Symmetry in reciprocal space

Shmueli, U.; Hall, S. R.; Grosse-Kunstleve, R. W.

doi:10.1107/97809553602060000552

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. B. ch. 1.4, pp. 120-149 | 1 | 2 |
https://doi.org/10.1107/97809553602060000552

Appendix A1.4.3. Structure-factor tables

U. Shmueli^a

Table A1.4.3.1| top | pdf |
Plane groups

The symbols appearing in this table are explained in Section 1.4.3 and in Tables A1.4.3.3 (monoclinic), A1.4.3.5 (tetragonal) and A1.4.3.6 (trigonal and hexagonal).

System	No.	Symbol	A	B
Oblique	1		c()	s()
Oblique	2		2c()	0
Rectangular	3		2c()c()	2c()s()
	4		2c()c()	2c()s()
			−2s()s()	2s()c()
	5		4c()c()	4c()s()
	6		4c()c()	0
	7		4c()c()	0
			−4s()s()	0
	8		4c()c()	0
			−4s()s()	0
	9		8c()c()	0
Square	10	p4	2[P(cc) − M(ss)]	0
	11		4P(cc)	0
	12		4P(cc)	0
			−4M(ss)	0
Hexagonal	13		C()	S()
	14		PH(cc)	MH(ss)
	15		PH(cc)	PH(ss)
	16		2C()	0
	17		2PH(cc)	0

Table A1.4.3.2| top | pdf |
Triclinic space groups

For the definition of the triple products ccc, csc etc., see Table A1.4.3.4.

P1 [No. 1]

	A	B
All	$[\cos2\pi]$ () = ccc − css − scs − ssc	$[\sin2\pi]$ () = scc csc $[\ +]$ ccs − sss

$[P\overline{1}]$ [No. 2]

	A	B
All	2(ccc − css − scs − ssc)	0

Table A1.4.3.3| top | pdf |
Monoclinic space groups

Each expression for A or B in the monoclinic system and for the space-group settings chosen in IT A is represented in terms of one of the following symbols: $[\hfill{\eqalign{{\rm c}(hl){\rm c}(ky)&=\cos[2\pi{}(hx+lz)]\cos(2\pi{}ky),\cr{\rm c}(hl){\rm s}(ky)&=\cos[2\pi{}(hx+lz)]\sin(2\pi{}ky),\cr{\rm s}(hl){\rm c}(ky)&=\sin[2\pi{}(hx+lz)]\cos(2\pi{}ky),\cr{\rm s}(hl){\rm s}(ky)&=\sin[2\pi{}(hx+lz)]\sin(2\pi{}ky),\cr}}\quad{\eqalign{{\rm c}(hk){\rm c}(lz)&=\cos[2\pi{}(hx+ky)]\cos(2\pi{}lz),\cr{\rm c}(hk){\rm s}(lz)&=\cos[2\pi{}(hx+ky)]\sin(2\pi{}lz),\cr{\rm s}(hk){\rm c}(lz)&=\sin[2\pi{}(hx+ky)]\cos(2\pi{}lz),\cr{\rm s}(hk){\rm s}(lz)&=\sin[2\pi{}(hx+ky)]\sin(2\pi{}lz),\cr}}\hfill{\eqalign{&\cr&\cr&\cr&({\rm A}1.4.3.1)\cr}}]$ where the left-hand column of expressions corresponds to space-group representations in the second setting, with b taken as the unique axis, and the right-hand column corresponds to representations in the first setting, with c taken as the unique axis.

The lattice types in this table are P, A, B, C and I, and are all explicit in the full space-group symbol only (see below). Note that s(hl), s(hk), s(ky) and s(lz) are zero for h = l = 0, h = k = 0, k = 0 and l = 0, respectively.

No.	Group symbol		Parity	A	B	Unique axis
No.	Short	Full	Parity	A	B	Unique axis
3				2c()c()	2c()s()	b
3				2c()c()	2c()s()	c
4	$[P2_{1}]$	$[P12_{1}1]$		2c()c()	2c()s()	b
				−2s()s()	2s()c()
4	$[P2_{1}]$	$[P112_{1}]$		2c()c()	2c()s()	c
				−2s()s()	2s()c()
5				4c()c()	4c()s()	b
5				4c()c()	4c()s()	b
5				4c()c()	4c()s()	b
5				4c()c()	4c()s()	c
5				4c()c()	4c()s()	c
5				4c()c()	4c()s()	c
6				2c()c()	2s()c()	b
6				2c()c()	2s()c()	c
7				2c()c()	2s()c()	b
				−2s()s()	2c()s()
7				2c()c()	2s()c()	b
				−2s()s()	2c()s()
7				2c()c()	2s()c()	b
				−2s()s()	2c()s()
7				2c()c()	2s()c()	c
				−2s()s()	2c()s()
7				2c()c()	2s()c()	c
				−2s()s()	2c()s()
7				2c()c()	2s()c()	c
				−2s()s()	2c()s()
8				4c()c()	4s()c()	b
8				4c()c()	4s()c()	b
8				4c()c()	4s()c()	b
8				4c()c()	4s()c()	c
8				4c()c()	4s()c()	c
8				4c()c()	4s()c()	c
9				4c()c()	4s()c()	b
				−4s()s()	4c()s()
9				4c()c()	4s()c()	b
				−4s()s()	4c()s()
9				4c()c()	4s()c()	b
				−4s()s()	4c()s()
9				4c()c()	4s()c()	c
				−4s()s()	4c()s()
9				4c()c()	4s()c()	c
				−4s()s()	4c()s()
9				4c()c()	4s()c()	c
				−4s()s()	4c()s()
10				4c()c()	0	b
10				4c()c()	0	c
11	$[P2_{1}/m]$	$[P12_{1}/m1]$		4c()c()	0	b
				−4s()s()	0
11	$[P2_{1}/m]$	$[P112_{1}/m]$		4c()c()	0	c
				−4s()s()	0
12				8c()c()	0	b
12				8c()c()	0	b
12				8c()c()	0	b
12				8c()c()	0	c
12				8c()c()	0	c
12				8c()c()	0	c
13				4c()c()	0	b
				−4s()s()	0
13				4c()c()	0	b
				−4s()s()	0
13				4c()c()	0	b
				−4s()s()	0
13				4c()c()	0	c
				−4s()s()	0
13				4c()c()	0	c
				−4s()s()	0
13				4c()c()	0	c
				−4s()s()	0
14	$[P2_{1}/c]$	$[P12_{1}/c1]$		4c()c()	0	b
				−4s()s()	0
14	$[P2_{1}/c]$	$[P12_{1}/n1]$		4c()c()	0	b
				−4s()s()	0
14	$[P2_{1}/c]$	$[P12_{1}/a1]$		4c()c()	0	b
				−4s()s()	0
14	$[P2_{1}/c]$	$[P112_{1}/a]$		4c()c()	0	c
				−4s()s()	0
14	$[P2_{1}/c]$	$[P112_{1}/n]$		4c()c()	0	c
				−4s()s()	0
14	$[P2_{1}/c]$	$[P112_{1}/b]$		4c()c()	0	c
				−4s()s()	0
15				8c()c()	0	b
				−8s()s()	0
15				8c()c()	0	b
				−8s()s()	0
15				8c()c()	0	b
				−8s()s()	0
15				8c()c()	0	c
				−8s()s()	0
15				8c()c()	0	c
				−8s()s()	0
15				8c()c()	0	c
				−8s()s()	0

Table A1.4.3.4| top | pdf |
Orthorhombic space groups

The expressions for A and B for the orthorhombic space groups in their standard settings [as in IT A (2005)] contain one, two or four terms of the form $[{\rm pqr}={\rm p}(2\pi{}hx){\rm q}(2\pi{}ky){\rm r}(2\pi{}lz)\eqno({\rm A}1.4.3.2)]$ preceded by a signed numerical constant, where p, q and r can each be either a sine or a cosine function, and the arguments of the functions in any product of the form (A1.4.3.2) are ordered as in (A1.4.3.2). These products are given in this table as ccc, ccs, csc, scc, ssc, scs, css and/or sss, where c and s are abbreviations for `sin' and `cos', respectively.

Note that pqr vanishes if at least one of p, q and r is a sine, and the corresponding index h, k or l is zero.

