Coloring and distortion variants of the bcc packing and for the aristotypes BaAl₄ and CeMg₂Si₂

1 Introduction

In the early times of X-ray structure determination, names for structure types were assigned on the basis of the Strukturbericht designation: A for elements, B for AB compounds, … D for A_mB_n compounds, … L for alloys, O for organic structures etc. [1], [2], [3], [4], [5], [6]. With the broad use of X-ray diffraction and the increasing number of compounds with complex compositions it soon became clear that this notation quickly reached its limits.

The Pearson data base [7], the TYPIX compilation [8] and The AFLOW Library of Crystallographic Prototypes [9], [10] use a pure crystallographic fingerprint; the space group symbol, the Pearson code (which includes the number of atoms within the unit cell) and the Wyckoff sequence. With this fingerprint it is possible to find all known compounds which have that given structural arrangement. Nevertheless, also this fingerprint has its limitations when it comes to the crystal-chemical interpretation. To give an example, CsCl and ErMg both crystallize in space group Pm3‾m, the Pearson code cP2 and the Wyckoff sequence ba. The crystal-chemical difference concerns the chemical bonding, i.e. salt-like CsCl versus intermetallic ErMg. Consequently, both compounds are rather isopointal [11], [12] than isotypic.

A further crystal-chemical feature concerns superstructures and distortion variants that derive from a structure type of higher symmetry. Drawing back to CsCl and ErMg as examples, these are the many bcc superstructures [13], [14]. Such superstructure and distortion variants are related by group-subgroup schemes. The compact and concise Bärnighausen formalism [15], [16], [17], [18] facilitates the crystal-chemical understanding of larger structural families. So-called Bärnighausen trees have been reported for the families of perovskite [19], [20] and rutile [21], [22] related structures in the field of ionic solids as well as for the intermetallic families of the aristotypes AlB₂ [23] and BaAl₄ [24], [25], [26], [27], [28], covering several thousand compounds that are crystal chemically related. Especially in the field of intermetallics [14, 29], these group-subgroup schemes are helpful for the understanding of coloring problems (introduced by the Miller group [30], [31], [32]; which Wyckoff position is occupied by which element?).

Herein we summarize some new crystal-chemical data that relate the structures of some inorganic (mainly intermetallic) compounds by group-subgroup schemes. These compounds are derived from the bcc packing or from the aristotypes BaAl₄ and CeMg₂Si₂.

2 Ordering variants of the bcc packing

The first examples stem from the bcc packing, one of the basic metal structure types. The simplest ordering variant is the CsCl type. The Cs⁺/Cl⁻ ordering induces a klassengleiche symmetry reduction from Im3‾m to Pm3‾m and this is detectable in the diffraction patterns through superstructure reflections. This is the textbook example [13, 29] for bcc superstructures.

Herein we concentrate on klassengleiche bcc superstructures of index 3 (k3). This ordering is possible for a combination of two elements (typical AB₂ compounds) that are different in size. The space group symmetry is then I4/mmm, with the Pearson code tI6 and the Wyckoff sequence ea. The Pearson data base [7] lists almost 200 entries for this crystallographic fingerprint; however, these compounds show distinctly different bonding peculiarities as a consequence of the size and the chemical nature of the atoms. A reasonable classification is possible via the c/a ratio of the unit cell. This was already reported by Parthé and coworkers in the early paper on the Structure Tidy program [12] and in a compilation on Elements of Inorganic Structural Chemistry [33].

From the 200 entries we present some examples in Figure 1. The respective structures are arranged with increasing c/a ratio. We start with the intermetallic compounds MoSi₂ [34], AlCr₂ [35], CuZr₂ [36], BePd₂ [37], and CdTi₂ [38]. Their c/a ratio increases from 2.46 for MoSi₂ to 4.68 for CdTi₂. The structural flexibility for these phases is achieved mainly by changes in the lattice parameters. For most phases (where diffraction data is available) the z parameters of the 4e site do not substantially deviate from the subcell value of 1/3. The drastic changes of the c/a ratios then influence the bonding peculiarities. To account for these differences, these AB₂ phases were devided into the MoSi₂ (c/a ratios <3) and CuZr₂ (c/a ratios >3) branches. Although these phases all have the same crystallographic fingerprint, those of the different branches are only isopointal [11], [12]. It is interesting to note that also LaI₂ [39] adopts this fingerprint. The excess electron of trivalent lanthanum leads to metallic behavior for LaI₂ [40].

