Physics and Educational Technology

SOME METHODS OF COUNTERING THE EFFECTS OF UNMANNED AIRCRAFT SYSTEMS AGAINST ELECTRONIC WARFARE DEVICES

Oleh VILIHURSKYI — 2025-12-30

One of the serious problems that has had to be solved constantly since the beginning of the modern Russian-Ukrainian war (since 2014) is the counteraction to electronic warfare (EW) means. Since there is a constant improvement of both aircraft (manned and unmanned) and means of combating them, this problem will not lose its relevance in the coming years. The main methods of counteraction: cryptographic signal protection, frequency aggregation, the use of physical barriers, autonomous flight modes (with the closure of all radio channels of communication with the operator), the creation of unique software for UAVs (Shkitov, 2024). This paper will consider one of the options for countering EW means, which aims to reduce the vulnerability to these means not of a single unmanned aerial vehicle (UAV), but of a properly configured system of many UAVs, which can be considered as a “swarm of drones” or “cloud of drones”. A system is proposed where individual combat UAVs are controlled by a central drone using radio signals or optical communication. Only the central drone has communication with the operator (via electromagnetic radiation in the radio range or via optical fiber). Within the “swarm”, the typical distances for signal transmission are small (of the order of meters), so signal transmission between aircraft is less vulnerable than in the case of direct communication with the operator. A variant of signal exchange is also possible when the aircraft are combined into a network; in this case, remote elements of the “swarm” may not be directly connected to the central device. That is, something like a two-dimensional crystalline structure with the interaction of the nearest (or close) neighbors may take place. A three-dimensional structure seems less appropriate from the point of view of the effectiveness of using means of impressing individual UAVs, but may be appropriate if the satellite drones are also assigned the function of protecting the central drone. Methods of protecting information exchange channels between the central drone and the operator are not considered in the work.

THERMODYNAMIC ORDERLINESS AS ONE OF THE ESSENCES OF THE INTERMOLECULAR STRUCTURE OF WATER

Vyacheslav VOLOSHYN — 2025-12-30

The paper considers possible estimates of the structure of water from the standpoint of its entropy indicators, understanding their growth as the state of disorder of the system and its disorganization. The possible ratios of individual water molecules and their structural accumulations in the form of organized clusters are taken into account. The main message consists in the fact that an always structured and ordered system has a lower entropy than an unordered and less organized system. dependence on its structuring and conditional ratio of individual molecules and their clusters. The technique is based on the combinatorial method of “gluing” n molecules into indistinguishable clusters and detecting mixing entropy taking into account dynamic factorization (time factor), as well as fine dynamics of bonds between molecules within the cluster. H2O 1 ⋅ 10-12 It is capable of reducing the etropy of the liquid by 14.7 % to its standard value. And in very short intervals of the cluster life cycle (10-13–10-14 s), the effect of thermodynamic ordering practically disappears. For arguments regarding such calculated clusters in the composition of water, publications with experimental data on the structure of water obtained by methods of quantum-chemical analysis of possible configurations of hydrogen networks in the cluster are referred to (H2O)20, including in the interface mode of the orientation of water molecules at the boundary of the interfacial field and the corresponding orientation entropy, the decrease of which is confirmed only in the nearsurface ordered layer of water. It is shown that structured water at the boundary with biological systems represents a more thermodynamically ordered medium, for which there are short-lived molecular clusters that are unconditionally stable near biomolecules and affect biochemistry, electrical processes and mechanics of organic interactions. Such and other indirect dependencies, despite their locality, should claim criterion abilities in favor of argumentation for cluster structuring of water systems.

