Isopointal intermetallics: the cP24, dca phases as a representative set of examples, along with their vacancy-ordered variants β-Mn and SrSi<subscript>2</subscript>.

The crystal-chemical relationship of the cubic Laves phase MgCu₂ (space group F d 3 ‾ m, cF24) with the ternary phases Cd₂Cu₃In, Na₂Au₃Al, Mg₂Rh₃P, Li₂Pd₃B, Ag₂Pd₃S, Cu₃Pt₂B, Mo₃Al₂C, Mo₃Ni₂N, and V₃Ga₂N (subgroup P4₁32, cP24, dca) is discussed based on a group-subgroup scheme. The course of the lattice parameters and the free positional parameters show substantially different distortions and thus clear differences in chemical bonding, classifying these phases as isopointal rather than isotypic (usually they are all assigned to the Mo₃Al₂C type). The group-subgroup scheme further shows that the β-Mn and SrSi₂ structures are vacancy-ordered variants of the cP24, dca phases. The structures of Mn₃IrSi and LaIrSi (space group types P2₁3; translationengleiche subgroups) are their ternary ordered versions.

Keywords: crystal chemistry; isopointal phases; intermetallics; vacancy ordering variants

Many of the basic structure types of crystalline ionic and intermetallic phases have huge numbers of representatives. Typical examples are the rock salt and fluorite-type structures with >7000 respectively >5000 entries in the Pearson data base [[1]]. These types are all classified by their individual crystallographic fingerprints, i.e., the space group symbol, their Pearson code and the Wyckoff sequence. This fingerprint is a sufficient prerequisite for a data base entry; however, it does not discriminate or differentiate different types of chemical bonding. This is nicely illustrated by some typical pairs of compounds with the same crystallographic fingerprint but a different bonding pattern: ionic NaCl and the metal carbide TiC (#225, cF8, ba), ionic CsCl and the brass phase CuZn (#221, cP2, ba), pyrite (FeS2) and carbon dioxide (#205, cP12, ca) or the intermetallic molybdenum silicide MoSi2 and molecular, covalently bound XeF2 (#139, tI6, ea). To account for the substantial differences in chemical bonding, those pairs of compounds are denoted isopointal rather than isotypic [[2]], [[3]], [[4]], [[5]], [[6]], [[7]]. The different site occupancies in these series, e.g., Ti for Na and C for Cl is often called coloring [[8]], [[9]], [[10]], [[11]], [[12]]. This issue is closely tied up with graph theory/vertex coloring [[13]].

Within a series of isopointal phases the parameters that induce changes in chemical bonding are (i) the coloring of Wyckoff sites with elements of different sizes and/or electronegativity, (ii) following from this, changes in the lattice parameters and (iii) shifts of the atoms to positions with free positional parameters. Such smooth changes towards another bonding situation are comparable to the well-known van Arkel–Ketelaar triangle [[14]]. The coloring can also change the dimensionality and/or bond order of/within the substructures. One of the striking families of compounds for those changes are the many BaAl4 related phases (#139, tI10, eda) [[1]], [[7]], [[15]], [[16]].

Herein we discuss the series of isopointal phases with the crystallographic fingerprint #213, cP24 and dca. The Pearson data base [[1]] lists 143 entries for this fingerprint, and the prototype Mo3Al2C [[17]] was assigned to all of these phases. This carbide gained broad interest as a 9 K non-centrosymmetric superconductor [[18]], [[19]]. In the following the group-subgroup scheme for the relation of Mo3Al2C with the aristotype MgCu2 is illustrated and discussed with respect to the different coloring variants along with their peculiar distortions and changes in chemical bonding. As an extension, the structures of β-Mn and SrSi2 (with substantially different crystal chemistry) are commented on, which can be derived from the #213, cP24, dca phases as vacancy ordering variants.