No.	Symbol	Origin	Parity	A	B
16	P222			4ccc	−4sss
17	$[P222_{1}]$			4ccc	−4sss
				−4css	4scc
18	$[P2_{1}2_{1}2]$			4ccc	−4sss
				−4ssc	4ccs
19	$[P2_{1}2_{1}2_{1}]$		;	4ccc	−4sss
			;	−4css	4scc
			;	−4scs	4csc
			;	−4ssc	4ccs
20	$[C222_{1}]$			8ccc	−8sss
				−8css	8scc
21	C222			8ccc	−8sss
22	F222			16ccc	−16sss
23	I222			8ccc	−8sss
24	$[I2_{1}2_{1}2_{1}]$		all even	8ccc	−8sss
			;	−8scs	8csc
			;	−8ssc	8ccs
			;	−8css	8scc
25	2			4ccc	4ccs
26	$[Pmc2_{1}]$			4ccc	4ccs
				−4css	4csc
27	2			4ccc	4ccs
				−4ssc	−4sss
28	2			4ccc	4ccs
				−4ssc	−4sss
29	$[Pca2_{1}]$		;	4ccc	4ccs
			;	−4scs	4scc
			;	−4ssc	−4sss
			;	−4css	4csc
30	2			4ccc	4ccs
				−4ssc	4sss
31	$[Pmn2_{1}]$			4ccc	4ccs
				−4css	4csc
32	2			4ccc	4ccs
				−4ssc	−4sss
33	$[Pna2_{1}]$		;	4ccc	4ccs
			;	−4scs	4scc
			;	−4ssc	−4sss
			;	−4css	4csc
34	2			4ccc	4ccs
				−4ssc	−4sss
35	2			8ccc	8ccs
36	$[Cmc2_{1}]$			8ccc	8ccs
				−8css	8csc
37	2			8ccc	8ccs
				−8ssc	−8sss
38	2			8ccc	8ccs
39	2			8ccc	8ccs
				−8ssc	−8sss
40	2			8ccc	8ccs
				−8ssc	−8sss
41	2			8ccc	8ccs
				−8ssc	−8sss
42	2			16ccc	16ccs
43	2			16ccc	16ccs
				8(ccc − ssc −ccs − sss)	8(ccs − sss ccc ssc)
				−16ssc	−16sss
				8(ccc − ssc ccs sss)	8(ccs − sss − ccc − ssc)
44	2			8ccc	8ccs
45	2			8ccc	8ccs
				−8ssc	−8sss
46	2			8ccc	8ccs
				−8ssc	−8sss
47				8ccc	0
48		(1)		8ccc	0
				0	−8sss
48		(2)	;	8ccc	0
			;	−8ssc	0
			;	−8css	0
			;	−8scs	0
49				8ccc	0
				−8ssc	0
50		(1)		8ccc	0
				0	−8sss
50		(2)	;	8ccc	0
			;	−8scs	0
			;	−8css	0
			;	−8ssc	0
51				8ccc	0
				−8scs	0
52			;	8ccc	0
			;	−8ssc	0
			;	−8css	0
			;	−8scs	0
53				8ccc	0
				−8css	0
54			;	8ccc	0
			;	−8ssc	0
			;	−8scs	0
			;	−8css	0
55				8ccc	0
				−8ssc	0
56			;	8ccc	0
			;	−8ssc	0
			;	−8css	0
			;	−8scs	0
57			;	8ccc	0
			;	−8css	0
			;	−8ssc	0
			;	−8scs	0
58				8ccc	0
				−8ssc	0
59		(1)		8ccc	0
				0	8ccs
59		(2)	;	8ccc	0
			;	−8css	0
			;	−8scs	0
			;	−8ssc	0
60			;	8ccc	0
			;	−8css	0
			;	−8scs	0
			;	−8ssc	0
61			;	8ccc	0
			;	−8css	0
			;	−8scs	0
			;	−8ssc	0
62			;	8ccc	0
			;	−8ssc	0
			;	−8scs	0
			;	−8css	0
63				16ccc	0
				−16css	0
64				16ccc	0
				−16css	0
65				16ccc	0
66				16ccc	0
				−16ssc	0
67				16ccc	0
				−16css	0
68		(1)		16ccc	0
				0	−16sss
68		(2)	;	16ccc	0
			;	−16ssc	0
			;	−16scs	0
			;	−16css	0
69				32ccc	0
70		(1)		32ccc	0
				16(ccc − sss)	A
				0	−32sss
				16(ccc sss)
70		(2)	; ;	32ccc	0
			; ;	−32ssc	0
			; ;	−32css	0
			; ;	−32scs	0
			; ;	− 16(ccc ssc scs css)	0
			; ;	16(ccc ssc − scs − css)	0
			; ;	16(ccc − ssc − scs css)	0
			; ;	16(ccc − ssc scs − css)	0
71				16ccc	0
72				16ccc	0
				−16ssc	0
73			;	16ccc	0
			;	−16scs	0
			;	−16ssc	0
			;	−16css	0
74				16ccc	0
				−16css	0

Table A1.4.3.5| top | pdf |
Tetragonal space groups

The symbols appearing in this table are based on the factorization of the scalar product appearing in equations (1.4.2.19) and (1.4.2.20) into its plane-group and unique-axis components. The symbols are $[\eqalignno{{\rm P}({\rm pq}) &={\rm p}(2\pi{}hx){\rm q}(2\pi{}ky) + {\rm p}(2\pi{}hy){\rm q}(2\pi{}kx)&\cr {\rm M}({\rm pq}) &={\rm p}(2\pi{}hx){\rm q}(2\pi{}ky)-{\rm p}(2\pi{}hy){\rm q}(2\pi{}kx),&({\rm A}1.4.3.3)\cr}]$ where p and q can each be a sine or a cosine.

Explicit trigonometric functions given in the table follow the convention $[{\rm c}(u)=\cos(2\pi{}u)\quad{\rm s}(u)=\sin(2\pi{}u).]$ Conditions for vanishing symbols: $[\displaylines{{\rm P}({\rm ss})={\rm M}({\rm ss})=0\;\;{\rm if\;\;}h=0{\;\; \rm or \;\;}k=0,\cr{\rm P}({\rm sc})={\rm M}({\rm sc})=0{\rm \;\;if\;\;}h=0,\cr{\rm P}({\rm cs})={\rm M}({\rm cs})=0{\rm \;\;if\;\;}k=0,\cr{\rm M}({\rm cc})={\rm M}({\rm ss})=0\;\;{\rm if\;\;}h=k{\;\; \rm or \;\;}h=-k,\cr}]$ and any explicit sine function vanishes if all the indices (h and k, or l) appearing in its argument are zero.

[P4] [No. 75]

	A	B
All	2[P(cc) − M(ss)]c()	2[P(cc) − M(ss)]s()

$[P4_{1}]$ [No. 76] (enantiomorphous to $[P4_{3}]$ [No. 78])

l	A	B
4n	2[P(cc) − M(ss)]c()	2[P(cc) − M(ss)]s()
4n 1	−2[s()s() − s()c()]	2[s()c() s()s()]
4n 2	2[M(cc) − P(ss)]c()	2[M(cc) − P(ss)]s()
4n 3	−2[s()s() s()c()]	2[s()c() − s()s()]

$[P4_{2}]$ [No. 77]

l	A	B
2n	2[P(cc) − M(ss)]c()	2[P(cc) − M(ss)]s()
2n 1	2[M(cc) − P(ss)]c()	2[M(cc) − P(ss)]s()

$[P4_{3}]$ [No. 78] (enantiomorphous to $[P4_{1}]$ [No. 76])

l	A	B
4n	2[P(cc) − M(ss)]c()	2[P(cc) − M(ss)]s()
4n 1	−2[s()s() s()c()]	2[s()c() − s()s()]
4n 2	2[M(cc) − P(ss)]c()	2[M(cc) − P(ss)]s()
4n 3	−2[s()s()− s()c()]	2[s()c() s()s()]

I4 [No. 79]

	A	B
All	4[P(cc) − M(ss)]c()	4[P(cc) − M(ss)]s()

$[I4_{1}]$ [No. 80]

	A	B
4n	4[P(cc) − M(ss)]c()	4[P(cc) − M(ss)]s()
4n 1	4[c()c() c()s()]	4[c()s() − c()c()]
4n 2	4[M(cc) − P(ss)]c()	4[M(cc) − P(ss)]s()
4n 3	4[c()c() − c()s()]	4[c()s() c()c()]

$[P\overline{4}]$ [No. 81]

	A	B
All	2[P(cc) − M(ss)]c()	2[M(cc) − P(ss)]s()

$[I\overline{4}]$ [No. 82]

	A	B
All	4[P(cc) − M(ss)]c()	4[M(cc) − P(ss)]s()

[P4/m] [No. 83]

	A	B
All	4[P(cc) − M(ss)]c()	0

$[P4_{2}/m]$ [No. 84] (B = 0 for all [h,k,l] )

l	A
2n	4[P(cc) − M(ss)]c()
2n 1	4[M(cc) − P(ss)]c()

[P4/n] [No. 85, Origin 1]

	A	B
2n	4[P(cc) − M(ss)]c()	0
2n 1	0	4[M(cc) − P(ss)]s()

[P4/n] [No. 85, Origin 2] (B = 0 for all [h,k,l] )

h	k	A
2n	2n	4[P(cc) − M(ss)]c()
2n	2n 1	−4[P(cs) M(sc)]s()
2n 1	2n	−4[M(cs) P(sc)]s()
2n 1	2n 1	4[M(cc) − P(ss)]c()

$[P4_{2}/n]$ [No. 86, Origin 1]

	A	B
2n	4[P(cc) − M(ss)]c()	0
2n 1	0	4[M(cc) − P(ss)]s()

$[P4_{2}/n]$ [No. 86, Origin 2] (B = 0 for all [h,k,l] )

			A
2n	2n	2n	4[P(cc) − M(ss)]c()
2n	2n 1	2n 1	4[M(cc) − P(ss)]c()
2n 1	2n 1	2n	−4[M(cs) P(sc)]s()
2n 1	2n	2n 1	−4[P(cs) M(sc)]s()

[I4/m] [No. 87]

	A	B
All	8[P(cc) − M(ss)]c()	0

$[I4_{1}/a]$ [No. 88, Origin 1]

	A	B
4n	8[P(cc) − M(ss)]c()	0
4n 1	4[P(cc) − M(ss)]c() [M(cc) − P(ss)]s()	A
4n 2	0	8[M(cc) − P(ss)]s()
4n 3	4[P(cc) − M(ss)]c() − [M(cc) − P(ss)]s()

$[I4_{1}/a]$ [No. 88, Origin 2] (B = 0 for all [h,k,l] )

h	k		A
2n	2n	4n	8[P(cc) − M(ss)]c()
2n	2n 1	4n	−8[s()s() − c()c()]
2n 1	2n	4n	8[c()c() − s()s()]
2n 1	2n 1	4n	−8[M(cs) P(sc)]s()
2n	2n	4n 2	8[M(cc) − P(ss)]c()
2n	2n 1	4n 2	−8[s()s() c()c()]
2n 1	2n	4n 2	8[c()c() s()s()]
2n 1	2n 1	4n 2	−8[P(cs) M(sc)]s()

