.. _guide_mclc: ****************************** Base-centred monoclinic (MCLC) ****************************** **Pearson symbol**: mS **Constructor**: :py:func:`.MCLC` It is defined by four parameter: :math:`a`, :math:`b`, :math:`c` and :math:`\alpha` with conventional cell: .. math:: \begin{matrix} \boldsymbol{a}_1 &=& (a, &0, &0)\\ \boldsymbol{a}_2 &=& (0, &b, &0)\\ \boldsymbol{a}_3 &=& (0, &c\cos\alpha, &c\sin\alpha) \end{matrix} And primitive cell: .. math:: \begin{matrix} \boldsymbol{a}_1 &=& (a/2, &b/2, &0)\\ \boldsymbol{a}_2 &=& (-a/2, &b/2, &0)\\ \boldsymbol{a}_3 &=& (0, &c\cos\alpha, &c\sin\alpha) \end{matrix} with .. math:: \boldsymbol{C} = \dfrac{1}{2} \begin{pmatrix} 1 & -1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix} \qquad \boldsymbol{C}^{-1} = \begin{pmatrix} 1 & 1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} Order of parameters: :math:`b \le c`, :math:`\alpha < 90^{\circ}`. K-path ====== MCLC\ :sub:`1` -------------- :math:`\mathrm{\Gamma-Y-F-L-I\vert I_1-Z-F_1\vert Y-X_1\vert X-\Gamma-N\vert M-\Gamma}` .. math:: \begin{matrix} \zeta = \dfrac{2 - b\cos\alpha/c}{4\sin^2\alpha} & \eta = \dfrac{1}{2} + \dfrac{2\zeta c\cos\alpha}{b} \\ \psi = \dfrac{3}{4} - \dfrac{a^2}{4b^2\sin^2\alpha} & \phi = \psi + \dfrac{(3/4-\psi)b\cos\alpha}{c} \end{matrix} ========================= ============================== ============================== ============================== Point :math:`\times\boldsymbol{b}_1` :math:`\times\boldsymbol{b}_2` :math:`\times\boldsymbol{b}_3` ========================= ============================== ============================== ============================== :math:`\mathrm{\Gamma}` :math:`0` :math:`0` :math:`0` :math:`\mathrm{N}` :math:`1/2` :math:`0` :math:`0` :math:`\mathrm{N_1}` :math:`0` :math:`-1/2` :math:`0` :math:`\mathrm{F}` :math:`1-\zeta` :math:`1-\zeta` :math:`1-\eta` :math:`\mathrm{F_1}` :math:`\zeta` :math:`\zeta` :math:`\eta` :math:`\mathrm{F_2}` :math:`-\zeta` :math:`-\zeta` :math:`1-\eta` :math:`\mathrm{F_3}` :math:`1-\zeta` :math:`-\zeta` :math:`1-\eta` :math:`\mathrm{I}` :math:`\phi` :math:`1-\phi` :math:`1/2` :math:`\mathrm{I_1}` :math:`1-\phi` :math:`\phi-1` :math:`1/2` :math:`\mathrm{L}` :math:`1/2` :math:`1/2` :math:`1/2` :math:`\mathrm{M}` :math:`1/2` :math:`0` :math:`1/2` :math:`\mathrm{X}` :math:`1-\psi` :math:`\psi-1` :math:`0` :math:`\mathrm{X_1}` :math:`\psi` :math:`1-\psi` :math:`0` :math:`\mathrm{X_2}` :math:`\psi-1` :math:`-\psi` :math:`0` :math:`\mathrm{Y}` :math:`1/2` :math:`1/2` :math:`0` :math:`\mathrm{Y_1}` :math:`-1/2` :math:`-1/2` :math:`0` :math:`\mathrm{Z}` :math:`0` :math:`0` :math:`1/2` ========================= ============================== ============================== ============================== MCLC\ :sub:`2` -------------- :math:`\mathrm{\Gamma-Y-F-L-I\vert I_1-Z-F_1\vert N-\Gamma-M}` .. math:: \begin{matrix} \zeta = \dfrac{2 - b\cos\alpha/c}{4\sin^2\alpha} & \eta = \dfrac{1}{2} + \dfrac{2\zeta c\cos\alpha}{b} \\ \psi = \dfrac{3}{4} - \dfrac{a^2}{4b^2\sin^2\alpha} & \phi = \psi + \dfrac{(3/4-\psi)b\cos\alpha}{c} \end{matrix} ========================= ============================== ============================== ============================== Point :math:`\times\boldsymbol{b}_1` :math:`\times\boldsymbol{b}_2` :math:`\times\boldsymbol{b}_3` ========================= ============================== ============================== ============================== :math:`\mathrm{\Gamma}` :math:`0` :math:`0` :math:`0` :math:`\mathrm{N}` :math:`1/2` :math:`0` :math:`0` :math:`\mathrm{N_1}` :math:`0` :math:`-1/2` :math:`0` :math:`\mathrm{F}` :math:`1-\zeta` :math:`1-\zeta` :math:`1-\eta` :math:`\mathrm{F_1}` :math:`\zeta` :math:`\zeta` :math:`\eta` :math:`\mathrm{F_2}` :math:`-\zeta` :math:`-\zeta` :math:`1-\eta` :math:`\mathrm{F_3}` :math:`1-\zeta` :math:`-\zeta` :math:`1-\eta` :math:`\mathrm{I}` :math:`\phi` :math:`1-\phi` :math:`1/2` :math:`\mathrm{I_1}` :math:`1-\phi` :math:`\phi-1` :math:`1/2` :math:`\mathrm{L}` :math:`1/2` :math:`1/2` :math:`1/2` :math:`\mathrm{M}` :math:`1/2` :math:`0` :math:`1/2` :math:`\mathrm{X}` :math:`1-\psi` :math:`\psi-1` :math:`0` :math:`\mathrm{X_1}` :math:`\psi` :math:`1-\psi` :math:`0` :math:`\mathrm{X_2}` :math:`\psi-1` :math:`-\psi` :math:`0` :math:`\mathrm{Y}` :math:`1/2` :math:`1/2` :math:`0` :math:`\mathrm{Y_1}` :math:`-1/2` :math:`-1/2` :math:`0` :math:`\mathrm{Z}` :math:`0` :math:`0` :math:`1/2` ========================= ============================== ============================== ============================== MCLC\ :sub:`3` -------------- :math:`\mathrm{\Gamma-Y-F-H-Z-I-F_1\vert H_1-Y_1-X-\Gamma-N\vert M-\Gamma}` .. math:: \begin{matrix} \mu = \dfrac{1+b^2/a^2}{4} & \delta = \dfrac{bc\cos\alpha}{2a^2} & \zeta = \mu - \dfrac{1}{4} + \dfrac{1 - b\cos\alpha/c}{4\sin^2\alpha} \\ \eta = \dfrac{1}{2} + \dfrac{2\zeta c \cos\alpha}{b} & \phi = 1 + \zeta - 2\mu & \psi = \eta - 2\delta \end{matrix} ========================= ============================== ============================== ============================== Point :math:`\times\boldsymbol{b}_1` :math:`\times\boldsymbol{b}_2` :math:`\times\boldsymbol{b}_3` ========================= ============================== ============================== ============================== :math:`\mathrm{\Gamma}` :math:`0` :math:`0` :math:`0` :math:`\mathrm{F}` :math:`1-\phi` :math:`1-\phi` :math:`1-\psi` :math:`\mathrm{F_1}` :math:`\phi` :math:`\phi-1` :math:`\psi` :math:`\mathrm{F_2}` :math:`1-\phi` :math:`-\phi` :math:`1-\psi` :math:`\mathrm{H}` :math:`\zeta` :math:`\zeta` :math:`\eta` :math:`\mathrm{H_1}` :math:`1-\zeta` :math:`-\zeta` :math:`1-\eta` :math:`\mathrm{H_2}` :math:`-\zeta` :math:`-\zeta` :math:`1-\eta` :math:`\mathrm{I}` :math:`1/2` :math:`-1/2` :math:`1/2` :math:`\mathrm{M}` :math:`1/2` :math:`0` :math:`1/2` :math:`\mathrm{N}` :math:`1/2` :math:`0` :math:`0` :math:`\mathrm{N_1}` :math:`0` :math:`-1/2` :math:`0` :math:`\mathrm{X}` :math:`1/2` :math:`-1/2` :math:`0` :math:`\mathrm{Y}` :math:`\mu` :math:`\mu` :math:`\delta` :math:`\mathrm{Y_1}` :math:`1-\mu` :math:`-\mu` :math:`-\delta` :math:`\mathrm{Y_2}` :math:`-\mu` :math:`-\mu` :math:`-\delta` :math:`\mathrm{Y_3}` :math:`\mu` :math:`\mu-1` :math:`\delta` :math:`\mathrm{Z}` :math:`0` :math:`0` :math:`1/2` ========================= ============================== ============================== ============================== MCLC\ :sub:`4` -------------- :math:`\mathrm{\Gamma-Y-F-H-Z-I\vert H_1-Y_1-X-\Gamma-N\vert M-\Gamma}` .. math:: \begin{matrix} \mu = \dfrac{1+b^2/a^2}{4} & \delta = \dfrac{bc\cos\alpha}{2a^2} & \zeta = \mu - \dfrac{1}{4} + \dfrac{1 - b\cos\alpha/c}{4\sin^2\alpha} \\ \eta = \dfrac{1}{2} + \dfrac{2\zeta c \cos\alpha}{b} & \phi = 1 + \zeta - 2\mu & \psi = \eta - 2\delta \end{matrix} ========================= ============================== ============================== ============================== Point :math:`\times\boldsymbol{b}_1` :math:`\times\boldsymbol{b}_2` :math:`\times\boldsymbol{b}_3` ========================= ============================== ============================== ============================== :math:`\mathrm{\Gamma}` :math:`0` :math:`0` :math:`0` :math:`\mathrm{F}` :math:`1-\phi` :math:`1-\phi` :math:`1-\psi` :math:`\mathrm{F_1}` :math:`\phi` :math:`\phi-1` :math:`\psi` :math:`\mathrm{F_2}` :math:`1-\phi` :math:`-\phi` :math:`1-\psi` :math:`\mathrm{H}` :math:`\zeta` :math:`\zeta` :math:`\eta` :math:`\mathrm{H_1}` :math:`1-\zeta` :math:`-\zeta` :math:`1-\eta` :math:`\mathrm{H_2}` :math:`-\zeta` :math:`-\zeta` :math:`1-\eta` :math:`\mathrm{I}` :math:`1/2` :math:`-1/2` :math:`1/2` :math:`\mathrm{M}` :math:`1/2` :math:`0` :math:`1/2` :math:`\mathrm{N}` :math:`1/2` :math:`0` :math:`0` :math:`\mathrm{N_1}` :math:`0` :math:`-1/2` :math:`0` :math:`\mathrm{X}` :math:`1/2` :math:`-1/2` :math:`0` :math:`\mathrm{Y}` :math:`\mu` :math:`\mu` :math:`\delta` :math:`\mathrm{Y_1}` :math:`1-\mu` :math:`-\mu` :math:`-\delta` :math:`\mathrm{Y_2}` :math:`-\mu` :math:`-\mu` :math:`-\delta` :math:`\mathrm{Y_3}` :math:`\mu` :math:`\mu-1` :math:`\delta` :math:`\mathrm{Z}` :math:`0` :math:`0` :math:`1/2` ========================= ============================== ============================== ============================== MCLC\ :sub:`5` -------------- :math:`\mathrm{\Gamma-Y-F-L-I\vert I_1-Z-H-F_1\vert H_1-Y_1-X-\Gamma-N\vert M-\Gamma}` .. math:: \begin{matrix} \zeta = \dfrac{b^2}{4a^2} + \dfrac{1 - b\cos\alpha/c}{4\sin^2\alpha} & \eta = \dfrac{1}{2} + \dfrac{2\zeta c\cos\alpha}{b} \\ \mu = \dfrac{\eta}{2} + \dfrac{b^2}{4a^2} - \dfrac{bc\cos\alpha}{2a^2} & \nu = 2\mu - \zeta \\ \omega = \dfrac{(4\nu - 1 - b^2\sin^2\alpha/a^2)c}{2b\cos\alpha} & \delta = \dfrac{\zeta c \cos\alpha}{b} + \dfrac{\omega}{2} - \dfrac{1}{4} & \rho = 1 - \dfrac{\zeta a^2}{b^2} \end{matrix} ========================= ============================== ============================== ============================== Point :math:`\times\boldsymbol{b}_1` :math:`\times\boldsymbol{b}_2` :math:`\times\boldsymbol{b}_3` ========================= ============================== ============================== ============================== :math:`\mathrm{\Gamma}` :math:`0` :math:`0` :math:`0` :math:`\mathrm{F}` :math:`\nu` :math:`\nu` :math:`\omega` :math:`\mathrm{F_1}` :math:`1-\nu` :math:`-\nu` :math:`1-\omega` :math:`\mathrm{F_2}` :math:`\nu` :math:`\nu-1` :math:`\omega` :math:`\mathrm{H}` :math:`\zeta` :math:`\zeta` :math:`\eta` :math:`\mathrm{H_1}` :math:`1-\zeta` :math:`-\zeta` :math:`1-\eta` :math:`\mathrm{H_2}` :math:`-\zeta` :math:`-\zeta` :math:`1-\eta` :math:`\mathrm{I}` :math:`\rho` :math:`1-\rho` :math:`1/2` :math:`\mathrm{I_1}` :math:`1-\rho` :math:`\rho-1` :math:`1/2` :math:`\mathrm{L}` :math:`1/2` :math:`1/2` :math:`1/2` :math:`\mathrm{M}` :math:`1/2` :math:`0` :math:`1/2` :math:`\mathrm{N}` :math:`1/2` :math:`0` :math:`0` :math:`\mathrm{N_1}` :math:`0` :math:`-1/2` :math:`0` :math:`\mathrm{X}` :math:`1/2` :math:`-1/2` :math:`0` :math:`\mathrm{Y}` :math:`\mu` :math:`\mu` :math:`\delta` :math:`\mathrm{Y_1}` :math:`1-\mu` :math:`-\mu` :math:`-\delta` :math:`\mathrm{Y_2}` :math:`-\mu` :math:`-\mu` :math:`-\delta` :math:`\mathrm{Y_3}` :math:`\mu` :math:`\mu-1` :math:`\delta` :math:`\mathrm{Z}` :math:`0` :math:`0` :math:`1/2` ========================= ============================== ============================== ============================== Variations ========== There are five variations for base-centered monoclinic lattice. Reciprocal :math:`\gamma` (:math:`k_{\gamma}`) is defined by the equation (for primitive lattice): .. math:: \cos(k_{\gamma}) = \frac{a^2 - b^2\sin^2(\alpha)}{a^2 + b^2\sin^2(\alpha)} For MCLC\ :sub:`2` :math:`k_{\gamma} = 90`, therefore :math:`a = b \sin(\alpha)`. For MCLC\ :sub:`1` we choose :math:`a < b \sin(\alpha)` and for MCLC\ :sub:`3`, MCLC\ :sub:`4` and MCLC\ :sub:`5` we choose :math:`a > b \sin(\alpha)`. For the variations 3-5 we define :math:`a = xb\sin(\alpha)`, where :math:`x > 1`. Then the condition for MCLC\ :sub:`4` gives: .. math:: c = \frac{x^2}{x^2 - 1}b\cos(\alpha) Where :math:`\cos(\alpha) > 0` (:math:`\alpha < 90^{\circ}`), since :math:`x > 1`. And the ordering condition :math:`b \le c` gives: .. math:: \cos(\alpha) \ge \frac{x^2 - 1}{x^2} For MCLC\ :sub:`3` (MCLC\ :sub:`5`) we choose parameters in a same way as for MCLC\ :sub:`4`, but with :math:`c > \frac{x^2}{x^2 - 1}b\cos(\alpha)` (:math:`c < \frac{x^2}{x^2 - 1}b\cos(\alpha)`) MCLC\ :sub:`1` -------------- :math:`k_{\gamma} > 90^{\circ}`, Predefined example: ``mclc1`` with :math:`a = \pi`, :math:`b = 1.4\cdot\pi`, :math:`c = 1.7\cdot\pi` and :math:`\alpha = 80^{\circ}` MCLC\ :sub:`2` -------------- :math:`k_{\gamma} = 90^{\circ}`, Predefined example: ``mclc2`` with :math:`a = 1.4\cdot\pi\cdot\sin(75^{\circ})`, :math:`b = 1.4\cdot\pi`, :math:`c = 1.7\cdot\pi` and :math:`\alpha=75^{\circ}` MCLC\ :sub:`3` -------------- :math:`k_{\gamma} < 90^{\circ}, \dfrac{b\cos(\alpha)}{c} + \dfrac{b^2\sin(\alpha)^2}{a^2} < 1` Predefined example with :math:`b = \pi`, :math:`x = 1.1`, :math:`\alpha = 78^{\circ}`, which produce: ``mclc4`` with :math:`a = 1.1\cdot\sin(78)\cdot\pi`, :math:`b = \pi`, :math:`c = 1.8\cdot 121\cdot\cos(65)\cdot\pi/21` and :math:`\alpha = 78^{\circ}` MCLC\ :sub:`4` -------------- :math:`k_{\gamma} < 90^{\circ}, \dfrac{b\cos(\alpha)}{c} + \dfrac{b^2\sin(\alpha)^2}{a^2} = 1` Predefined example with :math:`b = \pi`, :math:`x = 1.2`, :math:`\alpha = 65^{\circ}`, which produce: ``mclc4`` with :math:`a = 1.2\sin(65)\pi`, :math:`b = \pi`, :math:`c = 36\cos(65)\pi/11` and :math:`\alpha = 65^{\circ}` MCLC\ :sub:`5` -------------- :math:`k_{\gamma} < 90^{\circ}, \dfrac{b\cos(\alpha)}{c} + \dfrac{b^2\sin(\alpha)^2}{a^2} > 1` Predefined example with :math:`b = \pi`, :math:`x = 1.4`, :math:`\alpha = 53^{\circ}`, which produce: ``mclc5`` with :math:`a = 1.4\cdot\sin(53)\cdot\pi`, :math:`b = \pi`, :math:`c = 0.9\cdot 11\cdot\cos(53)\cdot\pi/6` and :math:`\alpha = 53^{\circ}` Examples ======== MCLC\ :sub:`1` -------------- Brillouin zone and default kpath -------------------------------- .. literalinclude:: mclc1_brillouin.py :language: py .. raw:: html :file: mclc1_brillouin.html Primitive and conventional cell ------------------------------- .. literalinclude:: mclc1_real.py :language: py .. raw:: html :file: mclc1_real.html Wigner-Seitz cell ----------------- .. literalinclude:: mclc1_wigner-seitz.py :language: py .. raw:: html :file: mclc1_wigner-seitz.html MCLC\ :sub:`2` -------------- Brillouin zone and default kpath -------------------------------- .. literalinclude:: mclc2_brillouin.py :language: py .. raw:: html :file: mclc2_brillouin.html Primitive and conventional cell ------------------------------- .. literalinclude:: mclc2_real.py :language: py .. raw:: html :file: mclc2_real.html Wigner-Seitz cell ----------------- .. literalinclude:: mclc2_wigner-seitz.py :language: py .. raw:: html :file: mclc2_wigner-seitz.html MCLC\ :sub:`3` -------------- Brillouin zone and default kpath -------------------------------- .. literalinclude:: mclc3_brillouin.py :language: py .. raw:: html :file: mclc3_brillouin.html Primitive and conventional cell ------------------------------- .. literalinclude:: mclc3_real.py :language: py .. raw:: html :file: mclc3_real.html Wigner-Seitz cell ----------------- .. literalinclude:: mclc3_wigner-seitz.py :language: py .. raw:: html :file: mclc3_wigner-seitz.html MCLC\ :sub:`4` -------------- Brillouin zone and default kpath -------------------------------- .. literalinclude:: mclc4_brillouin.py :language: py .. raw:: html :file: mclc4_brillouin.html Primitive and conventional cell ------------------------------- .. literalinclude:: mclc4_real.py :language: py .. raw:: html :file: mclc4_real.html Wigner-Seitz cell ----------------- .. literalinclude:: mclc4_wigner-seitz.py :language: py .. raw:: html :file: mclc4_wigner-seitz.html MCLC\ :sub:`5` -------------- Brillouin zone and default kpath -------------------------------- .. literalinclude:: mclc5_brillouin.py :language: py .. raw:: html :file: mclc5_brillouin.html Primitive and conventional cell ------------------------------- .. literalinclude:: mclc5_real.py :language: py .. raw:: html :file: mclc5_real.html Wigner-Seitz cell ----------------- .. literalinclude:: mclc5_wigner-seitz.py :language: py .. raw:: html :file: mclc5_wigner-seitz.html Cell standardization ==================== Standardization of the MCLC cell involves three steps and the second step had to be based on the conventional and not primitive cell. As a result the matrix :math:`\boldsymbol{S}` is not orthonormal in a general case (direct consequence of the non-orthogonality of the matrix :math:`\boldsymbol{C}`). Step 1 ------ First step ensures that the first two lattice vectors of the primitive cell are orthogonal to each other. We use lattice vectors of the primitive cell. * If :math:`\vert \boldsymbol{a}_1\vert = \vert \boldsymbol{a}_2\vert \ne \vert \boldsymbol{a}_3\vert`, then .. math:: (\boldsymbol{a}_1^1, \boldsymbol{a}_2^1, \boldsymbol{a}_3^1) = (\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3) and .. math:: \boldsymbol{S}_1 = \boldsymbol{S}_1^{-1} = \boldsymbol{S}_1^T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} * If :math:`\vert \boldsymbol{a}_2 \vert = \vert \boldsymbol{a}_3 \vert \ne \vert \boldsymbol{a}_1 \vert`, then .. math:: (\boldsymbol{a}_1^1, \boldsymbol{a}_2^1, \boldsymbol{a}_3^1) = (\boldsymbol{a}_2, \boldsymbol{a}_3, \boldsymbol{a}_1) and .. math:: \boldsymbol{S}_1 = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} \qquad \boldsymbol{S}_1^{-1} = \boldsymbol{S}_1^T = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} * If :math:`\vert \boldsymbol{a}_3 \vert = \vert \boldsymbol{a}_1 \vert \ne \vert \boldsymbol{a}_2 \vert`, then .. math:: (\boldsymbol{a}_1^1, \boldsymbol{a}_2^1, \boldsymbol{a}_3^1) = (\boldsymbol{a}_3, \boldsymbol{a}_1, \boldsymbol{a}_2) and .. math:: \boldsymbol{S}_1 = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \qquad \boldsymbol{S}_1^{-1} = \boldsymbol{S}_1^T = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} Step 2 ------ Second step ensures the condition :math:`b \le c`. We use lattice vectors of the primitive cell. .. dropdown:: Details Length of the lattice vectors for the primitive cell are: .. math:: \begin{matrix} \vert \boldsymbol{a}_1 \vert = \dfrac{\sqrt{a^2 + b^2}}{2} & \vert \boldsymbol{a}_2 \vert = \dfrac{\sqrt{a^2 + b^2}}{2} & \vert \boldsymbol{a}_3 \vert = c \end{matrix} angles between the lattice vectors for the primitive cell are: .. math:: \begin{matrix} \cos(\boldsymbol{a}_2\boldsymbol{a}_3) = \dfrac{2b}{\sqrt{a^2 + b^2}}\cos\alpha & \cos(\boldsymbol{a}_1\boldsymbol{a}_3) = \dfrac{2b}{\sqrt{a^2 + b^2}}\cos\alpha & \cos(\boldsymbol{a}_1\boldsymbol{a}_2) = \dfrac{b^2 - a^2}{b^2 + a^2} \end{matrix} Therefore, no simple condition can be formulated for the primitive cell, that will be equivalent to the condition :math:`b \le c` for the conventional cell. The actual condition is :math:`2\vert \boldsymbol{a}_1 \vert^2 (1 + \cos(\boldsymbol{a}_1\boldsymbol{a}_2)) \le \vert \boldsymbol{a}_3 \vert^2`. As a result, when this condition is not satisfied, simple