Figure 1:

Projections of several crystal structures with the crystallographic fingerprint of space group type I4/mmm, Pearson code tI6 and Wyckoff sequence ea. The structures are presented with increasing c/a ratio. For details see text.

We now turn to the structure of calcium acetylide CaC₂ [41]. This salt-like carbide has the same crystallographic fingerprint, but a drastically different bonding situation. The carbon atoms in the C22− acetylide unit have a C–C distance of 120 pm and this leads to a collapse of the c axis and thus a drastic decrease of the c/a ratio to 1.64. The whole series of AC₂, AEC₂, REC₂ (A = alkali cation, AE = alkaline earth cation, RE = rare earth cation) acetylides, the AO₂ superoxides, the AEO₂ peroxides and the AEN₂ diazenides [7] are isotypic (and only isopointal with the compounds discussed above!) with CaC₂ and form the third branch within the family of bcc superstructures.

A unique case is the XeF₂ structure [42], [43]. In contrast to the intermetallic compounds and the ionic compound CaC₂ discussed above, XeF₂ is a molecular compound with covalent intramolecular and van der Waals intermolecular bonding. The shift of the fluorine atoms in XeF₂ is opposite to the shift of the carbon atoms in CaC₂. This also leads to a collapse of the c axis and a small c/a ratio and to the formation of linear XeF₂ units. The low-temperature modification of KrF₂ [44] is isotypic with XeF₂.

The packing motif of the XeF₂ molecules is preserved in several ternary compounds which can be considered as ‘filled’ variants. This has first been observed for the oxidomercurate Na₂[HgO₂] [45] and later for other oxides, nitrides, hydrides and carbides [7]. As an example we present the packing of the linear polyanionic units [BC₂]⁵⁻, [PdD₂]²⁻ and [HgO₂]²⁻ in Sc₂BC₂ [46], Na₂PdD₂ [47] and Na₂HgO₂ [45], respectively, in Figure 2. The c/a ratio of these examples is again dominated by the nature of the atoms forming the respective structure. It is worthwhile to note that the structures of the cyanamides Na₂CN₂ [48] and K₂CN₂ [49] are monoclinic distortion variants of the Na₂HgO₂ type. The complete Bärnighausen tree is presented for Na₂CN₂ in ref. [48].

Figure 2:

The crystal structures of Sc₂BC₂ [46], Na₂PdD₂ [47] and Na₂HgO₂ [45]. The linear polyanionic units [BC₂]⁵⁻, [PdD₂]²⁻ and [HgO₂]²⁻ are emphasized.

The description of the ‘filled’ variants is only intuitive and simply guided by comparison with the XeF₂ molecules. A closer look at the unit cells of Sc₂BC₂, Na₂PdD₂ and Na₂HgO₂ (Figure 2) readily reveals that the substructures [Sc₂B], [Na₂Pd] and [Na₂Hg] correspond to a MoSi₂-type arrangement with the carbon, deuterium and oxygen atoms filling distorted C@Sc₅B, D@Na₅Pd and O@Na₅Hg octahedral voids in an ordered manner. The corresponding group-subgroup scheme for the isotypic carbide Th₂NiC₂ [50] is presented in Figure 3.

Figure 3:

Group-subgroup scheme in the Bärnighausen formalism [15], [16], [17], [18] for the structures of Th₂NiC₂ [50] and YCoC [51] along with the evolution of the atomic parameters for the octahedral voids (VAC denotes vacancies). In each carbide only one void (not shaded) is occupied.

The tungsten structure (the aristotype) exhibits six crystallographically equivalent distorted octahedral voids per unit cell (Wyckoff site 6b) which correspond to the middle of the edges and the faces. For steric reasons, only partial occupancy of the voids is possible and this can only be achieved through a symmetry reduction. In going to the protactinium structure (the compressed version of the tungsten type) with a translationengleiche symmetry reduction of index 3 (t3) to I4/mmm we get a 4 + 2 splitting of the 6b site. The subsequent tripling of the unit cell along the c axis (isomorphic transition of index 3) leads to a further 4 + 2 splitting of all three sites, and in total we obtain the two sites for thorium and nickel and four possible octahedral voids of which only the Wyckoff site 4e is occupied by carbon. This leads to the linear carbidonickelate units presented in Figure 4.

Figure 4:

The crystals structures of Th₂NiC₂ [50] and YCoC [51]. Thorium (yttrium), nickel (cobalt) and carbon atoms are drawn as light gray, blue and black circles, respectively. The [NiC₂] and [CoC] substructures are emphasized.

