Space Groups

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The PSCF programs that are designed to simulate periodic structures use algorithms that constrain the chemical potential fields to be symmetric under all operations of a specified crystallographic space group. Each possible space group is identified in PSCF by a unique identifer string.

Space Group Identifier Strings

The space group identifiers used in PSCF are ASCII strings with no white space that are constructed as mangled forms of the Hermann Mauguin space group names that are used in the international tables of crystallography. Each PSCF space group identifier is also the name of a file that contain a description of the space group. The international symbols cannot be used for this purpose because they contain typset elements such as subscripts and overbars that cannot be represented using an ascii character set. The following rules are used to convert Hermann Mauguin names into PSCF ascii space group identifiers:

• Logical elements of the Hermann Mauguin space group name are separated by underbars (_). For example, I_m_m_a is the PSCF identifier for space group \(Imma\).
• Each overbar symbol in the Hermann Mauguin name is replaced by a "-" that appears as prefix to the overbarred symbol. For example, I_a_-3_d is the PSCF identifier for space group \(Ia\overline{3}d\), which is the space group of the double gyroid phase of a diblock copolymer.
• Elements in which a single digit number appears as a subcript to another such number are listed as pairs of numbers with no intervening space, in which the second number in the pair is the subcript. For example, P_43_2_2 is the PSCF identifier for space group \(P4_{3}22\).
• Each slash ("/") in a Hermann Mauguin name is replaced by a percent sign ("%") symbol. For example P_4%m_m_m is the PSCF identifier for space group \(P4/mmm\).
• Different possible settings of a space group (e.g., due to different choices of origin) are indicated by a a colon (":") followed by a single digit or letter at the end of the space group name. For example F_d_-3_m:2 is the PSCF identifier for setting 2 of space group \(Fd\overline{3}m\), which has two settings labelled "1" and "2". Some space groups for crystals in the trigonal crystal system can be represented using either a hexagonal or rhombohedral Bravais lattice, which are identified with names that end in suffixes ":H" or ":R", respectively. Such setting labels are only used for space groups for which two or more setting are listed in the international tables.

Files that contain descriptions of all possible crystallographic space groups are stored in the data/groups directory of the main pscfpp directory. Descriptions of space groups for 1D, 2D and 3D crystals are stored in subdirectories named 1/, 2/ and 3/, respectively. The name of each such file is given by the PSCF identifier for the associated space group. One can thus scan through the names of files in these directories to find identifiers for all standard space groups.

Each file that describes a space group contains a list of descriptions for all the symmetry operations of the group. Each symmetry operation is represented by a matrix and a vector. A matrix with integer elements is used to represent a linear transformation of coordinates caused by a point group operation (e.g., a reflection, rotation, or inversion) in which coordinates are defined using a basis of Bravais lattice basis vectors. A vector with elements given by rational numbers (fractions) is used to represent a translation, in which each element represents a translation parallel to a Bravais lattice basis vector by some fraction of a unit cell.

Identifier strings for all of the standard 1D, 2D and 3D space groups are listed below:

1D Space Groups

There are only two possible symmetry groups for one-dimensionally periodic structures: Space group P_-1 has an inversion center at the origin, giving centrosymmetric lamellar structures, while space group P_1 does not, allowing non-centrosymmetric structures.

Group identifiers:

Number	Symbol	Comments
1	P_-1	Inversion symmetry
2	P_1	No symmetry

2D Space Groups

There are 17 possible 2D symmetry groups, also known as plane groups.

Group identifiers:

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

There are 230 possible 3D space groups. The international crystallographic tables lists 2 "settings" for some space groups, which differ in the convention for the placement of the origin. PSCF identifiers for space groups for which two or more settings are listed end in a colon followed by an integer id (1 or 2) for the setting. Trigonal groups that may be represented using either a hexagonal or rhombohedral Bravais lattice sometimes have two settings distinguished by subscripts "H" or "R".

Group identifiers:

