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

International Tables for Crystallography (2010). Vol. B, ch. 1.4, pp. 135-161   | 1 | 2 |
https://doi.org/10.1107/97809553602060000761

Appendix A1.4.3. Structure-factor tables

U. Shmuelia

[link][link][link][link][link][link][link]

Table A1.4.3.1| top | pdf |
Plane groups

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

SystemNo.SymbolParityAB
Oblique 1 [p1]   c([hk]) s([hk])
2 [p2]   2c([hk]) 0
Rectangular 3 [pm]   2c([hx])c([ky]) 2c([hx])s([ky])
4 [pg] [k=2n] 2c([hx])c([ky]) 2c([hx])s([ky])
    [k=2n+1] −2s([hx])s([ky]) 2s([hx])c([ky])
5 [cm]   4c([hx])c([ky]) 4c([hx])s([ky])
6 [p2mm]   4c([hx])c([ky]) 0
7 [p2mg] [h=2n] 4c([hx])c([ky]) 0
    [h=2n+1] −4s([hx])s([ky]) 0
8 [p2gg] [h+k=2n] 4c([hx])c([ky]) 0
    [h+k=2n+1] −4s([hx])s([ky]) 0
9 [c2mm]   8c([hx])c([ky]) 0
Square 10 p4   2[P(cc) − M(ss)] 0
11 [p4mm]   4P(cc) 0
12 [p4gm] [h+k=2n] 4P(cc) 0
    [h+k=2n+1] −4M(ss) 0
Hexagonal 13 [p3]   C([hki]) S([hki])
14 [p3m1]   PH(cc) MH(ss)
15 [p31m]   PH(cc) PH(ss)
16 [p6]   2C([hki]) 0
17 [p6mm]   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[link].

P1 [No. 1]

[hkl]AB
All [\cos2\pi]([hx+ky+lz]) = ccc − css − scs − ssc [\sin2\pi]([hx+ky+lz]) = scc [+] csc[\ +] ccs − sss

