The construction of the Gilmore–Perelomov coherent states for the Kratzer–Fues anharmonic oscillator with the use of the algebraic approach

By applying the algebraic approach and the displacement operator to the ground state, the unknown Gilmore–Perelomov coherent states for the rotating anharmonic Kratzer–Fues oscillator are constructed. In order to obtain the displacement operator the ladder operators have been applied. The deduced SU(1, 1) dynamical symmetry group associated with these operators enables us to construct this important class of the coherent states. Several important properties of these states are discussed. It is shown that the coherent states introduced are not orthogonal and form complete basis set in the Hilbert space. We have found that any vector of Hilbert space of the oscillator studied can be expressed in the coherent states basis set. It has been established that the coherent states satisfy the completeness relation. Also, we have proved that these coherent states do not possess temporal stability. The approach presented can be used to construct the coherent states for other anharmonic oscillators. The coherent states proposed can find applications in laser-matter interactions, in particular with regards to laser chemical processing, laser techniques, in micro-machinning and the patterning, coating and modification of chemical material surfaces.     

Exact solution of the Schrödinger equation with a new expansion of anharmonic potential with the use of the supersymmetric quantum mechanics and factorization method

The study involves finding exact eigenvalues of the radial Schrödinger equation for new expansion of the anharmonic potential energy function. All analytical calculations employ the mathematical formalism of the supersymmetric quantum mechanics. The novelty of this study is underlined by the fact that for the first time the recurrence formulas for rovibrational bound energy levels have been derived employing factorization method and algebraic approach. The ground state and the excited states have been determined by means of the hierarchy of the isospectral Hamiltonians. The Riccati nonlinear differential equation with superpotentials has been solved analytically. It has been shown that exact solutions exist when the potential and superpotential parameters satisfy certain supersymmetric constraints. The results obtained can be utilized both in computations of quantum chemistry and theoretical spectroscopy of diatomic molecules.     

Low-birefringence liquid crystal mixtures for photonic liquid crystal fibres application

ABSTRACT

Series of low-birefringence liquid crystal mixtures composed of alkyl alkylbicyclohexyl carbonates and modified mixtures with other compounds have been formulated, their refractive indices and electric permittivity measured upon temperature. They exhibit the ordinary refraction index n_o lower than the refractive index of silica glass in a different range of temperature. This enables to observe in photonic liquid crystal fibres (PLCFs) a change in the light propagation mechanism from photonic band gap guiding to modified total internal reflection at different temperatures. Selected applications of PLCFs are also discussed.     

In silico modeling of functionalized graphene oxide-metal cluster conjugates as Raman probe: Raman activity of pyridine

Functionalized graphene – metal nano-conjugates are used as Raman probes, in recent years, for trace level identification of materials having specific Raman active modes. In the present paper, model Raman probes were modeled through conjugation of Au₄ and Ag₄ clusters with functionalized graphene systems. In silico models of functionalized (5,5)-graphene sheets were designed at the density functional theory (DFT) level through attachments of epoxy, -OH and –NH(CH₃)₂SH/-CONH(CH₃)₂SH groups. Model Raman probes were designed through attachment of Au₄ and Ag₄ clusters to the functional sites. Full geometry optimizations followed by vibrational analysis were carried out to ensure that the designed Raman probes have acceptable geometric characteristics to attach Raman-active molecules to the metal site. Pyridine was used as a test system to investigate the functionality of such model Raman probes through attachment with the metal clusters. It was observed that the chemical effects due to such attachments increase the Raman intensities (RI) of specific Raman modes of pyridine (in-plane symmetric bending (1040 cm^-1) and asymmetric stretch-bend (1634 cm^-1)), which are too weak in the isolated molecule. Furthermore, the suggested in silico system could provide an important model for basic understanding of RI-enhancements of molecules through increase of the size of the metal clusters, as the observed enhancement was found to be dependent on the polarizability of the metal clusters attached to the molecule of interest.     

Copper Crystallization from Aqueous Solution: Initiation and Evolution of the Polynuclear Clusters

The initial steps of copper electrocrystallization process from aqueous solutions have been studied at DFT level of theory. It has been shown that Cu(H₂O) unit is the final product of Cu²⁺-ions electroreduction. From this particle clusters Cu_n·aq are formed and grow. Aggregation of copper atoms to the Cu_n·aq clusters consists of two steps. The first step includes condensation of Cu(H₂O) units to hydrated clusters Cu_n(H₂O)_n (n = 2–6). At the second step bonding of Cu(H₂O) particles is accompanied by dehydration of clusters yielding Cu_n(H₂O)_m structures (n > m). Cluster Cu₇·aq has been singled out as key structure based on calculated values of energies and Cu–Cu bond distances of Cu_n·aq clusters. This cluster is of D_5h symmetry which is typical for copper microcrystals formed from aqueous solutions in electrocrystallization processes on foreign surface. This key particle could be considered as a critical nucleus. Number of copper atoms therein matches average dimension of critical nucleus.     