AUTHORED PROGRAM “INFORMATION AND COMMUNICATION TECHNOLOGIES IN MILITARY AFFAIRS: UNMANNED AIRCRAFT (UAV) AND ELECTRONIC WARFARE (EW) EQUIPMENT” IN THE CONTEXT OF INITIAL MILITARY TRAINING

Vasyl VELYCHKO — 2025-12-30

In the conditions of a full-scale war ongoing in Ukraine, military-patriotic education is becoming an integral part of the educational process, acquiring strategic importance for the formation of national consciousness and defense capabilities of the younger generation. Responding to today’s demands, the Ministry of Education and Science of Ukraine (MES) is forming documents that define the goal, objectives and principles of national-patriotic education, clearly outlining the role of its military-patriotic component. The authors analyzed the Law of Ukraine “On Education”, the State Standards of Complete Secondary Education, orders of the Ministry of Education and Science of Ukraine (MES) concerning issues of national-patriotic education of children and youth. The regulatory and legal regulation of the creation of author’s programs and the methodology for their creation were considered. Methodological recommendations of the MES encourage teachers to integrate elements of military-patriotic education into all educational components, as well as to expand the variable component of educational subjects with an emphasis on the defense of the Fatherland. In this context, computer science appears as the most promising subject for integrating the study of modern military information technologies. According to the computer science program for grades 10–11, the basic module can be expanded by selective modules. The authors propose to focus attention on the role of modern technologies in military affairs, in particular, on the use of unmanned aerial vehicles (UAVs) and electronic warfare (EW) means, as well as on the role of artificial intelligence in their development. Given the key role of these devices in modern combat operations, the scope of the basic computer science module is not enough for an in-depth study of these topics. The purpose of the study is to develop a program for an elective module in computer science: “Information and communication technologies in the military sphere: unmanned aerial vehicles (UAVs) and electronic warfare (EW)”. The proposed author’s variable component of the computer science curriculum for grade 11 for 35 hours is a response to the current needs of today, allowing students to be more familiar with the principles of operation, characteristics and application of UAVs and EW. Its implementation will contribute not only to military-patriotic education, but also to the formation of practical skills necessary for future defenders of Ukraine. This program, after passing the examination in accordance with the established procedure, can be integrated into the educational process, significantly expanding the capabilities of the basic computer science module.

PRACTICAL APPLICATION OF ASTROPHYSICAL SCHOOL MATERIALS FOR TEACHING PHYSICS AND ASTRONOMY AT NUSH

Yurii DIATLOV — 2025-12-30

The article substantiates the possibilities of effective use of materials from international astrophysical schools for teaching physics and astronomy in the context of the New Ukrainian School (NUS). The authors emphasize that the modern educational paradigm involves a transition from reproductive learning to the development of key competencies, in particular critical thinking and research skills. The use of relevant scientific materials covering modern achievements in cosmology, astrophysics, radio astronomy and space technologies allows enriching school courses, making them more practice-oriented and integrating interdisciplinary approaches. The study shows that the materials of the astrophysical school are a valuable didactic tool that helps to implement the NUS competency-based approach, involving students in the analysis of modern experimental data. This contributes to the development of information skills, forms a research culture and serves as the basis for project activities. In addition, familiarization with the career trajectories of scientists and examples of real tasks contributes to a conscious choice of professions in the STEM field, increasing the intrinsic motivation of students. The article also highlights an important psychological aspect: working with new data stimulates cognitive interest, lowering the barrier between school science and modern research. The authors emphasize that the integration of astrophysics school materials allows not only to deepen the content of the training, but also contributes to career guidance, popularization of science and the formation of a positive attitude towards learning. The results of the study show that this is an effective tool for training a new generation capable of creatively applying knowledge. In the future, the development of methodological recommendations for teachers, the creation of educational projects and further analysis of the psychological impact of such work on student motivation are proposed.