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

[hkl]AB
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 symbolParityABUnique axis
ShortFull
3 [P2] [P121]   2c([hl])c([ky]) 2c([hl])s([ky]) b
3 [P2] [P112]   2c([hk])c([lz]) 2c([hk])s([lz]) c
4 [P2_{1}] [P12_{1}1] [k=2n] 2c([hl])c([ky]) 2c([hl])s([ky]) b
      [k=2n+1] −2s([hl])s([ky]) 2s([hl])c([ky])  
4 [P2_{1}] [P112_{1}] [l=2n] 2c([hk])c([lz]) 2c([hk])s([lz]) c
      [l=2n+1] −2s([hk])s([lz]) 2s([hk])c([lz])  
5 [C2] [C121]   4c([hl])c([ky]) 4c([hl])s([ky]) b
5 [C2] [A121]   4c([hl])c([ky]) 4c([hl])s([ky]) b
5 [C2] [I121]   4c([hl])c([ky]) 4c([hl])s([ky]) b
5 [C2] [A112]   4c([hk])c([lz]) 4c([hk])s([lz]) c
5 [C2] [B112]   4c([hk])c([lz]) 4c([hk])s([lz]) c
5 [C2] [I112]   4c([hk])c([lz]) 4c([hk])s([lz]) c
6 [Pm] [P1m1]   2c([hl])c([ky]) 2s([hl])c([ky]) b
6 [Pm] [P11m]   2c([hk])c([lz]) 2s([hk])c([lz]) c
7 [Pc] [P1c1] [l=2n] 2c([hl])c([ky]) 2s([hl])c([ky]) b
      [l=2n+1] −2s([hl])s([ky]) 2c([hl])s([ky])  
7 [Pc] [P1n1] [h+l=2n] 2c([hl])c([ky]) 2s([hl])c([ky]) b
      [h+l=2n+1] −2s([hl])s([ky]) 2c([hl])s([ky])  
7 [Pc] [P1a1] [h=2n] 2c([hl])c([ky]) 2s([hl])c([ky]) b
      [h=2n+1] −2s([hl])s([ky]) 2c([hl])s([ky])  
7 [Pc] [P11a] [h=2n] 2c([hk])c([lz]) 2s([hk])c([lz]) c
      [h=2n+1] −2s([hk])s([lz]) 2c([hk])s([lz])  
7 [Pc] [P11n] [h+k=2n] 2c([hk])c([lz]) 2s([hk])c([lz]) c
      [h+k=2n+1] −2s([hk])s([lz]) 2c([hk])s([lz])  
7 [Pc] [P11b] [k=2n] 2c([hk])c([lz]) 2s([hk])c([lz]) c
      [k=2n+1] −2s([hk])s([lz]) 2c([hk])s([lz])  
8 [Cm] [C1m1]   4c([hl])c([ky]) 4s([hl])c([ky]) b
8 [Cm] [A1m1]   4c([hl])c([ky]) 4s([hl])c([ky]) b
8 [Cm] [I1m1]   4c([hl])c([ky]) 4s([hl])c([ky]) b
8 [Cm] [A11m]   4c([hk])c([lz]) 4s([hk])c([lz]) c
8 [Cm] [B11m]   4c([hk])c([lz]) 4s([hk])c([lz]) c
8 [Cm] [I11m]   4c([hk])c([lz]) 4s([hk])c([lz]) c
9 [Cc] [C1c1] [l=2n] 4c([hl])c([ky]) 4s([hl])c([ky]) b
      [l=2n+1] −4s([hl])s([ky]) 4c([hl])s([ky])  
9 [Cc] [A1n1] [h+l=2n] 4c([hl])c([ky]) 4s([hl])c([ky]) b
      [h+l=2n+1] −4s([hl])s([ky]) 4c([hl])s([ky])  
9 [Cc] [I1a1] [h=2n] 4c([hl])c([ky]) 4s([hl])c([ky]) b
      [h=2n+1] −4s([hl])s([ky]) 4c([hl])s([ky])  
9 [Cc] [A11a] [h=2n] 4c([hk])c([lz]) 4s([hk])c([lz]) c
      [h=2n+1] −4s([hk])s([lz]) 4c([hk])s([lz])  
9 [Cc] [B11n] [h+k=2n] 4c([hk])c([lz]) 4s([hk])c([lz]) c
      [h+k=2n+1] −4s([hk])s([lz]) 4c([hk])s([lz])  
9 [Cc] [I11b] [k=2n] 4c([hk])c([lz]) 4s([hk])c([lz]) c
      [k=2n+1] −4s([hk])s([lz]) 4c([hk])s([lz])  
10 [P2/m] [P12/m1]   4c([hl])c([ky]) 0 b
10 [P2/m] [P112/m]   4c([hk])c([lz]) 0 c
11 [P2_{1}/m] [P12_{1}/m1] [k=2n] 4c([hl])c([ky]) 0 b
      [k=2n+1] −4s([hl])s([ky]) 0  
11 [P2_{1}/m] [P112_{1}/m] [l=2n] 4c([hk])c([lz]) 0 c
      [l=2n+1] −4s([hk])s([lz]) 0  
12 [C2/m] [C12/m1]   8c([hl])c([ky]) 0 b
12 [C2/m] [A12/m1]   8c([hl])c([ky]) 0 b
12 [C2/m] [I12/m1]   8c([hl])c([ky]) 0 b
12 [C2/m] [A112/m]   8c([hk])c([lz]) 0 c
12 [C2/m] [B112/m]   8c([hk])c([lz]) 0 c
12 [C2/m] [I112/m]   8c([hk])c([lz]) 0 c
13 [P2/c] [P12/c1] [l=2n] 4c([hl])c([ky]) 0 b
      [l=2n+1] −4s([hl])s([ky]) 0  
13 [P2/c] [P12/n1] [h+l=2n] 4c([hl])c([ky]) 0 b
      [h+l=2n+1] −4s([hl])s([ky]) 0  
13 [P2/c] [P12/a1] [h=2n] 4c([hl])c([ky]) 0 b
      [h=2n+1] −4s([hl])s([ky]) 0  
13 [P2/c] [P112/a] [h=2n] 4c([hk])c([lz]) 0 c
      [h=2n+1] −4s([hk])s([lz]) 0  
13 [P2/c] [P112/n] [h+k=2n] 4c([hk])c([lz]) 0 c
      [h+k=2n+1] −4s([hk])s([lz]) 0  
13 [P2/c] [P112/b] [k=2n] 4c([hk])c([lz]) 0 c
      [k=2n+1] −4s([hk])s([lz]) 0  
14 [P2_{1}/c] [P12_{1}/c1] [k+l=2n] 4c([hl])c([ky]) 0 b
      [k+l=2n+1] −4s([hl])s([ky]) 0  
14 [P2_{1}/c] [P12_{1}/n1] [h+k+l=2n] 4c([hl])c([ky]) 0 b
      [h+k+l=2n+1] −4s([hl])s([ky]) 0  
14 [P2_{1}/c] [P12_{1}/a1] [h+k=2n] 4c([hl])c([ky]) 0 b
      [h+k=2n+1] −4s([hl])s([ky]) 0  
14 [P2_{1}/c] [P112_{1}/a] [h+l=2n] 4c([hk])c([lz]) 0 c
      [h+l=2n+1] −4s([hk])s([lz]) 0  
14 [P2_{1}/c] [P112_{1}/n] [h+k+l=2n] 4c([hk])c([lz]) 0 c
      [h+k+l=2n+1] −4s([hk])s([lz]) 0  
14 [P2_{1}/c] [P112_{1}/b] [k+l=2n] 4c([hk])c([lz]) 0 c
      [k+l=2n+1] −4s([hk])s([lz]) 0  
15 [C2/c] [C12/c1] [l=2n] 8c([hl])c([ky]) 0 b
      [l=2n+1] −8s([hl])s([ky]) 0  
15 [C2/c] [A12/n1] [h+l=2n] 8c([hl])c([ky]) 0 b
      [h+l=2n+1] −8s([hl])s([ky]) 0  
15 [C2/c] [I12/a1] [h=2n] 8c([hl])c([ky]) 0 b
      [h=2n+1] −8s([hl])s([ky]) 0  
15 [C2/c] [A112/a] [h=2n] 8c([hk])c([lz]) 0 c
      [h=2n+1] −8s([hk])s([lz]) 0  
15 [C2/c] [B112/n] [h+k=2n] 8c([hk])c([lz]) 0 c
      [h+k=2n+1] −8s([hk])s([lz]) 0  
15 [C2/c] [I112/b] [k=2n] 8c([hk])c([lz]) 0 c
      [k=2n+1] −8s([hk])s([lz]) 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[link])] 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[link]) are ordered as in (A1.4.3.2[link]). 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.SymbolOriginParityAB
16 P222     4ccc −4sss
17 [P222_{1}]   [l=2n] 4ccc −4sss
      [l=2n+1] −4css 4scc
18 [P2_{1}2_{1}2]   [h+k=2n] 4ccc −4sss
      [h+k=2n+1] −4ssc 4ccs
19 [P2_{1}2_{1}2_{1}]   [h+k=2n]; [k+l=2n] 4ccc −4sss
      [h+k=2n]; [k+l=2n+1] −4css 4scc
      [h+k=2n+1]; [k+l=2n] −4scs 4csc
