.. _guide_mcl:

****************
Monoclinic (MCL)
****************

**Pearson symbol**: mP

**Constructor**:  :py:func:`.MCL`

It is defined by four parameters :math:`a`, :math:`b`, :math:`c` and :math:`\alpha` with
:math:`b \le c`, :math:`\alpha < 90^{\circ}`. Standardized primitive and conventional
cells in the default orientation are

.. math::

  \begin{matrix}
    \boldsymbol{a}_1^s &=& \boldsymbol{a}_1^{cs} &=& (a, &0, &0)\\
    \boldsymbol{a}_2^s &=& \boldsymbol{a}_2^{cs} &=& (0, &b, &0)\\
    \boldsymbol{a}_3^s &=& \boldsymbol{a}_3^{cs} &=& (0, &c\cos\alpha, &c\sin\alpha)
  \end{matrix}

Transformation matrix from standardized primitive cell to standardized conventional cell
is

.. include:: C_matrix.inc


K-path
======

:math:`\mathrm{\Gamma-Y-H-C-E-M_1-A-X-H_1\vert M-D-Z\vert Y-D}`

.. math::

  \begin{matrix}
    \eta = \dfrac{1 - b\cos\alpha / c}{2\sin^2\alpha} &
    \nu = \dfrac{1}{2} - \dfrac{\eta c\cos\alpha}{b}
  \end{matrix}

=========================  ================================  ================================  ================================
Point                      :math:`\times\boldsymbol{b}_1^s`  :math:`\times\boldsymbol{b}_2^s`  :math:`\times\boldsymbol{b}_3^s`
=========================  ================================  ================================  ================================
:math:`\mathrm{\Gamma}`    :math:`0`                         :math:`0`                         :math:`0`
:math:`\mathrm{A}`         :math:`1/2`                       :math:`1/2`                       :math:`0`
:math:`\mathrm{C}`         :math:`0`                         :math:`1/2`                       :math:`1/2`
:math:`\mathrm{D}`         :math:`1/2`                       :math:`0`                         :math:`1/2`
:math:`\mathrm{D_1}`       :math:`1/2`                       :math:`0`                         :math:`-1/2`
:math:`\mathrm{E}`         :math:`1/2`                       :math:`1/2`                       :math:`1/2`
:math:`\mathrm{H}`         :math:`0`                         :math:`\eta`                      :math:`1-\nu`
:math:`\mathrm{H_1}`       :math:`0`                         :math:`1-\eta`                    :math:`\nu`
:math:`\mathrm{H_2}`       :math:`0`                         :math:`\eta`                      :math:`-\nu`
:math:`\mathrm{M}`         :math:`1/2`                       :math:`\eta`                      :math:`1-\nu`
:math:`\mathrm{M_1}`       :math:`1/2`                       :math:`1-\eta`                    :math:`\nu`
:math:`\mathrm{M_2}`       :math:`1/2`                       :math:`\eta`                      :math:`-\nu`
:math:`\mathrm{X}`         :math:`0`                         :math:`1/2`                       :math:`0`
:math:`\mathrm{Y}`         :math:`0`                         :math:`0`                         :math:`1/2`
:math:`\mathrm{Y_1}`       :math:`0`                         :math:`0`                         :math:`-1/2`
:math:`\mathrm{Z}`         :math:`1/2`                       :math:`0`                         :math:`0`
=========================  ================================  ================================  ================================

Variations
==========

There are no variations for monoclinic lattice. One example is predefined: ``mcl`` with
:math:`a = \pi`, :math:`b = 1.3 \pi` :math:`c = 1.6 \pi` and :math:`\alpha = 75^{\circ}`.

Examples
========
Brillouin zone and default kpath
--------------------------------
.. literalinclude:: mcl_reciprocal.py
    :language: py

.. raw:: html
    :file: mcl_reciprocal.html

Primitive and Wigner-Seitz cells
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Click on the legend to hide a cell.

.. literalinclude:: mcl_real.py
    :language: py

.. raw:: html
    :file: mcl_real.html


Cell standardization
====================
First wulfric find two angles of :math:`90^{\circ}`, then conditions :math:`b \le c` and
:math:`\alpha < 90^{\circ}` are checked directly. Matrix :math:`\boldsymbol{S}` is
constructed in three steps.

