from __future__ import division from ase.utils import basestring import re import warnings from math import sin, cos import numpy as np from ase.geometry import cell_to_cellpar, crystal_structure_from_cell def monkhorst_pack(size): """Construct a uniform sampling of k-space of given size.""" if np.less_equal(size, 0).any(): raise ValueError('Illegal size: %s' % list(size)) kpts = np.indices(size).transpose((1, 2, 3, 0)).reshape((-1, 3)) return (kpts + 0.5) / size - 0.5 def get_monkhorst_pack_size_and_offset(kpts): """Find Monkhorst-Pack size and offset. Returns (size, offset), where:: kpts = monkhorst_pack(size) + offset. The set of k-points must not have been symmetry reduced.""" if len(kpts) == 1: return np.ones(3, int), np.array(kpts[0], dtype=float) size = np.zeros(3, int) for c in range(3): # Determine increment between k-points along current axis delta = max(np.diff(np.sort(kpts[:, c]))) # Determine number of k-points as inverse of distance between kpoints if delta > 1e-8: size[c] = int(round(1.0 / delta)) else: size[c] = 1 if size.prod() == len(kpts): kpts0 = monkhorst_pack(size) offsets = kpts - kpts0 # All offsets must be identical: if (offsets.ptp(axis=0) < 1e-9).all(): return size, offsets[0].copy() raise ValueError('Not an ASE-style Monkhorst-Pack grid!') def get_monkhorst_shape(kpts): warnings.warn('Use get_monkhorst_pack_size_and_offset()[0] instead.') return get_monkhorst_pack_size_and_offset(kpts)[0] def kpoint_convert(cell_cv, skpts_kc=None, ckpts_kv=None): """Convert k-points between scaled and cartesian coordinates. Given the atomic unit cell, and either the scaled or cartesian k-point coordinates, the other is determined. The k-point arrays can be either a single point, or a list of points, i.e. the dimension k can be empty or multidimensional. """ if ckpts_kv is None: icell_cv = 2 * np.pi * np.linalg.pinv(cell_cv).T return np.dot(skpts_kc, icell_cv) elif skpts_kc is None: return np.dot(ckpts_kv, cell_cv.T) / (2 * np.pi) else: raise KeyError('Either scaled or cartesian coordinates must be given.') def parse_path_string(s): """Parse compact string representation of BZ path. A path string can have several non-connected sections separated by commas. The return value is a list of sections where each section is a list of labels. Examples: >>> parse_path_string('GX') [['G', 'X']] >>> parse_path_string('GX,M1A') [['G', 'X'], ['M1', 'A']] """ paths = [] for path in s.split(','): names = [name if name != 'Gamma' else 'G' for name in re.split(r'([A-Z][a-z0-9]*)', path) if