The effective Lagrangians and amplitudes of Compton scattering on pion and nucleon with the account of electric and magnetic polarizabilities are obtained on the basis of the theoretical-field approach and solutions of electrodynamic equations by means of Green functions covariant method. Calculations of magnetic and electric quasi-static polarizabilities of spinor particle were evaluated on the basis of matrix elements calculation for Compton scattering amplitudes.

Keywords: polarizability, Lagrangian, Compton scattering, Green function.

V.V. Andreev – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

N.V. Maksimenko – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

It is shown that the group SU(3,3) is the complete continuous group of internal symmetry of the Lagrangian formulation of the theory of the massless Dirac fermion with three internal degrees of freedom. The color symmetry group SU(3) can be considered as a compact subgroup of the latter.

Keywords: transformations, internal symmetry, Dirac field, generators, group, invariance.

P.P. Andrusevich – Belarusian State University, Minsk, Belarus

V.A. Pletyukhov – A.S. Pushkin Brest State University, Brest, Belarus

V.I. Strazhev – Belarusian State University, Minsk, Belarus

State-of-the-art plasmonic structures can be splitted into two classes depending on whether they were obtained by top-down or bottom-up processes. The former have been extensively studied but they remain to be large and simple in terms of feature size and architecture, respectively. The latter have only been made available recently and need much investigation of their near-field optical properties, yet they push the integration limit further and provide a new approach to complex architectures. Self-assembling of nanoparticles in complex superstructures is one of the suggested bottom-up approaches. We discuss the challenging problem of the light evanescent wave optical addressing in such structures and introduce a concept of multi-scale plasmonic architectures.

Keywords: sub-wavelength optics, nanophotonics, nanostructures, metal colloid, near-field optics, plasmonics, self-assembled nanostructures.

S. Viarbitskaya – NanoSciences Group, CEMES/CNRS UPR 8011, 29 rue Jeanne Marvig, 31055 Toulouse Cedex 4, France

E. Dujardin – NanoSciences Group, CEMES/CNRS UPR 8011, 29 rue Jeanne Marvig, 31055 Toulouse Cedex 4, France

C. Girard – NanoSciences Group, CEMES/CNRS UPR 8011, 29 rue Jeanne Marvig, 31055 Toulouse Cedex 4, France

The unified formalism allowing to deduce common expressions for paraxial two-dimensional Gaussian-like light Kummer - Gaussian beams and discover correlations between them is developed. Ñonditions of their physical realizability are explored in details. New types of Kummer - Gaussian beams are discovered. Such beams are presented as Gaussian product on Kummer function of complex argument and a n integer index.

Keywords: paraxial beams, Hermite – Gaussian beams, Kummer – Gaussian beams, Gaussian-like beams, square integrability.

S.S. Girgel – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

The theory of propagation of non-uniform plane gravitational waves in the free space and in the continuous medium at the influence of the external gravitational field on the basis of relativistic gauge-invariant model of scalar gravitational field is developed.

Keywords: gravitational waves, gauge-invariant model, scalar field, wave equation, frequency dispersion, neutron star.

A.N. Godlevskaya – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

A.N. Yegorov – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

A.N. Serdyukov – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

The extinction and scattering cross sections for the case of the plane monochromatic circularly polarized electromagnetic wave scattered by a biisotropic sphere embedded in a biisotropic medium are calculated on the basis of the exact solution of the Mie problem. Scattering and extinction cross section dependences on the ratio of the sphere radius to the wavelength are analyzed numerically. The resonance structure of the scattering and extinction cross sections is revealed. These cross sections have narrow and very narrow peaks which are superimposed upon slowly oscillating diffraction background.

Keywords: biisotropic medium, constitutive relations, Mie theory, spherical electromagnetic waves, scattering cross section, extinction cross section, resonance structure.

