ISSN 10637745, Crystallography Reports, 2012, Vol. 57, No. 1, pp. 1–9. © Pleiades Publishing, Inc., 2012. Original Russian Text © A.A. Pankova, G.D. Ilyushin, V.A. Blatov, 2012, published in Kristallografiya, 2012, Vol. 57, No. 1, pp. 5–13.
THEORY OF CRYSTAL STRUCTURES
Nanoclusters Based on Pentagondodecahedra with Shells in the Form of D32, D42, and D50 Deltahedra in Crystal Structures of Intermetallic Compounds A. A. Pankovaa, G. D. Ilyushinb, and V. A. Blatova a
b
Samara State University, Samara, 443011 Russia Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskii pr. 59, Moscow, 119333 Russia email:
[email protected] Received January 25, 2011
Abstract—The TOPOS software package has been used to form a database of intermetallic compounds con taining pentagondodecahedral d clusters (528 crystal structures of intermetallic compounds, 111 topological types, and 47 space symmetry groups). On the whole, 606 atomic d configurations have been selected which are described by 14 point symmetry groups. Examples of nanoclusters are presented which are precursors of the crystal structures of intermetallic compounds with the outer shell in the form of deltahedra D, which are formed above dodecahedra. These nanoclusters are identified in the automatic mode of structural data pro cessing: D32 (K8In6Ge40, Cs30Na3Sn162), D42 (Ru3Be17, Y3Cd18, Ca3(Cd17Al)), and D50 (Yb3Zn18, Ce3(Au14Sn3), Pr3Cd18, Eu4Cd25), where 32, 42, and 50 are the numbers of atoms in the shell. Similar delta hedra were found previously in icosahedral nanoclusters (precursors of intermetallic compounds). Structures with the dodecahedral nanocluster precursors containing D42 and D50 deltahedra are approximants of MCd5.7 (M = Yb or Ca) quasicrystals and belong to the family of MCd6 (M = Ce, Pr, Nd, Sm, Eu, Gd, Dy, Yb, Y, or Ca). DOI: 10.1134/S1063774511040171
INTRODUCTION
balls; in all structural shell models developed by Casper and Klug, pentamers are surrounded by five hexamers [2].
Among many polyhedral structural units composed of molecules, clusters, or atoms, the quasispherical shells in the form of deltahedra with the iicosahedral (noncrystallographic) symmetry I (235) and Ih ( m35 ) are of particular interest [1]. These shells are typical of biological objects (viruses) [2, 3] and inorganic com pounds (alloys of metals with a 3D periodic structure and i quasicrystals) [4–13]. Currently, the structural details of many icosahe dral viruses have been established at the atomic level using Xray diffraction analysis [2, 3]. The number of structural units (ðåntamers and hexamers, which are composed of five and six protein molecules, respec tively) forming the outer shells of viruses is in complete agreement with the data of the geometric simulation of shells that was performed by Casper and Klug as early as in 1962; their structure was considered in detail by B.K. Vainshtein in [2]. Examples of shells of different viruses were reported in the review [3]. Casper and Klug established that, in the simplest representatives of the two icosahedral classes, the number of quasispherical units forming the shell are 12, 42, 92… (class Р = 1) and 32, 122, 272… (class Р = 3). In the structural models of viruses, the pen tamer and hexamer shells are depicted as twocolored
Among intermetallic compounds, nanoclusters in which the shells in the form of deltahedra with icosa hedral symmetry are located above the template (internal i icosahedron) are of particular interest. Such nanoclusters are typical periodic approximants of i quasicrystals [4–9]. The numbers of atoms forming the shells of nanoclusters that are Bergman approxi mants 1@12@32 [7–9] and Mackay approximants 1@12@42 [10] are, respectively, 32 and 42. The topo logical structure of the D32 and D42 deltahedra in nanocluster approximants corresponds to the simplest geometric versions of twocolored Casper and Klug shells; the D32 and D42 deltahedra belong to the classes Р = 3 and 1, respectively. In [11–13] we revealed for the first time two new highsymmetry icosahedral shells in the form of 50atom D50 deltahedron. They are characterized by compli cated structural topology because, along with atoms with coordination numbers of 5 and 6, atoms with a coordination number of 7 also arise in the shell; i.e., to describe them topologically, one must use models of threecolored shells. 1
