Tables for
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.4, pp. 800-824

Section 3.4.4. Applications of the lattice-complex concept

W. Fischera and E. Kocha*

aInstitut für Mineralogie, Petrologie und Kristallographie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail:

3.4.4. Applications of the lattice-complex concept

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To study the geometrical properties of all point configurations in three-dimensional space, it is not necessary to consider all Wyckoff positions of the space groups or all 1128 types of Wyckoff set. Instead, one may restrict the investigations to the characteristic Wyckoff positions of the 402 lattice complexes. The results can then be transferred to all non-characteristic Wyckoff positions of the lattice complexes, as listed in Tables[link] and[link].

The determination of all types of sphere packings with cubic and tetragonal symmetry forms an example for this kind of procedure (Fischer, 1973[link], 1974[link], 1991a[link],b[link], 1993[link]). The cubic lattice complex I4xxx, for example, allows two types of sphere packings within its characteristic Wyckoff position [I\bar{4}3m] 8c .3m. [x,x,x]. Sphere packings with three-membered rings and nine contacts per sphere are formed if [x = 3/16]. The parameter region [{3/16} \,\lt\, x \,\lt\, \textstyle{1 \over 4}] corresponds to sphere packings with four-membered rings and six contacts per sphere (cf. Fischer, 1973[link]). Ag3PO4 crystallizes with symmetry [P\bar{4}3n] (Deschizeaux-Cheruy et al., 1982[link]) and the oxygen atoms occupy Wyckoff position 8e .3. [x,x,x], which also belongs to lattice complex I4xxx. Comparison of the coordinate parameter x = 0.1491 for the oxygen atoms with the sphere-packing parameters listed for [I\bar{4}3m] c shows directly that the oxygen arrangement in this crystal structure does not form a sphere packing.

Other examples for this approach are the derivation of crystal potentials (Naor, 1958[link]), of coordinate restrictions in crystal structures (Smirnova, 1962[link]), of Patterson diagrams (Koch & Hellner, 1971[link]), of Dirichlet domains (Koch, 1973[link], 1984[link]) and of sphere packings for subperiodic groups (Koch & Fischer, 1978b[link]).

The 30 lattice complexes in two-dimensional space correspond uniquely to the `henomeric types of dot pattern' introduced by Grünbaum and Shephard (cf. e.g. Grünbaum & Shephard, 1981[link]; Grünbaum, 1983[link]). Relations between crystal structures

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Different crystal structures frequently show the same geometrical arrangement for some of their atoms, even though their space groups do not belong to the same type. In such cases, the corresponding Wyckoff positions either belong to the same lattice complex or there exists a close relationship between them, e.g. a limiting-complex relation.


  • (1) The Fe atoms in pyrite FeS2 occupy Wyckoff position [4a\ .\bar{3}.\ 0,0,0] of [Pa\bar{3}] (descriptive symbol F) that belongs to the invariant lattice complex [Fm\bar{3}m\ a]. Accordingly, the Fe atoms in pyrite form a face-centred cubic lattice, as do the Cu atoms in the element structure of copper.

  • (2) Cuprite Cu2O crystallizes with symmetry [Pn\bar{3}m]. The oxygen atoms occupy Wyckoff position [2a\ \bar{4}3m\ 0,0,0] (descriptive symbol I) and the copper atoms position [4b\ .\bar{3}m\ \textstyle{1 \over 4}, \textstyle{1 \over 4},\textstyle {1 \over 4}] (descriptive symbol [\textstyle{1 \over 4}\textstyle {1 \over 4}\textstyle {1 \over 4}\; F]). Position 2a belongs to lattice complex [Im\bar{3}m\ a] and position 4b to [Fm\bar{3}m\ a]. Therefore, the O atoms form a body-centred cubic lattice like the W atoms in the structure of tungsten, and the copper atoms form a face-centred cubic lattice. The tungsten configuration is shifted by [\textstyle{1 \over 4}, \textstyle{1 \over 4},\textstyle {1 \over 4}] with respect to the copper configuration.