V.N. Kapshai – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

A.A. Shamyna – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

V.V. Kondratjuk – Department of Automation and Information Systems, P.O. Sukhoi Gomel State Technical University, Gomel, Belarus

Method for determination of resonance states of the relativistic two-particle system and analysis of their influence on the cross section is presented. The method is based on the integral equations in the relativistic configurational representation. This method is applied for the identification of the scattering cross sections structures for a model potential.

Keywords: two-particle integral equations, relativistic configurational representation, complex scaling, resonance states, scattering amplitude, cross section.

V.N. Kapshai – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

K.P. Shilyaeva – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

Yu.A. Grishechkin – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

The article contains results of the research based on the scientific ideas and results of academician B.V. Bokut which were obtained using covariant methods of F.I. Fedorov. Results of the study of monoclinic optically active absorbing crystals of class 2 are presented. Comparison with rhombic crystals of class 222 is made. Peculiarities of polarization of eigenwaves near optical axes in crystals such as transparent optically active, absorbing, optically active and absorbing are consecutively considered. Dependencies of the azimuth of polarization of transmitted light with the incidence angle equal to zero are presented for these cases. Two cases for monoclinic crystals are considered. The first case is when there is absorption and symmetric axis number 2 lies in the same plane as optical axes and the second case is when axis number 2 is perpendicular to the optical axes. The main peculiarities are noticed when axis number 2 is perpendicular to the optical axes.

Keywords: covariant methods, transparent and absorbing crystals, biaxial crystals, monoclinic crystals of class 2, gyrotropy, azimuth of polarization, gyration tensor, eigen wave.

A.F. Konstantinova – A.V. Shubnikov Institute of Crystallography, Russian Academy of Sciences, Moscow, Russia

T.G. Golovina – A.V. Shubnikov Institute of Crystallography, Russian Academy of Sciences, Moscow, Russia

E.A. Evdischenko – A.V. Shubnikov Institute of Crystallography, Russian Academy of Sciences, Moscow, Russia

B.V. Nabatov – A.V. Shubnikov Institute of Crystallography, Russian Academy of Sciences, Moscow, Russia

K.K. Konstantinov – A.V. Shubnikov Institute of Crystallography, Russian Academy of Sciences, Moscow, Russia

Hydrogenated micromorph silicon layers (a-Si:H/µc-Si:H) were prepared on ZnO:B transparent conductive oxide (TCO) films with dif-ferent surface morphology (roughness and optical haze value). The required structure of zinc oxide films was obtained varying their thickness. This study reveals the dependence of structural and optical properties of Si layers on the ZnO topography. The minor influence of the substrate surface morphology on the silicon Raman crystallinity factor was observed. It was ~5% decreased, when the TCO haze value was increased from 2 to >40%, while the light trapping properties were significantly increased (the absorption coefficient of ZnO:B/a-Si:H/µc-Si:H was grown from 50 to 70%). The optimum combination of structural and optical properties was found at >20% TCO haze level.

Keywords: transparent conductive oxide (TCO), light trapping, Raman crystallinity factor, hydrogenated micromorph silicon (a-Si:H/µc-Si:H), optical haze.

F.V. Kurdzesau – Oerlikon Solar Ltd., Trubbach, Switzerland

We use extremely intense, ultrashort soft X-Ray pulses generated by the LCLS X-Ray Free Electron Laser to investigate the production of molecular double core-hole states by sequential two-photon X-Ray absorption. The effect of critical LCLS parameters such as the number of photons per pulse, the pulse duration, and the focal spot size on the photoelectron and Auger spectra is modeled in detail and the results of these simulations are used as an aid in the interpretation of the experimental spectra obtained. We review here the results from the two-photon experiments at LCLS.

Keywords: free electron laser, X-ray laser, polyatomic molecule, double core-hole state.