[P422] [No. 89]

	A	B
All	4P(cc)c()	−4M(ss)s()

$[P42_{1}2]$ [No. 90]

	A	B
2n	4P(cc)c()	−4M(ss)s()
2n 1	−4P(ss)c()	4M(cc)s()

$[P4_{1}22]$ [No. 91] (enantiomorphous to $[P4_{3}22]$ [No. 95])

l	A	B
4n	4P(cc)c()	−4M(ss)s()
4n 1	−4[s()c()s() − c()s()c()]	4[c()s()c() − s()c()s()]
4n 2	4M(cc)c()	−4P(ss)s()
4n 3	−4[s()c()s() c()s()c()]	4[c()s()c() s()c()s()]

$[P4_{1}2_{1}2]$ [No. 92] (enantiomorphous to $[P4_{3}2_{1}2]$ [No. 96])

	A	B
4n	4P(cc)c()	−4M(ss)s()
4n 1	2{[P(sc) − P(cs)]c() − [M(cs) − M(sc)]s()}	2{[P(sc) P(cs)]c() [M(cs) − M(sc)]s()}
4n 2	−4P(ss)c()	4M(cc)s()
4n 3	−2{[P(sc) − P(cs)]c() [M(cs) M(sc)]s()}	2{[P(sc) P(cs)]c() − [M(cs) − M(sc)s()}

$[P4_{2}22]$ [No. 93]

l	A	B
2n	4P(cc)c()	−4M(ss)s()
2n 1	4M(cc)c()	−4P(ss)s()

$[P4_{2}2_{1}2]$ [No. 94]

	A	B
2n	4P(cc)c()	−4M(ss)s()
2n 1	−4P(ss)c()	4M(cc)s()

$[P4_{3}22]$ [No. 95] (enantiomorphous to $[P4_{1}22]$ [No. 91])

l	A	B
4n	4P(cc)c()	−4M(ss)s()
4n 1	−4[s()c()s() c()s()c()]	4[c()s()c() s()c()c()]
4n 2	4M(cc)c()	−4P(ss)s()
4n 3	−4[s()c()s() − c()s()c()]	4[c()s()c() − s()c()c()]

$[P4_{3}2_{1}2]$ [No. 96] (enantiomorphous to $[P4_{1}2_{1}2]$ [No. 92])

	A	B
4n	4P(cc)c()	−4M(ss)s()
4n 1	−2{[P(sc) − P(cs)]c() [M(cs) M(sc)]s()}	2{[P(sc) P(cs)]c() − [M(cs) − M(sc)]s()}
4n 2	−4P(ss)c()	4M(cc)s()
4n 3	2{[P(sc) − P(cs)]c() − [M(cs) M(sc)]s()}	2{[P(sc) P(cs)]c() [M(cs) − M(sc)]s()}

[I422] [No. 97]

	A	B
All	8P(cc)c()	−8M(ss)s()

$[I4_{1}22]$ [No. 98]

	A	B
4n	8P(cc)c()	−8M(ss)s()
4n 1	4{[P(cc) − P(ss)]c() [M(cc) M(ss)]s()}	4{[P(cc) P(ss)]c() [M(cc) − M(ss)]s()}
4n 2	−8P(ss)c()	8M(cc)s()
4n 3	4{[P(cc) − P(ss)]c() − [M(cc) M(ss)]s()}	−4{[P(cc) P(ss)]c() − [M(cc) − M(ss)]s()}

[P4mm] [No. 99]

	A	B
All	4P(cc)c()	4P(cc)s()

[P4bm] [No. 100]

	A	B
2n	4P(cc)c()	4P(cc)s()
2n 1	−4M(ss)c()	−4M(ss)s()

$[P4_{2}cm]$ [No. 101]

l	A	B
2n	4P(cc)c()	4P(cc)s()
2n 1	−4P(ss)c()	−4P(ss)s()

$[P4_{2}nm]$ [No. 102]

	A	B
2n	4P(cc)c()	4P(cc)s()
2n 1	−4P(ss)c()	−4P(ss)s()

[P4cc] [No. 103]

l	A	B
2n	4P(cc)c()	4P(cc)s()
2n 1	−4M(ss)c()	−4M(ss)s()

P [4nc] [No. 104]

	A	B
2n	4P(cc)c()	4P(cc)s()
2n 1	−4M(ss)c()	−4M(ss)s()

$[P4_{2}mc]$ [No. 105]

l	A	B
2n	4P(cc)c()	4P(cc)s()
2n 1	4M(cc)c()	4M(cc)s()

$[P4_{2}bc]$ [No. 106]

	l	A	B
2n	2n	4P(cc)c()	4P(cc)s()
2n 1	2n	−4M(ss)c()	−4M(ss)s()
2n	2n 1	4M(cc)c()	4M(cc)s()
2n 1	2n 1	−4P(ss)c()	−4P(ss)s()

[I4mm] [No. 107]

	A	B
All	8P(cc)c()	8P(cc)s()

[I4cm] [No. 108]

l	A	B
2n	8P(cc)c()	8P(cc)s()
2n 1	−8M(ss)c()	−8M(ss)s()

$[I4_{1}md]$ [No. 109]

	A	B
4n	8P(cc)c()	8P(cc)s()
4n 1	8[c()c()c() − c()c()s()]	8[c()c()s() c()c()c()]
4n 2	8M(cc)c()	8M(cc)s()
4n 3	8[c()c()c() c()c()s()]	8[c()c()s() − c()c()c()]

$[I4_{1}cd]$ [No. 110]

	A	B
4n	8P(cc)c()	8P(cc)s()
4n 1	−8[s()s()c() s()s()s()]	−8[s()s()s() − s()s()c()]
4n 2	8M(cc)c()	8M(cc)s()
4n 3	−8[s()s()c() − s()s()s()]	−8[s()s()s() s()s()c()]

$[P\overline{4}2m]$ [No. 111]

	A	B
All	4P(cc)c()	−4P(ss)s()

$[P\overline{4}2c]$ [No. 112]

l	A	B
2n	4P(cc)c()	−4P(ss)s()
2n 1	−4M(ss)c()	4M(cc)s()

$[P\overline{4}2_{1}m]$ [No. 113]

	A	B
2n	4P(cc)c()	−4P(ss)s()
2n 1	−4M(ss)c()	4M(cc)s()

$[P\overline{4}2_{1}c]$ [No. 114]

	A	B
2n	4P(cc)c()	−4P(ss)s()
2n 1	−4M(ss)c()	4M(cc)s()

$[P\overline{4}m2]$ [No. 115]

	A	B
All	4P(cc)c()	4M(cc)s()

$[P\overline{4}c2]$ [No. 116]

l	A	B
2n	4P(cc)c()	4M(cc)s()
2n 1	−4M(ss)c()	−4P(ss)s()

$[P\overline{4}b2]$ [No. 117]

	A	B
2n	4P(cc)c()	4M(cc)s()
2n 1	−4M(ss)c()	−4P(ss)s()

$[P\overline{4}n2]$ [No. 118]

	A	B
2n	4P(cc)c()	4M(cc)s()
2n 1	−4M(ss)c()	−4P(ss)s()

$[I\overline{4}m2]$ [No. 119]

	A	B
All	8P(cc)c()	8M(cc)s()

$[I\overline{4}c2]$ [No. 120]

l	A	B
2n	8P(cc)c()	8M(cc)s()
2n 1	−8M(ss)c()	−8P(ss)s()

$[I\overline{4}2m]$ [No. 121]

	A	B
All	8P(cc)c()	−8P(ss)s()

$[I\overline{4}2d]$ [No. 122]

	A	B
4n	8P(cc)c()	−8P(ss)s()
4n 1	4{[P(cc) − M(ss)]c() − [M(cc) P(ss)]s()}	−4{[P(cc) M(ss)]c() − [M(cc) − P(ss)]s()}
4n 2	−8M(ss)c()	8M(cc)s()
4n 3	4{[P(cc) − M(ss)]c() [M(cc) P(ss)]s()}	4{[P(cc) M(ss)]c() [M(cc) − P(ss)]s()}

[P4/mmm] [No. 123]

	A	B
All	8P(cc)c()	0

[P4/mcc] [No. 124] (B = 0 for all [h,k,l] )

l	A
2n	8P(cc)c()
2n 1	−8M(ss)c()

[P4/nbm] [No. 125, Origin 1]

	A	B
2n	8P(cc)c()	0
2n 1	0	−8M(ss)s()

[P4/nbm] [No. 125, Origin 2] (B = 0 for all [h,k,l] )

h	k	A
2n	2n	8P(cc)c()
2n	2n 1	−8M(sc)s()
2n 1	2n	−8M(cs)s()
2n 1	2n 1	−8P(ss)c()

[P4/nnc] [No. 126, Origin 1]

	A	B
2n	8P(cc)c()	0
2n 1	0	−8M(ss)s()

[P4/nnc] [No. 126, Origin 2] (B = 0 for all [h,k,l] )

h	k	l	A
2n	2n	2n	8P(cc)c()
2n	2n	2n 1	−8M(ss)c()
2n	2n 1	2n	−8M(sc)s()
2n	2n 1	2n 1	−8P(cs)s()
2n 1	2n	2n	−8M(cs)s()
2n 1	2n	2n 1	−8P(sc)s()
2n 1	2n 1	2n	−8P(ss)c()
2n 1	2n 1	2n 1	8M(cc)c()

