Finite noncrystallographic groups, 11-vertex equi-edged triangulated clusters and polymorphic transformations in metals

Abstract

The one-to-one correspondence has been revealed between a set of cosets of the Mathieu group M 11 , a set of blocks of the Steiner system S(4, 5, 11) and 11vertex equi-edged triangulated clusters. The revealed correspondence provides the structure interpretation of the S(4, 5, 11) system: mapping of the biplane 2-(11, 5, 2) onto the Steiner system S(4, 5, 11) determines uniquely the 11-vertex tetrahedral cluster, and the automorphisms of the S(4, 5, 11) system determine uniquely transformations of the said 11-vertex tetrahedral cluster. The said transformations correspond to local reconstructions during polymorphic transformations in metals. The proposed symmetry description of polymorphic transformation in metals is consistent with experimental data.Automorphism of the S(4, 5, 11) Blocks corresponding to 'bottom' tetrahedra of cluster in Fig. 3 No mapping 39 562 34 987 35 40X 4 08 562 60 987 86 40X 4 À1 60 987 86 40X 08 562 2 4 À2 86 40X 08 562 60 987

Symmetry foundations of a polymer model for close-packed metallic liquids and glasses

“…We suggest describing the structure of metallic liquids and the related glasses using (i) a base set of three spirals made of seven-vortex clusters of four reg-ular tetrahedra and (ii) combinatorial permutations of the vertices of a certain set of coordination polyhedra. These permutations were successfully used to describe the polymorphic transformations in solid metals [5].…”

“…At k = 3, 4, and 5, the biplanes made of ν = 1 + k(k -1)/2 elements form a specific short series with ν = 4, 7, or 11, i.e., 2-(4, 3, 2), 2-(7, 4, 2), or 2- (11,5,2). Galois proved that there exists a specific series of four finite groups PSL(2, p), p = 3, 7, 11, and 5 [10].…”

“…The centers of the Bernal or Frank-Kasper polyhedra or their combinations can correspond to the vertices of spiral packings. 5,2), which determines the laws of assembling tetrablocks into tetrahedral spiral chains, is deduced from starting block 13459. This block is a set of nonzero Galois field squared on the order of 11 reduced to modulo 11.…”

“…For example, we have 10 2 = 1 since 100 -9 × 11 = 1. In turn biplane 2- (11,5,2) generates the scheme of block design 4-(11, 5, 1), which is called Steiner system S(4, 5, 11). As is seen from the designations of the Steiner system, 11 num- 2 The field is a closed set of numbers in which the laws of addition and multiplication are introduced.…”

Crystal size induced reduction in thermal hysteresis of Ni-Ti-Nb shape memory thin films

“…It is well known that phase transformations originate from the local reconstructions of a few atoms. 20 When the grain size becomes small enough, large amount of grain boundaries will promote the phase transformation since there are more defects and the phase transformation is more easily to occur there. 20 This would facilitate the martensitic transformation and the reverse transformation, resulting in a narrow thermal hysteresis, as indicated in the overlapped ER-T curves shown in Fig.…”