pymatgen.core.lattice module

Defines the classes relating to 3D lattices.

class Lattice(matrix: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]], pbc: tuple[bool, bool, bool] = (True, True, True))

Bases: MSONable

A lattice object. Essentially a matrix with conversion matrices. In general, it is assumed that length units are in Angstroms and angles are in degrees unless otherwise stated.

Create a lattice from any sequence of 9 numbers. Note that the sequence is assumed to be read one row at a time. Each row represents one lattice vector.

Parameters:

• matrix – Sequence of numbers in any form. Examples of acceptable input. i) An actual numpy array. ii) [[1, 0, 0], [0, 1, 0], [0, 0, 1]] iii) [1, 0, 0 , 0, 1, 0, 0, 0, 1] iv) (1, 0, 0, 0, 1, 0, 0, 0, 1) Each row should correspond to a lattice vector. E.g., [[10, 0, 0], [20, 10, 0], [0, 0, 30]] specifies a lattice with lattice vectors [10, 0, 0], [20, 10, 0] and [0, 0, 30].

• pbc – a tuple defining the periodic boundary conditions along the three axis of the lattice. If None periodic in all directions.

property a: float

a lattice parameter.

property abc: tuple[float, float, float]

Lengths of the lattice vectors, i.e. (a, b, c)

property alpha: float

Angle alpha of lattice in degrees.

property angles: tuple[float, float, float]

Returns the angles (alpha, beta, gamma) of the lattice.

as_dict(verbosity: int = 0) → dict

Json-serialization dict representation of the Lattice.

Parameters:

verbosity (int) – Verbosity level. Default of 0 only includes the matrix representation. Set to 1 for more details.

property b: float

b lattice parameter.

property beta: float

Angle beta of lattice in degrees.

property c: float

c lattice parameter.

copy()

Deep copy of self.

static cubic(a: float, pbc: tuple[bool, bool, bool] = (True, True, True)) → Lattice

Convenience constructor for a cubic lattice.

Parameters:

• a (float) – The a lattice parameter of the cubic cell.

• pbc (tuple) – a tuple defining the periodic boundary conditions along the three axis of the lattice. If None periodic in all directions.

Returns:

Cubic lattice of dimensions a x a x a.

d_hkl(miller_index: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]]) → float

Returns the distance between the hkl plane and the origin

Parameters:

miller_index ([h,k,l]) – Miller index of plane

Returns:

d_hkl (float)

dot(coords_a: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]], coords_b: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]], frac_coords: bool = False) → ndarray

Compute the scalar product of vector(s).

Parameters:

• coords_a – Array-like objects with the coordinates.

• coords_b – Array-like objects with the coordinates.

• frac_coords (bool) – Boolean stating whether the vector corresponds to fractional or Cartesian coordinates.

Returns:

one-dimensional numpy array.

find_all_mappings(other_lattice: Lattice, ltol: float = 1e-05, atol: float = 1, skip_rotation_matrix: bool = False) → Iterator[tuple[Lattice, np.ndarray | None, np.ndarray]]

Finds all mappings between current lattice and another lattice.

Parameters:

• other_lattice (Lattice) – Another lattice that is equivalent to this one.

• ltol (float) – Tolerance for matching lengths. Defaults to 1e-5.

• atol (float) – Tolerance for matching angles. Defaults to 1.

• skip_rotation_matrix (bool) – Whether to skip calculation of the rotation matrix

Yields:

(aligned_lattice, rotation_matrix, scale_matrix) if a mapping is found. aligned_lattice is a rotated version of other_lattice that has the same lattice parameters, but which is aligned in the coordinate system of this lattice so that translational points match up in 3D. rotation_matrix is the rotation that has to be applied to other_lattice to obtain aligned_lattice, i.e., aligned_matrix = np.inner(other_lattice, rotation_matrix) and op = SymmOp.from_rotation_and_translation(rotation_matrix) aligned_matrix = op.operate_multi(latt.matrix) Finally, scale_matrix is the integer matrix that expresses aligned_matrix as a linear combination of this lattice, i.e., aligned_matrix = np.dot(scale_matrix, self.matrix)

None is returned if no matches are found.

find_mapping(other_lattice: Lattice, ltol: float = 1e-05, atol: float = 1, skip_rotation_matrix: bool = False) → tuple[Lattice, np.ndarray | None, np.ndarray] | None

Finds a mapping between current lattice and another lattice. There are an infinite number of choices of basis vectors for two entirely equivalent lattices. This method returns a mapping that maps other_lattice to this lattice.

