Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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R3cC
6
3v
3m Trigonal
No. 161 R3c
Patterson symmetry R
¯
3m
RHOMBOHEDRAL AXES
Origin on 3c
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤ 1; 0 ≤ z ≤ 1; y ≤ x; z ≤ y
Vertices 0,0, 01,0, 01,1,01,1, 1
Symmetry operations
(1) 1 (2) 3
+
x,x, x (3) 3
−
x,x, x
(4) n(
1
2
,
1
2
,
1
2
) x,y,x (5) n(
1
2
,
1
2
,
1
2
) x,x,z (6) n(
1
2
,
1
2
,
1
2
) x,y, y
534
CONTINUED No. 161 R3c
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (4)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
6 b 1(1)x, y,z (2) z,x,y (3) y,z,x
(4) z +
1
2
,y +
1
2
,x +
1
2
(5) y +
1
2
,x +
1
2
,z +
1
2
(6) x +
1
2
,z +
1
2
,y +
1
2
hhl : l = 2n
hhh : h = 2n
Special: as above, plus
2 a 3 . x,x,xx+
1
2
,x +
1
2
,x +
1
2
hkl : h + k + l = 2n
Symmetry of special projections
Along [111] p31m
a
=
1
3
(2a − b− c) b
=
1
3
(−a + 2b − c)
Origin at x,x, x
Along [1
¯
10] p1
a
=
1
2
(a + b − 2c ) b
=
1
2
c
Origin at x, ¯x,0
Along [2
¯
1
¯
1] p1g1
a
=
1
2
(b− c) b
=
1
3
(a + b + c)
Origin at 2x, ¯x, ¯x
Maximal non-isomorphic subgroups
I
[2] R31 (R3, 146) 1; 2; 3
[3] R1c (Cc,9) 1; 4
[3] R1c (Cc,9) 1; 5
[3] R1c (Cc,9) 1; 6
IIa none
IIb [3] P3c1(a
= a − b,b
= b − c,c
= a + b + c) (158)
Maximal isomorphic subgroups of lowest index
IIc
[4] R3c (a
= −a + b + c,b
= a − b + c,c
= a + b − c) (161); [5] R3c (a
= a + 2b + 2c,b
= 2a + b + 2c,c
= 2a + 2b + c) (161)
Minimal non-isomorphic supergroups
I
[2] R
¯
3c (167); [4] P
¯
43n (218); [4] F
¯
43c (219); [4] I
¯
43d (220)
II [2] R3m (a
=
1
2
(−a + b+ c), b
=
1
2
(a − b + c),c
=
1
2
(a + b − c)) (160);
[3] P31c (a
=
1
3
(2a − b− c), b
=
1
3
(−a + 2b − c),c
=
1
3
(a + b + c)) (159)
535
P
¯
31mD
1
3d
¯
31m Trigonal
No. 162 P
¯
312/m
Patterson symmetry P
¯
31m
Origin at centre (
¯
31m)
Asymmetric unit 0 ≤ x ≤
2
3
;0≤ y ≤
1
2
;0≤ z ≤
1
2
; x ≤ (1 + y)/2; y ≤ min(1 − x,x)
Vertices 0,0, 0
1
2
,0, 0
2
3
,
1
3
,0
1
2
,
1
2
,0
0,0,
1
2
1
2
,0,
1
2
2
3
,
1
3
,
1
2
1
2
,
1
2
,
1
2
Symmetry operations
(1) 1 (2) 3
+
0,0,z (3) 3
−
0,0,z
(4) 2 x, ¯x,0(5)2x,2x,0(6)22x,x, 0
(7)
¯
10, 0,0(8)
¯
3
+
0,0,z;0,0,0(9)
¯
3
−
0,0,z;0,0,0
(10) mx,x,z (11) mx,0, z (12) m 0,y, z
536
International Tables for Crystallography (2006). Vol. A, Space group 162, pp. 536–537.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 162 P
¯
31m
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (4); (7)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
12 l 1(1)x, y,z (2) ¯y,x − y,z (3) ¯x+ y, ¯x,z
(4) ¯y, ¯x, ¯z (5) ¯x + y, y, ¯z (6) x,x − y, ¯z
(7) ¯x, ¯y, ¯z (8) y, ¯x + y, ¯z (9) x − y,x, ¯z
(10) y,x , z (11) x − y, ¯y,z (12) ¯x, ¯x +
y, z
no conditions
Special: no extra conditions
6 k ..mx,0,z 0,x,z ¯x, ¯x,z 0 , ¯x, ¯z ¯x,0, ¯zx,x, ¯z
6 j ..2 x, ¯x,
1
2
x,2x,
1
2
2¯x, ¯x,
1
2
¯x,x,
1
2
¯x,2¯x,
1
2
2x,x,
1
2
6 i ..2 x, ¯x,0 x,2x, 02¯x, ¯x,0¯x,x,0¯x,2¯x, 02x, x,0
4 h 3 ..
