INTRODUCTION

Crystallization of potassium intermetallic compounds was detected in binary systems K–M (with the participation of 15 chemical elements M), in 63 ternary systems K–M1–M2 (with the participation of 27 elements) and in 30 quaternary systems K–M1–M2–M3 (with the participation of 29 elements) [1, 2].

Twelve quaternary intermetallic compounds are formed in 11 systems K–Na–M1–M2 with the participation of atoms M1 and M2 = Au, Mg, Zn, Cd, Al, Ga, In, Sn, Sb, Bi) (Table 1, [3–14]). In the K–Na–Cd–Tl system, the formation of two compounds is detected; and in the other systems, of one compound (Table 1). With the participation of the large atoms K and Rb, and K and Cs in quaternary systems only 3 and 2 intermetallic compounds are formed [1, 2].

Table 1. Crystallographic data of intermetallic compounds formed in the systems K–Na–M1–M2, where M1, M2 = Au, Mg, Zn, Cd, Al, Ga, In, Sn, Sb, Bi
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In [15] for the quaternary intermetallic compound K23Na8Cd12In48-hP91 (a = b =17.114, c = 10.442 Å, spatial group P6/mmm), a new type of polyhedral precursor cluster was detected: K8 = 0@8(Na2In6) and K8 = 0@K2In6 in the form of a hexagonal bipyramid. The Na2In6 clusters were templates on the surfaces of which atomic shells of 36 atoms were formed. The composition of the two-layer templated cluster is K44 = 0@8(Na2In6)@36(In6Cd6K6)2.

The crystalline structure of the most complex quaternary intermetallic compound K52Na12Tl36In122-hP224 in terms of crystalline chemistry [13] is characterized by the high values of the parameters of the hexagonal cell: a = b =16.909 Å, c = 28.483 Å, V = 7 052 Å3, spatial group P-3m1(164), and 32 crystallographic independent atoms with unique Wyckoff positions j8i20d3c. Two Na atoms have coordination numbers (CN) = 13 and 15; eight K atoms have CN = 12, 14, 15, and 16; sixteen In atoms have CN = 9, 10, 11, 12, and 16; and six Tl atoms have CN = 10, 11, 13, and 14.

This work using the ToposPro software package [16] performs geometrical and topological analysis of the crystalline structure of the intermetallic compound K52Na12Tl36In122-hP224. The symmetrical and topological code of the self-assembly of the crystalline structure of the intermetallic compound made of nanoclusters K132 and K116 is reconstructed in the following form: primary chain → microlayer → microframe.

This work is a continuation of the studies [15, 17–23] in the field of simulation of the self-organization of systems at the supra-polyhedral level and the geometrical and topological analysis of crystalline structures by advanced computer methods.

METHODS OF COMPUTER ANALYSIS

The geometrical and topological analysis was performed using the ToposPro software package [16], which allows us to perform a multipurpose automatic study of the crystalline structure based on the representation of structures in the form of convoluted graphs (factor graphs). The data on the functional role of atoms upon the formation of a crystalline structure were obtained by computing the coordination sequences, that is, the sets of numbers {Nk}, where Nk is the number of atoms in the kth coordination sphere of the given atom.

The obtained values of the coordination sequence of atoms in 3D lattices are summarized in Table 2, where the number of neighboring atoms in the nearest surrounding area is highlighted in bold, that is, in the first coordination sphere of the atom. All atoms are characterized by various sets of coordination sequences {Nk}.

Table 2.   K52Na12Tl36In122-hP224. Local surrounding of Na, K, Tl, and In atoms and coordination sequences. Coordination number of atoms is highlighted in bold
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The automatic decomposition algorithm of the structure of any intermetallic compound was described elsewhere and implemented using the ToposPro software package [16].