Analytic monoids and factorization problems

The main goal of this paper is to initiate study of analytic monoids as a general framework for quantitative theory of factorization. So far the latter subject was developed either in concrete settings, for instance in orders of number fields, or abstractly, in an axiomatic way. Some of the abstract approaches are too general to address delicate problems concerning oscillatory nature of the related counting functions, or are too restrictive in the sense that they suffer from the lack of examples except classical ones i.e. the Hilbert monoids of algebraic integers and their products. The notion of an analytic monoid is enough flexible to allow constructions of many other examples, and also ensures sufficiently rich analytic structure. In particular, we construct examples of such monoids with the associated L-functions being products of classical Dirichlet L-functions and L-functions of twisted irreducible unitary cuspidal automorphic representations of \(GL_d({\mathbb {A}}_{\mathbb {Q}})\) satisfying the Ramanujan conjecture and having real coefficients. Finally, to illustrate how a typical problem from the quantitative theory of factorization can be studied in the framework of analytic monoids, we formulate several results concerning oscillations of the remainder term in the asymptotic formula for the number of irreducible elements with norms less or equal x, as x tends to infinity.     

The Properties and Activity of TiO2/beta‐SiC Nanocomposites in Organic Dyes Photodegradation

The TiO₂/beta‐SiC nanocomposites containing 0–25 wt. % of beta‐SiC were synthesized by the sol‐gel method and tested in the photodegradation of methylene blue and methyl orange water solutions. With the increase in SiC content, only a slight decrease in energy band gap was observed (3.19–3.12 eV), together with significant increase in the surface area of the catalysts (42.7–80.4 m² g^?1). In the synthesized material, the anatase phase of TiO₂ was present in the form of small agglomerates resulting from the mechanical mixing process. In the process conditions (catalyst concentration 0.5 g L^?1, initial dye concentration 100 ppm, light source 100 W UV‐Vis lamp), we have observed no signs of catalyst deactivation. The significantly higher photodegradation activity of methylene blue than methyl orange can be attributed to the preferable pH of the solution compared to pH_PZC and the cationic character of the first dye. In case of methyl orange, pH process conditions substantially limit the contact of the catalyst with the dye, as negatively charged surface of the catalysts repels the dissociated anionic dye molecules.     

Modeling Speciation and Solubility in Aqueous Systems Containing U(IV,VI), Np(IV,V, VI), Pu(III,IV, V,VI), Am(III), and Cm(III)

A comprehensive thermodynamic model, referred to as the Mixed-Solvent Electrolyte model, has been applied to calculate phase equilibria and chemical speciation in selected aqueous actinide systems. The solution chemistry of U(IV, VI), Np(IV, V, VI), Pu(III, IV, V, VI), Am(III), and Cm(III) has been analyzed to develop the parameters of the model. These parameters include the standard-state thermochemical properties of aqueous and solid actinide species as well as the ion interaction parameters that reflect the solution’s nonideality. The model reproduces the solubility behavior and accurately predicts the formation of competing solid phases as a function of pH (from 0 to 14 and higher), temperature (up to 573 K), partial pressure of CO₂ (up to \( p_{{{\text{CO}}_{2} }} \)  = 1 bar), and concentrations of acids (to 127 mol·kg^?1), bases (to 18 mol·kg^?1), carbonates (to 6 mol·kg^?1) and other ionic components (i.e., Na⁺, Ca²⁺, Mg²⁺, OH^?, Cl^?, \( {\text{ClO}}_{4}^{ - } \), and \( {\text{NO}}_{3}^{ - } \)). Redox effects on solubility and speciation have been incorporated into the model, as exemplified by the reductive and oxidative dissolution of Np(VI) and Pu(IV) solids, respectively. Thus, the model can be used to elucidate the phase and chemical equilibria for radionuclides in natural aquatic systems or in nuclear waste repository environments as a function of environmental conditions. Additionally, the model has been applied to systems relevant to nuclear fuel processing, in which nitric acid and nitrate salts of plutonium and uranium are present at high concentrations. The model reproduces speciation and solubility in the U(VI) + HNO₃ + H₂O and Pu(IV, VI) + HNO₃ + H₂O systems up to very high nitric acid concentrations (\( x_{{{\text{HNO}}_{3} }} \approx 0.70 \)). Furthermore, the similarities and differences in the solubility behavior of the actinides have been analyzed in terms of aqueous speciation.     

Generalizations of a property of orthogonal projectors

Generalizing the result in Lemma of Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Commutativity of projectors, Linear Algebra Appl. 341 (2002) 129-142], Baksalary et al. [J.K. Baksalary, O.M. Baksalary, T. Szulc, Linear Algebra Appl. 354 (2002) 35-39] have shown that if P₁ and P₂ are orthogonal projectors, then, in all nontrivial situations, a product of any length having P₁ and P₂ as its factors occurring alternately is equal to another such product if and only if P₁ and P₂ commute, in which case all products involving P₁ and P₂ reduce to the orthogonal projector P₁P₂ (= P₂P₁). In the present paper, further generalizations of this property are established. They consist in replacing a product of the type specified above, appearing on the left-hand side (say) of the equality under considerations, by an affine combination of two or three such products. Comments on the problem when the number of components in a combination exceeds three are also given.