MODELING AND OPTIMAL CONTROL OF AN UPPER-LIMB BIONIC PROSTHESIS IN THE SAGITTAL PLANE

Оksana ZAMURUIEVA — 2025-12-30

The study investigates the modelling and optimal control of a bionic upper-limb prosthesis in the sagittal plane. The aim is to develop a mathematical model of the bionic upper limb, simulate its motion in the sagittal plane, and design a robust linear–quadratic regulator (LQR) to ensure a physiologically natural response of the elbow joint, taking into account angular position, velocity, and acceleration. To describe upper-limb motions, the Lagrangian formalism is employed, enabling derivation of the equations of motion for a two-link arm model. The system of equations is linearized in the vicinity of operating points. Methods of optimal control theory, in particular the synthesis of an LQR controller, are applied to obtain the optimal control law. Computations and simulations are carried out in MATLAB using the open biomechanical arm model from OpenSim. The work combines nonlinear dynamic modelling with a state-space approximation to develop a robust controller. The proposed approach demonstrates the feasibility of effectively reproducing physiological upper-limb movements using an LQR regulator. The obtained results show the system’s ability to maintain stability and motion accuracy even under disturbances and measurement errors, bringing it closer to biological control principles. The designed LQR controller provides stable, accurate, and energy-efficient control of the bionic upper-limb prosthesis in the sagittal plane. The proposed model can serve as a basis for prototyping high-technology bionic devices and for further studies aimed at extending the model to three-dimensional space and integrating neural or EMG signals for more natural control.

GAME-THEORETIC MODELING OF SPACE COMPETITION: A DYNAMIC APPROACH WITH MAPLE IMPLEMENTATION

Оksana ZAMURUIEVA — 2025-12-30

Game theory is a universal tool for analyzing strategic interaction – from economic competition to military-political decision-making and science-and-technology races. This article is devoted to the development of a mathematical model of interstate competition in space exploration based on non-cooperative, non-zero-sum game theory and its numerical implementation in Maple. The object of the study is the competitive development of national space programs in a geopolitical context; the subject is a formal game-theoretic modeling of strategic rivalry under resource constraints, political priorities, and mutual influences. Theoretical foundations for applying game-theoretic approaches to space rivalry are synthesized; a system of differential equations is constructed for multiple players with logistic growth and mutual inhibition. The core analysis focuses on a three-player setting (the United States, China, and the European Space Agency), where internal growth is combined with bilateral “braking” effects. Drawing on historical and contemporary episodes of the “space race”, we build a model that specifies a system of equations describing players’ dynamics and the conditions for equilibrium states. The Maple implementation enabled numerical simulations and parameter-sensitivity assessment. The results confirm the suitability of game theory for forecasting states’ strategic behavior in the space domain and demonstrate the practical utility of the proposed approach for supporting strategic planning of national programs and shaping elements of international space policy.

PHYSICS AND RADIO ELECTRONICS: FROM FUNDAMENTAL LAWS TO MODERN TECHNOLOGIES

Andriy KEVSHYN — 2025-12-30

The article traces the close interconnection between physics as a fundamental science and radio electronics as its applied continuation. Physics, by studying the properties of matter, energy, space, and time, forms the theoretical foundation for understanding natural processes, while radio electronics applies this knowledge (particularly the principles of electrodynamics and electronics) to the creation, optimization, and practical implementation of electronic systems designed for the transmission, reception, and processing of information. Today, radio electronics encompasses almost all areas of human life – from household devices (smartphones, laptops, tablets) and wireless communication technologies (Wi-Fi, Bluetooth, mobile standards 3G–5G) to the Internet of Things (IoT), autonomous transport, medical diagnostic systems, and security technologies. Special attention is paid to its role in space technologies, particularly in the functioning of satellite navigation systems such as GPS and Galileo, as well as interplanetary communication systems. The paper provides a detailed analysis of the physical laws on which radio electronics is based. It emphasizes the fundamental importance of Ohm’s law, which serves as the foundation for calculating electrical circuits, and Kirchhoff’s laws, which describe the conservation of electric current at junctions and voltage in closed loops. These principles underpin the analysis and modeling of complex electronic circuits containing components such as transistors, resistors, diodes, and integrated microchips. A separate section is devoted to the historical development of radio electronics. The discovery of electromagnetic waves by James C. Maxwell and their experimental confirmation by Heinrich Hertz marked the starting point of wireless technology. Major contributions to the further advancement of the field were made by discoveries in quantum physics – the works of M. Planck, A. Einstein, and N. Bohr’s atomic model. Their findings led to an understanding of the band structure of solids and the creation of semiconductor devices – diodes, transistors, microchips, and processors – that formed the basis of modern digital electronics. The historical overview also covers the first wireless signal transmission experiments conducted by G. Marconi and A. Popov, as well as the development of vacuum tubes – J. Fleming’s diode and L. de Forest’s triode – which became the first active elements of electronics. A turning point came in 1947 with the invention of the transistor by J. Bardeen, W. Brattain, and W. Shockley, which launched the era of semiconductor technology and paved the way for microprocessor engineering and the information age. The modern stage of radio electronics is characterized by the miniaturization of components, the improvement of microprocessors, and the evolution of mobile communications (Wi-Fi, Bluetooth, 5G), which ensure high data transmission speeds and minimal signal delays. The paper also highlights the strategic role of radio electronic systems in navigation and satellite technologies used in civilian, scientific, and defense applications.