Number	Symbol	Crystal System
1	P_1	Triclinic
2	P_-1	Triclinic
3	P_1_2_1	Monoclinic
4	P_1_21_1	Monoclinic
5	C_1_2_1	Monoclinic
6	P_1_m_1	Monoclinic
7	P_1_c_1	Monoclinic
8	C_1_m_1	Monoclinic
9	C_1_c_1	Monoclinic
10	P_1_2%m_1	Monoclinic
11	P_1_21%m_1	Monoclinic
12	C_1_2%m_1	Monoclinic
13	P_1_2%c_1	Monoclinic
14	P_1_21%c_1	Monoclinic
15	C_1_2%c_1	Monoclinic
16	P_2_2_2	Orthorhombic
17	P_2_2_21	Orthorhombic
18	P_21_21_2	Orthorhombic
19	P_21_21_21	Orthorhombic
20	C_2_2_21	Orthorhombic
21	C_2_2_2	Orthorhombic
22	F_2_2_2	Orthorhombic
23	I_2_2_2	Orthorhombic
24	I_21_21_21	Orthorhombic
25	P_m_m_2	Orthorhombic
26	P_m_c_21	Orthorhombic
27	P_c_c_2	Orthorhombic
28	P_m_a_2	Orthorhombic
29	P_c_a_21	Orthorhombic
30	P_n_c_2	Orthorhombic
31	P_m_n_21	Orthorhombic
32	P_b_a_2	Orthorhombic
33	P_n_a_21	Orthorhombic
34	P_n_n_2	Orthorhombic
35	C_m_m_2	Orthorhombic
36	C_m_c_21	Orthorhombic
37	C_c_c_2	Orthorhombic
38	A_m_m_2	Orthorhombic
39	A b_m_2	Orthorhombic
40	A_m_a_2	Orthorhombic
41	A b_a_2	Orthorhombic
42	F_m_m_2	Orthorhombic
43	F_d_d_2	Orthorhombic
44	I_m_m_2	Orthorhombic
45	I_b_a_2	Orthorhombic
46	I_m_a_2	Orthorhombic
47	P_m_m_m	Orthorhombic
48	P_n_n_n:2	Orthorhombic
48	P_n_n_n:1	Orthorhombic
49	P_c_c_m	Orthorhombic
50	P_b_a_n:2	Orthorhombic
50	P_b_a_n:1	Orthorhombic
51	P_m_m_a	Orthorhombic
52	P_n_n_a	Orthorhombic
53	P_m_n_a	Orthorhombic
54	P_c_c_a	Orthorhombic
55	P_b_a_m	Orthorhombic
56	P_c_c_n	Orthorhombic
57	P_b_c_m	Orthorhombic
58	P_n_n_m	Orthorhombic
59	P_m_m_n:2	Orthorhombic
59	P_m_m_n:1	Orthorhombic
60	P_b_c_n	Orthorhombic
61	P_b_c_a	Orthorhombic
62	P_n_m_a	Orthorhombic
63	C_m_c_m	Orthorhombic
64	C_m_c_a	Orthorhombic
65	C_m_m_m	Orthorhombic
66	C_c_c_m	Orthorhombic
67	C_m_m_a	Orthorhombic
68	C_c_c_a:2	Orthorhombic
68	C_c_c_a:1	Orthorhombic
69	F_m_m_m	Orthorhombic
70	F_d_d_d_:2	Orthorhombic
70	F_d_d_d_:1	Orthorhombic
71	I_m_m_m	Orthorhombic
72	I_b_a_m	Orthorhombic
73	I_b_c_a	Orthorhombic
74	I_m_m_a	Orthorhombic
75	P_4	Tetragonal
76	P_41	Tetragonal
77	P_42	Tetragonal
78	P_43	Tetragonal
79	I_4	Tetragonal
80	I_41	Tetragonal
81	P_-4	Tetragonal
82	I_-4	Tetragonal
83	P_4%m	Tetragonal
84	P_42%m	Tetragonal
85	P_4%n:2	Tetragonal
85	P_4%n:1	Tetragonal
86	P_42%n:2	Tetragonal
86	P_42%n:1	Tetragonal
87	I_4%m	Tetragonal
88	I_41%a:2	Tetragonal
88	I_41%a:1	Tetragonal
89	P_4_2_2	Tetragonal
90	P_4_21_2	Tetragonal
91	P_41_2_2	Tetragonal
92	P_41_21_2	Tetragonal
93	P_42_2_2	Tetragonal
94	P_42_21_2	Tetragonal
95	P_43_2_2	Tetragonal
96	P_43_21_2	Tetragonal
97	I_4_2_2	Tetragonal
98	I_41_2_2	Tetragonal
99	P_4_m_m	Tetragonal
100	P_4_b_m	Tetragonal
101	P_42_c_m	Tetragonal
102	P_42_n_m	Tetragonal
103	P_4_c_c	Tetragonal
104	P_4_n_c	Tetragonal
105	P_42_m_c	Tetragonal
106	P_42_b_c	Tetragonal
107	I_4_m_m	Tetragonal
108	I_4_c_m	Tetragonal
109	I_41_m_d	Tetragonal
110	I_41_c_d	Tetragonal
111	P_-4_2_m	Tetragonal
112	P_-4_2_c	Tetragonal
113	P_-4_21_m	Tetragonal
114	P_-4_21_c	Tetragonal
115	P_-4_m_2	Tetragonal
116	P_-4_c_2	Tetragonal
117	P_-4 b 2	Tetragonal
118	P_-4_n_2	Tetragonal