Out of the 143 representatives for #213 (P4132), cP24 and dca listed in the Pearson data base [[1]] nine have been selected that reflect the substance classes and bonding peculiarities for this structural family: Cd2Cu3In [[20]], Na2Au3Al [[21]], Mg2Rh3P [[22]], Li2Pd3B [[23]], Ag2Pd3S [[24]], Cu3Pt2B [[25]], Mo3Al2C [[17]], Mo3Ni2N [[26]] and V3Ga2N [[27]]. The basic crystallographic data for these phases is summarized in Tables 1 and 2. In order to emphasize the crystal-chemical differences, the site occupancies of the 4a, 8c and 12d Wyckoff sites which reflect the different coloring are listed. An important degree of freedom is the x parameter of the 8c sites and the y parameter of the 12d sites, which are also included in Table 1. These two positional parameters determine the size of the octahedra, and therefore also the shortest interatomic distances between the atoms on sites 4a and 12d are listed. They also allow for the substantial distortions of these subunits, as described below. Additionally, one has to keep the course of the lattice parameters in mind which range from 775.52 pm for Na2Au3Al [[21]] via 732.5 pm for Cd2Cu3In [[20]] to 628.8 pm for Cr2.14Fe2.38W0.48C0.24 [[35]]. These are also free parameters that allow for a structural variability.

Table 1: Selected examples of coloring variants for cubic intermetallic compounds with the crystallographic fingerprint space group P4132, Pearson code cP24 and Wyckoff sequence dca and defect variants with Pearson code cP20 and Wyckoff sequence dc. The free coordinates of the 8c and 12d sites are listed. SrSi2 (cP12) is included as an additional vacancy-ordered variant.

Table 2: Selected distances and distortion angles for the cubic intermetallic compounds with the crystallographic fingerprint space group P4132, Pearson code cP24 and Wyckoff sequence dca. The values d(triangle) are the distances between the adjacent triangles of the dinuclear units shown in Figure 2.

A final issue concerns the nomenclature of the cP24 phases. Although the correct occupation sequences of the Wyckoff sites are given in Table 1, the nomenclature used for each individual phase in the original literature is kept, and that for good reasons. To give an example, with the formula Na2Au3Al the structural similarity with MgCu2 is emphasized with the gold and aluminum atoms forming the anionic tetrahedral network leaving sodium as the cationic component. A formula Au3Na2Al in conformity with the prototype Mo3Al2C would be misleading.

The crystal-chemical discussion starts with the aristotype MgCu2 [[36]], the cubic Laves phase with ideal, corner-sharing Cu4 tetrahedra. The group subgroup relation for MgCu2 → Mo3Al2C in the compact Bärnighausen formalism [[37]], [[38]], [[39]], [[40]], [[41]] is shown in Figure 1. In the first translationengleiche step of index 2 (t2), the space group F4132 is reached. This corresponds to a loss of the center of symmetry; however, neither free positional parameters for the atoms are gained, nor a splitting of the positions which is needed for a 3:1 ratio for the Mo/C ordering is achieved. Consequently, a second symmetry reduction, a klassengleiche transition of index 4 (k4) to space group P4132 is observed. The splitting of the 16d site allows a 3:1 atomic ordering and the loss of the face-centering leads to superstructure reflections through which the ordering is evident in powder X-ray diffraction patterns.

Graph: Figure 1: Group-subgroup relation in the Bärnighausen formalism [[37]], [[38]], [[39]], [[40]], [[41]] for the structures of MgCu2 (aristotype), β-Mn [[28]], [[29]], Mo3Al2C [[17]] and SrSi2 [[32]], [[33]], [[34]]. The indices for the translationengleiche (t) and klassengleiche (k) transitions, the unit cell shifts and the evolution of the atomic parameters are given. For details see text.

This model in the cubic subgroup P4132 is quasi the cubic pendant to the hexagonal Laves phase MgZn2 [[42]], [[43]], [[44]], space group P63/mmc. Here, already two crystallographically independent zinc sites (2a and 6h) are present. The latter has a free x parameter, and the c/a ratio is also flexible. Mg2Cu3Si [[45]], [[46]] is the prototype for the 3:1 ordering on the zinc substructure. An alternative to the P4132 variant described herein is the rhombohedral Mg2Ni3Si type (translationengleiche symmetry reduction of index 4: MgCu2 → Mg2Ni3Si) [[47]] which was recently also observed for the gallides RE2Rh3Ga [[48]] or the aluminum series RE2TiAl3 [[49]]. The size of the atoms, the electronegativity differences, and the valence electron count [[44]], [[50]] determine which of these three types is realized.