DEVELOPMENT OF STUDENTS’ EXPERIMENTAL COMPETENCIES THROUGH THE STUDY OF DAMPED OSCILLATIONS IN AN ELECTRIC CIRCUIT

Andriy KEVSHYN — 2025-12-30

The development of experimental skills occupies a central place in the modern education system, particularly in technical and natural science fields. When studying physical phenomena, especially oscillatory processes, the practical component is no less important than the theoretical one, as it enables a deeper understanding of the essence of the studied processes. The investigation of oscillations in an RLC circuit provides an opportunity to directly connect theory with practice. Observing damped oscillations, where the amplitude gradually decreases due to energy losses, allows students to clearly trace complex physical laws that can be studied under laboratory conditions. Performing such an experiment involves not only measurements but also data analysis, interpretation, and presentation of results, all of which contribute to the development of a wide range of professional skills. During the experiment, students acquire several key competencies: they learn to plan research (define objectives, select instruments and methodologies), gain hands-on experience working with measuring equipment such as oscilloscopes, multimeters, and power supplies. They also learn to organize and process data, plot graphs, determine oscillation parameters – period, frequency, and damping coefficient – compare experimental results with theoretical calculations, identify possible discrepancies, and draw generalized conclusions. An important aspect of the experiment is understanding how the parameters of the circuit (R, L, C) influence oscillatory behavior. Students experimentally verify the relationships between these quantities, which helps them better comprehend the underlying physical laws. For instance, changing the resistance R affects the damping rate of oscillations and allows observing the transition from oscillatory to aperiodic modes. Such work simultaneously fosters skills in evaluating measurement uncertainties and develops critical thinking in data analysis. The use of modern technologies – computer modeling and specialized software for data processing – significantly enhances the experimental experience. This approach enables not only physical measurements but also comparison with virtual models, deepening the understanding of the relationship between theory and practice. Thus, the study of damped oscillations in an electrical circuit serves as an effective means of comprehensive development of experimental competencies. It contributes not only to mastering the fundamentals of electrodynamics but also to forming analytical thinking, practical instrument-handling skills, and data-processing abilities. This approach ensures not mechanical memorization of facts, but a profound and conscious mastery of knowledge, forming the foundation of the professional competence of future specialists.

