# Source code for aiida.tools.data.array.kpoints.legacy

# -*- coding: utf-8 -*-
###########################################################################
# This file is part of the AiiDA code.                                    #
#                                                                         #
# The code is hosted on GitHub at https://github.com/aiidateam/aiida-core #
# For further information on the license, see the LICENSE.txt file        #
# For further information please visit http://www.aiida.net               #
###########################################################################

import numpy

_default_epsilon_length = 1e-5
_default_epsilon_angle = 1e-5

[docs]def change_reference(reciprocal_cell, kpoints, to_cartesian=True): """ Change reference system, from cartesian to crystal coordinates (units of b1,b2,b3) or viceversa. :param reciprocal_cell: a 3x3 array representing the cell lattice vectors in reciprocal space :param kpoints: a list of (3) point coordinates :return kpoints: a list of (3) point coordinates in the new reference """ if not isinstance(kpoints, numpy.ndarray): raise ValueError('kpoints must be a numpy.array') transposed_cell = numpy.transpose(numpy.array(reciprocal_cell)) if to_cartesian: matrix = transposed_cell else: matrix = numpy.linalg.inv(transposed_cell) # note: kpoints is a list Nx3, matrix is 3x3. # hence, first transpose kpoints, then multiply, finally transpose it back return numpy.transpose(numpy.dot(matrix, numpy.transpose(kpoints)))
[docs]def analyze_cell(cell=None, pbc=None): """ A function executed by the __init__ or by set_cell. If a cell is set, properties like a1, a2, a3, cosalpha, reciprocal_cell are set as well, although they are not stored in the DB. :note: units are Angstrom for the cell parameters, 1/Angstrom for the reciprocal cell parameters. """ if pbc is None: pbc = [True, True, True] dimension = sum(pbc) if cell is None: return { 'reciprocal_cell': None, 'dimension': dimension, 'pbc': pbc } the_cell = numpy.array(cell) reciprocal_cell = 2. * numpy.pi * numpy.linalg.inv(the_cell).transpose() a1 = numpy.array(the_cell[0, :]) # units = Angstrom a2 = numpy.array(the_cell[1, :]) # units = Angstrom a3 = numpy.array(the_cell[2, :]) # units = Angstrom a = numpy.linalg.norm(a1) # units = Angstrom b = numpy.linalg.norm(a2) # units = Angstrom c = numpy.linalg.norm(a3) # units = Angstrom b1 = reciprocal_cell[0, :] # units = 1/Angstrom b2 = reciprocal_cell[1, :] # units = 1/Angstrom b3 = reciprocal_cell[2, :] # units = 1/Angstrom cosalpha = numpy.dot(a2, a3) / b / c cosbeta = numpy.dot(a3, a1) / c / a cosgamma = numpy.dot(a1, a2) / a / b result = { 'a1': a1, 'a2': a2, 'a3': a3, 'a': a, 'b': b, 'c': c, 'b1': b1, 'b2': b2, 'b3': b3, 'cosalpha': cosalpha, 'cosbeta': cosbeta, 'cosgamma': cosgamma, 'dimension': dimension, 'reciprocal_cell': reciprocal_cell, 'pbc': pbc, } return result
[docs]def get_explicit_kpoints_path(value=None, cell=None, pbc=None, kpoint_distance=None, cartesian=False, epsilon_length=_default_epsilon_length, epsilon_angle=_default_epsilon_angle): """ Set a path of kpoints in the Brillouin zone. :param value: description of the path, in various possible formats. None: automatically sets all irreducible high symmetry paths. Requires that a cell was set or:: [('G','M'), (...), ...] [('G','M',30), (...), ...] [('G',(0,0,0),'M',(1,1,1)), (...), ...] [('G',(0,0,0),'M',(1,1,1),30), (...), ...] :param cell: 3x3 array representing the structure cell lattice vectors :param pbc: 3-dimensional array of booleans signifying the periodic boundary conditions along each lattice vector :param float kpoint_distance: parameter controlling the distance between kpoints. Distance is given in crystal coordinates, i.e. the distance is computed in the space of b1,b2,b3. The distance set will be the closest possible to this value, compatible with the requirement of putting equispaced points between two special points (since extrema are included). :param bool cartesian: if set to true, reads the coordinates eventually passed in value as cartesian coordinates. Default: False. :param float epsilon_length: threshold on lengths comparison, used to get the bravais lattice info. It has to be used if the user wants to be sure the right symmetries are recognized. :param float epsilon_angle: threshold on angles comparison, used to get the bravais lattice info. It has to be used if the user wants to be sure the right symmetries are recognized. :returns: point_coordinates, path, bravais_info, explicit_kpoints, labels """ bravais_info = find_bravais_info( cell=cell, pbc=pbc, epsilon_length=epsilon_length, epsilon_angle=epsilon_angle ) analysis = analyze_cell(cell, pbc) dimension = analysis['dimension'] reciprocal_cell = analysis['reciprocal_cell'] pbc = list(analysis['pbc']) if dimension == 0: # case with zero dimension: only gamma-point is set return [[0., 0., 0.]], None, bravais_info def _is_path_1(path): try: are_two = all([len(i) == 2 for i in path]) if not are_two: return False for i in path: are_str = all([isinstance(b, str) for b in i]) if not are_str: return False except IndexError: return False return True def _is_path_2(path): try: are_three = all([len(i) == 3 for i in path]) if not are_three: return False are_good = all([all([isinstance(b[0], str), isinstance(b[1], str), isinstance(b[2], int)]) for b in path]) if not are_good: return False # check that at least two points per segment (beginning and end) points_num = [int(i[2]) for i in path] if any([i < 2 for i in points_num]): raise ValueError('Must set at least two points per path ' 'segment') except IndexError: return False return True def _is_path_3(path): # [('G',(0,0,0),'M',(1,1,1)), (...), ...] try: _ = len(path) are_four = all([len(i) == 4 for i in path]) if not are_four: return False have_labels = all(all([isinstance(i[0], str), isinstance(i[2], str)]) for i in path) if not have_labels: return False for i in path: coord1 = [float(j) for j in i[1]] coord2 = [float(j) for j in i[3]] if len(coord1) != 3 or len(coord2) != 3: return False except (TypeError, IndexError): return False return True def _is_path_4(path): # [('G',(0,0,0),'M',(1,1,1),30), (...), ...] try: _ = len(path) are_five = all([len(i) == 5 for i in path]) if not are_five: return