Role of interactions of triplets of atoms in the formation of elastic properties of cubic crystals

It is widely known that the interaction of triplets of particles (which are not reducible to the sum of pair interactions) play an important role in formation the nonlinear and anisotropic properties of crystals. The aim of this work is to establish a general form of the dependence of these properties on interactions of triplets of atoms that is dependent only on the structure of the crystals. To separate the interactions of triplets of atoms from those of quadruples, we assume that the energy of triplets of interacting atoms depends only on the characteristics of a triangle whose vertices are the centers of gravity of interacting atoms. In this model, the internal energy must therefore be invariant upon rearrangement of the numbers of interacting atoms and any rotation of the triangle in isotropic space, P(3) × O(3). To specify the form of energy dependence on characteristics of selected triangles, we constructed an integral rational basis of invariants in dependence on the components of the vectors connecting the vertices of each triangle. Using the example of a simple dependence of the energy on basic invariants, we show how to move from the assumed energy dependence on invariants to the dependence of energy on lattice sums.     

Planar detectors for multielement systems of millimeter-wave imaging

We discuss the concept of developing a millimeter-wave multielement matrix receiver on the basis of planar antennas with directly coupled low-barrier Schottky diodes. Three main problems are considered which involve choosing and developing a design of the planar antenna coupled with the low-barrier Schottky diode, optimizing the parameters of the low-barrier Schottky diodes for obtaining maximum sensitivity, and ensuring compact arrangement with weak mutual influence of the planar detectors in a two-dimensional array. We propose a design of the slot antenna with an active resistance of about 800 Ω at a resonant frequency of 94 GHz. The detection characteristics of diodes with the differential Schottky-barrier resistance in the range R_j = 0.4–1000 kΩ for a zero bias are studied experimentally. The mutual influence of the neighboring antennas is examined for developing the multielement radio-imaging system. The conditions of weak cross influence of the closely located planar detectors are determined. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 50, No. 12, pp. 1077–1087, December 2007.     

A new method for calculating three-particle interaction in the theory of modulus elasticity

We propose a new method for calculating the potential of multiparticle interaction. Our method considers the energy symmetry for clusters that contain N identical particles with respect to permutation of the number of atoms and free rotation in three-dimensional space. As an example, we calculate moduli of third-order rigidity for copper considering only the three-particle interaction. We analyze nine models of energy dependence on the polynomials that form the integral rational basis of invariants (IRBI) for the group G ₃ = O(3) ? P ₃. In this work, we use only the simplest relation between energy and the invariants forming the IRBI: \(\varepsilon \left( {\left. {i,k,l} \right|j} \right) = \sum\nolimits_{i,k,l} {\left[ { - A_1 r_{ik}^{ - 6} + A_2 r_{ik}^{ - 12} + Q_j I_j^{ - n} } \right]}\), where I _j is the invariant number j (j = 1, 2,..., 9). The results are in good agreement with the experimental values. The best agreement is observed at n = 2, j = 4: \(I_4 = \left( {\vec r_{ik} \vec r_{kl} } \right)\left( {\vec r_{kl} \vec r_{li} } \right) + \left( {\vec r_{kl} \vec r_{li} } \right)\left( {\vec r_{li} \vec r_{ik} } \right) + \left( {\vec r_{li} \vec r_{ik} } \right)\left( {\vec r_{ik} \vec r_{kl} } \right)\).     

Electron microscopy of pyrocarbon and pyroboroncarbon layers

We present the results from a comparative analysis of the structures of isotropic pyrocarbon and pyroboroncarbon performed by means of transmission electronic microscopy. It is shown that the structure and properties of both materials depend on the mutual alignment of the hexagonal carbon networks forming a pyrocarbon phase. The formation of a pyrocarbon phase in pyroboroncarbon is influenced by monocrystal inclusions with sizes of 10 to 20 nm.     

Groups of Exponential Algebras

In this paper we study groups corresponding a free exponential algebra, a free product of exponential algebras and a HNN-extension of exponential algebras over a field F of characteristic 0.     

Unsolvable algorithmic problems for semigroups,groups, and rings

A survey of the subject outlined in the heading (with many proof s sketched) is given. A special focus is on the original proofs of the unsolvability theorems of Markov, Post, and Novikov for word problems in semigroups and groups. A method of Shirshov is described, which has led to proof of the main unsolvability theorems for Lie algebras.Translated from Itogi Nauki I Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 25, pp. 3–66, 1987.The Russian names in this survey are given according to standard transliteration. Some Russian authors, however, are also known in the Western literature in a different spelling. The corresponding equivalents are given below, in alphabetical order: