kristal.geometry¶

Various utilities for working with atoms’ equivalence positions.

class kristal.geometry.equiv.FractionalCoordinateSystem(a, b, c, alpha, beta, gamma)¶

Class representing fractional coordinates system.

Variables:	a (float) – length of first periodic vector. b (float) – length of second periodic vector. c (float) – length of second periodic vector. alpha (float) – an angle between second and third periodic vector. beta (float) – an angle between first and third periodic vector. gamma (float) – an angle between first and second periodic vector.
Parameters:	a (float) – length of first periodic vector. b (float) – length of second periodic vector. c (float) – length of second periodic vector. alpha (float) – an angle between second and third periodic vector. beta (float) – an angle between first and third periodic vector. gamma (float) – an angle between first and second periodic vector.

change_of_basis_matrix()¶

Change of basis matrix from fractional to Cartesian coordinates.

Let \((u, v, w)\) be components of arbitrary vector in fractional coordinates. Components of this vector in Cartesian coordinates are given by

\[\begin{split}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = M \cdot \begin{bmatrix} u \\ v \\ w \end{bmatrix}\end{split}\]

where \(M\) is given by

\[\begin{split}M = \begin{bmatrix} a & b \cos(\gamma) & c \cos(\beta) \\ 0 & b \sin(\gamma) & c \frac{\cos(\alpha) - \cos(\beta)\cos(\gamma)}{\sin(\gamma)} \\ 0 & 0 & \frac{V}{ab \sin(\gamma)} \end{bmatrix}\end{split}\]

This method returns \(M\) from the above equation.

inv_change_of_basis_matrix()¶

“Matrix changing basis from this system to Cartesian.

This is an inverse of a matrix \(M\) returned by change_of_basis_matrix().

unit_cell_volume¶

Volume of the cell in this coordinates system.

Volume of the unit cell is defined as a volume of parallelepiped determined by periodic vectors and is given by:

\[V = abc \sqrt{1 - \cos^2(\alpha) - \cos^2(\beta) - \cos^2(\gamma) + 2\cos(\alpha)\cos(\beta) \cos(\gamma)}\]

kristal.geometry.equiv.cos2(x)¶: Shortcut for computing np.cos(x) ** 2.

kristal.geometry.equiv.sin2(x)¶: Shortcut for computing np.sin(x) ** 2.