Step 1
------

* If :math:`\beta = \gamma = \frac{\pi}{2}` and :math:`\alpha \ne \frac{\pi}{2}` 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
    =
    \begin{pmatrix}
      1 & 0 & 0 \\
      0 & 1 & 0 \\
      0 & 0 & 1
    \end{pmatrix}
    \qquad
    \boldsymbol{S}_1^{-1}
    =
    \begin{pmatrix}
      1 & 0 & 0 \\
      0 & 1 & 0 \\
      0 & 0 & 1
    \end{pmatrix}

* If :math:`\alpha = \gamma = \frac{\pi}{2}` and :math:`\beta \ne \frac{\pi}{2}` 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 & 0 & 1 \\
      1 & 0 & 0 \\
      0 & 1 & 0
    \end{pmatrix}
    \qquad
    \boldsymbol{S}_1^{-1}
    =
    \begin{pmatrix}
      0 & 1 & 0 \\
      0 & 0 & 1 \\
      1 & 0 & 0
    \end{pmatrix}

* If :math:`\alpha = \beta = \frac{\pi}{2}` and :math:`\gamma \ne \frac{\pi}{2}` 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 & 1 & 0 \\
      0 & 0 & 1 \\
      1 & 0 & 0
    \end{pmatrix}
    \qquad
    \boldsymbol{S}_1^{-1}
    =
    \begin{pmatrix}
      0 & 0 & 1 \\
      1 & 0 & 0 \\
      0 & 1 & 0
    \end{pmatrix}

Step 2
------
* If :math:`b^{(1)} \le c^{(1)}`, 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
    =
    \begin{pmatrix}
      1 & 0 & 0 \\
      0 & 1 & 0 \\
      0 & 0 & 1
    \end{pmatrix}
    \qquad
    \boldsymbol{S}_2^{-1}
    =
    \begin{pmatrix}
      1 & 0 & 0 \\
      0 & 1 & 0 \\
      0 & 0 & 1
    \end{pmatrix}

* If :math:`b^{(1)} > c^{(1)}` then

  .. math::

    (\boldsymbol{a}_1^{(2)}, \boldsymbol{a}_2^{(2)}, \boldsymbol{a}_3^{(2)})
    =
    (-\boldsymbol{a}_1^{(1)}, \boldsymbol{a}_3^{(1)}, \boldsymbol{a}_2^{(1)})

  and

  .. math::

    \boldsymbol{S}_2
    =
    \begin{pmatrix}
      -1 & 0 & 0 \\
      0 & 0 & 1 \\
      0 & 1 & 0
    \end{pmatrix}
    \qquad
    \boldsymbol{S}_2^{-1}
    =
    \begin{pmatrix}
      -1 & 0 & 0 \\
      0 & 0 & 1 \\
      0 & 1 & 0
    \end{pmatrix}

Step 3
------

* If :math:`\alpha^{(2)} < \frac{\pi}{2}` 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
    =
    \begin{pmatrix}
      1 & 0 & 0 \\
      0 & 1 & 0 \\
      0 & 0 & 1
    \end{pmatrix}
    \qquad
    \boldsymbol{S}_3^{-1}
    =
    \begin{pmatrix}
      1 & 0 & 0 \\
      0 & 1 & 0 \\
      0 & 0 & 1
    \end{pmatrix}

* If :math:`\alpha^{(2)} > \frac{\pi}{2}` 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
    =
    \begin{pmatrix}
      -1 & 0 & 0 \\
      0 & -1 & 0 \\
      0 & 0 & 1
    \end{pmatrix}
    \qquad
    \boldsymbol{S}_3^{-1}
    =
    \begin{pmatrix}
      -1 & 0 & 0 \\
      0 & -1 & 0 \\
      0 & 0 & 1
    \end{pmatrix}

Finally
-------

.. math::

    \boldsymbol{S}
    =
    \boldsymbol{S}_1 \boldsymbol{S}_2 \boldsymbol{S}_3
    \qquad
    \boldsymbol{S}^{-1}
    =
    \boldsymbol{S}_3^{-1} \boldsymbol{S}_2^{-1} \boldsymbol{S}_1^{-1}

.. note::

    All changes of the cell preserve handiness of the original one.

Edge cases
==========

If (:math:`\alpha = 60^{\circ}` or :math:`\alpha = 120^{\circ}`) and :math:`b = c`,
then the lattice is :ref:`guide_hex`.

If (:math:`\alpha = 30^{\circ}` or :math:`\alpha = 150^{\circ}`
or :math:`\alpha = 45^{\circ}` or :math:`\alpha = 145^{\circ}`) and :math:`b = c`,
then the lattice is :ref:`guide_orcc`.

If (:math:`\alpha = 60^{\circ}` or :math:`\alpha = 120^{\circ}`) and :math:`a \ne b = c/2`,
then the lattice is :ref:`guide_orc`.

If :math:`a \ne b \ne c` and :math:`\alpha = 90^{\circ}`, then the lattice is :ref:`guide_orc`.

If (:math:`\alpha = 60^{\circ}` or :math:`\alpha = 120^{\circ}`) and :math:`a = b = c/2`,
then the lattice is :ref:`guide_tet`.

If (:math:`a = b \ne c` or :math:`a = c \ne b` or :math:`b = c \ne a`) and :math:`\alpha = 90^{\circ}`, then the lattice is :ref:`guide_tet`.

If :math:`a = b = c` and :math:`\alpha = 90^{\circ}`, then the lattice is :ref:`guide_cub`.