reordering of vectors of the primitive cell will not be enough. The recipe, that we follow instead is to * Calculate conventional cell as .. math:: (\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3) = (\boldsymbol{a}_1^c, \boldsymbol{a}_2^c, \boldsymbol{a}_3^c) \boldsymbol{C} * Find a standardization matrix for the **conventional** cell .. math:: (\boldsymbol{a}_1^c, \boldsymbol{a}_2^c, \boldsymbol{a}_3^c) = (\boldsymbol{a}_1^{cs}, \boldsymbol{a}_2^{cs}, \boldsymbol{a}_3^{cs}) \boldsymbol{S^c} * Calculate standardized primitive cell as .. math:: (\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (\boldsymbol{a}_1^{cs}, \boldsymbol{a}_2^{cs}, \boldsymbol{a}_3^{cs}) \boldsymbol{C} Then the standardization matrix for the primitive cell is .. math:: \boldsymbol{S} = \boldsymbol{C}^{-1} \boldsymbol{S^c} \boldsymbol{C} * If :math:`2\vert \boldsymbol{a}_1 \vert^2 (1 + \cos(\boldsymbol{a}_1\boldsymbol{a}_2)) \le \vert \boldsymbol{a}_3 \vert^2`, then .. math:: (\boldsymbol{a}_1^2, \boldsymbol{a}_2^2, \boldsymbol{a}_3^2) = (\boldsymbol{a}_1^1, \boldsymbol{a}_2^1, \boldsymbol{a}_3^1) and .. math:: \boldsymbol{S}_2 = \boldsymbol{S}_2^{-1} = \boldsymbol{S}_2^T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} * If :math:`2\vert \boldsymbol{a}_1 \vert^2 (1 + \cos(\boldsymbol{a}_1\boldsymbol{a}_2)) > \vert \boldsymbol{a}_3 \vert^2`, then .. math:: (\boldsymbol{a}_1^{c,2}, \boldsymbol{a}_2^{c,2}, \boldsymbol{a}_3^{c,2}) = (-\boldsymbol{a}_1^{c,1}, \boldsymbol{a}_3^{c,1}, \boldsymbol{a}_2^{c,1}) .. math:: \boldsymbol{S^s}_2 = (\boldsymbol{S^s}_2)^{-1} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} and .. math:: \boldsymbol{S}_2 = \boldsymbol{S}_2^{-1} = \boldsymbol{C}^{-1} \boldsymbol{S^s}_2 \boldsymbol{C} = \begin{pmatrix} -0.5 & 0.5 & 1 \\ 0.5 & -0.5 & 1 \\ 0.5 & 0.5 & 0 \end{pmatrix} .. note:: :math:`\boldsymbol{S}_2^T \ne \boldsymbol{S}_2^{-1}` Step 3 ------ Last step ensures that :math:`\alpha < \frac{\pi}{2}`. Translated to the primitive cell, this condition reads as :math:`\boldsymbol{a}_2\boldsymbol{a}_3 > 0`. We use lattice vectors of the primitive cell. * If :math:`\alpha < \frac{\pi}{2}` (i.e. :math:`\boldsymbol{a}_2\cdot\boldsymbol{a}_3 > 0`), then .. math:: (\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (\boldsymbol{a}_1^2, \boldsymbol{a}_2^2, \boldsymbol{a}_3^2) and .. math:: \boldsymbol{S}_3 = \boldsymbol{S}_3^{-1} = \boldsymbol{S}_3^T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} * If :math:`\alpha > \frac{\pi}{2}` (i.e. :math:`\boldsymbol{a}_2\cdot\boldsymbol{a}_3 < 0`), then .. math:: (\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s) = (-\boldsymbol{a}_1^2, -\boldsymbol{a}_2^2, \boldsymbol{a}_3^2) and .. math:: \boldsymbol{S}_3 = \boldsymbol{S}_3^{-1} = \boldsymbol{S}_3^T = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix} Finally ------- .. math:: \boldsymbol{S} = \boldsymbol{S}_3 \boldsymbol{S}_2 \boldsymbol{S}_1 \qquad \boldsymbol{S}^{-1} = \boldsymbol{S}_1^{-1} \boldsymbol{S}_2^{-1} \boldsymbol{S}_3^{-1}