Drawing back to the P4132 variants, our discussion starts with the Laves phases Cd2Cu3In [[20]] and Na2Au3Al [[21]]. Note that single crystal studies of the latter showed a small homogeneity range Na2Au3−xAl1+x. For better readability we keep the ideal composition in the following discussion. These two superstructures have by far the smallest deviations from the ideal MgCu2 arrangement. As is evident from Table 1, the x values for the 8c site of 0.0087 for Cd2Cu3In and 0.0170 for Na2Au3Al are very small and thus only slightly distorted Cu3In (3 × 264 pm Cu–In and 3 × 254 pm Cu–Cu) respectively Au3Al (3 × 261 pm Au–Al and 3 × 287 pm Au–Au) tetrahedra are observed. The short Cu–In and Au–Al distances compare well with the sums of the covalent radii [[51]] of 267 pm for Cu + In and 259 pm Au + Al. This is in full agreement with ICOHP data that point to substantial covalent Cu–Cu and Cu–In bonding in Cd2Cu3In [[20]], respectively Au–Al and Au–Au bonding in Na2Au3Al [[21]]. Especially the directed bonding of the central indium respectively aluminum atoms linking adjacent tetrahedra induces the small distortions. Double units of two corner-sharing tetrahedra as the basic building units are presented in Figure 2 for selected examples of the Mo3Al2C family. The distortions within the double units are not only a small contraction or elongation, but additionally a small screwing of adjacent triangular faces with respect to each other (the ideal 60° angle shrinks to 56.0° for Cd2Cu3In and 50.8° for Na2Au3Al). As an example, the Na2Au3Al [[21]] unit cell is presented in Figure 3. The structure reminds of the highly symmetric cubic Laves phase structure since the distortion of the tetrahedral substructure is small.

Graph: Figure 2: The basic building units in the structures of MgCu2 [[36]], Na2Au2.46Al1.54 [[21]], Mg2Rh3P [[22]], Mo3Al2C [[17]], and Ag2Pd3S [[24]]. The upper line shows units of two corner-sharing tetrahedra. These units are rotated by ca. 90° in the lower row to allow a view through the triangular faces. For details see text.

Graph: Figure 3: Unit cell of Na2Au2.46Al1.54 [[21]]. Sodium, gold and aluminum atoms are drawn as medium grey, blue and magenta circles, respectively. The mixed occupied site is emphasized by segments. The network of corner-sharing tetrahedra is outlined.

The next example is the metal-rich phosphide Mg2Rh3P [[22]], [[52]]. Magnesium deficiency induces superconductivity at a critical temperature of TC = 3.9 K. For Mg2Rh3P the first stronger distortion (when compared with the Laves phases Cd2Cu3In and Na2Au3Al) with an increase of the 8c x parameter to 0.0397 is observed. This corresponds to a substantial flattening of the condensed Rh3P tetrahedra (Figure 2), a consequence of the strong covalent Rh–P bonding. Two of the Rh3P tetrahedra are condensed via a common phosphorus atom, and the strong distortion along with a twist of the triangular Rh3 faces lead to a distorted octahedral coordination for the phosphorus atoms. The P–Rh distances of 6 × 228 pm are shorter than the sum of the covalent radii [[49]] for P + Rh of 235 pm. The Rh–P distances in the phosphides MgRh6P4 (231–246 pm) [[53]] and La4Rh8P9 (229–254 pm) [[54]] are in a similar range.

Based on their electronic structure calculations, Hase et al. [[52]] proposed an electron precise formula splitting (Mg2+)2[Rh3P]4− with full charge transfer from magnesium to the [Rh3P] network. This agrees with the course of the Pauling electronegativities (Mg: 1.31, Rh: 2.28 and P: 2.19) [[49]] and with the Mg–Mg distances of 278 pm which are drastically shortened with respect to hcp magnesium (6 × 320 and 6 × 321 pm) [[55]]. The structural relationship of Mg2Rh3P with the Laves phases Cd2Cu3In and Na2Au3Al and the aristotype MgCu2 is only of isopointal nature.

