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Abstract. The mechanism of electrochemical deposition of silver-tungsten thin films from the ammonium sulfate solution has been studied. The dependencies of the coating formation parameters (exchange current, transfer coefficient and nucleation parameters) on a sodium tungstate concentration in the electrolyte and deposition conditions have been defined. It was shown that the introduction of tungsten reduces the crystallization overvoltage and critical nucleus radius as well as reduces the average crystal grain size in the coating with the increase in the rate of nucleation. Coating structure has been researched, the elemental analysis has been done and grain size distribution density in a silver-tungsten coating has been calculated using energy dispersive X-ray spectroscopy.

Keywords: silver, tungsten, composite electrochemical coatings, nucleation, crystallization.

For citation: Kinetics of electrochemical formation silver-tungsten coatings from the ammonium sulfate electrolyte / N.V. Bogush, I.I. Kyzmar, L.K. Kushner, A.A. Khmyl // Problems of Physics, Mathematics and Technics. – 2023. – N1 (54). – P. 7–12. – DOI: https://doi.org/10.54341/20778708_2023_1_54_7 (in Russian). – EDN: BLQMAA

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Abstract. A model is proposed and the calculation of a pulsed ionizer of dense gas fluxes is made, based on the absorption of intense, characteristic, X-ray radiation from a copper anode of the transmission type, covered with a layer of tungsten or molybdenum. The pulse power of the ionizer is from 250 to 1000 kW, which makes it possible to ionize to a high degree of ionization close to ~100% the dense gas flow of atoms and molecules in the combustion chamber of a small-sized, liquid, jet engine or for other applications in science and technology.

Keywords: extrapolated mileage, fluorescence of primary X-rays, photoluminescence of secondary X-rays.

For citation: Voronin, S.T. Ionization of hot gas flows of high density by X-ray radiation during photoluminescence of the transmission-type anode / S.T. Voronin // Problems of Physics, Mathematics and Technics. – 2023. – N1 (54). – P. 13-19. – DOI: https://doi.org/10.54341/20778708_2023_1_54_13 (in Russian). – EDN: EIPASY

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Abstract. The analytical expressions in the closed form for vector circular 3D light Kummer beams with nonhomogeneous polarization (TM-modes) are offered and investigated. At certain restrictions on free parameters such Kummer beams transfer terminat-ing power. The pictures of intensity and crossflows of energy of such beams are calculated and are graphically investigated.

Keywords: paraxial beams, vector circular beams, Kummer beams, nonhomogeneous polarization, TM-modes.

For citation: Girgel, S.S. Energy properties of the vector circular Kummer beams with terminating power. II. The nonhomogeneous polarization / S.S. Girgel // Problems of Physics, Mathematics and Technics. – 2023. – N1 (54). – P. 20-24. – DOI: https://doi.org/10.54341/20778708_2023_1_54_20 (in Russian). – EDN: DVBYBL

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Abstract. Partial Green's functions for d-states are defined in the relativistic configurational representation and expressed in terms of elementary functions. For the Green's functions obtained the asymptotics are found for large values of the coordinate, and their nonrelativistic limit is determined. Four quasipotential partial equations in the relativistic configuration representation for scattering states are solved exactly in cases of “delta-sphere potential” and “superposition of two delta-sphere potentials”. The partial amplitudes and the scattering cross sections are determined.

Keywords: quasipotential approach, relativistic configurational representation, Green’s functions, scattering state, d-state, delta function potential.

For citation: Kapshai, V.N. Solution of relativistic partial equations for scattering d-states / V.N. Kapshai, A.A. Grishechkina // Problems of Physics, Mathematics and Technics. – 2023. – N1 (54). – P. 25-30. – DOI: https://doi.org/ 10.54341/20778708_2023_1_54_25 (in Russian). – EDN: GOLFMT

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Abstract. A method for determining the contact zone, stresses and displacements during the interaction of gear teeth made of composite and functionally gradient materials (FGM), based on the theory of anisotropic elasticity theory is presented. The graphs of the depth dependence of stresses on the contact surface for various materials are presented, and a good correspondence is shown for the limiting case – the isotropy of the material.

Keywords: gears, contact tension, contact displacement, contact zone, functional gradient material, composite.