      [h+k=2n+1]; [k+l=2n+1] −4ssc 4ccs
20 [C222_{1}]   [l=2n] 8ccc −8sss
      [l=2n+1] −8css 8scc
21 C222     8ccc −8sss
22 F222     16ccc −16sss
23 I222     8ccc −8sss
24 [I2_{1}2_{1}2_{1}]   [h,k,l] all even 8ccc −8sss
      [h=2n]; [k,l=2n+1] −8scs 8csc
      [k=2n]; [l,h=2n+1] −8ssc 8ccs
      [l=2n]; [h,k=2n+1] −8css 8scc
25 [Pmm]2     4ccc 4ccs
26 [Pmc2_{1}]   [l=2n] 4ccc 4ccs
      [l=2n+1] −4css 4csc
27 [Pcc]2   [l=2n] 4ccc 4ccs
      [l=2n+1] −4ssc −4sss
28 [Pma]2   [h=2n] 4ccc 4ccs
      [h=2n+1] −4ssc −4sss
29 [Pca2_{1}]   [h=2n]; [l=2n] 4ccc 4ccs
      [h=2n]; [l=2n+1] −4scs 4scc
      [h=2n+1]; [l=2n] −4ssc −4sss
      [h=2n+1]; [l=2n+1] −4css 4csc
30 [Pnc]2   [k+l=2n] 4ccc 4ccs
      [k+l=2n+1] −4ssc 4sss
31 [Pmn2_{1}]   [h+l=2n] 4ccc 4ccs
      [h+l=2n+1] −4css 4csc
32 [Pba]2   [h+k=2n] 4ccc 4ccs
      [h+k=2n+1] −4ssc −4sss
33 [Pna2_{1}]   [h+k=2n]; [l=2n] 4ccc 4ccs
      [h+k=2n]; [l=2n+1] −4scs 4scc
      [h+k=2n+1]; [l=2n] −4ssc −4sss
      [h+k=2n+1]; [l=2n+1] −4css 4csc
34 [Pnn]2   [h+k+l=2n] 4ccc 4ccs
      [h+k+l=2n+1] −4ssc −4sss
35 [Cmm]2     8ccc 8ccs
36 [Cmc2_{1}]   [l=2n] 8ccc 8ccs
      [l=2n+1] −8css 8csc
37 [Ccc]2   [l=2n] 8ccc 8ccs
      [l=2n+1] −8ssc −8sss
38 [Amm]2     8ccc 8ccs
39 [Abm]2   [k=2n] 8ccc 8ccs
      [k=2n+1] −8ssc −8sss
40 [Ama]2   [h=2n] 8ccc 8ccs
      [h=2n+1] −8ssc −8sss
41 [Aba]2   [h+k=2n] 8ccc 8ccs
      [h+k=2n+1] −8ssc −8sss
42 [Fmm]2     16ccc 16ccs
43 [Fdd]2   [h+k+l=4n] 16ccc 16ccs
      [h+k+l=4n+1] 8(ccc − ssc −ccs − sss) 8(ccs − sss [+] ccc [+]ssc)
      [h+k+l=4n+2] −16ssc −16sss
      [h+k+l=4n+3] 8(ccc − ssc [+] ccs [+] sss) 8(ccs − sss − ccc − ssc)
44 [Imm]2     8ccc 8ccs
45 [Iba]2   [l=2n] 8ccc 8ccs
      [l=2n+1] −8ssc −8sss
46 [Iam]2   [h=2n] 8ccc 8ccs
      [h=2n+1] −8ssc −8sss
47 [Pmmm]     8ccc 0
48 [Pnnn] (1) [h+k+l=2n] 8ccc 0
      [h+k+l=2n+1] 0 −8sss
48 [Pnnn] (2) [h+k=2n]; [k+l=2n] 8ccc 0
      [h+k=2n]; [k+l=2n+1] −8ssc 0
      [h+k=2n+1]; [k+l=2n] −8css 0
      [h+k=2n+1]; [k+l=2n+1] −8scs 0
49 [Pccm]   [l=2n] 8ccc 0
      [l=2n+1] −8ssc 0
50 [Pban] (1) [h+k=2n] 8ccc 0
      [h+k=2n+1] 0 −8sss
50 [Pban] (2) [h=2n]; [k=2n] 8ccc 0
      [h=2n]; [k=2n+1] −8scs 0
      [h=2n+1]; [k=2n] −8css 0
      [h=2n+1]; [k=2n+1] −8ssc 0
51 [Pmma]   [h=2n] 8ccc 0
      [h=2n+1] −8scs 0
52 [Pnna]   [h=2n]; [k+l=2n] 8ccc 0
      [h=2n]; [k+l=2n+1] −8ssc 0
      [h=2n+1]; [k+l=2n] −8css 0
      [h=2n+1]; [k+l=2n+1] −8scs 0
53 [Pmna]   [h+l=2n] 8ccc 0
      [h+l=2n+1] −8css 0
54 [Pcca]   [h=2n]; [l=2n] 8ccc 0
      [h=2n]; [l=2n+1] −8ssc 0
      [h=2n+1]; [l=2n] −8scs 0
      [h=2n+1]; [l=2n+1] −8css 0
55 [Pbam]   [h+k=2n] 8ccc 0
      [h+k=2n+1] −8ssc 0
56 [Pccn]   [h+k=2n]; [h+l=2n] 8ccc 0
      [h+k=2n]; [h+l=2n+1] −8ssc 0
      [h+k=2n+1]; [h+l=2n] −8css 0
      [h+k=2n+1]; [h+l=2n+1] −8scs 0
57 [Pbcm]   [k=2n]; [l=2n] 8ccc 0
      [k=2n]; [l=2n+1] −8css 0
      [k=2n+1]; [l=2n] −8ssc 0
      [k=2n+1]; [l=2n+1] −8scs 0
58 [Pnnm]   [h+k+l=2n] 8ccc 0
      [h+k+l=2n+1] −8ssc 0
59 [Pmmn] (1) [h+k=2n] 8ccc 0
      [h+k=2n+1] 0 8ccs
59 [Pmmn] (2) [h=2n]; [k=2n] 8ccc 0
      [h=2n]; [k=2n+1] −8css 0
      [h=2n+1]; [k=2n] −8scs 0
      [h=2n+1]; [k=2n+1] −8ssc 0
60 [Pbcn]   [h+k=2n]; [l=2n] 8ccc 0
      [h+k=2n]; [l=2n+1] −8css 0
      [h+k=2n+1]; [l=2n] −8scs 0
      [h+k=2n+1]; [l=2n+1] −8ssc 0
61 [Pbca]   [h+k=2n]; [k+l=2n] 8ccc 0
      [h+k=2n]; [k+l=2n+1] −8css 0
      [h+k=2n+1]; [k+l=2n] −8scs 0
      [h+k=2n+1]; [k+l=2n+1] −8ssc 0
62 [Pnma]   [h+l=2n]; [k=2n] 8ccc 0
      [h+l=2n]; [k=2n+1] −8ssc 0
      [h+l=2n+1]; [k=2n] −8scs 0
      [h+l=2n+1]; [k=2n+1] −8css 0
63 [Cmcm]   [l=2n] 16ccc 0
      [l=2n+1] −16css 0
64 [Cmca]   [k+l=2n] 16ccc 0
      [k+l=2n+1] −16css 0
65 [Cmmm]     16ccc 0
66 [Cccm]   [l=2n] 16ccc 0
      [l=2n+1] −16ssc 0
67 [Cmma]   [h=2n] 16ccc 0
      [h=2n+1] −16css 0
68 [Ccca] (1) [h+l=2n] 16ccc 0
      [h+l=2n+1] 0 −16sss
68 [Ccca] (2) [k=2n]; [l=2n] 16ccc 0
      [k=2n]; [l=2n+1] −16ssc 0
      [k=2n+1]; [l=2n] −16scs 0
      [k=2n+1]; [l=2n+1] −16css 0
69 [Fmmm]     32ccc 0
70 [Fddd] (1) [h+k+l=4n] 32ccc 0
      [h+k+l=4n+1] 16(ccc − sss) A
      [h+k+l=4n+2] 0 −32sss
      [h+k+l=4n+3] 16(ccc [+] sss) [-A]
70 [Fddd] (2) [h+k=4n]; [k+l=4n]; [l+h=4n] 32ccc 0
      [h+k=4n]; [k+l=4n+2]; [l+h=4n+2] −32ssc 0
      [h+k=4n+2]; [k+l=4n]; [l+h=4n+2] −32css 0
      [h+k=4n+2]; [k+l=4n+2]; [l+h=4n] −32scs 0
      [h+k=4n+2]; [k+l=4n+2]; [l+h=4n+2] − 16(ccc [+] ssc [+] scs [+] css) 0
      [h+k=4n+2]; [k+l=4n]; [l+h=4n] 16(ccc [+] ssc − scs − css) 0
      [h+k=4n]; [k+l=4n+2]; [l+h=4n] 16(ccc − ssc − scs [+] css) 0
      [h+k=4n]; [k+l=4n]; [l+h=4n+2] 16(ccc − ssc [+] scs − css) 0
71 [Immm]     16ccc 0
72 [Ibam]   [l=2n] 16ccc 0
      [l=2n+1] −16ssc 0
73 [Ibca]   [h=2n]; [k=2n] 16ccc 0
      [h=2n]; [k=2n+1] −16scs 0
      [h=2n+1]; [k=2n] −16ssc 0
      [h=2n+1]; [k=2n+1] −16css 0
74 [Imma]   [k=2n] 16ccc 0
      [k=2n+1] −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[link]) and (1.4.2.20[link]) 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]