name] paths.append(names) return paths def bandpath(path, cell, npoints=50): """Make a list of kpoints defining the path between the given points. path: list or str Can be: * a string that parse_path_string() understands: 'GXL' * a list of BZ points: [(0, 0, 0), (0.5, 0, 0)] * or several lists of BZ points if the the path is not continuous. cell: 3x3 Unit cell of the atoms. npoints: int Length of the output kpts list. Return list of k-points, list of x-coordinates and list of x-coordinates of special points.""" if isinstance(path, basestring): cellinfo = get_cellinfo(cell) special = cellinfo.special_points paths = [] for names in parse_path_string(path): for name in names: if name not in special: msg = ('Invalid k-point label {} for {} cell. ' 'Valid labels are {}.' .format(name, cellinfo.lattice, ', '.join(sorted(special)))) raise ValueError(msg) paths.append([special[name] for name in names]) elif np.array(path[0]).ndim == 1: paths = [path] else: paths = path points = np.concatenate(paths) dists = points[1:] - points[:-1] lengths = [np.linalg.norm(d) for d in kpoint_convert(cell, skpts_kc=dists)] i = 0 for path in paths[:-1]: i += len(path) lengths[i - 1] = 0 length = sum(lengths) kpts = [] x0 = 0 x = [] X = [0] for P, d, L in zip(points[:-1], dists, lengths): n = max(2, int(round(L * (npoints - len(x)) / (length - x0)))) for t in np.linspace(0, 1, n)[:-1]: kpts.append(P + t * d) x.append(x0 + t * L) x0 += L X.append(x0) kpts.append(points[-1]) x.append(x0) return np.array(kpts), np.array(x), np.array(X) get_bandpath = bandpath # old name def labels_from_kpts(kpts, cell, eps=1e-5): """Get an x-axis to be used when plotting a band structure. The first of the returned lists can be used as a x-axis when plotting the band structure. The second list can be used as xticks, and the third as xticklabels. Parameters: kpts: list List of scaled k-points. cell: list Unit cell of the atomic structure. Returns: Three arrays; the first is a list of cumulative distances between k-points, the second is x coordinates of the special points, the third is the special points as strings. """ special_points = get_special_points(cell) points = np.asarray(kpts) diffs = points[1:] - points[:-1] kinks = abs(diffs[1:] - diffs[:-1]).sum(1) > eps N = len(points) indices = [0] indices.extend(np.arange(1, N - 1)[kinks]) indices.append(N - 1) labels = [] for kpt in points[indices]: for label, k in special_points.items(): if abs(kpt - k).sum() < eps: break else: # No exact match. Try modulus 1: for label, k in special_points.items(): if abs((kpt - k) % 1).sum() < eps: break else: label = '?' labels.append(label) jump = False # marks a discontinuity in the path xcoords = [0] for i1, i2 in zip(indices[:-1], indices[1:]): if not jump and i1 + 1 == i2: length = 0 jump = True # we don't want two jumps in a row else: diff = points[i2] - points[i1] length = np.linalg.norm(kpoint_convert(cell, skpts_kc=diff)) jump = False xcoords.extend(np.linspace(0, length, i2 - i1 + 1)[1:] + xcoords[-1]) xcoords = np.array(xcoords) return xcoords, xcoords[indices], labels special_points = { 'cubic': {'G': [0, 0, 0], 'M': [1 / 2, 1 / 2, 0], 'R': [1 / 2, 1 / 2, 1 / 2], 'X': [0, 1 / 2, 0]}, 'fcc': {'G': [0, 0, 0], 'K': [3 / 8, 3 / 8, 3 / 4], 'L': [1 / 2, 1 / 2, 1 / 2], 'U': [5 / 8, 1 / 4, 5 / 8], 'W': [1 / 2, 1 / 4, 3 / 4], 'X': [1 / 2, 0, 1 / 2]}, 'bcc': {'G': [0, 0, 0], 'H': [1 / 2, -1 / 2, 1 / 2], 'P': [1 / 4, 1 / 4, 1 / 4], 'N': [0, 0, 1 / 2]}, 'tetragonal': {'G': [0, 0, 0], 'A': [1 / 2, 1 / 2, 1 / 2], 'M': [1 / 2, 1 / 2, 0], 'R': [0, 1 / 2, 1 / 2], 'X': [0, 1 / 2, 0], 'Z': [0, 0, 1 / 2]}, 'orthorhombic': {'G': [0, 0, 0], 'R': [1 / 2, 1 / 2, 1 / 2], 'S': [1 / 2, 1 / 2, 0], 'T': [0, 1 / 2, 1 / 2], 'U': [1 / 2, 0, 1 / 2], 'X': [1 / 2, 0, 0], 'Y': [0, 1 / 2, 0], 'Z': [0, 0, 1 / 2]}, 'hexagonal': {'G': [0, 0, 0], 'A': [0, 0, 1 / 2], 'H': [1 / 3, 1 / 3, 1 / 2], 'K': [1 / 3, 1 / 3, 0], 'L': [1 / 2, 0, 1 / 2], 'M': [1 / 2, 0, 0]}} special_paths = { 'cubic': 'GXMGRX,MR', 'fcc': 'GXWKGLUWLK,UX', 'bcc': 'GHNGPH,PN', 'tetragonal': 'GXMGZRAZXR,MA', 'orthorhombic': 'GXSYGZURTZ,YT,UX,SR', 'hexagonal': 'GMKGALHA,LM,KH', 'monoclinic': 'GYHCEM1AXH1,MDZ,YD', 'rhombohedral type 1': 'GLB1,BZGX,QFP1Z,LP', 'rhombohedral type 2': 'GPZQGFP1Q1LZ'} class CellInfo: def __init__(self, rcell, lattice, special_points): self.rcell = rcell self.lattice = lattice self.special_points = special_points def get_cellinfo(cell, lattice=None, eps=2e-4): from ase.build.tools import niggli_reduce_cell rcell, M = niggli_reduce_cell(cell) latt = crystal_structure_from_cell(rcell, niggli_reduce=False) if lattice: assert latt == lattice.lower(), latt if latt == 'monoclinic': # Transform From Niggli to Setyawana-Curtarolo cell: a, b, c, alpha, beta, gamma = cell_to_cellpar(rcell, radians=True) if abs(beta - np.pi / 2) > eps: T = np.array([[0, 1, 0], [-1, 0, 0], [0, 0, 1]]) scell = np.dot(T, rcell) elif abs(gamma - np.pi / 2) > eps: T = np.array([[0, 0, 1], [1, 0, 0], [0, -1, 0]]) else: raise ValueError('You are using a badly oriented ' + 'monoclinic unit cell. Please choose one with ' + 'either beta or gamma != pi/2') scell = np.dot(np.dot(T, rcell), T.T) a, b, c, alpha, beta, gamma = cell_to_cellpar(scell, radians=True) assert alpha < np.pi / 2, 'Your monoclinic angle has to be < pi / 2' M = np.dot(M, T.T) eta = (1 - b * cos(alpha) / c) / (2 * sin(alpha)**2) nu = 1 / 2 - eta * c * cos(alpha) / b points = {'G': [0, 0, 0], 'A': [1 / 2, 1 / 2, 0], 'C': [0, 1 / 2, 1 / 2], 'D': [1 / 2, 0, 1 / 2], 'D1': [1 / 2, 0, -1 / 2], 'E': [1 / 2, 1 / 2, 1 / 2], 'H': [0, eta, 1 - nu], 'H1': [0, 1 - eta, nu], 'H2': [0, eta, -nu], 'M': [1 / 2, eta, 1 - nu], 'M1': [1 / 2, 1 - eta, nu], 'M2': [1 / 2, eta, -nu], 'X': [0, 1 / 2, 0], 'Y': [0, 0, 1 / 2], 'Y1': [0, 0, -1 / 2], 'Z': [1 / 2, 0, 0]} elif latt == 'rhombohedral type 1': a, b, c, alpha, beta, gamma = cell_to_cellpar(cell=cell, radians=True) eta = (1 + 4 * np.cos(alpha)) / (2 + 4 * np.cos(alpha)) nu = 3 / 4 - eta / 2 points = {'G': [0, 0, 0], 'B': [eta, 1 / 2, 1 - eta], 'B1': [1 / 2, 1 - eta, eta - 1], 'F': [1 / 2, 1 / 2, 0], 'L': [1 / 2, 0, 0], 'L1': [0, 0, - 1 / 2], '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]} else: points = special_points[latt] myspecial_points = {label: np.dot(M, kpt) for label, kpt in points.items()} return CellInfo(rcell=rcell, lattice=latt, special_points=myspecial_points) def get_special_points(cell, lattice=None, eps=2e-4): """Return dict of special points. The definitions are from a paper by Wahyu Setyawana and Stefano Curtarolo:: http://dx.doi.org/10.1016/j.commatsci.2010.05.010 cell: 3x3 ndarray Unit cell. lattice: str Optionally check that the cell is one of the following: cubic, fcc, bcc, orthorhombic, tetragonal, hexagonal or monoclinic. eps: float Tolerance for cell-check. """ if isinstance(cell, str): warnings.warn('Please call this function with cell as the first ' 'argument') lattice, cell = cell, lattice cellinfo = get_cellinfo(cell=cell, lattice=lattice, eps=eps) return cellinfo.special_points def monkhorst_pack_interpolate(path, values, icell, bz2ibz, size, offset=(0, 0, 0)): """Interpolate values from Monkhorst-Pack sampling. path: (nk, 3) array-like Desired path in units of reciprocal lattice vectors. values: (nibz, ...) array-like Values on Monkhorst-Pack grid. icell: (3, 3) array-like Reciprocal lattice vectors. bz2ibz: (nbz,) array-like of int Map from nbz points in BZ to nibz reduced points in IBZ. size: (3,) array-like of int Size of Monkhorst-Pack grid. offset: (3,) array-like Offset of Monkhorst-Pack grid. Returns *values* interpolated to *path*. """ from scipy.interpolate