  • (3) K2NaAlF6 (elpasolite, cf. Morss, 1974[link]) and K2PbNi(NO2)6 (cf. Takagi et al., 1975[link]) crystallize with symmetry [Fm\bar{3}m] and [Fm\bar{3}], respectively.[\displaylines{\quad\quad\quad{\rm K}_{2}{\rm NaAlF}_{6}\hfill\cr \quad\matrix{\hbox{Al}\hfill &4a \hfill &m\bar{3}m \hfill &0,0,0 \hfill &F\hfill\cr \hbox{Na} \hfill &4b \hfill &m\bar{3}m \hfill &\textstyle{1 \over 2},\textstyle{1 \over 2},\textstyle{1 \over 2} \hfill &\textstyle{1 \over 2}\textstyle{1 \over 2}\textstyle{1 \over 2} F\cr \hbox{K} \hfill &8c \hfill &\bar{4}3m \hfill &\textstyle{1 \over 4},\textstyle{1 \over 4},\textstyle{1 \over 4} \hfill &\textstyle{1 \over 4}\textstyle{1 \over 4}\textstyle{1 \over 4} P_{2}\cr \hbox{F} \hfill &24e \hfill &4m.m \hfill &x,0,0 \hfill &F6z\hfill\cr & & &x = 0.219\hfill \cr}}][\displaylines{\quad\quad\quad {\rm K}_{2}\hbox{\rm PbNi(NO}_{2}{\rm )}_{6}\hfill\cr \quad\matrix{\hbox{Ni} \hfill &4a \hfill &m\bar{3}. \hfill &0,0,0 \hfill &F\hfill\cr \hbox{Pb} \hfill &4b \hfill &m\bar{3}. \hfill &\textstyle{1 \over 2},\textstyle{1 \over 2},\textstyle{1 \over 2} \hfill &\textstyle{1 \over 2}{1 \over 2}\textstyle{1 \over 2} F\hfill\cr \hbox{K} \hfill &8c \hfill &23. \hfill &\textstyle{1 \over 4},\textstyle{1 \over 4},\textstyle{1 \over 4} \hfill &\textstyle{1 \over 4}\textstyle{1 \over 4}\textstyle{1 \over 4} P_{2} \hfill\cr \hbox{N} \hfill &24e \hfill &mm2.. \hfill &x,0,0 \hfill &F6z\hfill\cr & & &x = 0.1966\cr \hbox{O} \hfill &48h \hfill &m.. \hfill &0,y,z \hfill &F6z2x \hfill \cr}}]As the descriptive lattice-complex symbols for the various atomic positions show immediately, the two crystal structures are very similar. The only difference originates from the replacement of the fluorine atoms in elpasolite by NO2 groups in K2PbNi(NO2)6, which causes the symmetry reduction from [Fm\bar{3}m] to [Fm\bar{3}].

  • (4) The crystal structure of CoU (Baenziger et al., 1950[link]) may be interpreted as a slightly distorted CsCl (or β-brass, CuZn)-type structure. CsCl corresponds to Wyckoff positions 1a and 1b of [Pm\bar{3}m] with descriptive symbols P and [\textstyle{1 \over 2}\textstyle{1 \over 2}\textstyle{1 \over 2}\, P], respectively; Co and U both occupy Wyckoff position 8a .3. [x,x,x] of [I2_{1}3] with x = 0.0347 for U and x = 0.294 for Co. As the descriptive symbol [2_{1}2_{1}..\ P_{2}Y^{*}1xxx] shows, this Wyckoff position belongs to a Weissenberg complex with two invariant limiting complexes, namely P ([Pm\bar{3}m\ a]) and [Y^{*}\ (I4_{1}32\ a)]. x = 0 corresponds to [P_{2}], [x = \textstyle{1 \over 4}] to [\textstyle{1 \over 4}\textstyle{1 \over 4}\textstyle{1 \over 4}\, P_{2}], [x = \textstyle{1 \over 8}] to [^{+}Y^{*}] and [x = \textstyle{7 \over 8}] to [^{-}Y^{*}]. Consequently, the uranium and cobalt atoms form approximately a [P_{2}] and a [\textstyle{1 \over 4}\textstyle{1 \over 4}\textstyle{1 \over 4}\, P_{2}] configuration, respectively.

Publications by Hellner (1965[link], 1976a[link],b[link],c[link], 1977[link], 1979[link]), Loeb (1970[link]), Smirnova & Vasserman (1971[link]), Sakamoto & Takahasi (1971[link]), Niggli (1971[link]), Fischer & Koch (1974b[link]), Hellner et al. (1981[link]) and Hellner & Sowa (1985[link]) refer to this aspect. Reflection conditions

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Wyckoff positions belonging to the same lattice complex show analogous reflection conditions. Therefore, lattice complexes have also been used to check the reflection conditions for all Wyckoff positions in the space-group tables of this volume.