A similar ordered occupancy of a bcc derived metal substructure has been observed for YCoC [51], SrLiN [52] and CaNiN [53]. The symmetry reduction is explained for the prototype YCoC in Figure 3. The first step is the decentering of the lattice in going to the CsCl structure which leads to a 3 + 3 splitting of the 6b site. This is the decisive difference to the body-centered branch discussed before. The subsequent translationengleiche step of index 3 (t3) leads to the tetragonal CuAu structure (P4/mmm) and in the final step the unit cell is doubled along the c axis via a klassengleiche symmetry reduction of index 2 (k2) to space group P4₂/mmc. This model leaves five crystallographically independent octahedral voids of which only the Wyckoff site 2c is filled with carbon atoms. This leads to linear [CoC] chains which change their orientation by 90° in every other layer (Figure 4) as a consequence of the 4₂ screw axis. The isotypic nitrides contain [LiN], respectively [NiN] chains.

Especially the [CoC] and [NiN] chains have attracted considerable interest by theoreticians [54], [55], [56] with respect to a possible Peierls distortion. Keeping the [NiC₂] units in Th₂NiC₂ and the infinite [CoC] chains in YCoC in mind, the most plausible Peierls distortion would by a pairwise shift of the carbon atoms towards the cobalt atoms as emphasized by red arrows in Figure 5. This ordering leaves two crystallographically independent cobalt sites, emphasized by blue and reddish color in Figure 5. The resulting structural model for a Peierls-distorted YCoC variant would crystallize in space group P4₂/mnm (Figure 6). The klassengleiche symmetry reduction of index 2 (k2) leads to the splitting of the cobalt site and leaves a free x parameter for the carbon atoms, allowing the formation of [CoC₂] pairs, similar to [NiC₂] in Th₂NiC₂.

Figure 5:

Cutouts of the structures of YCoC [51] and a possible Peierls-distorted variant (see the Bärnighausen tree in Figure 6). Only one layer is shown. The red arrows indicate the shift of the carbon atoms during Peierls distortion. The two crystallographically independent cobalt atoms in the distorted model are shaded in the same manner as in the Bärnighausen tree.

Figure 6:

Group-subgroup scheme in the Bärnighausen formalism [15], [16], [17], [18] for a possible Peierls distorted variant of YCoC [51] and the nitridometalate Sr₂[CoLiN₂] [57], [58]. The index for the klassengleiche (k) symmetry reduction and the evolution of the atomic parameters are given. The differently shaded cobalt positions correspond to the ordering presented in this Figure.

The plausibility of this model is nicely underlined by the structure of isotypic nitrodometalate Sr₂[CoLiN₂] [57], [58], where ordered chains –N–Co–N–Li– are embedded in the strontium matrix. It is remarkable that the Co⁺ cations show ferromagnetic ordering at a comparatively high Curie temperature of 44 K [58]. Ca₂[Li_1.25Cu_0.75N₂], Ca₂[Li_0.93Cu_1.07N₂] [59] and Ca₂[Li_1.18Fe_0.82N₂] [60] are further representatives of this structure type.

A look into the Pearson data base [7] for the fingerprint #136, tP12, gdba yields further representatives. The structure of Li₃[BN₂] [61], [62] shows linear [BN₂] nitridoborate units that are linked via one kind of lithium cations to linear chains. This is similar to the oxidonickelates(I) A₃[NiO₂] with A = K, Rb and Cs [63], [64], where one of the alkali cations forms linear chains with the [NiO₂] units. It is interesting to note that K₃[NiO₂] exhibits a phase transition at around 423 K. α-K₃[NiO₂] crystallizes in the polar space group P4₁2₁2. The complete Bärnighausen tree is presented in ref. [64].

Keeping the temperature-induced phase transition of the nickelate in mind we redraw to the xenon difluoride structure which shows two different pressure-induced phase transitions [65]. The polymorphs HP1-XeF₂ (Immm) and HP2-XeF₂ (Pnma) are derived from the normal-pressure modification NP-XeF₂ (I4/mmm) via two different symmetry reductions (Figure 7); I4/mmm is the common supergroup of Immm and Pnma. HP1-XeF₂ shows just an orthorhombic distortion of the unit cell (loss of the four-fold axis) and this proceeds via a translationengleiche transition of index 2 (t2) from I4/mmm to Immm along with a splitting of the axis to a = 329.5 and b = 325.7 pm. The XeF₂ units remain in parallel alignment in the HP1-XeF₂ polymorph (Figure 8). The intermetallic compounds ReSi₂ [66] and TiPd₂ [67] are isopointal [11], [12] with HP1-XeF₂; however, as discussed above, with larger c/a and c/b ratios.

