import itertools
from fractions import Fraction
from typing import Dict, List, Tuple
import numpy as np
from numpy.typing import NDArray
from ase import Atoms
[docs]def get_supercell_qpoints_along_path(
path: List[Tuple[str, str]],
coordinates: Dict[str, NDArray[float]],
primitive_cell: NDArray[float],
super_cell: NDArray[float]) -> List[NDArray[float]]:
r"""
Returns the q-points commensurate with the given supercell along the specific path.
Parameters
----------
path
list of pairs of q-point labels
coordinates
dict with q-point labels and coordinates as keys and values, respectively;
there must be one entry for each q-point label used in :attr:`path`
primitive_cell
cell metric of the primitive cell with lattice vectors as rows
super_cell
cell metric of the supercell with lattice vectors as rows
Returns
-------
supercell_paths
A list of the accessible q-point coordinates along the specified segment
Example
--------
The following example illustrates how to retrieve the q-points that
can be sampled using a supercell comprising :math:`6 \times 6 \times 6`
conventional (4-atom) unit cells of FCC Al along the path X-:math:`\Gamma`-L.
>>> import numpy as np
>>> from ase.build import bulk
>>> from dynasor.qpoints import get_supercell_qpoints_along_path
>>> prim = bulk('Al', 'fcc', a=4.0)
>>> supercell = bulk('Al', 'fcc', a=4.0, cubic=True).repeat(6)
>>> path = [('X', 'G'), ('G', 'L'), ('L', 'W')]
>>> coordinates = dict(X=[0.5, 0.5, 0], G=[0, 0, 0],
... L=[0.5, 0.5, 0.5], W=[0.5, 0.25, 0.75])
>>> qpoints = get_supercell_qpoints_along_path(path, coordinates, prim.cell, supercell.cell)
"""
from .lattice import Lattice
lat = Lattice(primitive_cell, super_cell)
for lbl in np.array(path).flatten():
if lbl not in coordinates:
raise ValueError(f'Unknown point in path: {lbl}')
# build the segments
supercell_paths = []
for k, (l1, l2) in enumerate(path):
q1 = np.array(coordinates[l1], dtype=float)
q2 = np.array(coordinates[l2], dtype=float)
dynasor_path, _ = lat.make_path(q1, q2)
supercell_paths.append(dynasor_path)
return supercell_paths
def find_on_line(start: NDArray, stop: NDArray, P: NDArray):
"""Find fractional distances between start and stop combatible with P
A supercell is defined by P @ c = S for some repetition matrix P and we
want to find fractions so that
[start + f * (stop - start)] @ P = n
Parameters
----------
start
start of line in reduced supercell coordinates
stop
end of line in reduced supercell coordinates
P
repetion matrix defining the supercell
"""
if np.allclose(start, stop):
return [Fraction(0, 1)]
start = np.array([Fraction(s).limit_denominator() for s in start])
stop = np.array([Fraction(s).limit_denominator() for s in stop])
A = start @ P
B = (stop - start) @ P
fracs = None
for a, b in zip(A, B):
fs = solve_Diophantine(a, b)
if fs is None: # "inf" solutions
continue
elif fs == []: # No solutions
return []
fracs = set(fs) if fracs is None else fracs.intersection(fs)
return sorted(fracs)
def solve_Diophantine(a: Fraction, b: Fraction) -> List[Fraction]:
"""Solve n = a + xb for all n in Z and a,b in Q such that 0 <= x <= 1"""
if b == 0:
if a.denominator == 1:
return None
else:
return []
if b < 0:
right = np.ceil(a)
left = np.floor(a + b)
else:
left = np.floor(a)
right = np.ceil(a + b)
ns = np.arange(left, right + 1)
fracs = [Fraction(n - a, b) for n in ns]
fracs = [f for f in fracs if 0 <= f <= 1]
return fracs
def det(A):
"""Determinant of an integer matrix using Laplace cofactor expansion"""
if len(A) == 2:
return A[0, 0] * A[1, 1] - A[0, 1] * A[1, 0]
d = 0
for i, B in enumerate(A[0]): # along first row
minor = np.hstack([A[1:, :i], A[1:, i+1:]])
d += (-1)**i * B * det(minor)
assert np.isclose(d, np.linalg.det(A))
return d
def inv(A):
"""Takes the inverse of an integer 3x3 matrix based on Cayley-Hamilton"""
detx2 = det(A) * 2 # Denominator "determinant times two"
# Numerator
numerator = ((np.trace(A)**2 - np.trace(A @ A)) * np.diag([1, 1, 1])
- 2 * A * np.trace(A)
+ 2 * A @ A)
# We want the sign to be in the Numerator
if detx2 < 0:
detx2 = -detx2
numerator = -numerator
inverse = numerator / detx2
assert np.allclose(inverse, np.linalg.inv(A))
# Return inverse, numerator (int matrix) and denominator (int)
return inverse, numerator, detx2
def get_P_matrix(c, S):
""" P c = S -> c.T P.T = S.T
The P matrix must be an integer matrix
"""
PT = np.linalg.solve(c.T, S.T)
P_float = PT.T
P = np.round(P_float).astype(int)
if not np.allclose(P_float, P) or not np.allclose(P @ c, S):
raise ValueError(
f'Please check that the supercell metric ({S}) is related to the'
f' the primitive cell {c} by an integer transformation matrix.')
return P
def get_commensurate_lattice_points(P: NDArray) -> NDArray:
"""Return commensurate points for a supercell defined by repetition matrix P
Finds all n such that n = f P where f is between 0 and 1
Parameters
----------
P
the repetion matrix relating the primitive and supercell
Returns
-------
lattice_points
the commensurate lattice points
"""
n_max = np.where(P > 0, P, 0).sum(axis=0) + 1
n_min = np.where(P < 0, P, 0).sum(axis=0)
ranges = [np.arange(*n) for n in zip(n_min, n_max)]
inv_P_matrix, num, den = inv(P)
lattice_points = []
for lp in itertools.product(*ranges):
s = lp @ num # here we skip the denominator to keep everything integer
# the denominator is also integer so no numerics here
if np.all(s >= 0) and np.all(s < den):
lattice_points.append(lp)
lattice_points = np.array(lattice_points)
# Begin sane checks...
# No duplicates
assert len(lattice_points) == len(np.unique(lattice_points, axis=0))
# Did we get everyone?
assert len(lattice_points) == abs(det(P))
return lattice_points
def get_index_offset(supercell: Atoms, prim: Atoms, atol=1e-3, rtol=0.0):
"""
Get the basis index and primitive cell offsets for a supercell
"""
index, offset = [], []
for pos in supercell.positions:
spos = np.linalg.solve(prim.cell.T, pos)
for i, spos2 in enumerate(prim.get_scaled_positions()):
off = spos - spos2
off_round = np.round(off)
if not np.allclose(off, off_round, atol=atol, rtol=rtol):
continue
index.append(i)
off = off_round.astype(int)
assert np.allclose(off, off_round)
offset.append(off)
break
else:
raise ValueError('prim not compatible with atoms')
index, offset = np.array(index), np.array(offset)
return index, offset