[P4/mbm] [No. 127] (B = 0 for all [h,k,l] )

	A
2n	8P(cc)c()
2n 1	−8M(ss)c()

[P4/mnc] [No. 128] (B = 0 for all [h,k,l] )

	A
2n	8P(cc)c()
2n 1	−8M(ss)c()

[P4/nmm] [No. 129, Origin 1]

	A	B
2n	8P(cc)c()	0
2n 1	0	8M(cc)s()

[P4/nmm] [No. 129, Origin 2] (B = 0 for all [h,k,l] )

h	k	A
2n	2n	8P(cc)c()
2n	2n 1	−8P(cs)s()
2n 1	2n	−8P(sc)s()
2n 1	2n 1	−8P(ss)c()

[P4/ncc] [No. 130, Origin 1]

	l	A	B
2n	2n	8P(cc)c()	0
2n	2n 1	−8M(ss)c()	0
2n 1	2n	0	8M(cc)s()
2n 1	2n 1	0	−8P(ss)s()

[P4/ncc] [No. 130, Origin 2] (B = 0 for all [h,k,l] )

h	k	l	A
2n	2n	2n	8P(cc)c()
2n	2n	2n 1	−8M(ss)c()
2n	2n 1	2n	−8P(cs)s()
2n	2n 1	2n 1	−8M(sc)s()
2n 1	2n	2n	−8P(sc)s()
2n 1	2n	2n 1	−8M(cs)s()
2n 1	2n 1	2n	−8P(ss)c()
2n 1	2n 1	2n 1	8M(cc)c()

$[P4_{2}/mmc]$ [No. 131] (B = 0 for all [h,k,l] )

l	A
2n	8P(cc)c()
2n 1	8M(cc)c()

$[P4_{2}/mcm]$ [No. 132] (B = 0 for all [h,k,l] )

l	A
2n	8P(cc)c()
2n 1	−8P(ss)c()

$[P4_{2}/nbc]$ [No. 133, Origin 1]

	l	A	B
2n	2n	8P(cc)c()	0
2n	2n 1	−8M(ss)c()	0
2n 1	2n	0	−8P(ss)s()
2n 1	2n 1	0	8M(cc)s()

$[P4_{2}/nbc]$ [No. 133, Origin 2] (B = 0 for all [h,k,l] )

h	k	l	A
2n	2n	2n	8P(cc)c()
2n	2n	2n 1	8M(cc)c()
2n	2n 1	2n	−8M(sc)s()
2n	2n 1	2n 1	−8P(sc)s()
2n 1	2n	2n	−8M(cs)s()
2n 1	2n	2n 1	−8P(cs)s()
2n 1	2n 1	2n	−8P(ss)c()
2n 1	2n 1	2n 1	−8M(ss)c()

$[P4_{2}/nnm]$ [No. 134, Origin 1]

	A	B
2n	8P(cc)c()	0
2n 1	0	−8P(ss)s()

$[P4_{2}/nnm]$ [No. 134, Origin 2] (B = 0 for all [h,k,l] )

			A
2n	2n	2n	8P(cc)c()
2n	2n 1	2n 1	−8P(ss)c()
2n 1	2n 1	2n	−8M(sc)s()
2n 1	2n	2n 1	−8M(cs)s()

$[P4_{2}/mbc]$ [No. 135] (B = 0 for all [h,k,l] )

	l	A
2n	2n	8P(cc)c()
2n	2n 1	8M(cc)c()
2n 1	2n	−8M(ss)c()
2n 1	2n 1	−8P(ss)c()

$[P4_{2}/mnm]$ [No. 136] (B = 0 for all [h,k,l] )

	A
2n	8P(cc)c()
2n 1	−8P(ss)c()

$[P4_{2}/nmc]$ [No. 137, Origin 1]

	A	B
2n	8P(cc)c()	0
2n 1	0	8M(cc)s()

$[P4_{2}/nmc]$ [No. 137, Origin 2] (B = 0 for all [h,k,l] )

h	k	l	A
2n	2n	2n	8P(cc)c()
2n	2n	2n 1	8M(cc)c()
2n	2n 1	2n	−8P(cs)s()
2n	2n 1	2n 1	−8M(cs)s()
2n 1	2n	2n	−8P(sc)s()
2n 1	2n	2n 1	−8M(sc)s()
2n 1	2n 1	2n	−8P(ss)c()
2n 1	2n 1	2n 1	−8M(ss)c()

$[P4_{2}/ncm]$ [No. 138, Origin 1]

	l	A	B
2n	2n	8P(cc)c()	0
2n 1	2n 1	−8M(ss)c()	0
2n 1	2n	0	8M(cc)s()
2n	2n 1	0	−8P(ss)s()

$[P4_{2}/ncm]$ [No. 138, Origin 2] (B = 0 for all [h,k,l] )

			A
2n	2n	2n	8P(cc)c()
2n	2n 1	2n 1	−8P(ss)c()
2n 1	2n 1	2n	−8P(cs)s()
2n 1	2n	2n 1	−8P(sc)s()

[I4/mmm] [No. 139]

	A	B
All	16P(cc)c()	0

[I4/mcm] [No. 140] (B = 0 for all [h,k,l] )

l	A
2n	16P(cc)c()
2n 1	−16M(ss)c()

$[I4_{1}/amd]$ [No. 141, Origin 1]

	A	B
4n	16P(cc)c()	0
4n 1	8[P(cc)c() − M(cc)s()]
4n 2	0	16M(cc)s()
4n 3	8[P(cc)c() M(cc)s()]	A

$[I4_{1}/amd]$ [No. 141, Origin 2] (B = 0 for all [h,k,l] )

h	k		A
2n	2n	4n	16P(cc)c()
2n	2n 1	4n	−16[c()s()s() c()c()c()]
2n 1	2n	4n	16[c()c()c() c()s()s()]
2n 1	2n 1	4n	−16[c()s()s() c()s()s()]
2n	2n	4n 2	16M(cc)c()
2n	2n 1	4n 2	−16[c()s()s() − c()c()c()]
2n 1	2n	4n 2	16[c()c()c() − c()s()s()]
2n 1	2n 1	4n 2	−16[c()s()s() − c()s()s()]

$[I4_{1}/acd]$ [No. 142, Origin 1]

	A	B
4n	16P(cc)c()	0
4n 1	−8[M(ss)c() − P(ss)s()]
4n 2	0	16M(cc)s()
4n 3	−8[M(ss)c() P(ss)s()]	A

$[I4_{1}/acd]$ [No. 142, Origin 2] (B = 0 for all [h,k,l] )

h	k		A
2n	2n	4n	16P(cc)c()
2n	2n 1	4n	−16[s()c()s() s()s()c()]
2n 1	2n	4n	−16[s()s()c() s()c()s()]
2n 1	2n 1	4n	−16[c()s()s() c()s()s()]
2n	2n	4n 2	16M(cc)c()
2n	2n 1	4n 2	−16[s()c()s() − s()s()c()]
2n 1	2n	4n 2	−16[s()s()c() − s()c()s()]
2n 1	2n 1	4n 2	−16[c()s()s() − c()s()s()]

Table A1.4.3.6| top | pdf |
Trigonal and hexagonal space groups

The table lists the expressions for A and B for the space groups belonging to the hexagonal family. For the space groups that are referred to hexagonal axes the expressions are given in terms of symbols related to the decomposition of the scalar products into their plane-group and unique-axis components [cf. equations (1.4.3.10)–(1.4.3.12) ]. The symbols for the seven rhombohedral space groups in their rhombohedral-axes representation are the same as those used for the cubic space groups [cf. equations (1.4.3.4) and (1.4.3.5), and the notes at the start of Table A1.4.3.7]. Factors of the forms $[\cos(2\pi{}x)]$ and $[\sin(2\pi{}x)]$ are abbreviated by c(x) and s(x), respectively. All the symbols used in this table are repeated below. Most expressions are given in terms of $[\eqalignno{{\rm C}(hki) &= {\rm c}(p_1)+{\rm c}(p_2)+{\rm c}(p_3),\cr{\rm C}(khi)&={\rm c}(q_1)+{\rm c}(q_2)+{\rm c}(q_3)\;\; {\rm and}&\cr{\rm S}(hki) &= {\rm s}(p_1)+{\rm s}(p_2)+{\rm s}(p_3),\cr {\rm S}(khi)&={\rm s}(q_1)+{\rm s}(q_2)+{\rm s}(q_3),&({\rm A}1.4.3.4)\cr}]$ where $[\eqalignno{p_1&=hx+ky,\;\;p_2=kx+iy,\;\;p_3=ix+hy,&\cr q_1&=kx+hy,\;\;q_2=hx+iy,\;\;q_3=ix+ky,&({\rm A}1.4.3.5)\cr}]$ and the abbreviations $[\eqalignno{{\rm PH}({\rm cc})&={\rm C}(hki)+{\rm C}(khi),\cr{\rm PH}({\rm ss})&={\rm S}(hki)+{\rm S}(khi),&\cr {\rm MH}({\rm cc})&={\rm C}(hki)-{\rm C}(khi)\;\; {\rm and}\cr{\rm MH}({\rm ss})&={\rm S}(hki)-{\rm S}(khi).&({\rm A}1.4.3.6)\cr}]$ In addition, the following abbreviations are employed for some space groups: $[u_1=lz,\;\;u_2=lz+\textstyle{1 \over 3}\;\;{\rm and}\;\;u_3=lz-{1 \over 3}.]$ Conditons for vanishing symbols: $[\displaylines{{\rm S}(hki)={\rm S}(khi)=0\;\;{\rm if}\;\;h=k=0,\cr{\rm PH}({\rm ss})=0\;\;{\rm if}\;\;h=-k\;\;({\rm or}\;\;k=-i\;\;{\rm or}\;\;i=-h),\cr{\rm MH}({\rm cc})=0\;\;{\rm if}\;\;|h|=|k|\;\;({\rm or}\;\;|k|=|i|\;\;{\rm or}\;\;|i|=|h|)\cr}]$ and any explicit sine function vanishes if all the indices (h and k, or l) appearing in its argument are zero.