Figure 7:

Group-subgroup scheme in the Bärnighausen formalism [15], [16], [17], [18] for the structures of the normal-pressure (NP) and high-pressure (HP) polymorphs of XeF₂ [65]. The index for the klassengleiche (k) symmetry reduction and the evolution of the atomic parameters are given. Note that the relationship between the XeF₂ modifications and the intermetallic phases is solely isopointal [11], [12]. For further details see text.

Figure 8:

Arrangement of the molecules in the normal-pressure (NP) and high-pressure (HP) polymorphs of XeF₂ [65]. Xenon and fluorine atoms are drawn as blue and magenta circles.

The HP2-XeF₂ structure (Figure 8) shows substantial tilting of the XeF₂ units. The symmetry reduction proceeds in three steps (Figure 7) via the space groups Fmmm and Cmcm. The essential step is the unit cell enlargement during the transition from I4/mmm to Fmmm. A search in the Pearson data base [7] for the intermediate space groups Fmmm and Cmcm revealed the representatives TiSi₂ (observed in thin films) [68] and VAu₂ [62, 69]. Again, these two intermetallics are only isopointal [11], [12], since the bonding peculiarities are different.

We now turn to the equiatomic compounds ReGaSi and the recently reported isotypic compound ReAlSi [70], [71]. These silicides attracted attention with respect to their low valence electron count of 14 and the occurrence of superconductivity in the solid solution ReAl_1+xSi_1−x (T_C = 3.5 K for ReAl_1.2Si_0.8). Samples of that solid solution with a slightly higher aluminum content showed structural disorder [72]. A structure refinement of ReAl_1.28Si_0.72 revealed the MoSi₂-type structure with a mixed occupancy (64% Al + 36% Si) for the 4e site (Figure 9). A 1:1 ordering of aluminum and silicon atoms in ReAlSi and of gallium and silicon atoms in ReGaSi requires a symmetry reduction and a splitting of the 4e subcell site into two two-fold sites. A look into the International Table A1 [73] for space group I4/mmm reveals two possibilities for klassengleiche transitions of index 2 which allow that kind of splitting. The first possibility is the transition to the primitive space group P4/mmm. Here, also the 2a site splits into two one-fold sites, and this superstructure model is realized for the series RE₂RuMg₃ and RE₃Ru₂Mg [74]. For ReAlSi and ReGaSi, the second possibility is realized. The space group symmetry is reduced to P4/nmm and the 4e site splits into two sites 2c (Figure 10). The 1:1 ordering (the ReAlSi structure is presented as an example in Figure 9) was unambiguously determined from neutron diffraction data (aluminum and silicon have too close scattering factors for a reliable refinement from X-ray diffraction data).

Figure 9:

The crystal structures of ReAl_1.28Si_0.72 and ReAlSi [70], [71]. Rhenium, aluminum and silicon atoms are drawn as light gray, blue and magenta circles, respectively. The mixed occupied site in ReAl_1.28Si_0.72 is emphasized by segments. The heights of the differently compressed cubes are given in units of pm.

Figure 10:

Group-subgroup scheme in the Bärnighausen formalism [15], [16], [17], [18] for the structures of ReAl_1.28Si_0.72 and ReAlSi [70], [71]. The index for the klassengleiche (k) symmetry reduction and the evolution of the atomic parameters are given. Note that the setting is for origin choice 1.

The ordering in space group P4/nmm allows the stacking of three different cubes (Figure 9) with edge lengths of 262.7, 257.1 and 283.9 pm. The a lattice parameter of ReAlSi is 317.7 pm, thus all cubes are compressed. In the P4/mmm superstructure of RE₂RuMg₃ (RE = Dy, Ho, Er, Tm, Lu) and RE₃Ru₂Mg (RE = Sc, Er, Lu) [74] the mirror planes are kept perpendicular to the c axis leaving only two crystallographically independent cubes; a decisive difference to the P4/nmm superstructure.

ReAlSi and ReGaSi are simple ternary stacking sequences of three ordered and distorted cubes. They belong to a larger family of bcc superstructures which show a stacking in c direction with up to 13 cubes. Selected representatives of this family are summarized in Table 1. For an overview of more complex bcc superstructures (which are also enlarged in the a and b directions) we refer to the textbook by Müller [13] and two review articles [14, 82].

