A simple and direct periodic solution of the Korteweg-de vries equations

The solutionq(x, t) of one of the KdV hierarchy is assumed to be a potential in the Schrödinger equation as usual. We differentiate this equation with respect to the time variable and solve it with the aid of the Green function. The obtained equation relatesw _t (x, t, λ)=φ _t (x + c, x, t, λ) withq _t (x, t). The functionφ(x, x ₀,t, λ) obeys the Schrödinger equation and the boundary conditionsφ(x ₀,x ₀,t, λ)=0,φ _x (x ₀,x ₀;t, λ)=1. The shiftingc is equal to the period. We differentiatew _t (x, t, λ) three times with respect to thex coordinate and obtain the time derivative of the Milne equation. The integration of this equation with respect tox allows to solve simply the inverse problem. The reconstructed periodic potential is given by means of the well known formula for the root functions ofw(x, t, λ). The time behaviour of this function, i.e. the solution of the KdV equation, is obtained when one replacesq _t (x, t) by an expression of the KdV hiearchy in the relation betweenq _t (x, t) andw _t (x, t, λ) and transforms it. We estimated also the limit, whenc → ∞, i.e. the possible relation of the periodic solutions with the soliton ones.     

Thermal properties and structures of chlorocadmate, bromocadmate, and bromochlorocadmate glasses

Glass transition temperatures (T_g), crystallization temperatures (T_x), and ¹¹³Cd magic-angle spinning (MAS) NMR spectra of the chlorocadmate, bromocadmate, and bromochlorocadmate glasses and the CdCl₂, CdBrCl, and CdBr₂ crystals have been measured. The T_g and T_x, and thermal stabilities (T_x−T_g) decreased with increasing Br-content. No enhancement of (T_x−T_g) of bromochlorocadmate glasses by anion-mixing was observed. The CdCl₂ and CdBr₂ crystals gave the isotropic chemical shifts of 183 and −24 ppm, which are assigned to the CdCl₆ and CdBr₆ octahedra, respectively. The isotropic chemical shifts of CdBrCl crystal (186, 166, 135, 101, 63, 25 and −21 ppm) were assigned to CdBr_nCl_(6−n) (n=0-6) octahedra, respectively. It was found that the Br and Cl atoms in CdBrCl randomly occupy chlorine sites in the CdCl₂ structure. The ¹¹³Cd MAS NMR spectra revealed that the chlorocadmate, bromocadmate, and bromochlorocadmate glasses are composed of the CdCl₆, CdBr₆ and CdBr_nCl_(6−n) octahedra (n=0-6), respectively, and that the Cd²⁺ in all the bromochlorocadmate glasses was more preferentially bonded to Br⁻ than Cl⁻. The relationship between the thermal properties and the anion coordination environments around Cd²⁺ in these glasses is discussed.     

Thermoelectric performance in pseudo-ternary (PbTe)0.95-x(Sb2Se3)x(PbS)0.05 system with ultra-low thermal conductivity via multi-scale phonon scattering

Among the intermediate temperature (500–900 K) thermoelectric materials, both the p- and n-type lead telluride (PbTe) compounds have attracted extensive interests. Till date various approaches were adopted to enhance the thermoelectric performance of n-type PbTe-based materials as they show greater potential space for further improvement compared to p-type ones. Herein, a pseudo-ternary n-type (PbTe)_0.95-x(Sb₂Se₃)_x(PbS)_0.05 system was designed and a large value of ZT = 1.61 at 850 K in case of x = 0.01 was achieved. Two factors are responsible for the improved thermoelectric performance. The incorporation of lead sulfide is the key factor to maintain a high level of power factor above 700 K and the multi-scale hierarchical architectures yield ultra-low thermal conductivity.     

