Lattices (erlab.lattice)

Tools related to the real and reciprocal lattice.

Functions

angle_between(v1, v2)	Return the angle between two vectors.
abc2avec(a, b, c, alpha, beta, gamma)	Construct lattice vectors from lattice parameters.
avec2abc(avec)	Determine lattice parameters from lattice vectors.
to_reciprocal(avec)	Construct the reciprocal lattice vectors from real lattice vectors.
to_real(bvec)	Construct the real lattice vectors from reciprocal lattice vectors.

erlab.lattice.abc2avec(a, b, c, alpha, beta, gamma)

Construct lattice vectors from lattice parameters.

Parameters:

• a (floating) – Lattice parameter \(a\).

• b (floating) – Lattice parameter \(b\).

• c (floating) – Lattice parameter \(c\).

• alpha (floating) – Lattice parameter \(\alpha\) in degrees.

• beta (floating) – Lattice parameter \(\beta\) in degrees.

• gamma (floating) – Lattice parameter \(\gamma\) in degrees.

Returns:

avec – Real lattice vectors, given as a 3 by 3 numpy array with each basis vector given in each row.

Return type:

ndarray[tuple[int, …], dtype[floating]]

erlab.lattice.angle_between(v1, v2)

Return the angle between two vectors.

Parameters:

• v1 (array-like) – 1D array of length 3, specifying a vector.

• v2 (array-like) – 1D array of length 3, specifying a vector.

Returns:

float – The angle in degrees.

Return type:

float

erlab.lattice.avec2abc(avec)

Determine lattice parameters from lattice vectors.

Parameters:

avec (ndarray[tuple[int, ...], dtype[floating]]) – Real lattice vectors, given as a 3 by 3 numpy array with each basis vector given in each row.

Returns:

a, b, c, alpha, beta, gamma

Return type:

tuple[float, float, float, float, float, float]

erlab.lattice.get_2d_vertices(basis, *, reciprocal=True, rotate=0.0, offset=(0.0, 0.0))

Get the vertices of a 2D Brillouin zone.

Unlike get_bz_edge(), this function only returns the vertices of the BZ. The vertices are ordered such that they can be used to create a polygon. Also, this function allows for rotation and offset of the BZ.

Parameters:

• basis (ndarray[tuple[int, ...], dtype[floating]]) – A 2D or 3D numpy array with shape (N, N) where N = 2 or 3, containing the basis vectors of the lattice. If N is 3, only the upper left 2x2 submatrix is used.

• reciprocal (bool, default: True) – If True, the basis are treated as reciprocal lattice vectors. If False, the basis are treated as real space lattice vectors.

• rotate (float, default: 0.0) – Rotation angle in degrees to apply to the BZ.

• offset (tuple[float, float], default: (0.0, 0.0)) – Offset for the Brillouin zone center in the form of a tuple (x, y).

Returns:

vertices (array-like) – Vertices of the BZ.

Return type:

ndarray[tuple[int, …], dtype[floating]]

erlab.lattice.get_bz_edge(basis, reciprocal=True, extend=None)

Calculate the edge of the first Brillouin zone (BZ) from lattice vectors.

Parameters:

• basis (ndarray[tuple[int, ...], dtype[floating]]) – (N, N) numpy array where N = 2 or 3 with each row containing the lattice vectors.

• reciprocal (bool, default: True) – If False, the basis are given in real space lattice vectors.

• extend (tuple[int, ...] | None, default: None) – Tuple of positive integers specifying the number of times to extend the BZ in each direction. If None, only the first BZ is returned (equivalent to (1,) * N).

Returns:

• lines (array-like) – (M, 2, N) array that specifies the endpoints of the M lines that make up the BZ edge, where N = len(basis).

• vertices (array-like) – Vertices of the BZ.

Return type:

tuple[ndarray[tuple[int, …], dtype[floating]], ndarray[tuple[int, …], dtype[floating]]]

erlab.lattice.to_real(bvec)

Construct the real lattice vectors from reciprocal lattice vectors.

Parameters:

bvec (ndarray[tuple[int, ...], dtype[floating]]) – Reciprocal lattice vectors, given as a 3 by 3 numpy array with each basis vector given in each row.

Returns:

avec – The real lattice vectors.

Return type:

ndarray[tuple[int, …], dtype[floating]]

erlab.lattice.to_reciprocal(avec)

Construct the reciprocal lattice vectors from real lattice vectors.

Parameters:

avec (ndarray[tuple[int, ...], dtype[floating]]) – Real lattice vectors, given as a 3 by 3 numpy array with each basis vector given in each row.

Returns:

bvec – The reciprocal lattice vectors.

Return type:

ndarray[tuple[int, …], dtype[floating]]