Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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P
¯
4m2 D
5
2d
¯
4m2 Tetragonal
No. 115 P
¯
4m2
Patterson symmetry P4/mmm
Origin at
¯
4m2
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
Symmetry operations
(1) 1 (2) 2 0, 0,z (3)
¯
4
+
0,0,z;0,0,0(4)
¯
4
−
0,0,z;0,0,0
(5) mx,0,z (6) m 0,y, z (7) 2 x,x,0(8)2x, ¯x, 0
414
International Tables for Crystallography (2006). Vol. A, Space group 115, pp. 414–415.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 115 P
¯
4m2
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
8 l 1(1)x,y,z (2) ¯x, ¯y,z (3) y, ¯x, ¯z (4) ¯y,x, ¯z
(5) x, ¯y,z (6) ¯x,y, z (7) y,x, ¯z (8) ¯y, ¯x, ¯z
no conditions
Special:
4 k . m . x,
1
2
,z ¯x,
1
2
,z
1
2
, ¯x, ¯z
1
2
,x, ¯z no extra conditions
4 j . m . x,0,z ¯x,0, z 0, ¯x, ¯z 0,x, ¯z no extra conditions
4 i ..2 x,x,
1
2
¯x, ¯x,
1
2
x, ¯x,
1
2
¯x,x,
1
2
no extra conditions
4 h ..2 x,x, 0¯x, ¯x,0 x, ¯x,0¯x,x,0 no extra conditions
2 g 2 mm. 0,
1
2
,z
1
2
,0, ¯zhk0: h + k = 2n
2 f 2 mm.
1
2
,
1
2
,z
1
2
,
1
2
, ¯z no extra conditions
2 e 2 mm. 0,0,z 0,0, ¯z no extra conditions
1 d
¯
4 m 20,0,
1
2
no extra conditions
1 c
¯
4 m 2
1
2
,
1
2
,
1
2
no extra conditions
1 b
¯
4 m 2
1
2
,
1
2
,0 no extra conditions
1 a
¯
4 m 20,0,0 no extra conditions
Symmetry of special projections
Along [001] p4mm
a
= ab
= b
Origin at 0,0,z
Along [100] p1m1
a
= bb
= c
Origin at x,0,0
Along [110] p2mm
a
=
1
2
(−a + b) b
= c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] P
¯
411 (P
¯
4, 81) 1; 2; 3; 4
[2] P2m1(Pmm2, 25) 1; 2; 5; 6
[2] P212 (C222, 21) 1; 2; 7; 8
IIa none
IIb [2] P
¯
4c2(c
= 2c) (116); [2] C
¯
4m2
1
(a
= 2a,b
= 2b)(P
¯
42
1
m, 113); [2] C
¯
4m2(a
= 2a,b
= 2b)(P
¯
42m, 111);
[2] F
¯
4m2(a
= 2a,b
= 2b,c
= 2c)(I
¯
42m, 121)
Maximal isomorphic subgroups of lowest index
IIc
[2] P
¯
4m2(c
= 2c) (115); [9] P
¯
4m2(a
= 3a,b
= 3b) (115)
Minimal non-isomorphic supergroups
I
[2] P4/mmm(123); [2] P4/nmm(129); [2] P4
2
/mmc (131); [2] P4
2
/nmc(137)
II [2] C
¯
4m2(P
¯
42m, 111); [2] I
¯
4m2 (119)
415
P
¯
4c2 D
6
2d
¯
4m2 Tetragonal
No. 116 P
¯
4c2
Patterson symmetry P4/mmm
Origin at
¯
4c1
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤ 1; 0 ≤ z ≤
1
4
Symmetry operations
(1) 1 (2) 2 0, 0,z (3)
¯
4
+
0,0,z;0,0,0(4)
¯
4
−
0,0,z;0,0,0
(5) cx,0,z (6) c 0,y,z (7) 2 x, x,
1
4
(8) 2 x, ¯x,
1
4
416
International Tables for Crystallography (2006). Vol. A, Space group 116, pp. 416–417.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 116 P
¯
4c2
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
8 j 1(1)x,y, z (2) ¯x, ¯y,z (3) y, ¯x, ¯z (4) ¯y,x, ¯z
(5) x, ¯y,z +
1
2
(6) ¯x,y, z +
1
2
(7) y,x, ¯z +
1
2
(8) ¯y, ¯x, ¯z +
1
2
0kl : l = 2n
00l : l = 2n
Special: as above, plus
4 i 2 .. 0,
1
2
,z
1
2
,0, ¯z 0,
1
2
,z +
1
2
1
2
,0, ¯z +
1
2
hkl : l = 2n
hk0: h + k = 2n
4 h 2 ..