Optical and magneto-optical properties of room-temperature ferromagnetic In(Ga)MnAs layers, deposited by pulse laser ablation

Spectral dependences of refractive and absorption indices n(hν), k(hν) (hν=1.2-4.4 eV) and the transversal Kerr effect δ(hν) (hν=0.5-4.4 eV) in In(Ga)MnAs layers fabricated by laser deposition have been investigated. Spectra of the diagonal and off-diagonal components of the dielectric permittivity tensor of these layers have been calculated. Comparison of the spectral dependences δ(hν), ε′(hν) and ε′₂×(hν)² of the In(Ga)MnAs layers with similar spectra for MnAs has been carried out. Particular features in the spectra of the In(Ga)MnAs layers have been explained by a competition of contributions of the In_1−x(Ga_1−x)Mn_xAs host and MnAs inclusions.     

Luminescence and chromaticity of alkaline earth aluminate MxSr1−xAl2O4:Eu (M: Ca,Ba)

In order to prepare fluorescent material for UV-LED used as illumination light source, two series of Eu²⁺ doped (1 mol%) alkaline earth aluminate phosphors Ca_xSr_1−xAl₂O₄ and Ba_xSr_1−xAl₂O₄ were prepared. The crystal structure, relative quantum efficiency(Qr), peak wavelength(λ_p), color tuning and chromaticity were investigated by XRD patterns and photoluminescence (PL) on samples prepared by solid solution system (s series) and powder mixing system (m series) respectively. For the s series, the synthesized Ca_xSr_1−xAl₂O₄:Eu²⁺ powders show that the structure transforms from monoclinic to hexagonal at x?0.5, and λ_p increases from 442.3 to 529.7 nm with decreasing x. For the Ba_xSr_1−xAl₂O₄:Eu²⁺ system, the structure transforms from monoclinic to hexagonal at x?0.3, and λ_p decreases from 520.5 to 502.2 nm continuously from x=0 to 1. The shift in λ_p could be explained by the crystal field effect, which is affected by different coulomb attractive forces due to the various fraction of alkaline earth cation in the host lattice. Different phosphor properties prepared by either solid solution or powder mixing methods were characterized by chromaticity measurements for both reflective and transmissive modes.     

Schrödinger operator with a nonlocal potential whose absolutely continuous and point spectra coexist

We consider the Schrödinger-like operatorH in which the role of a potential is played by the lattice sum of rank 1 operators \(|\left. {v_n } \right\rangle \left\langle {v_n |} \right.\) multiplied by g tan π[(α,n)+ω],g>0, α∈? ^d ,n∈? ^d , ω∈[0, 1]. We show that if the vector α satisfies the Diophantine condition and the Fourier transform support of the functionsv _n (x)=v(x-n),x∈? ^d ,n∈? ^d , is small then the spectrum ofH consists of a dense point component coinciding with? and an absolutely continuous component coinciding with [?, ∞), where ? is the radius of the mentioned support. Besides, we find the integrated density of statesN(λ) (it has a jump at λ=?) and zero temperature a.c. conductivityσ _λ (v), that also has a jump at λ=? and vanishes faster than any power of the external field frequency ν as ν→0 and λ≠?.     

The encapsulation of nucleoside monophosphates into Cd0.8PS3

The biological molecules cytidine monophosphate (CMP) monohydrate, and adenosine monophosphate (AMP) monohydrate, have been encapsulated using exfoliated single layer Cd_0.8PS₃, to form restacked Cd_0.8PS₃(NMP)_x(H₂O)_y, where N stands for cytidine or adenosine. The layer spacings of these inorganic/organic hybrid materials depend strongly on x and y. We have found a range of Cd_0.8PS₃ layer expansions up to 21 Å for CMP, and up to 31 Å for AMP. For one sample with AMP (x≈1.5), 15 orders of diffraction are observed, indicating a well-stacked system with an interlayer expansion of 16.4 Å. For many samples, no significant changes in layer expansion are observed for relative humidities less than 40%, though at higher humidities, the absorption of water causes the layers to expand greatly.We have also found that the acid form of these molecules was needed for successful encapsulation, because in the salt form the positive counterions of CMP and AMP salts bind preferentially to the negatively charged Cd_0.8PS₃ single layers, thereby inhibiting the encapsulation of the CMP or AMP.     

