Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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P4/mC
1
4h
4/m Tetragonal
No. 83 P4 /m
Patterson symmetry P4/m
Origin at centre (4/m)
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 (4) 4
−
0,0,z
(5)
¯
10, 0,0(6)mx,y,0(7)
¯
4
+
0,0,z;0,0, 0(8)
¯
4
−
0,0,z;0,0, 0
344
International Tables for Crystallography (2006). Vol. A, Space group 83, pp. 344–345.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 83 P4/m
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 km.. x,y,
1
2
¯x, ¯y,
1
2
¯y,x,
1
2
y, ¯x,
1
2
no extra conditions
4 jm.. x, y,0¯x, ¯y,0¯y,x,0 y, ¯x,0 no extra conditions
4 i 2 .. 0,
1
2
,z
1
2
,0, z 0,
1
2
, ¯z
1
2
,0, ¯zhkl: h + k = 2n
2 h 4 ..
1
2
,
1
2
,z
1
2
,
1
2
, ¯z no extra conditions
2 g 4 .. 0,0,z 0,0, ¯z no extra conditions
2 f 2/m .. 0,
1
2
,
1
2
1
2
,0,
1
2
hkl : h + k = 2n
2 e 2/m .. 0,
1
2
,0
1
2
,0, 0 hkl : h + k = 2n
1 d 4/m ..
1
2
,
1
2
,
1
2
no extra conditions
1 c 4/m ..
1
2
,
1
2
,0 no extra conditions
1 b 4/ m .. 0,0,
1
2
no extra conditions
1 a 4/ m .. 0,0,0 no extra conditions
Symmetry of special projections
Along [001] p4
a
= ab
= b
Origin at 0,0,z
Along [100] p2mm
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
¯
4 (81) 1; 2; 7; 8
[2] P4 (75) 1; 2; 3; 4
[2] P2/m (10) 1; 2; 5; 6
IIa none
IIb [2] P4
2
/m (c
= 2c) (84); [2] C4/e (a
= 2a,b
= 2b)(P4/n, 85); [2] F 4/m (a
= 2a,b
= 2b,c
= 2c)(I 4/m, 87)
Maximal isomorphic subgroups of lowest index
IIc
[2] P4/m (c
= 2c) (83); [2] C4/m (a
= 2a,b
= 2b)(P4/m, 83)
Minimal non-isomorphic supergroups
I
[2] P4/mmm(123); [2] P4/mcc (124); [2] P4 /mbm(127); [2] P4/mnc (128)
II [2] I 4/m (87)
345
P4
2
/mC
2
4h
4/m Tetragonal
No. 84 P4
2
/m Patterson symmetry P4/m
Origin at centre (2/m) on 4
2
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,
1
2
) 0,0, z (4) 4
−
(0,0,
1
2
) 0,0,z
(5)
¯
10, 0,0(6)mx,y,0(7)
¯
4
+
0,0,z;0,0,
1
4
(8)
¯
4
−
0,0,z;0,0,
1
4
346
International Tables for Crystallography (2006). Vol. A, Space group 84, pp. 346–347.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 84 P4
2
/m
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 k 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯y,x, z +
1
2
(4) y, ¯x, z +
1
2
(5) ¯x, ¯y, ¯z (6) x, y, ¯z (7) y, ¯x, ¯z+
1
2
(8) ¯y,x, ¯z +
1
2
00l : l = 2n
Special: as above, plus
4 jm.. x, y,0¯x, ¯y,0¯y,x,
1
2
y, ¯x,
1
2
no extra conditions
4 i 2 .. 0,
1
2
,z
1
2
,0, z +
1
2
0,
1
2
, ¯z
1
2
,0, ¯z +
1
2
hkl : h + k + l = 2n
4 h 2 ..
1
2
,
1
2
,z
1
2
,
1
2
,z +
1
2
1
2
,
1
2
, ¯z
1
2
,
1
2
, ¯z +
1
2
hkl : l = 2n
4 g 2 .. 0,0,z 0,0,z +
1
2
0,0, ¯z 0,0 , ¯z +
1
2
hkl : l = 2n
2 f
¯
4 ..
1
2
,
1
2
,
1
4
1
2
,
1
2
,
3
4
hkl : l = 2n
2 e
¯
4 .. 0,0,
1
4
0,0,
3
4
hkl : l = 2n
2 d 2/m .. 0,
1
2
,
1
2
1
2
,0, 0 hkl : h + k + l = 2n
2 c 2/m .. 0,
1
2
,0
1
2
,0,
1
2
hkl : h + k + l = 2n
2 b 2/ m ..