False have_labels = all(all([isinstance(i[0], str), isinstance(i[2], str)]) for i in path) if not have_labels: return False have_points_num = all([isinstance(i[4], int) for i in path]) if not have_points_num: return False # check that at least two points per segment (beginning and end) points_num = [int(i[4]) for i in path] if any([i < 2 for i in points_num]): raise ValueError('Must set at least two points per path ' 'segment') for i in path: coord1 = [float(j) for j in i[1]] coord2 = [float(j) for j in i[3]] if len(coord1) != 3 or len(coord2) != 3: return False except (TypeError, IndexError): return False return True def _num_points_from_coordinates(path, point_coordinates, kpoint_distance=None): # NOTE: this way of creating intervals ensures equispaced objects # in crystal coordinates of b1,b2,b3 distances = [numpy.linalg.norm(numpy.array(point_coordinates[i[0]]) - numpy.array(point_coordinates[i[1]]) ) for i in path] if kpoint_distance is None: # Use max_points_per_interval as the default guess for automatically # guessing the number of points max_point_per_interval = 10 max_interval = max(distances) try: points_per_piece = [max(2, max_point_per_interval * i // max_interval) for i in distances] except ValueError: raise ValueError('The beginning and end of each segment in the ' 'path should be different.') else: points_per_piece = [max(2, int(distance // kpoint_distance)) for distance in distances] return points_per_piece if cartesian: if cell is None: raise ValueError('To use cartesian coordinates, a cell must ' 'be provided') if kpoint_distance is not None: if kpoint_distance <= 0.: raise ValueError('kpoints_distance must be a positive float') if value is None: if cell is None: raise ValueError('Cannot set a path not even knowing the ' 'kpoints or at least the cell') point_coordinates, path, bravais_info = get_kpoints_path( cell=cell, pbc=pbc, cartesian=cartesian, epsilon_length=epsilon_length, epsilon_angle=epsilon_angle) num_points = _num_points_from_coordinates(path, point_coordinates, kpoint_distance) elif _is_path_1(value): # in the form [('X','M'),(...),...] if cell is None: raise ValueError('Cannot set a path not even knowing the ' 'kpoints or at least the cell') path = value point_coordinates, _, bravais_info = get_kpoints_path( cell=cell, pbc=pbc, cartesian=cartesian, epsilon_length=epsilon_length, epsilon_angle=epsilon_angle) num_points = _num_points_from_coordinates(path, point_coordinates, kpoint_distance) elif _is_path_2(value): # [('G','M',30), (...), ...] if cell is None: raise ValueError('Cannot set a path not even knowing the ' 'kpoints or at least the cell') path = [(i[0], i[1]) for i in value] point_coordinates, _, bravais_info = get_kpoints_path( cell=cell, pbc=pbc, cartesian=cartesian, epsilon_length=epsilon_length, epsilon_angle=epsilon_angle) num_points = [i[2] for i in value] elif _is_path_3(value): # [('G',(0,0,0),'M',(1,1,1)), (...), ...] path = [(i[0], i[2]) for i in value] point_coordinates = {} for piece in value: if piece[0] in point_coordinates: if point_coordinates[piece[0]] != piece[1]: raise ValueError('Different points cannot have the same label') else: if cartesian: point_coordinates[piece[0]] = change_reference( reciprocal_cell, numpy.array([piece[1]]), to_cartesian=False)[0] else: point_coordinates[piece[0]] = piece[1] if piece[2] in point_coordinates: if point_coordinates[piece[2]] != piece[3]: raise ValueError('Different points cannot have the same label') else: if cartesian: point_coordinates[piece[2]] = change_reference( reciprocal_cell, numpy.array([piece[3]]), to_cartesian=False)[0] else: point_coordinates[piece[2]] = piece[3] num_points = _num_points_from_coordinates(path, point_coordinates, kpoint_distance) elif _is_path_4(value): # [('G',(0,0,0),'M',(1,1,1),30), (...), ...] path = [(i[0], i[2]) for i in value] point_coordinates = {} for piece in value: if piece[0] in point_coordinates: if point_coordinates[piece[0]] != piece[1]: raise ValueError('Different points cannot have the same label') else: if cartesian: point_coordinates[piece[0]] = change_reference( reciprocal_cell, numpy.array([piece[1]]), to_cartesian=False)[0] else: point_coordinates[piece[0]] = piece[1] if piece[2] in point_coordinates: if point_coordinates[piece[2]] != piece[3]: raise ValueError('Different points cannot have the same label') else: if cartesian: point_coordinates[piece[2]] = change_reference( reciprocal_cell, numpy.array([piece[3]]), to_cartesian=False)[0] else: point_coordinates[piece[2]] = piece[3] num_points = [i[4] for i in value] else: raise ValueError('Input format not recognized') explicit_kpoints = [tuple(point_coordinates[path[0][0]])] labels = [(0, path[0][0])] for count_piece, i in enumerate(path): ini_label = i[0] end_label = i[1] ini_coord = point_coordinates[ini_label] end_coord = point_coordinates[end_label] path_piece = list(zip(numpy.linspace(ini_coord[0], end_coord[0], num_points[count_piece]), numpy.linspace(ini_coord[1], end_coord[1], num_points[count_piece]), numpy.linspace(ini_coord[2], end_coord[2], num_points[count_piece]), )) for count, j in enumerate(path_piece): if all(numpy.array(explicit_kpoints[-1]) == j): continue # avoid duplcates else: explicit_kpoints.append(j) # add labels for the first and last point if count == 0: labels.append((len(explicit_kpoints) - 1, ini_label)) if count == len(path_piece) - 1: labels.append((len(explicit_kpoints) - 1, end_label)) # I still have some duplicates in the labels: eliminate them sorted(set(labels), key=lambda x: x[0]) return point_coordinates, path, bravais_info, explicit_kpoints, labels