OPTICAL PROPERTIES OF Ag3AsS3 CRYSTAL DOPED WITH Ho

Maxym KYRYCHENKO — 2025-12-30

This paper presents a comprehensive study of the effect of rare-earth holmium (Ho) doping on the electronic structure, crystal lattice, and optical properties of the semiconductor proustite crystal (Ag3AsS3). The investigated samples with impurity concentrations of 0.3, 0.6, and 0.9 wt% were synthesized by melting high-purity components followed by homogenizing annealing. X-ray diffraction analysis confirmed the single-phase nature of the obtained materials and the correspondence of their structure to the R3c space group symmetry. Theoretical calculations of the electronic energy structure were performed within the framework of Density Functional Theory (DFT) using the CASTEP code. The Generalized Gradient Approximation (GGA) with Perdew-Burke-Ernzerhof (PBE) parameterization and Vanderbilt ultrasoft pseudopotentials were applied to describe the exchange-correlation interaction. Geometric optimization of the crystal lattice was carried out using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. Supercell calculations revealed that the incorporation of Ho atoms, substituting Ag sites, leads to the appearance of narrow bands of localized f-electrons near the top of the valence band formed by sulfur p-states. It was established that doping does not alter the indirect nature of the Ag3AsS3 matrix band gap. The spectral distribution of the absorption coefficient was experimentally investigated in the temperature range of 100–300 K. The optical band gap (Eg) was determined using the Tauc method for indirect transitions. It was found that increasing the Ho concentration from 0.3 to 0.9 wt% results in a monotonic decrease of Eg at room temperature from 2.09 eV to 2.06 eV. This narrowing is attributed to local lattice distortions arising from the difference in ionic radii between Ag+ and Ho3+. The temperature dependence of the absorption edge was analyzed within the Varshni model. The parameter related to the Debye temperature remained stable for all concentrations (~180 K), indicating the preservation of the vibrational properties of the host lattice. The obtained results demonstrate that holmium doping is an effective tool for the controlled modification of the optical parameters of Ag3AsS3 crystals for prospective applications in optoelectronics and nonlinear optics.

DETERMINATION OF THE MAGNETIC FIELD ENERGY OF A TOROIDAL COIL WITH A FERROMAGNETIC CORE

Igor LINCHEVSKYI — 2025-12-30

The article demonstrates a method for taking into account the nonlinear dependence of the relative magnetic permeability of a ferromagnet on the magnetic field strength when determining the field energy and inductance of a coil on a toroidal core. Unlike the simplified physical model of the coil, where the authors use the average value of the relative magnetic permeability 〈μ〉, it is shown that the calculation result is determined by the function ( ) , H H μ and is more correct than by the average value of the relative magnetic permeability. In the examples for calculating the coil parameters, reference dependences of the magnetic field induction on its strength for the ARMCO ferromagnetic material are used. The differences in the results obtained for the field energy and inductance of the coil according to our method and according to the classical method using the average value of the relative magnetic permeability significantly depend on the boundaries between the maximum and minimum values of the magnetic field strength in the core. It was found that the differences in the determination of the magnetic field energy in the core of the coil according to the proposed method and in comparison with the results according to the simplified method, where the magnetic permeability is considered unchanged, can reach 30 %., At the same time, the sign of such differences also depends on the current strength, or on the limits of change in the magnetic field strength in the cross-sectional area of the core. The differences in both the field energy values and the coil inductance increase to 34 % with an increase in the ratio of the outer and inner radii of the core, compared with the results where the magnetic permeability is considered unchanged. The use of software when determining the area of figures under function graphs helps to simplify individual calculations and increase the accuracy of the results obtained. The direction of using the method of determining the energy and inductance of a coil in computational and graphical work has been determined, for which the student will be able to use the magnetization curve obtained during laboratory work.

