Symmorphism and antimorphism

Wilson, A. J. C.; Karen, V. L.; Mighell, A.

doi:10.1107/97809553602060000623

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 9.7, pp. 897-899

Section 9.7.1.2. Symmorphism and antimorphism

A. J. C. Wilson,^a ^‡ V. L. Karen^b and A. Mighell^b

^a St John's College, Cambridge CB2 1TP, England, and ^bNIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

9.7.1.2. Symmorphism and antimorphism

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Wilson (1993d) classified the space groups by degree of symmorphism. A fully symmorphic space group contains only the `syntropic' symmetry elements $[2,\,3,\,4,\,6\semi\,\overline {2}=m,\,\overline {3}=3+\overline {1}\,,\overline{4}, \overline {6}=3/m]$ and a fully antimorphic space group contains only the `antitropic' elements $[ 2_1,\,3_{1,2},\,4_{1,3},\,6_{1,5}\semi \hbox {{ any glide plane}}. ]$ The remaining symmetry elements $[ 1,\,\overline {1}\semi \,4_2\semi\,\bar4\semi\, 6_{2,4}=2+3_{1,2}\semi\,6_3=2_1+3 ]$ are `atropic'. The two triclinic space groups, P1 and $[P\overline {1}]$ , contain only `atropic' elements, and are thus not classified by these criteria. The rest are divided into five groups, in accordance with the balance of symmetry elements within the unit cell. For the 71 non-triclinic space groups symmorphic in the strict sense (Wilson, 1993d; Subsection 1.4.2.1 ), the classification gives:

(1) Fully symmorphic (only syntropic elements): 14.
(2) Tending to symmorphism (mainly syntropic elements): 28.
(3) Equally balanced (equal numbers of syntropic and antitropic elements): 20.
(4) Tending to antimorphism (mainly antitropic elements): 9.
(5) Fully antimorphic (only antitropic elements): 0.

The distribution of the 230 space groups (11 enantiomorphic couples merged) by arithmetic crystal class and degree of symmorphism is given in Table 9.7.1.2.

Table 9.7.1.2| top | pdf |
Space groups arranged by arithmetic crystal class and degree of symmorphism (Wilson, 1993d), as frequented by homomolecular structures with one molecule in the general position (in superscript numerals; according to Belsky, Zorkaya & Zorky, 1995)

(a) Triclinic, monoclinic and orthorhombic systems. The triclinic space groups are a special case, with `degree of symmorphism' undefined, and they are not assigned to any particular column. For *, † see Subsection 9.7.4.1.

Arithmetic crystal class	Fully symmorphic	Tending to symmorphism	Equally balanced	Tending to antimorphism	Fully antimorphic
1P	*P1†⁽⁹⁰⁾
$[\overline {1}P]$	*P $[{\overline 1}]$ †⁽¹⁷⁹⁶⁾
2P	*P2⁽⁰⁾	⋯	⋯	⋯	P2₁†⁽¹³²⁷⁾
2C	⋯	⋯	*C2 †⁽¹⁰⁹⁾	⋯	⋯
mP	*Pm⁽⁰⁾	⋯	⋯	⋯	Pc†⁽⁵⁸⁾
mC	⋯	⋯	*Cm⁽⁰⁾	⋯	Cc†⁽¹⁴⁴⁾
2/mP	*P2/m⁽⁰⁾	⋯	P2₁/m⁽⁰⁾	⋯	P2₁/c†⁽⁵⁹⁵¹⁾
2/mP			P2/c⁽¹¹⁾
2/mC	⋯	⋯	*C2/m⁽⁰⁾	C2/c †⁽⁵⁸⁷⁾	⋯
222P	*P222⁽⁰⁾	P222₁⁽⁰⁾	⋯	P2₁2₁2⁽³⁰⁾	P2₁2₁2₁†⁽²⁷⁹⁵⁾
222C	⋯	*C222⁽⁰⁾	⋯	C222₁†⁽¹¹⁾	⋯
222F	⋯	⋯	*F222⁽⁰⁾	⋯	⋯
222I	⋯	⋯	*I222⁽⁰⁾	⋯	⋯
222I			I2₁2₁2₁†⁽⁰⁾
mm2P	*Pmm2⁽⁰⁾	Pma2⁽⁰⁾	⋯	Pmc2₁⁽⁰⁾	Pca2₁⁽¹⁵³⁾
				Pcc2⁽¹⁾	Pna2₁†⁽³⁶⁷⁾
				Pnc2⁽¹⁾
				Pmn2₁⁽⁰⁾
				Pba2⁽¹⁾
				Pnn2⁽¹⁾
mm2C	⋯	*Cmm2⁽⁰⁾	⋯	Cmc2₁†⁽⁰⁾	⋯
mm2C				Ccc2⁽⁰⁾
2mmC	⋯	*Amm2⁽⁰⁾	⋯	Abm2⁽⁰⁾	⋯
				Ama2⁽⁰⁾
				Aba2† ⁽¹¹⁾
mm2F			*Fmm2⁽⁰⁾	Fdd2†⁽³⁵⁾
mm2I	⋯	⋯	*Imm2⁽⁰⁾	Iba2†⁽¹⁴⁾
mm2I				Ima2⁽⁰⁾
mmmP	*Pmmm⁽⁰⁾	Pccm⁽⁰⁾	Pnnn⁽⁰⁾	Pnna⁽¹⁾	Pbca†⁽⁸²⁷⁾
		Pmma⁽⁰⁾	Pban⁽⁰⁾	Pcca⁽³⁾
			Pmna⁽⁰⁾	Pbam⁽⁰⁾
			Pmmn⁽⁰⁾	Pccn⁽³⁷⁾
				Pbcm⁽⁰⁾
				Pnnm⁽⁰⁾
				Pbcn⁽⁶⁰⁾
				Pnma⁽⁰⁾
mmmC	⋯	*Cmmm⁽⁰⁾	Cmma⁽⁰⁾	Cmcm⁽⁰⁾	⋯
				Cmca⁽⁰⁾
				Cccm⁽⁰⁾
				Ccca†⁽⁰⁾
mmmF	⋯	⋯	*Fmmm⁽⁰⁾	Fddd†⁽²⁾	⋯
mmmI	⋯	⋯	*Immm⁽⁰⁾	Ibam⁽⁰⁾
				Ibca⁽⁰⁾
				Imma⁽⁰⁾