[P3] [No. 143]

	A	B
All	C()c() − S()s()	C()s() S()c()

$[P3_{1}]$ [No. 144] (enantiomorphous to $[P3_{2}]$ [No. 145])

l	A	B
3n	as for [No. 143]
3n 1	c( $[p_{1}+u_{1}]$ ) c( $[p_{2}+u_{2}]$ ) c( $[p_{3}+u_{3}]$ )	s( $[p_{1}+u_{1}]$ ) s( $[p_{2}+u_{2}]$ ) s( $[p_{3}+u_{3}]$ )
3n 2	c( $[p_{1}+u_{1}]$ ) c( $[p_{2}+u_{3}]$ ) c( $[p_{3}+u_{2}]$ )	s( $[p_{1}+u_{1}]$ ) s( $[p_{2}+u_{3}]$ ) s( $[p_{3}+u_{2}]$ )

$[P3_{2}]$ [No. 145] (enantiomorphous to $[P3_{1}]$ [No. 144])

l
3n	as for [No. 143]
3n 1	as for l = 3n 2 in $[P3_{1}]$ [No. 144]
3n 2	as for l = 3n 1 in $[P3_{1}]$ [No. 144]

[R3] [No. 146] (rhombohedral axes)

	A	B
All	c() c() c()	s() s() s()

[R3] [No. 146] (hexagonal axes)

	A	B
All	3[C()c() − S()s()]	3[C()s() S()c()]

$[P\overline{3}]$ [No. 147]

	A	B
All	2[C()c() − S()s()]	0

$[R\overline{3}]$ [No. 148] (rhombohedral axes)

	A	B
All	2[c() c() c()]	0

$[R\overline{3}]$ [No. 148] (hexagonal axes)

	A	B
All	6[C()c() − S()s()]	0

[P312] [No. 149]

	A	B
All	PH(cc)c() − PH(ss)s()	MH(cc)s() MH(ss)c()

[P321] [No. 150]

	A	B
All	PH(cc)c() − MH(ss)s()	PH(ss)c() MH(cc)s()

$[P3_{1}12]$ [No. 151] (enantiomorphous to $[P3_{2}12]$ [No. 153])

l	A	B
3n	as for [No. 149]
3n 1	$[\hbox{c}(p_{1}+u_{1}) + \hbox{c}(p_{2}+u_{2}) + \hbox{c}(p_{3}+u_{3}) + \hbox{c}(q_{1}+u_{2}) ]$ $[+\;\hbox{c}(q_{2}+u_{3}) +\hbox{c}(q_{3}+u_{1})]]$	$[\hbox{s}(p_{1}+u_{1}) + \hbox{s}(p_{2}+u_{2}) + \hbox{s}(p_{3}+u_{3}) -\hbox{s}(q_{1}+u_{2}) ]$ $[-\;\hbox{s}(q_{2}+u_{3}) - \hbox{s}(q_{3}+u_{1})]$
3n 2	$[\hbox{c}(p_{1}+u_{1}) + \hbox{c}(p_{2}+u_{3}) + \hbox{c}(p_{3}+u_{2}) + \hbox{c}(q_{1}+u_{3}) ]$ $[+\; \hbox{c}(q_{2}+u_{2}) + \hbox{c}(q_{3}+u_{1})]$	$[\hbox{s}(p_{1}+u_{1}) + \hbox{s}(p_{2}+u_{3}) + \hbox{s}(p_{3}+u_{2}) - \hbox{s}(q_{1}+u_{3})]$ $[ -\;\hbox{s}(q_{2}+u_{2}) - \hbox{s}(q_{3}+u_{1})]$

$[P3_{1}21]$ [No. 152] (enantiomorphous to $[P3_{2}21]$ [No. 154])

l	A	B
3n	as for [No. 150]
3n 1	$[\hbox{c}(p_{1}+u_{1}) + \hbox{c}(p_{2}+u_{2}) + \hbox{c}(p_{3}+u_{3}) + \hbox{c}(q_{1} - u_{1})]$ $[ +\; \hbox{c}(q_{2}-u_{2}) +\hbox{c}(q_{3} - u_{3})]$	s( $[p_{1}+u_{1}]$ ) s( $[p_{2}+u_{2}]$ ) s( $[p_{3}+u_{3}]$ ) s( $[q_{1}]$ − $[u_{1}]$ ) $[+\hbox{ s}(q_{2}-u_{2}]$ ) s( $[q_{3}]$ − $[u_{3}]$ )
3n 2	$[\hbox{c}(p_{1}+u_{1}) + \hbox{c}(p_{2}+u_{3}) + \hbox{c}(p_{3}+u_{2}) + \hbox{c}(q_{1} - u_{1})]$ $[ +\hbox{ c}(q_{2} -\; u_{3}) + \hbox{c}(q_{3} -u_{2})]$	$[\hbox{s}(p_{1}+u_{1}) + \hbox{s}(p_{2}+u_{3}) + \hbox{s}(p_{3}+u_{2}) + \hbox{s}(q_{1} - u_{1}) ]$ $[+\; \hbox{s}(q_{2} - u_{3}) + \hbox{s}(q_{3} - u_{2})]$

$[P3_{2}12]$ [No. 153] (enantiomorphous to $[P3_{1}12]$ [No. 151])

l
3n	as for [No. 149]
3n 1	as for l = 3n 2 in $[P3_{1}12]$ [No. 151]
3n 2	as for l = 3n 1 in $[P3_{1}12]$ [No. 151]

$[P3_{2}21]$ [No. 154] (enantiomorphous to $[P3_{1}21]$ [No. 152])

l
3n	as for [No. 150]
3n 1	as for l = 3n 2 in $[P3_{1}21]$ [No. 152]
3n 2	as for l = 3n 1 in $[P3_{1}21]$ [No. 152]

[R32] [No. 155] (rhombohedral axes)

	A	B
All	Eccc − Ecss − Escs − Essc Occc − Ocss − Oscs − Ossc	Escc Ecsc Eccs − Esss − Oscc − Ocsc − Occs Osss

[R32] [No. 155] (hexagonal axes)

	A	B
All	3[PH(cc)c() − MH(ss)s()]	3[PH(ss)c() MH(cc)s()]

[P3m1] [No. 156]

	A	B
All	PH(cc)c() − MH(ss)s()	PH(cc)s() MH(ss)c()

[P31m] [No. 157]

	A	B
All	PH(cc)c() − PH(ss)s()	PH(cc)s() PH(ss)c()

[P3c1] [No. 158]

l	A	B
2n	PH(cc)c() − MH(ss)s()	PH(cc)s() MH(ss)c()
2n 1	MH(cc)c() − PH(ss)s()	PH(ss)c() MH(cc)s()

[P31c] [No. 159]

l	A	B
2n	PH(cc)c() − PH(ss)s()	PH(cc)s() PH(ss)c()
2n 1	MH(cc)c() − MH(ss)s()	MH(cc)s() MH(ss)c()

[R3m] [No. 160] (rhombohedral axes)

	A	B
All	Eccc − Ecss − Escs − Essc Occc − Ocss − Oscs − Ossc	Escc Ecsc Eccs − Esss Oscc Ocsc Occs − Osss

[R3m] [No. 160] (hexagonal axes)

	A	B
All	3[PH(cc)c() − MH(ss)s()]	3[PH(cc)s() MH(ss)c()]

[R3c] [No. 161] (rhombohedral axes)

	A	B
2n	Eccc − Ecss − Escs − Essc Occc − Ocss − Oscs − Ossc	Escc Ecsc Eccs − Esss Oscc Ocsc Occs − Osss
2n 1	Eccc − Ecss − Escs − Essc − Occc Ocss Oscs Ossc	Escc Ecsc Eccs − Esss − Oscc − Ocsc − Occs Osss

[R3c] [No. 161] (hexagonal axes)

l	A	B
2n	3[PH(cc)c() − MH(ss)s()]	3[PH(cc)s() MH(ss)c()]
2n 1	3[MH(cc)c() − PH(ss)s()]	3[PH(ss)c() MH(cc)s()]

$[P\overline{3}1m]$ [No. 162] ( [B=0] for all [h,k,l] )

A
2[PH(cc)c() − PH(ss)s()]

$[P\overline{3}1c]$ [No. 163] ( [B=0] for all [h,k,l] )

l	A
2n	2[PH(cc)c() − PH(ss)s()]
2n 1	2[MH(cc)c() − MH(ss)s()]

$[P\overline{3}m1]$ [No. 164] ( [B=0] for all [h,k,l] )

A
2[PH(cc)c() − MH(ss)s()]

$[P\overline{3}c1]$ [No. 165] ( [B=0] for all [h,k,l] )

l	A
2n	2[PH(cc)c() − MH(ss)s()]
2n 1	2[MH(cc)c() − PH(ss)s()]

$[R\overline{3}m]$ [No. 166] (rhombohedral axes, [B=0] for all [h,k,l] )

A
2(Eccc − Ecss − Escs − Essc Occc − Ocss − Oscs − Ossc)

$[R\overline{3}m]$ [No. 166] (hexagonal axes, [B=0] for all [h,k,l] )

