# Source code for pymatgen.core.lattice

# coding: utf-8
# Copyright (c) Pymatgen Development Team.

import math
import itertools
import warnings

from functools import reduce
from math import gcd

from fractions import Fraction

from typing import List, Union, Dict, Tuple, Iterator, Optional
from pymatgen.util.typing import Vector3Like

import numpy as np
from numpy.linalg import inv
from numpy import pi, dot, transpose, radians

from monty.json import MSONable
from pymatgen.util.coord import pbc_shortest_vectors
from pymatgen.util.num import abs_cap

"""
This module defines the classes relating to 3D lattices.
"""

__author__ = "Shyue Ping Ong, Michael Kocher"
__version__ = "1.0"
__maintainer__ = "Shyue Ping Ong"
__email__ = "shyuep@gmail.com"
__status__ = "Production"
__date__ = "Sep 23, 2011"

[docs]class Lattice(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.
"""

# Properties lazily generated for efficiency.

def __init__(self, matrix: Union[List[float], List[List[float]], np.ndarray]):
"""
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.

Args:
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].
"""
m = np.array(matrix, dtype=np.float64).reshape((3, 3))
lengths = np.sqrt(np.sum(m ** 2, axis=1))
angles = np.zeros(3)
for i in range(3):
j = (i + 1) % 3
k = (i + 2) % 3
angles[i] = abs_cap(dot(m[j], m[k]) / (lengths[j] * lengths[k]))

self._angles = np.arccos(angles) * 180.0 / pi
self._lengths = lengths

m.setflags(write=False)
self._matrix = m

self._inv_matrix = None
self._metric_tensor = None
self._diags = None
self._lll_matrix_mappings = {}
self._lll_inverse = None
self.is_orthogonal = all([abs(a - 90) < 1e-5 for a in self._angles])

def __format__(self, fmt_spec=""):
"""
Support format printing. Supported formats are:

1. "l" for a list format that can be easily copied and pasted, e.g.,
".3fl" prints something like
"[[10.000, 0.000, 0.000], [0.000, 10.000, 0.000], [0.000, 0.000, 10.000]]"
2. "p" for lattice parameters ".1fp" prints something like
"{10.0, 10.0, 10.0, 90.0, 90.0, 90.0}"
3. Default will simply print a 3x3 matrix form. E.g.,
10.000 0.000 0.000
0.000 10.000 0.000
0.000 0.000 10.000
"""
m = self.matrix.tolist()
if fmt_spec.endswith("l"):
fmt = "[[{}, {}, {}], [{}, {}, {}], [{}, {}, {}]]"
fmt_spec = fmt_spec[:-1]
elif fmt_spec.endswith("p"):
fmt = "{{{}, {}, {}, {}, {}, {}}}"
fmt_spec = fmt_spec[:-1]
m = self.lengths_and_angles
else:
fmt = "{} {} {}\n{} {} {}\n{} {} {}"
return fmt.format(*[format(c, fmt_spec) for row in m for c in row])

[docs]    def copy(self):
"""Deep copy of self."""
return self.__class__(self.matrix.copy())

@property
def matrix(self) -> np.ndarray:
"""Copy of matrix representing the Lattice"""
return self._matrix

@property
def inv_matrix(self) -> np.ndarray:
"""
Inverse of lattice matrix.
"""
if self._inv_matrix is None:
self._inv_matrix = inv(self._matrix)
self._inv_matrix.setflags(write=False)
return self._inv_matrix

@property
def metric_tensor(self) -> np.ndarray:
"""
The metric tensor of the lattice.
"""
if self._metric_tensor is None:
self._metric_tensor = np.dot(self._matrix, self._matrix.T)
return self._metric_tensor

[docs]    def get_cartesian_coords(self, fractional_coords: Vector3Like) -> np.ndarray:
"""
Returns the cartesian coordinates given fractional coordinates.

Args:
fractional_coords (3x1 array): Fractional coords.

Returns:
Cartesian coordinates
"""
return dot(fractional_coords, self._matrix)

[docs]    def get_fractional_coords(self, cart_coords: Vector3Like) -> np.ndarray:
"""
Returns the fractional coordinates given cartesian coordinates.

Args:
cart_coords (3x1 array): Cartesian coords.

Returns:
Fractional coordinates.
"""
return dot(cart_coords, self.inv_matrix)

[docs]    def get_vector_along_lattice_directions(self, cart_coords: Vector3Like):
"""
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.