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Noncrystallographic icosahedral point symmetry groups I (235) and Ih ( m35 ) were considered in detail in [1], where is was noted that dual i icosahedron and d dodecahedron (20atom dodecahedron with faces in the form of regular pentagons) are described by the same point symmetry group m35 . One would expect that, because of the absence of symmetry limitations, d dodecahedra (both empty and filled) in intermetallic compounds play the role of templates on which D32, D42, and D50 deltahedra (with icosahedral symmetry) are formed. Examples of three spontaneously formed unoccupied shells with icosahedral symmetry, along with the i icosahedron В12, are C20H20 molecules (with a d dodecahedron formed of C atoms) and fullerenes С60, which have the maximally possible crystallographic symmetry m3 (with a group order of 24) in crystal structures with the sp. gr. Fm3 (no. 202). The selfassembly of В12, С20, and С60 clusters from microclusters was simulated in [14, 15]. In this study we performed a systematic analysis of all known structures of binary and more complex (in composition) intermetallic compounds containing local regions in the form of empty and filled dodeca hedra in which atoms are bound by chemical bonds. The symmetry space groups and the crystallographic positions occupied by d dodecahedra in unit cells were determined for these crystal structures of intermetallic compounds. We present examples of nanocluster pre cursors of crystal structures with shells in the form of D32, D42, and D50 deltahedra, which were identified in the automatic mode of structural data processing using the TOPOS package [16, 17]. The nanocluster precursors of crystal structures are considered to be the main types of clusters which, being linked, form the primary chain of crystal structure and determine the moduli of translation vectors [18–20]. This study continues the investigations [8, 9, 11– 15, 18–20] in the field of geometrical and topological analyses of the structure of crystalline phases and sim ulation of the selforganization of chemical systems, as well as the development of new methods for analyz ing crystal structures. OBJECTS AND METHODS OF ANALYSIS In this study we developed a data bank containing 2001 topological types of crystal structures of interme tallic compounds. Note that a topological type includes atomic nets with the same topology, regard less of the space symmetry of the net. In this context, one topological type may include several structure types. The bank was developed based on the TOPOS package [16] and the atomicstructure data from the CRYSTMET and ICSD databases, which contain information about more than 27 000 completely inter
preted structures of alloys and intermetallic com pounds. The algorithm of automatic geometric and topo logical analysis with the use of the TOPOS package included the following stages. (i) Calculation of the adjacency matrix of the struc ture and selection of the simplest polyhedral structural units using the AutoCN program. We took into account the interatomic interactions corresponding to the main faces of the Voronoi–Dirichlet polyhedra of atoms. As a result, the structures of intermetallic com pounds were represented in the form of a 3D unori ented graph and atoms and interatomic bonds were associated with the graph vertices and edges, respec tively. (ii) Calculation of the coordination sequences for all crystallographically independent atoms in a crystal structure using the IsoTest program. (iii) Search for dodecahedral fragments in atomic nets applying the algorithm for selecting finite sub graphs of arbitrary complexity in infinite periodic graphs. Local configurations corresponding to each crys tallographically independent atom were determined for all crystal structures of intermetallic compounds, and point symmetry was found for each established d configuration. The presence of local regions in the form of multi layer d nanoclusters was revealed for structures with large unitcell parameters. In some cases these multi layer d nanoclusters correspond to nanocluster pre cursors (primary nanoclusters), which can be identi fied in the automatic mode of structural data process ing using the TOPOS package. To determine the composition and structure of the clusters forming an intermetallic compound, we used the nanocluster model [11–15], which is based on the following principles: (i) The structure is formed by multilayer primary nanoclusters, both centered and lacking the central atom. The number of different primary nanoclusters generally does not exceed two, and the number of lay ers in them varies in the range of 1–3. (ii) The centers of nanoclusters are in the most symmetric structural positions. (iii) Nanoclusters should not have common inter nal atoms (i.e., they are not interpenetrating) but may share surface atoms. (iv) Along with primary nanoclusters, the structure may contain spacers (clusters of smaller atoms or sin gle atoms) which fill voids between primary nanoclus ters. (v) The set of primary nanoclusters and spacers should include all atoms of the structure. As a result, we developed a database of intermetal lic compounds containing linked atomic fragments in the form of d dodecahedra (606 d clusters, 111 topo