1
3
,
2
3
,z
1
3
,
2
3
, ¯z
2
3
,
1
3
, ¯z
2
3
,
1
3
,z
3 g ..2/m
1
2
,0,
1
2
0,
1
2
,
1
2
1
2
,
1
2
,
1
2
3 f ..2/m
1
2
,0, 00,
1
2
,0
1
2
,
1
2
,0
2 e 3 . m 0,0,z 0,0, ¯z
2 d 3 . 2
1
3
,
2
3
,
1
2
2
3
,
1
3
,
1
2
2 c 3 . 2
1
3
,
2
3
,0
2
3
,
1
3
,0
1 b
¯
3 . m 0,0,
1
2
1 a
¯
3 . m 0,0,0
Symmetry of special projections
Along [001] p6mm
a
= ab
= b
Origin at 0,0,z
Along [100] p2mm
a
=
1
2
(a + 2b) b
= c
Origin at x,0,0
Along [210] p2
a
=
1
2
bb
= c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] P31m (157) 1; 2; 3; 10; 11; 12
[2] P312 (149) 1; 2; 3; 4; 5; 6
[2] P
¯
311 (P
¯
3, 147) 1; 2; 3; 7; 8; 9
[3] P112/m (C2/m, 12) 1; 4; 7; 10
[3] P112/m (C2/m, 12) 1; 5; 7; 11
[3] P112/m (C2/m, 12) 1; 6; 7; 12
IIa none
IIb [2] P
¯
31c (c
= 2c) (163); [3] H
¯
31m (a
= 3a,b
= 3b)(P
¯
3m1, 164); [3] R
¯
3m (a
= a − b,b
= a + 2b,c
= 3c) (166);
[3] R
¯
3m (a
= 2a + b,b
= −a + b,c
= 3c) (166)
Maximal isomorphic subgroups of lowest index
IIc
[2] P
¯
31m (c
= 2c) (162); [4] P
¯
31m (a
= 2a,b
= 2b) (162)
Minimal non-isomorphic supergroups
I
[2] P6/mmm(191); [2] P6
3
/mcm (193)
II [3] H
¯
31m (P
¯
3m1, 164)
537
P
¯
31cD
2
3d
¯
31m Trigonal
No. 163 P
¯
312/c
Patterson symmetry P
¯
31m
Origin at centre (
¯
3) at
¯
31c
Asymmetric unit 0 ≤ x ≤
2
3
;0≤ y ≤
2
3
;0≤ z ≤
1
4
; x ≤ (1 + y)/2; y ≤ min(1 − x,(1 + x)/2)
Vertices 0,0, 0
1
2
,0, 0
2
3
,
1
3
,0
1
3
,
2
3
,00,
1
2
,0
0,0,
1
4
1
2
,0,
1
4
2
3
,
1
3
,
1
4
1
3
,
2
3
,
1
4
0,
1
2
,
1
4
Symmetry operations
(1) 1 (2) 3
+
0,0,z (3) 3
−
0,0,z
(4) 2 x, ¯x,
1
4
(5) 2 x,2x,
1
4
(6) 2 2x,x,
1
4
(7)
¯
10, 0,0(8)
¯
3
+
0,0,z;0,0,0(9)
¯
3
−
0,0,z;0,0,0
(10) cx,x,z (11) cx,0, z (12) c 0,y,z
538
International Tables for Crystallography (2006). Vol. A, Space group 163, pp. 538–539.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 163 P
¯
31c
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (4); (7)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
12 i 1(1)x, y,z (2) ¯y,x − y, z (3) ¯x + y, ¯x,z
(4) ¯y, ¯x, ¯z +
1
2
(5) ¯x + y,y, ¯z +
1
2
(6) x,x − y, ¯z +