M. Larsson – Stockholm University, Physics Department, 106 91 Stockholm, Sweden

P. Salen – Stockholm University, Physics Department, 106 91 Stockholm, Sweden

P. van der Meulen – Stockholm University, Physics Department, 106 91 Stockholm, Sweden

R.D. Thomas – Stockholm University, Physics Department, 106 91 Stockholm, Sweden

H.T. Schmidt – Stockholm University, Physics Department, 106 91 Stockholm, Sweden

R. Feifel – Uppsala University, Department of Physics and Astronomy, 751 20 Uppsala, Sweden

M.N. Piancastelli – Uppsala University, Department of Physics and Astronomy, 751 20 Uppsala, Sweden

L. Fang – Western Michigan University, Physics Department, Kalamazoo, MI 49008, USA

T. Osipov – Western Michigan University, Physics Department, Kalamazoo, MI 49008, USA

B. Murphy – Western Michigan University, Physics Department, Kalamazoo, MI 49008, USA

P. Juranic – Western Michigan University, Physics Department, Kalamazoo, MI 49008, USA

N. Berrah – Western Michigan University, Physics Department, Kalamazoo, MI 49008, USA

E. Kukk – University of Turku, Department of Physics and Astronomy, 20014 Turku, Finland

K. Ueda – IMRAM, Tohoku University, Sendai 980-8577, Japan

R. Richter – Sincrotrone Trieste, 34149 Basovizza, Trieste, Italy

K.C. Prince – Sincrotrone Trieste, 34149 Basovizza, Trieste, Italy

J.D. Bozek – SLAC, LCLS, Menlo Park, CA 94025, USA

C. Bostedt – SLAC, LCLS, Menlo Park, CA 94025, USA

S. Wada – SLAC, LCLS, Menlo Park, CA 94025, USA

M. Tashiro – Institute for Molecular Science, Okazaki 444-8585, Japan

M. Ehara – Institute for Molecular Science, Okazaki 444-8585, Japan

F. Tarantelli –Chemistry Department, University of Perugia and ISTM-CNR, 06123 Perugia, Italy

Means and methods of amplitude-polarization modulation of plane-polarized light are considered. These methods can be realized by means of the splitting plane-polarized light beam on two beams, their phase modulation and subsequent summation.

Keywords: electrooptical effect, amplitude modulation, phase modulation, polarization splitter, birefringence, isotropic angle.

V.A. Pilipovich – B.I. Stepanov Institute of Physics NAS of Belarus, Minsk, Belarus

A.I. Konojko – B.I. Stepanov Institute of Physics NAS of Belarus, Minsk, Belarus

A.M. Polikanin – B.I. Stepanov Institute of Physics NAS of Belarus, Minsk, Belarus

Use of pulses of a current of high density, electric and magnetic fields, ionic implantation allowed intensifying plastic deformation of metals, thus, giving basic opportunity for management of plastic deformation twinning with the help of forces of the nonmechanical nature, influencing on conditions and character of hardening of a material by means of controllable twinning.

Keywords: twinning, twinning dispositions, metals, electroplastic effect, deformation, disclosing, indentor.

V.S. Savenko – I.P. Shamyakin Mozyr State Pedagogical University, Mozyr, Belarus

The analytical study of refractive and absorbing properties of a sample of the artificial medium consisting of the optimal helices with equal permittivity and permeability has been carried out. Significant advantage of this medium in comparison with «pure» dielectric and magnetic materials was shown on the basis of the obtained results.

Keywords: composite materials, chirality, helical element, permittivity, permeability.

I.V. Semchenko – Prorector of F. Skorina Gomel State University, Gomel, Belarus

S.A. Khakhomov – Prorector of F. Skorina Gomel State University, Gomel, Belarus

A.L. Samofalov – Department of Physics, F. Skorina Gomel State University, Gomel, Belarus

S.A. Tretyakov – Aalto University School of Electrical Engineering, Finland

We designed and made a compact laser radiation converter on the basis of a biaxial crystal oriented along its binormal. The deviñe is de-signed for apodization and decreasing laser beam divergence, as well as for shaping hollow light beams with a wave front dislocation. Optical elements of the converter allow to transform both pulsed, and continuous powerful laser radiation. It is possible to transfer converted beams at large distance in transparent media. The device holds much promise for manipulation by particles on the principle of a laser tweezer, for laser processing of materials, for influence on organic cells and its components, and in interferometry and profilometry.