We now turn to boride chemistry. Li2Pd3B, Li2Pt3B [[23]] and Pt2Cu3B [[25]] are the sole boride representatives within this structural family. Especially Li2Pd3B and Li2Pt3B gained considerable attention since they belong to the small class of non-centrosymmetric superconductors with critical temperatures TC of 7.2 K for Li2Pd3B and of 2.6 K for Li2Pt3B [[56]], [[57]], [[58]]. The boride structures show stronger distortions than the phosphide structure discussed above. The x parameters of the 8c site further increase to 0.0572 for Li2Pd3B and to 0.0576 for Pt2Cu3B (Table 1). The small size of the boron atoms causes a substantial displacement of the palladium (platinum, copper) triangles, and these distortions lead towards an octahedral coordination. The course of the interatomic distances is exemplarily discussed for the Li2Pd3B structure. The B–Pd distances (6 × 213 pm) within the B@Pd6 units are close to the sum of the covalent radii [[49]] of 216 pm for B + Pd. This is comparable to the B–Pd distances within the B@Pd6 octahedra in Pd6B (202–205 pm) [[59]] and in the BPd6 trigonal prisms in Cu5Pd5B2 (213–220 pm) [[60]]. Each distorted B@Pd6 octahedron is condensed with neighboring octahedra via common corners (Figure 4), leading to a densely packed network. This network is penetrated by the lithium substructure (Figure 5), where the 41 screw-axis forms narrow spirals. Here, each lithium atom has three lithium neighbors at Li–Li distances of 257 pm, much shorter than the Li–Li distance of 304 pm in bcc lithium [[55]]. This clearly follows from a comparison of the individual Pauling electronegativities [[51]]: 1.0 for Li, 2.2 for Pd and 2.0 for B. The less electronegative lithium atoms transferred their valence electrons to the palladium and boron atoms, establishing the covalently bonded [Pd3B] substructure. This bonding situation is similar to that in Mg2Rh3P as discussed above. Considering this charge transfer one can propose the formula (2Li) δ+[Pd3B]δ−.

Graph: Figure 4: Cutout of the Li2Pd3B structure [[23]] emphasizing the condensation of the distorted B@Pd6 octahedra (6 × 213 pm B–Pd).

Graph: Figure 5: The lithium substructure of Li2Pd3B [[23]]. Each lithium atom has three lithium neighbors at 256 pm.

In going from the borides to carbides and nitrides, the shift of the x parameters of the 8c sites further increases with a maximum value observed for V3Ga2N [[27]]. As a consequence of the stronger distortions, the 4a–12d distances decrease, as e.g., to 215 pm Mo–C for Mo3Al2C [[17]], 208 pm Mo–N for Mo3Ni2N [[26]] and 207 pm V–N for V3Ga2N [[27]]. The carbon and nitrogen atoms always fill the voids of the higher melting element, a general crystal-chemical rule in the large family of hard materials with electron-poor transition metals (T) [[61]]. The short metal-carbon and metal-nitrogen distances in these phases reflect the strong covalent bonding within the [T3C] respectively [T3N] substructures. In the three examples listed in Table 1, the nickel, aluminum respectively gallium atoms form substructures similar to those observed for lithium in Li2Pd3B. The carbides and nitrides are of importance in the field of hard materials since they exhibit significant bulk moduli [[26]], [[62]], [[63]]. Mo3Al2C shows a bulk superconducting transition at a TC of ∼9.2 K [[64]].

When turning from the almost undistorted Laves phases Cd2Cu3In and Na2Au3Al to the hard materials, e.g., V3Ga2N, one observes a continuous increase of covalent bonding within the substructures formed by the atoms on the Wyckoff sites 4a and 12d. The crystal-chemical discussion of the phases listed in Tables 1 and 2 nicely reveals the changes in the coordination environment and chemical bonding and proves that the individual phases are rather isopointal than isotypic.