For citation: Mozharovsky, V.V. Calculation of the stress-strain state of gear teeth made of composite and functional gradient materials / V.V. Mozharovsky, S.V. Kirhintsava // Problems of Physics, Mathematics and Technics. – 2023. – N1 (54). – P. 31-37. – DOI: https://doi.org/10.54341/20778708_2023_1_54_31 (in Russian). – EDN: GTXFSB

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Abstract. The results of the formation of Cu-Sn-Ni precursors by electrochemical deposition on glass substrates with a molybdenum sublayer are presented. The structural and morphological properties of precursors have been studied. The X-ray diffraction patterns showed characteristics of CuSn hexagonal phase, CuNi cubic phase, Ni₃Sn cubic phase, and Ni cubic phase. The peaks corresponding to the substrate material, molybdenum, were also found on the X-ray diffraction pattern of the precursors.

Keywords: thin films, Cu₂NiSn(S,Se)₄, surface morphology, X-ray phase analysis.

For citation: Osmolovskaya, T.N. Formation, structural and morphological properties of thin films of Cu-Sn-Ni precursors / T.N. Osmolovskaya, A.A. Feschenko, A.V. Stanchik // Problems of Physics, Mathematics and Technics. – 2023. – N1 (54). – P. 38-42. – DOI: https://doi.org/10.54341/20778708_2023_1_54_38 (in Russian). – EDN: GYQICK

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Abstract. A static and dynamic problem is solved for the boundaries of a martensitic layer in a mechanically unloaded ferromagnetic single crystal with a shape memory effect located in a magnetic field and fixed in a rigid embodiment. It is shown that to ensure static equilibrium, it is necessary to have compensatory at the interfaces that balance the action of the magnetic field.

Keywords: martensitic layer, ferromagnetic single crystal with shape memory, interface.

For citation: Ostrikov, V.O. A static and dynamic problem for a single martensite layer in a ferromagnetic single crystal with a shape memory effect in a magnetic field in a rigid embodiment / V.O. Ostrikov, O.M. Ostrikov // Problems of Physics, Mathematics and Technics. – 2023. – N1 (54). – P. 43-46. – DOI: https://doi.org/10.54341/20778708_2023_1_54_43 (in Russian). – EDN: MJNIAY

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Abstract. Carbon coatings doped with nitrogen and aluminum were deposited from combined flows of pulsed carbon plasma and aluminum ions generated by a DC arc discharge at a nitrogen partial pressure of 6×10^-2 Pa. The regularities of the influence of conditions and modes of deposition on the phase composition of the carbon matrix and the mechanical properties of the coatings are determined. In the deposition mode (6 Hz, 60 A), the coatings are characterized by the smallest size of Csp² clusters and their high concentration, which determined their maximum microhardness, high plastic and tribological properties.

Keywords: carbon coating, aluminum, nitrogen, structure, mechanical properties, alloying, vacuum pulsed cathode-arc evaporation method.

For citation: Structure and mechanical properties of carbon coatings alloyed with nitrogen and aluminum / D.G. Piliptsou, Bing Zhou, A.V. Rogachev, N.N. Fedosenko, E.A. Kulesh // Problems of Physics, Mathematics and Technics. – 2023. – N1 (54). – P. 47-53. – DOI: https://doi.org/10.54341/20778708_2023_1_54_47 (in Russian). – EDN: OCMFGB

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Abstract. The features of the influence of the modes of formation and heat treatment of coatings based on starch and copper acetate on their morphology and molecular composition are determined. The influence of the intensity of laser radiation and heat treatment on the distribution of individual structural formations has been established. It is shown that heat treatment of coatings based on starch and cop-per acetate promotes an increase in the degree of dispersion of individual structural formations.

Keywords: starch, acetate, copper, laser, morphology, structure.

