}_3 = \mathbf{a}_3\end{aligned}
\end{aligned}$$

  BZ variation   k-point line
  -------------- ------------------------------------------------------------------------------
  CUB            $\Gamma$-X-M-$\Gamma$-R-X+M-R
  FCC            $\Gamma$-X-W-K-$\Gamma$-L-U-W-L-K+U-X
  BCC            $\Gamma$-H-N-$\Gamma$-P-H+P-N
  RHL1           B-Z-$\Gamma$-X+$\Gamma$-L-B~1~+Q-F-P~1~-Z+L-P
  RHL2           $\Gamma$-P-Z-Q-$\Gamma$-F-P~1~-Q~1~-L-Z
  BCT1           $\Gamma$-X-M-$\Gamma$-Z-P-N-Z~1~-M+X-P
  BCT2           $\Gamma$-X-Y-$\Sigma$-$\Gamma$-Z-$\Sigma_1$-N-P-Y~1~-Z+X-P
  HEX            $\Gamma$-M-K-$\Gamma$-A-L-H-A+L-M+K-H
  TET            $\Gamma$-X-M-$\Gamma$-Z-R-A-Z, X-R, M-A
  ORC            $\Gamma$-X-S-Y-$\Gamma$-Z-U-R-T-Z+Y-T+U-X+S-R
  ORCF1          $\Gamma$-Y-T-Z-$\Gamma$-X-A~1~-Y+T-X~1~+X-A-Z+L-$\Gamma$
  ORCF2          $\Gamma$-Y-T-Z-$\Gamma$-X-A~1~-Y+X-A-Z+L-$\Gamma$
  ORCF3          $\Gamma$-Y-C-D-X-$\Gamma$-Z-D~1~-H-C+C~1~-Z+X-H~1~+H-Y+L-$\Gamma$
  ORCI           $\Gamma$-X-L-T-W-R-X~1~-Z-$\Gamma$-Y-S-W+L~1~-Y+Y~1~-Z
  ORCC           $\Gamma$-X-S-R-A-Z-$\Gamma$-Y-X~1~-A~1~-T-Y+Z-T
  MCL            $\Gamma$-Y-H-C-E-M~1~-A-X-$\Gamma$-Z-D-M+Z-A+D-Y+X-H~1~
  MCLC           $\Gamma$-Y-F-L-I+I~1~-Z-$\Gamma$-X+X~1~-Y+M-$\Gamma$-N+Z-F~1~
  MCLC2          $\Gamma$-Y-F-L-I+I~1~-Z-F~1~+N-$\Gamma$-M
  MCLC3          $\Gamma$-Y-F-H-Z-I-X-$\Gamma$-Z+M-$\Gamma$-N+X-Y~1~-H~1~+I-F~1~
  MCLC4          $\Gamma$-Y-F-H-Z-I+H~1~-Y~1~-X-$\Gamma$-N+M-$\Gamma$
  MCLC5          $\Gamma$-Y-F-L-I+I~1~-Z-$\Gamma$-X-Y~1~-H~1~+H-F~1~+F$_2$-X+M-$\Gamma$-N+H-Z

  : Brillouin zone variations recognized by RESCU and default k-point
  lines. A $-$ symbol indicates that two points are linked. A $+$ symbol
  indicates the beginning of a new line.

Once the Brillouin zone variation is identified, the corners of the
irreducible Brillouin zone are labelled according to
[\[SC10\]](#SC10){.citation}; we do not reproduce the definitions here.
The paths covering the irreducible Brillouin zones edges are different
in certain cases, so they are listed in table
`tab:bsLines`{.interpreted-text role="numref"}. It is generally not
possible to visit all edges using a single non-redundant path, and hence
default paths generally include many k-point lines.

::: {#citations}

[SC10]{#SC10 .citation-label}

:   Wahyu Setyawan and Stefano Curtarolo.
    `High-throughput electronic band structure calculations: Challenges and tools. <http://dx.doi.org/10.1016/j.commatsci.2010.05.010>`{.interpreted-text
    role="doi"} Computational Materials Science 49.2 (2010), pp. 299
    --312.
:::