SELF-ASSEMBLY OF THE K52Na12Tl36In122-hP224 CRYSTALLINE STRUCTURE

The applied simulation method of the crystalline structure is based on the determination of the hierarchical sequence of its self-assembly in crystallographic space [15, 17]. At the first level of the system’s self-organization, the mechanism of the primary chain formation of nanoclusters of the zero level formed at the template stage of the system’s chemical evolution is determined, then the mechanism of self-assembly of the layer chain (the 2nd level), and then of the layer of the structure’s 3D frame (the 3rd level).

Crystallographic data. The spatial group P-3m1 (no. 164) is characterized by the elements with point symmetry: g = –3m (1a, 1b), 3m (2c, 2d), 2/m (3e, 3f), 2 (6g, 6h), and m (6i). The group’s order is 12.

Table 2 shows the local surrounding of K, Na, Tl, and In atoms and their coordination sequences in a 3D atomic lattice.

The number of variants of representing a 3D atomic lattice with the number of structural units ranging from 4 to 10 was 1649 (Table 3).

Table 3.   K52Na12Tl36In122-hP224. Variants of cluster representation of crystalline structure with 4 and 10 structural units. Central atom or void center of polyhedral cluster is shown together with the number of its shells (the first brackets) and number of atoms in each shell (the second and the third brackets). Crystallographic positions corresponding to void centers of polyhedral clusters are denoted as ZA1 and ZA2
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Two variants of different crystallographic icosahedral clusters ico-In6Tl6(0@12) were detected with the symmetry –3m, occupying the highly symmetric positions 4a and 4b.

Icosahedrons ico-In6Tl6(0@12) are templates on which three-layer clusters K132 and K116 are formed with the size of 17 Å (Figs. 1, 2).

Fig. 1.
figure 1

Two-layer clusters 0@12@26 (a) and (0@12@30 (b).

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Fig. 2.
figure 2

Three-layer clusters K116 (a) and K132 (b).

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The chemical composition of the shells of the nanocluster K132 is 0@12(In6Tl6)@30(In6Na6-K18)@-90(In72Na12K6). The chemical composition of the shells of the nanocluster K116 is 0@12(In6Tl6)@26(In12K14)@78(In36Tl20K12).

Self-assembly of crystalline structure. The nanoclusters K132 (Table 4) and K116 (Table 5) are the frame-forming clusters.

Table 4.   Atoms forming 132–atom nanocluster K132. Na and K atoms are highlighted in bold
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Table 5.   Atoms forming 116-atom nanocluster K116. K atoms are highlighted in bold
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Layer. The \(S_{3}^{2}\) basic layer is formed of clusters K132 upon the binding of the primary chains with a shift (Fig. 3). The distance between the centers of clusters K132 in the primary chain and in the layer determines the values of the translation vectors a = b = 16.909 Å.

Fig. 3.
figure 3

Binding of clusters K132 (a) and K116 (b) upon layer formation.

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Packet. The layer of clusters K116 is formed on the surface of the layer of clusters K132 (Fig. 4). In the voids of the layer of clusters K116, spacer atoms K8 and Tl10 occupy positions 2d (1/3, 2/3, 0.174) and Tl10 (2/3, 1/3, 0.185) with symmetry 3m. The thickness of the two-layer packet corresponds to the absolute value of the translation vector c = 28.483 Å.

Fig. 4.
figure 4

Binding of layers of clusters K116 and K132 upon formation of packet.

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Self-assembly of shell. The 3D framework structure \(S_{3}^{3}\) is formed upon the binding of two-layer packets in the Z direction.

CONCLUSIONS

Two frame-forming nanoclusters have been detected using the decomposition of the 3D atomic lattice: K132 and K116 with symmetry g = –3m. The voids of the layer of K116 clusters are occupied by spacer atoms K and Tl.

The chemical composition of shells of the nanocluster K132 is 0@12(In6Tl6)@30(In6Na6K18)-@90(In72Na12K6) and that of the nanocluster K116 is 0@12(In6Tl6)@26(In12K14)@78(In36Tl20K12).

The symmetrical and topological code of the self-assembly of the 3D structure from the precursor nanoclusters K132 and K116 has been reconstructed in the following form: primary chain → layer → frame.