Space Groups¶

The symbol for a space group may be entered as the value of “space_group” in the BASIS section of the parameter file. The tables below list the allowed space group symbols.

1D Space Groups¶

The only possible nontrivial symmetry for a one-dimensional lamellar phase is inversion symmetry. There are thus only two possible groups: The centrosymmetric group (group -1) and the non-centrosymmetric group (group 1). Fields for a centrosymmetric lamellar phase are expanded using a basis of cosine waves, while fields for a non-centrosymmetric phase are expanded using a basis that contains both cosines and sine waves.

Number	Symbol	Comments
1	-1	Inversion symmetry
2	1	No symmetry

2D Space Groups¶

The names of all 17 possible 2D plane groups are given below in the text format expected by PSCF. The format used in PSCF for both 2D and 3D space group names is based on the names used in the international tables of crystallography, but allows space group names to be written as simple ascii text strings, which contain spaces between elements of the space group name.

Number Symbol Lattice System

1 p 1 oblique

2 p 2 oblique

3 p m rectangular

4 p g rectangular

5 c m rectangular

6 p 2 m m rectangular

7 p 2 m g rectangular

8 p 2 g g rectangular

9 c 2 m m rectangular

10 p 4 square

11 p 4 m m square

12 p 4 g m square

13 p 3 hexagonal

14 p 3 m 1 hexagonal

15 p 3 1 m hexagonal

16 p 6 hexagonal

17 p 6 m m hexagonal

3D Space Groups¶

The names of all possible 3D space groups are given below in the text format expected by PSCF. These names are based on the names given in Hermann-Mauguin or “international” notation used the international tables of crystallography, but are given in a format that allows space group names to be written as simple ascii text strings, with no special symbols or subscripts. In this format, for example, the space group \(Ia\overline{3}d\) of the gyroid phase (space group 230) is written as “I a -3 d”.

Name conventions

The following conventions are used to convert standard Hermann-Mauguin space group symbols into text strings:

A single space is introduced between different elements of the space group name, with a few exceptions described below.

Integers with overbars in the Hermann-Mauguin symbol, which indicate inversion (\(\overline{1}\)) or a 3-, 4- or 6-fold rotoinversion axis (\(\overline{3}\), \(\overline{4}\), or \(\overline{6}\)), are indicated in the PSCF text string by placing a “-” sign before the integer. Thus for, example, \(\overline{3}\) is replaced by “-3” in the ascii identifier “I a -3 d”.

Integers with subscripts, such as \(4_2\), which represent screw axes, are indicated in the text representation by placing the two integers directly adjacent, with no intervening white space. Thus, for example, \(4_2\) is replaced by “42”.

Symbols that are separated by a slash appear with no space on either side of the slash.

Different “settings” of the same space group, which correspond to different definitions of the origin of space in the definition of the symmetry elements, are indicated by a colon followed by an integer label at the end of the space group.

