Niggli cell¶

Usage of the code¶

The code is written in C and a python wrapper is prepared in 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,

$(a_x, b_x, c_x, a_y, b_y, c_y, a_z, b_z, c_z).$

In python, the input array will be fattened in the module. Therefore, e.g., the following $3\times 3$ shape of a numpy array or a python list is accepted:

$\begin{pmatrix} a_x & b_x & c_x \\ a_y & b_y & c_y \\ a_z & b_z & c_z \\ \end{pmatrix}.$

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 following section.

Test¶

The test is found in python directory as a python code. A set of lattice parameters is found in lattices.dat and the references, which are the reduced lattice parameter made in the version 0.1.1, are stored in reduced_lattices.dat.

Example¶

An example is found in python directory as a python code.

Algorithm to determine Niggli cell¶

Reference¶

A Unified Algorithm for Determining the Reduced (Niggli) Cell, I. Krivý and B. Gruber, Acta Cryst., A32, 297-298 (1976)
The Relationship between Reduced Cells in a General Bravais lattice, B. Gruber, Acta Cryst., A29, 433-440 (1973)
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

$&A = \mathbf{a}\cdot\mathbf{a}\\ &B = \mathbf{b}\cdot\mathbf{b}\\ &C = \mathbf{c}\cdot\mathbf{c}\\ &\xi=2\mathbf{b}\cdot\mathbf{c}\\ &\eta=2\mathbf{c}\cdot\mathbf{a}\\ &\zeta=2\mathbf{a}\cdot\mathbf{b}.$

They are elements of metric tensor where the off-diagonal elements are doubled. Therefore the metric tensor $\mathbf{G}$ is represented as

$\mathbf{G} = \begin{pmatrix} A & \zeta/2 & \eta/2 \\ \zeta/2 & B & \xi/2 \\ \eta/2 & \xi/2 & C \end{pmatrix}.$

$\xi$ , $\eta$ , $\zeta$ are sorted by their ranges of angles as shown below.

Angle	value
Acute	1
Obtuse	-1
Right	0

These values are stored in variables $l, m, n$ as follows.

Set initially $l=m=n=0$ .
If $\xi<-\varepsilon$ , $l=-1$ .
If $\xi>\varepsilon$ , $l=1$ .
If $\eta<-\varepsilon$ , $m=-1$ .
If $\eta>\varepsilon$ , $m=1$ .
If $\zeta<-\varepsilon$ , $n=-1$ .
If $\zeta>\varepsilon$ , $n=1$ .

$\mathbf{C}$ found in each step is the transformation matrix that is applied to basis vectors:

$(\mathbf{a}', \mathbf{b}', \mathbf{c}') = (\mathbf{a}, \mathbf{b}, \mathbf{c})\mathbf{C}.$

A1¶

If $A > B + \varepsilon$ or ( $\overline{|A-B|>\varepsilon}$ and $|\xi|>|\eta| + \varepsilon$ ),

$\mathbf{C} = \begin{pmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & -1 \\ \end{pmatrix}.$

A2¶

If $B > C + \varepsilon$ or ( $\overline{|B-C|>\varepsilon}$ and $|\eta|>|\zeta| + \varepsilon$ ),

$\mathbf{C} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0 \\ \end{pmatrix}.$

Go to A1.

A3¶

If $lmn = 1$ :

$i=-1$ if $l=-1$ else $i=1$
$j=-1$ if $m=-1$ else $j=1$
$k=-1$ if $n=-1$ else $k=1$

$\mathbf{C} = \begin{pmatrix} i & 0 & 0 \\ 0 & j & 0 \\ 0 & 0 & k \\ \end{pmatrix}.$

A4¶

If $l=-1$ , $m=-1$ , and $n=-1$ , do nothing in A4.

If $lmn = 0$ or $lmn = -1$ :

Set $i=j=k=1$ . $r$ is used as a reference to $i$ , $j$ , or $k$ , and is initially undefined.

$i=-1$ if $l=1$
$r\rightarrow i$ if $l=0$
$j=-1$ if $m=1$
$r\rightarrow j$ if $j=0$
$k=-1$ if $n=1$
$r\rightarrow k$ if $k=0$

If $ijk=-1$ :

$i$ , $j$ , or $k$ refered by $r$ is set to $-1$ .

$\mathbf{C} = \begin{pmatrix} i & 0 & 0 \\ 0 & j & 0 \\ 0 & 0 & k \\ \end{pmatrix}.$

A5¶

If $|\xi|>B + \varepsilon$ or $(\overline{|B - \xi| > \varepsilon}$ and $2\eta< \zeta -\varepsilon)$ or $(\overline{|B + \xi| > \varepsilon}$ and $\zeta< -\varepsilon)$ :

$\mathbf{C} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & -\mathrm{sign}(\xi) \\ 0 & 0 & 1 \\ \end{pmatrix}.$

Go to A1.

A6¶

If $|\eta|>A + \varepsilon$ or $(\overline{|A - \eta| > \varepsilon}$ and $2\xi < \zeta -\varepsilon)$ or $(\overline{|A + \eta| > \varepsilon}$ and $\zeta< -\varepsilon)$ :

$\mathbf{C} = \begin{pmatrix} 1 & 0 & -\mathrm{sign}(\eta) \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}.$

Go to A1.

A7¶

If $|\zeta|>A + \varepsilon$ or $(\overline{|A - \zeta| > \varepsilon}, 2\xi < \eta -\varepsilon)$ or $(\overline{|A + \zeta| > \varepsilon}$ and $\eta< -\varepsilon)$ :