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Table 1. Distributions of 607 dodecahedral (D20) and 8892 icosahedral (I12) configurations in intermetallic compounds over the point symmetry groups Point group m 3 (Th) mmm (D2h) 3 m (D3d) 3m (C3v) 2/m (C2h) m (Cs) 3 (C3) 23 (T)
Number of configu Number of configu rations I12 rations D20 322 73 53 45 32 25 23 9
732 441 2269 805 1380 1382 83 9
3 (C3i) 1 (Ci) 1 (C1) mm2 (C2v) 32 (D3) 2 (C2) 222 (D2)
logical types, and 528 crystal structures of intermetal lic compounds). SYMMETRY OF DODECAHEDRAL CLUSTERS IN THE CRYSTAL STRUCTURES OF INTERMETALLIC COMPOUNDS A dodecahedron (d), which is the base for selected d nanoclusters, is a 20vertex polyhedron and its faces are pentagons. This pentagonal dodecahedron has 20 vertices, 30 edges, and 12 faces. The maximum symmetry of a d dodecahedron, as well as that of i icosahedron, is m35 ; the point group order is 120. In crystal structures, both dodecahedra and icosahedra occupy positions corresponding to crystallographic point groups that are subgroups of either m35 Th ( m3 ) (order 24) or D3d ( 3m ) (order 12) groups or the positions corresponding to their sub groups. The point symmetry elements of d and i poly hedra are three and twofold axes, m planes, and the center of symmetry. In total, 47 symmetry space groups were estab lished for the crystal structures of intermetallic com pounds. The distribution of dodecahedra over the point symmetry groups is given in Table 1. For com parison, this table also contains a similar distribution for 8892 icosahedral configurations in 5691 structures of intermetallic compounds. This analysis showed that the symmetry of dodeca hedral clusters in structures of intermetallic com pounds is described by 14 crystallographic point groups, which are subgroups of the symmetry group m35 . Dodecahedra with the D2 (222) symmetry, which also corresponds to a subgroup of the m35 group, were not revealed. CRYSTALLOGRAPHY REPORTS
Point group
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Number of configu Number of configu rations I12 rations D20 9 5 3 3 2 2
22 134 780 725 7 122
The following structural features of the crystal structures with dodecahedral configurations were established: (i) The crystallographic symmetry Th ( m3 ) of d dodecahedron is implemented most often (in more than 50% of cases). The D3d ( 3m ) point symmetry comes in third place (8.7%). For icosahedral configu rations it was found that the D3d point symmetry is most widespread, while Th is in the sixth place. (ii) The practical absence of completely desymme trized clusters (point group C1) is due to the fact that structures contain generally only one dodecahedral configuration. This is the main difference between dodecahedral structures and icosahedral ones, which often contain several crystallographically independent icosahedra (with a successively lowering symmetry). (iii) No examples of implementing nonequivalent centered dodecahedral configurations in the same structure were found. An example of crystal structure with two noncentered dodecahedral configurations is Eu4Cd25 (sp. gr. Fd3 (no. 203), in which the centers of nanoclusters are in the 8a and 8b positions with the symmetry 23). CRYSTAL STRUCTURES OF INTERMETALLIC COMPOUNDS CONTAINING SHELLS IN THE FORM OF D32, D42, AND D50DELTAHEDRA Note the main structural features of intermetallic compounds containing dodecahedral clusters. The structures of intermetallic compounds are known to contain icosahedra of two types: in the form of 12atom (empty) 0@B12 clusters and 13atom C@B12 clusters (centered by smaller C atoms). Dodecahedral clusters can also be empty; one exam ple is 0@B20 (Fig. 1). However, when filled, their cen ters are generally occupied by either the largest А atoms, which enter the compound composition (А@B20, Fig. 2a, version 1); the simplest polyhedra,
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(b)
(a)
Fig. 1. Dodecahedral 0@Be20 cluster in the Ru3Be17 structure in the form of (a) bound atoms and (b) a polyhedron.