The lattice complex oF consists of all face-centred point lattices with orthorhombic symmetry. For its characteristic Wyckoff position Fmmm 4a, only the general conditions for reflections hkl in space group Fmmm are valid, namely h + k, h + l, k + l = 2n (cf. Chapter 2.3[link] ). The non-characteristic Wyckoff position Ccce 4a also belongs to this lattice complex. The general reflection condition for Ccce is hkl: h + k = 2n. This has to be combined with k + l = 2n, the special condition for Wyckoff position a. Together the two conditions produce h + l = 2n, the third condition for a face-centred point lattice.

The descriptive symbols may supply information on the reflection conditions. If the symbol does not contain any distribution-symmetry part, the reflection conditions of the Wyckoff position are indicated by the symbol of the invariant lattice complex in the central part (e.g. P4/nmm g: C4xx shows that the reflection condition is that of a C lattice, hkl: h + k = 2n). In cases where the site set consists of only one point, i.e. the Wyckoff position belongs to a Weissenberg complex, all conditions for general reflections hkl that may arise from special choices of the coordinates can be read from the central part of the symbol (e.g. [P4/nmm\ c\hbox{:}\ 0\textstyle{1 \over 2}0\; .. 2\ CI1z] indicates that, by special choice of z, either hkl: h + k = 2n or hkl: h + k + l = 2n may be produced). Phase transitions

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If a crystal undergoes a phase transition from a high- to a low-symmetry modification, the transition may be connected with a group–subgroup transition. In such cases, the comparison of the lattice complexes corresponding to the Wyckoff positions of the original space group on the one hand and of its various subgroups on the other hand very often shows which of these subgroups are suitable for the low-symmetry modification.

This kind of procedure will be demonstrated with the aid of the space group [R\bar{3}m] and its three translationengleiche subgroups with index 2, namely R32, [R\bar{3}] and R3m. In the course of the restriction to a subgroup, the Wyckoff positions of [R\bar{3}m] behave differently:

The descriptive symbols R and [00\textstyle{1 \over 2}\,R] refer to Wyckoff positions [R\bar{3}m\ 3a] and 3b as well as to Wyckoff positions R32 3a and 3b and [R\bar{3}] 3a and 3b. Therefore, all corresponding point configurations and atomic arrangements remain unchanged in these subgroups. In subgroup R3m, however, the respective Wyckoff position is 3a with descriptive symbol R[z], i.e. a shift parallel to [001] of the entire point configuration is allowed.

The descriptive symbol R2z for [R\bar{3}m\ 6c] also occurs for R32 6c and [R\bar{3}\ 6c]. Again, neither subgroup allows any deformations of the corresponding point configurations or atomic arrangements. Symmetry reduction to R3m, however, yields a splitting of each R2z configuration into two R[z] configurations. The two z parameters may be chosen independently.

As M and [00\textstyle{1 \over 2}\ M] are the descriptive symbols not only of [R\bar{3}m] 9e and 9d but also of [R\bar{3}] 9e and 9d, [R\bar{3}] does not enable any deformation of the corresponding atomic arrangements. In R32 and in R3m, however, the respective point configurations may be differently deformed, as the descriptive symbols show: R3x and [00\textstyle{1 \over 2}\, R3x] (R32 9d and 9e), [R3x\bar{x}[z]] (R3m 9b).

Wyckoff positions [R\bar{3}m\ 18f] and 18g (R6x and [00\textstyle{1 \over 2}\, R6x]) correspond to R32 9d and 9e (R3x and [00\textstyle{1 \over 2}\, R3x]), to [R\bar{3}] 18f (R6xyz), and to R3m 18c [(R3x\bar{x}2y[z])]. In R32, the hexagons 6x around the points of the R lattice are split into two oppositely oriented triangles 3x, which may have different size. In [R\bar{3}] and in R3m, the hexagons may be differently deformed.

Wyckoff position [R\bar{3}m\ 18h\ (R6x\bar{x}z)] corresponds to sets of trigonal antiprisms around the points of an R lattice. These antiprisms may be distorted in R32 18f (R3x2yz) or rotated in [R\bar{3}] 18f (R6xyz). In R3m 9b [(R3x\bar{x}[z])], each antiprism is split into two parallel triangles that may differ in size.

In each of the three subgroups, any point configuration belonging to the general position [R\bar{3}m] 36i splits into two parts. Each of these parts may be differently deformed. Incorrect space-group assignment

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In the literature, some crystal structures are still described within space groups that are only subgroups of the correct symmetry groups. Many such mistakes (but not all of them) could be avoided by simply looking at the lattice complexes (and their descriptive symbols) that correspond to the Wyckoff positions of the different kinds of atoms. Whenever the same (or an analogous) lattice-complex description of a crystal structure is also possible within a supergroup, then the crystal structure has at least that symmetry.