Parameters:

• other_lattice (Lattice) – Another lattice that is equivalent to this one.

• ltol (float) – Tolerance for matching lengths. Defaults to 1e-5.

• atol (float) – Tolerance for matching angles. Defaults to 1.

Returns:

(aligned_lattice, rotation_matrix, scale_matrix) if a mapping is found. aligned_lattice is a rotated version of other_lattice that has the same lattice parameters, but which is aligned in the coordinate system of this lattice so that translational points match up in 3D. rotation_matrix is the rotation that has to be applied to other_lattice to obtain aligned_lattice, i.e., aligned_matrix = np.inner(other_lattice, rotation_matrix) and op = SymmOp.from_rotation_and_translation(rotation_matrix) aligned_matrix = op.operate_multi(latt.matrix) Finally, scale_matrix is the integer matrix that expresses aligned_matrix as a linear combination of this lattice, i.e., aligned_matrix = np.dot(scale_matrix, self.matrix)

None is returned if no matches are found.

classmethod from_dict(d: dict, fmt: Optional[str] = None, **kwargs)

Create a Lattice from a dictionary containing the a, b, c, alpha, beta, and gamma parameters if fmt is None.

If fmt == “abivars”, the function build a Lattice object from a dictionary with the Abinit variables acell and rprim in Bohr. If acell is not given, the Abinit default is used i.e. [1,1,1] Bohr

Example

Lattice.from_dict(fmt=”abivars”, acell=3*[10], rprim=np.eye(3))

classmethod from_parameters(a: float, b: float, c: float, alpha: float, beta: float, gamma: float, vesta: bool = False, pbc: tuple[bool, bool, bool] = (True, True, True))

Create a Lattice using unit cell lengths (in Angstrom) and angles (in degrees).

Parameters:

• a (float) – a lattice parameter.

• b (float) – b lattice parameter.

• c (float) – c lattice parameter.

• alpha (float) – alpha angle in degrees.

• beta (float) – beta angle in degrees.

• gamma (float) – gamma angle in degrees.

• vesta – True if you import Cartesian coordinates from VESTA.

• pbc (tuple) – a tuple defining the periodic boundary conditions along the three axis of the lattice. If None periodic in all directions.

Returns:

Lattice with the specified lattice parameters.

property gamma: float

Angle gamma of lattice in degrees.

get_all_distances(fcoords1: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]], fcoords2: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]]) → ndarray

Returns the distances between two lists of coordinates taking into account periodic boundary conditions and the lattice. Note that this computes an MxN array of distances (i.e. the distance between each point in fcoords1 and every coordinate in fcoords2). This is different functionality from pbc_diff.

Parameters:

• fcoords1 – First set of fractional coordinates. e.g., [0.5, 0.6, 0.7] or [[1.1, 1.2, 4.3], [0.5, 0.6, 0.7]]. It can be a single coord or any array of coords.

• fcoords2 – Second set of fractional coordinates.

Returns:

2d array of Cartesian distances. E.g the distance between fcoords1[i] and fcoords2[j] is distances[i,j]

get_brillouin_zone() → list[list[numpy.ndarray]]

Returns the Wigner-Seitz cell for the reciprocal lattice, aka the Brillouin Zone.

Returns:

A list of list of coordinates. Each element in the list is a “facet” of the boundary of the Brillouin Zone. For instance, a list of four coordinates will represent a square facet.

get_cartesian_coords(fractional_coords: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]]) → ndarray

Returns the Cartesian coordinates given fractional coordinates.

Parameters:

fractional_coords (3x1 array) – Fractional coords.

Returns:

Cartesian coordinates

get_distance_and_image(frac_coords1: ArrayLike, frac_coords2: ArrayLike, jimage: ArrayLike | None = None) → tuple[float, np.ndarray]

Gets distance between two frac_coords assuming periodic boundary conditions. If the index jimage is not specified it selects the j image nearest to the i atom and returns the distance and jimage indices in terms of lattice vector translations. If the index jimage is specified it returns the distance between the frac_coords1 and the specified jimage of frac_coords2, and the given jimage is also returned.