Space Groups

Number Symbol

1 P 1

2 P -1

3 P 1 2 1

4 P 1 21 1

5 C 1 2 1

6 P 1 m 1

7 P 1 c 1

8 C 1 m 1

9 C 1 c 1

10 P 1 2/m 1

11 P 1 21/m 1

12 C 1 2/m 1

13 P 1 2/c 1

14 P 1 21/c 1

15 C 1 2/c 1

16 P 2 2 2

17 P 2 2 21

18 P 21 21 2

19 P 21 21 21

20 C 2 2 21

21 C 2 2 2

22 F 2 2 2

23 I 2 2 2

24 I 21 21 21

25 P m m 2

26 P m c 21

27 P c c 2

28 P m a 2

29 P c a 21

30 P n c 2

31 P m n 21

32 P b a 2

33 P n a 21

34 P n n 2

35 C m m 2

36 C m c 21

37 C c c 2

38 A m m 2

39 A b m 2

40 A m a 2

41 A b a 2

42 F m m 2

43 F d d 2

44 I m m 2

45 I b a 2

46 I m a 2

47 P m m m

48 P n n n : 2

48 P n n n : 1

49 P c c m

50 P b a n : 2

50 P b a n : 1

51 P m m a

52 P n n a

53 P m n a

54 P c c a

55 P b a m

56 P c c n

57 P b c m

58 P n n m

59 P m m n : 2

59 P m m n : 1

60 P b c n

61 P b c a

62 P n m a

63 C m c m

64 C m c a

65 C m m m

66 C c c m

67 C m m a

68 C c c a : 2

68 C c c a : 1

69 F m m m

70 F d d d : 2

70 F d d d : 1

71 I m m m

72 I b a m

73 I b c a

74 I m m a

75 P 4

76 P 41

77 P 42

78 P 43

79 I 4

80 I 41

81 P -4

82 I -4

83 P 4/m

84 P 42/m

85 P 4/n : 2

85 P 4/n : 1

86 P 42/n : 2

86 P 42/n : 1

87 I 4/m

88 I 41/a : 2

88 I 41/a : 1

89 P 4 2 2

90 P 4 21 2

91 P 41 2 2

92 P 41 21 2

93 P 42 2 2

94 P 42 21 2

95 P 43 2 2

96 P 43 21 2

97 I 4 2 2

98 I 41 2 2

99 P 4 m m

100 P 4 b m

101 P 42 c m

102 P 42 n m

103 P 4 c c

104 P 4 n c

105 P 42 m c

106 P 42 b c

107 I 4 m m

108 I 4 c m

109 I 41 m d

110 I 41 c d

111 P -4 2 m

112 P -4 2 c

113 P -4 21 m

114 P -4 21 c

115 P -4 m 2

116 P -4 c 2

117 P -4 b 2

118 P -4 n 2

119 I -4 m 2

120 I -4 c 2

121 I -4 2 m

122 I -4 2 d

123 P 4/m m m

124 P 4/m c c

125 P 4/n b m : 2

125 P 4/n b m : 1

126 P 4/n n c : 2

126 P 4/n n c : 1

127 P 4/m b m

128 P 4/m n c

129 P 4/n m m : 2

129 P 4/n m m : 1

130 P 4/n c c : 2

130 P 4/n c c : 1

131 P 42/m m c

132 P 42/m c m

133 P 42/n b c : 2

133 P 42/n b c : 1

134 P 42/n n m : 2

134 P 42/n n m : 1

135 P 42/m b c

136 P 42/m n m

137 P 42/n m c : 2

137 P 42/n m c : 1

138 P 42/n c m : 2

138 P 42/n c m : 1

139 I 4/m m m

140 I 4/m c m

141 I 41/a m d : 2

141 I 41/a m d : 1

142 I 41/a c d : 2

142 I 41/a c d : 1

143 P 3

144 P 31

145 P 32

146 R 3 : H

146 R 3 : R

147 P -3

148 R -3 : H

148 R -3 : R

149 P 3 1 2

150 P 3 2 1

151 P 31 1 2

152 P 31 2 1

153 P 32 1 2

154 P 32 2 1

155 R 3 2 : H

155 R 3 2 : R

156 P 3 m 1

157 P 3 1 m

158 P 3 c 1

159 P 3 1 c

160 R 3 m : H

160 R 3 m : R

161 R 3 c : H

161 R 3 c : R

162 P -3 1 m

163 P -3 1 c

164 P -3 m 1

165 P -3 c 1

166 R -3 m : H

166 R -3 m : R

167 R -3 c : H

167 R -3 c : R

168 P 6

169 P 61

170 P 65

171 P 62

172 P 64

173 P 63

174 P -6

175 P 6/m

176 P 63/m

177 P 6 2 2

178 P 61 2 2

179 P 65 2 2

180 P 62 2 2

181 P 64 2 2

182 P 63 2 2

183 P 6 m m

184 P 6 c c

185 P 63 c m

186 P 63 m c

187 P -6 m 2

188 P -6 c 2

189 P -6 2 m

190 P -6 2 c

191 P 6/m m m

192 P 6/m c c

193 P 63/m c m

194 P 63/m m c

195 P 2 3

196 F 2 3

197 I 2 3

198 P 21 3

199 I 21 3

200 P m -3

201 P n -3 : 2

201 P n -3 : 1

202 F m -3

203 F d -3 : 2

203 F d -3 : 1

204 I m -3

205 P a -3

206 I a -3

207 P 4 3 2

208 P 42 3 2

209 F 4 3 2

210 F 41 3 2

211 I 4 3 2

212 P 43 3 2

213 P 41 3 2

214 I 41 3 2

215 P -4 3 m

216 F -4 3 m

217 I -4 3 m

218 P -4 3 n

219 F -4 3 c

220 I -4 3 d

221 P m -3 m

222 P n -3 n : 2

222 P n -3 n : 1

223 P m -3 n

224 P n -3 m : 2

224 P n -3 m : 1

225 F m -3 m

226 F m -3 c

227 F d -3 m : 2

227 F d -3 m : 1

228 F d -3 c : 2

228 F d -3 c : 1

229 I m -3 m

230 I a -3 d

Symmetry Elements¶

A list of all of the symmetry elements of any space group can be output to file by placing a “OUTPUT_GROUP” command in the parameter file at any point after the “BASIS” section.

