Universal relationships for the phonon spectra in BCC,FCC, and HCP crystals with a short-range interatomic interaction

The frequencies of the phonon branches that correspond to the vibrations of the close-packed atomic planes in bcc, fcc, and hcp crystals with short-range interatomic interaction are shown to be described by a universal relationship, which only contains two parameters for each branch, for any polarization λ. These phonon branches correspond to the (ξ, ξ, 0) direction in bcc crystals, the (ξ, ξ, ξ) direction in fcc crystals, and the (0, 0, ξ) direction in hcp crystals. This universal relationship can only be violated by long-range interactions, namely, the interactions outside the sixth coordination shell in a bcc crystal, the fifth coordination shell in an fcc crystal, and the eleventh or tenth coordination shell in an hcp crystal. The effect of these long-range interactions for each phonon branch can be quantitatively characterized by certain parameters Δ _nλ, which are simply expressed in terms of the frequencies of three phonons of the branch. The values of these parameters are presented for all bcc, fcc, and hcp metals whose phonon spectra are measured. In most cases, the proposed relationships for the frequencies are found to be fulfilled accurate to several percent. In the cases where the Δ _nλ parameters are not small, they can give substantial information on the type and scale of long-range interaction effects in various metals.     

Specific Selectivity of Modified Ion-Exchange Membranes

Effective transfer numbers through homogeneous membranes MK-100 and polyethylene diamine-modified membranes MK-100M are measured. A mathematical model for the membranes is considered. The presence of a modified surface is accounted for by introducing phenomenological conductivity coefficients. The effect of the solution composition, overall concentration, and current density on competitive transfer is analyzed. The modified layer has a pronounced effect on the membrane specific selectivity at low current densities and only slightly affects it in a limiting state.     

Self-induced transparency of circularly polarized femtosecond pulses

The phenomenon of self-induced transparency in a two-level medium is studied using a new integrable set of evolution equations for the optical pulses with a duration on the order of the energy-transition oscillation period. A mathematical apparatus is developed for the inverse scattering problem and used to obtain solitonic solutions to the model. The characteristics of linearly and circularly polarized pulses are compared with each other.     

Generalization and prognostication of mass exchange characteristics of electrodialyzers operating in overlimiting current regimes with use made of similarity theory and compartmentation method

A semiempirical approach, which allows one to perform generalization, prognostication, and scaling of the mass exchange characteristics of electrodialyzers intended for the desalination of dilute solutions in the overlimiting current regimes, is proposed. The approach is based on modern theoretical notions concerning the dependence of mass transfer on conjugated effects of concentration polarization, such as electroconvection and the exaltation effect and makes use of the similarity theory principles. As a result, there are obtained sufficiently simple equations that describe to within a satisfactory accuracy characteristics of the process of electrodialysis (mass transfer coefficient, Sherwood number, solution desalination degree, and so on) as functions of four parameters, namely, the input concentration, the solution flow rate, the potential drop across the paired chamber of the membrane stack of an electrodialyzer, and the length of the desalination channel. For scaling the obtained dependences with respect to a channel length a compartmentation method is used. Results of experimental investigations and calculations of electrodialyzers with desalination channels without a filler, or containing an inert separator, a monolayer of an ion-exchange resin, a profiled membrane, are presented.     

Regular growth of various characteristics of entire functions of order zero

We study the relationship between the strongly regular growth of an entire function f of order zero, the existence of the angular density of its zeros, the behavior of the Fourier coefficients of the logarithm of f, and the regular growth of the logarithm of the modulus and the argument of f in the L^p[0, 2π]-metric, p ≥ 1.     

Mathematical model for the overlimiting state of an ion-exchange membrane system

A three-layered mathematical model is proposed for describing the overlimiting state in an ion-exchange membrane system. The model’s prominent feature is the allowance for the space-charge region; the water dissociation reaction, which is catalyzed by active ionogenic groups; and the coupled gravitational and electroosmotic convection, which leads to the emergence of dependence of the effective diffusion layer thickness on the electric current density. The model is used for calculating, on the basis of known initial current-voltage curves and dependences of effective transport numbers on the current density, such internal characteristics of the system as the diffusion layer thickness, distribution of concentration of ions, space charge, and electric-field strength at various current densities.     