1
2
,
1
2
,z
1
2
,
1
2
, ¯z
1
2
,
1
2
,z +
1
2
1
2
,
1
2
, ¯z +
1
2
hkl : l = 2n
4 g 2 .. 0,0,z 0,0, ¯z 0,0,z +
1
2
0,0, ¯z +
1
2
hkl : l = 2n
4 f ..2 x,x,
3
4
¯x, ¯x,
3
4
x, ¯x,
1
4
¯x,x,
1
4
no extra conditions
4 e ..2 x,x,
1
4
¯x, ¯x,
1
4
x, ¯x,
3
4
¯x,x,
3
4
no extra conditions
2 d
¯
4 ..
1
2
,
1
2
,0
1
2
,
1
2
,
1
2
hkl : l = 2n
2 c
¯
4 .. 0,0, 00,0,
1
2
hkl : l = 2n
2 b 2 . 22
1
2
,
1
2
,
1
4
1
2
,
1
2
,
3
4
hkl : l = 2n
2 a 2 . 22 0,0,
1
4
0,0,
3
4
hkl : l = 2n
Symmetry of special projections
Along [001] p4mm
a
= ab
= b
Origin at 0,0,z
Along [100] p1m1
a
= bb
=
1
2
c
Origin at x,0,0
Along [110] p2mm
a
=
1
2
(−a + b) b
= c
Origin at x,x ,
1
4
Maximal non-isomorphic subgroups
I
[2] P
¯
411 (P
¯
4, 81) 1; 2; 3; 4
[2] P2c1(Pcc2, 27) 1; 2; 5; 6
[2] P212 (C222, 21) 1; 2; 7; 8
IIa none
IIb [2] C
¯
4c2
1
(a
= 2a,b
= 2b)(P
¯
42
1
c, 114); [2] C
¯
4c2(a
= 2a,b
= 2b)(P
¯
42c, 112)
Maximal isomorphic subgroups of lowest index
IIc
[3] P
¯
4c2(c
= 3c) (116); [9] P
¯
4c2(a
= 3a,b
= 3b) (116)
Minimal non-isomorphic supergroups
I
[2] P4/mcc(124); [2] P4/ncc(130); [2] P4
2
/mcm (132); [2] P4
2
/ncm(138)
II [2] C
¯
4c2(P
¯
42c, 112); [2] I
¯
4c2 (120); [2] P
¯
4m2(c
=
1
2
c) (115)
417
P
¯
4b2 D
7
2d
¯
4m2 Tetragonal
No. 117 P
¯
4b2
Patterson symmetry P4/mmm
Origin at
¯
412
1
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
Symmetry operations
(1) 1 (2) 2 0,0,z (3)
¯
4
+
0,0,z;0,0,0(4)
¯
4
−
0,0,z;0,0,0
(5) ax,
1
4
,z (6) b
1
4
,y,z (7) 2(
1
2
,
1
2
,0) x,x,0(8)2x, ¯x +
1
2
,0
418
International Tables for Crystallography (2006). Vol. A, Space group 117, pp. 418–419.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 117 P
¯
4b2
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
8 i 1(1)x,y, z (2) ¯x, ¯y,z (3) y, ¯x, ¯z (4) ¯y,x, ¯z
(5) x +
1
2
, ¯y +
1
2
,z (6) ¯x +
1
2
,y +
1
2
,z (7) y +
1
2
,x +
1
2
, ¯z (8) ¯y +
1
2
, ¯x +
1
2
, ¯z
0kl : k = 2n
h00 : h = 2n
Special: as above, plus
4 h ..2 x,x +
1
2
,
1
2
¯x, ¯x +
1
2
,
1
2
x +
1
2
, ¯x,
1
2
¯x+
1
2
,x,
1
2
no extra conditions
4 g ..2 x,x +
1
2
,0¯x, ¯x +
1
2
,0 x +
1
2
, ¯x, 0¯x +
1
2
,x,0 no extra conditions
4 f 2 .. 0,
1
2
,z
1
2
,0, ¯z
1
2
,0, z 0,
1
2
, ¯zhkl: h + k = 2n
4 e 2 .. 0,0,z 0,0, ¯z
1
2
,
1
2
,z
1
2
,
1
2
, ¯zhkl: h + k = 2n
2 d 2 . 22 0,
1
2
,
1
2
1
2
,0,
1
2
hkl : h + k = 2n
2 c 2 . 22 0,
1
2
,0
1
2
,0, 0 hkl : h + k = 2n
2 b
¯
4 .. 0,0,
1
2
1
2
,
1
2
,
1
2
hkl : h + k = 2n
2 a
¯
4 .. 0,0, 0
1
2
,
1
2
,0 hkl : h + k = 2n
Symmetry of special projections