SYNTHESIS AND PROPERTIES OF Ag3SbS3, CRYSTALS DOPED WITH PRASEODYMIUM

Denis MYRONCHUK — 2025-12-30

The article presents the results of a study on the synthesis, structural, and optical properties of praseodymium (Pr)- doped pyrargyrite Ag3SbS3 crystals. This work belongs to the field of chalcogenide semiconductor materials, which, due to their unique electrophysical and optical characteristics, have found wide application in optoelectronics, sensor technology, thermoelectrics, and nonlinear optics. Ternary compounds, particularly Ag3SbS3, are known for combining chemical stability with high sensitivity to doping, making them suitable for targeted modification of electronic properties. Ag3SbS3 crystals were synthesized from high-purity elements-silver, antimony, and sulfur-using the Bridgman– Stockbarger method under a controlled temperature regime. Additionally, a series of samples doped with praseodymium at concentrations of 0.3, 0.6, and 0.9 at.% were obtained. The synthesis was carried out using precision vacuum equipment and prolonged homogenizing annealing to ensure material uniformity. X-ray phase analysis (DRON-4-13, CuKα radiation) confirmed the single-phase nature of all doped samples, the absence of secondary phases, and the structural stability of pyrargyrite, indicating the successful incorporation of Pr ions into the Ag3SbS3 crystal lattice. Doping with the rare-earth element led to significant changes in the optical properties of the compound. Absorption spectra measurements showed a consistent decrease in the optical band gap with increasing Pr concentration: from 1.84 eV for 0.3 at.% to 1.81 eV for 0.9 at.%. This trend indicates the formation of additional energy levels near the valence and conduction band edges, caused by the partially filled 4f orbitals of praseodymium. These states act as intermediate energy levels that reduce the interband transition energy and shift the fundamental absorption edge toward lower photon energies. Simultaneously, doping is accompanied by the formation of point defects, vacancies, and local lattice distortions, which further affect the material’s electronic structure. Thus, the combined effect of electronic and structural factors results in the narrowing of the band gap. The reduction of the band gap directly influences the nonlinear optical properties of the material. The study of second harmonic generation (SHG), conducted using the Kurtz–Perry powder technique, demonstrated an increase in SHG intensity with rising praseodymium concentration, i.e., with a decreasing Eg value. This observation agrees with theoretical predictions that link enhanced electronic polarizability to materials with smaller interband transition energies. In such cases, valence band electrons are more easily perturbed by the electromagnetic field of laser radiation, leading to an increase in the second-order nonlinear susceptibility (χ2) and enhancement of second harmonic generation. The obtained results prove that doping Ag3SbS3 with praseodymium is an effective way to control its optical parameters. The reduction of the band gap and the enhancement of SHG efficiency make these crystals promising candidates for use in nonlinear optical components, radiation modulators, infrared laser frequency conversion systems, and next-generation sensor devices. Given their structural stability, reproducible synthesis, and predictable electronic modifications under doping, the Ag3SbS3–Pr system can be considered a promising basis for developing new functional materials with tunable optoelectronic properties.

SOME FEATURES OF MODELLING DIFFUSIVE PROCESSES OF RELAXED OPTICS

Petro TROKHIMCHUCK — 2025-12-30

Some problems of modelling the diffusive processes in Relaxed Optics are discussed. Diffusion processes in Relaxed Optics can occur at different stages. We will focus on processes that are caused by the absorption of optical radiation with an intensity slightly greater than the value required for the generation of maximum photokinetic defect in the nearsurface region of the irradiated material, and less than the intensity required for the destruction of the irradiated material. The simulation was performed for antiomnide and indium arsenide irradiated with 20 nanosecond pulses of a ruby laser. The model was based on a photoionization model: the generation of donor centers occurs due to the direct photoionization of two of the three chemical bonds of the two-dimensional sphalerite lattice. The rupture of the third bond leads to the generation of diffusion processes. Two diffusion models are presented. The one-diffusion model is based on the behavior and evolution of laser-induced donor centers as separate objects. The disadvantage of this model is that it cannot explain the tails of the donor center distribution profiles in the diffusion approximation. In this regard, a two-diffusion model was proposed, which is based on the idea of non-uniform photostimulated diffusion of atoms of the components of the irradiated material: indium and antimony for indium antimonide and indium and arsenic for indium arsenide. This is confirmed by experimental results for laser-irradiated cadmium telluride. The two-diffusion model allows us to explain the diffusion profiles of the distribution of donor centers in indium antimonide and indium arsenide in a consistent manner. Based on this model, it is concluded that such modeling methods can be extended to all binary compounds and it is worth expanding these methods to more complex materials (ternary, quaternary, etc.).