For citation: Rudenkov, A.S. Effect of heat treatment on the morphology and structure of nanocomposite coatings based on starch and copper acetate / A.S. Rudenkov, M.A. Yarmolenko, S.A. Frolov // Problems of Physics, Mathematics and Technics. – 2023. – N1 (54). – P. 54-59. – DOI: https://doi.org/10.54341/20778708_2023_1_54_54 (in Russian). – EDN: OMDUPR

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Abstract. At present, photocatalytic water treatment technology has important potential application value in the field of green energy. It is very attractive to prepare efficient photocatalysts by simple and green methods. In this study, a simple, low temperature, low cost and green low temperature water bath stirring method was designed to synthesize the well-dispersed Zn²⁺-doped CuS hollow sphere photocatalyst in one step. The morphology, structure and composition of the obtained products were ex-amined by field emission scanning electron microscopy, transmission electron microscopy and X-ray diffraction. Zn²⁺ doping forms carrier traps, which improve the transfer rate of electrons to the surface of the catalyst, significantly enhance its visible light response, and reduce the recombination rate of electron holes. Combined with the special interconnection channel and large specific surface area of the hollow sphere structure, it provides more surface active sites for adsorption and makes photocatalysis more efficient. The reaction rate of the prepared hollow spherical Zn²⁺-doped CuS catalyst reached 0,04736 min^-1, which was 12 times that of pure CuS.

Keywords: photocatalysis, CuS, hollow sphere, visible light, Zn²⁺ doped.

For citation: Preparation of Zn²⁺-doped CuS hollow spheres by one-step synthesis and its photocatalytic performance in visible light / Gaopeng Yang, Tiantian Wu, Jinxing Cao, Xiaohong Jiang // Problems of Physics, Mathematics and Technics. – 2023. – N1 (54). – P. 60-68. – DOI: https://doi.org/10.54341/20778708_2023_1_54_60. – EDN: PYOQLH

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Abstract. A subgroup H of a group G is called P_π-subnormal in G if either H=G or from H to G there exists a chain of subgroups, whose every index is either a prime in π or a π'-number (π is some set of primes). For a finite π-closed group with given P_π-subnormal subgroups, the necessary and sufficient conditions of π-supersolvability are obtained.

Keywords: π-soluble group, π-supersoluble group, P_π-subnormal subgroup, normalizers of Sylow subgroups.

For citation: Vasilyeva, T.I. On π-supersolvability of finite groups / T.I. Vasilyeva, A.G. Koranchuk // Problems of Physics, Mathematics and Technics. – 2023. – N1 (54). – P. 69-74. – DOI: https://doi.org/10.54341/20778708_2023_1_54_69 (in Russian). – EDN: QVPXWQ

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Abstract. Let σ={σ_i:i∈I} be some partition of the set of all primes P, i. e. P=∪_i∈I and σ_i∩σ_j=∅ for all i≠j. Finite group G is σ-soluble, if every chief factor H/K of G is a σ_i-group for some σ_i∈σ. Fitting class is said to be σ-class Hartley. In this paper we prove the existence and conjugacy of -injectors of G and describe their characterization in the terminal of the radicals.

Keywords: σ-soluble group, σ-class Hartley, injector.

For citation: Vorob’ev, N.T. Injectors of finite σ-soluble groups / N.T. Vorob'ev, E.D. Volkova // Problems of Physics, Mathematics and Technics. – 2023. – N1 (54). – P. 75-84. – DOI: https://doi.org/10.54341/20778708_2023_1_54_75 (in Russian). – EDN: RSKBWR

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Abstract. The study on the identities and their generalizations in polyadic groupoids of special form is carried on.

Keywords: polyadic operation, n-ary groupoid, identity, substitution.

For citation: Gal'mak, A.M. On identities and their generalizations in polyadic groupoids of special form. II / A.M. Gal'mak // Problems of Physics, Mathematics and Technics. – 2023. – N1 (54). – P. 85-88. – DOI: https://doi.org/10.54341/ 20778708_2023_1_54_85 (in Russian). – EDN: TFEBFR

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Abstract. The sufficient conditions are found under which rational Hermite – Jacobi approximations exist for a system of functions. It is shown that under these conditions, the Hermite – Jacobi approximations coincide with the corresponding Hermite – Padå approximations. The main result in the particular case when the system consists of one function is well-known Jacobi's theorem.

Keywords: Hadamard determinants, perfect systems of functions, Hermite – Padå approximants, Hermite – Jacobi approximants.

For citation: Osnach, T.M. The analogue of Jacobi’s theorem for simultaneous Hermitian interpolation of several functions / T.M. Osnach, N.V. Ryabchenko, A.P. Starovoitov // Problems of Physics, Mathematics and Technics. – 2023. – N1 (54). – P. 89-92. – DOI: https://doi.org/10.54341/20778708_2023_1_54_89 (in Russian). – EDN: ZVWNLP

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