(b) Tetragonal space groups. For *, † see Subsection 9.7.4.1.

Arithmetic crystal class	Fully symmorphic	Tending to symmorphism	Equally balanced	Tending to antimorphism	Fully antimorphic
4P	*P4⁽⁰⁾	P4₂⁽¹⁾	⋯	⋯	P4_1,3†⁽⁴⁰⁾
4I	⋯	⋯	I4₁†⁽³⁾	*I4⁽³⁾	⋯
$[\overline {4}]$ P	*P $[\overline {4}]$ †⁽⁰⁾	⋯	⋯	⋯	⋯
$[\overline {4}]$ I	⋯	⋯	*I $[\overline {4}]$ †⁽⁷⁾	⋯	⋯
4/mP	*P4/m⁽⁰⁾	P4₂/m⁽⁰⁾ P4/n⁽¹⁾	P4₂/n†⁽²⁰⁾	⋯	⋯
4/mI	⋯	⋯	⋯	*I4/m⁽⁰⁾ I4₁/a†⁽²⁹⁾	⋯
422P	⋯	*P422⁽⁰⁾	P42₁2⁽⁰⁾	P4_1,32₁2†⁽⁴⁹⁾	⋯
422P		P4₂22⁽⁰⁾	P4_1,322⁽¹⁾	P4₂2₁2⁽¹⁾
422I	⋯	⋯	I4₁22†⁽⁰⁾	*I422⁽⁰⁾	⋯
4mmP	⋯	*P4mm⁽⁰⁾	P4bm⁽⁰⁾	P4₂cm⁽⁰⁾	⋯
				P4₂nm⁽⁰⁾
				P4cc⁽⁰⁾
				P4nc⁽⁰⁾
				P4₂mc⁽⁰⁾
				P4₂bc†⁽¹⁾
4mmI	⋯	⋯	⋯	*I4mm⁽⁰⁾	⋯
				I4cm⁽⁰⁾
				I4₁md⁽⁰⁾
				I4₁cd†⁽⁵⁾
$[\overline {4}]$ 2mP	⋯	*P $[\overline {4}]$ 2m⁽⁰⁾	P $[\overline {4}]$ 2c⁽⁰⁾	P $[\overline {4}]$ 2₁c†⁽¹²⁾	⋯
$[\overline {4}]$ 2mP			P $[\overline {4}]$ 2₁m⁽⁰⁾
$[\overline {4}]$ m2P	⋯	*P $[\overline {4}]$ m2⁽⁰⁾	P $[\overline {4}]$ c2⁽⁰⁾	⋯	⋯
			P $[\overline {4}]$ b2⁽⁰⁾
			P $[\overline {4}]$ n2⁽⁰⁾
$[\overline {4}]$ m2I	⋯	⋯	*I $[\overline {4}]$ m2⁽⁰⁾	I $[\overline {4}]$ c2†⁽⁰⁾	⋯
$[\overline {4}]$ 2mI	⋯	⋯	*I $[\overline {4}]$ 2m⁽⁰⁾	I $[\overline {4}]$ 2d†⁽⁰⁾	⋯
4/mmmP	⋯	P4/mmm*⁽⁰⁾	P4/mcc⁽⁰⁾	P4/nbm⁽⁰⁾	⋯
		P4₂/mmc⁽⁰⁾	P4/nmm⁽⁰⁾	P4/nnc⁽⁰⁾
		P4₂/mcm⁽⁰⁾		P4/mbm⁽⁰⁾
				P4/mnc⁽⁰⁾
				P4/ncc⁽⁰⁾
				P4₂/nbc⁽⁰⁾
				P4₂/nnm⁽⁰⁾
				P4₂/mbc⁽⁰⁾
				P4₂/mnm⁽⁰⁾
				P4₂/nmc⁽⁰⁾
				P4₂/ncm⁽⁰⁾
4/mmmI	⋯	I4/mmm*⁽⁰⁾	⋯	I4/mcm⁽⁰⁾
				I4₁/amd⁽⁰⁾
				I4₁/acd†⁽⁰⁾