INFLUENCE OF PRECURSOR CONCENTRATION ON STRUCTURAL AND MORPHOLOGICAL CHARACTERISTICS OF ELECTROCHEMICALLY SYNTHESIZED CDS NANOCRYSTALS

Yurii KHMARUK — 2025-12-30

The paper presents the results of a comprehensive study on the influence of sodium chloride concentration in the electrolyte on the structural, morphological, and nonlinear optical properties of cadmium sulfide (CdS) nanocrystals. A2B6 semiconductor compounds are promising materials for modern optoelectronics and photonics due to unique physicochemical properties arising from quantum confinement effects. The synthesis of nanostructures was carried out by electrolysis of an aqueous solution containing thiourea and sodium chloride, using a soluble cadmium anode. The process was conducted at a temperature of 363 K and a constant current density of 0.192 A/cm2. A key feature of the work is the analysis of the NaCl concentration influence in a wide range from 0.2 to 2.0 M on the formation of the precipitate crystal structure. Using scanning electron microscopy (SEM), it was established that the synthesized nanocrystals possess predominantly flake-like morphology. Size analysis revealed the polydispersity of the samples: the particle diameter varies within 5–70 nm, with an average size of 35–40 nm, and the flake thickness is 5–25 nm. An important experimental result is that the change in precursor salt concentration does not exert a significant influence on the average geometric parameters of nanoparticles, indicating the stability of nucleation mechanisms under the chosen conditions. X-ray diffraction (XRD) showed that all obtained samples are two-phase systems containing a mixture of metastable cubic phase (sphalerite) and thermodynamically stable hexagonal phase (wurtzite). A nonlinear dependence of the phase composition on NaCl concentration was revealed: the content of the acentric wurtzite phase varies within 74–90 %, reaching a maximum at a concentration of 1.4 M. Particular attention is paid to the study of the nonlinear optical properties of the synthesized nanopowders using the Second Harmonic Generation (SHG) method under laser radiation. High efficiency of second optical harmonic generation was experimentally confirmed. A clear correlation between the phase composition of nanocrystals and the magnitude of the nonlinear optical response was established. The sample with the maximum content of the noncentrosymmetric wurtzite phase (90 %) demonstrated the highest frequency conversion efficiency (output signal 1150 mV), whereas samples with a lower content of this phase showed lower results. The obtained data prove the possibility of targeted control of the functional properties of nanomaterials by optimizing the electrolyte composition, which is important for creating new effective media for nonlinear optics.

ASPECTS VIRTUAL PROBLEMS OF PHYSICS FOR STUDENTS OF HIGHER TECHNICAL EDUCATIONAL INSTITUTIONS

Myroslava CHERNOVA — 2025-12-30

The possibility of using the latest technologies in the study of physics is considered, in particular such a direction in the educational program as problem solving. The relevance lies in the ability to use virtual tasks during the absence of visual-tactile contact with the student due to military actions and the consequences associated with this. The main goal is to identify the most favorable opportunities and find means to provide higher education students with the opportunity for continuous learning in today’s extreme conditions. The research combines analysis the application and perception at virtual problems for physics by students. The effectiveness of the impact on the level of knowledge by comparing the results of sessional control over the last 5 years (this includes the time of the pandemic, martial law, and previously peacetime). A three-year study was conducted on higher education students at a higher technical educational institution, where the student begins solving a problem, but at a certain point is unable to complete this. The number of students analyzed is 150 in each of the 2 semesters. The method of uncertainty in face-to-face and distance learning, which arises independently and dictates the specifics of teaching work, is investigated. The importance of using a methodology based on the implementation of virtual the problems for physics in such uncertainty is emphasized. An important conclusion was made that the use of virtual the problems physics, as a supplement to similar virtual laboratory work in times of uncertainty, for full-time and distance learning is justified. This approach does not reduce the student’s averagestatistical score in physics. The proposed teaching technology is viable and necessary in times of spatial uncertainty. Such an innovative strategy will ensure the accessibility of education and will not cause a leakage of higher on the education students in Ukrain.