1
2
,
1
2
,0
1
2
,
1
2
,
1
2
hkl : l = 2n
2 a 2/ m .. 0,0,00, 0,
1
2
hkl : l = 2n
Symmetry of special projections
Along [001] p4
a
= ab
= b
Origin at 0,0,z
Along [100] p2mm
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
¯
4 (81) 1; 2; 7; 8
[2] P4
2
(77) 1; 2; 3; 4
[2] P2/m (10) 1; 2; 5; 6
IIa none
IIb [2] C 4
2
/e (a
= 2a,b
= 2b)(P4
2
/n, 86)
Maximal isomorphic subgroups of lowest index
IIc
[2] C 4
2
/m (a
= 2a,b
= 2b)(P4
2
/m, 84); [3] P4
2
/m (c
= 3c) (84)
Minimal non-isomorphic supergroups
I
[2] P4
2
/mmc (131); [2] P4
2
/mcm (132); [2] P4
2
/mbc (135); [2] P4
2
/mnm (136)
II [2] I 4/m (87); [2] P4/m (c
=
1
2
c) (83)
347
P4/nC
3
4h
4/m Tetragonal
No. 85 P4 /n
Patterson symmetry P4/m
ORIGIN CHOICE 1
Origin at
¯
4onn,at−
1
4
,
1
4
,0 from
¯
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,
1
2
,z (4) 4
−
1
2
,0, z
(5)
¯
1
1
4
,
1
4
,0(6)n(
1
2
,
1
2
,0) x, y,0(7)
¯
4
+
0,0,z;0,0, 0(8)
¯
4
−
0,0,z;0,0, 0
348
International Tables for Crystallography (2006). Vol. A, Space group 85, pp. 348–351.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 85 P4/n
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 g 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯y +
1
2
,x +
1
2
,z (4) y +
1
2
, ¯x +
1
2
,z
(5) ¯x+
1
2
, ¯y +
1
2
, ¯z (6) x +
1
2
,y +
1
2
, ¯z (7) y, ¯x, ¯z (8) ¯y,x, ¯z
hk0: h + k = 2n
h00 : h = 2n
Special: as above, plus
4 f 2 .. 0,0, z
1
2
,
1
2
,z
1
2
,
1
2
, ¯z 0, 0, ¯zhkl: h + k = 2n
4 e
¯
1
1
4
,
1
4
,
1
2
3
4
,
3
4
,
1
2
1
4
,
3
4
,
1
2
3
4
,
1
4
,
1
2
hkl : h,k = 2n
4 d
¯
1
1
4
,
1
4
,0
3
4
,
3
4
,0
1
4
,
3
4
,0
3
4
,
1
4
,0 hkl : h, k = 2n
2 c 4 .. 0,
1
2
,z
1
2
,0, ¯z no extra conditions
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] p4
a
=
1
2
(a − b) b
=
1
2
(a + b)
Origin at 0,0,z
Along [100] p2mg
a
= bb
= c
Origin at x,
1
4
,0
Along [110] p2mm
a
=
1
2
(−a + b) b
= c
Origin at x, x,0
Maximal non-isomorphic subgroups
I
[2] P
¯
4 (81) 1; 2; 7; 8
[2] P4 (75) 1; 2; 3; 4
[2] P2/n (P2/c , 13) 1; 2; 5; 6
IIa none
IIb [2] P4
2
/n (c
= 2c) (86)
Maximal isomorphic subgroups of lowest index
IIc
[2] P4/n (c
= 2c) (85); [5] P4/n (a
= a + 2b,b
= −2a + b or a
= a − 2b,b
= 2a + b) (85)
Minimal non-isomorphic supergroups
I
[2] P4/nbm(125); [2] P4/nnc(126); [2] P4/nmm(129); [2] P4/ncc (130)
II [2] C 4/m (P4/m, 83); [2] I 4/m (87)
349
P4/nC
3
4h
4/m Tetragonal
No. 85 P4 /n
Patterson symmetry P4/m
ORIGIN CHOICE 2
Origin at
¯
1onn,at
1
4
,−
1
4
,0 from
¯
4
Asymmetric unit −
1
4
≤ x ≤
1
4
; −
1
4
≤ y ≤
1
4
;0≤ z ≤
1
2
Symmetry operations
(1) 1 (2) 2
1
4
,
1
4
,z (3) 4
+
1
4
,
1
4
,z (4) 4
−
1
4
,
1
4
,z
(5)
¯
10, 0,0(6)n(
1
2
,
1
2
,0) x,y,0(7)
¯
4
+
1
4
,−
1
4
,z;
1
4
,−
1
4
,0(8)
¯
4
−
−
1
4
,
1
4
,z; −
1
4
,
1
4
,0
350
CONTINUED No. 85 P4/n
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 g 1(1)x, y,z (2) ¯x+
1
2
, ¯y +
1
2
,z (3) ¯y +
1
2
,x,z (4) y, ¯x +
1
2
,z
(5) ¯x, ¯y, ¯z (6) x +
1
2
,y +
1
2
, ¯z (7) y +
1
2
, ¯x, ¯z (8) ¯y,x +
1
2
, ¯z
hk0: h + k = 2n
h00 : h = 2n
Special: as above, plus
4 f 2 ..