Parameters:

• frac_coords1 (3x1 array) – Reference fcoords to get distance from.

• frac_coords2 (3x1 array) – fcoords to get distance from.

• jimage (3x1 array) – Specific periodic image in terms of lattice translations, e.g., [1,0,0] implies to take periodic image that is one a-lattice vector away. If jimage is None, the image that is nearest to the site is found.

Returns:

distance and periodic lattice translations of the other site for which the distance applies. This means that the distance between frac_coords1 and (jimage + frac_coords2) is equal to distance.

Return type:

(distance, jimage)

get_frac_coords_from_lll(lll_frac_coords: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]]) → ndarray

Given fractional coordinates in the lll basis, returns corresponding fractional coordinates in the lattice basis.

get_fractional_coords(cart_coords: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]]) → ndarray

Returns the fractional coordinates given Cartesian coordinates.

Parameters:

cart_coords (3x1 array) – Cartesian coords.

Returns:

Fractional coordinates.

get_lll_frac_coords(frac_coords: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]]) → ndarray

Given fractional coordinates in the lattice basis, returns corresponding fractional coordinates in the lll basis.

get_lll_reduced_lattice(delta: float = 0.75) → Lattice

Parameters:

delta – Delta parameter.

Returns:

LLL reduced Lattice.

get_miller_index_from_coords(coords: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]], coords_are_cartesian: bool = True, round_dp: int = 4, verbose: bool = True) → tuple[int, int, int]

Get the Miller index of a plane from a list of site coordinates.

A minimum of 3 sets of coordinates are required. If more than 3 sets of coordinates are given, the best plane that minimises the distance to all points will be calculated.

Parameters:

• coords (iterable) – A list or numpy array of coordinates. Can be Cartesian or fractional coordinates. If more than three sets of coordinates are provided, the best plane that minimises the distance to all sites will be calculated.

• coords_are_cartesian (bool, optional) – Whether the coordinates are in Cartesian space. If using fractional coordinates set to False.

• round_dp (int, optional) – The number of decimal places to round the miller index to.

• verbose (bool, optional) – Whether to print warnings.

Returns:

The Miller index.

Return type:

(tuple)

get_niggli_reduced_lattice(tol: float = 1e-05) → Lattice

Get the Niggli reduced lattice using the numerically stable algo proposed by R. W. Grosse-Kunstleve, N. K. Sauter, & P. D. Adams, Acta Crystallographica Section A Foundations of Crystallography, 2003, 60(1), 1-6. doi:10.1107/S010876730302186X

Parameters:

tol (float) – The numerical tolerance. The default of 1e-5 should result in stable behavior for most cases.

Returns:

Niggli-reduced lattice.

get_points_in_sphere(frac_points: ArrayLike, center: ArrayLike, r: float, zip_results=True) → list[tuple[np.ndarray, float, int, np.ndarray]] | list[np.ndarray] | list

Find all points within a sphere from the point taking into account periodic boundary conditions. This includes sites in other periodic images.

Algorithm:

1. place sphere of radius r in crystal and determine minimum supercell (parallelpiped) which would contain a sphere of radius r. for this we need the projection of a_1 on a unit vector perpendicular to a_2 & a_3 (i.e. the unit vector in the direction b_1) to determine how many a_1”s it will take to contain the sphere.

   Nxmax = r * length_of_b_1 / (2 Pi)

2. keep points falling within r.

Parameters:

• frac_points – All points in the lattice in fractional coordinates.

• center – Cartesian coordinates of center of sphere.

• r – radius of sphere.

• zip_results (bool) – Whether to zip the results together to group by point, or return the raw fcoord, dist, index arrays

Returns:

[(fcoord, dist, index, supercell_image) …] since most of the time, subsequent

processing requires the distance, index number of the atom, or index of the image

else:

fcoords, dists, inds, image

Return type:

if zip_results

get_points_in_sphere_old(**kwargs)

get_points_in_sphere_py(frac_points: ArrayLike, center: ArrayLike, r: float, zip_results=True) → list[tuple[np.ndarray, float, int, np.ndarray]] | list[np.ndarray]

Find all points within a sphere from the point taking into account periodic boundary conditions. This includes sites in other periodic images.