Along [001] p4gm
a
= ab
= b
Origin at 0,0,z
Along [100] p1m1
a
=
1
2
bb
= c
Origin at x, 0,0
Along [110] p2mm
a
=
1
2
(−a + b) b
= c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] P
¯
411 (P
¯
4, 81) 1; 2; 3; 4
[2] P2b1(Pba2, 32) 1; 2; 5; 6
[2] P212 (C222, 21) 1; 2; 7; 8
IIa none
IIb [2] P
¯
4n2(c
= 2c) (118)
Maximal isomorphic subgroups of lowest index
IIc
[2] P
¯
4b2(c
= 2c) (117); [9] P
¯
4b2(a
= 3a,b
= 3b) (117)
Minimal non-isomorphic supergroups
I
[2] P4/nbm(125); [2] P4/mbm(127); [2] P4
2
/nbc(133); [2] P4
2
/mbc(135)
II [2] C
¯
4m2(P
¯
42m, 111); [2] I
¯
4c2 (120)
419
P
¯
4n2 D
8
2d
¯
4m2 Tetragonal
No. 118 P
¯
4n2
Patterson symmetry P4/mmm
Origin at
¯
4
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤ 1; 0 ≤ z ≤
1
4
Symmetry operations
(1) 1 (2) 2 0, 0,z (3)
¯
4
+
0,0,z;0,0,0(4)
¯
4
−
0,0,z;0,0,0
(5) n(
1
2
,0,
1
2
) x,
1
4
,z (6) n(0,
1
2
,
1
2
)
1
4
,y,z (7) 2 (
1
2
,
1
2
,0) x,x,
1
4
(8) 2 x, ¯x +
1
2
,
1
4
420
International Tables for Crystallography (2006). Vol. A, Space group 118, pp. 420–421.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 118 P
¯
4n2
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
8 i 1(1)x,y,z (2) ¯x, ¯y,z (3) y, ¯x, ¯z (4) ¯y,x, ¯z
(5) x +
1
2
, ¯y +
1
2
,z +
1
2
(6) ¯x+
1
2
,y +
1
2
,z +
1
2
(7) y +
1
2
,x +
1
2
, ¯z +
1
2
(8) ¯y +
1
2
, ¯x +
1
2
, ¯z +
1
2
0kl : k + l = 2 n
00l : l = 2n
h00 : h = 2n
Special: as above, plus
4 h 2 .. 0,
1
2
,z
1
2
,0, ¯z
1
2
,0, z +
1
2
0,
1
2
, ¯z +
1
2
hkl : h + k + l = 2n
4 g ..2 x,x +
1
2
,
1
4
¯x, ¯x +
1
2
,
1
4
x +
1
2
, ¯x,
3
4
¯x+
1
2
,x,
3
4
no extra conditions
4 f ..2 x, ¯x +
1
2
,
1
4
¯x,x +
1
2
,
1
4
¯x +
1
2
, ¯x,
3
4
x +
1
2
,x,
3
4
no extra conditions
4 e 2 .. 0,0,z 0,0, ¯z
1
2
,
1
2
,z +
1
2
1
2
,
1
2
, ¯z +
1
2
hkl : h + k + l = 2n
2 d 2 . 22 0,
1
2
,
3
4
1
2
,0,
1
4
hkl : h + k + l = 2n
2 c 2 . 22 0,
1
2
,
1
4
1
2
,0,
3
4
hkl : h + k + l = 2n
2 b
¯
4 .. 0,0,
1
2
1
2
,
1
2
,0 hkl : h + k + l = 2n
2 a
¯
4 .. 0,0, 0
1
2
,
1
2
,
1
2
hkl : h + k + l = 2n
Symmetry of special projections
Along [001] p4gm
a
= ab
= b
Origin at 0,0,z
Along [100] c1m1
a
= bb
= c
Origin at x,0,0
Along [110] p2mm
a
=
1
2
(−a + b) b
= c
Origin at x,x ,
1
4
Maximal non-isomorphic subgroups
I
[2] P
¯
411 (P
¯
4, 81) 1; 2; 3; 4
[2] P2n1(Pnn2, 34) 1; 2; 5; 6
[2] P212 (C222, 21) 1; 2; 7; 8
IIa none
IIb [2] F
¯
4d 2(a
= 2a,b
= 2b,c
= 2c)(I
¯
42d, 122)
Maximal isomorphic subgroups of lowest index
IIc
[3] P
¯
4n2(c
= 3c) (118); [9] P
¯
4n2(a
= 3a,b
