Table A1.4.3.6

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.4, pp. 137-143

Table A1.4.3.6

U. Shmueli,^a ^* S. R. Hall^b and R. W. Grosse-Kunstleve^c

^aSchool of Chemistry, Tel Aviv University, 69 978 Tel Aviv, Israel, ^bCrystallography Centre, University of Western Australia, Nedlands 6907, WA, Australia, and ^cLawrence Berkeley National Laboratory, 1 Cyclotron Road, Mailstop 4-230, Berkeley, CA 94720, USA
Correspondence e-mail: ushmueli@post.tau.ac.il

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()