.. _Niggli_cell: Niggli cell ============ Usage of the code ------------------ The code is written in C and a python wrapper is prepared in :file:`python` directory. The C code is used to compile with your code. Usual library interface is not prepared. The python code is used as a module. To use this, ``numpy`` is required. In both C and python codes, there are two input arguments, ``lattice`` and ``eps``. ``lattice`` is a ``double`` array with nine elements, .. math:: (a_x, b_x, c_x, a_y, b_y, c_y, a_z, b_z, c_z). In python, the input array will be fattened in the module. Therefore, e.g., the following :math:`3\times 3` shape of a numpy array or a python list is accepted: .. math:: \begin{pmatrix} a_x & b_x & c_x \\ a_y & b_y & c_y \\ a_z & b_z & c_z \\ \end{pmatrix}. The ``double`` variable of ``eps`` is used as the tolerance parameter. The value should be much smaller than lattice parameters, e.g., 1e-8. How it works is shown in the :ref:`following section `. Test ^^^^^ The test is found in :file:`python` directory as a python code. A set of lattice parameters is found in :file:`lattices.dat` and the references, which are the reduced lattice parameter made in the version 0.1.1, are stored in :file:`reduced_lattices.dat`. Example ^^^^^^^^ An example is found in :file:`python` directory as a python code. .. _niggli_algorithm: Algorithm to determine Niggli cell ----------------------------------- Reference ^^^^^^^^^^ 1. A Unified Algorithm for Determining the Reduced (Niggli) Cell, I. Krivý and B. Gruber, Acta Cryst., A32, 297-298 (1976) 2. The Relationship between Reduced Cells in a General Bravais lattice, B. Gruber, Acta Cryst., A29, 433-440 (1973) 3. Numerically stable algorithms for the computation of reduced unit cells, R. W. Grosse-Kunstleve, N. K. Sauter and P. D. Adams, Acta Cryst., A60, 1-6 (2004) Algorithm ^^^^^^^^^^ Update variables ~~~~~~~~~~~~~~~~~~ The following variables used in this algorithm are initialized at the beginning of the algorithm and updated at the every end of A1-8 steps. Define following variables as .. math:: &A = \mathbf{a}\cdot\mathbf{a}\\ &B = \mathbf{b}\cdot\mathbf{b}\\ &C = \mathbf{c}\cdot\mathbf{c}\\ &\xi=2\mathbf{b}\cdot\mathbf{c}\\ &\eta=2\mathbf{c}\cdot\mathbf{a}\\ &\zeta=2\mathbf{a}\cdot\mathbf{b}. They are elements of metric tensor where the off-diagonal elements are doubled. Therefore the metric tensor :math:`\mathbf{G}` is represented as .. math:: \mathbf{G} = \begin{pmatrix} A & \zeta/2 & \eta/2 \\ \zeta/2 & B & \xi/2 \\ \eta/2 & \xi/2 & C \end{pmatrix}. :math:`\xi`, :math:`\eta`, :math:`\zeta` are sorted by their ranges of angles as shown below. ======= ===== Angle value ======= ===== Acute 1 Obtuse -1 Right 0 ======= ===== These values are stored in variables :math:`l, m, n` as follows. * Set initially :math:`l=m=n=0`. * If :math:`\xi<-\varepsilon`, :math:`l=-1`. * If :math:`\xi>\varepsilon`, :math:`l=1`. * If :math:`\eta<-\varepsilon`, :math:`m=-1`. * If :math:`\eta>\varepsilon`, :math:`m=1`. * If :math:`\zeta<-\varepsilon`, :math:`n=-1`. * If :math:`\zeta>\varepsilon`, :math:`n=1`. :math:`\mathbf{C}` found in each step is the transformation matrix that is applied to basis vectors: .. math:: (\mathbf{a}', \mathbf{b}', \mathbf{c}') = (\mathbf{a}, \mathbf{b}, \mathbf{c})\mathbf{C}. A1 ~~ If :math:`A > B + \varepsilon` or (:math:`\overline{|A-B|>\varepsilon}` and :math:`|\xi|>|\eta| + \varepsilon`), .. math:: \mathbf{C} = \begin{pmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & -1 \\ \end{pmatrix}. A2 ~~ If :math:`B > C + \varepsilon` or (:math:`\overline{|B-C|>\varepsilon}` and :math:`|\eta|>|\zeta| + \varepsilon`), .. math:: \mathbf{C} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0 \\ \end{pmatrix}. Go to A1. A3 ~~ If :math:`lmn = 1`: * :math:`i=-1` if :math:`l=-1` else :math:`i=1` * :math:`j=-1` if :math:`m=-1` else :math:`j=1` * :math:`k=-1` if :math:`n=-1` else :math:`k=1` .. math:: \mathbf{C} = \begin{pmatrix} i & 0 & 0 \\ 0 & j & 0 \\ 0 & 0 & k \\ \end{pmatrix}. A4 ~~ If :math:`l=-1`, :math:`m=-1`, and :math:`n=-1`, do nothing in A4. If :math:`lmn = 0` or :math:`lmn = -1`: Set :math:`i=j=k=1`. :math:`r` is used as a reference to :math:`i`, :math:`j`, or :math:`k`, and is initially undefined. * :math:`i=-1` if :math:`l=1` * :math:`r\rightarrow i` if :math:`l=0` * :math:`j=-1` if :math:`m=1` * :math:`r\rightarrow j` if :math:`j=0` * :math:`k=-1` if :math:`n=1` * :math:`r\rightarrow k` if :math:`k=0` If :math:`ijk=-1`: * :math:`i`, :math:`j`, or :math:`k` refered by :math:`r` is set to :math:`-1`. .. math:: \mathbf{C} = \begin{pmatrix} i & 0 & 0 \\ 0 & j & 0 \\ 0 & 0 & k \\ \end{pmatrix}. A5 ~~ If :math:`|\xi|>B + \varepsilon` or :math:`(\overline{|B - \xi| > \varepsilon}` and :math:`2\eta< \zeta -\varepsilon)` or :math:`(\overline{|B + \xi| > \varepsilon}` and :math:`\zeta< -\varepsilon)`: .. math:: \mathbf{C} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & -\mathrm{sign}(\xi) \\ 0 & 0 & 1 \\ \end{pmatrix}. Go to A1. A6 ~~ If :math:`|\eta|>A + \varepsilon` or :math:`(\overline{|A - \eta| > \varepsilon}` and :math:`2\xi < \zeta -\varepsilon)` or :math:`(\overline{|A + \eta| > \varepsilon}` and :math:`\zeta< -\varepsilon)`: .. math:: \mathbf{C} = \begin{pmatrix} 1 & 0 & -\mathrm{sign}(\eta) \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}. Go to A1. A7 ~~ If :math:`|\zeta|>A + \varepsilon` or :math:`(\overline{|A - \zeta| > \varepsilon}, 2\xi < \eta -\varepsilon)` or :math:`(\overline{|A + \zeta| > \varepsilon}` and :math:`\eta< -\varepsilon)`: .. math:: \mathbf{C} = \begin{pmatrix} 1 & -\mathrm{sign}(\zeta) & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}. Go to A1. A8 ~~ If :math:`\xi + \eta + \zeta + A + B < -\varepsilon` or :math:`(\overline{|\xi + \eta + \zeta + A + B| > \varepsilon}` and :math:`2(A + \eta) + \zeta > \varepsilon)`: .. math:: \mathbf{C} = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{pmatrix}. Go to A1.