A mathematical model describing voltammograms and transport numbers under intensive electrodialysis modes

A three-layer mathematical model of overlimiting state is developed. A reactive layer with a thickness depending on the current density is introduced into the model. A decrease in the thickness of diffusion layer, which donates the counterions, with increasing current density as a result of electroconvection is also taken into account. A boundary-value problem is formulated within the Nernst-Planck and Poisson’s model in the three-layer region with the boundary conditions of constant concentrations in the bulk solution. It is shown that an increase in the reactive layer thickness with increasing current density determines the behavior of effective transport numbers in the overlimiting state of ion-exchange electromembrane system. In the current range under consideration (from 1 to 20 limiting currents), the reactive layer thickness is several tens nanometers and reaches 70 nm at a 100-fold excess over the limiting current. To calculate the voltammograms, the dependence of effective thickness δ_N of diffusion layer on the current density is required. This dependence can be obtained by solving an inverse problem, from the laser interferometry experiments, or calculated by the Navier-Stokes hydrodynamic model. The model enables one to calculate the distribution of electric field strength, potential, concentrations in the diffusion layers and membrane.     

Electric mass transport through homogeneous and surface-modified heterogeneous ion-exchange membranes at a rotating membrane disk

Polarization characteristics of the homogeneous MF-4SK perfluorinated sulfonated cation-exchange membrane and the heterogeneous MK-40 sulfonic acid membrane with its surface modified by a homogeneous film of Nafion are studied at a rotating membrane disk in 0.1 and 0.001 M sodium chloride solutions. Partial current-voltage curves (CVC) are obtained for sodium and hydrogen ions, and limiting current densities in the electromembrane systems (EMS) under study are calculated as a function of the rotation rate of the membrane disk. Contribution from different mechanisms (electrodiffusion, electroconvection, dissociation of water, and the effect of the limiting-current exaltation) to the total ion flow is estimated experimentally and theoretically under conditions that the diffusion layer in the EMS has stabilized in thickness. It is established that surface modification of the heterogeneous MK-40 membrane with a 7 μm layer of a modifying agent almost completely eliminates the dissociation of water molecules, and the properties of the heterogeneous MK-40 membrane approximate those of the homogeneous Nafion membrane. From IR spectra and potentiometric titration curves of the MK-40 and MF-4SK membranes, it is shown that the acidity of the sulfonate groups in these membranes is nearly identical, but a difference in the dissociation rate of water at these membranes is determined by a different character of charge-density distribution and potential near the membrane-solution interphase boundary. By means of the theory of the overlimiting state in EMS, the internal parameters of the systems under investigation are calculated: distribution of space-charge density and electric-field potential in the diffusion layer and in the membrane. Partial CVC are calculated for H⁺ ions for the space-charge region in the phase of the MF-4SK and MK-40/Nafion ion-exchange membranes. Partial CVC with similar characteristics are compared for the heterogeneous monopolar MK-40 and the bipolar MB-2 membranes, which contain sulfonate groups. It is concluded that the membrane surface layer, where the space charge is localized, plays a dominant role in speeding up the dissociation of water in EMS.     

Valiron-type and valiron-titchmarsh-type theorems for entire functions of order zero

By using known asymptotics of the counting function of zeros of an entire functionf of order zero, we determine the asymptotics of Inf under the condition that all zeros off lie on the same ray. The inverse problem is also analyzed.     

Single-ion approach to the interpretation of the x-ray photoelectron spectra of the valence bands of monoxides of 3d elements

The single-ion approach taking account of only the intra-atomic Coulomb interaction in free d ⁿ ions and in d ⁿ ions in an O _h crystal field is used to analyze the fine structure of the x-ray photoelectron spectra of the valence bands of monoxides of 3d elements. The x-ray photoelectron spectra were studied as a collection of d ⁿ⁻¹ and d ⁿ L multiplets representing the unscreened and screened parts, respectively, of the final state. The unscreened part of the final state can be described by the distribution of the line strengths of the photoelectronic transition d ⁿ → d ⁿ⁻¹ and the screened part can be described as a partially relaxed distribution of the statistical weights of the d ⁿ ions. Fiz. Tverd. Tela (St. Petersburg) 39, 1056–1063 (June 1997)