(a)
(b) Cd4
Cd4
Cd7
Cd8 Cd4
Fig. 2. Filled dodecahedral nanoclusters: (a) K@Ge20 in the K8In6Ge40 structure and (b) Cd4@Cd20 with an internal tetrahe dron in the YbCd6 structure.
for example, tetrahedra formed by smaller atoms (C4@B20) or composed of the same atoms that form the dodecahedral shell (В4@B20, Fig. 2b, version 2); or both versions of shell filling are implemented simul taneously (version 3). Dodecahedral clusters can be isolated in the form of a shell of 20 chemically bound large А atoms formed above an i icosahedron of any type: in the form of an empty 12atom 0@B12 cluster or 13atom C@B12 cluster. The А atoms of the dodecahedral shell are located above the 20 faces of the icosahedron, and all fiveatom windows of this shell are centered by 12 B atoms. This version corresponds to Bergman nano clusters with the D32 deltahedron (version 4); a series of such structures was analyzed in [8]. Dodecahedral clusters, both empty and filled (ver sions 1, 2, and 3), as well as icosahedral clusters, are templates; on their surface, three types of highsym metry shells are built in the form of D32, D42, and D50 deltahedra. Bergman nanoclusters with the D32 deltahedron (version 4) are also templates; on their surface, differ
ent types of highsymmetry shells are formed. One example of a Bergman nanocluster with the D92 delta hedron (corresponding to class Р = 1 in the Casper and Klug notation) was considered in [9]. Below we consider examples of crystal structures containing D32, D42, and D50 deltahedra formed on template dodecahedra (versions 1, 2, and 3); the main structural data for these compounds are listed in Table 2. One can see that the crystal structures have mainly cubic symmetry and are characterized by a wide range of variation in the linear cell parameters (from a = 11.337 to a = 31.871 Å) and in the cell vol ume (from 1457.12 to 32 375.74 Å3) in the Ru3Be17 and Eu4Cd25 intermetallic compounds. Note that the Eu4Cd25 structure with the Pearson index cF1416 and the Wyckoff sequence g12fe6c (with 20 independent atoms) is one of the most complex structures of inter metallic compounds. Table 3 contains the coordination sequences {Nk} of atoms for three structures with D32, D42, and D50 deltahedra. The values of the coordination sequences
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Table 2. Main structural data for intermetallic compounds with shells in the form of D32, D42, and D50 deltahedra Compound
Space group
Pearson index
Wyckoff sequence
Cell parameters a, c, Å
Volume, Å3
10.977
1322.67
23.556, 12.104
6716.67
11.337
1457.12
15.482
3710.92
15.638 22.134, 27.108
3824.23 11501.33
14.299
2923.6
15.118
3455.28
Structures with shells D32 K8In6Ge40 [21]
cP54
Cs30Na3Sn162 [22]
Pm 3 P42/mnm
Ru3Be17 [23]
Im 3
cI160
Y3Cd18 [24]
Im 3 I23 R3
cI162
cI194
Ce3(Au14Sn3) [28]
Im 3 I23
cI159
hg4fedca f5edc3
Pr3Cd18 [29]
Im 3
cI258
h2g4f2edc
15.643
3827.90
Eu4Cd25 [30]
Fd 3
cF1416
g12fe6c
31.872
32375.74
YbCd6 [25] Ca3Cd17Al [26] Yb3Zn18 [27]
kidca 647 22
tP206 k j i g f db Structures with shells D42 hg3fed
hg4fed cI158 f5dc3 hR168 b54a6 Structures with shells D50
of atoms at k = 1 and 2, which are 20 and 32 for the K8In6Ge40 structure and 20 and 50 for the Yb3Zn18 structure, indicate the presence of a nanocluster formed on the dodecahedron and containing a large atom (K or Yb) at the center. For the YbCd6 interme tallic compound, we should note the topological sym metry of Cd7 and Cd8 atoms (in the dodecahedron shell) and Cd1 and Cd2 atoms (in the shell of the D42 deltahedron), which have the same values of coordi nation sequences. For the entire isolated group of crystal structures, the geometric and topological char acteristics of the D32, D42, and D50 deltahedra are listed in Table 4.