[hkl]AB
All 2[P(cc) − M(ss)]c([lz]) 2[P(cc) − M(ss)]s([lz])

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

lAB
4n 2[P(cc) − M(ss)]c([lz]) 2[P(cc) − M(ss)]s([lz])
4n [+] 1 −2[s([hx+ky])s([lz]) − s([hy-kx])c([lz])] 2[s([hx+ky])c([lz]) [+] s([hy-kx])s([lz])]
4n [+] 2 2[M(cc) − P(ss)]c([lz]) 2[M(cc) − P(ss)]s([lz])
4n [+] 3 −2[s([hx+ky])s([lz]) [+] s([hy-kx])c([lz])] 2[s([hx+ky])c([lz]) − s([hy-kx])s([lz])]

[P4_{2}] [No. 77]

lAB
2n 2[P(cc) − M(ss)]c([lz]) 2[P(cc) − M(ss)]s([lz])
2n [+] 1 2[M(cc) − P(ss)]c([lz]) 2[M(cc) − P(ss)]s([lz])

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

lAB
4n 2[P(cc) − M(ss)]c([lz]) 2[P(cc) − M(ss)]s([lz])
4n [+] 1 −2[s([hx+ky])s([lz]) [+] s([hy-kx])c([lz])] 2[s([hx+ky])c([lz]) − s([hy-kx])s([lz])]
4n [+] 2 2[M(cc) − P(ss)]c([lz]) 2[M(cc) − P(ss)]s([lz])
4n [+] 3 −2[s([hx+ky])s([lz])− s([hy-kx])c([lz])] 2[s([hx+ky])c([lz]) [+] s([hy-kx])s([lz])]

I4 [No. 79]

[hkl]AB
All 4[P(cc) − M(ss)]c([lz]) 4[P(cc) − M(ss)]s([lz])

[I4_{1}] [No. 80]

[2h+l]AB
4n 4[P(cc) − M(ss)]c([lz]) 4[P(cc) − M(ss)]s([lz])
4n [+] 1 4[c([hx+ky])c([lz]) [+] c([hy-kx])s([lz])] 4[c([hx+ky])s([lz]) − c([hy-kx])c([lz])]
4n [+] 2 4[M(cc) − P(ss)]c([lz]) 4[M(cc) − P(ss)]s([lz])
4n [+] 3 4[c([hx+ky])c([lz]) − c([hy-kx])s([lz])] 4[c([hx+ky])s([lz]) [+] c([hy-kx])c([lz])]

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

[hkl]AB
All 2[P(cc) − M(ss)]c([lz]) 2[M(cc) − P(ss)]s([lz])

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

[hkl]AB
All 4[P(cc) − M(ss)]c([lz]) 4[M(cc) − P(ss)]s([lz])

[P4/m] [No. 83]

[hkl]AB
All 4[P(cc) − M(ss)]c([lz]) 0

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

lA
2n 4[P(cc) − M(ss)]c([lz])
2n [+] 1 4[M(cc) − P(ss)]c([lz])

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

[h+k]AB
2n 4[P(cc) − M(ss)]c([lz]) 0
2n [+] 1 0 4[M(cc) − P(ss)]s([lz])

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

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

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

[h+k+l]AB
2n 4[P(cc) − M(ss)]c([lz]) 0
2n [+] 1 0 4[M(cc) − P(ss)]s([lz])

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

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

[I4/m] [No. 87]

[hkl]AB
All 8[P(cc) − M(ss)]c([lz]) 0

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

[2k+l]AB
4n 8[P(cc) − M(ss)]c([lz]) 0
4n [+] 1 4[P(cc) − M(ss)]c([lz]) [+] [M(cc) − P(ss)]s([lz]) A
4n [+] 2 0 8[M(cc) − P(ss)]s([lz])
4n [+] 3 4[P(cc) − M(ss)]c([lz]) − [M(cc) − P(ss)]s([lz]) [-A]

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

hk[h+k+l]A
2n 2n 4n 8[P(cc) − M(ss)]c([lz])
2n 2n [+] 1 4n −8[s([hx+ky])s([lz]) − c([hy-kx])c([lz])]
2n [+] 1 2n 4n 8[c([hx+ky])c([lz]) − s([hy-kx])s([lz])]
2n [+] 1 2n [+] 1 4n −8[M(cs) [+] P(sc)]s([lz])
2n 2n 4n [+] 2 8[M(cc) − P(ss)]c([lz])
2n 2n [+] 1 4n [+] 2 −8[s([hx+ky])s([lz]) [+] c([hy-kx])c([lz])]
2n [+] 1 2n 4n [+] 2 8[c([hx+ky])c([lz]) [+] s([hy-kx])s([lz])]
2n [+] 1 2n [+] 1 4n [+] 2 −8[P(cs) [+] M(sc)]s([lz])

[P422] [No. 89]

[hkl]AB
All 4P(cc)c([lz]) −4M(ss)s([lz])

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

[h+k]AB
2n 4P(cc)c([lz]) −4M(ss)s([lz])
2n [+] 1 −4P(ss)c([lz]) 4M(cc)s([lz])

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

lAB
4n 4P(cc)c([lz]) −4M(ss)s([lz])
4n [+] 1 −4[s([hx])c([ky])s([lz]) − c([kx])s([hy])c([lz])] 4[c([hx])s([ky])c([lz]) − s([kx])c([hy])s([lz])]
4n [+] 2 4M(cc)c([lz]) −4P(ss)s([lz])
4n [+] 3 −4[s([hx])c([ky])s([lz]) [+] c([kx])s([hy])c([lz])] 4[c([hx])s([ky])c([lz]) [+] s([kx])c([hy])s([lz])]

