.. _guide_orci:

********************************
Body-centred orthorhombic (ORCI)
********************************

**Pearson symbol**: oI

**Constructor**:  :py:func:`.ORCI`

It is defined by three parameters :math:`a`, :math:`b` and :math:`c` with
:math:`a < b < c`. Standardized primitive and conventional cells in the default
orientation are

.. math::

  \begin{matrix}
    \boldsymbol{a}_1^s &=& (-a/2, &b/2, &c/2)\\
    \boldsymbol{a}_2^s &=& (a/2, &-b/2, &c/2)\\
    \boldsymbol{a}_3^s &=& (a/2, &b/2, &-c/2)
  \end{matrix}

.. math::

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

Transformation matrix from standardized primitive cell to standardized conventional cell
is

.. include:: C_matrix.inc

K-path
======

:math:`\mathrm{\Gamma-X-L-T-W-R-X_1-Z-\Gamma-Y-S-W\vert L_1-Y\vert Y_1-Z}`

.. math::

    \begin{matrix}
    \zeta = \dfrac{1 + a^2/c^2}{4} &
    \eta = \dfrac{1 + b^2/c^2}{4} &
    \delta = \dfrac{b^2 - a^2}{4c^2} &
    \mu = \dfrac{a^2 + b^2}{4c^2}
    \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{L}`         :math:`-\mu`                      :math:`\mu`                       :math:`1/2 - \delta`
:math:`\mathrm{L_1}`       :math:`\mu`                       :math:`-\mu`                      :math:`1/2 + \delta`
:math:`\mathrm{L_2}`       :math:`1/2-\delta`                :math:`1/2+\delta`                :math:`-\mu`
:math:`\mathrm{R}`         :math:`0`                         :math:`1/2`                       :math:`0`
:math:`\mathrm{S}`         :math:`1/2`                       :math:`0`                         :math:`0`
:math:`\mathrm{T}`         :math:`0`                         :math:`0`                         :math:`1/2`
:math:`\mathrm{W}`         :math:`1/4`                       :math:`1/4`                       :math:`1/4`
:math:`\mathrm{X}`         :math:`-\zeta`                    :math:`\zeta`                     :math:`\zeta`
:math:`\mathrm{X_1}`       :math:`\zeta`                     :math:`1-\zeta`                   :math:`-\zeta`
:math:`\mathrm{Y}`         :math:`\eta`                      :math:`-\eta`                     :math:`\eta`
:math:`\mathrm{Y_1}`       :math:`1-\eta`                    :math:`\eta`                      :math:`-\eta`
:math:`\mathrm{Z}`         :math:`1/2`                       :math:`1/2`                       :math:`-1/2`
=========================  ================================  ================================  ================================


Variations
==========

There are no variations for body-centered orthorombic.
One example is predefined: ``orci`` with
:math:`a = \pi`, :math:`b  = 1.3\pi` and :math:`c = 1.7\pi`.

Examples
========

Brillouin zone and default kpath
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.. literalinclude:: orci_reciprocal.py
    :language: py

.. raw:: html
    :file: orci_reciprocal.html

Primitive, Wigner-Seitz and conventional cells
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Click on the legend to hide a cell.

.. literalinclude:: orci_real.py
    :language: py

.. raw:: html
    :file: orci_real.html


Cell standardization
====================

Condition :math:`a < b < c` implies :math:`\gamma^s < \beta^s < \alpha^s` for the
primitive cell in a standard form. Therefore, wulfric uses angles of the primitive cell
for standardization.


* If :math:`\gamma < \beta < \alpha`,
  then

  .. math::

    (\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s)
    =
    (\boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3)

  and

  .. math::

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

* If :math:`\gamma < \alpha < \beta`,
  then

  .. math::

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

  and

  .. math::

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

* If :math:`\beta < \gamma < \alpha`,
  then

  .. math::

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

  and

  .. math::

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

* If :math:`\beta < \alpha < \gamma`,
  then

  .. math::

    (\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s)
    =
    (\boldsymbol{a}_3, \boldsymbol{a}_1, \boldsymbol{a}_2)

  and

  .. math::

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

* If :math:`\alpha < \gamma < \beta`,
  then

  .. math::

    (\boldsymbol{a}_1^s, \boldsymbol{a}_2^s, \boldsymbol{a}_3^s)
    =
    (\boldsymbol{a}_2, \boldsymbol{a}_3, \boldsymbol{a}_1)

  and

  .. math::

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


* If :math:`\alpha < \beta < \gamma`,
  then

  .. math::

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

  and

  .. math::

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

.. note::

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

Edge cases
==========
If :math:`a = b \ne c` or :math:`a = c \ne b` or :math:`b = c \ne a`,
then the lattice is :ref:`guide_bct`.

If :math:`a = b = c`, then the lattice is :ref:`guide_bcc`.