Structures with Shells in the Form of D32 Deltahedra In the K8In6Ge40 and Cs30Na3Sn162 structures, dodecahedral K@Ge20 and Cs@Sn20 clusters with shells in the form of D32 deltahedra were established (Fig. 3). The cluster centers occupy the most symmet ric positions (2a and 2b in the cubic and tetragonal cells, respectively) and are characterized by the m 3 and mmm symmetries (Table 4). In the K8In6Ge40 structure, singlelayer dodecahe dral K@Ge20 clusters are primary nanoclusters and atoms from the D32 shell (K and In) are located between linked K@Ge20 nanoclusters. (b)
(a)
Fig. 3. Outer D32 shells above dodecahedral clusters: (a) the K8In6Ge40 structure (in the shell 12 atoms (white spheres) are located above the dodecahedron faces and the Ge and In atoms (black and gray spheres, respectively) are above the dodecahedron vertices) and (b) the Cs30Na3Sn162 structure (in the shell 12 atoms (white spheres) are above the dodecahedron faces and the Sn atoms (black spheres) are above the dodecahedron vertices). CRYSTALLOGRAPHY REPORTS
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Table 3. Coordination sequences of atoms in structures with D32, D42, and D50 deltahedra Coordination sequences Atom N1
N2
N3
K1 K2 Ge1 Ge2 In1
20 24 8 8 8
32 38 68 67 70
152 160 98 103 92
Cd1, Cd2 Cd2 Cd3 Cd4 Cd5 Cd6 Cd7, Cd8 Yb1 Cdtetraedr
12 12 12 10 12 15 10 16 20
47 47 47 51 45 45 52 47 42
112 112 109 107 92 107 109 109 114
Zn2 Zn3 Zn4 Zn5 Zn6 Zn7 Zn8 Yb1 Yb2
15 13 10 12 11 12 8 16 20
49 49 53 49 53 49 38 49 50
111 116 113 115 116 96 110 113 122
N4 K8In6Ge40 230 278 254 246 250 YbCd6 194 194 197 189 207 195 193 208 224 Yb3Zn18 203 205 197 203 206 211 188 220 224
N5
N6
N7
N8
344 324 410 421 414
560 658 497 509 452
818 824 827 799 846
992 946 1067 1091 1058
321 321 320 328 308 324 320 316 314
478 478 482 460 462 465 466 467 476
642 642 637 633 665 661 650 648 654
858 858 840 854 836 853 853 861 854
340 334 344 336 332 332 308 330 362
481 502 490 504 497 470 464 491 500
689 677 667 667 689 705 674 678 686
913 896 890 884 896 884 848 907 950
Note: The numbers of neighboring atoms in the nearest environment (k = 1) in 3D nets are bolded.
Table 4. Geometric and topological characteristics of D32, D42, and D50 deltahedra in crystal structures Compound
Coordination numbers of atoms in the shell
Deltahedron type
Space group
Position
Point symmetry
K8In6Ge40
D32
2a
D32
2b
m3 mmm
5, 6
Cs30Na3Sn162
Pm 3 P42/mnm
Ru3Be17
D42
Im 3
2a
m3
5, 6
Y3Cd18
D42
2a
D42
2a
m3 23
5, 6
YbCd6
Im 3 I23
5, 6
Ca3(Cd17Al)
D42
R3
3a
3
5, 6
Yb3Zn18
D50
2a
Ce3(Au14Sn3)
D50
Im 3 I23
2a
m3 23
5, 6, 7
Pr3Cd18
D50
Im 3
2a
m3
5, 6, 7
Eu4Cd25
D50
Fd 3
8а
23
5, 6, 7
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5, 6, 7
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(a)
Fig. 4. Outer D42 shells above dodecahedral clusters: (a) the Ru3Be17 structure (in the shell Ru atoms (white spheres) are located above the dodecahedron faces and Be atoms (black spheres) are above the dodecahedron edges) and (b) the YbCd6 structure (in the shell Yb atoms (white spheres) are located above the dodecahedron faces and Cd atoms (black spheres) are above the dodeca hedron edges).