Args:
cart_coords (3x1 array): Cartesian coords.

Returns:
Lattice coordinates.
"""
return self.lengths_and_angles[0] * self.get_fractional_coords(cart_coords)

[docs]    def d_hkl(self, miller_index: Vector3Like) -> float:
"""
Returns the distance between the hkl plane and the origin

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

Returns:
d_hkl (float)
"""

gstar = self.reciprocal_lattice_crystallographic.metric_tensor
hkl = np.array(miller_index)
return 1 / ((np.dot(np.dot(hkl, gstar), hkl.T)) ** (1 / 2))

[docs]    @staticmethod
def cubic(a: float):
"""
Convenience constructor for a cubic lattice.

Args:
a (float): The *a* lattice parameter of the cubic cell.

Returns:
Cubic lattice of dimensions a x a x a.
"""
return Lattice([[a, 0.0, 0.0], [0.0, a, 0.0], [0.0, 0.0, a]])

[docs]    @staticmethod
def tetragonal(a: float, c: float):
"""
Convenience constructor for a tetragonal lattice.

Args:
a (float): *a* lattice parameter of the tetragonal cell.
c (float): *c* lattice parameter of the tetragonal cell.

Returns:
Tetragonal lattice of dimensions a x a x c.
"""
return Lattice.from_parameters(a, a, c, 90, 90, 90)

[docs]    @staticmethod
def orthorhombic(a: float, b: float, c: float):
"""
Convenience constructor for an orthorhombic lattice.

Args:
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.

Returns:
Orthorhombic lattice of dimensions a x b x c.
"""
return Lattice.from_parameters(a, b, c, 90, 90, 90)

[docs]    @staticmethod
def monoclinic(a: float, b: float, c: float, beta: float):
"""
Convenience constructor for a monoclinic lattice.

Args:
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.

Returns:
Monoclinic lattice of dimensions a x b x c with non right-angle
beta between lattice vectors a and c.
"""
return Lattice.from_parameters(a, b, c, 90, beta, 90)

[docs]    @staticmethod
def hexagonal(a: float, c: float):
"""
Convenience constructor for a hexagonal lattice.

Args:
a (float): *a* lattice parameter of the hexagonal cell.
c (float): *c* lattice parameter of the hexagonal cell.

Returns:
Hexagonal lattice of dimensions a x a x c.
"""
return Lattice.from_parameters(a, a, c, 90, 90, 120)

[docs]    @staticmethod
def rhombohedral(a: float, alpha: float):
"""
Convenience constructor for a rhombohedral lattice.

Args:
a (float): *a* lattice parameter of the rhombohedral cell.
alpha (float): Angle for the rhombohedral lattice in degrees.

Returns:
Rhombohedral lattice of dimensions a x a x a.
"""
return Lattice.from_parameters(a, a, a, alpha, alpha, alpha)

[docs]    @staticmethod
def from_lengths_and_angles(abc: List[float], ang: List[float]):
"""
Create a Lattice using unit cell lengths and angles (in degrees).

Args:
abc (3x1 array): Lattice parameters, e.g. (4, 4, 5).
ang (3x1 array): Lattice angles in degrees, e.g., (90,90,120).

Returns:
A Lattice with the specified lattice parameters.
"""
return Lattice.from_parameters(abc[0], abc[1], abc[2], ang[0], ang[1], ang[2])

[docs]    @staticmethod
def from_parameters(
a: float,
b: float,
c: float,
alpha: float,
beta: float,
gamma: float,
vesta: bool = False,
):
"""
Create a Lattice using unit cell lengths and angles (in degrees).

Args:
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.

Returns:
Lattice with the specified lattice parameters.
"""

if vesta:
cos_alpha = np.cos(alpha_r)
cos_beta = np.cos(beta_r)
cos_gamma = np.cos(gamma_r)
sin_gamma = np.sin(gamma_r)

c1 = c * cos_beta
c2 = (c * (cos_alpha - (cos_beta * cos_gamma))) / sin_gamma

vector_a = [float(a), 0.0, 0.0]
vector_b = [b * cos_gamma, b * sin_gamma, 0]
vector_c = [c1, c2, math.sqrt(c ** 2 - c1 ** 2 - c2 ** 2)]

else:
val = (np.cos(alpha_r) * np.cos(beta_r) - np.cos(gamma_r)) / (
np.sin(alpha_r) * np.sin(beta_r)
)
# Sometimes rounding errors result in values slightly > 1.
val = abs_cap(val)
gamma_star = np.arccos(val)

vector_a = [a * np.sin(beta_r), 0.0, a * np.cos(beta_r)]
vector_b = [
-b * np.sin(alpha_r) * np.cos(gamma_star),
b * np.sin(alpha_r) * np.sin(gamma_star),
b * np.cos(alpha_r),
]
vector_c = [0.0, 0.0, float(c)]

return Lattice([vector_a, vector_b, vector_c])