[docs]def find_bravais_info(cell, pbc, epsilon_length=_default_epsilon_length, epsilon_angle=_default_epsilon_angle): """ Finds the Bravais lattice of the cell passed in input to the Kpoint class :note: We assume that the cell given by the cell property is the primitive unit cell. .. note:: in 3D, this implementation expects that the structure is already standardized according to the Setyawan paper. If this is not the case, the kpoints and band structure returned will be incorrect. The only case that is dealt correctly by the library is the case when axes are swapped, where the library correctly takes this swapping/rotation into account to assign kpoint labels and coordinates. :param cell: 3x3 array representing the structure cell lattice vectors :param pbc: 3-dimensional array of booleans signifying the periodic boundary conditions along each lattice vector passed in value as cartesian coordinates. Default: False. :param float epsilon_length: threshold on lengths comparison, used to get the bravais lattice info. It has to be used if the user wants to be sure the right symmetries are recognized. :param float epsilon_angle: threshold on angles comparison, used to get the bravais lattice info. It has to be used if the user wants to be sure the right symmetries are recognized. :return: a dictionary, with keys short_name, extended_name, index (index of the Bravais lattice), and sometimes variation (name of the variation of the Bravais lattice) and extra (a dictionary with extra parameters used by the get_kpoints_path method) """ if cell is None: return None analysis = analyze_cell(cell, pbc) a1 = analysis['a1'] a2 = analysis['a2'] a3 = analysis['a3'] a = analysis['a'] b = analysis['b'] c = analysis['c'] cosa = analysis['cosalpha'] cosb = analysis['cosbeta'] cosc = analysis['cosgamma'] dimension = analysis['dimension'] pbc = list(pbc) # values of cosines at various angles _90 = 0. _60 = 0.5 _30 = numpy.sqrt(3.) / 2. _120 = -0.5 # NOTE: in what follows, I'm assuming the textbook order of alfa, beta and gamma # TODO: Maybe additional checks to see if the "correct" primitive # cell is used ? (there are other equivalent primitive # unit cells to the one expected here, typically for body-, c-, and # face-centered lattices) def l_are_equals(a, b): # function to compare lengths return abs(a - b) <= epsilon_length def a_are_equals(a, b): # function to compare angles (actually, cosines) return abs(a - b) <= epsilon_angle if dimension == 3: # =========================================# # 3D case -> 14 possible Bravais lattices # # =========================================# comparison_length = [l_are_equals(a, b), l_are_equals(b, c), l_are_equals(c, a)] comparison_angles = [a_are_equals(cosa, cosb), a_are_equals(cosb, cosc), a_are_equals(cosc, cosa)] if comparison_length.count(True) == 3: # needed for the body centered orthorhombic: orci_a = numpy.linalg.norm(a2 + a3) orci_b = numpy.linalg.norm(a1 + a3) orci_c = numpy.linalg.norm(a1 + a2) orci_the_a, orci_the_b, orci_the_c = sorted([orci_a, orci_b, orci_c]) bco1 = - (-orci_the_a ** 2 + orci_the_b ** 2 + orci_the_c ** 2) / (4. * a ** 2) bco2 = - (orci_the_a ** 2 - orci_the_b ** 2 + orci_the_c ** 2) / (4. * a ** 2) bco3 = - (orci_the_a ** 2 + orci_the_b ** 2 - orci_the_c ** 2) / (4. * a ** 2) # ======================# # simple cubic lattice # # ======================# if comparison_angles.count(True) == 3 and a_are_equals(cosa, _90): bravais_info = {'short_name': 'cub', 'extended_name': 'cubic', 'index': 1, 'permutation': [0, 1, 2] } # =====================# # face centered cubic # # =====================# elif comparison_angles.count(True) == 3 and a_are_equals(cosa, _60): bravais_info = {'short_name': 'fcc', 'extended_name': 'face centered cubic', 'index': 2, 'permutation': [0, 1, 2] } # =====================# # body centered cubic # # =====================# elif comparison_angles.count(True) == 3 and a_are_equals(cosa, -1. / 3.): bravais_info = {'short_name': 'bcc', 'extended_name': 'body centered cubic', 'index': 3, 'permutation': [0, 1, 2] } # ==============# # rhombohedral # # ==============# elif comparison_angles.count(True) == 3: # logical order is important, this check must come after the cubic cases bravais_info = {'short_name': 'rhl', 'extended_name': 'rhombohedral', 'index': 11, 'permutation': [0, 1, 2] } if cosa > 0.: bravais_info['variation'] = 'rhl1' eta = (1. + 4. * cosa) / (2. + 4. * cosa) bravais_info['extra'] = {'eta': eta, 'nu': 0.75 - eta / 2., } else: bravais_info['variation'] = 'rhl2' eta = 1. / (2. * (1. - cosa) / (1. + cosa)) bravais_info['extra'] = {'eta': eta, 'nu': 0.75 - eta / 2., } # ==========================# # body centered tetragonal # # ==========================# elif comparison_angles.count(True) == 1: # two angles are the same bravais_info = {'short_name': 'bct', 'extended_name': 'body centered tetragonal', 'index': 5, } if comparison_angles.index(True) == 0: # alfa=beta ref_ang = cosa bravais_info['permutation'] = [0, 1, 2] elif comparison_angles.index(True) == 1: # beta=gamma ref_ang = cosb bravais_info['permutation'] = [2, 0, 1] else: # comparison_angles.index(True)==2: # gamma = alfa ref_ang = cosc bravais_info['permutation'] = [1, 2, 0] if ref_ang >= 0.: raise ValueError('Problems on the definition of ' 'body centered tetragonal lattices') the_c = numpy.sqrt(-4. * ref_ang * (a ** 2)) the_a = numpy.sqrt(2. * a ** 2 - (the_c ** 2) / 2.) if the_c < the_a: bravais_info['variation'] = 'bct1' bravais_info['extra'] = {'eta': (1. + (the_c / the_a) ** 2) / 4.} else: bravais_info['variation'] = 'bct2' bravais_info['extra'] = {'eta': (1. + (the_a / the_c) ** 2) / 4., 'csi': ((the_a / the_c) ** 2) / 2., } # ============================# # body centered orthorhombic # # ============================# elif (any([a_are_equals(cosa, bco1), a_are_equals(cosb, bco1), a_are_equals(cosc, bco1)]) and any([a_are_equals(cosa, bco2), a_are_equals(cosb, bco2), a_are_equals(cosc, bco2)]) and any([a_are_equals(cosa, bco3), a_are_equals(cosb, bco3), a_are_equals(cosc, bco3)]) ): bravais_info = {'short_name': 'orci', 'extended_name': 'body centered orthorhombic', 'index': 8, } if a_are_equals(cosa, bco1) and a_are_equals(cosc, bco3): bravais_info['permutation'] = [0, 1, 2] if a_are_equals(cosa, bco1) and a_are_equals(cosc, bco2): bravais_info['permutation'] = [0, 2, 1] if a_are_equals(cosa, bco3) and a_are_equals(cosc, bco2): bravais_info['permutation'] = [1, 2, 0] if a_are_equals(cosa, bco2) and a_are_equals(cosc, bco3): bravais_info['permutation'] = [1, 0, 2] if a_are_equals(cosa, bco2) and a_are_equals(cosc, bco1): bravais_info['permutation'] = [2, 0, 1] if a_are_equals(cosa, bco3) and a_are_equals(cosc, bco1): bravais_info['permutation'] = [2, 1, 0] bravais_info['extra'] = {'csi': (1. + (orci_the_a / orci_the_c) ** 2) / 4., 'eta': (1. + (orci_the_b / orci_the_c) ** 2) / 4., 'dlt': (orci_the_b ** 2 - orci_the_a ** 2) / (4. * orci_the_c ** 2), 'mu': (orci_the_a ** 2 + orci_the_b ** 2) / (4. * orci_the_c ** 2), } # if it doesn't fall in the above, is triclinic else: bravais_info = {'short_name': 'tri', 'extended_name': 'triclinic', 'index': 14, } # the check for triclinic variations is at the end of the method elif comparison_length.count(True) == 1: # ============# # tetragonal # # ============# if comparison_angles.count(True) == 3 and a_are_equals(cosa, _90): bravais_info = {'short_name': 'tet', 'extended_name': 'tetragonal', 'index': 4, } if comparison_length[0] == True: bravais_info['permutation'] = [0, 1, 2] if comparison_length[1] == True: bravais_info['permutation'] = [2, 0, 1] if comparison_length[2] == True: bravais_info['permutation'] = [1, 2, 0] # ====================================# # c-centered orthorombic + hexagonal # # ====================================# # alpha/=beta=gamma=pi/2 elif (comparison_angles.count(True) == 1 and any([a_are_equals(cosa, _90), a_are_equals(cosb, _90), a_are_equals(cosc, _90)]) ): if any([a_are_equals(cosa, _120), a_are_equals(cosb, _120), a_are_equals(cosc, _120)]): bravais_info = {'short_name': 'hex', 'extended_name': 'hexagonal', 'index': 10, } else: bravais_info = {'short_name': 'orcc', 'extended_name': 'c-centered orthorhombic', 'index': 9, } if comparison_length[0] == True: the_a1 = a1 the_a2 = a2 elif comparison_length[1] == True: the_a1 = a2 the_a2 = a3 else: # comparison_length[2]==True: the_a1 = a3 the_a2 = a1 the_a = numpy.linalg.norm(the_a1 + the_a2) the_b = numpy.linalg.norm(the_a1 - the_a2) bravais_info['extra'] = {'csi': (1. + (the_a / the_b) ** 2) / 4., } # TODO : re-check this case, permutations look weird if comparison_length[0] == True: bravais_info['permutation'] = [0, 1, 2] if comparison_length[1] == True: bravais_info['permutation'] = [2, 0, 1] if comparison_length[2] == True: bravais_info['permutation'] = [1, 2, 0] # =======================# # c-centered monoclinic # # =======================# elif comparison_angles.count(True) == 1: bravais_info = {'short_name': 'mclc', 'extended_name': 'c-centered monoclinic', 'index': 13, } # TODO : re-check this case, permutations look weird if comparison_length[0] == True: bravais_info['permutation'] = [0, 1, 2] the_ka = cosa the_a1 = a1 the_a2 = a2 the_c = c if comparison_length[1] == True: bravais_info['permutation'] = [2, 0, 1] the_ka = cosb the_a1 = a2 the_a2 = a3 the_c = a if comparison_length[2] == True: bravais_info['permutation'] = [1, 2, 0] the_ka = cosc the_a1 = a3 the_a2 = a1 the_c = b the_b = numpy.linalg.norm(the_a1 + the_a2) the_a = numpy.linalg.norm(the_a1 - the_a2) the_cosa = 2. * numpy.linalg.norm(the_a1) / the_b * the_ka if a_are_equals(the_ka, _90): # order matters: has to be before the check on mclc1 bravais_info['variation'] = 'mclc2' csi = (2. - the_b * the_cosa / the_c) / (4. * (1. - the_cosa ** 2)) psi = 0.75 - the_a ** 2 / (4. * the_b * (1. - the_cosa ** 2)) bravais_info['extra'] = {'csi': csi, 'eta': 0.5 + 2. * csi * the_c * the_cosa / the_b, 'psi': psi, 'phi': psi + (0.75 - psi) * the_b * the_cosa / the_c, } elif the_ka < 0.: bravais_info['variation'] = 'mclc1' csi = (2. - the_b * the_cosa / the_c) / (4. * (1. - the_cosa ** 2)) psi = 0.75 - the_a ** 2 / (4. * the_b * (1. - the_cosa ** 2)) bravais_info['extra'] = {'csi': csi, 'eta': 0.5 + 2. * csi * the_c * the_cosa / the_b, 'psi': psi, 'phi': psi + (0.75 - psi) * the_b * the_cosa / the_c, } else: # if the_ka>0.: x = the_b * the_cosa / the_c + the_b ** 2 * (1. - the_cosa ** 2) / the_a ** 2 if a_are_equals(x, 1.): bravais_info['variation'] = 'mclc4' # order matters here too mu = (1. + (the_b / the_a) ** 2) / 4. dlt = the_b * the_c * the_cosa / (2. * the_a ** 2) csi = mu - 0.25 + (1. - the_b * the_cosa / the_c) / (4. * (1. - the_cosa ** 2)) eta = 0.5 + 2. * csi * the_c * the_cosa / the_b phi = 1. + eta - 2. * mu psi = eta - 2. * dlt bravais_info['extra'] = {'mu': mu, 'dlt': dlt, 'csi': csi, 'eta': eta, 'phi': phi, 'psi': psi, } elif x < 1.: bravais_info['variation'] = 'mclc3' mu = (1. + (the_b / the_a) ** 2) / 4. dlt = the_b * the_c * the_cosa / (2. * the_a ** 2) csi = mu - 0.25 + (1. - the_b * the_cosa / the_c) / (4. * (1. - the_cosa ** 2)) eta = 0.5 + 2. * csi * the_c * the_cosa / the_b phi = 1. + eta - 2. * mu psi = eta - 2. * dlt bravais_info['extra'] = {'mu': mu, 'dlt': dlt, 'csi': csi, 'eta': eta, 'phi': phi, 'psi': psi, } elif x > 1.: bravais_info['variation'] = 'mclc5' csi = ((the_b / the_a) ** 2 + (1. - the_b * the_cosa / the_c) / (1. - the_cosa ** 2)) / 4. eta = 0.5 + 2. * csi * the_c * the_cosa / the_b mu = eta / 2. + the_b ** 2 / 4. / the_a ** 2 - the_b * the_c * the_cosa / 2. / the_a ** 2 nu = 2. * mu - csi omg = (4. * nu - 1. - the_b ** 2 * (1. - the_cosa ** 2) / the_a ** 2) * the_c / ( 2. * the_b * the_cosa) dlt = csi * the_c * the_cosa / the_b + omg / 2. - 0.25 rho = 1. - csi * the_a ** 2 / the_b ** 2 bravais_info['extra'] = {'mu': mu, 'dlt': dlt, 'csi': csi, 'eta': eta, 'rho': rho, } # if it doesn't fall in the above, is triclinic else: bravais_info = {'short_name': 'tri', 'extended_name': 'triclinic', 'index': 14, } # the check for triclinic variations is at the end of the method else: # if comparison_length.count(True)==0: fco1 = c ** 2 / numpy.sqrt((a ** 2 + c ** 2) * (b ** 2 + c ** 2)) fco2 = a ** 2 / numpy.sqrt((a ** 2 + b ** 2) * (a ** 2 + c ** 2)) fco3 = b ** 2 / numpy.sqrt((a ** 2 + b ** 2) * (b ** 2 + c ** 2)) # ==============# # orthorhombic # # ==============# if comparison_angles.count(True) == 3: bravais_info = {'short_name': 'orc', 'extended_name': 'orthorhombic', 'index': 6, } lens = [a, b, c] ind_a = lens.index(min(lens)) ind_c = lens.index(max(lens)) if ind_a == 0 and ind_c == 1: bravais_info['permutation'] = [0, 2, 1] if ind_a == 0 and ind_c == 2: bravais_info['permutation'] = [0, 1, 2] if ind_a == 1 and ind_c == 0: bravais_info['permutation'] = [1, 2, 0] if ind_a == 1 and ind_c == 2: bravais_info['permutation'] = [1, 0, 2] if ind_a == 2 and ind_c == 0: bravais_info['permutation'] = [2, 1, 0] if ind_a == 2 and ind_c == 1: bravais_info['permutation'] = [2, 0, 1] # ============# # monoclinic # # ============# elif (comparison_angles.count(True) == 1 and any([a_are_equals(cosa, _90), a_are_equals(cosb, _90), a_are_equals(cosc, _90)])): bravais_info = {'short_name': 'mcl', 'extended_name': 'monoclinic', 'index': 