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

[2h+2k+l]AB
4n 4P(cc)c([lz]) −4M(ss)s([lz])
4n [+] 1 2{[P(sc) − P(cs)]c([lz]) − [M(cs) − M(sc)]s([lz])} 2{[P(sc) [+] P(cs)]c([lz]) [+] [M(cs) − M(sc)]s([lz])}
4n [+] 2 −4P(ss)c([lz]) 4M(cc)s([lz])
4n [+] 3 −2{[P(sc) − P(cs)]c([lz]) [+] [M(cs) [+] M(sc)]s([lz])} 2{[P(sc) [+] P(cs)]c([lz]) − [M(cs) − M(sc)s([lz])}

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

lAB
2n 4P(cc)c([lz]) −4M(ss)s([lz])
2n [+]1 4M(cc)c([lz]) −4P(ss)s([lz])

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

[h+k+l]AB
2n 4P(cc)c([lz]) −4M(ss)s([lz])
2n [+] 1 −4P(ss)c([lz]) 4M(cc)s([lz])

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

lAB
4n 4P(cc)c([lz]) −4M(ss)s([lz])
4n [+] 1 −4[s([hx])c([ky])s([lz]) [+] c([kx])s([hy])c([lz])] 4[c([hx])s([ky])c([lz]) [+] s([kx])c([hy])c([lz])]
4n [+] 2 4M(cc)c([lz]) −4P(ss)s([lz])
4n [+] 3 −4[s([hx])c([ky])s([lz]) − c([kx])s([hy])c([lz])] 4[c([hx])s([ky])c([lz]) − s([kx])c([hy])c([lz])]

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

[2h+2k+l]AB
4n 4P(cc)c([lz]) −4M(ss)s([lz])
4n [+] 1 −2{[P(sc) − P(cs)]c([lz]) [+] [M(cs) [+] M(sc)]s([lz])} 2{[P(sc) [+] P(cs)]c([lz]) − [M(cs) − M(sc)]s([lz])}
4n [+] 2 −4P(ss)c([lz]) 4M(cc)s([lz])
4n [+] 3 2{[P(sc) − P(cs)]c([lz]) − [M(cs) [+] M(sc)]s([lz])} 2{[P(sc) [+] P(cs)]c([lz]) [+] [M(cs) − M(sc)]s([lz])}

[I422] [No. 97]

[hkl]AB
All 8P(cc)c([lz]) −8M(ss)s([lz])

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

[2k+l]AB
4n 8P(cc)c([lz]) −8M(ss)s([lz])
4n [+] 1 4{[P(cc) − P(ss)]c([lz]) [+] [M(cc) [+] M(ss)]s([lz])} 4{[P(cc) [+] P(ss)]c([lz]) [+] [M(cc) − M(ss)]s([lz])}
4n [+] 2 −8P(ss)c([lz]) 8M(cc)s([lz])
4n [+] 3 4{[P(cc) − P(ss)]c([lz]) − [M(cc) [+] M(ss)]s([lz])} −4{[P(cc) [+] P(ss)]c([lz]) − [M(cc) − M(ss)]s([lz])}

[P4mm] [No. 99]

[hkl]AB
All 4P(cc)c([lz]) 4P(cc)s([lz])

[P4bm] [No. 100]

[h+k]AB
2n 4P(cc)c([lz]) 4P(cc)s([lz])
2n [+] 1 −4M(ss)c([lz]) −4M(ss)s([lz])

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

lAB
2n 4P(cc)c([lz]) 4P(cc)s([lz])
2n [+] 1 −4P(ss)c([lz]) −4P(ss)s([lz])

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

[h+k+l]AB
2n 4P(cc)c([lz]) 4P(cc)s([lz])
2n [+]1 −4P(ss)c([lz]) −4P(ss)s([lz])

[P4cc] [No. 103]

lAB
2n 4P(cc)c([lz]) 4P(cc)s([lz])
2n [+] 1 −4M(ss)c([lz]) −4M(ss)s([lz])

P[4nc] [No. 104]

[h+k+l]AB
2n 4P(cc)c([lz]) 4P(cc)s([lz])
2n [+] 1 −4M(ss)c([lz]) −4M(ss)s([lz])

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

lAB
2n 4P(cc)c([lz]) 4P(cc)s([lz])
2n [+] 1 4M(cc)c([lz]) 4M(cc)s([lz])

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

[h+k]lAB
2n 2n 4P(cc)c([lz]) 4P(cc)s([lz])
2n [+] 1 2n −4M(ss)c([lz]) −4M(ss)s([lz])
2n 2n [+] 1 4M(cc)c([lz]) 4M(cc)s([lz])
2n [+] 1 2n [+] 1 −4P(ss)c([lz]) −4P(ss)s([lz])

[I4mm] [No. 107]

[hkl]AB
All 8P(cc)c([lz]) 8P(cc)s([lz])

[I4cm] [No. 108]

lAB
2n 8P(cc)c([lz]) 8P(cc)s([lz])
2n [+] 1 −8M(ss)c([lz]) −8M(ss)s([lz])

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

[2k+l]AB
4n 8P(cc)c([lz]) 8P(cc)s([lz])
4n [+] 1 8[c([hx])c([ky])c([lz]) − c([kx])c([hy])s([lz])] 8[c([hx])c([ky])s([lz]) [+] c([kx])c([hy])c([lz])]
4n [+] 2 8M(cc)c([lz]) 8M(cc)s([lz])
4n [+] 3 8[c([hx])c([ky])c([lz]) [+] c([kx])c([hy])s([lz])] 8[c([hx])c([ky])s([lz]) − c([kx])c([hy])c([lz])]

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

[2k+l]AB
4n 8P(cc)c([lz]) 8P(cc)s([lz])
4n [+] 1 −8[s([hx])s([ky])c([lz]) [+] s([kx])s([hy])s([lz])] −8[s([hx])s([ky])s([lz]) − s([kx])s([hy])c([lz])]
4n [+] 2 8M(cc)c([lz]) 8M(cc)s([lz])
4n [+] 3 −8[s([hx])s([ky])c([lz]) − s([kx])s([hy])s([lz])] −8[s([hx])s([ky])s([lz]) [+] s([kx])s([hy])c([lz])]

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

[hkl]AB
All 4P(cc)c([lz]) −4P(ss)s([lz])

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

lAB
2n 4P(cc)c([lz]) −4P(ss)s([lz])
2n [+] 1 −4M(ss)c([lz]) 4M(cc)s([lz])

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

[h+k]AB
2n 4P(cc)c([lz]) −4P(ss)s([lz])
2n [+] 1 −4M(ss)c([lz]) 4M(cc)s([lz])

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

[h+k+l]AB
2n 4P(cc)c([lz]) −4P(ss)s([lz])
2n [+] 1 −4M(ss)c([lz]) 4M(cc)s([lz])

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

[hkl]AB
All 4P(cc)c([lz]) 4M(cc)s([lz])

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

lAB
2n 4P(cc)c([lz]) 4M(cc)s([lz])
2n [+] 1 −4M(ss)c([lz]) −4P(ss)s([lz])

