Two main concepts of infinity (Aristotle's and Cantor's) are known in the history of mathematics. The last one, prevailing at present, was formulated by founder of the set theory Cantor about a century and a half ago. Cantor used (1) the diagonal method to compare the powers of the set of infinite rows of digits 0 and 1 and natural number series; (2) the Cantor's theorem about prevalence of the power of the set of all subsets of a set A over the power of A: |P(A)|>|A|. In this work it is shown by use of specific examples that Cantor's reasons can't be considered as strict proofs. Therefore, the concept of the common potential (Aristotelian) infinity seems to be more acceptable.

This work presents the results of experimental and theoretical studies of the magnetic properties of the LiErF4 single crystal and powder samples at low temperatures and applied field range of 0-9 T. The magnetization was calculated in the framework of the exchange-charge model taking into account dipole-dipole and electron-deformation interactions, with the calculation of the electron-deformation parameters. The theoretical analysis presents uantitative agreement in the temperature range of 2-300 K with the magnetization measurements of the LiErF₄ samples.

Within the framework of the DFT approach, with a hybrid functional PBE0 that takes into account the contribution of nonlocal exchange in the Hartree-Fock formalism, the structure of impurity centers Gd₂Ti₂O₇:Eu³⁺, Gd₂Ti₂O₇ :Dy³⁺ and Tb₂Ti₂O₇:Eu³⁺ was calculated. It has been shown that there is practically no lattice distortion in these impurity centers.

The results of laser site selective and Zeeman spectroscopy studies of ZnWO₄ single crystal doped with Er³⁺ ions are reported. Three types of Er³⁺ sites have been discovered. The energies of the levels of the ⁴I_15/2 and ⁴I_13/2 multiplets and the g-factors of several states were determined for three orientations of the crystal relative to the magnetic field. The possible structure of three types of sites and further research techniques are discussed.

The spectroscopic study of dysprosium chain nickelate, Dy₂BaNiO₅, was performed. New information on the crystal-field (CF) levels of the ground ⁶H_15/2 multiplet of Dy³⁺ ion was obtained. The evaluation of the magnitude of the Dy³⁺ magnetic moment using spectroscopic data gives a value of 9μ_B. Spectroscopic study of Dy₂BaNiO₅:Er (1 at.%) has shown that the g_x component of the magnetic $g$ factor of dysprosium is zero. These findings are in agreement with predictions based on CF calculations performed by professor Boris Malkin[Phys. Rev. B 71, 024414 (2005)].

Using the stationary perturbation theory the equations for small polaron energies are obtained. This polaron has small spatial dimensions - only three equal atoms of a linear chain. The initial Hamiltonian has the contributions related to electron hoppings between the atoms, interaction of electron of the central atom with its oscillations and anharmonic contributions to the energy of its oscillations of the third and the fourth orders. The obtained analytical expressions give evidence that the polaron states with the fully filled (or significantly filled) atomic orbitals have the lowest energies that is in agreement with the results of numerical calculations available in the literature.

We demonstrate coherent electron and electron-nuclear spin manipulations using the impurity trivalent gadolinium ion and the nearby ⁷Li and ¹⁹F nuclei incorporated in the LiYF₄ host crystal. In particular, we present the electronic Rabi oscillations corresponding to -1/2 ⇔ 1/2 and 5/2 ⇔ 7/2 transitions between the projections of the Gd³⁺ spin S=7/2 of the lowest manifold ⁸S_7/2, together with the spin-lattice and spin-spin coherence times of these transitions. High-resolution pulsed electron-nuclear double resonance spectra involving -1/2 ⇔ 1/2 or 5/2 ⇔ 7/2 gadolinium transitions and either ⁷Li (nuclear spin I=3/2) or ¹⁹F (I=1/2) nuclear spin transition are obtained. The results suggest that the particular system can be potentially used for the implementation of hybrid quantum calculations utilizing both the electronic high-spin gadolinium states and the nuclear spin states of the adjacent ions.

Defects (color centers) in wide-gap semiconductors are considered as the basis for the realization of highly sensitive sensors, single-photon sources, and for the implementation of quantum technologies. Silicon carbide (SiC) crystal can serve as a reliable solid-state matrix for the range of high-spin (electron spin S = 1) color centers to become an alternative to the diamond with the widely-known nitrogen-vacancy (NV^-) centers. This paper reviews the electron paramagnetic resonance (EPR) and electron-nuclear double resonance (ENDOR) studies of the divacancies (VV) and negatively charged NV^- centers in different SiC polytypes. The main spin Hamiltonian components of non-equivalent spin defects in SiC are presented depending on their structural features (positions) and local environment: the zero-field splitting (D ≈ 1.3 GHz), hyperfine (A ≈ 1.1 MHz) and quadrupole (P ≈ 1.8 MHz) interaction values. The luminescence spectrum of the color center in SiC (λ = 1.1-1.25μm) in near-IR range is favorable for fiber-optic channels (O-band) and biological objects study, which brings these defects to a higher level of practical application.

To measure magnetostriction in LiTmF₄ and LiDyF₄ single crystals, the acoustic resonance method was used. It is shown that the combination of capacitive dilatometry and the acoustic resonance method makes it possible to measure not only the field dependence of the crystal dimensions but also the field dependence of the sound speed.

In this study, we demonstrate a detection of three phase transitions in a GdFe3 (BO3 )4 sin­gle crystal using the photoluminescence of Er³⁺ ions introduced in a small amount into the crystal. A first-order R32^P3₁21 structural phase transition is seen as a sharp spectral line shift at T_s = 127 K. Magnetic ordering of the iron subsystem at T_N = 37 K causes a splitting of Er³⁺ Kramers doublets and is detected by a splitting of spectral lines. A first-order spin­reorientation phase transition at TR = 9 K manifests itself in a sharp change of the line splitting pattern. A comparison of the measured ratio of luminescence line intensities with the ratio of the Boltzmann populations of the involved levels is proposed to control the possible heating of the GdFe3(BO3)4:Er³⁺ sample by exciting radiation.

The EPR spectroscopy and magnetization measurements were used to study the effect of annealing conditions on the local structure of Er³⁺ ions in CeO₂ : 1% Er³⁺ nanoparticles. The nanoparticles were synthesized using the coprecipitation technique from an aqueous solution of cerium nitrate and hexamethylenetetramine.

A correlation was found between the EPR spectra of the Er³⁺ ions, magnetization and luminescence, depending on the annealing atmosphere, which proved the crucial role of oxygen vacancies in the origin of magnetism in CeO₂ nanoparticles.

EPR lines due to trigonal centers were clearly detected in CeO₂ : 1% Er³⁺ nanoparticles annealed in a vacuum, while no such lines were found for similar nanoparticles annealed under argon or air atmospheres.

This paper discusses the features of the structure of splitting of energy sublevels of Kramers ion pairs connected by Coulomb interaction. The structure energy sublevels of non-Kramers ion pairs in the presence of a perturbation V is also discussed. Section 2 of this work represent the review of the theoretical part of our previous work [1]. The “Davydov splitting” in cluster composed of two Kramers rare-earth ions was theoretically analyzed for the first time.

In CsCdBr₃ crystals, the EPR spectra of Fe²⁺ ion replacing Cd²⁺ ion in a trigonally distorted octahedron were recorded and studied in the subterahertz frequency range. The zero-field splitting parameter and the value of the g-factor are determined. A comparative analysis of the obtained experimental results with literature data is made.

Analytical expressions for generating function and form function of absorption spectrum of a paramagnetic ion interacting with an oscillator at zero temperature are derived in the adiabatic and Condon approximations taking into account frequency effect: quadratic dependence of adiabatic potential on the vibrational coordinate is obtained by considering linear electron-vibrational interaction in the second order of perturbation theory. Approximations made throughout the paper and their limitations are discussed in detail.

Electron paramagnetic resonance is one of the most informative methods for studying the local structure and magnetic properties of impurity paramagnetic centers in crystals. This minireview presents the results of a study of paramagnetic centers formed by rare earth ions Ho³⁺, Tm³⁺, Er³⁺ and Yb³⁺ in single crystals of synthetic forsterite (Mg₂SiO₄). The structural features and magnetic properties of the studied centers are presented. It was found that a wellpronounced effect of dimer self-organization is observed for impurity trivalent ions in forsterite. As a result, concentration of dimer associates significantly exceeds the concentration expected with a statistically uniform distribution of impurity ions across the nodes of the crystal lattice.

The purpose of this short review article is to discuss at a simple qualitative level some key requirements the mixed-valence (MV) molecules should meet to be potentially applicable as cells of quantum cellular automata (QCA), and also how different interactions affect their fulfillment. We focus on two requirements, which are closely related to encoding and propagating of binary information within the electronic circuits and power dissipation caused by the logical operations. The physical features behind these requirements are the following: the ability of MV molecules to be efficiently switched between two logical binary states which assumes high polarizability manifesting itself in a strong non-linear cell-cell response and a low heat release caused by molecular rearrangements accompanying logical operations. We discuss the role of such electronic interactions as intramolecular electron transfer, intramolecular interelectronic Coulomb repulsion and the interaction of the excess electrons of a molecular cell with the electric field produced by the neighboring polarized cell. The pivotal role of the interaction of the excess electrons with the molecular vibrations (pseudo Jahn-Teller vibronic coupling) is discussed as well. Finally, the optimal conditions expressed as a parametric regime ensuring simultaneous fulfillment of the aforenamed requirements are discussed.

In the Y₂SiO₅ single crystal doped with iron, transitions of three triclinic Fe³⁺ centers localized in silicon positions were detected near (g = 4.3). The positions of these transitions weakly depend on the orientation of the magnetic field. The orientation behavior of other transitions of these centers has been studied. The parameters of the constructed spin Hamiltonians of two of them in the main axes satisfy the conditions: (b₂⁰ ≡ D) is close to (b₂² ≡ 3E) while (b₂⁰ ≫ gβB). These centers can be attributed to Fe³⁺ ions localized in silicon positions and compensated for both locally and non-locally by oxygen vacancies.

In SiC crystal enriched by ²⁸Si isotope with nuclear spin I = 0, two negatively charged silicon vacancy centers V_Si^-, V_k1 and V_k2 were investigated using X-band CW electron paramagnetic resonance (EPR) spectroscopy combined with tunable Ti-sapphire laser excitation. For the V_k1 and V_k2 centers, the EPR line position depends on the optical excitation energy which demonstrates inhomogeneous broadening of the optical transition correlated with variations in the zero field splitting.