12, } lens = [a, b, c] # find the angle different from 90 # then order (if possible) a<b<c if not a_are_equals(cosa, _90): the_cosa = cosa the_a = min(a, b) the_b = max(a, b) the_c = c if lens.index(the_a) == 0: bravais_info['permutation'] = [0, 1, 2] else: bravais_info['permutation'] = [1, 0, 2] elif not a_are_equals(cosb, _90): the_cosa = cosb the_a = min(a, c) the_b = max(a, c) the_c = b if lens.index(the_a) == 0: bravais_info['permutation'] = [0, 2, 1] else: bravais_info['permutation'] = [1, 2, 0] else: # if not _are_equals(cosc,_90): the_cosa = cosc the_a = min(b, c) the_b = max(b, c) the_c = a if lens.index(the_a) == 1: bravais_info['permutation'] = [2, 0, 1] else: bravais_info['permutation'] = [2, 1, 0] eta = (1. - the_b * the_cosa / the_c) / (2. * (1. - the_cosa ** 2)) bravais_info['extra'] = {'eta': eta, 'nu': 0.5 - eta * the_c * the_cosa / the_b, } # ============================# # face centered orthorhombic # # ============================# elif (any([a_are_equals(cosa, fco1), a_are_equals(cosb, fco1), a_are_equals(cosc, fco1)]) and any([a_are_equals(cosa, fco2), a_are_equals(cosb, fco2), a_are_equals(cosc, fco2)]) and any([a_are_equals(cosa, fco3), a_are_equals(cosb, fco3), a_are_equals(cosc, fco3)]) ): bravais_info = {'short_name': 'orcf', 'extended_name': 'face centered orthorhombic', 'index': 7, } lens = [a, b, c] ind_a1 = lens.index(max(lens)) ind_a3 = lens.index(min(lens)) if ind_a1 == 0 and ind_a3 == 2: bravais_info['permutation'] = [0, 1, 2] the_a1 = a1 the_a2 = a2 the_a3 = a3 elif ind_a1 == 0 and ind_a3 == 1: bravais_info['permutation'] = [0, 2, 1] the_a1 = a1 the_a2 = a3 the_a3 = a2 elif ind_a1 == 1 and ind_a3 == 2: bravais_info['permutation'] = [1, 0, 2] the_a1 = a2 the_a2 = a1 the_a3 = a3 elif ind_a1 == 1 and ind_a3 == 0: bravais_info['permutation'] = [2, 0, 1] the_a1 = a3 the_a2 = a1 the_a3 = a2 elif ind_a1 == 2 and ind_a3 == 1: bravais_info['permutation'] = [1, 2, 0] the_a1 = a2 the_a2 = a3 the_a3 = a1 else: # ind_a1 == 2 and ind_a3 == 0: bravais_info['permutation'] = [2, 1, 0] the_a1 = a3 the_a2 = a2 the_a3 = a1 the_a = numpy.linalg.norm(- the_a1 + the_a2 + the_a3) the_b = numpy.linalg.norm(+ the_a1 - the_a2 + the_a3) the_c = numpy.linalg.norm(+ the_a1 + the_a2 - the_a3) fco4 = 1. / the_a ** 2 - 1. / the_b ** 2 - 1. / the_c ** 2 # orcf3 if a_are_equals(fco4, 0.): bravais_info['variation'] = 'orcf3' # order matters bravais_info['extra'] = {'csi': (1. + (the_a / the_b) ** 2 - (the_a / the_c) ** 2) / 4., 'eta': (1. + (the_a / the_b) ** 2 + (the_a / the_c) ** 2) / 4., } # orcf1 elif fco4 > 0.: bravais_info['variation'] = 'orcf1' bravais_info['extra'] = {'csi': (1. + (the_a / the_b) ** 2 - (the_a / the_c) ** 2) / 4., 'eta': (1. + (the_a / the_b) ** 2 + (the_a / the_c) ** 2) / 4., } # orcf2 else: bravais_info['variation'] = 'orcf2' bravais_info['extra'] = {'eta': (1. + (the_a / the_b) ** 2 - (the_a / the_c) ** 2) / 4., 'dlt': (1. + (the_b / the_a) ** 2 + (the_b / the_c) ** 2) / 4., 'phi': (1. + (the_c / the_b) ** 2 - (the_c / the_a) ** 2) / 4., } else: bravais_info = {'short_name': 'tri', 'extended_name': 'triclinic', 'index': 14, } # ===========# # triclinic # # ===========# # still miss the variations of triclinic if bravais_info['short_name'] == 'tri': lens = [a, b, c] ind_a = lens.index(min(lens)) ind_c = lens.index(max(lens)) if ind_a == 0 and ind_c == 1: the_a = a the_b = c the_c = b the_cosa = cosa the_cosb = cosc the_cosc = cosb bravais_info['permutation'] = [0, 2, 1] if ind_a == 0 and ind_c == 2: the_a = a the_b = b the_c = c the_cosa = cosa the_cosb = cosb the_cosc = cosc bravais_info['permutation'] = [0, 1, 2] if ind_a == 1 and ind_c == 0: the_a = b the_b = c the_c = a the_cosa = cosb the_cosb = cosc the_cosc = cosa bravais_info['permutation'] = [1, 0, 2] if ind_a == 1 and ind_c == 2: the_a = b the_b = a the_c = c the_cosa = cosb the_cosb = cosa the_cosc = cosc bravais_info['permutation'] = [1, 0, 2] if ind_a == 2 and ind_c == 0: the_a = c the_b = b the_c = a the_cosa = cosc the_cosb = cosb the_cosc = cosa bravais_info['permutation'] = [2, 1, 0] if ind_a == 2 and ind_c == 1: the_a = c the_b = a the_c = b the_cosa = cosc the_cosb = cosa the_cosc = cosb bravais_info['permutation'] = [2, 0, 1] if the_cosa < 0. and the_cosb < 0.: if a_are_equals(the_cosc, 0.): bravais_info['variation'] = 'tri2a' elif the_cosc < 0.: bravais_info['variation'] = 'tri1a' else: raise ValueError('Structure erroneously fell into the triclinic (a) case') elif the_cosa > 0. and the_cosb > 0.: if a_are_equals(the_cosc, 0.): bravais_info['variation'] = 'tri2b' elif the_cosc > 0.: bravais_info['variation'] = 'tri1b' else: raise ValueError('Structure erroneously fell into the triclinic (b) case') else: raise ValueError('Structure erroneously fell into the triclinic case') elif dimension == 2: # ========================================# # 2D case -> 5 possible Bravais lattices # # ========================================# # find the two in-plane lattice vectors out_of_plane_index = pbc.index(False) # the non-periodic dimension in_plane_indexes = list(set(range(3)) - set([out_of_plane_index])) # in_plane_indexes are the indexes of the two dimensions (e.g. [0,1]) # build a length-2 list with the 2D cell lattice vectors list_vectors = ['a1', 'a2', 'a3'] vectors = [eval(list_vectors[i]) for i in in_plane_indexes] # build a length-2 list with the norms of the 2D cell lattice vectors lens = [numpy.linalg.norm(v) for v in vectors] # cosine of the angle between the two primitive vectors list_angles = ['cosa', 'cosb', 'cosc'] cosphi = eval(list_angles[out_of_plane_index]) comparison_length = l_are_equals(lens[0], lens[1]) comparison_angle_90 = a_are_equals(cosphi, _90) # ================# # square lattice # # ================# if comparison_angle_90 and comparison_length: bravais_info = {'short_name': 'sq', 'extended_name': 'square', 'index': 1, } # =========================# # (primitive) rectangular # # =========================# elif comparison_angle_90: bravais_info = {'short_name': 'rec', 'extended_name': 'rectangular', 'index': 2, } # set the order such that first_vector < second_vector in norm if lens[0] > lens[1]: in_plane_indexes.reverse() # ===========# # hexagonal # # ===========# # this has to be put before the centered-rectangular case elif (l_are_equals(lens[0], lens[1]) and a_are_equals(cosphi, _120)): bravais_info = {'short_name': 'hex', 'extended_name': 'hexagonal', 'index': 4, } # ======================# # centered rectangular # # ======================# elif (comparison_length and l_are_equals(numpy.dot(vectors[0] + vectors[1], vectors[0] - vectors[1]), 0.)): bravais_info = {'short_name': 'recc', 'extended_name': 'centered rectangular', 'index': 3, } # =========# # oblique # # =========# else: bravais_info = {'short_name': 'obl', 'extended_name': 'oblique', 'index': 5, } # set the order such that first_vector < second_vector in norm if lens[0] > lens[1]: in_plane_indexes.reverse() # the permutation is set such that p[2]=out_of_plane_index (third # new axis is always the non-periodic out-of-plane axis) # TODO: check that this (and the special points permutation of # coordinates) works also when the out-of-plane axis is not aligned # with one of the cartesian axis (I suspect that it doesn't...) permutation = in_plane_indexes + [out_of_plane_index] bravais_info['permutation'] = permutation elif dimension <= 1: # ====================================================# # 0D & 1D cases -> only one possible Bravais lattice # # ====================================================# if dimension == 1: # TODO: check that this (and the special points permutation of # coordinates) works also when the 1D axis is not aligned # with one of the cartesian axis (I suspect that it doesn't...) in_line_index = pbc.index(True) # the only periodic dimension # the permutation is set such that p[0]=in_line_index (the 2 last # axes are always the non-periodic ones) permutation = [in_line_index] + list(set(range(3)) - set([in_line_index])) else: permutation = [0, 1, 2] bravais_info = { 'short_name': '{}D'.format(dimension), 'extended_name': '{}D'.format(dimension), 'index': 1, 'permutation': permutation, } return bravais_info
[docs]def get_kpoints_path(cell, pbc=None, cartesian=False, epsilon_length=_default_epsilon_length, epsilon_angle=_default_epsilon_angle): """ Get the special point and path of a given structure. .. note:: in 3D, this implementation expects that the structure is already standardized according to the Setyawan paper. If this is not the case, the kpoints and band structure returned will be incorrect. The only case that is dealt correctly by the library is the case when axes are swapped, where the library correctly takes this swapping/rotation into account to assign kpoint labels and coordinates. - In 2D, coordinates are based on the paper: R. Ramirez and M. C. Bohm, Int. J. Quant. Chem., XXX, pp. 391-411 (1986) - In 3D, coordinates are based on the paper: W. Setyawan, S. Curtarolo, Comp. Mat. Sci. 49, 299 (2010) :param cell: 3x3 array representing the structure cell lattice vectors :param pbc: 3-dimensional array of booleans signifying the periodic boundary conditions along each lattice vector :param cartesian: If true, returns points in cartesian coordinates. Crystal coordinates otherwise. Default=False :param epsilon_length: threshold on lengths comparison, used to get the bravais lattice info :param epsilon_angle: threshold on angles comparison, used to get the bravais lattice info :return special_points,path: special_points: a dictionary of point_name:point_coords key,values. path: the suggested path which goes through all high symmetry lines. A list of lists for all path segments. e.g. ``[('G','X'),('X','M'),...]`` It's not necessarily a continuous line. :note: We assume that the cell given by the cell property is the primitive unit cell """ # recognize which bravais lattice we are dealing with bravais_info = find_bravais_info( cell=cell, pbc=pbc, epsilon_length=epsilon_length, epsilon_angle=epsilon_angle ) analysis = analyze_cell(cell, pbc) dimension = analysis['dimension'] reciprocal_cell = analysis['reciprocal_cell'] # pick the information about the special k-points. # it depends on the dimensionality and the Bravais lattice number. if dimension == 3: # 3D case: 14 Bravais lattices # simple cubic if bravais_info['index'] == 1: special_points = {'G': [0., 0., 0.], 'M': [0.5, 0.5, 0.], 'R': [0.5, 0.5, 0.5], 'X': [0., 0.5, 0.], } path = [('G', 'X'), ('X', 'M'), ('M', 'G'), ('G', 'R'), ('R', 'X'), ('M', 'R'), ] # face centered cubic elif bravais_info['index'] == 2: special_points = {'G': [0., 0., 0.], 'K': [3. / 8., 3. / 8., 0.75], 'L': [0.5, 0.5, 0.5], 'U': [5. / 8., 0.25, 5. / 8.], 'W': [0.5, 0.25, 0.75], 'X': [0.5, 0., 0.5], } path = [('G', 'X'), ('X', 'W'), ('W', 'K'), ('K', 'G'), ('G', 'L'), ('L', 'U'), ('U', 'W'), ('W', 'L'), ('L', 'K'), ('U', 'X'), ] # body centered cubic elif bravais_info['index'] == 3: special_points = {'G': [0., 0., 0.], 'H': [0.5, -0.5, 0.5], 'P': [0.25, 0.25, 0.25], 'N': [0., 0., 0.5], } path = [('G', 'H'), ('H', 'N'), ('N', 'G'), ('G', 'P'), ('P', 'H'), ('P', 'N'), ] # Tetragonal elif bravais_info['index'] == 4: special_points = {'G': [0., 0., 0.], 'A': [0.5, 0.5, 0.5], 'M': [0.5, 0.5, 0.], 'R': [0., 0.5, 0.5], 'X': [0., 0.5, 0.], 'Z': [0., 0., 0.5], } path = [('G', 'X'), ('X', 'M'), ('M', 'G'), ('G', 'Z'), ('Z', 'R'), ('R', 'A'), ('A', 'Z'), ('X', 'R'), ('M', 'A'), ] # body centered tetragonal elif bravais_info['index'] == 5: if bravais_info['variation'] == 'bct1': # Body centered tetragonal bct1 eta = bravais_info['extra']['eta'] special_points = {'G': [0., 0., 0.], 'M': [-0.5, 0.5, 0.5], 'N': [0., 0.5, 0.], 'P': [0.25, 0.25, 0.25], 'X': [0., 0., 0.5], 'Z': [eta, eta, -eta], 'Z1': [-eta, 1. - eta, eta], } path = [('G', 'X'), ('X', 'M'), ('M', 'G'), ('G', 'Z'), ('Z', 'P'), ('P', 'N'), ('N', 'Z1'), ('Z1', 'M'), ('X', 'P'), ] else: # bct2 # Body centered tetragonal bct2 eta = bravais_info['extra']['eta'] csi = bravais_info['extra']['csi'] special_points = { 'G': [0., 0., 0.], 'N': [0., 0.5, 0.], 'P': [0.25, 0.25, 0.25], 'S': [-eta, eta, eta], 'S1': [eta, 1 - eta, -eta], 'X': [0., 0., 0.5], 'Y': [-csi, csi, 0.5], 'Y1': [0.5, 0.5, -csi], 'Z': [0.5, 0.5, -0.5], } path = [('G', 'X'), ('X', 'Y'), ('Y', 'S'), ('S', 'G'), ('G', 'Z'), ('Z', 'S1'), ('S1', 'N'), ('N', 'P'), ('P', 'Y1'), ('Y1', 'Z'), ('X', 'P'), ] # orthorhombic elif bravais_info['index'] == 6: special_points = {'G': [0., 0., 0.], 'R': [0.5, 0.5, 0.5], 'S': [0.5, 0.5, 0.], 'T': [0., 0.5, 0.5], 'U': [0.5, 0., 