Properties of bismuth selenide intercalated with lithium from a solution of n-C4H9Li

The lithium-selective intercalation of bismuth selenide (Bi₂Se₃) samples is accomplished by treating them in a solution of n-butyllithium (n-C₄H₉Li) in hexane. It is found that, upon incorporation of lithium atoms, the interlayer spacing increases and the resistivity in the direction normal to the layers (ρ_⊥) increases by a factor of more than two. The dependence ρ_⊥(x) acquires the form of an oscillating function of the degree of guest “load” (the intercalant content) and the frequency-independent portion of the ρ_⊥ curve (100 mHz≤ω≤1 MHz) substantially decreases upon completion of the process.     

Nonlinear optical susceptibilities χ of nitridosilicate powders

Nitridosilicates are of interest as novel nonlinear materials due to their extraordinary chemical and thermal stability. Unfortunately, large nitridosilicate single crystals are presently not available for the investigation of their nonlinear optical properties. The first experiments are presented in which an averaged nonlinear susceptibility χ⁽²⁾ for several nitridosilicates is studied by using two different powder techniques, the Kurtz Perry method and the SHEW method (Second Harmonic Wave generated by an Evanescent Wave). We observe nonlinearities of the new materials which are comparable to those of LiIO₃. The highest averaged M_eff=(χ_eff⁽²⁾)²/4n_2ωn_ω² values found are ∼0.9 pm²/V². The refractive indices of the materials are determined to be between n=2 and 3.     

Magnetorefractive effect in (Co<Subscript>50</Subscript>Fe<Subscript>50</Subscript>)<Subscript>x</Subscript>(Al<Subscript>2</Subscript>O<Subscript>3</Subscript>)<Subscript>1−<Emphasis Type="Italic">x</Emphasis></Subscript> granular films

The reflectivity spectra and the magnetorefractive effect (MRE) of (Co₅₀Fe₅₀)_x(Al₂O₃)_1?x metal-dielectric granular films (0.07<x<0.52) are analyzed in the IR spectral range λ=2.5–25 µm. It is revealed that the specific features observed in the spectra at λ≈8.5 and 20 µm are associated with the excitation of phonon modes in the dielectric matrix. The magnetorefractive effect in the films is observed below the percolation thresh-old only in p-polarized light and above the percolation threshold for both the p and s polarizations. It is demonstrated that the optical properties of (Co₅₀Fe₅₀)_x(Al₂O₃)_1?x films in the IR spectral range, to a first approximation, can be interpreted in the framework of the effective-medium theory and the magnetorefractive effect can be explained in terms of the modified Hagen-Rubens relation.     

Spectral resonances which become eigenvalues

The stationary Schrödinger equation is ? _x ² φ + λV(x)φ=zφ for φ∈?²(R ⁺,dx). If the potential is bounded below, singular only atx=0, negative on some compact interval and behaves likeV(x)～1/x ^μ asx→∞ with 2≧μ>0, then the system admits shape resonances which continuously become eigenvalues as λ increases. Here λ>0 and for μ=2 a sufficiently large λ is required. Exponential bounds are obtained on Im(z) as λ approaches a threshold. The group velocity near threshold is also estimated.     

A compounded rare-earth iron garnet single crystal exhibiting stable Faraday rotation against wavelength and temperature variation in the 1.55 μm band