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

[h+k]AB
2n 4P(cc)c([lz]) 4M(cc)s([lz])
2n [+] 1 −4M(ss)c([lz]) −4P(ss)s([lz])

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

[h+k+l]AB
2n 4P(cc)c([lz]) 4M(cc)s([lz])
2n [+] 1 −4M(ss)c([lz]) −4P(ss)s([lz])

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

[hkl]AB
All 8P(cc)c([lz]) 8M(cc)s([lz])

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

lAB
2n 8P(cc)c([lz]) 8M(cc)s([lz])
2n [+] 1 −8M(ss)c([lz]) −8P(ss)s([lz])

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

[hkl]AB
All 8P(cc)c([lz]) −8P(ss)s([lz])

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

[2h+l]AB
4n 8P(cc)c([lz]) −8P(ss)s([lz])
4n [+] 1 4{[P(cc) − M(ss)]c([lz]) − [M(cc) [+] P(ss)]s([lz])} −4{[P(cc) [+] M(ss)]c([lz]) − [M(cc) − P(ss)]s([lz])}
4n [+] 2 −8M(ss)c([lz]) 8M(cc)s([lz])
4n [+] 3 4{[P(cc) − M(ss)]c([lz]) [+] [M(cc) [+] P(ss)]s([lz])} 4{[P(cc) [+] M(ss)]c([lz]) [+] [M(cc) − P(ss)]s([lz])}

[P4/mmm] [No. 123]

[hkl]AB
All 8P(cc)c([lz]) 0

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

lA
2n 8P(cc)c([lz])
2n [+] 1 −8M(ss)c([lz])

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

[h+k]AB
2n 8P(cc)c([lz]) 0
2n [+] 1 0 −8M(ss)s([lz])

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

hkA
2n 2n 8P(cc)c([lz])
2n 2n [+] 1 −8M(sc)s([lz])
2n [+] 1 2n −8M(cs)s([lz])
2n [+] 1 2n [+] 1 −8P(ss)c([lz])

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

[h+k+l]AB
2n 8P(cc)c([lz]) 0
2n [+] 1 0 −8M(ss)s([lz])

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

hklA
2n 2n 2n 8P(cc)c([lz])
2n 2n 2n [+] 1 −8M(ss)c([lz])
2n 2n [+] 1 2n −8M(sc)s([lz])
2n 2n [+] 1 2n [+] 1 −8P(cs)s([lz])
2n [+] 1 2n 2n −8M(cs)s([lz])
2n [+] 1 2n 2n [+] 1 −8P(sc)s([lz])
2n [+] 1 2n [+] 1 2n −8P(ss)c([lz])
2n [+] 1 2n [+] 1 2n [+] 1 8M(cc)c([lz])

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

[h+k]A
2n 8P(cc)c([lz])
2n [+] 1 −8M(ss)c([lz])

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

[h+k+l]A
2n 8P(cc)c([lz])
2n [+] 1 −8M(ss)c([lz])

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

[h+k]AB
2n 8P(cc)c([lz]) 0
2n [+] 1 0 8M(cc)s([lz])

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

hkA
2n 2n 8P(cc)c([lz])
2n 2n [+] 1 −8P(cs)s([lz])
2n [+] 1 2n −8P(sc)s([lz])
2n [+] 1 2n [+] 1 −8P(ss)c([lz])

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

[h+k]lAB
2n 2n 8P(cc)c([lz]) 0
2n 2n [+] 1 −8M(ss)c([lz]) 0
2n [+]1 2n 0 8M(cc)s([lz])
2n [+] 1 2n [+] 1 0 −8P(ss)s([lz])

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

hklA
2n 2n 2n 8P(cc)c([lz])
2n 2n 2n [+] 1 −8M(ss)c([lz])
2n 2n [+] 1 2n −8P(cs)s([lz])
2n 2n [+] 1 2n [+] 1 −8M(sc)s([lz])
2n [+] 1 2n 2n −8P(sc)s([lz])
2n [+] 1 2n 2n [+] 1 −8M(cs)s([lz])
2n [+] 1 2n [+] 1 2n −8P(ss)c([lz])
2n [+] 1 2n [+] 1 2n [+] 1 8M(cc)c([lz])

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

lA
2n 8P(cc)c([lz])
2n [+] 1 8M(cc)c([lz])

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

lA
2n 8P(cc)c([lz])
2n [+] 1 −8P(ss)c([lz])

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

[h+k+l]lAB
2n 2n 8P(cc)c([lz]) 0
2n 2n [+] 1 −8M(ss)c([lz]) 0
2n [+]1 2n 0 −8P(ss)s([lz])
2n [+] 1 2n [+] 1 0 8M(cc)s([lz])

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

hklA
2n 2n 2n 8P(cc)c([lz])
2n 2n 2n [+] 1 8M(cc)c([lz])
2n 2n [+] 1 2n −8M(sc)s([lz])
2n 2n [+] 1 2n [+] 1 −8P(sc)s([lz])
2n [+] 1 2n 2n −8M(cs)s([lz])
2n [+] 1 2n 2n [+] 1 −8P(cs)s([lz])
2n [+] 1 2n [+] 1 2n −8P(ss)c([lz])
2n [+] 1 2n [+] 1 2n [+] 1 −8M(ss)c([lz])

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

[h+k+l]AB
2n 8P(cc)c([lz]) 0
2n [+] 1 0 −8P(ss)s([lz])

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

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

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

[h+k]lA
2n 2n 8P(cc)c([lz])
2n 2n [+] 1 8M(cc)c([lz])
2n [+] 1 2n −8M(ss)c([lz])
2n [+] 1 2n [+] 1 −8P(ss)c([lz])

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

[h+k+l]A
2n 8P(cc)c([lz])
2n [+] 1 −8P(ss)c([lz])

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

[h+k+l]AB
2n 8P(cc)c([lz]) 0
2n [+] 1 0 8M(cc)s([lz])

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

hklA
2n 2n 2n 8P(cc)c([lz])
2n 2n 2n [+] 1 8M(cc)c([lz])
2n 2n [+] 1 2n −8P(cs)s([lz])
2n 2n [+] 1 2n [+] 1 −8M(cs)s([lz])
2n [+] 1 2n 2n −8P(sc)s([lz])
2n [+] 1 2n 2n [+] 1 −8M(sc)s([lz])
2n [+] 1 2n [+] 1 2n −8P(ss)c([lz])
2n [+] 1 2n [+] 1 2n [+] 1 −8M(ss)c([lz])

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

[h+k]lAB
2n 2n 8P(cc)c([lz]) 0
2n [+] 1 2n [+] 1 −8M(ss)c([lz]) 0
2n [+] 1 2n 0 8M(cc)s([lz])
2n 2n [+] 1 0 −8P(ss)s([lz])