0.5], 'X': [0.5, 0., 0.], 'Y': [0., 0.5, 0.], 'Z': [0., 0., 0.5], } path = [('G', 'X'), ('X', 'S'), ('S', 'Y'), ('Y', 'G'), ('G', 'Z'), ('Z', 'U'), ('U', 'R'), ('R', 'T'), ('T', 'Z'), ('Y', 'T'), ('U', 'X'), ('S', 'R'), ] # face centered orthorhombic elif bravais_info['index'] == 7: if bravais_info['variation'] == 'orcf1': csi = bravais_info['extra']['csi'] eta = bravais_info['extra']['eta'] special_points = {'G': [0., 0., 0.], 'A': [0.5, 0.5 + csi, csi], 'A1': [0.5, 0.5 - csi, 1. - csi], 'L': [0.5, 0.5, 0.5], 'T': [1., 0.5, 0.5], 'X': [0., eta, eta], 'X1': [1., 1. - eta, 1. - eta], 'Y': [0.5, 0., 0.5], 'Z': [0.5, 0.5, 0.], } path = [('G', 'Y'), ('Y', 'T'), ('T', 'Z'), ('Z', 'G'), ('G', 'X'), ('X', 'A1'), ('A1', 'Y'), ('T', 'X1'), ('X', 'A'), ('A', 'Z'), ('L', 'G'), ] elif bravais_info['variation'] == 'orcf2': eta = bravais_info['extra']['eta'] dlt = bravais_info['extra']['dlt'] phi = bravais_info['extra']['phi'] special_points = {'G': [0., 0., 0.], 'C': [0.5, 0.5 - eta, 1. - eta], 'C1': [0.5, 0.5 + eta, eta], 'D': [0.5 - dlt, 0.5, 1. - dlt], 'D1': [0.5 + dlt, 0.5, dlt], 'L': [0.5, 0.5, 0.5], 'H': [1. - phi, 0.5 - phi, 0.5], 'H1': [phi, 0.5 + phi, 0.5], 'X': [0., 0.5, 0.5], 'Y': [0.5, 0., 0.5], 'Z': [0.5, 0.5, 0.], } path = [('G', 'Y'), ('Y', 'C'), ('C', 'D'), ('D', 'X'), ('X', 'G'), ('G', 'Z'), ('Z', 'D1'), ('D1', 'H'), ('H', 'C'), ('C1', 'Z'), ('X', 'H1'), ('H', 'Y'), ('L', 'G'), ] else: csi = bravais_info['extra']['csi'] eta = bravais_info['extra']['eta'] special_points = {'G': [0., 0., 0.], 'A': [0.5, 0.5 + csi, csi], 'A1': [0.5, 0.5 - csi, 1. - csi], 'L': [0.5, 0.5, 0.5], 'T': [1., 0.5, 0.5], 'X': [0., eta, eta], 'X1': [1., 1. - eta, 1. - eta], 'Y': [0.5, 0., 0.5], 'Z': [0.5, 0.5, 0.], } path = [('G', 'Y'), ('Y', 'T'), ('T', 'Z'), ('Z', 'G'), ('G', 'X'), ('X', 'A1'), ('A1', 'Y'), ('X', 'A'), ('A', 'Z'), ('L', 'G'), ] # Body centered orthorhombic elif bravais_info['index'] == 8: csi = bravais_info['extra']['csi'] dlt = bravais_info['extra']['dlt'] eta = bravais_info['extra']['eta'] mu = bravais_info['extra']['mu'] special_points = {'G': [0., 0., 0.], 'L': [-mu, mu, 0.5 - dlt], 'L1': [mu, -mu, 0.5 + dlt], 'L2': [0.5 - dlt, 0.5 + dlt, -mu], 'R': [0., 0.5, 0.], 'S': [0.5, 0., 0.], 'T': [0., 0., 0.5], 'W': [0.25, 0.25, 0.25], 'X': [-csi, csi, csi], 'X1': [csi, 1. - csi, -csi], 'Y': [eta, -eta, eta], 'Y1': [1. - eta, eta, -eta], 'Z': [0.5, 0.5, -0.5], } path = [('G', 'X'), ('X', 'L'), ('L', 'T'), ('T', 'W'), ('W', 'R'), ('R', 'X1'), ('X1', 'Z'), ('Z', 'G'), ('G', 'Y'), ('Y', 'S'), ('S', 'W'), ('L1', 'Y'), ('Y1', 'Z'), ] # C-centered orthorhombic elif bravais_info['index'] == 9: csi = bravais_info['extra']['csi'] special_points = {'G': [0., 0., 0.], 'A': [csi, csi, 0.5], 'A1': [-csi, 1. - csi, 0.5], 'R': [0., 0.5, 0.5], 'S': [0., 0.5, 0.], 'T': [-0.5, 0.5, 0.5], 'X': [csi, csi, 0.], 'X1': [-csi, 1. - csi, 0.], 'Y': [-0.5, 0.5, 0.], 'Z': [0., 0., 0.5], } path = [('G', 'X'), ('X', 'S'), ('S', 'R'), ('R', 'A'), ('A', 'Z'), ('Z', 'G'), ('G', 'Y'), ('Y', 'X1'), ('X1', 'A1'), ('A1', 'T'), ('T', 'Y'), ('Z', 'T'), ] # Hexagonal elif bravais_info['index'] == 10: special_points = {'G': [0., 0., 0.], 'A': [0., 0., 0.5], 'H': [1. / 3., 1. / 3., 0.5], 'K': [1. / 3., 1. / 3., 0.], 'L': [0.5, 0., 0.5], 'M': [0.5, 0., 0.], } path = [('G', 'M'), ('M', 'K'), ('K', 'G'), ('G', 'A'), ('A', 'L'), ('L', 'H'), ('H', 'A'), ('L', 'M'), ('K', 'H'), ] # rhombohedral elif bravais_info['index'] == 11: if bravais_info['variation'] == 'rhl1': eta = bravais_info['extra']['eta'] nu = bravais_info['extra']['nu'] special_points = {'G': [0., 0., 0.], 'B': [eta, 0.5, 1. - eta], 'B1': [0.5, 1. - eta, eta - 1.], 'F': [0.5, 0.5, 0.], 'L': [0.5, 0., 0.], 'L1': [0., 0., -0.5], 'P': [eta, nu, nu], 'P1': [1. - nu, 1. - nu, 1. - eta], 'P2': [nu, nu, eta - 1.], 'Q': [1. - nu, nu, 0.], 'X': [nu, 0., -nu], 'Z': [0.5, 0.5, 0.5], } path = [('G', 'L'), ('L', 'B1'), ('B', 'Z'), ('Z', 'G'), ('G', 'X'), ('Q', 'F'), ('F', 'P1'), ('P1', 'Z'), ('L', 'P'), ] else: # Rhombohedral rhl2 eta = bravais_info['extra']['eta'] nu = bravais_info['extra']['nu'] special_points = {'G': [0., 0., 0.], 'F': [0.5, -0.5, 0.], 'L': [0.5, 0., 0.], 'P': [1. - nu, -nu, 1. - nu], 'P1': [nu, nu - 1., nu - 1.], 'Q': [eta, eta, eta], 'Q1': [1. - eta, -eta, -eta], 'Z': [0.5, -0.5, 0.5], } path = [('G', 'P'), ('P', 'Z'), ('Z', 'Q'), ('Q', 'G'), ('G', 'F'), ('F', 'P1'), ('P1', 'Q1'), ('Q1', 'L'), ('L', 'Z'), ] # monoclinic elif bravais_info['index'] == 12: eta = bravais_info['extra']['eta'] nu = bravais_info['extra']['nu'] special_points = {'G': [0., 0., 0.], 'A': [0.5, 0.5, 0.], 'C': [0., 0.5, 0.5], 'D': [0.5, 0., 0.5], 'D1': [0.5, 0., -0.5], 'E': [0.5, 0.5, 0.5], 'H': [0., eta, 1. - nu], 'H1': [0., 1. - eta, nu], 'H2': [0., eta, -nu], 'M': [0.5, eta, 1. - nu], 'M1': [0.5, 1. - eta, nu], 'M2': [0.5, eta, -nu], 'X': [0., 0.5, 0.], 'Y': [0., 0., 0.5], 'Y1': [0., 0., -0.5], 'Z': [0.5, 0., 0.], } path = [('G', 'Y'), ('Y', 'H'), ('H', 'C'), ('C', 'E'), ('E', 'M1'), ('M1', 'A'), ('A', 'X'), ('X', 'H1'), ('M', 'D'), ('D', 'Z'), ('Y', 'D'), ] elif bravais_info['index'] == 13: if bravais_info['variation'] == 'mclc1': csi = bravais_info['extra']['csi'] eta = bravais_info['extra']['eta'] psi = bravais_info['extra']['psi'] phi = bravais_info['extra']['phi'] special_points = {'G': [0., 0., 0.], 'N': [0.5, 0., 0.], 'N1': [0., -0.5, 0.], 'F': [1. - csi, 1. - csi, 1. - eta], 'F1': [csi, csi, eta], 'F2': [csi, -csi, 1. - eta], 'F3': [1. - csi, -csi, 1. - eta], 'I': [phi, 1. - phi, 0.5], 'I1': [1. - phi, phi - 1., 0.5], 'L': [0.5, 0.5, 0.5], 'M': [0.5, 0., 0.5], 'X': [1. - psi, psi - 1., 0.], 'X1': [psi, 1. - psi, 0.], 'X2': [psi - 1., -psi, 0.], 'Y': [0.5, 0.5, 0.], 'Y1': [-0.5, -0.5, 0.], 'Z': [0., 0., 0.5], } path = [('G', 'Y'), ('Y', 'F'), ('F', 'L'), ('L', 'I'), ('I1', 'Z'), ('Z', 'F1'), ('Y', 'X1'), ('X', 'G'), ('G', 'N'), ('M', 'G'), ] elif bravais_info['variation'] == 'mclc2': csi = bravais_info['extra']['csi'] eta = bravais_info['extra']['eta'] psi = bravais_info['extra']['psi'] phi = bravais_info['extra']['phi'] special_points = {'G': [0., 0., 0.], 'N': [0.5, 0., 0.], 'N1': [0., -0.5, 0.], 'F': [1. - csi, 1. - csi, 1. - eta], 'F1': [csi, csi, eta], 'F2': [csi, -csi, 1. - eta], 'F3': [1. - csi, -csi, 1. - eta], 'I': [phi, 1. - phi, 0.5], 