[docs]    @classmethod
def from_dict(cls, d: Dict, fmt: 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))
"""
if fmt == "abivars":
from pymatgen.io.abinit.abiobjects import lattice_from_abivars

kwargs.update(d)
return lattice_from_abivars(cls=cls, **kwargs)

if "matrix" in d:
return cls(d["matrix"])
else:
return cls.from_parameters(
d["a"], d["b"], d["c"], d["alpha"], d["beta"], d["gamma"]
)

@property
def angles(self) -> Tuple[float]:
"""
Returns the angles (alpha, beta, gamma) of the lattice.
"""
return tuple(self._angles)

@property
def a(self) -> float:
"""
*a* lattice parameter.
"""
return self._lengths[0]

@property
def b(self) -> float:
"""
*b* lattice parameter.
"""
return self._lengths[1]

@property
def c(self) -> float:
"""
*c* lattice parameter.
"""
return self._lengths[2]

@property
def abc(self) -> Tuple[float]:
"""
Lengths of the lattice vectors, i.e. (a, b, c)
"""
return tuple(self._lengths)

@property
def alpha(self) -> float:
"""
Angle alpha of lattice in degrees.
"""
return self._angles[0]

@property
def beta(self) -> float:
"""
Angle beta of lattice in degrees.
"""
return self._angles[1]

@property
def gamma(self) -> float:
"""
Angle gamma of lattice in degrees.
"""
return self._angles[2]

@property
def volume(self) -> float:
"""
Volume of the unit cell.
"""
m = self._matrix
return abs(np.dot(np.cross(m[0], m[1]), m[2]))

@property
def lengths_and_angles(self) -> Tuple[Tuple[float, float, float], Tuple[float, float, float]]:
"""
Returns (lattice lengths, lattice angles).
"""
return tuple(self._lengths), tuple(self._angles)

@property
def reciprocal_lattice(self) -> "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.
"""
try:
return self._reciprocal_lattice
except AttributeError:
v = np.linalg.inv(self._matrix).T
self._reciprocal_lattice = Lattice(v * 2 * np.pi)
return self._reciprocal_lattice

@property
def reciprocal_lattice_crystallographic(self) -> "Lattice":
"""
Returns the *crystallographic* reciprocal lattice, i.e., no factor of
2 * pi.
"""
return Lattice(self.reciprocal_lattice.matrix / (2 * np.pi))

@property
def lll_matrix(self) -> np.ndarray:
if 0.75 not in self._lll_matrix_mappings:
self._lll_matrix_mappings[0.75] = self._calculate_lll()
return self._lll_matrix_mappings[0.75][0]

@property
def lll_mapping(self) -> np.ndarray:
if 0.75 not in self._lll_matrix_mappings:
self._lll_matrix_mappings[0.75] = self._calculate_lll()
return self._lll_matrix_mappings[0.75][1]

@property
def lll_inverse(self) -> np.ndarray:
if self._lll_inverse is not None:
return self._lll_inverse
else:
self._lll_inverse = np.linalg.inv(self.lll_mapping)
return self._lll_inverse

def __repr__(self):
outs = [
"Lattice",
"    abc : " + " ".join(map(repr, self._lengths)),
" angles : " + " ".join(map(repr, self._angles)),
" volume : " + repr(self.volume),
"      A : " + " ".join(map(repr, self._matrix[0])),
"      B : " + " ".join(map(repr, self._matrix[1])),
"      C : " + " ".join(map(repr, self._matrix[2])),
]
return "\n".join(outs)

def __eq__(self, other):
"""
A lattice is considered to be equal to another if the internal matrix
representation satisfies np.allclose(matrix1, matrix2) to be True.
"""
if other is None:
return False
# shortcut the np.allclose if the memory addresses are the same
# (very common in Structure.from_sites)
return self is other or np.allclose(self.matrix, other.matrix)

def __ne__(self, other):
return not self.__eq__(other)

def __hash__(self):
return 7

def __str__(self):
return "\n".join([" ".join(["%.6f" % i for i in row]) for row in self._matrix])

[docs]    def as_dict(self, verbosity: int = 0) -> Dict:
"""""
Json-serialization dict representation of the Lattice.