Finally we draw back to the crystallographic fingerprint (space group P4/nmm, Pearson code oP6 and Wyckoff sequence c³) of ReGaSi. In Figure 11 we present projections of the ReGaSi [70], Ta₂Se [83], EuSnP [84], NbCrN [85], and CaGaN [86], [87] structures along the a axes. As discussed above, the ReGaSi structure shows a stacking of three slightly compressed cubes and is classified as an intermetallic phase, similar to the intermetallics discussed in Figure 1. The three cubes are emphasized by differently colored shadings. The Ta₂Se structure shows one significantly stretched cube which contains the selenium atoms (van der Waals bonding between adjacent selenium layers). This is also the case for the isotypic members of the solid solution Ta_2−xNb_xS [88], [89] and Ta₂Se_0.66Te_0.34 [81]. The remaining three compounds presented in Figure 11 show distinctly different chemical bonding and are only isopointal with ReGaSi. The differences are briefly discussed. The superstructure model in space group P4/nmm leaves free z parameters for all atoms, and in the cases of EuSnP (SrSnP [90]), NbCrN (NbMoN [91], TaMoN [92]), and CaGaN we observe substantial shifts of the phosphorus and nitrogen atoms off the sub-cube centers towards the sub-cube faces. These positions now correspond to a distorted octahedral coordination: P@Eu₅Sn, N@Nb₅Cr and N@Ca₅Ga. The comparison of these five structures nicely underlines, that we need to divide the P4/nmm, tP6, c³ phases into three branches: (i) ReGaSi, (ii) Ta₂Se and (iii) EuSnP, NbCrN and CaGaN, in analogy to the I4/mmm, tI6, ea phases discussed above. These branches account for the differences in chemical bonding and also underline the limits of structure comparison based on group-subgroup schemes.

Figure 11:

Projection of the crystal structures of ReGaSi [70], [71], Ta₂Se [83], EuSnP [84], NbCrN [85] and CaGaN [86], [87] which all have the crystallographic fingerprint of space group type P4/nmm, Pearson code oP6 and Wyckoff sequence c³. For details see text.

3 Ordering variants of BaAl₄

The structures of more than four thousand intermetallic compounds are derived from the BaAl₄ type [93]. This simple binary structure, space group I4/mmm, tI10 and Wyckoff sequence eda allows for a coloring on the e and d sites, leading to a manifold of ternary representatives. These ordering variants are listed in the Pearson data base [7] under the prototypes CeAl₂Ga₂ [94]/ThCr₂Si₂ [95] (3445 entries) and TlCu₂Se₂ [96] (627 entries). Herein we discuss some coloring variants from the so-called 1:2:2 pnictide family.

Depending on the size of the atoms and the valence electron count (VEC), the pnictogen atoms form homoatomic bonds or one observes pronounced two-dimensional substructures. These structural features have repeatedly been discussed [8, 24], [25], [26], [27], [28, 97], [98]. As an example we present the EuFe₂As₂ structure [99], [100] in Figure 12. The electron-precise description Eu²⁺(Fe²⁺)₂(As³⁻)₂ is compatible with the As–As distance of 333.4 pm, which is much longer than the As–As single bond of 252 pm in α-arsenic [101], thus underlining isolated arsenide anions.

Figure 12:

The crystal structures of EuFe₂As₂ [100] and EuRbFe₄As₄ [109]. Europium, rubidium, iron and arsenic atoms are drawn as light gray, green, blue and magenta circles, respectively. The tetrahedral [FeAs] substructures are emphasized and relevant interatomic distances are given in units of pm.

The field of BaAl₄ related materials rapidly grew especially in the last 10 years in the context of pnictide based superconductors [102], [103], [104], [105], [106], [107], [108]. In the present subchapter we focus on some new superstructure variants which extend the previously described Bärnighausen trees [24], [25], [26], [27], [28]. The first ordering variant concerns the quaternary phases EuRbFe₄As₄ (Figure 12) [109], CaAFe₄As₄ (A = K, Rb, Cs) and SrAFe₄As₄ (A = Rb, Cs) [110], [111], [112] which contain a 1:1 distribution of mono- and divalent cations. The ordering is only possible through a loss of the body-centering. The corresponding group-subgroup relation is shown in Figure 13 for the pair EuFe₂As₂ and EuRbFe₄As₄. Due to the klassengleiche transition of index 2 (k2) from I4/mmm to P4/mmm (Figure 13), the X-ray powder patterns reveal superstructure reflections (so-called primitive reflections) which indicate the ordering. The 2a subcell site splits into two one-fold sites for europium and rubidium. The complete ordering is a consequence of the distinctly different ionic radii [113] of smaller Eu²⁺ (125 pm for coordination number 8) and larger Rb⁺ (161 pm for coordination number 8). The unit cell drastically expands from EuFe₂As₂ (c = 1212.47 pm) to EuRbFe₄As₄ (c = 1331.09 pm) and this significantly enlarges the As–As distances around the rubidium atoms (Figure 12). The lowering of the space group symmetry also affects the iron and arsenic atoms. The iron atoms lose their subcell mirror plane and are shifted to z = 0.2351 while the 4e arsenic site splits into two two-fold arsenic sites which independently coordinate to europium and rubidium.