1
2
(7) ¯x, ¯y, ¯z (8) y, ¯x+ y, ¯z (9) x − y,x, ¯z
(10) y,x,z +
1
2
(11) x − y, ¯y,z +
1
2
(12) ¯x, ¯x + y, z +
1
2
hh2hl : l = 2n
000l : l = 2n
Special: as above, plus
6 h ..2 x, ¯x,
1
4
x,2x,
1
4
2¯x, ¯x,
1
4
¯x,x,
3
4
¯x,2¯x,
3
4
2x,x,
3
4
no extra conditions
6 g
¯
1
1
2
,0, 00,
1
2
,0
1
2
,
1
2
,00,
1
2
,
1
2
1
2
,0,
1
2
1
2
,
1
2
,
1
2
hkil : l = 2n
4 f 3 ..
1
3
,
2
3
,z
1
3
,
2
3
, ¯z +
1
2
2
3
,
1
3
, ¯z
2
3
,
1
3
,z +
1
2
hkil : l = 2n
or h − k = 3n + 1
or h − k = 3n + 2
4 e 3 .. 0,0,z 0,0, ¯z +
1
2
0,0, ¯z 0,0 , z +
1
2
hkil : l = 2n
2 d 3 . 2
2
3
,
1
3
,
1
4
1
3
,
2
3
,
3
4
hkil : l = 2n
or h − k = 3n + 1
or h − k = 3n + 2
2 c 3 . 2
1
3
,
2
3
,
1
4
2
3
,
1
3
,
3
4
hkil : l = 2n
or h − k = 3n + 1
or h − k = 3n + 2
2 b
¯
3 .. 0,0 , 00,0,
1
2
hkil : l = 2n
2 a 3 . 20, 0,
1
4
0,0,
3
4
hkil : l = 2n
Symmetry of special projections
Along [001] p6mm
a
= ab
= b
Origin at 0,0,z
Along [100] p2gm
a
=
1
2
(a + 2b) b
= c
Origin at x,0,0
Along [210] p2
a
=
1
2
bb
=
1
2
c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] P31c (159) 1; 2; 3; 10; 11; 12
[2] P312 (149) 1; 2; 3; 4; 5; 6
[2] P
¯
311 (P
¯
3, 147) 1; 2; 3; 7; 8; 9
[3] P112/c (C2/c, 15) 1; 4; 7; 10
[3] P112/c (C2/c, 15) 1; 5; 7; 11
[3] P112/c (C2/c, 15) 1; 6; 7; 12
IIa none
IIb [3] H
¯
31c (a
= 3a,b
= 3b)(P
¯
3c1, 165); [3] R
¯
3c (a
= a − b,b
= a + 2b,c
= 3c) (167);
[3] R
¯
3c (a
= 2a + b,b
= −a + b,c
= 3c) (167)
Maximal isomorphic subgroups of lowest index
IIc
[3] P
¯
31c (c
= 3c) (163); [4] P
¯
31c (a
= 2a,b
= 2b) (163)
Minimal non-isomorphic supergroups
I
[2] P6/mcc(192); [2] P6
3
/mmc (194)
II [3] H
¯
31c (P
¯
3c1, 165); [2] P
¯
31m (c
=
1
2
c) (162)
539
P
¯
3m1 D
3
3d
¯
3m1 Trigonal
No. 164 P
¯
32/m1
Patterson symmetry P
¯
3m1
Origin at centre (
¯
3m1)
Asymmetric unit 0 ≤ x ≤
2
3
;0≤ y ≤
1
3
;0≤ z ≤ 1; x ≤ (1 + y)/2; y ≤ x/2
Vertices 0,0, 0
1
2
,0, 0
2
3
,
1
3
,0
0,0,1
1
2
,0, 1
2
3
,
1
3
,1
Symmetry operations
(1) 1 (2) 3
+
0,0,z (3) 3
−
0,0,z
(4) 2 x,x,0(5)2x,0, 0(6)20,y, 0
(7)
¯
10, 0,0(8)
¯
3
+
0,0,z;0,0,0(9)