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

d = {
"@module": self.__class__.__module__,
"@class": self.__class__.__name__,
"matrix": self._matrix.tolist(),
}
if verbosity > 0:
d.update(
{
"a": float(self.a),
"b": float(self.b),
"c": float(self.c),
"alpha": float(self.alpha),
"beta": float(self.beta),
"gamma": float(self.gamma),
"volume": float(self.volume),
}
)

return d

[docs]    def find_all_mappings(
self,
other_lattice: "Lattice",
ltol: float = 1e-5,
atol: float = 1,
skip_rotation_matrix: bool = False,
) -> Iterator[Tuple["Lattice", Optional[np.ndarray], np.ndarray]]:
"""
Finds all mappings between current lattice and another lattice.

Args:
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.
"""
(lengths, angles) = other_lattice.lengths_and_angles
(alpha, beta, gamma) = angles

frac, dist, _, _ = self.get_points_in_sphere(
[[0, 0, 0]], [0, 0, 0], max(lengths) * (1 + ltol), zip_results=False
)
cart = self.get_cartesian_coords(frac)
# this can't be broadcast because they're different lengths
inds = [
np.logical_and(dist / l < 1 + ltol, dist / l > 1 / (1 + ltol))
for l in lengths
]
c_a, c_b, c_c = (cart[i] for i in inds)
f_a, f_b, f_c = (frac[i] for i in inds)
l_a, l_b, l_c = (np.sum(c ** 2, axis=-1) ** 0.5 for c in (c_a, c_b, c_c))

def get_angles(v1, v2, l1, l2):
x = np.inner(v1, v2) / l1[:, None] / l2
x[x > 1] = 1
x[x < -1] = -1
angles = np.arccos(x) * 180.0 / pi
return angles

alphab = np.abs(get_angles(c_b, c_c, l_b, l_c) - alpha) < atol
betab = np.abs(get_angles(c_a, c_c, l_a, l_c) - beta) < atol
gammab = np.abs(get_angles(c_a, c_b, l_a, l_b) - gamma) < atol

for i, all_j in enumerate(gammab):
inds = np.logical_and(
all_j[:, None], np.logical_and(alphab, betab[i][None, :])
)
for j, k in np.argwhere(inds):
scale_m = np.array((f_a[i], f_b[j], f_c[k]), dtype=np.int)
if abs(np.linalg.det(scale_m)) < 1e-8:
continue

aligned_m = np.array((c_a[i], c_b[j], c_c[k]))

if skip_rotation_matrix:
rotation_m = None
else:
rotation_m = np.linalg.solve(aligned_m, other_lattice.matrix)

yield Lattice(aligned_m), rotation_m, scale_m

[docs]    def find_mapping(
self,
other_lattice: "Lattice",
ltol: float = 1e-5,
atol: float = 1,
skip_rotation_matrix: bool = False,
) -> Optional[Tuple["Lattice", Optional[np.ndarray], np.ndarray]]:
"""
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.

Args:
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.
"""
for x in self.find_all_mappings(
other_lattice, ltol, atol, skip_rotation_matrix=skip_rotation_matrix
):
return x

[docs]    def get_lll_reduced_lattice(self, delta: float = 0.75) -> "Lattice":
if delta not in self._lll_matrix_mappings:
self._lll_matrix_mappings[delta] = self._calculate_lll()
return Lattice(self._lll_matrix_mappings[delta][0])

def _calculate_lll(self, delta: float = 0.75) -> Tuple[np.ndarray, np.ndarray]:
"""
Performs a Lenstra-Lenstra-Lovasz lattice basis reduction to obtain a
c-reduced basis. This method returns a basis which is as "good" as
possible, with "good" defined by orthongonality of the lattice vectors.

This basis is used for all the periodic boundary condition calculations.

Args:
delta (float): Reduction parameter. Default of 0.75 is usually
fine.