Algorithm:

1. place sphere of radius r in crystal and determine minimum supercell (parallelpiped) which would contain a sphere of radius r. for this we need the projection of a_1 on a unit vector perpendicular to a_2 & a_3 (i.e. the unit vector in the direction b_1) to determine how many a_1”s it will take to contain the sphere.

   Nxmax = r * length_of_b_1 / (2 Pi)

2. keep points falling within r.

Parameters:

• frac_points – All points in the lattice in fractional coordinates.

• center – Cartesian coordinates of center of sphere.

• r – radius of sphere.

• zip_results (bool) – Whether to zip the results together to group by point, or return the raw fcoord, dist, index arrays

Returns:

[(fcoord, dist, index, supercell_image) …] since most of the time, subsequent

processing requires the distance, index number of the atom, or index of the image

else:

fcoords, dists, inds, image

Return type:

if zip_results

get_recp_symmetry_operation(symprec: float = 0.01) → list

Find the symmetric operations of the reciprocal lattice, to be used for hkl transformations :param symprec: default is 0.001

get_vector_along_lattice_directions(cart_coords: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]]) → ndarray

Returns the coordinates along lattice directions given Cartesian coordinates.

Note, this is different than a projection of the Cartesian vector along the lattice parameters. It is simply the fractional coordinates multiplied by the lattice vector magnitudes.

For example, this method is helpful when analyzing the dipole moment (in units of electron Angstroms) of a ferroelectric crystal. See the Polarization class in pymatgen.analysis.ferroelectricity.polarization.

Parameters:

cart_coords (3x1 array) – Cartesian coords.

Returns:

Lattice coordinates.

get_wigner_seitz_cell() → list[list[numpy.ndarray]]

Returns the Wigner-Seitz cell for the given lattice.

Returns:

A list of list of coordinates. Each element in the list is a “facet” of the boundary of the Wigner Seitz cell. For instance, a list of four coordinates will represent a square facet.

static hexagonal(a: float, c: float, pbc: tuple[bool, bool, bool] = (True, True, True)) → Lattice

Convenience constructor for a hexagonal lattice.

Parameters:

• a (float) – a lattice parameter of the hexagonal cell.

• c (float) – c lattice parameter of the hexagonal cell.

• pbc (tuple) – a tuple defining the periodic boundary conditions along the three axis of the lattice. If None periodic in all directions.

Returns:

Hexagonal lattice of dimensions a x a x c.

property inv_matrix: ndarray

Inverse of lattice matrix.

property is_3d_periodic: bool

True if the Lattice is periodic in all directions

is_hexagonal(hex_angle_tol: float = 5, hex_length_tol: float = 0.01) → bool

Parameters:

• hex_angle_tol – Angle tolerance

• hex_length_tol – Length tolerance

Returns:

Whether lattice corresponds to hexagonal lattice.

property is_orthogonal: bool

Whether all angles are 90 degrees.

Type:

return

property lengths: tuple[float, float, float]

The lengths (a, b, c) of the lattice.

Type:

return

property lll_inverse: ndarray

Inverse of self.lll_mapping.

Type:

return

property lll_mapping: ndarray

The mapping between the LLL reduced lattice and the original lattice.

Type:

return

property lll_matrix: ndarray

The matrix for LLL reduction

Type:

return

property matrix: ndarray

Copy of matrix representing the Lattice

property metric_tensor: ndarray

The metric tensor of the lattice.

static monoclinic(a: float, b: float, c: float, beta: float, pbc: tuple[bool, bool, bool] = (True, True, True)) → Lattice

Convenience constructor for a monoclinic lattice.

Parameters:

• a (float) – a lattice parameter of the monoclinc cell.

• b (float) – b lattice parameter of the monoclinc cell.

• c (float) – c lattice parameter of the monoclinc cell.

• beta (float) – beta angle between lattice vectors b and c in degrees.

• pbc (tuple) – a tuple defining the periodic boundary conditions along the three axis of the lattice. If None periodic in all directions.

Returns:

Monoclinic lattice of dimensions a x b x c with non right-angle beta between lattice vectors a and c.

norm(coords: Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]], frac_coords: bool = True) → ndarray

Compute the norm of vector(s).

Parameters:

• coords – Array-like object with the coordinates.

• frac_coords – Boolean stating whether the vector corresponds to fractional or Cartesian coordinates.