Arithmetic crystal class	Fully symmorphic	Tending to symmorphism	Equally balanced	Tending to antimorphism	Fully antimorphic
3P	*P3⁽⁰⁾	⋯	⋯	⋯	P3_1,2†⁽³³⁾
3R	⋯	⋯	⋯	*R3†⁽¹¹⁾	⋯
$[\overline {3}]$ P	*P $[\overline {3}]$ †⁽¹⁾	⋯	⋯	⋯	⋯
$[\overline {3}]$ R	⋯	⋯	⋯	*R $[\overline {3}]$ †⁽³⁰⁾	⋯
312P 321P	⋯	P312⁽⁰⁾ P321⁽⁰⁾	⋯	P3_1,212†⁽⁰⁾ P3_1,221†⁽¹⁰⁾	⋯
32R	⋯	⋯	⋯	*R32†⁽⁰⁾	⋯
3m1P 31mP	⋯	P3m1⁽⁰⁾ P31m⁽⁰⁾	⋯	P3c1†⁽⁰⁾ P31c†⁽⁰⁾	⋯
3mR	⋯	⋯	⋯	*R3m⁽⁰⁾ R3c†⁽⁷⁾	⋯
$[\overline {3}]$ m1P $[\overline {3}]$ 1mP	⋯	P $[\overline {3}]$ m1⁽⁰⁾ P $[\overline {3}]$ 1m⁽⁰⁾	⋯	P $[\overline {3}]$ c1†⁽⁰⁾ P $[\overline {3}]$ 1c†⁽⁰⁾	⋯
$[\overline {3}]$ mR	⋯	⋯	⋯	*R $[\overline {3}]$ m⁽⁰⁾ R $[\overline {3}]$ c†⁽⁰⁾	$[\cdots ]$

(d) Hexagonal space groups. For *, † see Subsection 9.7.4.1.

Arithmetic crystal class	Fully symmorphic	Tending to symmorphism	Equally balanced	Tending to antimorphism	Fully antimorphic
6P	*P6⁽⁰⁾	⋯	P6_2,4⁽¹⁾ P6₃⁽⁰⁾	⋯	P6_1,5†⁽²²⁾
$[\overline {6}]$ P	*P $[\overline {6}]$ †⁽⁰⁾	⋯	⋯	⋯	⋯
6/mP	*P6/m⁽⁰⁾	⋯	P6₃/m†⁽⁰⁾	⋯	⋯
622P	⋯	*P622⁽⁰⁾ P6_2,422⁽⁰⁾	⋯	P6₃22⁽¹⁾ P6_1,522†⁽²⁾	⋯
6mmP	⋯	*P6mm⁽⁰⁾	⋯	P6cc⁽⁰⁾	⋯
6mmP				P6₃cm⁽⁰⁾ P6₃mc⁽⁰⁾
$[\overline {6}]$ m2P $[\overline {6}]$ 2mP	⋯	P $[\overline {6}]$ m2⁽⁰⁾ P $[\overline {6}]$ 2m⁽⁰⁾	⋯	P $[\overline {6}]$ c2†⁽⁰⁾ P $[\overline {6}]$ 2c†⁽⁰⁾
6/mmmP	⋯	P6/mmm*⁽⁰⁾	⋯	P6/mcc†⁽⁰⁾	⋯
6/mmmP				P6₃/mcm⁽⁰⁾ P6₃/mmc⁽⁰⁾

(e) Cubic space groups. For *, †, see Subsection 9.7.4.1. No examples with one molecule in general position were found, so the frequencies are omitted.