A
6[PH(cc)c() − MH(ss)s()]

$[R\overline{3}c]$ [No. 167] (rhombohedral axes, [B=0] for all [h,k,l] )

	A
2n	2(Eccc − Ecss − Escs − Essc Occc − Ocss − Oscs − Ossc)
2n 1	2(Eccc − Ecss − Escs − Essc − Occc Ocss Oscs Ossc)

$[R\overline{3}c]$ [No. 167] (hexagonal axes, [B=0] for all [h,k,l] )

l	A
2n	6[PH(cc)c() − MH(ss)s()]
2n 1	6[MH(cc)c() − PH(ss)s()]

[P6] [No. 168]

	A	B
All	2C()c()	2C()s()

$[P6_{1}]$ [No. 169] (enantiomorphous to $[P6_{5}]$ [No. 170])

l	A	B
6n	as for [No.168]
6n 1	−2[s( $[p_{1}]$ )s( $[u_{1}]$ ) s( $[p_{2}]$ )s( $[u_{2}]$ ) s( $[p_{3}]$ )s( $[u_{3}]$ )]	2[s( $[p_{1}]$ )c( $[u_{1}]$ ) s( $[p_{2}]$ )c( $[u_{2}]$ ) s( $[p_{3}]$ )c( $[u_{3}]$ )]
6n 2	2[c( $[p_{1}]$ )c( $[u_{1}]$ ) c( $[p_{2}]$ )c( $[u_{3}]$ ) c( $[p_{3}]$ )c( $[u_{2}]$ )]	2[c( $[p_{1}]$ )s( $[u_{1}]$ ) c( $[p_{2}]$ )s( $[u_{3}]$ ) c( $[p_{3}]$ )s( $[u_{2}]$ )]
6n 3	−2S()s()	2S()c()
6n 4	2[c( $[p_{1}]$ )c( $[u_{1}]$ ) c( $[p_{2}]$ )c( $[u_{2}]$ ) c( $[p_{3}]$ )c( $[u_{3}]$ )]	2[c( $[p_{1}]$ )s( $[u_{1}]$ ) c( $[p_{2}]$ )s( $[u_{2}]$ ) c( $[p_{3}]$ )s( $[u_{3}]$ )]
6n 5	−2[s( $[p_{1}]$ )s( $[u_{1}]$ ) s( $[p_{2}]$ )s( $[u_{3}]$ ) s( $[p_{3}]$ )s( $[u_{2}]$ )]	2[s( $[p_{1}]$ )c( $[u_{1}]$ ) s( $[p_{2}]$ )c( $[u_{3}]$ ) s( $[p_{3}]$ )c( $[u_{2}]$ )]

$[P6_{5}]$ [No. 170] (enantiomorphous to $[P6_{1}]$ [No. 169])

l
6n	as for [No. 168]
6n 1	as for l = 6n 5 in $[P6_{1}]$ [No. 169]
6n 2	as for l = 6n 4 in $[P6_{1}]$ [No. 169]
6n 3	as for l = 6n 3 in $[P6_{1}]$ [No. 169]
6n 4	as for l = 6n 2 in $[P6_{1}]$ [No. 169]
6n 5	as for l = 6n 1 in $[P6_{1}]$ [No. 169]

$[P6_{2}]$ [No. 171] (enantiomorphous to $[P6_{4}]$ [No. 172])

l
3n	as for [No. 168]
3n 1	as for l = 6n 2 in $[P6_{1}]$ [No. 169]
3n 2	as for l = 6n 4 in $[P6_{1}]$ [No. 169]

$[P6_{4}]$ [No. 172] (enantiomorphous to $[P6_{2}]$ [No. 171])

l
3n	as for [No. 168]
3n 1	as for l = 6n 4 in $[P6_{1}]$ [No.169]
3n 2	as for l = 6n 2 in $[P6_{1}]$ [No. 169]

$[P6_{3}]$ [No. 173]

l
2n	as for [No. 168]
2n 1	as for l = 6n 3 in $[P6_{1}]$ [No. 169]

$[P\overline{6}]$ [No. 174]

	A	B
All	2C()c()	2S()c()

[P6/m] [No. 175]

	A	B
All	4C()c()	0

$[P6_{3}/m]$ [No. 176]

l	A	B
2n	4C()c()	0
2n 1	−4S()s()	0

[P622] [No. 177]

	A	B
All	2PH(cc)c()	2MH(cc)s()

$[P6_{1}22]$ [No. 178] (enantiomorphous to $[P6_{5}22]$ [No. 179])

l	A	B
6n	as for [No. 177]
6n 1	$[-2[\hbox{s}(p_{1})\hbox{s}(u_{1}) + \hbox{s}(p_{2})\hbox{s}(u_{2}) + \hbox{s}(p_{3})\hbox{s}(u_{3}) - \hbox{s}(q_{1})\hbox{s}(u_{3})]$ $[ -\; \hbox{s}(q_{2})\hbox{s}(u_{1}) - \hbox{s}(q_{3})\hbox{s}(u_{2})]]$	$[2[\hbox{s}(p_{1})\hbox{c}(u_{1}) + \hbox{s}(p_{2})\hbox{c}(u_{2}) + \hbox{s}(p_{3})\hbox{c}(u_{3}) + \hbox{s}(q_{1})\hbox{c}(u_{3}) ]$ $[+\; \hbox{s}(q_{2})\hbox{c}(u_{1}) + \hbox{s}(q_{3})\hbox{c}(u_{2})]]$
6n 2	$[2[\hbox{c}(p_{1})\hbox{c}(u_{1}) + \hbox{c}(p_{2})\hbox{c}(u_{3}) + \hbox{c}(p_{3})\hbox{c}(u_{2}) + \hbox{c}(q_{1})\hbox{c}(u_{2}) ]$ $[+\; \hbox{c}(q_{2})\hbox{c}(u_{1}) + \hbox{c}(q_{3})\hbox{c}(u_{3})]]$	$[2[\hbox{c}(p_{1})\hbox{s}(u_{1}) + \hbox{c}(p_{2})\hbox{s}(u_{3}) + \hbox{c}(p_{3})\hbox{s}(u_{2}) - \hbox{c}(q_{1})\hbox{s}(u_{2}) ]$ $[-\; \hbox{c}(q_{2})\hbox{s}(u_{1}) - \hbox{c}(q_{3})\hbox{s}(u_{3})]]$
6n 3	−2MH(ss)s()	2PH(ss)c()
6n 4	$[2[\hbox{c}(p_{1})\hbox{c}(u_{1}) + \hbox{c}(p_{2})\hbox{c}(u_{2}) + \hbox{c}(p_{3})\hbox{c}(u_{3}) + \hbox{c}(q_{1})\hbox{c}(u_{3}) ]$ $[+\; \hbox{c}(q_{2})\hbox{c}(u_{1}) + \hbox{c}(q_{3})\hbox{c}(u_{2})]]$	$[2[\hbox{c}(p_{1})\hbox{s}(u_{1}) + \hbox{c}(p_{2})\hbox{s}(u_{2}) + \hbox{c}(p_{3})\hbox{s}(u_{3}) - \hbox{c}(q_{1})\hbox{s}(u_{3})]$ $[ -\; \hbox{c}(q_{2})\hbox{s}(u_{1}) - \hbox{c}(q_{3})\hbox{s}(u_{2})]]$
6n 5	$[-2[\hbox{s}(p_{1})\hbox{s}(u_{1}) + \hbox{s}(p_{2})\hbox{s}(u_{3}) + \hbox{s}(p_{3})\hbox{s}(u_{2}) - \hbox{s}(q_{1})\hbox{s}(u_{2})]$ $[ -\; \hbox{s}(q_{2})\hbox{s}(u_{1}) - \hbox{s}(q_{3})\hbox{s}(u_{3})]]$	$[2[\hbox{s}(p_{1})\hbox{c}(u_{1}) + \hbox{s}(p_{2})\hbox{c}(u_{3}) + \hbox{s}(p_{3})\hbox{c}(u_{2}) + \hbox{s}(q_{1})\hbox{c}(u_{2})]$ $[ +\; \hbox{s}(q_{2})\hbox{c}(u_{1}) + \hbox{s}(q_{3})\hbox{c}(u_{3})]]$

$[P6_{5}22]$ [No. 179] (enantiomorphous to $[P6_{1}22]$ [No. 178])

l
6n	as for [No. 177]
6n 1	as for l = 6n 5 in $[P6_{1}22]$ [No. 178]
6n 2	as for l = 6n 4 in $[P6_{1}22]$ [No. 178]
6n 3	as for l = 6n 3 in $[P6_{1}22]$ [No. 178]
6n 4	as for l = 6n 2 in $[P6_{1}22]$ [No. 178]
6n 5	as for l = 6n 1 in $[P6_{1}22]$ [No. 178]

$[P6_{2}22]$ [No. 180] (enantiomorphous to $[P6_{4}22]$ [No. 181])

l
n	as for [No. 177]
3n 1	as for l = 6n 2 in $[P6_{1}22]$ [No. 178]
3n 2	as for l = 6n 4 in $[P6_{1}22]$ [No.178]

$[P6_{4}22]$ [No. 181] (enantiomorphous to $[P6_{2}22]$ [No. 180])

l
3n	as for [No. 177]
3n 1	as for l = 6n 4 in $[P6_{1}22]$ [No. 178]
3n 2	as for l = 6n 2 in $[P6_{1}22]$ [No. 178]

$[P6_{3}22]$ [No. 182]

l
2n	as for [No. 177]
2n 1	as for l = 6n 3 in $[P6_{1}22]$ [No. 178]