'I1': [1. - phi, phi - 1., 0.5], 'L': [0.5, 0.5, 0.5], 'M': [0.5, 0., 0.5], 'X': [1. - psi, psi - 1., 0.], 'X1': [psi, 1. - psi, 0.], 'X2': [psi - 1., -psi, 0.], 'Y': [0.5, 0.5, 0.], 'Y1': [-0.5, -0.5, 0.], 'Z': [0., 0., 0.5], } path = [('G', 'Y'), ('Y', 'F'), ('F', 'L'), ('L', 'I'), ('I1', 'Z'), ('Z', 'F1'), ('N', 'G'), ('G', 'M'), ] elif bravais_info['variation'] == 'mclc3': mu = bravais_info['extra']['mu'] dlt = bravais_info['extra']['dlt'] csi = bravais_info['extra']['csi'] eta = bravais_info['extra']['eta'] phi = bravais_info['extra']['phi'] psi = bravais_info['extra']['psi'] special_points = { 'G': [0., 0., 0.], 'F': [1. - phi, 1 - phi, 1. - psi], 'F1': [phi, phi - 1., psi], 'F2': [1. - phi, -phi, 1. - psi], 'H': [csi, csi, eta], 'H1': [1. - csi, -csi, 1. - eta], 'H2': [-csi, -csi, 1. - eta], 'I': [0.5, -0.5, 0.5], 'M': [0.5, 0., 0.5], 'N': [0.5, 0., 0.], 'N1': [0., -0.5, 0.], 'X': [0.5, -0.5, 0.], 'Y': [mu, mu, dlt], 'Y1': [1. - mu, -mu, -dlt], 'Y2': [-mu, -mu, -dlt], 'Y3': [mu, mu - 1., dlt], 'Z': [0., 0., 0.5], } path = [('G', 'Y'), ('Y', 'F'), ('F', 'H'), ('H', 'Z'), ('Z', 'I'), ('I', 'F1'), ('H1', 'Y1'), ('Y1', 'X'), ('X', 'F'), ('G', 'N'), ('M', 'G'), ] elif bravais_info['variation'] == 'mclc4': mu = bravais_info['extra']['mu'] dlt = bravais_info['extra']['dlt'] csi = bravais_info['extra']['csi'] eta = bravais_info['extra']['eta'] phi = bravais_info['extra']['phi'] psi = bravais_info['extra']['psi'] special_points = {'G': [0., 0., 0.], 'F': [1. - phi, 1 - phi, 1. - psi], 'F1': [phi, phi - 1., psi], 'F2': [1. - phi, -phi, 1. - psi], 'H': [csi, csi, eta], 'H1': [1. - csi, -csi, 1. - eta], 'H2': [-csi, -csi, 1. - eta], 'I': [0.5, -0.5, 0.5], 'M': [0.5, 0., 0.5], 'N': [0.5, 0., 0.], 'N1': [0., -0.5, 0.], 'X': [0.5, -0.5, 0.], 'Y': [mu, mu, dlt], 'Y1': [1. - mu, -mu, -dlt], 'Y2': [-mu, -mu, -dlt], 'Y3': [mu, mu - 1., dlt], 'Z': [0., 0., 0.5], } path = [('G', 'Y'), ('Y', 'F'), ('F', 'H'), ('H', 'Z'), ('Z', 'I'), ('H1', 'Y1'), ('Y1', 'X'), ('X', 'G'), ('G', 'N'), ('M', 'G'), ] else: csi = bravais_info['extra']['csi'] mu = bravais_info['extra']['mu'] omg = bravais_info['extra']['omg'] eta = bravais_info['extra']['eta'] nu = bravais_info['extra']['nu'] dlt = bravais_info['extra']['dlt'] rho = bravais_info['extra']['rho'] special_points = { 'G': [0., 0., 0.], 'F': [nu, nu, omg], 'F1': [1. - nu, 1. - nu, 1. - omg], 'F2': [nu, nu - 1., omg], 'H': [csi, csi, eta], 'H1': [1. - csi, -csi, 1. - eta], 'H2': [-csi, -csi, 1. - eta], 'I': [rho, 1. - rho, 0.5], 'I1': [1. - rho, rho - 1., 0.5], 'L': [0.5, 0.5, 0.5], 'M': [0.5, 0., 0.5], 'N': [0.5, 0., 0.], 'N1': [0., -0.5, 0.], 'X': [0.5, -0.5, 0.], 'Y': [mu, mu, dlt], 'Y1': [1. - mu, -mu, -dlt], 'Y2': [-mu, -mu, -dlt], 'Y3': [mu, mu - 1., dlt], 'Z': [0., 0., 0.5], } path = [('G', 'Y'), ('Y', 'F'), ('F', 'L'), ('L', 'I'), ('I1', 'Z'), ('Z', 'H'), ('H', 'F1'), ('H1', 'Y1'), ('Y1', 'X'), ('X', 'G'), ('G', 'N'), ('M', 'G'), ] # triclinic elif bravais_info['index'] == 14: if bravais_info['variation'] == 'tri1a' or bravais_info['variation'] == 'tri2a': special_points = {'G': [0.0, 0.0, 0.0], 'L': [0.5, 0.5, 0.0], 'M': [0.0, 0.5, 0.5], 'N': [0.5, 0.0, 0.5], 'R': [0.5, 0.5, 0.5], 'X': [0.5, 0.0, 0.0], 'Y': [0.0, 0.5, 0.0], 'Z': [0.0, 0.0, 0.5], } path = [('X', 'G'), ('G', 'Y'), ('L', 'G'), ('G', 'Z'), ('N', 'G'), ('G', 'M'), ('R', 'G'), ] else: special_points = {'G': [0.0, 0.0, 0.0], 'L': [0.5, -0.5, 0.0], 'M': [0.0, 0.0, 0.5], 'N': [-0.5, -0.5, 0.5], 'R': [0.0, -0.5, 0.5], 'X': [0.0, -0.5, 0.0], 'Y': [0.5, 0.0, 0.0], 'Z': [-0.5, 0.0, 0.5], } path = [('X', 'G'), ('G', 'Y'), ('L', 'G'), ('G', 'Z'), ('N', 'G'), ('G', 'M'), ('R', 'G'), ] elif dimension == 2: # 2D case: 5 Bravais lattices if bravais_info['index'] == 1: # square special_points = {'G': [0., 0., 0.], 'M': [0.5, 0.5, 0.], 'X': [0.5, 0., 0.], } path = [('G', 'X'), ('X', 'M'), ('M', 'G'), ] elif bravais_info['index'] == 2: # (primitive) rectangular special_points = {'G': [0., 0., 0.], 'X': [0.5, 0., 0.], 'Y': [0., 0.5, 0.], 'S': [0.5, 0.5, 0.], } path = [('G', 'X'), ('X', 'S'), ('S', 'Y'), ('Y', 'G'), ] elif bravais_info['index'] == 3: # centered rectangular (rhombic) # TODO: this looks quite different from the in-plane part of the # 3D C-centered orthorhombic lattice, which is strange... # NOTE: special points below are in (b1, b2) fractional # coordinates (primitive reciprocal cell) as for the rest. # Ramirez & Bohn gave them initially in (s1=b1+b2, s2=-b1+b2) # coordinates, i.e. using the conventional reciprocal cell. special_points = {'G': [0., 0., 0.], 'X': [0.5, 0.5, 0.], 'Y1': [0.25, 0.75, 0.], 'Y': [-0.25, 0.25, 0.], # typo in p. 404 of Ramirez & Bohm (should be Y=(0,1/4)) 'C': [0., 0.5, 0.], } path = [('Y1', 'X'), ('X', 'G'), ('G', 'Y'), ('Y', 'C'), ] elif bravais_info['index'] == 4: # hexagonal special_points = {'G': [0., 0., 0.], 'M': [0.5, 0., 0.], 'K': [1. / 3., 1. / 3., 0.], } path = [('G', 'M'), ('M', 'K'), ('K', 'G'), ] elif bravais_info['index'] == 5: # oblique # NOTE: only end-points are high-symmetry points (not the path # in-between) special_points = {'G': [0., 0., 0.], 'X': [0.5, 0., 0.], 'Y': [0., 0.5, 0.], 'A': [0.5, 0.5, 0.], } path = [('X', 'G'), ('G', 'Y'), ('A', 'G'), ] elif dimension == 1: # 1D case: 1 Bravais lattice special_points = {'G': [0., 0., 0.], 'X': [0.5, 0., 0.], } path = [('G', 'X'), ] elif dimension == 0: # 0D case: 1 Bravais lattice, only Gamma point, no path special_points = {'G': [0., 0., 0.], } path = [('G', 'G'), ] permutation = bravais_info['permutation'] def permute(x, permutation): # return new_x such that new_x[i]=x[permutation[i]] return [x[int(p)] for p in permutation] def invpermute(permutation): # return the inverse of permutation return [permutation.index(i) for i in range(3)] the_special_points = {} for k in special_points.keys(): # NOTE: this originally returned the inverse of the permutation, but was later changed to permutation the_special_points[k] = permute(special_points[k], permutation) # output crystal or cartesian if cartesian: the_abs_special_points = {} for k in the_special_points.keys(): the_abs_special_points[k] = change_reference( reciprocal_cell, numpy.array(the_special_points[k]), to_cartesian=True ) return the_abs_special_points, path, bravais_info else: return the_special_points, path, bravais_info