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

[h+k][k+l][h+l]A
2n 2n 2n 8P(cc)c([lz])
2n 2n [+] 1 2n [+] 1 −8P(ss)c([lz])
2n [+] 1 2n [+] 1 2n −8P(cs)s([lz])
2n [+] 1 2n 2n [+] 1 −8P(sc)s([lz])

[I4/mmm] [No. 139]

[hkl]AB
All 16P(cc)c([lz]) 0

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

lA
2n 16P(cc)c([lz])
2n [+] 1 −16M(ss)c([lz])

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

[2h+l]AB
4n 16P(cc)c([lz]) 0
4n [+] 1 8[P(cc)c([lz]) − M(cc)s([lz])] [-A]
4n [+] 2 0 16M(cc)s([lz])
4n [+] 3 8[P(cc)c([lz]) [+] M(cc)s([lz])] A

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

hk[h+k+l]A
2n 2n 4n 16P(cc)c([lz])
2n 2n [+] 1 4n −16[c([hx])s([ky])s([lz]) [+] c([kx])c([hy])c([lz])]
2n [+] 1 2n 4n 16[c([hx])c([ky])c([lz]) [+] c([kx])s([hy])s([lz])]
2n [+] 1 2n [+] 1 4n −16[c([hx])s([ky])s([lz]) [+] c([kx])s([hy])s([lz])]
2n 2n 4n [+] 2 16M(cc)c([lz])
2n 2n [+] 1 4n [+] 2 −16[c([hx])s([ky])s([lz]) − c([kx])c([hy])c([lz])]
2n [+] 1 2n 4n [+] 2 16[c([hx])c([ky])c([lz]) − c([kx])s([hy])s([lz])]
2n [+] 1 2n [+] 1 4n [+] 2 −16[c([hx])s([ky])s([lz]) − c([kx])s([hy])s([lz])]

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

[2h+l]AB
4n 16P(cc)c([lz]) 0
4n [+] 1 −8[M(ss)c([lz]) − P(ss)s([lz])] [-A]
4n [+] 2 0 16M(cc)s([lz])
4n [+] 3 −8[M(ss)c([lz]) [+] P(ss)s([lz])] A

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

hk[h+k+l]A
2n 2n 4n 16P(cc)c([lz])
2n 2n [+]1 4n −16[s([hx])c([ky])s([lz]) [+] s([kx])s([hy])c([lz])]
2n [+] 1 2n 4n −16[s([hx])s([ky])c([lz]) [+] s([kx])c([hy])s([lz])]
2n [+] 1 2n [+] 1 4n −16[c([hx])s([ky])s([lz]) [+] c([kx])s([hy])s([lz])]
2n 2n 4n [+] 2 16M(cc)c([lz])
2n 2n [+] 1 4n [+] 2 −16[s([hx])c([ky])s([lz]) − s([kx])s([hy])c([lz])]
2n [+] 1 2n 4n [+] 2 −16[s([hx])s([ky])c([lz]) − s([kx])c([hy])s([lz])]
2n [+] 1 2n [+] 1 4n [+] 2 −16[c([hx])s([ky])s([lz]) − c([kx])s([hy])s([lz])]

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)[link]–(1.4.3.12)[link][link]]. 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)[link] and (1.4.3.5)[link], and the notes at the start of Table A1.4.3.7[link]]. 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]

[hkl]AB
All C([hki])c([lz]) − S([hki])s([lz]) C([hki])s([lz]) [+] S([hki])c([lz])

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

lAB
3n as for [P3] [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])

lA, B
3n as for [P3] [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)

[hkl]AB
All c([hx+ky+lz]) [+] c([kx+ly+hz]) [+] c([lx+hy+kz]) s([hx+ky+lz]) [+] s([kx+ly+hz]) [+] s([lx+hy+kz])

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

[hkl]AB
All 3[C([hki])c([lz]) − S([hki])s([lz])] 3[C([hki])s([lz]) [+] S([hki])c([lz])]

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

[hkl]AB
All 2[C([hki])c([lz]) − S([hki])s([lz])] 0

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

[hkl]AB
All 2[c([hx+ky+lz]) [+] c([kx+ly+hz]) [+] c([lx+hy+kz])] 0

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

[hkl]AB
All 6[C([hki])c([lz]) − S([hki])s([lz])] 0

[P312] [No. 149]

[hkl]AB
All PH(cc)c([lz]) − PH(ss)s([lz]) MH(cc)s([lz]) [+] MH(ss)c([lz])

[P321] [No. 150]

[hkl]AB
All PH(cc)c([lz]) − MH(ss)s([lz]) PH(ss)c([lz]) [+] MH(cc)s([lz])

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

lAB
3n as for [P312] [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])

lAB
3n as for [P321] [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})] [\hbox{s}(p_{1}+u_{1}) +\hbox{s}(p_{2}+u_{2}) + \hbox{s}(p_{3}+u_{3}) +\hbox{s}(q_{1}-u_{1})] [+\hbox{ s}(q_{2}-u_{2})+ \hbox{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])

lA, B
3n as for [P312] [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])

lA, B
3n as for [P321] [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)

[hkl]AB
All Eccc − Ecss − Escs − Essc [+] Occc − Ocss − Oscs − Ossc Escc [+] Ecsc [+] Eccs − Esss − Oscc − Ocsc − Occs [+] Osss

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

[hkl]AB
All 3[PH(cc)c([lz]) − MH(ss)s([lz])] 3[PH(ss)c([lz]) [+] MH(cc)s([lz])]

[P3m1] [No. 156]

[hkl]AB
All PH(cc)c([lz]) − MH(ss)s([lz]) PH(cc)s([lz]) [+] MH(ss)c([lz])

[P31m] [No. 157]

[hkl]AB
All PH(cc)c([lz]) − PH(ss)s([lz]) PH(cc)s([lz]) [+] PH(ss)c([lz])

[P3c1] [No. 158]

lAB
2n PH(cc)c([lz]) − MH(ss)s([lz]) PH(cc)s([lz]) [+] MH(ss)c([lz])
2n [+] 1 MH(cc)c([lz]) − PH(ss)s([lz]) PH(ss)c([lz]) [+] MH(cc)s([lz])

[P31c] [No. 159]

lAB
2n PH(cc)c([lz]) − PH(ss)s([lz]) PH(cc)s([lz]) [+] PH(ss)c([lz])
2n [+] 1 MH(cc)c([lz]) − MH(ss)s([lz]) MH(cc)s([lz]) [+] MH(ss)c([lz])

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

[hkl]AB
All Eccc − Ecss − Escs − Essc [+] Occc − Ocss − Oscs − Ossc Escc [+] Ecsc [+] Eccs − Esss [+] Oscc [+] Ocsc [+] Occs − Osss

