INTRODUCTION

At present, there are several types of two-layered icosahedral clusters known in the chemistry of intermetallics that differ by the number of atoms in the second shell: 1@12@32 (Bergman cluster [1]), 1@12@42 (Mackay cluster [2]), and 1@12@50 [1].

Intermetallics containing Bergman nanoclusters are formed in multiple binary and ternary systems. Up to 150 crystalline structures have been investigated [3]. The chemical composition and structure of binary intermetallics with Bergman nanoclusters-precursors were investigated in [1].

Ternary and binary intermetallics containing local regions in the forms of Bergman icosahedral clusters represent the most complex crystalline structures in terms of crystal chemistry. Table 1 demonstrates intermetallics with a unit cell volume exceeding 10 000 Å3 containing up to 1100 atoms [4–12].

Table 1.   Crystal structures with Bergman clusters
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The crystal structure of (Ba,Ca)46Li102 (Li33.3Ba13.1Ca3, [12]) exhibits the largest values of the unit cell volume V = 30 979 Å3 and the translation vector с = 90.213 Å (Table 1). The CN values of 15 (for one atom), 16 (for seven atoms), and 17 (for two atoms) were found for 10 crystallographically independent Ba atoms. In the crystal structure of (Ba,Ca)46Li102, the Ca atoms are statistically distributed along the crystallographic positions of 8 Ba atoms [12].

The crystal structure of (Ba,Ca)46Li102 was described in the form of bonded Li(Li4Ba8) icosahedra with Li(Li12) and Li(Li9Ba3) icosahedra located between them [12].

The present study carries out a geometrical and topological analysis of the crystal structure of the (Ba,Ca)46Li102 intermetallic system by the ToposPro software package [13]. The symmetry and topological code of the process of cluster self-assembly from the 143-atom nanocluster in the form of the primary chain \({\text{S}}_{3}^{1}\) → layer \({\text{S}}_{3}^{2}\) → framework \({\text{S}}_{3}^{3}\) was established.

This study represents a follow-up of [1, 2, 14–22] in the simulation of processes of self-the assembly of systems on the suprapolyhedral level and the geometric and topological analysis of crystal structures by advanced computer methods.

TECHNIQUES USED IN COMPUTER ANALYSIS

The geometric and topological analysis was carried out by the ToposPro [13] software package that allows performing a multipurpose automated investigation of the crystal structure built on the presentation of structures as “folded graphs” (quotient graphs). The data on the functional role of the atoms at the formation of the crystal structure were obtained by calculating the coordination sequences, i.e., the number sets {Nk}, where Nk is the number of atoms in the k-coordination sphere of a specific atom.

The obtained values of the coordination sequences of atoms in 3D-lattices are provided in Table 2, where the number of neighboring atoms in the adjacent region, i.e., in the first coordination sphere, are given in bold. All the atoms exhibited various sets of the coordination sequences {Nk}, so that they are all different topologically (and functionally).

Table 2.   Local surroundings of Li, Ba atom and coordination sequence values
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The algorithm of the automated decomposition of any intermetallic structure, represented as a folded graph, on the cluster units was based on the following principles. A structure is formed as a result of the self-assembly from clusters-precursors. Here, the clusters-precursors formed a framework of the structure, where voids were occupied by clusters-spacers (consisting of a small number of atoms). Nanoclusters-precursors did not have shared internal atoms, although shared surface atoms could be observed. The clusters-precursors occupied highly symmetric positions. A set of nanoclusters-precursors and clusters-spacers consisted of all the atoms of the structure. The algorithm was implemented in the ToposPro software package [13].

SELF-ASSEMBLY OF THE CRYSTAL STRUCTURE OF (Ba,Ca)46Li102

The applied method of the simulation of the crystalline structure was based on the determination of the hierarchic sequence of its self-assembly in the crystallographic dimension [14, 15]. At the first level of the system’s self-assembly, the mechanism was determined by the formation of the primary chain of the structure from nanoclusters of the 0-level, formed at the template stage of the system’s chemical evolution, then the mechanism of self-assembly from the chain of the level was initiated (2nd level) and, thereafter, from the layer a 3D framework of the structure (3rd level) was initiated.

Crystallographic data on(Ba,Ca)46Li102. The space group R\(\bar {3}\)c is characterized by elements with the point symmetry g = 32 (6a), g = –3 (6b), 3 (12c), –1 (18d), and 2 (18e).

Table 2 demonstrates the local surroundings of the Li and Ba atoms and the coordination sequence values in the 3D atomic network.

The number of variations of the decomposition of a 3D atomic lattice with of 3, 4, 5, 6, and 7 structural units equaled 3, 16, 53, 44, and 17 (total 133 variations), respectively.