  • (1) The crystal structure of β-LiRhO2 has been refined in space group [F4_{1}32] (cf. Hobbie & Hoppe, 1986[link]).[\openup3pt\matrix{\hbox{Rh} \hfill &16c \hfill &.32 \hfill &\textstyle{1 \over 8},\textstyle{1 \over 8},\textstyle{1 \over 8} \hfill &T\hfill \cr \hbox{Li} \hfill &16d \hfill &.32 \hfill &\textstyle{5 \over 8},\textstyle{5 \over 8},\textstyle{5 \over 8} \hfill &\textstyle{1 \over 2}\textstyle{1 \over 2}\textstyle{1 \over 2}\, T\hfill\cr \hbox{O} \hfill &32e \hfill &.3. \hfill &x,x,x \hfill &..2\ D4xxx \hfill\cr}]The same atomic arrangement is possible in the supergroup [Fd\bar{3}m] of [F4_{1}32], as can easily be read from Table[link]:[\openup3pt\matrix{\hbox{Rh} \hfill &16c \hfill &.\bar{3}m \hfill &\textstyle{1 \over 8},\textstyle{1 \over 8},\textstyle{1 \over 8} \hfill &T\hfill\cr \hbox{Li} \hfill &16d \hfill &.\bar{3}m \hfill &\textstyle{5 \over 8},\textstyle{5 \over 8},\textstyle{5 \over 8} \hfill &\textstyle{1 \over 2}\textstyle{1 \over 2}\textstyle{1 \over 2}\, T\hfill\cr \hbox{O} \hfill &32e \hfill &.3m \hfill &x,x,x \hfill &..2\ D4xxx\hfill\cr}]Therefore, β-LiRhO2 should be described in [Fd\bar{3}m].

  • (2) KIAs4O6 (Pertlik, 1988[link]) has been described with symmetry P622.[\openup3pt\matrix{\hbox{I} \hfill &1a \hfill &622 \hfill &0,0,0 \hfill &P\hfill\cr \hbox{K} \hfill &1b \hfill &622 \hfill &0,0,\textstyle{1 \over 2} \hfill &00\textstyle{1 \over 2}\, P\hfill \cr \hbox{As} \hfill &4h \hfill &3.. \hfill &\textstyle{1 \over 3},\textstyle{2 \over 3},z \hfill &G2z\hfill \cr \hbox{O} \hfill &6i \hfill &2.. \hfill &\textstyle{1 \over 2},0,z \hfill &N2z\hfill \cr}]Space group [P6/mmm] allows the same atomic arrangement:[\openup3pt\matrix{\hbox{I} \hfill &1a \hfill &6/mmm \hfill &0,0,0 \hfill &P\hfill\cr \hbox{K} \hfill &1b \hfill &6/mmm \hfill &0,0,\textstyle{1 \over 2} \hfill &00\textstyle{1 \over 2}\, P\hfill\cr \hbox{As} \hfill &4h \hfill &3m. \hfill &\textstyle{1 \over 3},\textstyle{2 \over 3},z \hfill &G2z\hfill\cr \hbox{O} \hfill &6i \hfill &2mm \hfill &\textstyle{1 \over 2},0,z \hfill &N2z\hfill \cr}]Therefore, KIAs4O6 should be described in [P6/mmm]. Application of descriptive lattice-complex symbols

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Descriptive symbols of lattice complexes – at least those of the invariant lattice complexes – have been used for the description of crystal structures (cf. Section[link] and the literature cited there), for the nomenclature of three-periodic surfaces (von Schnering & Nesper, 1987[link]) and in connection with orbifolds of space groups (Johnson et al., 2001[link]). Weissenberg complexes

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In general, each lattice complex involves point configurations that cannot be related to any crystal structure because the shortest distances between the atoms in a corresponding arrangement would become too small. Only the 67 Weissenberg complexes (cf. Section[link]) form an exception from this rule. Assuming that the metrical parameters are chosen adequately, each point configuration stemming from a Weissenberg complex may, in principle, refer to the arrangement of some atoms in a crystal structure. In case of the 36 invariant lattice complexes this property is immediately evident. The further 31 Weissenberg complexes have one or more degrees of freedom (cf. Section[link] and Table[link]). Nevertheless, varying the corresponding free coordinate parameters never results in point configurations with infinitesimally small distances.


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