Returns:
Reduced lattice matrix, mapping to get to that lattice.
"""
# Transpose the lattice matrix first so that basis vectors are columns.
# Makes life easier.
a = self._matrix.copy().T

b = np.zeros((3, 3))  # Vectors after the Gram-Schmidt process
u = np.zeros((3, 3))  # Gram-Schmidt coeffieicnts
m = np.zeros(3)  # These are the norm squared of each vec.

b[:, 0] = a[:, 0]
m[0] = dot(b[:, 0], b[:, 0])
for i in range(1, 3):
u[i, 0:i] = dot(a[:, i].T, b[:, 0:i]) / m[0:i]
b[:, i] = a[:, i] - dot(b[:, 0:i], u[i, 0:i].T)
m[i] = dot(b[:, i], b[:, i])

k = 2

mapping = np.identity(3, dtype=np.double)
while k <= 3:
# Size reduction.
for i in range(k - 1, 0, -1):
q = round(u[k - 1, i - 1])
if q != 0:
# Reduce the k-th basis vector.
a[:, k - 1] = a[:, k - 1] - q * a[:, i - 1]
mapping[:, k - 1] = mapping[:, k - 1] - q * mapping[:, i - 1]
uu = list(u[i - 1, 0 : (i - 1)])
uu.append(1)
# Update the GS coefficients.
u[k - 1, 0:i] = u[k - 1, 0:i] - q * np.array(uu)

# Check the Lovasz condition.
if dot(b[:, k - 1], b[:, k - 1]) >= (
delta - abs(u[k - 1, k - 2]) ** 2
) * dot(b[:, (k - 2)], b[:, (k - 2)]):
# Increment k if the Lovasz condition holds.
k += 1
else:
# If the Lovasz condition fails,
# swap the k-th and (k-1)-th basis vector
v = a[:, k - 1].copy()
a[:, k - 1] = a[:, k - 2].copy()
a[:, k - 2] = v

v_m = mapping[:, k - 1].copy()
mapping[:, k - 1] = mapping[:, k - 2].copy()
mapping[:, k - 2] = v_m

# Update the Gram-Schmidt coefficients
for s in range(k - 1, k + 1):
u[s - 1, 0 : (s - 1)] = (
dot(a[:, s - 1].T, b[:, 0 : (s - 1)]) / m[0 : (s - 1)]
)
b[:, s - 1] = a[:, s - 1] - dot(
b[:, 0 : (s - 1)], u[s - 1, 0 : (s - 1)].T
)
m[s - 1] = dot(b[:, s - 1], b[:, s - 1])

if k > 2:
k -= 1
else:
# We have to do p/q, so do lstsq(q.T, p.T).T instead.
p = dot(a[:, k:3].T, b[:, (k - 2) : k])
q = np.diag(m[(k - 2) : k])
result = np.linalg.lstsq(q.T, p.T, rcond=None)[0].T
u[k:3, (k - 2) : k] = result

return a.T, mapping.T

[docs]    def get_lll_frac_coords(self, frac_coords: Vector3Like) -> np.ndarray:
"""
Given fractional coordinates in the lattice basis, returns corresponding
fractional coordinates in the lll basis.
"""
return np.dot(frac_coords, self.lll_inverse)

[docs]    def get_frac_coords_from_lll(self, lll_frac_coords: Vector3Like) -> np.ndarray:
"""
Given fractional coordinates in the lll basis, returns corresponding
fractional coordinates in the lattice basis.
"""
return np.dot(lll_frac_coords, self.lll_mapping)

[docs]    def get_niggli_reduced_lattice(self, tol: float = 1e-5) -> "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

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

Returns:
Niggli-reduced lattice.
"""
# lll reduction is more stable for skewed cells
matrix = self.lll_matrix
a = matrix[0]
b = matrix[1]
c = matrix[2]
e = tol * self.volume ** (1 / 3)

# Define metric tensor
G = [
[dot(a, a), dot(a, b), dot(a, c)],
[dot(a, b), dot(b, b), dot(b, c)],
[dot(a, c), dot(b, c), dot(c, c)],
]
G = np.array(G)