import LinearNDInterpolator path = (np.asarray(path) + 0.5) % 1 - 0.5 path = np.dot(path, icell) # Fold out values from IBZ to BZ: v = np.asarray(values)[bz2ibz] v = v.reshape(tuple(size) + v.shape[1:]) # Create padded Monkhorst-Pack grid: size = np.asarray(size) i = np.indices(size + 2).transpose((1, 2, 3, 0)).reshape((-1, 3)) k = (i - 0.5) / size - 0.5 + offset k = np.dot(k, icell) # Fill in boundary values: V = np.zeros(tuple(size + 2) + v.shape[3:]) V[1:-1, 1:-1, 1:-1] = v V[0, 1:-1, 1:-1] = v[-1] V[-1, 1:-1, 1:-1] = v[0] V[:, 0, 1:-1] = V[:, -2, 1:-1] V[:, -1, 1:-1] = V[:, 1, 1:-1] V[:, :, 0] = V[:, :, -2] V[:, :, -1] = V[:, :, 1] interpolate = LinearNDInterpolator(k, V.reshape((-1,) + V.shape[3:])) return interpolate(path) # ChadiCohen k point grids. The k point grids are given in units of the # reciprocal unit cell. The variables are named after the following # convention: cc+''+_+'shape'. For example an 18 k point # sq(3)xsq(3) is named 'cc18_sq3xsq3'. cc6_1x1 = np.array([ 1, 1, 0, 1, 0, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 0, 1, 0]).reshape((6, 3)) / 3.0 cc12_2x3 = np.array([ 3, 4, 0, 3, 10, 0, 6, 8, 0, 3, -2, 0, 6, -4, 0, 6, 2, 0, -3, 8, 0, -3, 2, 0, -3, -4, 0, -6, 4, 0, -6, -2, 0, -6, -8, 0]).reshape((12, 3)) / 18.0 cc18_sq3xsq3 = np.array([ 2, 2, 0, 4, 4, 0, 8, 2, 0, 4, -2, 0, 8, -4, 0, 10, -2, 0, 10, -8, 0, 8, -10, 0, 2, -10, 0, 4, -8, 0, -2, -8, 0, 2, -4, 0, -4, -4, 0, -2, -2, 0, -4, 2, 0, -2, 4, 0, -8, 4, 0, -4, 8, 0]).reshape((18, 3)) / 18.0 cc18_1x1 = np.array([ 2, 4, 0, 2, 10, 0, 4, 8, 0, 8, 4, 0, 8, 10, 0, 10, 8, 0, 2, -2, 0, 4, -4, 0, 4, 2, 0, -2, 8, 0, -2, 2, 0, -2, -4, 0, -4, 4, 0, -4, -2, 0, -4, -8, 0, -8, 2, 0, -8, -4, 0, -10, -2, 0]).reshape((18, 3)) / 18.0 cc54_sq3xsq3 = np.array([ 4, -10, 0, 6, -10, 0, 0, -8, 0, 2, -8, 0, 6, -8, 0, 8, -8, 0, -4, -6, 0, -2, -6, 0, 2, -6, 0, 4, -6, 0, 8, -6, 0, 10, -6, 0, -6, -4, 0, -2, -4, 0, 0, -4, 0, 4, -4, 0, 6, -4, 0, 10, -4, 0, -6, -2, 0, -4, -2, 0, 0, -2, 0, 2, -2, 0, 6, -2, 0, 8, -2, 0, -8, 0, 0, -4, 0, 0, -2, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, -8, 2, 0, -6, 2, 0, -2, 2, 0, 0, 2, 0, 4, 2, 0, 6, 2, 0, -10, 4, 0, -6, 4, 0, -4, 4, 0, 0, 4, 0, 2, 4, 0, 6, 4, 0, -10, 6, 0, -8, 6, 0, -4, 6, 0, -2, 6, 0, 2, 6, 0, 4, 6, 0, -8, 8, 0, -6, 8, 0, -2, 8, 0, 0, 8, 0, -6, 10, 0, -4, 10, 0]).reshape((54, 3)) / 18.0 cc54_1x1 = np.array([ 2, 2, 0, 4, 4, 0, 8, 8, 0, 6, 8, 0, 4, 6, 0, 6, 10, 0, 4, 10, 0, 2, 6, 0, 2, 8, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, -2, 6, 0, -2, 4, 0, -4, 6, 0, -6, 4, 