[P6mm] [No. 183]

	A	B
All	2PH(cc)c()	2PH(cc)s()

[P6cc] [No. 184]

l	A	B
2n	2PH(cc)c()	2PH(cc)s()
2n 1	2MH(cc)c()	2MH(cc)s()

$[P6_{3}cm]$ [No. 185]

l	A	B
2n	2PH(cc)c()	2PH(cc)s()
2n 1	−2PH(ss)s()	2PH(ss)c()

$[P6_{3}mc]$ [No. 186]

l	A	B
2n	2PH(cc)c()	2PH(cc)s()
2n 1	−2MH(ss)s()	2MH(ss)c()

$[P\overline{6}m2]$ [No. 187]

	A	B
All	2PH(cc)c()	2MH(ss)c()

$[P\overline{6}c2]$ [No. 188]

l	A	B
2n	2PH(cc)c()	2MH(ss)c()
2n 1	−2PH(ss)s()	2MH(cc)s()

$[P\overline{6}2m]$ [No. 189]

	A	B
All	2PH(cc)c()	2PH(ss)c()

$[P\overline{6}2c]$ [No. 190]

l	A	B
2n	2PH(cc)c()	2PH(ss)c()
2n 1	−2MH(ss)s()	2MH(cc)s()

[P6/mmm] [No. 191]

	A	B
All	4PH(cc)c()	0

[P6/mcc] [No. 192] ( [B=0] for all [h,k,l] )

l	A
2n	4PH(cc)c()
2n 1	4MH(cc)c()

$[P6_{3}/mcm]$ [No. 193] ( [B=0] for all [h,k,l] )

l	A
2n	4PH(cc)c()
2n 1	−4PH(ss)s()

$[P6_{3}/mmc]$ [No. 194] ( [B=0] for all [h,k,l] )

l	A
2n	4PH(cc)c()
2n 1	−4MH(ss)s()

Table A1.4.3.7| top | pdf |
Cubic space groups

The symbols appearing in this table are related to the pqr representation used with the orthorhombic space groups as follows: Each of the symbols defined below is a sum of three pqr terms, where the order of hkl is fixed in each of the three terms and that of xyz is permuted.

This table and parts of Table A1.4.3.6 (rhombohedral space groups referred to rhombohedral axes) are given in terms of the following two symbols: $[{{\rm Epqr}={\rm p}(hx){\rm q}(ky){\rm r}(lz)+{\rm p}(hy){\rm q}(kz){\rm r}(lx)+{\rm p}(hz){\rm q}(kx){\rm r}(ly)}\eqno({\rm A}1.4.3.7)]$ and $[{{\rm Opqr}={\rm p}(hx){\rm q}(kz){\rm r}(ly)+{\rm p}(hz){\rm q}(ky){\rm r}(lx)+{\rm p}(hy){\rm q}(kx){\rm r}(lz),}\eqno({\rm A}1.4.3.8)]$ where p, q and r can each be a sine or a cosine, and the factor $[2\pi{}]$ has been absorbed in the abbreviations (see text). As in Tables A1.4.3.1–A1.4.3.6 , cosine and sine are abbreviated by c and s, respectively. The permutation of the coordinates is even in Epqr and odd in Opqr.

Conditions for vanishing symbols:

Epqr = Opqr = 0 if at least one of p, q, r is a sine and the index h, k or l in its argument is zero, $[\displaylines{{\rm Eccc}-{\rm Occc}=0\;\;{\rm if}\;\;|h|=|k|\;\;({\rm or}\;\;|k|=|l|\;\;{\rm or}\;\;|l|=|h|),\cr{\rm Esss}-{\rm Osss}=0\;\;{\rm if}\;\;|h|=|k|\;\;({\rm or}\;\;|k|=|l|\;\;{\rm or}\;\;|l|=|h|),\cr{\rm Ecss}-{\rm Ocss}={\rm Escc}-{\rm Oscc}=0\;\;{\rm if}\;\;|k|=|l|,\cr{\rm Escs}-{\rm Oscs}={\rm Ecsc}-{\rm Ocsc}=0\;\;{\rm if}\;\;|l|=|h|\;\;{\rm and}\cr{\rm Essc}-{\rm Ossc}={\rm Eccs}-{\rm Occs}=0\;\;{\rm if}\;\;|h|=|k|.}]$

[P23] [No. 195]

	A	B
All	4Eccc	−4Esss

[F23] [No. 196]

	A	B
All	16Eccc	−16Esss

[I23] [No. 197]

	A	B
All	8Eccc	−8Esss

$[P2_{1}3]$ [No. 198]

			A	B
2n	2n	2n	4Eccc	−4Esss
2n	2n 1	2n 1	−4Ecss	4Escc
2n 1	2n	2n 1	−4Escs	4Ecsc
2n 1	2n 1	2n	−4Essc	4Eccs

$[I2_{1}3]$ [No. 199]

			A	B
2n	2n	2n	8Eccc	−8Esss
2n 1	2n	2n 1	−8Escs	8Ecsc
2n 1	2n 1	2n	−8Essc	8Eccs
2n	2n 1	2n 1	−8Ecss	8Escc

$[Pm\overline{3}]$ [No. 200]

	A	B
All	8Eccc	0

$[Pn\overline{3}]$ (Origin 1) [No. 201]

	A	B
2n	8Eccc	0
2n 1	0	−8Esss

$[Pn\overline{3}]$ (Origin 2) [No. 201] (B = 0 for all [h,k,l] )

			A
2n	2n	2n	8Eccc
2n	2n 1	2n 1	−8Essc
2n 1	2n	2n 1	−8Ecss
2n 1	2n 1	2n	−8Escs

$[Fm\overline{3}]$ [No. 202]

	A	B
All	32Eccc	0

$[Fd\overline{3}]$ (Origin 1) [No. 203]

	A	B
4n	32Eccc	0
4n 1	16(Eccc − Esss)	A
4n 2	0	−32Esss
4n 3	16(Eccc Esss)

$[Fd\overline{3}]$ (Origin 2) [No. 203] (B = 0 for all [h,k,l] )

			A
4n	4n	4n	32Eccc
4n	4n 2	4n 2	−32Essc
4n 2	4n	4n 2	−32Ecss
4n 2	4n 2	4n	−32Escs
4n 2	4n 2	4n 2	−16(Eccc Ecss Escs Essc)
4n 2	4n	4n	16(Eccc − Ecss − Escs Essc)
4n	4n 2	4n	16(Eccc Ecss − Escs − Essc)
4n	4n	4n 2	16(Eccc − Ecss Escs −Essc)

$[Im\overline{3}]$ [No. 204]

	A	B
All	16Eccc	0

$[Pa\overline{3}]$ [No. 205] (B = 0 for all [h,k,l] )

			A
2n	2n	2n	8Eccc
2n	2n 1	2n 1	−8Ecss
2n 1	2n	2n 1	−8Escs
2n 1	2n 1	2n	−8Essc

$[Ia\overline{3}]$ [No. 206] (B = 0 for all [h,k,l] )

			A
2n	2n	2n	16Eccc
2n	2n 1	2n 1	−16Ecss
2n 1	2n	2n 1	−16Escs
2n 1	2n 1	2n	−16Essc

[P432] [No. 207]

	A	B
All	4(Eccc Occc)	−4(Esss − Osss)

$[P4_{2}32]$ [No. 208]

	A	B
2n	4(Eccc Occc)	−4(Esss − Osss)
2n 1	4(Eccc − Occc)	−4(Esss Osss)

[F432] [No. 209]

	A	B
All	16(Eccc Occc)	−16(Esss − Osss)

$[F4_{1}32]$ [No. 210]

	A	B
4n	16(Eccc Occc)	−16(Esss − Osss)
4n 1	16(Eccc − Osss)	−16(Esss − Occc)
4n 2	16(Eccc − Occc)	−16(Esss Osss)
4n 3	16(Eccc Osss)	−16(Esss Occc)

[I432] [No. 211]

	A	B
All	8(Eccc Occc)	−8(Esss − Osss)

$[P4_{3}32]$ [No. 212] (enantiomorphous to $[P4_{1}32]$ [No. 213])

				A	B
2n	2n	2n	4n	4(Eccc Occc)	−4(Esss − Osss)
2n	2n 1	2n 1	4n	−4(Ecss Oscs)	4(Escc − Ocsc)
2n 1	2n	2n 1	4n	−4(Escs Ossc)	4(Ecsc − Occs)
2n 1	2n 1	2n	4n	−4(Essc Ocss)	4(Eccs − Oscc)
2n	2n	2n	4n 1	4(Eccc − Osss)	−4(Esss − Occc)
2n	2n 1	2n 1	4n 1	−4(Ecss − Ocsc)	4(Escc − Oscs)
2n 1	2n	2n 1	4n 1	−4(Escs − Occs)	4(Ecsc − Ossc)
2n 1	2n 1	2n	4n 1	−4(Essc − Oscc)	4(Eccs − Ocss)
2n	2n	2n	4n 2	4(Eccc − Occc)	−4(Esss Osss)
2n	2n 1	2n 1	4n 2	−4(Ecss − Oscs)	4(Escc Ocsc)
2n 1	2n	2n 1	4n 2	−4(Escs − Ossc)	4(Ecsc Occs)
2n 1	2n 1	2n	4n 2	−4(Essc − Ocss)	4(Eccs Oscc)
2n	2n	2n	4n 3	4(Eccc Osss)	−4(Esss Occc)
2n	2n 1	2n 1	4n 3	−4(Ecss Ocsc)	4(Escc Oscs)
2n 1	2n	2n 1	4n 3	−4(Escs Occs)	4(Ecsc Ossc)
2n 1	2n 1	2n	4n 3	−4(Essc Oscc)	4(Eccs Ocss)