The last phase of this structural family is the metal-rich sulfide Ag2Pd3S [[24]], which occurs also naturally as the mineral coldwellite, e.g., in the Marathon deposit of the Coldwell Complex in Ontario, Canada [[65]]. Besides the single crystal growth employing the self-flux technique [[66]], its synthesis is also possible by a low-temperature solution-based alternative [[67]]. Taking silver as monovalent, a first simple electron count leads to a description (Ag+)2[Pd3S]2−, implying that the transfer of two valence electrons is sufficient for the reduction of sulfur to sulfide. Within the [Pd3S] substructure the corner-sharing S@Pd6 octahedra show short S–Pd distances (6 × 229 pm), somewhat shorter than the sum of the covalent radii [[51]] for S + Pd of 232 pm. This is in agreement with covalent S–Pd bonding. The edges of the S@Pd6 octahedra show the shortest Pd–Pd distances at 297 pm, slightly longer than the Pd–Pd distances in fcc palladium (12 × 275 pm) [[55]]. The silver substructure is similar to the lithium substructure discussed for Li2Pd3B (vide ultra). Again, short Ag–Ag distances of 274 pm are observed which are shortened with respect to fcc silver (12 × 289 pm) [[55]] and confirm the formation of cationic species by the charge transfer from the Ag atoms to the [Pd3S] substructure.

The excess valence electrons provided by the Pd atoms that cannot be accommodated by the sulfur atoms lead to a band structure giving rise to metallic conductivity, a golden silvery appearance and a metallic luster. Early resistivity measurements by Fischmeister [[68]] showed a specific resistivity of 1.93 × 10−4 Ω cm at room temperature. The metallic behavior was confirmed by Khan et al. [[69]] via resistivity measurements on polycrystalline Ag2Pd3S. Their sample additionally showed a superconducting transition at TC = 1.13 ± 0.02 K. The flux-grown single crystalline material prepared by Yoshida et al. [[66]] exhibited a slightly higher critical temperature of TC = 2.25 K.

The cP24, dca phases discussed in Chapter 3 were all assigned to the prototype Mo3Al2C [[17]], [[27]] in the Pearson data base [[1]]. Careful inspection of the atomic parameters reveals distinct differences in chemical bonding, depending on the size and electronegativity of the respective elements. Therefore, we focused on their isopointal structural relationship (vide infra). It is worthwhile to note that already in the original work on Mo3Al2C and V3Ga2N [[17]] Jeitschko et al. claimed that the ternary phases are stuffed variants (filling of the 4a distorted octahedral sites) of the β-Mn structure. According to the group-subgroup scheme (Figure 1), at least from a purely geometrical point of view the β-Mn structure should rather be classified as a vacancy-ordered derivative of the cubic Laves phase [[30]].

In that view it is interesting to briefly review the structural work on β-Mn. In his comprehensive review Bonding Patterns in Intermetallic Compounds, Nesper [[70]] pointed to the unusual coordination patterns in the α- and β-Mn structures with four (α-Mn) respectively two (β-Mn) crystallographically independent manganese sites. These sites have coordination numbers (12–16 in α-Mn; 12 and 14 for β-Mn) with distinctly different Mn–Mn distances. Nesper claimed that this goes along with a kind of charge redistribution between the individual manganese sites. For the β-Mn structure this suggestion requires that the 12d site would have cationic and the 8c site anionic characteristics. This is nicely paralleled by the binary compound Mg3Ru2 [[31]] which shows a clear charge transfer from magnesium to ruthenium, classifying Mg3Ru2 as a coloring variant of the β-Mn structure, similar to the Co8+xZn12−x phases [[30]]. The Pearson data base [[1]] lists many solid solutions with β-Mn structure. Based on these charge arguments one can assume ordering within these phases based on the different electronegativities of the constituent elements.

Before passing over to the Zintl phase SrSi2, we briefly summarize three further aspects of the β-Mn structure which are of crystal-chemical importance:

• (i) Single crystals of β -Mn and Cu 3 Pt 2 B were thoroughly studied with respect to the assignment of enantiomorphs by diffraction techniques [[71]]. Kikuchi patterns for dynamical electron diffraction have allowed the spatially resolved assignment of the β -Mn enantiomorphs.
• (ii) β -Mn as a tetrahedrally close packed derivative of the cubic Laves phase MgCu 2 (structures with truncated tetrahedra) can furthermore by itself perform as a substructure and deliver 100 tetrahedral and four metaprismatic voids [[72]], [[73]]. Partial filling of these voids leads to a manifold of complex chalcogenides and halides along with a further lowering of the space group symmetry, creating the R 32 and P 3 1 21 variants. Especially the tellurides (e.g., Na 2 Ge 6 Te 10) within this structural family gained special interest in the field of thermoelectrics, when searching for materials with ultralow lattice thermal conductivity [[74]]. These phases are out of the scope of this article; however, they are impressive examples for the broad crystal-chemical importance of the β -Mn type.
• (iii) Molecules with a somewhat spherical shape often show packing motifs that resemble close or closest packed structures of various element combinations [[39]]. The corresponding example for the β -Mn packing is a high-temperature modification of P 4 S 3 [[70]], [[75]].

