1# Copyright (C) 2010, Jesper Friis 

2# (see accompanying license files for details). 

3 

4# XXX bravais objects need to hold tolerance eps, *or* temember variant 

5# from the beginning. 

6# 

7# Should they hold a 'cycle' argument or other data to reconstruct a particular 

8# cell? (E.g. rotation, niggli transform) 

9# 

10# Implement total ordering of Bravais classes 1-14 

11 

12import numpy as np 

13from numpy import pi, sin, cos, arccos, sqrt, dot 

14from numpy.linalg import norm 

15 

16 

17def unit_vector(x): 

18 """Return a unit vector in the same direction as x.""" 

19 y = np.array(x, dtype='float') 

20 return y / norm(y) 

21 

22 

23def angle(x, y): 

24 """Return the angle between vectors a and b in degrees.""" 

25 return arccos(dot(x, y) / (norm(x) * norm(y))) * 180. / pi 

26 

27 

28def cell_to_cellpar(cell, radians=False): 

29 """Returns the cell parameters [a, b, c, alpha, beta, gamma]. 

30 

31 Angles are in degrees unless radian=True is used. 

32 """ 

33 lengths = [np.linalg.norm(v) for v in cell] 

34 angles = [] 

35 for i in range(3): 

36 j = i - 1 

37 k = i - 2 

38 ll = lengths[j] * lengths[k] 

39 if ll > 1e-16: 

40 x = np.dot(cell[j], cell[k]) / ll 

41 angle = 180.0 / pi * arccos(x) 

42 else: 

43 angle = 90.0 

44 angles.append(angle) 

45 if radians: 

46 angles = [angle * pi / 180 for angle in angles] 

47 return np.array(lengths + angles) 

48 

49 

50def cellpar_to_cell(cellpar, ab_normal=(0, 0, 1), a_direction=None): 

51 """Return a 3x3 cell matrix from cellpar=[a,b,c,alpha,beta,gamma]. 

52 

53 Angles must be in degrees. 

54 

55 The returned cell is orientated such that a and b 

56 are normal to `ab_normal` and a is parallel to the projection of 

57 `a_direction` in the a-b plane. 

58 

59 Default `a_direction` is (1,0,0), unless this is parallel to 

60 `ab_normal`, in which case default `a_direction` is (0,0,1). 

61 

62 The returned cell has the vectors va, vb and vc along the rows. The 

63 cell will be oriented such that va and vb are normal to `ab_normal` 

64 and va will be along the projection of `a_direction` onto the a-b 

65 plane. 

66 

67 Example: 

68 

69 >>> cell = cellpar_to_cell([1, 2, 4, 10, 20, 30], (0, 1, 1), (1, 2, 3)) 

70 >>> np.round(cell, 3) 

71 array([[ 0.816, -0.408, 0.408], 

72 [ 1.992, -0.13 , 0.13 ], 

73 [ 3.859, -0.745, 0.745]]) 

74 

75 """ 

76 if a_direction is None: 

77 if np.linalg.norm(np.cross(ab_normal, (1, 0, 0))) < 1e-5: 

78 a_direction = (0, 0, 1) 

79 else: 

80 a_direction = (1, 0, 0) 

81 

82 # Define rotated X,Y,Z-system, with Z along ab_normal and X along 

83 # the projection of a_direction onto the normal plane of Z. 

84 ad = np.array(a_direction) 

85 Z = unit_vector(ab_normal) 

86 X = unit_vector(ad - dot(ad, Z) * Z) 

87 Y = np.cross(Z, X) 

88 

89 # Express va, vb and vc in the X,Y,Z-system 

90 alpha, beta, gamma = 90., 90., 90. 

91 if isinstance(cellpar, (int, float)): 

92 a = b = c = cellpar 

93 elif len(cellpar) == 1: 

94 a = b = c = cellpar[0] 

95 elif len(cellpar) == 3: 

96 a, b, c = cellpar 

97 else: 

98 a, b, c, alpha, beta, gamma = cellpar 

99 

100 # Handle orthorhombic cells separately to avoid rounding errors 

101 eps = 2 * np.spacing(90.0, dtype=np.float64) # around 1.4e-14 

102 # alpha 

103 if abs(abs(alpha) - 90) < eps: 

104 cos_alpha = 0.0 

105 else: 

106 cos_alpha = cos(alpha * pi / 180.0) 

107 # beta 

108 if abs(abs(beta) - 90) < eps: 

109 cos_beta = 0.0 

110 else: 

111 cos_beta = cos(beta * pi / 180.0) 

112 # gamma 

113 if abs(gamma - 90) < eps: 

114 cos_gamma = 0.0 

115 sin_gamma = 1.0 

116 elif abs(gamma + 90) < eps: 

117 cos_gamma = 0.0 

118 sin_gamma = -1.0 

119 else: 

120 cos_gamma = cos(gamma * pi / 180.0) 

121 sin_gamma = sin(gamma * pi / 180.0) 

122 

123 # Build the cell vectors 

124 va = a * np.array([1, 0, 0]) 

125 vb = b * np.array([cos_gamma, sin_gamma, 0]) 

126 cx = cos_beta 

127 cy = (cos_alpha - cos_beta * cos_gamma) / sin_gamma 

128 cz_sqr = 1. - cx * cx - cy * cy 

129 assert cz_sqr >= 0 

130 cz = sqrt(cz_sqr) 

131 vc = c * np.array([cx, cy, cz]) 

132 

133 # Convert to the Cartesian x,y,z-system 

134 abc = np.vstack((va, vb, vc)) 

135 T = np.vstack((X, Y, Z)) 

136 cell = dot(abc, T) 

137 

138 return cell 

139 

140 

141def metric_from_cell(cell): 

142 """Calculates the metric matrix from cell, which is given in the 

143 Cartesian system.""" 

144 cell = np.asarray(cell, dtype=float) 

145 return np.dot(cell, cell.T) 

146 

147 

148def complete_cell(cell): 

149 """Calculate complete cell with missing lattice vectors. 

150 

151 Returns a new 3x3 ndarray. 

152 """ 

153 

154 cell = np.array(cell, dtype=float) 

155 missing = np.nonzero(~cell.any(axis=1))[0] 

156 

157 if len(missing) == 3: 

158 cell.flat[::4] = 1.0 

159 if len(missing) == 2: 

160 # Must decide two vectors: 

161 V, s, WT = np.linalg.svd(cell.T) 

162 sf = [s[0], 1, 1] 

163 cell = (V @ np.diag(sf) @ WT).T 

164 if np.sign(np.linalg.det(cell)) < 0: 

165 cell[missing[0]] = -cell[missing[0]] 

166 elif len(missing) == 1: 

167 i = missing[0] 

168 cell[i] = np.cross(cell[i - 2], cell[i - 1]) 

169 cell[i] /= np.linalg.norm(cell[i]) 

170 

171 return cell 

172 

173 

174def is_orthorhombic(cell): 

175 """Check that cell only has stuff in the diagonal.""" 

176 return not (np.flatnonzero(cell) % 4).any() 

177 

178 

179def orthorhombic(cell): 

180 """Return cell as three box dimensions or raise ValueError.""" 

181 if not is_orthorhombic(cell): 

182 raise ValueError('Not orthorhombic') 

183 return cell.diagonal().copy() 

184 

185 

186# We make the Cell object available for import from here for compatibility 

187from ase.cell import Cell # noqa