The Bi, Tb and Yb partially substituted iron garnet bulk single crystals of Tb_3−x−yYb_yBi_xFe₅O₁₂ were grown by using Bi₂O₃/B₂O₃ as flux and accelerated crucible rotation technique for single-crystal growth. Faraday rotation (FR) spectra showed that the specific FR of the (Tb_0.91Yb_1.38Bi_0.71)Fe₅O₁₂ crystal under magnetic field at saturation was measured to be about −1617°/cm at λ=1.55 μm, Faraday rotation wavelength coefficient (FWC, 0.009%/nm) in the wavelength range of 1.50–1.62 μm and Faraday rotation temperature coefficient (FTC, 3.92×10⁻⁵/K) at λ=1.55 μm were even smaller than that of YIG. It is proven that through combining two types of Bi-substituted rare-earth iron garnets with opposite FWC and FTC signs, the compound rare-earth iron garnets with low FWC and FTC may be obtained due to the compensation effect. The saturation magnetization of (Tb_0.91Yb_1.38Bi_0.71) Fe₅O₁₂ crystal is 0.48×10⁶ A/M and is also much smaller than that of YIG. We have found empirically that there is a simple relationship between the FR θ_f(x) and Bi content x for Tb_3−x−yYb_yBi_xFe₅O₁₂, which is given by θ_f(x)=(−2759x+400)°/cm.     

High-frequency permeability and permittivity of NixZn(1−x)Fe2O4 thick film

Magnetic materials such as Ni_xZn_(1−x)Fe₂O₄ have resonant frequency in high frequency; therefore, they are more useful especially in microwaves. The Ni_xZn_(1−x)Fe₂O₄ was prepared by the chemical coprecipitation method using citrate precursors, and the fritless thick film was screen printed on alumina substrates. The composition-dependent permeability and permittivity in the high frequency 8–12 GHz are investigated. Using the overlay technique on Ag-thick-film patch antenna, the change in reflectance and transmittance has been measured. The Ni_xZn_(1−x)Fe₂O₄ thick film, when used as overlay on Ag-thick-film patch antenna, changes the resonance characteristics. The changes in resonance frequency, reflectance and transmittance have been used to calculate the permeability and permittivity of the thick film. Zinc-concentration-dependent changes are obtained.     

Analytical solution of the double-logarithmic integral equation with QCD running coupling

The analytical solution of the double-logarithmic integral equation with QCD running coupling describing small-x behaviour of the non-singlet structure function ? _NS(x,Q ²) has been found for any cut-off parameter μ. Analytical properties of the solution and a position of the right-most singularity in the complex ρ-plane which determines the asymptotics of ? _NS(x,Q ²) at small x have been studied. The asymptotical formula ? _NS(x,Q ²) = C ₁ x ^-λ1{ln^κ1(Q ²/Λ²) —ln^κ1 (μ ²/Λ²) + κ ₁ ln^κ1-1(Q ²/Λ²)[ψ(1) - ψ(λ₁)]} valid if x ? 1 and ln(Q ²/Λ²) ? 1 has been obtained where C ₁, λ₁ are constants, κ ₁ = g/λ₁, λ₁ < g = 8/(33 - 2gh _f), gh _f is a number of active flavours and ψ(ξ) denotes the digamma function.     

The magnetic state of diamagnetically diluted antiferromagnetic cobalt and nickel monoxide

The nearest neighbour J₁(x) and the next-neighbour super-exchange J₂(x) interactions are evaluated by using the mean field theory for Mg1−xBxO (B=Co and Ni) systems. The magnetic energy E(x) is obtained. A magnetic phase diagram of the Mg1−xBxO (B=Co and Ni) solid solutions with 0?x?1 is drawn by high-temperature series expansions (HTSE) combined with the Padé approximants method (PA). The critical exponents associated with the magnetic susceptibility (γ) and with the correlation length (ν) are deduced in order phase.     

Structure transition of single-texture CoSi2 nanolayer grown by refractory-interlayer-mediated epitaxy method

In this investigation, the crystalline structure of a nanometric CoSi₂ layer, formed in heat treated Co/W_xTa_(1−x)/Si(1 0 0) systems, has been studied by XRD analysis. Careful measurements of the diffraction intensities revealed that temporary formation of a metastable diamond cubic structure of CoSi₂ phase, rather than its usual CaF₂ structure, was occurred. It has been shown that formation of this metastable structure depends on the kind of the applied interlayer in addition to the annealing temperature. Among the studied systems with x = 0, 0.25, 0.5, 0.75 and 1, the second and the last systems resulted in growing a (1 0 0) single-texture CoSi₂ layer with the preferred usual CaF₂ structure, a strained lattice parameter, and the best thermal stability (900-1000 °C).     