# This sets an upper limit on the number of iterations.
for count in range(100):
# The steps are labelled as Ax as per the labelling scheme in the
# paper.
(A, B, C, E, N, Y) = (
G[0, 0],
G[1, 1],
G[2, 2],
2 * G[1, 2],
2 * G[0, 2],
2 * G[0, 1],
)

if A > B + e or (abs(A - B) < e and abs(E) > abs(N) + e):
# A1
M = [[0, -1, 0], [-1, 0, 0], [0, 0, -1]]
G = dot(transpose(M), dot(G, M))
if (B > C + e) or (abs(B - C) < e and abs(N) > abs(Y) + e):
# A2
M = [[-1, 0, 0], [0, 0, -1], [0, -1, 0]]
G = dot(transpose(M), dot(G, M))
continue

l = 0 if abs(E) < e else E / abs(E)
m = 0 if abs(N) < e else N / abs(N)
n = 0 if abs(Y) < e else Y / abs(Y)
if l * m * n == 1:
# A3
i = -1 if l == -1 else 1
j = -1 if m == -1 else 1
k = -1 if n == -1 else 1
M = [[i, 0, 0], [0, j, 0], [0, 0, k]]
G = dot(transpose(M), dot(G, M))
elif l * m * n == 0 or l * m * n == -1:
# A4
i = -1 if l == 1 else 1
j = -1 if m == 1 else 1
k = -1 if n == 1 else 1

if i * j * k == -1:
if n == 0:
k = -1
elif m == 0:
j = -1
elif l == 0:
i = -1
M = [[i, 0, 0], [0, j, 0], [0, 0, k]]
G = dot(transpose(M), dot(G, M))

(A, B, C, E, N, Y) = (
G[0, 0],
G[1, 1],
G[2, 2],
2 * G[1, 2],
2 * G[0, 2],
2 * G[0, 1],
)

# A5
if (
abs(E) > B + e
or (abs(E - B) < e and 2 * N < Y - e)
or (abs(E + B) < e and Y < -e)
):
M = [[1, 0, 0], [0, 1, -E / abs(E)], [0, 0, 1]]
G = dot(transpose(M), dot(G, M))
continue

# A6
if (
abs(N) > A + e
or (abs(A - N) < e and 2 * E < Y - e)
or (abs(A + N) < e and Y < -e)
):
M = [[1, 0, -N / abs(N)], [0, 1, 0], [0, 0, 1]]
G = dot(transpose(M), dot(G, M))
continue

# A7
if (
abs(Y) > A + e
or (abs(A - Y) < e and 2 * E < N - e)
or (abs(A + Y) < e and N < -e)
):
M = [[1, -Y / abs(Y), 0], [0, 1, 0], [0, 0, 1]]
G = dot(transpose(M), dot(G, M))
continue

# A8
if E + N + Y + A + B < -e or (abs(E + N + Y + A + B) < e < Y + (A + N) * 2):
M = [[1, 0, 1], [0, 1, 1], [0, 0, 1]]
G = dot(transpose(M), dot(G, M))
continue

break

A = G[0, 0]
B = G[1, 1]
C = G[2, 2]
E = 2 * G[1, 2]
N = 2 * G[0, 2]
Y = 2 * G[0, 1]
a = math.sqrt(A)
b = math.sqrt(B)
c = math.sqrt(C)
alpha = math.acos(E / 2 / b / c) / math.pi * 180
beta = math.acos(N / 2 / a / c) / math.pi * 180
gamma = math.acos(Y / 2 / a / b) / math.pi * 180

latt = Lattice.from_parameters(a, b, c, alpha, beta, gamma)

mapped = self.find_mapping(latt, e, skip_rotation_matrix=True)
if mapped is not None:
if np.linalg.det(mapped[0].matrix) > 0:
return mapped[0]
else:
return Lattice(-mapped[0].matrix)

raise ValueError("can't find niggli")

[docs]    def scale(self, 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.

Args:
new_volume:
New volume to scale to.