0, -4, 2, 0, -6, 2, 0, -2, 0, 0, -4, 0, 0, -8, 0, 0, -8, -2, 0, -6, -2, 0, -10, -4, 0, -10, -6, 0, -6, -4, 0, -8, -6, 0, -2, -2, 0, -4, -4, 0, -8, -8, 0, 4, -2, 0, 6, -2, 0, 6, -4, 0, 2, 0, 0, 4, 0, 0, 6, 2, 0, 6, 4, 0, 8, 6, 0, 8, 0, 0, 8, 2, 0, 10, 4, 0, 10, 6, 0, 2, -4, 0, 2, -6, 0, 4, -6, 0, 0, -2, 0, 0, -4, 0, -2, -6, 0, -4, -6, 0, -6, -8, 0, 0, -8, 0, -2, -8, 0, -4, -10, 0, -6, -10, 0]).reshape((54, 3)) / 18.0 cc162_sq3xsq3 = np.array([ -8, 16, 0, -10, 14, 0, -7, 14, 0, -4, 14, 0, -11, 13, 0, -8, 13, 0, -5, 13, 0, -2, 13, 0, -13, 11, 0, -10, 11, 0, -7, 11, 0, -4, 11, 0, -1, 11, 0, 2, 11, 0, -14, 10, 0, -11, 10, 0, -8, 10, 0, -5, 10, 0, -2, 10, 0, 1, 10, 0, 4, 10, 0, -16, 8, 0, -13, 8, 0, -10, 8, 0, -7, 8, 0, -4, 8, 0, -1, 8, 0, 2, 8, 0, 5, 8, 0, 8, 8, 0, -14, 7, 0, -11, 7, 0, -8, 7, 0, -5, 7, 0, -2, 7, 0, 1, 7, 0, 4, 7, 0, 7, 7, 0, 10, 7, 0, -13, 5, 0, -10, 5, 0, -7, 5, 0, -4, 5, 0, -1, 5, 0, 2, 5, 0, 5, 5, 0, 8, 5, 0, 11, 5, 0, -14, 4, 0, -11, 4, 0, -8, 4, 0, -5, 4, 0, -2, 4, 0, 1, 4, 0, 4, 4, 0, 7, 4, 0, 10, 4, 0, -13, 2, 0, -10, 2, 0, -7, 2, 0, -4, 2, 0, -1, 2, 0, 2, 2, 0, 5, 2, 0, 8, 2, 0, 11, 2, 0, -11, 1, 0, -8, 1, 0, -5, 1, 0, -2, 1, 0, 1, 1, 0, 4, 1, 0, 7, 1, 0, 10, 1, 0, 13, 1, 0, -10, -1, 0, -7, -1, 0, -4, -1, 0, -1, -1, 0, 2, -1, 0, 5, -1, 0, 8, -1, 0, 11, -1, 0, 14, -1, 0, -11, -2, 0, -8, -2, 0, -5, -2, 0, -2, -2, 0, 1, -2, 0, 4, -2, 0, 7, -2, 0, 10, -2, 0, 13, -2, 0, -10, -4, 0, -7, -4, 0, -4, -4, 0, -1, -4, 0, 2, -4, 0, 5, -4, 0, 8, -4, 0, 11, -4, 0, 14, -4, 0, -8, -5, 0, -5, -5, 0, -2, -5, 0, 1, -5, 0, 4, -5, 0, 7, -5, 0, 10, -5, 0, 13, -5, 0, 16, -5, 0, -7, -7, 0, -4, -7, 0, -1, -7, 0, 2, -7, 0, 5, -7, 0, 8, -7, 0, 11, -7, 0, 14, -7, 0, 17, -7, 0, -8, -8, 0, -5, -8, 0, -2, -8, 0, 1, -8, 0, 4, -8, 0, 7, -8, 0, 10, -8, 0, 13, -8, 0, 16, -8, 0, -7, -10, 0, -4, -10, 0, -1, -10, 0, 2, -10, 0, 5, -10, 0, 8, -10, 0, 11, -10, 0, 14, -10, 0, 17, -10, 0, -5, -11, 0, -2, -11, 0, 1, -11, 0, 4, -11, 0, 7, -11, 0, 10, -11, 0, 13, -11, 0, 16, -11, 0, -1, -13, 0, 2, -13, 0, 5, -13, 0, 8, -13, 0, 11, -13, 0, 14, -13, 0, 1, -14, 0, 4, -14, 0, 7, -14, 0, 10, -14, 0, 13, -14, 0, 5, -16, 0, 8, -16, 0, 11, -16, 0, 7, -17, 0, 10, -17, 0]).reshape((162, 3)) / 27.0 cc162_1x1 = np.array([ -8, -16, 0, -10, -14, 0, -7, -14, 0, -4, -14, 0, -11, -13, 0, -8, -13, 0, -5, -13, 0, -2, -13, 0, -13, -11, 0, -10, -11, 0, -7, -11, 0, -4, -11, 0, -1, -11, 0, 2, -11, 0, -14, -10, 0, -11, -10, 0, -8, -10, 0, -5, -10, 0, -2, -10, 0, 1, -10, 0, 4, -10, 0, -16, -8, 0, -13, -8, 0, -10, -8, 0, -7, -8, 0, -4, -8, 