Keywords: laser beam, laser radiation converter, crystal, apodization, multiring light beam.

N.A. Khilo – B.I. Stepanov Institute of Physics NAS of Belarus, Minsk, Belarus

N.S. Kazak – B.I. Stepanov Institute of Physics NAS of Belarus, Minsk, Belarus

A.A. Ryzhevich – B.I. Stepanov Institute of Physics NAS of Belarus, Minsk, Belarus

S.V. Solonevich – B.I. Stepanov Institute of Physics NAS of Belarus, Minsk, Belarus

Necessary and sufficient conditions for nonautonomous differential system with cubic right-hand side to be equivalent to embedable system in sense of Mironenko were obtained.

Keywords: reflecting function, embedable system, differential system, two-point boundary-value problem.

M.S. Belokursky – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

In this paper the method of constructing of Abel differential equations possessing the same Mironenko reflecting function is studied. For another equation with the same reflecting function to be constructed, the necessary and sufficient conditions for given Abel equation are achieved.

Keywords: Abel equation, reflecting function, equivalence of differential equations, polynomial perturbations.

V.A. Belsky – Gomel Engineering Institute of the Ministry for Emergency Situations of the Republic of Belarus, Gomel, Belarus

V.I. Mironenko – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

Let be a non-empty formation. A subgroup H of group G is called -subnormal in G if either H=G or there is a chain of subgroups such that for every i = 1, … , n. In the work the class of groups and every Sylow subgroup of G is -subnormal in G) are studied. Properties of the class are obtained. In particular, for hereditary saturated formation it is proved that the class is a hereditary saturated formation. Necessary and sufficient conditions are found, at which

Keywords: finite group, Sylow subgroup, -subnormal subgroup, hereditary formation, saturated formation.

A.F. Vasilyev – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

T.I. Vasilyeva – Belarusion State University of Transport, Gomel, Belarus

The properties of -compatible vector-matrices are studied in this paper.

Keywords: matrice, vector-matrice, semigroup, group, ring, n-ary group.

Gal’mak A.M. – Department of Economic, Mogilev State University of Food Technologies, Mogilev, Belarus

Let A be an RG-module, where R is a commutative ring with the unit, is not an artinian R-module, and G is a locally finite group. Let be a system of all subgroups such that quotient modules are not artinian R-modules. The author studies RG-module A, such that satisfies either weak minimal condition or weak maximal condition as an ordered set. The properties of the locally finite group G with these conditions are described. Some properties of a locally soluble periodic group G under consideration are obtained if R is a dedekind ring.

Keywords: an artinian R-module, a group ring, a locally finite group.

O.Yu. Dashkova – O. Honchar Dnepropetrovsk National University, Dnepropetrovsk, Ukraine

The upper estimations of the derivative length of the Frattini quotient group depending on the indices of the maximal subgroups which do not contain a Fitting subgroup in the solvable group have been obtained.

Keywords: finite group, solvable group, maximal subgroup, Fitting subgroup, Frattini subgroup, derivative length.

D.A. Hodanovich – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

The description of the computer system PALS performing the method of the probability-algebraic simulation of complex systems is presented. It allows estimating in dynamics the probability characteristics of the investigated property of complex systems with many states. The assessment of probability properties of structurally-simple and structurally-complex systems is shown in the examples.

Keywords: probability-algebraic simulation, reliability of complex system, structurally-simple system, structurally-complex system.

E.I. Sukach – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus

D.V. Ratobylskaya – Department of Mathematics, F. Skorina Gomel State University, Gomel, Belarus