119	I_-4_m_2	Tetragonal
120	I_-4_c_2	Tetragonal
121	I_-4_2_m	Tetragonal
122	I_-4_2_d	Tetragonal
123	P_4%m_m_m	Tetragonal
124	P_4%m_c_c	Tetragonal
125	P_4%n_b_m:2	Tetragonal
125	P_4%n_b_m:1	Tetragonal
126	P_4%n_n_c:2	Tetragonal
126	P_4%n_n_c:1	Tetragonal
127	P_4%m_b_m	Tetragonal
128	P_4%m_n_c	Tetragonal
129	P_4%n_m_m:2	Tetragonal
129	P_4%n_m_m:1	Tetragonal
130	P_4%n_c_c:2	Tetragonal
130	P_4%n_c_c:1	Tetragonal
131	P_42%m_m_c	Tetragonal
132	P_42%m_c_m	Tetragonal
133	P_42%n_b_c:2	Tetragonal
133	P_42%n_b_c:1	Tetragonal
134	P_42%n_n_m:2	Tetragonal
134	P_42%n_n_m:1	Tetragonal
135	P_42%m_b_c	Tetragonal
136	P_42%m_n_m	Tetragonal
137	P_42%n_m_c:2	Tetragonal
137	P_42%n_m_c:1	Tetragonal
138	P_42%n_c_m:2	Tetragonal
138	P_42%n_c_m:1	Tetragonal
139	I_4%m_m_m	Tetragonal
140	I_4%m_c_m	Tetragonal
141	I_41%a_m_d:2	Tetragonal
141	I_41%a_m_d:1	Tetragonal
142	I_41%a_c_d:2	Tetragonal
142	I_41%a_c_d:1	Tetragonal
143	P_3	Trigonal
144	P_31	Trigonal
145	P_32	Trigonal
146	R_3:H	Trigonal
146	R_3:R	Trigonal
147	P_-3	Trigonal
148	R_-3:H	Trigonal
148	R_-3:R	Trigonal
149	P_3_1_2	Trigonal
150	P_3_2_1	Trigonal
151	P_31_1_2	Trigonal
152	P_31_2_1	Trigonal
153	P_32_1_2	Trigonal
154	P_32_2_1	Trigonal
155	R_3_2:H	Trigonal
155	R_3_2:R	Trigonal
156	P_3_m_1	Trigonal
157	P_3_1_m	Trigonal
158	P_3_c_1	Trigonal
159	P_3_1_c	Trigonal
160	R_3_m:H	Trigonal
160	R_3_m:R	Trigonal
161	R_3_c:H	Trigonal
161	R_3_c:R	Trigonal
162	P_-3_1_m	Trigonal
163	P_-3_1_c	Trigonal
164	P_-3_m_1	Trigonal
165	P_-3_c_1	Trigonal
166	R_-3_m:H	Trigonal
166	R_-3_m:R	Trigonal
167	R_-3_c:H	Trigonal
167	R_-3_c:R	Trigonal
168	P_6	Hexagonal
169	P_61	Hexagonal
170	P_65	Hexagonal
171	P_62	Hexagonal
172	P_64	Hexagonal
173	P_63	Hexagonal
174	P_-6	Hexagonal
175	P_6%m	Hexagonal
176	P_63%m	Hexagonal
177	P_6_2_2	Hexagonal
178	P_61_2_2	Hexagonal
179	P_65_2_2	Hexagonal
180	P_62_2_2	Hexagonal
181	P_64_2_2	Hexagonal
182	P_63_2_2	Hexagonal
183	P_6_m_m	Hexagonal
184	P_6_c_c	Hexagonal
185	P_63_c_m	Hexagonal
186	P_63_m_c	Hexagonal
187	P_-6_m_2	Hexagonal
188	P_-6_c_2	Hexagonal
189	P_-6 2_m	Hexagonal
190	P_-6 2_c	Hexagonal
191	P_6%m_m_m	Hexagonal
192	P_6%m_c_c	Hexagonal
193	P_63%m_c_m	Hexagonal
194	P_63%m_m_c	Hexagonal
195	P_2_3	Cubic
196	F_2_3	Cubic
197	I_2_3	Cubic
198	P_21 3	Cubic
199	I_21 3	Cubic
200	P_m_-3	Cubic
201	P_n_-3:2	Cubic
201	P_n_-3:1	Cubic
202	F_m_-3	Cubic
203	F_d_-3:2	Cubic
203	F_d_-3:1	Cubic
204	I_m_-3	Cubic
205	P_a_-3	Cubic
206	I_a_-3	Cubic
207	P_4_3_2	Cubic
208	P_42_3_2	Cubic
209	F_4_3_2	Cubic
210	F_41_3_2	Cubic
211	I_4_3_2	Cubic
212	P_43_3_2	Cubic
213	P_41_3_2	Cubic
214	I_41_3_2	Cubic
215	P_-4_3_m	Cubic
216	F_-4_3_m	Cubic
217	I_-4_3_m	Cubic
218	P_-4_3_n	Cubic
219	F_-4_3_c	Cubic
220	I_-4_3_d	Cubic
221	P_m_-3_m	Cubic
222	P_n_-3_n:2	Cubic
222	P_n_-3_n:1	Cubic
223	P_m_-3_n	Cubic
224	P_n_-3_m:2	Cubic
224	P_n_-3_m:1	Cubic
225	F_m_-3_m	Cubic
226	F_m_-3_c	Cubic
227	F_d_-3_m:2	Cubic
227	F_d_-3_m:1	Cubic
228	F_d_-3_c:2	Cubic
228	F_d_-3_c:1	Cubic
229	I_m_-3_m	Cubic
230	I a_-3_d	Cubic

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