The variant of describing the crystal structure of (Ba,Ca)46Li102 suggested in [12] implying isolation of the layer from the bonded Li(Li4Ba8) icosahedra with the Li(Li12) and Li(Li9Ba3) icosahedra located between them corresponds to the variation with the largest number of elementary structural units of 7 (Table 3). Upon the selection of such a type of intermetallics simulation, apart from the three investigated elementary polyhedral clusters, the functional role of another three isolated elementary polyhedral clusters (Ba1(1)(1@16), Li3(1)(1@12), and Ba4(1)(1@17)) and atoms-spacers (Li5(0)(1)) must be considered.

Table 3.   Variations of cluster representation of crystal structure. Displayed: central atoms of polyhedral cluster (Li and Ba) and total number of shells (first braces) and number of atoms in shells (second braces)
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Quasis-pherical clusters the iсo-K143 of Li(Li12)(Li12Ba20)(Ba32Li66) composition, Ba3Li9 clusters-spacers, and Li atoms-spacers were established as the framework-forming nanoclusters (Table 4, Figs. 1, 2).

Table 4.   Atoms forming icosahedral cluster 1@12 and 32-, 98-atom shells
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Fig. 1.
figure 1

Li(Li12) cluster (left) and Li9Ba3 cluster (right). Numbers indicate distances of atomic bonds, Å.

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Fig. 2.
figure 2

Two-layer Li @12(Li12) @32(Li12Ba20) cluster.

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The iсo-K143 nanocluster appeared to be three-layered of composition 1@12@32@98 with the center in the position 6b (1/3, 2/3, 1/6) and the point symmetry g = –3.

The center of the Ba3Li9 cluster-spacer with the point symmetry g = 32 was located in the position 6a (2/3, 1/3, 0.583) (Fig. 1).

The second 32-atom icosahedral shell-deltohedron corresponded to the Bergman cluster with 32, 90, and 60 peaks, edges, and faces, respectively (Fig. 2). The second shell contained 20 Ba atoms located on 20 edges of an icosahedron and 12 Li atoms bonded to 12 Li atoms (peaks of the icosahedrons).

The third new type of 98-atom icosahedral shell-deltohedron exhibited of 98, 288, and 192 peaks, edges, and faces. The chemical composition of the shell is Ba32 Li66. In the third shell, all 32 Ba atoms were located above 32 atoms of the Berman shell (Fig. 3).

Fig. 3.
figure 3

Three-layer Li @12(Li12) @32(Li12Ba20) @98(Ba32Li66) cluster.

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The basic 3D lattice for the framework-forming iсo-K143 clusters corresponded to the hexagonal densest packing of clusters with CN = 12 = 6 (in the layer) + 3 (above the layer) + 3 (under the layer).

Self-assembly of the(Ba,Ca)46Li102crystalline structure. Primary chain. The self-assembly of the primary chains from the iсo-K143 cluster proceeded along the X axis (Fig. 4). Here, the localization of two atoms-spacers (Li2) occurred between iсo-K143 clusters.

Fig. 4.
figure 4

(а) Mechanism of iсo-K143 clusters bonding upon formation of primary chain and layer, (b) mechanism of clusters iсo-K143 bonding from adjacent layers upon formation of framework.

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The distance between the centers of iсo-K143 clusters corresponds to the translation vector value a = 19.913 Å.

Self-assembly of the layer. The formation of the \(S_{3}^{2}\) layer proceeded upon the complementary bonding of the clusters from the adjacent primary chains with a shift (Fig. 5). At this stage of self-assembly, localization occurs in the voids of the Ba3Li9 cluster-spacer.

Fig. 5.
figure 5

Layer of two primary chains. Ba3Li9 clusters-spacers and Li atoms-spacers are displayed.

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The distance between the centers of iсo-K143 clusters from the adjacent primary chains corresponded to the translation vector value b = 19.913 Å.

Self-assembly of the framework. The structure of the framework of \(S_{3}^{3}\) is formed upon the bonding of two basic layers with a shift. The distance between layers along the Z axis determines the translation vector value c/6 = 90.213 Å/6.

CONCLUSIONS

The framework-forming 143-atom icosahedral nanocluster with a diameter of 20 Å has been established via the method of the complete decomposition of the 3D atomic lattice into cluster structures. ico-K143 nanoclusters have been determined as three-layered Li @12(Li12) @32(Li12Ba20) @98(Ba32Li66) with a symmetry of g = \(\bar {3}\).

The symmetrical and topological code of the self-assembly processes of 3D structures from iсo-K143 nanoclusters-precursors in the primary chain → layer → framework form has been reconstructed. iсo-K143 nanoclusters form densely packed 2D layers 36 located with a shift along the c axis.

The 12-atom Ba3Li9 clusters-spacers with a symmetry of g = 32 that occupied voids in the 3D framework structure and Li atoms-spacers that participated in the bonding of iсo-K143 upon the formation of the primary chain and layer have been identified.