Superconducting and magnetic properties of Nb/Pd 1-xFex/Nb triple layers

The superconducting and magnetic properties of Nb/Pd_1-xFe_x/Nb triple layers with constant Nb layer thickness d_Nb=200 ? and different interlayer thicknesses 3 ?≤ d_PdFe ≤ ? are investigated. The thickness dependence of the magnetization and of the superconducting transition temperature shows that for small iron concentration x the Pd_1-xFe_x layer is likely to be in the paramagnetic state for very thin films whereas ferromagnetic order is established for x ≥ 0.13. The parallel critical field B_c2||(T){B_{c2||}}(T) exhibits a transition from two-dimensional (2D) behavior where the Nb films are coupled across the interlayer, towards a 2D behavior of decoupled Nb films with increasing d_PdFeand/or x. This transition allows a determination of the penetration depth x_F{\xi _F} of Cooper pairs into the Pd_1-xFe_x layer as a function of x. For samples with a ferromagnetic interlayer x_F{\xi _F} is found to be independent of x.     

Pressure dependence of dielectric constant,elastic constants,and lattice parameters of the Y3(Ga,Al)5O12 host

Atomistic simulations were performed to investigate the lattice parameters, dielectric constant, and elastic constants of Y₃(Ga_xAl_5−x)O₁₂ (x = 1, 2, 3, 4, 5) structures. The calculated lattice parameters and elastic constants are in good agreement with those in available experimental results. The pressure dependence of all studied quantities was investigated. In general, a change in the behavior of all studied quantities is found when the Ga concentration becomes more than that of the aluminum (Al) in Y₃(Ga_xAl_5−x)O₁₂ (x = 1, 2, 3, 4, 5) structures.     

Remarks concerning the connection between properties of the 4-point-function and the Wilson-Zimmermann expansion

This paper concerns an investigation of the Wilson-Zimmermann (or “short distance”) expansion forA(x)A(y) withx →y whereA(x) is a real scalar field fulfilling Wightman's axioms. If one assumes that such an expansion exists, where the terms of the expansion are operators relatively local toA(x), then the singularities arising in the 4-point-function forx ₃ →x ₄ must control the singularities of then-point functions (n=4, 5, 6, ...) arising forx _j →x _x+1,j=1,2,...,n?1. A similar consequence can be drawn if the terms of the expansion are assumed to exist only as bilinear-forms (Section 2). For certain classes of fields one can show that this condition necessary for the short distance expansion is indeed fulfilled (Section 3). The result of the last section is that the above mentioned condition is also sufficient for the Wilson-Zimmermann expansion, interpreted as an expansion into bilinear forms, and also as an operator expansion in a somewhat modified sense.     

Transformation mechanism of n-butyl terminated Si nanoparticles embedded into Si1−xCx nanocomposites mixed with Si nanoparticles and C atoms

Bright-field transmission electron microscopy (TEM) images, high-resolution TEM (HRTEM) images, and fast-Fourier transformed electron-diffraction patterns showed that n-butyl terminated Si nanoparticles were aggregated. The formation of Si_1−xC_x nanocomposites was mixed with Si nanoparticles and C atoms embedded in a SiO₂ layer due to the diffusion of C atoms from n-butyl termination shells into aggregated Si nanoparticles. Atomic force microscopy (AFM) images showed that the Si_1−xC_x nanocomposites mixed with Si nanoparticles and C atoms existed in almost all regions of the SiO₂ layer. The formation mechanism of Si nanoparticles and the transformation mechanism of n-butyl terminated Si nanoparticles embedded into Si_1−xC_x nanocomposites mixed with Si nanoparticles and C atoms are described on the basis of the TEM, HRTEM, and AFM results. These results can help to improve the understanding of the formation mechanism of Si nanoparticles.