Shymanovskyi A. Symmetry, consolidation, modulated structures of order parameter

Shymanovskyi, A. R. Symmetry, consolidation, and modulated structures of the order parameter. – Qualifying research paper in the form of a manuscript. Thesis for the degree of Doctor of Philosophy in the speciality 105 – Applied Physics and Nanomaterials (Field of Knowledge 10 – Natural Sciences). – V. N. Karazin Kharkiv National University of the Ministry of Education and Science of Ukraine, Kharkiv, 2026. This thesis is devoted to a comprehensive theoretical and phenomenological analysis of the properties of spatially modulated structures of various physical natures in solids (in particular, ferroelectrics and metals). The main focus is on the evolution and stability of these objects in the vicinity of phase transition points. The introduction justifies the relevance of the topic and defines the aim, object, subject and methods of the research. The scientific novelty and practical significance of the results are formulated, and details regarding the validation and structure of the work are provided. Chapter 1 contains a review of the literature on the physics of spatially inhomogeneous states in solids. Systems with a single-component order parameter, the nature of incommensurate phases in type I and II ferroelectrics, and modern generalisations of Landau’s theory are considered. Particular attention is paid to the effect of intense radiation on phase transitions in molybdenum. Сhapter 2 describes the procedure for irradiating molybdenum targets using the TEMP-A accelerator and details the comprehensive methods used to study their structure and properties (SEM, XRD, and microhardness). A mathematical model of thermal processes in cylindrical coordinates was developed and implemented in the software using Python. The model is based on the numerical solution of the nonlinear heat equation, accounting for phase transitions and temperature-dependent material properties. Chapter 3 is devoted to the development of a phenomenological model of phase transitions involving the emergence of modulated structures. The physical meaning of the gradient terms in the thermodynamic potential is analysed. Scenarios for the emergence of Bell-soliton and kink-soliton structures are investigated, conditions for their thermodynamic stability are determined, and corresponding phase diagrams are constructed. The peculiarities of the relaxation of the order parameter in organic media are examined. In Chapter 4, exact and approximate analytical solutions are obtained for the spatial distributions of the order parameter, expressed in terms of elliptic Jacobian functions (sine, cosine, delta-amplitude). Conditions for the existence of phases are also derived, as well as conditions for the complete evolution of phases from ordered to non-commensurate to commensurate. Chapter 5 is devoted to the analysis of phase transformations and symmetry-breaking effects in molybdenum targets under the action of concentrated ionising radiation fluxes. Results of modelling structural changes in the material under extreme conditions are presented. The scientific novelty of the work lies in the following: 1. The phenomenological description of phase transitions for systems with modulated structures has been refined through a detailed analysis of gradient contributions. 2. The conditions for the realisation and stability of new types of soliton solutions (bell- and kink-type) have been identified, depending on the parameters of the thermodynamic potential. 3. Group analysis (methods of Lie theory) has been applied to variational equations describing incommensurate states, which has enabled the discovery of various types of solutions. 4. A family of exact analytical solutions for the spatial distribution of the order parameter in the form of elliptic Jacobi functions has been obtained. 5. Patterns of symmetry breaking and phase instability in molybdenum under the influence of intense irradiation have been established. The results are included in the implementation programme under the departmental theme of the Institute of Electrophysics and Radiation Technologies of the National Academy of Sciences of Ukraine: ‘Nanoscale structures in metals and alloys as a factor influencing radiation resistance, electrophysical and mechanical properties’.