Figure 13:

Group-subgroup scheme in the Bärnighausen formalism [15], [16], [17], [18] for the structures of EuFe₂As₂ [100] and EuRbFe₄As₄ [109]. The index for the klassengleiche (k) symmetry reduction and the evolution of the atomic parameters are given.

The incorporation of a monovalent cation has an influence on the electronic structure of the quaternary arsenides. In order to keep an electron precise description, one iron cation per formula unit needs to be oxidized, leading to an overall oxidation state of +2.25 for iron. Magnetic susceptibility and resistivity measurements indicated superconductivity for these quaternary phases with transition temperatures in the range of 31–36 K [110]. Keeping the huge number of CeAl₂Ga₂/ThCr₂Si₂ intermetallics in mind, the 1:1 ordering on the cation site should be possible for many other element combinations, allowing for a number of new phases with modified magnetic properties.

At this point we should note that P4/mmm-type ordering was also considered for the uranium compounds UPd₂Si₂, UPd₂Ge₂, URh₂Si₂ and URh₂Ge₂ [114], [115]; however, the primitive reflections result rather from a CaBe₂Ge₂-type arrangement, and these tetrelides exhibit space group P4/nmm.

We now turn to a stacking of BaAl₄ related cells along the c axis. First examples were the structures of BaMg₂Sn₂ (i2 of P4/nmm) [116] with a doubling the c axis, and the structure of BaCu₂Sb₂ (i3 of I4/mmm) [117] with a tripling of the c axis (Figures 14 and 15). These two superstructure variants were already discussed in the first Bärnighausen tree of BaAl₄ superstructures in 1999 [24]. An X-ray-pure sample of BaCu₂Sb₂ was obtained by a reaction of BaSb₃, Ba₅Sb₃ and Cu₂Sb in a NaCl/KCl flux at 900 °C. BaCu₂Sb₂ has recently been reinvestigated with respect to its physical properties [118], [119]. A lead flux synthesis with a final annealing temperature of 550 °C led to platelets of α-BaCu₂Sb₂. Single crystal X-ray diffraction data proved the CaBe₂Ge₂-type structure, space group P4/nmm [123]. Reinvestigation of the high-temperature phase β-BaCu₂Sb₂ indicated lowering of the space group symmetry to Immm, C2/m or probably even to P2₁/c [124]. This situation might be similar to that observed for SrRh₂As₂ [120]; however, the quality of the diffraction data was not sufficient. The β-BaCu₂Sb₂ structure deserves further investigations in order to clarify possible phase transitions along with symmetry reductions in going to lower temperature.

Figure 14:

Group-subgroup scheme in the Bärnighausen formalism [15], [16], [17], [18] for the structures of BaFe₂As₂ [121]/α-BaCu₂As₂ [117], α-BaCu₂Sb₂ [118], β-BaCu₂Sb₂ [117]/Sr₃Cd₈Ge₄ [122], and β-BaCu₂As₂ [119, 123]. For details see text.

Figure 15:

Group-subgroup scheme in the Bärnighausen formalism [15], [16], [17], [18] for the structures of BaFe₂As₂ [121], β-BaCu₂Sb₂ [117] and Sr₃Cd₈Ge₄ [122]. The index for the isomorphic (i) symmetry reduction and the evolution of the atomic parameters are given. The shaded areas emphasize the change in coloring between copper and antimony in β-BaCu₂Sb₂, respectively cadmium and germanium in Sr₃Cd₈Ge₄.

We can thus conclude that α- and β-BaCu₂Sb₂ have the common supergroup I4/mmm (ThCr₂Si₂ type) and the phase transition is of a reconstructive type. There is no direct group-subgroup relation between the space group types of α- and β-BaCu₂Sb₂ (Figure 14). We start the crystal-chemical discussion of the congener BaCu₂As₂ [117] with the α modification which adopts the ThCr₂Si₂ type, space group I4/mmm, similar to the well know spin density wave material BaFe₂As₂ [121]. The copper atoms have tetrahedral arsenic coordination (Figure 16) and adjacent layers of condensed tetrahedra are related by mirror planes. Thus, these layers are crystallographically identical and we call them I layers in Figure 16. Due to the mirror symmetry, the arsenic atoms of adjacent layers point towards each other.