$[P4_{1}32]$ [No. 213] (enantiomorphous to $[P4_{3}32]$ [No. 212])

h	k	l		A	B
2n	2n	2n	4n	4(Eccc Occc)	−4(Esss − Osss)
2n	2n 1	2n 1	4n	−4(Escs Ossc)	4(Ecsc − Occs)
2n 1	2n	2n 1	4n	−4(Essc Ocss)	4(Eccs − Oscc)
2n 1	2n 1	2n	4n	−4(Ecss Oscs)	4(Escc − Ocsc)
2n 1	2n 1	2n 1	4n 1	4(Eccc Osss)	−4(Esss Occc)
2n	2n	2n 1	4n 1	−4(Ecss Ocsc)	4(Escc Oscs)
2n 1	2n	2n	4n 1	−4(Escs Occs)	4(Ecsc Ossc)
2n	2n 1	2n	4n 1	−4(Essc Oscc)	4(Eccs Ocss)
2n	2n	2n	4n 2	4(Eccc − Occc)	−4(Esss Osss)
2n	2n 1	2n 1	4n 2	−4(Escs − Ossc)	4(Ecsc Occs)
2n 1	2n	2n 1	4n 2	−4(Essc − Ocss)	4(Eccs Oscc)
2n 1	2n 1	2n	4n 2	−4(Ecss − Oscs)	4(Escc Ocsc)
2n 1	2n 1	2n 1	4n 3	4(Eccc − Osss)	−4(Esss − Occc)
2n	2n	2n 1	4n 3	−4(Ecss − Ocsc)	4(Escc − Oscs)
2n 1	2n	2n	4n 3	−4(Escs − Occs)	4(Ecsc − Ossc)
2n	2n 1	2n	4n 3	−4(Essc − Oscc)	4(Eccs − Ocss)

$[I4_{1}32]$ [No. 214]

h	k	l		A	B
2n	2n	2n	4n	8(Eccc Occc)	−8(Esss − Osss)
2n	2n 1	2n 1	4n	−8(Escs Ossc)	8(Ecsc − Occs)
2n 1	2n	2n 1	4n	−8(Essc Ocss)	8(Eccs − Oscc)
2n 1	2n 1	2n	4n	−8(Ecss Oscs)	8(Escc − Ocsc)
2n	2n	2n	4n 2	8(Eccc − Occc)	−8(Esss Osss)
2n	2n 1	2n 1	4n 2	−8(Escs − Ossc)	8(Ecsc Occs)
2n 1	2n	2n 1	4n 2	−8(Essc − Ocss)	8(Eccs Oscc)
2n 1	2n 1	2n	4n 2	−8(Ecss − Oscs)	8(Escc Ocsc)

$[P\overline{4}3m]$ [No. 215]

	A	B
All	4(Eccc Occc)	−4(Esss Osss)

$[F\overline{4}3m]$ [No. 216]

	A	B
All	16(Eccc Occc)	−16(Esss Osss)

$[I\overline{4}3m]$ [No. 217]

	A	B
All	8(Eccc Occc)	−8(Esss Osss)

$[P\overline{4}3n]$ [No. 218]

	A	B
2n	4(Eccc Occc)	−4(Esss Osss)
2n 1	4(Eccc − Occc)	−4(Esss − Osss)

$[F\overline{4}3c]$ [No. 219]

	A	B
2n	16(Eccc Occc)	−16(Esss Osss)
2n 1	16(Eccc − Occc)	−16(Esss − Osss)

$[I\overline{4}3d]$ [No. 220]

h	k	l		A	B
2n	2n	2n	4n	8(Eccc Occc)	−8(Esss Osss)
2n	2n 1	2n 1	4n	−8(Escs Ossc)	8(Ecsc Occs)
2n 1	2n	2n 1	4n	−8(Essc Ocss)	8(Eccs Oscc)
2n 1	2n 1	2n	4n	−8(Ecss Oscs)	8(Escc Ocsc)
2n	2n	2n	4n 2	8(Eccc − Occc)	−8(Esss − Osss)
2n	2n 1	2n 1	4n 2	−8(Escs − Ossc)	8(Ecsc − Occs)
2n 1	2n	2n 1	4n 2	−8(Essc − Ocss)	8(Eccs − Oscc)
2n 1	2n 1	2n	4n 2	−8(Ecss − Oscs)	8(Escc −Ocsc)

$[Pm\overline{3}m]$ [No. 221]

	A	B
All	8(Eccc Occc)	0

$[Pn\overline{3}n]$ (Origin 1) [No. 222]

	A	B
2n	8(Eccc Occc)	0
2n 1	0	−8(Esss − Osss)

$[Pn\overline{3}n]$ (Origin 2) [No. 222] (B = 0 for all [h,k,l] )

h	k	l	A
2n	2n	2n	8(Eccc Occc)
2n	2n 1	2n 1	−8(Ecss Ocss)
2n 1	2n	2n 1	−8(Escs Oscs)
2n 1	2n 1	2n	−8(Essc Ossc)
2n 1	2n 1	2n 1	8(Eccc − Occc)
2n 1	2n	2n	−8(Ecss − Ocss)
2n	2n 1	2n	−8(Escs − Oscs)
2n	2n	2n 1	−8(Essc − Ossc)

$[Pm\overline{3}n]$ [No. 223] (B = 0 for all [h,k,l] )

	A
2n	8(Eccc Occc)
2n 1	8(Eccc − Occc)

$[Pn\overline{3}m]$ (Origin 1) [No. 224]

	A	B
2n	8(Eccc $[+\ ]$ Occc)	0
2n 1	0	−8(Esss Osss)

$[Pn\overline{3}m]$ (Origin 2) [No. 224] (B = 0 for all [h,k,l] )

			A
2n	2n	2n	8(Eccc Occc)
2n	2n 1	2n 1	−8(Essc Ossc)
2n 1	2n	2n 1	−8(Ecss Ocss)
2n 1	2n 1	2n	−8(Escs Oscs)

$[Fm\overline{3}m]$ [No. 225]

	A	B
All	32(Eccc Occc)	0

$[Fm\overline{3}c]$ [No. 226] (B = 0 for all [h,k,l] )

	A
2n	32(Eccc Occc)
2n 1	32(Eccc − Occc)

$[Fd\overline{3}m]$ (Origin 1) [No. 227]

	A	B
4n	32(Eccc Occc)	0
4n 1	16(Eccc − Esss Occc − Osss)	A
4n 2	0	−32(Esss Osss)
4n 3	16(Eccc Esss Occc Osss)

$[Fd\overline{3}m]$ (Origin 2) [No. 227] (B = 0 for all [h,k,l] )

			A
4n	4n	4n	32(Eccc Occc)
4n	4n 2	4n 2	−32(Essc Ossc)
4n 2	4n	4n 2	−32(Ecss Ocss)
4n 2	4n 2	4n	−32(Escs Oscs)
4n 2	4n 2	4n 2	−16(Eccc Ecss Escs Essc Occc Ocss Oscs Ossc)
4n 2	4n	4n	16(Eccc − Ecss − Escs Essc Occc − Ocss − Oscs Ossc)
4n	4n 2	4n	16(Eccc Ecss − Escs − Essc Occc Ocss − Oscs − Ossc)
4n	4n	4n 2	16(Eccc − Ecss Escs − Essc Occc − Ocss Oscs − Ossc)

$[Fd\overline{3}c]$ (Origin 1) [No. 228]

	A	B
4n	32(Eccc Occc)	0
4n 1	16(Eccc Esss − Occc − Osss)
4n 2	0	−32(Esss Osss)
4n 3	16(Eccc − Esss − Occc Osss)	A

$[Fd\overline{3}c]$ (Origin 2) [No. 228] (B = 0 for all [h,k,l] )

			A
4n	4n	4n	32(Eccc Occc)
4n	4n 2	4n 2	−32(Essc Ossc)
4n 2	4n	4n 2	−32(Ecss Ocss)
4n 2	4n 2	4n	−32(Escs Oscs)
4n 2	4n 2	4n 2	−16(Eccc Ecss Escs Essc − Occc − Ocss − Oscs −Ossc)
4n 2	4n	4n	16(Eccc − Ecss − Escs Essc − Occc Ocss Oscs − Ossc)
4n	4n 2	4n	16(Eccc Ecss − Escs − Essc − Occc − Ocss Oscs Ossc)
4n	4n	4n 2	16(Eccc − Ecss Escs − Essc −Occc Ocss − Oscs Ossc)

$[Im\overline{3}m]$ [No. 229]

	A	B
All	16(Eccc Occc)	0

$[Ia\overline{3}d]$ [No. 230] (B = 0 for all [h,k,l] )

h	k	l		A
2n	2n	2n	4n	16(Eccc Occc)
2n	2n 1	2n 1	4n	−16(Escs Ossc)
2n 1	2n	2n 1	4n	−16(Essc Ocss)
2n 1	2n 1	2n	4n	−16(Ecss Oscs)
2n	2n	2n	4n 2	16(Eccc − Occc)
2n	2n 1	2n 1	4n 2	−16(Escs − Ossc)
2n 1	2n	2n 1	4n 2	−16(Essc − Ocss)
2n 1	2n 1	2n	4n 2	−16(Ecss − Oscs)

References

International Tables for Crystallography (2005). Vol. A. Space-group symmetry, edited by Th. Hahn, 5th ed., corrected reprint. Heidelberg: Springer.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.4, pp. 120-149
https://doi.org/10.1107/97809553602060000552