In the Cs30Na3Sn162 structure, the primary nanoclus ters are twolayer dodecahedral Cs@Sn20@Cs12Sn20 clusters with D32 shells. Nanoclusters form primary chains which are oriented in the [001] direction and spaced by c/2. In both cases the largest atoms entering the com pound composition (K and Cs) are located at the cen ters of dodecahedra and in the D32 shell above the dodecahedron faces, whereas Ge, In, and Sn atoms are above the dodecahedron vertices. The coordina tion numbers of atoms in the shell are 5 and 6. Structures with Shells in the Form of D42 Deltahedra Dodecahedral clusters with shells in the form of D42 deltahedra were established in the cubic structure of types Ru3Be17 (Fig. 4a) and Y3Cd18 (YCd6) (Fig. 4b), as well as in the trigonal structure Ca3(Cd17Al). The symmetries of nanoclusters (m 3 and 3 ) are also maximally possible for the corresponding space groups (Table 4). In the shells of D42 deltahedra, Cd1 and Cd2 atoms are equivalent to Be6 atoms (Figs. 4a, 4b). In the cubic structure of Ru3Be17, the dodecahedral cluster is not filled (Fig. 1), whereas in the Y3Cd18 cubic structure, dodecahedral clusters contain tetra hedra composed of Cd atoms, which occupy three dif ferent positions; the same atoms form the dodecahe dral shell. In the Ca3(Cd17Al) trigonal structure, dodecahedral clusters contain tetrahedra composed of Al atoms, which occupy a fixed position. In all clusters the largest atoms entering the com pounds (Ru, Y, Ca) are located above the dodecahe dron faces, whereas the other atoms (Be and Cd) are above the midpoints of dodecahedron edges. The coordination numbers of atoms in the shell are 5 and 6. CRYSTALLOGRAPHY REPORTS
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In all three structures, the twolayer dodecahedral clusters with D42 deltahedra are primary nanoclusters and form a bcc packing. Note that all the structures containing shells of this type are known to be approx imants of quasicrystals or types related to quasicrys tals. Structures with Shells in the Form of D50 Deltahedra The third type of the deltahedral shell (D50) occurs in the cubic structures Yb3Zn18 (Fig. 5a), Ce3(Au14Sn3) (Fig.5b), Eu4Cd25, and Pr3Cd18, which are related to the quasicrystal approximants MCd5.7 (M = Yb or Ca) and belong to the set of approximants MCd6 (M = Ce, Pr, Nd, Sm, Eu, Gd, Dy, Yb, Y, or Ca). Nanoclusters have maximally possible symme tries m 3 and 23 and occupy 2a positions in unit cells. The dodecahedral cluster in the Ce3(Au14Sn3) structure is not filled (Fig. 5b). In Yb3Zn18 and the Aldoped compound Yb3(Zn,Al)18, filled dodecahe dral clusters contain equiprobably both the largest atoms, which enter the compound composition (Yb), and the simplest polyhedra of atoms forming the icosahedral shell. In all structures the twolayer dodecahedral clusters with a D50 shell are primary nanoclusters. As in the aboveconsidered deltahedral shells, the largest atoms of the D50 shell are located above the dodecahedron faces and the other atoms are above both the vertices and the midpoints of dodecahedron edges. The 50atom ε shell, found previously in icosa hedral nanoclusters, has the same symmetry and topo logical structure [11–13].