Figure 16:

The crystal structures of α- and β-BaCu₂As₂ [117, 119, 123] α- and β-BaCu₂Sb₂ [117], [118] and Sr₃Cd₈Ge₄ [122]. Strontium (barium), copper (cadmium) and arsenic (antimony, germanium) atoms are drawn as medium gray, blue and magenta circles, respectively. The tetrahedral substructures are emphasized. I and P denote the slabs that are derived from α-BaCu₂As₂ (ThCr₂Si₂ type, I4/mmm) and α-BaCu₂Sb₂ (CaBe₂Ge₂ type, P4/nmm). For details see text.

In the structure of α-BaCu₂Sb₂ (CaBe₂Ge₂ type, space group P4/nmm), the coloring within each other [Cu₂Sb₂] layer is inverted to Sb@Cu₄ tetrahedra and we now observe Cu–Sb (i.e. heteroatomic) bonding between adjacent layers. These layers are called P layers in Figure 16. The β-polymorph of BaCu₂Sb₂ shows a tripling of the unit cell (isomorphic transition of index 3). The decisive point is the inverted Cu/Sb coloring of the two Wyckoff sites shaded in Figure 15. This allows for the different stacking of P and I slabs (4 + 2 per unit cell), and one observes homoatomic (Sb–Sb) as well as heteratomic (Cu–Sb) interactions between the slabs. In other words, the β-BaCu₂Sb₂ structure is an intergrowth variant of ThCr₂Si₂ and CaBe₂Ge₂ related slabs.

The longest stacking period of a BaAl₄/ThCr₂Si₂ superstructure has been observed for β-BaCu₂As₂ [119]. The original work on BaCu₂As₂ powders and single crystals (now called α-BaCu₂As₂) [117, 119] revealed the ThCr₂Si₂ type, space group I4/mmm. X-ray-pure samples of BaCu₂As₂ were obtained from an adequate mixture of Ba + CuCl + As. The resulting BaCl₂ was extracted with diluted acetic acid and water.

Recent flux growth experiments [119, 123] with tin, lead, copper arsenide and gallium as flux agents at different temperatures resulted in a new superstructure variant. Only the Sn flux with a final annealing temperature of 500 or 550 °C led to crystals of β-BaCu₂As₂, which is an i5 superstructure of α-BaCu₂As₂. Similar to β-BaCu₂Sb₂ described above, we observe a switch in coloring (blue shading in Figure 17) with a different stacking of P and I slabs (4 + 6 per unit cell). The quintupled unit cell was unambiguously determined from a combination of single crystal X-ray diffraction data and high resolution electron microscopy.

Figure 17:

Group-subgroup scheme in the Bärnighausen formalism [15], [16], [17], [18] for the structures of α-BaCu₂As₂ and β-BaCu₂As₂ [117, 119, 123]. The index for the isomorphic (i) symmetry reduction and the evolution of the atomic parameters are given. The shaded areas emphasize the change in coloring between copper and arsenic.

An extension of the i3 superstructure variant was reported for the germanides Sr₃Cd₈Ge₄ and Eu₃Cd₈Ge₄ [122]. The Wyckoff sequence (Figure 15) is the same as for β-BaCu₂Sb₂; however, the coloring is different. Thus, Sr₃Cd₈Ge₄ and β-BaCu₂Sb₂ are isopointal [11], [12], not isotypic. The occupancy of three Wyckoff positions with cadmium leads to a pronounced cadmium substructure; exemplarily shown for Sr₃Cd₈Ge₄ in Figure 16. The two crystallographically independent germanium atoms are isolated (Ge2) and paired (251 pm Ge1–Ge1). For a detailed discussion of chemical bonding in these ternary germanides we refer to the original work.

Further compounds with the i3 superstructure variant are SrPt₂Ge₂ and BaPt₂Ge₂ [125], [126], [127], and the germanium-rich phases SrAu_1.65Ge_2.35 and SrAu_1.60Ge_2.40 [128]. So far only powder diffraction data is available for these three germanides. For SrPt₂Ge₂ and BaPt₂Ge₂ an inverse coloring has been reported and furthermore, monoclinic distortions [126], [127] might be possible. The structures of these three phases need to be studied on the basis of precise single crystal diffraction data in the future.

