A priori
           classifications of space groups

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. 897-900

Section 9.7.1. A priori classifications of space groups

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. A priori classifications of space groups

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The space group P2₁/c accounts for about 1/3 of all known molecular organic structures, whereas the space group P2/m has no certain example. Why? Ultimately, the space group of a crystal of a particular substance is determined by the minimum (or a local minimum) of the thermodynamic potential (Gibbs free energy) of the van der Waals and other forces, but a very simple model goes a long way towards `explaining' the relative frequency of the various space groups within a crystal class or larger grouping. Nowacki (1943) discussed the basic idea that space-group frequency is determined by packing considerations, and had earlier (1942) given statistics for the structures known at the time. Nowacki's statistics were used by Kitajgorodskij¹ (1945), and recent writers tend to cite Kitajgorodskij as having originated the idea. Kitajgorodskij pointed out that the most frequent space groups are those that permit the close packing of triaxial ellipsoids. Later, Kitajgorodskij (1955) showed that the same space groups allowed close packing of objects of any reasonable shape, `close packing' meaning packing with 12-point contact. Wilson (1988, 1990, 1993d) used the complementary idea that space groups are rare when they contain symmetry elements – notably mirror planes and rotation axes – that prevent the molecules from freely choosing their positions within the unit cell. A twofold axis excludes molecular centres from a column of diameter equal to some molecular diameter (say M), a mirror plane from a layer of thickness M, and a centre of symmetry from a sphere of diameter M. The volumes excluded by screw axes and glide planes are small in comparison.

9.7.1.1. Kitajgorodskij's categories

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In his book² Organicheskaya Kristallokhimiya, Kitajgorodskij (1955) treated the triclinic, monoclinic and orthorhombic space groups in considerable detail, analysing the possibility of (a) forming close-packed layers (six-point contact), and (b) close stacking of the layers. On this basis, he divided the layers and the space groups into four categories each. For the layers they are:

(1) Coordination close-packed layers. A coordination close-packed layer is one in which molecules of arbitrary shape and symmetry can be packed with six-point coordination.
(2) Closest-packed layers. A closest-packed layer is one in which one can select the orientation of molecules of given shape and symmetry so as to produce a cell of minimal dimensions.
(3) Limitingly close-packed layers. A limitingly close-packed layer for a given symmetry is a closest-packed layer in which a molecule retains inherent symmetry; in other words, in which it occupies a special position.
(4) Permissible layers. A permissible layer is coordination close packed but neither closest packed nor limitingly close packed.

The categories of space groups are:

(1) Closest-packed space groups are those that permit the closest stacking of closest-packed layers – the packing can be made no denser by varying the cell parameters and the orientation of the molecules. Closest stackings can be made by a monoclinic displacement (a translation making an arbitrary angle with the layer plane), a centre of symmetry, a glide plane, or a screw axis.
(2) Limitingly close-packed space groups are those that contain limitingly close-packed layers stacked as closely as possible.
(3) Permissible space groups fall into three subcategories: (a) Those containing closest-packed layers that can be closely stacked if the layer relief is suitable; this group contains layers stacked by centring (C, I, F) or by diad axes. (b) Those containing limitingly close-packed layers that can be most closely stacked if the layer relief is suitable. (c) Those containing permissible layers stacked in the densest fashion.
(4) Impossible space groups fall into two subcategories: (a) Those containing any layers (even closest-packed layers) that are related by mirror planes and translations normal to the layer plane. (b) Those containing permissible coordination close-packed layers not stacked in the densest possible way.

Kitajgorodskij expected the frequency of space groups to decrease in the order (1) > (2) > (3 > (4). In particular, `permissible space groups should be found but rarely, as exceptions'. The categorization is summarized in Table 9.7.1.1, based on Table 8 of Kitajgorodskij (1955).

Table 9.7.1.1| top | pdf |
Kitajgorodskij's categorization of the triclinic, monoclinic and orthorhombic space groups, as modified by Wilson (1993a)

Wilson's additions are enclosed in square brackets [⋯] and the original positions of space groups transferred by him by round brackets (⋯). Space groups not listed belong to the `impossible' category.

Molecular symmetry	1	$[\overline {1}]$	2	m	2/m	222	mm	mmm
Closest packed	P $[\overline {1} ]$	P $[\overline {1} ]$
	P2₁
	P2₁/c	P2₁/c
	[C2/c]	C2/c	[C2/c]
	P2₁2₁2₁
	Pca2₁
	Pna2₁
	[Pbca]	Pbca
Limitingly close packed			(C2/c)		C2/m
	[P2₁2₁2]		P2₁2₁2			C222
						F222
						I222
				Pmc2₁			Fmm2
				Cmc2₁
	[Pbcn]		Pbcn	Pnma			Pmma, Pmmn
					Cmca	Ccca		Cmmm
						^†		Fmmm
								Immm
Permissible	P1
	C2		C2
	[Pc]
	Cc			Cm	Pbam
	[P2/c]		[P2/c]	P2₁/m
	(P2₁2₁2)
	[C222₁]		[C222₁]
				Pmn2₁
			Aba2	Ama2
			[Fdd2]	Ima2
	(Pbca)			Pbcm^‡
	[Pccn]	Pccn

^†Kitajgorodskij (1961

) includes Pnnm at this position, but this is inconsistent with the text of either the Russian or the English version.
^‡Kitajgorodskij (1961

) correctly includes Pbcm at this position.

Kitajgorodskij's categorization proved very successful in broad outline, but Wilson's (1993b,c) detailed statistics revealed about a dozen anomalous space-group types. The anomalies were of two kinds. The first was the frequent occurrence of molecules in general positions in space groups in which Kitajgorodskij expected molecules to use inherent symmetry in special positions. Wilson (1993a) pointed out that in such cases structural dimers³ can be formed, with two molecules in general positions related by the required symmetry elements – both enantiomers would be required if the element were $[\overline {1}]$ or m. Such space groups could therefore be added to Kitajgorodskij's table, in the column for `molecular symmetry 1'. The second kind of anomaly was the fairly frequent occurrence of structures with the `impossible' space groups Pc and P2/c. These could be transferred from `impossible' to `permissible', subgroup (a), by the same packing argument that Kitajgorodskij had used for P1. These and a few other reclassifications are indicated in Table 9.7.1.1, the new entries being enclosed in square brackets for distinction. Where the change is a transfer to a higher category, the original position of the space group is indicated in round brackets.

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'.

9.7.1.3. Comparison of Kitajgorodskij's and Wilson's classifications

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Since both Kitajgorodskij's and Wilson's classifications were made with a view to `explaining' the empirically observed frequencies of space groups – though neither makes any use of empirical frequencies – it would be expected that there should be considerable correlation between them. All `closest-packed' space groups are also `fully antimorphic', and most of the `limitingly close packed' and `permissible' are `tending to antimorphism'; a few requiring high molecular symmetry (222, mm2, mmm) and a couple of others are `equally balanced'. Two `fully antimorphic' groups, Pc and Cc, are merely `permissible'. All `fully symmorphic' space groups are `impossible'.

9.7.1.4. Relation to structural classes

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Structural classes (Belsky & Zorky, 1977, and papers cited there and below) are not an a priori classification of space groups but are a classification of structures within a space-group type in accordance with the number and kind of Wyckoff positions occupied by the molecules. As a considerable knowledge of the structures is required before their structural classes can be assigned, they form an a posteriori classification, and will be described (Section 9.7.5 below) after the empirical frequencies of space groups have been discussed.

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