¯
3
−
0,0,z;0,0,0
(10) mx, ¯x,z (11) mx,2x, z (12) m 2x, x,z
540
International Tables for Crystallography (2006). Vol. A, Space group 164, pp. 540–541.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 164 P
¯
3m1
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (4); (7)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
12 j 1(1)x, y,z (2) ¯y,x − y,z (3) ¯x+ y, ¯x,z
(4) y,x, ¯z (5) x − y, ¯y, ¯z (6) ¯x, ¯x + y, ¯z
(7) ¯x, ¯y, ¯z (8) y, ¯x + y, ¯z (9) x − y,x, ¯z
(10) ¯y, ¯x,z (11) ¯x + y, y,z (12) x,x −
y, z
no conditions
Special: no extra conditions
6 i . m . x, ¯x,zx, 2x,z 2¯x, ¯x, z ¯x,x, ¯z 2x,x, ¯z ¯x, 2¯x, ¯z
6 h . 2 . x,0,
1
2
0,x,
1
2
¯x, ¯x,
1
2
¯x,0,
1
2
0, ¯x,
1
2
x,x,
1
2
6 g . 2 . x,0,00,x,0¯x, ¯x, 0¯x,0,00, ¯x,0 x,x,0
3 f . 2/m .
1
2
,0,
1
2
0,
1
2
,
1
2
1
2
,
1
2
,
1
2
3 e . 2/m .
1
2
,0, 00,
1
2
,0
1
2
,
1
2
,0
2 d 3 m .
1
3
,
2
3
,z
2
3
,
1
3
, ¯z
2 c 3 m . 0,0,z 0,0, ¯z
1 b
¯
3 m . 0,0,
1
2
1 a
¯
3 m . 0,0,0
Symmetry of special projections
Along [001] p6mm
a
= ab
= b
Origin at 0,0,z
Along [100] p2
a
=
1
2
(a + 2b) b
= c
Origin at x,0,0
Along [210] p2mm
a
=
1
2
bb
= c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] P3m1 (156) 1; 2; 3; 10; 11; 12
[2] P321 (150) 1; 2; 3; 4; 5; 6
[2] P
¯
311 (P
¯
3, 147) 1; 2; 3; 7; 8; 9
[3] P12/m1(C 2 /m, 12) 1; 4; 7; 10
[3] P12/m1(C 2 /m, 12) 1; 5; 7; 11
[3] P12/m1(C 2 /m, 12) 1; 6; 7; 12
IIa none
IIb [2] P
¯
3c1(c
= 2c) (165); [3] H
¯
3m1(a
= 3a,b
= 3b)(P
¯
31m, 162)
Maximal isomorphic subgroups of lowest index
IIc
[2] P
¯
3m1(c
= 2c) (164); [4] P
¯
3m1(a
= 2a,b
= 2b) (164)
Minimal non-isomorphic supergroups
I
[2] P6/mmm(191); [2] P6
3
/mmc (194)
II [3] H
¯
3m1(P
¯
31m, 162); [3] R
¯
3m (obverse) (166); [3] R
¯
3m (reverse) (166)
541
P
¯
3c1 D
4
3d
¯
3m1 Trigonal
No. 165 P
¯
32/c1
Patterson symmetry P
¯
3m1
Origin at centre (
¯
3) at
¯
3c1
Asymmetric unit 0 ≤ x ≤
2
3
;0≤ y ≤
2
3
;0≤ z ≤
1
4
; x ≤ (1 + y)/2; y ≤ min(1 − x,(1 + x)/2)
Vertices 0,0, 0
1
2
,0, 0
2
3
,
1
3
,0
1
3
,
2
3
,00,
1
2
,0
0,0,
1
4
1
2