Further examples for BaAl₄ superstructures are summarized in Table 2. The underlying group-subgroup relations are documented in the respective references.

4 Ordering variants that are derived from the CeMg₂Si₂ type

The third group of compounds is derived from the CeMg₂Si₂-type structure [136], [137], [138], [139], [140], space group P4/mmm, Pearson code tP5 and Wyckoff sequence hea. The CeMg₂Si₂ type is realized for only few of the so-called 122 phases. Besides isotypic LaMg₂Si₂ [141], the pnictides BaPd₂P₂ [142] and BaPd₂As₂ [142], [143], [144] have been reported.

The relatively simple CeMg₂Si₂ structure is presented in Figure 18. The cerium and magnesium atoms form a three-dimensional network of corner-sharing Ce₂Mg₄ octahedra. The large cavities within this network (formed by eight triangular faces of the octahedra) are filled by Si₂ dumbbells with 257 pm Si–Si. The chemical bonding in CeMg₂Si₂ has been studied on the basis of extended Hückel calculations along with a discussion on the valence electron concentration with respect to the competing structure types BaAl₄/ThCr₂Si₂ [138].

Figure 18:

The crystal structures of CeMg₂Si₂ [136], SrAu₃Ge [146] and LaRu₂Al₂B [156]. Cerium (strontium, lanthanum), magnesium (gold, ruthenium) and silicon (germanium, aluminum) atoms are drawn as medium gray, blue and magenta circles, respectively. The octahedral building units Ce₂Mg₄, Sr₂Au₄ and B@La₂Ru₄ and the Si₂, Au–Ge and Al₂ dumbbells are emphasized. For details see text.

Superstructure formation was first reported for BaAu₃Ge [145] and later also for SrAu₃Ge [146], BaAu_3+xSi_1−x [147] and SrPt₃P [148], [149], [150], and recently for CePt₃P [151]. We exemplarily discuss SrAu₃Ge [146]. The superstructure formation concerns coloring within the Si₂ dumbbells sites of the aristotype CeMg₂Si₂. This deserves a symmetry reduction and a splitting of the 2h site. The corresponding group-subgroup scheme is shown in Figure 19. In the original version [146], the group-subgroup scheme has inconsistencies with respect to the space group symbols and the Wyckoff sites. The splitting is realized through a klassengleiche transition of index 2 (k2) from P4/mmm to P4/nmm (√2a × √2a × c cell). The Au–Ge dumbbells in SrAu₃Ge (Figure 18) point alternatingly in +c and −c direction.

Figure 19:

Group-subgroup scheme in the Bärnighausen formalism [15], [16], [17], [18] for the structures of CeMg₂Si₂ [136] and SrAu₃Ge [146]. The index for the klassengleiche (k) symmetry reduction and the evolution of the atomic parameters are given.

The network of corner-sharing octahedra reminds of the Cu₃Au and perovskite structures; however, the Ce₂Mg₄ octahedra in CeMg₂Si₂ are empty. It is interesting to note, that these octahedral voids can indeed be filled with small atoms, leading to quaternary phases. This was first observed for CeCr₂Si₂C [152] and then for other RECr₂Si₂C (RE = Y, La–Nd, Sm, Gd–Ho) silicide carbides [153], [154], [155]. Later, also borides, e.g. LaRu₂Al₂B [156] and CeRu₂Ga₂B [157], as well as the oxides BaTi₂As₂O [158] and BaTi₂Sb₂O [159] with this structure type have been discovered. The structure of LaRu₂Al₂B is presented in Figure 18. Such phases have repeatedly been discussed as anti-perovskite variants. The ‘LaRu₂B’ substructure of LaRu₂Al₂B corresponds to the ‘TiO₃’ network of perovskite. The cavities are filled by the calcium cations in perovskite, and by Al₂ dumbbells aligned parallel to c in LaRu₂Al₂B, causing a tetragonal distortion.

5 Conclusion

The present overview discussed coloring and distortion variants of the bcc packing of the aristotypes BaAl₄ and CeMg₂Si₂ on the basis of group-subgroup schemes. The many ordering variants include examples from intermetallics, ionic compounds and also some molecules. The Bärnighausen trees are a compact tool for structure systemization and nicely merge crystal chemistry and group theory. However, the many ordering variants discussed in the present overview also manifest the limits of comparability. The coloring with different atom types and/or the resulting distortions often drastically modify the bonding pattern. Thus, coloring variants are often only isopointal and not isotypic.