Returns:
New lattice with desired volume.
"""
versors = self.matrix / self.abc

geo_factor = abs(np.dot(np.cross(versors[0], versors[1]), versors[2]))

ratios = self.abc / self.c

new_c = (new_volume / (geo_factor * np.prod(ratios))) ** (1 / 3.0)

return Lattice(versors * (new_c * ratios))

[docs]    def get_wigner_seitz_cell(self) -> List[List[np.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.
"""
vec1 = self.matrix[0]
vec2 = self.matrix[1]
vec3 = self.matrix[2]

list_k_points = []
for i, j, k in itertools.product([-1, 0, 1], [-1, 0, 1], [-1, 0, 1]):
list_k_points.append(i * vec1 + j * vec2 + k * vec3)
from scipy.spatial import Voronoi

tess = Voronoi(list_k_points)
to_return = []
for r in tess.ridge_dict:
if r[0] == 13 or r[1] == 13:
to_return.append([tess.vertices[i] for i in tess.ridge_dict[r]])

[docs]    def get_brillouin_zone(self) -> List[List[np.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.
"""
return self.reciprocal_lattice.get_wigner_seitz_cell()

[docs]    def dot(
self, coords_a: Vector3Like, coords_b: Vector3Like, frac_coords: bool = False
) -> np.ndarray:
"""
Compute the scalar product of vector(s).

Args:
coords_a, 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.
"""
coords_a, coords_b = (
np.reshape(coords_a, (-1, 3)),
np.reshape(coords_b, (-1, 3)),
)

if len(coords_a) != len(coords_b):
raise ValueError("")

if np.iscomplexobj(coords_a) or np.iscomplexobj(coords_b):
raise TypeError("Complex array!")

if not frac_coords:
cart_a, cart_b = coords_a, coords_b
else:
cart_a = np.reshape(
[self.get_cartesian_coords(vec) for vec in coords_a], (-1, 3)
)
cart_b = np.reshape(
[self.get_cartesian_coords(vec) for vec in coords_b], (-1, 3)
)

return np.array([np.dot(a, b) for a, b in zip(cart_a, cart_b)])

[docs]    def norm(self, coords: Vector3Like, frac_coords: bool = True) -> float:
"""
Compute the norm of vector(s).

Args:
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.
"""
return np.sqrt(self.dot(coords, coords, frac_coords=frac_coords))

[docs]    def get_points_in_sphere(
self,
frac_points: List[Vector3Like],
center: Vector3Like,
r: float,
zip_results=True,
) -> Union[
List[Tuple[np.ndarray, float, int, np.ndarray]],
Tuple[List[np.ndarray], List[float], List[int], 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.

Args:
frac_points: All points in the lattice in fractional coordinates.
center: Cartesian coordinates of center of sphere.
zip_results (bool): Whether to zip the results together to group by
point, or return the raw fcoord, dist, index arrays

Returns:
if zip_results:
[(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
"""
# TODO: refactor to use lll matrix (nmax will be smaller)
# Determine the maximum number of supercells in each direction
#  required to contain a sphere of radius n
recp_len = np.array(self.reciprocal_lattice.abc) / (2 * pi)
nmax = float(r) * recp_len + 0.01

# Get the fractional coordinates of the center of the sphere
pcoords = self.get_fractional_coords(center)
center = np.array(center)

# Prepare the list of output atoms
n = len(frac_points)
fcoords = np.array(frac_points) % 1
indices = np.arange(n)

# Generate all possible images that could be within r of center
mins = np.floor(pcoords - nmax)
maxes = np.ceil(pcoords + nmax)
arange = np.arange(start=mins[0], stop=maxes[0], dtype=np.int)
brange = np.arange(start=mins[1], stop=maxes[1], dtype=np.int)
crange = np.arange(start=mins[2], stop=maxes[2], dtype=np.int)
arange = arange[:, None] * np.array([1, 0, 0], dtype=np.int)[None, :]
brange = brange[:, None] * np.array([0, 1, 0], dtype=np.int)[None, :]
crange = crange[:, None] * np.array([0, 0, 1], dtype=np.int)[None, :]
images = arange[:, None, None] + brange[None, :, None] + crange[None, None, :]

# Generate the coordinates of all atoms within these images
shifted_coords = fcoords[:, None, None, None, :] + images[None, :, :, :, :]

# Determine distance from center
cart_coords = self.get_cartesian_coords(fcoords)
cart_images = self.get_cartesian_coords(images)
coords = cart_coords[:, None, None, None, :] + cart_images[None, :, :, :, :]
coords -= center[None, None, None, None, :]
coords **= 2
d_2 = np.sum(coords, axis=4)

# Determine which points are within r of center
within_r = np.where(d_2 <= r ** 2)
#  within_r now contains the coordinates of each image that is
#    inside of the cutoff distance. It has 4 coordinates:
#   0 - index of the image within frac_points
#   1,2,3 - index of the supercell which holds the images in the x, y, z directions

if zip_results:
return list(
zip(
shifted_coords[within_r],
np.sqrt(d_2[within_r]),
indices[within_r[0]],
images[within_r[1:]],
)
)
else:
return (
shifted_coords[within_r],
np.sqrt(d_2[within_r]),
indices[within_r[0]],
images[within_r[1:]],
)

[docs]    def get_all_distances(
self,
fcoords1: Union[Vector3Like, List[Vector3Like]],
fcoords2: Union[Vector3Like, List[Vector3Like]],
) -> np.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.