1
4
,
3
4
,z
3
4
,
1
4
,z
3
4
,
1
4
, ¯z
1
4
,
3
4
, ¯zhkl: h + k = 2n
4 e
¯
10, 0,
1
2
1
2
,
1
2
,
1
2
1
2
,0,
1
2
0,
1
2
,
1
2
hkl : h,k = 2n
4 d
¯
10, 0,0
1
2
,
1
2
,0
1
2
,0, 00,
1
2
,0 hkl : h,k = 2n
2 c 4 ..
1
4
,
1
4
,z
3
4
,
3
4
, ¯z no extra conditions
2 b
¯
4 ..
1
4
,
3
4
,
1
2
3
4
,
1
4
,
1
2
hkl : h + k = 2n
2 a
¯
4 ..
1
4
,
3
4
,0
3
4
,
1
4
,0 hkl : h + k = 2n
Symmetry of special projections
Along [001] p4
a
=
1
2
(a − b) b
=
1
2
(a + b)
Origin at
1
4
,
1
4
,z
Along [100] p2mg
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
¯
4 (81) 1; 2; 7; 8
[2] P4 (75) 1; 2; 3; 4
[2] P2/n (P2/c , 13) 1; 2; 5; 6
IIa none
IIb [2] P4
2
/n (c
= 2c) (86)
Maximal isomorphic subgroups of lowest index
IIc
[2] P4/n (c
= 2c) (85); [5] P4/n (a
= a + 2b,b
= −2a + b or a
= a − 2b,b
= 2a + b) (85)
Minimal non-isomorphic supergroups
I
[2] P4/nbm(125); [2] P4/nnc(126); [2] P4/nmm(129); [2] P4/ncc (130)
II [2] C 4/m (P4/m, 83); [2] I 4/m (87)
351
P4
2
/nC
4
4h
4/m Tetragonal
No. 86 P4
2
/n Patterson symmetry P4/m
ORIGIN CHOICE 1
Origin at
¯
4, at −
1
4
,−
1
4
,−
1
4
from
¯
1
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,
1
2
) 0,
1
2
,z (4) 4
−
(0,0,
1
2
)
1
2
,0, z
(5)
¯
1
1
4
,
1
4
,
1
4
(6) n(
1
2
,
1
2
,0) x, y,
1
4
(7)
¯
4
+
0,0,z;0,0, 0(8)
¯
4
−
0,0,z;0,0, 0
352
International Tables for Crystallography (2006). Vol. A, Space group 86, pp. 352–355.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 86 P4
2
/n
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 g 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯y +
1
2
,x +
1
2
,z +
1
2
(4) y +
1
2
, ¯x +
1
2
,z +
1
2
(5) ¯x +
1
2
, ¯y +
1
2
, ¯z +
1
2
(6) x +
1
2
,y +
1
2
, ¯z +
1
2
(7) y, ¯x, ¯z (8) ¯y,x, ¯z
hk0: h + k = 2n
00l : l = 2n
h00 : h = 2n
Special: as above, plus
4 f 2 .. 0,0, z
1
2
,
1
2
,z +
1
2
1
2
,
1
2
, ¯z +
1
2
0,0, ¯zhkl: h + k + l = 2n
4 e 2 .. 0,
1
2
,z 0,
1
2
,z +
1
2
1
2
,0, ¯z +
1
2
1
2
,0, ¯zhkl: l = 2n
4 d
¯
1
1
4
,
1
4
,
3
4
3
4
,
3
4
,
3
4
1
4
,
3
4
,
1
4
3
4
,
1
4
,
1
4
hkl : h + k,h + l,k + l = 2n
4 c
¯
1
1
4
,
1
4
,
1
4
3
4
,
3
4
,
1
4
1
4
,
3
4
,
3
4
3
4
,
1
4
,
3
4
hkl : h + k,h + l,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] p4
a
=
1
2
(a − b) b
=
1
2
(a + b)
Origin at 0,0,z
Along [100] p2mg
a
= bb
= c
Origin at x,
1
4
,
1
4
Along [110] p2mm
a
=
1
2
(−a + b) b
= c
Origin at x, x,
1
4
Maximal non-isomorphic subgroups
I
[2] P
¯
4 (81) 1; 2; 7; 8
[2] P4
2
(77) 1; 2; 3; 4
[2] P2/n (P2/c , 13) 1; 2; 5; 6
IIa none
IIb [2] F 4
1
/d (a
= 2a,b
= 2b,c
= 2c)(I 4
1
/a, 88)
Maximal isomorphic subgroups of lowest index
IIc
[3] P4
2
/n (c
= 3c) (86); [5] P4
2
/n (a
= a + 2b,b
= −2a + b or a
= a − 2b,b
= 2a + b) (86)
Minimal non-isomorphic supergroups
I
[2] P4
2
/nbc(133); [2] P4
2
/nnm(134); [2] P4
2
/nmc(137); [2] P4
2
/ncm(138)
II [2] C 4
2
/m (P4
2
/m, 84); [2] I 4/m (87); [2] P4/n (c
=
1
2
c) (85)
353