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

[hkl]AB
All 3[PH(cc)c([lz]) − MH(ss)s([lz])] 3[PH(cc)s([lz]) [+] MH(ss)c([lz])]

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

[h+k+l]AB
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)

lAB
2n 3[PH(cc)c([lz]) − MH(ss)s([lz])] 3[PH(cc)s([lz]) [+] MH(ss)c([lz])]
2n [+] 1 3[MH(cc)c([lz]) − PH(ss)s([lz])] 3[PH(ss)c([lz]) [+] MH(cc)s([lz])]

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

A
2[PH(cc)c([lz]) − PH(ss)s([lz])]

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

lA
2n 2[PH(cc)c([lz]) − PH(ss)s([lz])]
2n [+] 1 2[MH(cc)c([lz]) − MH(ss)s([lz])]

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

A
2[PH(cc)c([lz]) − MH(ss)s([lz])]

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

lA
2n 2[PH(cc)c([lz]) − MH(ss)s([lz])]
2n [+] 1 2[MH(cc)c([lz]) − PH(ss)s([lz])]

[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([lz]) − MH(ss)s([lz])]

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

[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])

lA
2n 6[PH(cc)c([lz]) − MH(ss)s([lz])]
2n [+] 1 6[MH(cc)c([lz]) − PH(ss)s([lz])]

[P6] [No. 168]

[hkl]AB
All 2C([hki])c([lz]) 2C([hki])s([lz])

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

lAB
6n as for [P6] [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([hki])s([lz]) 2S([hki])c([lz])
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])

lA, B
6n as for [P6] [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])

lA, B
3n as for [P6] [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])

lA, B
3n as for [P6] [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]

lA, B
2n as for [P6] [No. 168]
2n [+] 1 as for l = 6n [+] 3 in [P6_{1}] [No. 169]

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

[hkl]AB
All 2C([hki])c([lz]) 2S([hki])c([lz])

[P6/m] [No. 175]

[hkl]AB
All 4C([hki])c([lz]) 0

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

lAB
2n 4C([hki])c([lz]) 0
2n [+] 1 −4S([hki])s([lz]) 0

[P622] [No. 177]

[hkl]AB
All 2PH(cc)c([lz]) 2MH(cc)s([lz])

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

lAB
6n as for [P622] [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([lz]) 2PH(ss)c([lz])
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])

lA, B
6n as for [P622] [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])

lA, B
n as for [P622] [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])

lA, B
3n as for [P622] [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]

lA, B
2n as for [P622] [No. 177]
2n [+] 1 as for l = 6n [+] 3 in [P6_{1}22] [No. 178]

[P6mm] [No. 183]

[hkl]AB
All 2PH(cc)c([lz]) 2PH(cc)s([lz])

[P6cc] [No. 184]

lAB
2n 2PH(cc)c([lz]) 2PH(cc)s([lz])
2n [+] 1 2MH(cc)c([lz]) 2MH(cc)s([lz])

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

lAB
2n 2PH(cc)c([lz]) 2PH(cc)s([lz])
2n [+] 1 −2PH(ss)s([lz]) 2PH(ss)c([lz])

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

lAB
2n 2PH(cc)c([lz]) 2PH(cc)s([lz])
2n [+] 1 −2MH(ss)s([lz]) 2MH(ss)c([lz])

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

[hkl]AB
All 2PH(cc)c([lz]) 2MH(ss)c([lz])

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

lAB
2n 2PH(cc)c([lz]) 2MH(ss)c([lz])
2n [+] 1 −2PH(ss)s([lz]) 2MH(cc)s([lz])

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

[hkl]AB
All 2PH(cc)c([lz]) 2PH(ss)c([lz])

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

lAB
2n 2PH(cc)c([lz]) 2PH(ss)c([lz])
2n [+] 1 −2MH(ss)s([lz]) 2MH(cc)s([lz])

[P6/mmm] [No. 191]

[hkl]AB
All 4PH(cc)c([lz]) 0

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

lA
2n 4PH(cc)c([lz])
2n [+] 1 4MH(cc)c([lz])

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

lA
2n 4PH(cc)c([lz])
2n [+] 1 −4PH(ss)s([lz])

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

lA
2n 4PH(cc)c([lz])
2n [+] 1 −4MH(ss)s([lz])

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[link] (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[link]–A1.4.3.6[link][link][link][link][link], 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]

[hkl]AB
All 4Eccc −4Esss

[F23] [No. 196]

[hkl]AB
All 16Eccc −16Esss

[I23] [No. 197]

[hkl]AB
All 8Eccc −8Esss

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

[h+k][k+l][h+l]AB
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]

[h+k][k+l][h+l]AB
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]

[hkl]AB
All 8Eccc 0

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

[h+k+l]AB
2n 8Eccc 0
2n [+] 1 0 −8Esss

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

[h+k][k+l][h+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]

[hkl]AB
All 32Eccc 0

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

[h+k+l]AB
4n 32Eccc 0
4n [+] 1 16(Eccc − Esss) A
4n [+] 2 0 −32Esss
4n [+] 3 16(Eccc [+] Esss) [-A]

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

[h+k][k+l][h+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]

[hkl]AB
All 16Eccc 0

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

[h+k][k+l][h+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])

[h+k][k+l][h+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]

[hkl]AB
All 4(Eccc [+] Occc) −4(Esss − Osss)

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

[h+k+l]AB
2n 4(Eccc [+] Occc) −4(Esss − Osss)
2n [+] 1 4(Eccc − Occc) −4(Esss [+] Osss)

[F432] [No. 209]

[hkl]AB
All 16(Eccc [+] Occc) −16(Esss − Osss)

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

[h+k+l]AB
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]

[hkl]AB
All 8(Eccc [+] Occc) −8(Esss − Osss)

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

[h+k][k+l][h+l][h+k+l]AB
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])

hkl[h+k+l]AB
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]

hkl[h+k+l]AB
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]

[hkl]AB
All 4(Eccc [+] Occc) −4(Esss [+] Osss)

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

[hkl]AB
All 16(Eccc [+] Occc) −16(Esss [+] Osss)

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

[hkl]AB
All 8(Eccc [+] Occc) −8(Esss [+]Osss)

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

[h+k+l]AB
2n 4(Eccc [+] Occc) −4(Esss [+] Osss)
2n [+] 1 4(Eccc − Occc) −4(Esss − Osss)

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

[h+k+l]AB
2n 16(Eccc [+] Occc) −16(Esss [+] Osss)
2n [+] 1 16(Eccc − Occc) −16(Esss − Osss)

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

hkl[h+k+l]AB
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]

[hkl]AB
All 8(Eccc [+] Occc) 0

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

[h+k+l]AB
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])

hklA
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