Returns:

one-dimensional numpy array.

static orthorhombic(a: float, b: float, c: float, pbc: tuple[bool, bool, bool] = (True, True, True)) → Lattice

Convenience constructor for an orthorhombic lattice.

Parameters:

• a (float) – a lattice parameter of the orthorhombic cell.

• b (float) – b lattice parameter of the orthorhombic cell.

• c (float) – c lattice parameter of the orthorhombic cell.

• pbc (tuple) – a tuple defining the periodic boundary conditions along the three axis of the lattice. If None periodic in all directions.

Returns:

Orthorhombic lattice of dimensions a x b x c.

property parameters: tuple[float, float, float, float, float, float]

(a, b, c, alpha, beta, gamma).

Type:

Returns

property pbc: tuple[bool, bool, bool]

Tuple defining the periodicity of the Lattice

property reciprocal_lattice: Lattice

Return the reciprocal lattice. Note that this is the standard reciprocal lattice used for solid state physics with a factor of 2 * pi. If you are looking for the crystallographic reciprocal lattice, use the reciprocal_lattice_crystallographic property. The property is lazily generated for efficiency.

property reciprocal_lattice_crystallographic: Lattice

Returns the crystallographic reciprocal lattice, i.e., no factor of 2 * pi.

static rhombohedral(a: float, alpha: float, pbc: tuple[bool, bool, bool] = (True, True, True)) → Lattice

Convenience constructor for a rhombohedral lattice.

Parameters:

• a (float) – a lattice parameter of the rhombohedral cell.

• alpha (float) – Angle for the rhombohedral lattice in degrees.

• pbc (tuple) – a tuple defining the periodic boundary conditions along the three axis of the lattice. If None periodic in all directions.

Returns:

Rhombohedral lattice of dimensions a x a x a.

scale(new_volume: float) → Lattice

Return a new Lattice with volume new_volume by performing a scaling of the lattice vectors so that length proportions and angles are preserved.

Parameters:

new_volume – New volume to scale to.

Returns:

New lattice with desired volume.

selling_dist(other)

Returns the minimum Selling distance between two lattices.

property selling_vector: ndarray

Returns the (1,6) array of Selling Scalars.

static tetragonal(a: float, c: float, pbc: tuple[bool, bool, bool] = (True, True, True)) → Lattice

Convenience constructor for a tetragonal lattice.

Parameters:

• a (float) – a lattice parameter of the tetragonal cell.

• c (float) – c lattice parameter of the tetragonal cell.

• pbc (tuple) – a tuple defining the periodic boundary conditions along the three axis of the lattice. If None periodic in all directions.

Returns:

Tetragonal lattice of dimensions a x a x c.

property volume: float

Volume of the unit cell.

find_neighbors(label: ndarray, nx: int, ny: int, nz: int) → list[numpy.ndarray]

Given a cube index, find the neighbor cube indices

Parameters:

• label – (array) (n,) or (n x 3) indice array

• nx – (int) number of cells in y direction

• ny – (int) number of cells in y direction

• nz – (int) number of cells in z direction

Returns: neighbor cell indices

get_integer_index(miller_index: Sequence[float], round_dp: int = 4, verbose: bool = True) → tuple[int, int, int]

Attempt to convert a vector of floats to whole numbers.

Parameters:

• miller_index (list of float) – A list miller indexes.

• round_dp (int, optional) – The number of decimal places to round the miller index to.

• verbose (bool, optional) – Whether to print warnings.

Returns:

The Miller index.

Return type:

(tuple)

get_points_in_spheres(all_coords: np.ndarray, center_coords: np.ndarray, r: float, pbc: bool | list[bool] | tuple[bool, bool, bool] = True, numerical_tol: float = 1e-08, lattice: Lattice = None, return_fcoords: bool = False) → list[list[tuple[np.ndarray, float, int, np.ndarray]]]

For each point in center_coords, get all the neighboring points in all_coords that are within the cutoff radius r.

Parameters:

• all_coords – (list of Cartesian coordinates) all available points

• center_coords – (list of Cartesian coordinates) all centering points

• r – (float) cutoff radius

• pbc – (bool or a list of bool) whether to set periodic boundaries

• numerical_tol – (float) numerical tolerance

• lattice – (Lattice) lattice to consider when PBC is enabled

• return_fcoords – (bool) whether to return fractional coords when pbc is set.

Returns:

List[List[Tuple[coords, distance, index, image]]]