.. _guide_orcc: ******************************** Base-centred orthorhombic (ORCC) ******************************** **Pearson symbol**: oS **Constructor**: :py:func:`.ORCC` It is defined by three parameters :math:`a`, :math:`b` and :math:`c` with :math:`a < b`. Standardized primitive and conventional cells in the default orientation are .. math:: \begin{matrix} \boldsymbol{a}_1^s &=& (a/2, &-b/2, &0)\\ \boldsymbol{a}_2^s &=& (a/2, &b/2, &0)\\ \boldsymbol{a}_3^s &=& (0, &0, &c) \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-S-R-A-Z-\Gamma-Y-X_1-A_1-T-Y\vert Z-T}` .. math:: \zeta = \dfrac{1 + a^2/b^2}{4} ========================= ================================ ================================ ================================ 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:`\zeta` :math:`\zeta` :math:`1/2` :math:`\mathrm{A_1}` :math:`-\zeta` :math:`1-\zeta` :math:`1/2` :math:`\mathrm{R}` :math:`0` :math:`1/2` :math:`1/2` :math:`\mathrm{S}` :math:`0` :math:`1/2` :math:`0` :math:`\mathrm{T}` :math:`-1/2` :math:`1/2` :math:`1/2` :math:`\mathrm{X}` :math:`\zeta` :math:`\zeta` :math:`0` :math:`\mathrm{X_1}` :math:`-\zeta` :math:`1-\zeta` :math:`0` :math:`\mathrm{Y}` :math:`-1/2` :math:`1/2` :math:`0` :math:`\mathrm{Z}` :math:`0` :math:`0` :math:`1/2` ========================= ================================ ================================ ================================ Variations ========== There are no variations for base-centered orthorombic. One example is predefined: ``orcc`` with :math:`a = \pi`, :math:`b = 1.3\pi` and :math:`c = 1.7\pi`. Examples ======== Brillouin zone and default kpath ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. literalinclude:: orcc_reciprocal.py :language: py .. raw:: html :file: orcc_reciprocal.html Primitive, Wigner-Seitz and conventional cells ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Click on the legend to hide a cell. .. literalinclude:: orcc_real.py :language: py .. raw:: html :file: orcc_real.html Cell standardization ==================== Length of third vector of the primitive cell has to be different from the lengths of the first two vectors of the primitive cell. Together with the :math:`a < b` we arrive at the following condition of the angles of the primitive cell in a standard form: :math:`\alpha^s = \beta^s = 90^{\circ}` and :math:`\gamma^s > 90^{\circ}`. Wulfric uses angles of the primitive cell for standardization. * If :math:`\alpha = \beta = \frac{\pi}{2}` and :math:`\gamma > \frac{\pi}{2}` 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:`\alpha = \beta = \frac{\pi}{2}` and :math:`\gamma < \frac{\pi}{2}` 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 = \frac{\pi}{2}` and :math:`\alpha > \frac{\pi}{2}` 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:`\beta = \gamma = \frac{\pi}{2}` and :math:`\alpha < \frac{\pi}{2}` 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} \qquad \boldsymbol{S}^{-1} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & 0 \end{pmatrix} * If :math:`\alpha = \gamma = \frac{\pi}{2}` and :math:`\beta > \frac{\pi}{2}` 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 = \frac{\pi}{2}` and :math:`\beta < \frac{\pi}{2}` 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} .. note:: All six changes of the cell preserve handiness of the original one. Edge cases ========== If :math:`a = b`, then the lattice is :ref:`guide_tet`. If :math:`a = b = \sqrt{2} c`, then the lattice is :ref:`guide_cub`.