Arithmetic crystal class	Fully symmorphic	Tending to symmorphism	Equally balanced	Tending to antimorphism	Antimorphic except for 3
23P	⋯	*P23	⋯	⋯	P2₁3†
23F	⋯	⋯	*F23†	⋯	⋯
23I	⋯	⋯	*I23 I2₁3†	⋯	⋯
m $[\overline 3]$ P	⋯	*Pm $[\overline 3]$	Pn $[\overline 3]$	⋯	Pa $[\overline 3 ]$ †
m $[\overline 3]$ F	⋯	⋯	*Fm $[\overline 3]$	Fd $[\overline 3]$ †	⋯
m $[\overline 3]$ I	⋯	⋯	*Im $[\overline 3]$	Ia $[\overline 3]$ †	⋯
432P	⋯	*P432	⋯	P4₂32† P4_1,332†	⋯
432F	⋯	⋯	*F432	F4₁32†	⋯
432I	⋯	⋯	*I432	I4₁32†	⋯
$[\overline {4}]$ 3mP	⋯	*P $[\overline {4}]$ 3m	⋯	P $[\overline {4}]$ 3n†	⋯
$[\overline {4}]$ 3mF	⋯	⋯	*F $[\overline {4}]$ 3m	F $[\overline {4}]$ 3c†	⋯
$[\overline {4}]$ 3mI	⋯	⋯	*I $[\overline {4}]$ 3m	I $[\overline {4}]$ 3d†	⋯
m $[\overline {3}]$ mP	⋯	*Pm $[\overline {3}]$ m	Pm $[\overline {3}]$ n Pn $[\overline {3}]$ m	Pn $[\overline {3}]$ n†	⋯
m $[\overline {3}]$ mF	⋯	*Fm $[\overline {3}]$ m	⋯	Fm $[\overline {3}]$ c Fd $[\overline {3}]$ m	Fd $[\overline {3}]$ c†
$[m\overline {3}]$ mI	⋯	*Im $[\overline {3}]$ m	⋯	Ia $[\overline {3}]$ d†	⋯

A few points about the symmorphic groups are worth noting. The 14 `fully symmorphic' space groups are those that (i) have primitive cells and (ii) have no secondary or tertiary axes (three each monoclinic, orthorhombic, tetragonal, hexagonal; two trigonal; no cubic). Secondary axes, even though syntropic in the conventional space-group notation, generate additional antitropic axes in accordance with the principles set out by Bertaut (2005, Chap. 4.1 ). These additional axes are not indicated in the `full' Hermann–Mauguin space-group symbol, but should appear in the `extended' symbol (Bertaut, 2005, Table 4.3.2.1 ⁴). As a result of the additional axes, 21 symmorphic space groups with primitive cells are shifted to the `tending to symmorphism' column (five tetragonal, six trigonal, five hexagonal, five cubic). Lattice centring has a similar or greater effect; the 36 centred symmorphic space groups are spread over the three columns `tending to symmorphism' (seven), `equally balanced' (20) and `tending to antimorphism' (nine).

From the nature of the definitions, no symmorphic space group can be `fully antimorphic'. The 14 groups under the latter heading consist of (i) 12 space groups with no special positions (Subsection 9.7.4.1), and (ii) two space groups whose only special positions have symmetry $[\overline {1}]$ (Subsection 9.7.4.2). On these criteria, the two triclinic groups excluded from discussion would fall naturally into the column `fully antimorphic'. The remaining space groups have no obvious outstanding characteristics. Most of them fall under the heading `tending to antimorphism', though there are some in each of the columns `tending to symmorphism' and `equally balanced'.

References

Belsky, V. K., Zorkaya, O. N. & Zorky, P. M. (1995). Structural classes and space groups of organic homomolecular crystals: new statistical data. Acta Cryst. A51, 473–481.Google Scholar

Bertaut, E. F. (2005). International tables for crystallography, Vol. A, fifth edition, Chap. 4.1. Heidelberg: Springer.Google Scholar

Wilson, A. J. C. (1993d). Space groups rare for organic structures. III. Symmorphism and inherent molecular symmetry. Acta Cryst. A49, 795–806.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 9.7, pp. 897-899