INTRODUCTION

The crystal chemistry family of double Hg intermetallides includess about 200 compounds [1]. The formation of Hg intermetallides was established in 45 A–Hg systems. The largest number of crystallochemically different intermetallides (eleven) was found in the binary system Ca–Hg, with a wide range of changes in the Hg : A atomic ratio from 11 to 0.33 (Table 1, [2–6]).

Table 1. Crystallographic data
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The crystal structures of six Hg-intermetallides were included in the crystal chemistry families of the most common types of crystal structures, with several hundred specimen each. Three Hg intermetallides belonged to the crystal chemistry families with more than forty species (Table 1). The authors of [7, 8] simulated the cluster self-assembly of widely spread types of crystal structures established for Hg intermetallides.

The most crystallochemically complex family of intermetallides Ca11Hg54–hP65 included two more compounds: Sr11Hg54–hP65 [5] and Yb11Hg54–hP [9]. The crystal structure of the intermetallide Ca11Hg54–hP65 [5] with the spatial group P-6 (no. 174) and V =1492.7 Å3 was characterized by 18 crystallographically independent atoms with the Wyckoff sequence l6k3j4ihgfa. For Hg atoms, a wide spectrum of coordination number (CN) values was established: 11 (5 atoms), 12 (8 atoms), and 13 (1 atom). Ca atoms exhibited the CN of 14 (1 atom), 15 (2 atoms), and 16 (1 atom).

In the present study, we performed the geometric and topological analysis of the crystal structure of the intermetallide Ca11Hg54–hP65 by the ToposPro software package [10]. The symmetry and topological code for the processes of self-assembly of the 3D structure from precursor clusters K46 were simulated in the following form: primary chain → microlayer → microframework.

The present study represents a followup to the works [7, 8, 11–16] in the field of simulation of the processes of self-assembly of systems on the suprapolyhedral scale and the geometrical and topological analysis of crystal structures by advanced computer methods.

COMPUTER ANALYSIS METHODS

The geometrical and topological analysis was performed by means of the ToposPro software package [10], which made it possible to conduct a multipurpose study of the crystal structure automatically, using the representation of structures in the form of “folded graphs” (factor graphs). The data on the functional role of atoms in the formation of a crystalline structure were obtained by calculating coordination sequences, i.e., sets of numbers {Nk}, where Nk was the number of atoms in the kth coordination sphere of a given atom.

The obtained values of the coordination sequences of atoms in the 3D lattices are provided in Table 2, in which the number of neighboring atoms in the most proximate surrounding, i.e., in the first coordination sphere of the atom, was separated. All atoms were characterized by different sets of the coordination sequences {Nk}; therefore, all the atoms were topologically (and functionally) different.

Table 2.   Local surrounding of Ca and Hg atoms and values of coordination sequences
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The algorithm for automatic decomposition of a structure of any intermetallide, represented in the form of a folded graph, onto cluster units was based on the following principles: a structure was formed as a result of self-assembly from precursor clusters. Here, the precursor clusters formed a structure framework, voids in which were filled with spacer clusters (consisting of a small number of atoms), the precursor nanoclusters did not have shared internal atoms; however, they could have shared atoms on the surface, the precursor clusters occupied highly symmetric positions and the set of precursor nanoclusters and spacer clusters included all atoms of the structure.

The algorithm was implemented by the ToposPro software package [10].

SELF-ASSEMBLY OF THE CRYSTAL STRUCTURE OF Ca11Hg54–hP65

The method of simulation of the crystal structure that we used was based on determining the hierarchical sequence of its self-assembly in the crystallographic space [11]. At the first level of the system self-assembly, the mechanism of formation of a primary chain of the structure from 0-level nanoclusters formed at the template stage of the chemical evolution of the system is determined, then the mechanism of self-assembly from the layer chain (2nd level), and, thereafter, from the layer of three-dimensional framework (3rd level).

Crystallographic data. Parameters of the hexagonal cell: a = b = 13.389, c = 9.615 Å.

Space group P-6 (no. 174) with elements of the point symmetry: g = –6 (1a, 1b, 1c, 1d, 1e, 1f), 3 (2g, 2h, 2i), m (3j, 3k). The order of the group was 6.

Polyhedral clusters K8, K11 and K12. 184 variants of the cluster representation of the 3D atomic lattice with a number of structural units from 3 to 7 was established.

The polyhedral clusters K8 = 0@Ca2Hg6 which appeared to be hexagonal bipyramids, the polyhedral clusters K11 = 0@Ca3Hg8 and the polyhedral clusters with the central Hg atom K12 = Hg (Ca3Hg8) were determined (Fig. 1).

Fig. 1.
figure 1

Polyhedral clusters.

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The centers of the clusters Ca2Hg6, 0@Ca3Hg8 and Hg(Ca3Hg8) occupied the most highly symmetric positions 1c, 1b, and 1f with a –6 symmetry.

Suprapolyhedral precursor clusters K46. The clusters Ca2Hg6 represented templates, on the surface of which atomic shells were formed of 38 atoms (Fig. 2). The composition of the two-layered cluster K46 was 0@8(Ca2Hg6)@38(Hg6 + CaHg6)2(Ca6Hg6).

Fig. 2.
figure 2

Cluster K46 (two projections).

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Self-assembly of the crystalline structure. Primary chain. The formation of primary chains \(S_{3}^{1}\) occured upon binding of K46 clusters by triple Hg3 rings towards the direction [001]. The distance between the centers of the K46 in the primary chain determined the value of the translation vector c = 9.818 Å.

2D layer. The layer \(S_{3}^{2}\) was formed upon binding of the primary chains \(S_{3}^{1}\) towards the direction [100] (Fig. 3). A distance between the axes of the primary chains determined the value of the translation vector a = 13.602 Å. The voids in the framework were occupied by the polyhedral clusters 0@Ca3Hg8 and Hg(Ca3Hg8) and the spacer Hg(14) atoms (Fig. 3).

Fig. 3.
figure 3

Framework structure (two projections).

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Self-assembly of the framework. The 3D framework structure \(S_{3}^{3}\) was formed upon binding of the 2D layers (with a shift) towards the direction [010] (Fig. 3). In the 3D framework, the distance between the equivalent 2D layers determined the value of the vector b = 13.602 Å.

CONCLUSIONS

The combinatory-topological analysis and simulation of cluster self-assembly of the Ca11Hg54–hP65 crystal structure have been performed.

The polyhedral clusters K8 = 0@Ca2Hg6, which had appeared to be hexagonal bipyramids, the polyhedral clusters K11 = 0@Ca3Hg8, and the polyhedral clusters with the central Hg atom K12 = Hg(Ca3Hg8) have been determined.

The clusters Ca2Hg6 represented templates, on the surface of which atomic shells of 38 atoms have been formed. The composition of the two-layered cluster K46 was 0@8(Ca2Hg6)@38(Hg6 + 6)2(Ca6Hg6).

The symmetry and topological code of the processes of self-assembly of the 3D structures from K46 precursor nanoclusters in the form primary chain → layer → framework have been simulated. The voids between the primary chains have been occupied by the polyhedral clusters 0@Ca3Hg8 and Hg(Ca3Hg8), as well as by the Hg spacer atoms.