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(a)
Fig. 5. Outer D50 shells above dodecahedral clusters: (a) the Yb3Zn18 (Yb Zn6) structure (in the shell Yb atoms (white spheres) are above the dodecahedron faces and Zn atoms (black spheres) are above the dodecahedron vertices and edges) and (b) the Ce3(Au14Sn3) structure (in the shell Ce atoms (white spheres) are above the dodecahedron faces and Au atoms (black spheres) and Sn atoms (gray spheres) are above the dodecahedron vertices and edges).
CONCLUSIONS We performed a systematic analysis of all known structures of intermetallic compounds containing local regions in the form of dodecahedra. Examples of nanocluster precursors with shells in the form of D32 and D42 deltahedra with coordination numbers of 5 and 6 were considered. These deltahedra correspond to the simplest geometric versions of the structure of twocolored Casper and Klug shells; the D32 and D42 deltahedra are related to the classes P = 3 and 1, respectively. In addition, we revealed an icosahedral shell in the form of 50atom D50 deltahedron with a complicated structural topology related to the pres ence of atoms with coordination numbers of not only 5 and 6 but also 7 in the shell. These deltahedra were previously found in icosahedral nanocluster precur sors of intermetallic compounds. The reason is that the dual polyhedra, i icosahedron and d dodecahe dron, which play the role of templates, are described by the same point symmetry group m35 and they occupy the same positions in crystal structures, which correspond to crystallographic point groups Th ( m3 ) (order 24) and D3d ( 3m ) (order 12), as well as to their subgroups. Note that a change in the symmetry of template polyhedron will lead to the occurrence of new types of shells in the form of deltahedra. For example, nanocluster precursors of intermetallic com pounds based on the Friauf polyhedron, which is a deltahedron with 16 vertices and 28 faces and has the 43m point symmetry (group order 24), revealed 44 and 52atom deltahedra with conventional coordina tion numbers of 5 and 6; these nanocluster precursors have compositions 1@16@44 and 1@16@52.
ACKNOWLEDGMENTS This study was supported by the Russian Founda tion for Basic Research, project no. 090201269. REFERENCES 1. B. K. Vainshtein, Modern Crystallography. Symmetry of Crystals. Methods of Structural Crystallography (Nauka, Moscow, 1979), Vol. 1 [in Russian]. 2. B. K. Vainshtein, Modern Crystallography (Nauka, Mos cow, 1979), Vol. 2 [in Russian]. 3. V. A. Kostyuchenkov and V. V. Mesyanzhinov, Usp. Biol. Khim. 42, 177 (2002). 4. W. Steurer and S. Deloudi, Acta Crystallogr. A 64, 1 (2008). 5. Yu. Kh. Vekilov and M. A. Chernikov, Usp. Fiz. Nauk 180 (6), 561 (2010). 6. V. E. Dmitrienko and V. A. Chizhikov, Kristallografiya 52, 1177 (2007) [Crystallogr. Rep. 52, 1040 (2007)]. 7. G. Bergman, J. L. T. Waugh, and L. Pauling, Acta Crys tallogr. 10, 254 (1957). 8. V. A. Blatov and G. D. Ilyushin, Zh. Neorg. Khim. 56, 5 (2011). 9. G. D. Ilyushin and V. A. Blatov, Crystallogr. Rep. 55 (Suppl. 1), 1093 (2010). 10. A. L. Mackay, Acta Crystallogr. 15, 916 (1962). 11. V. A. Blatov, G. D. Ilyushin, and D. M. Proserpio, Inorg. Chem. 49, 1811 (2010). 12. V. A. Blatov and G. D. Ilyushin, XIV Nat. Conf. on Crys tal Growth NKRK2010, Moscow, 2010 (IK RAN, Mos cow), Vol. 2, p. 180. 13. G. D. Ilyushin and V. A. Blatov, Kristallografiya 56 (1), 80 (2011) [Crystallogr. Rep. 56 (1), 75 (2011)]. 14. G. D. Ilyushin, Modeling of SelfOrganization Processes in CrystalForming Systems (URSS, Moscow, 2003) [in Russian]. 15. G. D. Ilyushin, Crystallogr. Rep. 49 (Suppl. 1), S5 (2004).
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Translated by A. Madonov