Every space group symmetry can be expressed mathematically as an operation

\[\textbf{r} \rightarrow \textbf{A}\textbf{r} + \textbf{t}\]

Here, \(\textbf{r} = [r_{1}, \ldots, r_{D}]^{T}\) is a dimensionless D-element column vector containing the components of a position within the unit cell in a basis of Bravais lattice vectors, \(\textbf{A}\) is a \(D \times D\) matrix that represents a point group symmetry operation (e.g., identity, inversion, rotation about an axis, or reflection through a plane), and \(\textbf{t}\) is a dimenionless D-element colummn vector that (if not zero) represents a translation by a fraction of a unit cell. Every group contains an identity element in which \(\textbf{A}\) is the identity matrix and \(\textbf{t}=0\).

The elements of the column vectors \(\textbf{r}\) and \(\textbf{t}\) in the above are dimensionless components defined using a basis of Bravais basis vectors. The position \(\textbf{r} = [1/2, 1/2, 1/2]^{T}\) thus always represents the center of a 3D unit cell. The Cartesian representation of a position vector is instead given by a sum

\[\sum_{i=1}^{D} r_{i}\textbf{a}_{i}\]

in which \(\textbf{a}_{i}\) is the Cartesian representation of Bravais lattice vector number i. The elements of the dimensionless translation vector \(\textbf{t}\) are always either zero or simple fractions such as 1/2, 1/4, or 1/3. For example, a symmetry element in a 3D BCC lattice in which \(\textbf{A}\) is the identity matrix and \(\textbf{t} = [1/2, 1/2, 1/2]^{T}\) represents the purely translational symmetry that relates the two equivalent positions per cubic unit cell in a BCC lattice. Similarly, a glide plane in a 3D crystal is represented by a diagonal \(\textbf{A}\) matrix with values of \(\pm 1\) on the diagonal that represents inversion through a plane and a translation vector that represents a translation by half a unit cell within that plane.

The OUTPUT_GROUP command outputs a list of symmetry elements in which each element is displayed by showing the elements of the matrix \(\textbf{A}\) followed by elements of the associated column vector \(\textbf{t}\).

The Bravais lattice vectors used internally by PSCF for cubic, tetragonal, and orthorhombic 3D systems are orthogonal basis vectors for the simple cubic, tetragonal, or orthorhombic unit cells, which are aligned along the x, y, and z axes of a Cartesian coordinate system. Similarly, the basis vectors used for the 2D square and rectangular space groups are orthogonal vectors which form a basis for a cubic or rectangular unit cell. The grid used to solve the modified diffusion equation is based on the same choice of unit cell and, thus for example, uses a regular grid within a cubic unit cell to represent fields in a BCC or FCC lattice. For body-centered and space-centered lattice systems, it is worth nothing that this unit cell not a primitive (minimum size) unit cell of the crystal: For example, a cubic unit cell actually contains 2 equivalent primitive unit cells of a BCC lattice or 4 primitive cells of an FCC lattice.

One consequence of the fact that PSCF does not always use a primitive unit cell is that, in the Fourier expansion of the omega and rho fields, the Fourier coefficients associated with some sets of symmetry-related wavevectors (some “stars”) are required to vanish in order to satisfy the requirement that the field be invariant under all elements of the specified space group. The rules regarding which stars must have vanishing Fourier coefficients are the same as the rules for systematic cancellations of Bragg reflections in X-ray or neutron scattering from a crystal of the specified space group. The procedure used by PSCF to construct symmetry adapted basis functions automatially identifies and accounts for these systematic cancellations.