= 3b) (118)
Minimal non-isomorphic supergroups
I
[2] P4/nnc(126); [2] P4/mnc(128); [2] P4
2
/nnm(134); [2] P4
2
/mnm (136)
II [2] C
¯
4c2(P
¯
42c, 112); [2] I
¯
4m2 (119); [2] P
¯
4b2(c
=
1
2
c) (117)
421
I
¯
4m2 D
9
2d
¯
4m2 Tetragonal
No. 119 I
¯
4m2
Patterson symmetry I 4/mmm
Origin at
¯
4m2
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
4
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2 0, 0,z (3)
¯
4
+
0,0,z;0,0,0(4)
¯
4
−
0,0,z;0,0,0
(5) mx,0,z (6) m 0,y,z (7) 2 x,x, 0(8)2x, ¯x,0
For (
1
2
,
1
2
,
1
2
)+ set
(1) t(
1
2
,
1
2
,
1
2
) (2) 2(0, 0,
1
2
)
1
4
,
1
4
,z (3)
¯
4
+
1
2
,0, z;
1
2
,0,
1
4
(4)
¯
4
−
0,
1
2
,z;0,
1
2
,
1
4
(5) n(
1
2
,0,
1
2
) x,
1
4
,z (6) n(0,
1
2
,
1
2
)
1
4
,y,z (7) 2(
1
2
,
1
2
,0) x,x,
1
4
(8) 2 x, ¯x +
1
2
,
1
4
422
International Tables for Crystallography (2006). Vol. A, Space group 119, pp. 422–423.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 119 I
¯
4m2
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
1
2
,
1
2
,
1
2
); (2); (3); (5)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,
1
2
,
1
2
)+
Reflection conditions
General:
16 j 1(1)x,y,z (2) ¯x, ¯y,z (3) y, ¯x, ¯z (4) ¯y,x, ¯z
(5) x, ¯y,z (6) ¯x,y, z (7) y,x, ¯z (8) ¯y, ¯x, ¯z
hkl : h + k + l = 2n
hk0: h + k = 2n
0kl : k + l = 2n
hhl : l = 2n
00l : l = 2
n
h00 : h = 2n
Special: no extra conditions
8 i . m . x,0,z ¯x, 0,z 0, ¯x, ¯z 0,x, ¯z
8 h ..2 x,x +
1
2
,
1
4
¯x, ¯x +
1
2
,
1
4
x +
1
2
, ¯x,
3
4
¯x +
1
2
,x,
3
4
8 g ..2 x,x,0¯x, ¯x,0 x, ¯x, 0¯x,x, 0
4 f 2 mm. 0,
1
2
,z
1
2
,0, ¯z
4 e 2 mm. 0,0,z 0,0, ¯z
2 d
¯
4 m 20,
1
2
,
3
4
2 c
¯
4 m 20,
1
2
,
1
4
2 b
¯
4 m 20,0,
1
2
2 a
¯
4 m 20,0,0
Symmetry of special projections
Along [001] p4mm
a
=
1
2
(a − b) b
=
1
2
(a + b)
Origin at 0,0,z
Along [100] c1m1
a
= bb
= c
Origin at x,0,0
Along [110] p2mm
a
=
1
2
(−a + b) b
=
1
2
c
Origin at x,x , 0
Maximal non-isomorphic subgroups
I
[2] I
¯
411(I
¯
4, 82) (1; 2; 3; 4)+
[2] I 2m1(Imm2, 44) (1; 2; 5; 6)+
[2] I 212(F 222, 22) (1; 2; 7; 8)+
IIa [2] P
¯
4n2 (118) 1; 2; 3; 4; (5; 6; 7; 8)+(
1
2
,
1
2
,
1
2
)
[2] P
¯
4n2 (118) 1; 2; 7; 8; (3; 4; 5; 6)+(
1
2
,
1
2
,
1
2
)
[2] P
¯
4m2 (115) 1; 2; 3; 4; 5; 6; 7; 8
[2] P
¯
4m2 (115) 1; 2; 5; 6; (3; 4; 7; 8)+(
1
2
,
1
2
,
1
2
)
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] I
¯
4m2(c
= 3c) (119); [9] I
¯
4m2(a
= 3a,b
= 3b) (119)
Minimal non-isomorphic supergroups
I
[2] I 4/mmm(139); [2] I 4
1
/amd (141); [3] F
¯
43m (216)
II [2] C
¯
4m2(c
=
1
2
c)(P
¯
42m, 111)
423