,0,
1
4
2
3
,
1
3
,
1
4
1
3
,
2
3
,
1
4
0,
1
2
,
1
4
Symmetry operations
(1) 1 (2) 3
+
0,0,z (3) 3
−
0,0,z
(4) 2 x,x,
1
4
(5) 2 x,0,
1
4
(6) 2 0, y,
1
4
(7)
¯
10, 0,0(8)
¯
3
+
0,0,z;0,0,0(9)
¯
3
−
0,0,z;0,0,0
(10) cx, ¯x,z (11) cx,2x, z (12) c 2x,x, z
542
International Tables for Crystallography (2006). Vol. A, Space group 165, pp. 542–543.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 165 P
¯
3c1
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (4); (7)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
12 g 1(1)x,y, z (2) ¯y,x − y,z (3) ¯x + y, ¯x,z
(4) y,x, ¯z +
1
2
(5) x − y, ¯y, ¯z +
1
2
(6) ¯x, ¯x + y, ¯z +
1
2
(7) ¯x, ¯y, ¯z (8) y, ¯x+ y, ¯z (9) x − y,x, ¯z
(10) ¯y, ¯x,z +
1
2
(11) ¯x + y, y,z +
1
2
(12) x,x − y, z +
1
2
h
¯
h0l : l = 2n
000l : l = 2n
Special: as above, plus
6 f . 2 . x,0,
1
4
0,x,
1
4
¯x, ¯x,
1
4
¯x,0,
3
4
0, ¯x,
3
4
x,x,
3
4
no extra conditions
6 e
¯
1
1
2
,0, 00,
1
2
,0
1
2
,
1
2
,00,
1
2
,
1
2
1
2
,0,
1
2
1
2
,
1
2
,
1
2
hkil : l = 2n
4 d 3 ..
1
3
,
2
3
,z
2
3
,
1
3
, ¯z +
1
2
2
3
,
1
3
, ¯z
1
3
,
2
3
,z +
1
2
hkil : l = 2n
4 c 3 .. 0,0,z 0,0, ¯z +
1
2
0,0, ¯z 0,0 , z +
1
2
hkil : l = 2n
2 b
¯
3 .. 0,0 , 00,0,
1
2
hkil : l = 2n
2 a 32. 0,0,
1
4
0,0,
3
4
hkil : l = 2n
Symmetry of special projections
Along [001] p6mm
a
= ab
= b
Origin at 0,0,z
Along [100] p2
a
=
1
2
(a + 2b) b
=
1
2
c
Origin at x,0,0
Along [210] p2gm
a
=
1
2
bb
= c
Origin at x,
1
2
x,0
Maximal non-isomorphic subgroups
I
[2] P3c1 (158) 1; 2; 3; 10; 11; 12
[2] P321 (150) 1; 2; 3; 4; 5; 6
[2] P
¯
311 (P
¯
3, 147) 1; 2; 3; 7; 8; 9
[3] P12/c1(C 2/c, 15) 1; 4; 7; 10
[3] P12/c1(C 2/c, 15) 1; 5; 7; 11
[3] P12/c1(C 2/c, 15) 1; 6; 7; 12
IIa none
IIb [3] H
¯
3c1(a
= 3a,b
= 3b)(P
¯
31c, 163)
Maximal isomorphic subgroups of lowest index
IIc
[3] P
¯
3c1(c
= 3c) (165); [4] P
¯
3c1(a
= 2a,b
= 2b) (165)
Minimal non-isomorphic supergroups
I
[2] P6/mcc(192); [2] P6
3
/mcm (193)
II [3] H
¯
3c1(P
¯
31c, 163); [3] R
¯
3c (obverse) (167); [3] R
¯
3c (reverse) (167); [2] P
¯
3m1(c
=
1
2
c) (164)
543