The lithium substructure of Li2Pd3B presented in Figure 5 reminds the silicon substructure of the Zintl phase SrSi2 [[32]], [[33]], [[34]]. As is evident from Figure 1, the SrSi2 structure can also be derived as a vacancy-ordered variant of the cP24, dca phases. The negatively charged silicon atoms reside on the 8c site, compatible with the anionic character discussed above as well as with a description with periodic nodal surfaces [[76]]. Although there is a concrete group-subgroup relation between these structures, their bonding patterns differ distinctly, e.g., half of the sites (Wyckoff position 12d) remain unoccupied for the vacancy ordering variant SrSi2 (Figure 1).

Finally, ordering variants of the SrSi2 and β-Mn structures are briefly mentioned here. With translationengleiche symmetry reductions (Figures 6 and 7) to space groups P213 one can split the 8c sites to two 4a sites, enabling an ordering of two different atoms. This coloring leads to a 1:1 ordering on the spiral substructure shown in Figure 5. The first examples for these two coloring variants are LaIrSi [[77]] and Mn3IrSi [[78]]. Again, this relationship relies on the subgroup ordering, and the bonding patterns are distinctly different. Especially the family of LaIrSi related phases [[79]] has branches with rather ionic covalent (e.g., ZrOS) and rather covalent-metallic (e.g., NiSbS) bonding. This is a striking feature, keeping in mind that only three steps of symmetry reduction are to be made from the cubic Laves phase to these phases in space group type P213.

Graph: Figure 6: Group-subgroup relation in the Bärnighausen formalism [[37]], [[38]], [[39]], [[40]], [[41]] for the structures of SrSi2 [[32]], [[33]], [[34]] and LaIrSi [[77]]. The index for the translationengleiche (t) transition and the evolution of the atomic parameters are given.

Graph: Figure 7: Group-subgroup relation in the Bärnighausen formalism [[37]], [[38]], [[39]], [[40]], [[41]] for the structures of β-Mn [[28]], [[29]] and Mn3IrSi [[78]]. The index for the translationengleiche (t) transition and the evolution of the atomic parameters are given.

This mini review focusses on the structural relationship of the cubic Laves phase MgCu2 and the non-centrosymmetric compounds of the Mo3Al2C type. By a two-step symmetry reduction (t2 followed by k4), which can be illustrated by a group-subgroup scheme according to the Bärnighausen formalism, one can easily explain the structural peculiarities of the different isopointal Mo3Al2C-type representatives. The symmetry reduction leads to a free positional parameter x for the atoms occupying the 8c position as well as a free y parameter for the 12d site. Especially the latter is of key importance, since it allows for a distortion of the tetrahedral network. Depending on the chemical composition, the structures become adapted to the respective bonding peculiarities, leading away from two corner sharing tetrahedra towards distorted octahedral coordination environments. The different bonding patterns are nicely illustrated by the twisting and distortion of the coordination environment of the 4a site.

We have furthermore illustrated that the β-Mn and the SiS2-type structures can be understood as vacancy ordering variants of the Mo3Al2C type which can lead, with a further symmetry reduction, to a convincing description of the LaIrSi and Mn3IrSi-type structures.

Finally, it should be noted that direct structural relationships not only exist between MgCu2 and other Laves phases [[44]], but also for the compounds of the CeCr2Al20-type structure which can be described based on the cubic Laves phase. Here, the Ce and Cr atoms formally adapt the MgCu2-type structure (same Wyckoff positions), however, the Al atoms separate these elements leading to the structure of the CeCr2Al20 prototype [[80]].