LatticeBase

class sknano.core.crystallography.LatticeBase(nd=None, cell_matrix=None, orientation_matrix=None, offset=None)

Bases: sknano.core.strings.TabulateMixin, sknano.core.meta.BaseClass

Base class for crystallographic lattice objects.

Parameters:

• nd (int) –
• cell_matrix (array_like) –
• orientation_matrix (array_like, optional) –
• offset (array_like, optional) –

Attributes

bounding_box	region bounding_box.
cell	Alias for cell_matrix.
cell_matrix	Matrix of lattice row vectors.
centroid	Region centroid.
fmtstr	Format string.
fractional_matrix	Transformation matrix to convert from cartesian coordinates to fractional coordinates.
matrix	Alias for cell_matrix.
metric_tensor	Metric tensor.
offset	Lattice offset.
ortho_matrix	Transformation matrix to convert from fractional coordinates to cartesian coordinates.
region	Parallelepiped defined by lattice vectors.

Methods

cartesian_to_fractional(ccoords)	Convert cartesian coordinate to fractional coordinate.
fractional_diff(fcoords1, fcoords2)	Compute difference between fractional coordinates.
fractional_to_cartesian(fcoords)	Convert fractional coordinate to cartesian coordinate.
rotate(**kwargs)	Rotate unit cell.
todict()	Return dict of constructor parameters.
translate(t)	Translate lattice.
wrap_cartesian_coordinate(p[, epsilon, pbc])	Wrap cartesian coordinate to lie within unit cell.
wrap_cartesian_coordinates(points[, ...])	Wrap array of cartesian coordinates to lie within unit cell.
wrap_fractional_coordinate(p[, epsilon, pbc])	Wrap fractional coordinate to lie within unit cell.
wrap_fractional_coordinates(points[, ...])	Wrap array of fractional coordinates to lie within unit cell.

Attributes Summary

bounding_box	region bounding_box.
cell	Alias for cell_matrix.
cell_matrix	Matrix of lattice row vectors.
centroid	Region centroid.
fractional_matrix	Transformation matrix to convert from cartesian coordinates to fractional coordinates.
matrix	Alias for cell_matrix.
metric_tensor	Metric tensor.
offset	Lattice offset.
ortho_matrix	Transformation matrix to convert from fractional coordinates to cartesian coordinates.
region	Parallelepiped defined by lattice vectors.

Methods Summary

cartesian_to_fractional(ccoords)	Convert cartesian coordinate to fractional coordinate.
fractional_diff(fcoords1, fcoords2)	Compute difference between fractional coordinates.
fractional_to_cartesian(fcoords)	Convert fractional coordinate to cartesian coordinate.
rotate(**kwargs)	Rotate unit cell.
translate(t)	Translate lattice.
wrap_cartesian_coordinate(p[, epsilon, pbc])	Wrap cartesian coordinate to lie within unit cell.
wrap_cartesian_coordinates(points[, ...])	Wrap array of cartesian coordinates to lie within unit cell.
wrap_fractional_coordinate(p[, epsilon, pbc])	Wrap fractional coordinate to lie within unit cell.
wrap_fractional_coordinates(points[, ...])	Wrap array of fractional coordinates to lie within unit cell.

Attributes Documentation

bounding_box

region bounding_box.

cell

Alias for cell_matrix.

cell_matrix

Matrix of lattice row vectors.

Same as the matrix transpose of the matrix product LatticeBase.orientation_matrix $\times$ LatticeBase.ortho_matrix:

$$\begin{split}[A] = ([R][M])^T \begin{pmatrix} \mathbf{a}\\ \mathbf{b}\\ \mathbf{c} \end{pmatrix} =([R][\mathbf{a}\,\mathbf{b}\,\mathbf{c}])^T\end{split}$$

centroid

Region centroid.

fractional_matrix

Transformation matrix to convert from cartesian coordinates to fractional coordinates.

The fractional matrix $[Q]$ is given by the inverse of the orthogonal matrix $[M]$ (see: ortho_matrix), i.e. $[Q]=[M]^{-1}$

$$\begin{split}[Q] = [M]^{-1} =\begin{pmatrix} 1/a & -1/a\tan\gamma & (\cos\alpha\cos\gamma - \cos\beta)/ a\phi\sin\gamma\\ 0 & 1/b\sin\gamma & (\cos\beta\cos\gamma - \cos\alpha)/ b\phi\sin\gamma\\ 0 & 0 & \sin\gamma/c\phi \end{pmatrix}\end{split}$$

where:

$$\begin{split}\begin{align*} \phi &= \frac{V}{abc}\\ &=\sqrt{1 - \cos^2\alpha - \cos^2\beta - \cos^2\gamma + 2\cos\alpha\cos\beta\cos\gamma}\\ &=\sin\alpha\sin\beta\sin\gamma^* \end{align*}\end{split}$$

where $V$ is the volume of the unit cell.

matrix

Alias for cell_matrix.

metric_tensor

Metric tensor.

offset

Lattice offset.

ortho_matrix

Transformation matrix to convert from fractional coordinates to cartesian coordinates.

$$\begin{split}[M] = \begin{pmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{pmatrix} = \begin{pmatrix} a & b\cos\gamma & c\cos\beta\\ 0 & b\sin\gamma & c(\cos\alpha - \cos\beta\cos\gamma)/\sin\gamma\\ 0 & 0 & c\sin\alpha\sin\beta\sin\gamma^*/\sin\gamma \end{pmatrix}\end{split}$$

region

Parallelepiped defined by lattice vectors.

Methods Documentation

cartesian_to_fractional(ccoords)

Convert cartesian coordinate to fractional coordinate.

Parameters:ccoords (array_like) – Returns: Return type:ndarray

fractional_diff(fcoords1, fcoords2)

Compute difference between fractional coordinates.

See also

core.crystallography.pbc_diff

fractional_to_cartesian(fcoords)

Convert fractional coordinate to cartesian coordinate.

Parameters:fcoords (array_like) – Returns: Return type:ndarray

rotate(**kwargs)

Rotate unit cell.

Parameters:

• angle (float) –
• axis (Vector, optional) –
• anchor_point (Point, optional) –
• rot_point (Point, optional) –
• to_vector (from_vector,) –
• degrees (bool, optional) –
• transform_matrix (ndarray) –

See also

core.math.rotate

translate(t)

Translate lattice.

Parameters:t (Vector) –

See also

core.math.translate

wrap_cartesian_coordinate(p, epsilon=1e-08, pbc=None)

Wrap cartesian coordinate to lie within unit cell.

Parameters:p (array_like) – Returns: Return type:ndarray

wrap_cartesian_coordinates(points, epsilon=1e-08, pbc=None)

Wrap array of cartesian coordinates to lie within unit cell.

Parameters:p (array_like) – Returns: Return type:ndarray

wrap_fractional_coordinate(p, epsilon=1e-08, pbc=None)

Wrap fractional coordinate to lie within unit cell.

Parameters:p (array_like) – Returns: Return type:ndarray

wrap_fractional_coordinates(points, epsilon=1e-08, pbc=None)

Wrap array of fractional coordinates to lie within unit cell.

Parameters:p (array_like) – Returns: Return type:ndarray