.. _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 lattice: .. 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 lattice: .. 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} Order of parameters: :math:`b \le c`, :math:`\alpha < 90^{\circ}`. Cell standardization ==================== Length of the third vector of the primitive lattice has to be different from the length of the other two lattice vectors. Angle between second and third lattice vectors of conventional lattice (:math:`\alpha`) has to be less then :math:`90^{\circ}`. Length of the second lattice vector of the conventional lattice has to be less or equal to the length of the third lattice vector. If these conditions are not satisfied, then the lattice is transformed to the standard form: First we ensure the length of the third vector is different from the length of the other two vectors. For this step we use vectors of the primitive lattice: * If :math:`\vert\boldsymbol{a}_1\vert = \vert\boldsymbol{a}_3\vert`: .. math:: (\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3) \rightarrow (\boldsymbol{a}_3, \boldsymbol{a}_1, \boldsymbol{a}_2) * If :math:`\vert\boldsymbol{a}_2\vert = \vert\boldsymbol{a}_3\vert`: .. math:: (\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3) \rightarrow (\boldsymbol{a}_2, \boldsymbol{a}_3, \boldsymbol{a}_1) Then we ensure the :math:`\alpha < 90^{\circ}`. For this step we use vectors of the conventional lattice: * If :math:`\alpha > 90^{\circ}`: .. math:: (\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3) \rightarrow (\boldsymbol{a}_1, \boldsymbol{a}_3, -\boldsymbol{a}_2) Finally, we ensure the :math:`b \le c`. For this step we use vectors of the conventional lattice: * If :math:`b > c`: .. math:: (\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3) \rightarrow (-\boldsymbol{a}_1, \boldsymbol{a}_3, \boldsymbol{a}_2) .. note:: First and second lattice vectors are multiplied by :math:`-1` in some cases to preserve the handedness of the cell. 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