0, -1, -8, 0, 2, -8, 0, 5, -8, 0, 8, -8, 0, -14, -7, 0, -11, -7, 0, -8, -7, 0, -5, -7, 0, -2, -7, 0, 1, -7, 0, 4, -7, 0, 7, -7, 0, 10, -7, 0, -13, -5, 0, -10, -5, 0, -7, -5, 0, -4, -5, 0, -1, -5, 0, 2, -5, 0, 5, -5, 0, 8, -5, 0, 11, -5, 0, -14, -4, 0, -11, -4, 0, -8, -4, 0, -5, -4, 0, -2, -4, 0, 1, -4, 0, 4, -4, 0, 7, -4, 0, 10, -4, 0, -13, -2, 0, -10, -2, 0, -7, -2, 0, -4, -2, 0, -1, -2, 0, 2, -2, 0, 5, -2, 0, 8, -2, 0, 11, -2, 0, -11, -1, 0, -8, -1, 0, -5, -1, 0, -2, -1, 0, 1, -1, 0, 4, -1, 0, 7, -1, 0, 10, -1, 0, 13, -1, 0, -10, 1, 0, -7, 1, 0, -4, 1, 0, -1, 1, 0, 2, 1, 0, 5, 1, 0, 8, 1, 0, 11, 1, 0, 14, 1, 0, -11, 2, 0, -8, 2, 0, -5, 2, 0, -2, 2, 0, 1, 2, 0, 4, 2, 0, 7, 2, 0, 10, 2, 0, 13, 2, 0, -10, 4, 0, -7, 4, 0, -4, 4, 0, -1, 4, 0, 2, 4, 0, 5, 4, 0, 8, 4, 0, 11, 4, 0, 14, 4, 0, -8, 5, 0, -5, 5, 0, -2, 5, 0, 1, 5, 0, 4, 5, 0, 7, 5, 0, 10, 5, 0, 13, 5, 0, 16, 5, 0, -7, 7, 0, -4, 7, 0, -1, 7, 0, 2, 7, 0, 5, 7, 0, 8, 7, 0, 11, 7, 0, 14, 7, 0, 17, 7, 0, -8, 8, 0, -5, 8, 0, -2, 8, 0, 1, 8, 0, 4, 8, 0, 7, 8, 0, 10, 8, 0, 13, 8, 0, 16, 8, 0, -7, 10, 0, -4, 10, 0, -1, 10, 0, 2, 10, 0, 5, 10, 0, 8, 10, 0, 11, 10, 0, 14, 10, 0, 17, 10, 0, -5, 11, 0, -2, 11, 0, 1, 11, 0, 4, 11, 0, 7, 11, 0, 10, 11, 0, 13, 11, 0, 16, 11, 0, -1, 13, 0, 2, 13, 0, 5, 13, 0, 8, 13, 0, 11, 13, 0, 14, 13, 0, 1, 14, 0, 4, 14, 0, 7, 14, 0, 10, 14, 0, 13, 14, 0, 5, 16, 0, 8, 16, 0, 11, 16, 0, 7, 17, 0, 10, 17, 0]).reshape((162, 3)) / 27.0 # The following is a list of the critical points in the 1. Brillouin zone # for some typical crystal structures. # (In units of the reciprocal basis vectors) # See http://en.wikipedia.org/wiki/Brillouin_zone ibz_points = {'cubic': {'Gamma': [0, 0, 0], 'X': [0, 0 / 2, 1 / 2], 'R': [1 / 2, 1 / 2, 1 / 2], 'M': [0 / 2, 1 / 2, 1 / 2]}, 'fcc': {'Gamma': [0, 0, 0], 'X': [1 / 2, 0, 1 / 2], 'W': [1 / 2, 1 / 4, 3 / 4], 'K': [3 / 8, 3 / 8, 3 / 4], 'U': [5 / 8, 1 / 4, 5 / 8], 'L': [1 / 2, 1 / 2, 1 / 2]}, 'bcc': {'Gamma': [0, 0, 0], 'H': [1 / 2, -1 / 2, 1 / 2], 'N': [0, 0, 1 / 2], 'P': [1 / 4, 1 / 4, 1 / 4]}, 'hexagonal': {'Gamma': [0, 0, 0], 'M': [0, 1 / 2, 0], 'K': [-1 / 3, 1 / 3, 0], 'A': [0, 0, 1 / 2], 'L': [0, 1 / 2, 1 / 2], 'H': [-1 / 3, 1 / 3, 1 / 2]}, 'tetragonal': {'Gamma': [0, 0, 0], 'X': [1 / 2, 0, 0], 'M': [1 / 2, 1 / 2, 0], 'Z': [0, 0, 1 / 2], 'R': [1 / 2, 0, 1 / 2], 'A': [1 / 2, 1 / 2, 1 / 2]}, 'orthorhombic': {'Gamma': [0, 0, 0], 'R': [1 / 2, 1 / 2, 1 / 2], 'S': [1 / 2, 1 / 2, 0], 'T': [0, 1 / 2, 1 / 2], 'U': [1 / 2, 0, 1 / 2], 'X': [1 / 2, 0, 0], 'Y': [0, 1 / 2, 0], 'Z': [0, 0, 1 / 2]}}