Args:
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]
"""
v, d2 = pbc_shortest_vectors(self, fcoords1, fcoords2, return_d2=True)
return np.sqrt(d2)

[docs]    def is_hexagonal(
self, hex_angle_tol: float = 5, hex_length_tol: float = 0.01
) -> bool:
lengths, angles = self.lengths_and_angles
right_angles = [i for i in range(3) if abs(angles[i] - 90) < hex_angle_tol]
hex_angles = [
i
for i in range(3)
if abs(angles[i] - 60) < hex_angle_tol
or abs(angles[i] - 120) < hex_angle_tol
]

return (
len(right_angles) == 2
and len(hex_angles) == 1
and abs(lengths[right_angles[0]] - lengths[right_angles[1]])
< hex_length_tol
)

[docs]    def get_distance_and_image(
self,
frac_coords1: Vector3Like,
frac_coords2: Vector3Like,
jimage: Optional[Union[List[int], np.ndarray]] = 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.

Args:
fcoords1 (3x1 array): Reference fcoords to get distance from.
fcoords2 (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, jimage): 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.
"""
if jimage is None:
v, d2 = pbc_shortest_vectors(
self, frac_coords1, frac_coords2, return_d2=True
)
fc = self.get_fractional_coords(v[0][0]) + frac_coords1 - frac_coords2
fc = np.array(np.round(fc), dtype=np.int)
return np.sqrt(d2[0, 0]), fc

jimage = np.array(jimage)
mapped_vec = self.get_cartesian_coords(jimage + frac_coords2 - frac_coords1)
return np.linalg.norm(mapped_vec), jimage

[docs]    def get_miller_index_from_coords(
self,
coords: Vector3Like,
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.

Args:
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:
(tuple): The Miller index.
"""
if coords_are_cartesian:
coords = [self.get_fractional_coords(c) for c in coords]

coords = np.asarray(coords)
g = coords.sum(axis=0) / coords.shape[0]

# run singular value decomposition
_, _, vh = np.linalg.svd(coords - g)

# get unitary normal vector
u_norm = vh[2, :]
return get_integer_index(u_norm, round_dp=round_dp, verbose=verbose)

[docs]def get_integer_index(
miller_index: bool, round_dp: int = 4, verbose: bool = True
) -> Tuple[int, int, int]:
"""
Attempt to convert a vector of floats to whole numbers.

Args:
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:
(tuple): The Miller index.
"""
miller_index = np.asarray(miller_index)

# deal with the case we have small irregular floats
# that are all equal or factors of each other
miller_index /= min([m for m in miller_index if m != 0])
miller_index /= np.max(np.abs(miller_index))

# deal with the case we have nice fractions
md = [Fraction(n).limit_denominator(12).denominator for n in miller_index]
miller_index *= reduce(lambda x, y: x * y, md)
int_miller_index = np.int_(np.round(miller_index, 1))
miller_index /= np.abs(reduce(gcd, int_miller_index))

# round to a reasonable precision
miller_index = np.array([round(h, round_dp) for h in miller_index])

# need to recalculate this after rounding as values may have changed
int_miller_index = np.int_(np.round(miller_index, 1))
if np.any(np.abs(miller_index - int_miller_index) > 1e-6) and verbose:
warnings.warn("Non-integer encountered in Miller index")
else:
miller_index = int_miller_index

# minimise the number of negative indexes
miller_index += 0  # converts -0 to 0

def n_minus(index):
return len([h for h in index if h < 0])

if n_minus(miller_index) > n_minus(miller_index * -1):
miller_index *= -1

# if only one index is negative, make sure it is the smallest
# e.g. (-2 1 0) -> (2 -1 0)
if (
sum(miller_index != 0) == 2
and n_minus(miller_index) == 1
and abs(min(miller_index)) > max(miller_index)
):
miller_index *= -1

return tuple(miller_index)