ELECTRICAL PROPERTIES OF Pb4Ga4GeSe12–Pb4Ga4GeS12 SOLID SOLUTION CRYSTALS

Oleg SHYGORIN — 2025-12-30

The paper presents the results of studies on the electrical and optical properties of Pb4Ga4GeS12–Pb4Ga4GeSе12 crystals. The Pb4Ga4GeSe12–Pb4Ga4GeS12 crystals corresponded to the compositional content of 10, 20, 30, 40 and 50 mol.% Pb4Ga4GeSе12. The semiconductor compounds Pb4Ga4GeSе12 and Pb4Ga4GeS12, as well as their solid solutions Pb4Ga4GeS12– Pb4Ga4GeSе12, crystallize in a tetragonal space group Р–421с and are promising materials for research in advanced areas of semiconductor materials science. These compounds combine the properties of several classes of semiconductors: classical semiconductors, thermoelectric materials, and nonlinear optical materials. Single crystals of Pb4Ga4GeS12 exhibit p-type conductivity, which is primarily associated with the presence of defects such as Pb, Ga, or Ge vacancies (VPb, VGa, VGe), or substitutional defects like PbGa and PbGe. In contrast, the solid solutions Pb4Ga4GeS12–Pb4Ga4GeSе12 containing 10–50 mol.% of Pb4Ga4GeSе12 demonstrate n-type conductivity. The main factors responsible for this conductivity type inversion are the decrease in band gap (Eg) and the increase in selenium vacancy (VSe) concentration with rising Pb4Ga4GeSе12 content. The non-monotonic dependence of the electrical resistivity of Pb4Ga4GeS12–Pb4Ga4GeSе12 on the Se concentration results from the dominance of different conduction mechanisms at various compositions. The initial increase in resistivity (from 0 to about 10 mol.% Pb4Ga4GeSе12) occurs because some sulfur atoms are replaced by selenium atoms, increasing the crystal lattice defect density and reducing carrier mobility. At concentrations around 20 mol.% Pb4Ga4GeSе12 and higher, the crystal lattice gradually transforms into a more thermodynamically stable structure similar to Pb4Ga4GeSе12, which leads to a decrease in resistivity. Moreover, the reduction of the band gap (Eg) further lowers the thermal activation energy of donor and acceptor centers, enhancing conductivity.

INVESTIGATION OF THE ANGULAR DISTRIBUTION OF LAGRANGE POINTS

Pavlo SHYGORIN — 2025-12-30

This work presents a theoretical investigation of the Lagrange points for the restricted three-body problem using the polar coordinate system. Employing the effective gravitational potential, relationships were derived to describe the angular and radial positions of the Lagrange points as functions of the characteristic parameters of the system. To obtain analytical expressions, perturbation theory was applied, with the small parameter being the mass ratio of the primary components of the system. The use of polar coordinates allowed the explicit description of the angular distribution of the equilibrium points. It was shown that the angular position of the Lagrange points L4 and L5, depending on the mass ratio of the two primary components, varies within the range from π/3 to π/2. The obtained result was confirmed with examples of the stellar systems Groombridge 34, HD 155358, and HD 69830, which exhibit different mass ratios in the range from 0,443 ⋅ 10-5 to 0,383. The relationship describing the angular distribution of the Lagrange points derived in this work was applied to the study of the exoplanetary system PDS 70. Based on the analysis of images of the PDS 70 system, the angular position of a gas–dust cloud located on the orbit of the exoplanet PDS 70b was determined. Additionally, the angular position of this cloud was calculated using the known mass of the central star in the PDS 70 system and the mass of the exoplanet PDS 70b. The results show that the dust cloud in which a new planet is forming is located at the L5 Lagrange point of the PDS 70 – PDS 70b system. This finding supports the hypothesis that another “Trojan” exoplanet is forming in the orbit of PDS 70b. It also allows us to assert that co-orbital configurations, known in the Solar System (e.g., Jupiter’s Trojan asteroids), are a universal phenomenon capable of arising in exoplanetary systems.