Renormalization group functions of the φ<Superscript>4</Superscript> theory in the strong coupling limit: Analytical results

The previous attempts of reconstructing the Gell-Mann-Low function β(g) of the φ⁴ theory by summing perturbation series give the asymptotic behavior β(g) = β_∞ g in the limit g→∞, where α = 1 for the space dimensions d = 2, 3, 4. It can be hypothesized that the asymptotic behavior is β(g) ～ g for all d values. The consideration of the zero-dimensional case supports this hypothesis and reveals the mechanism of its appearance: it is associated with vanishing of one of the functional integrals. The generalization of the analysis confirms the asymptotic behavior β(g) ～ g in the general d-dimensional case. The asymptotic behaviors of other renormalization group functions are constant. The connection with the zero-charge problem and triviality of the φ⁴ theory is discussed.     

Nuclear asymptotic normalization coefficients for <Superscript>14</Superscript>N → <Superscript>13</Superscript>C + <Emphasis Type="Italic">p</Emphasis> configurations and astrophysical <Emphasis Type="Italic">S</Emphasis> factor for radiative proton capture

Empirical values of the asymptotic normalization coefficient for proton bound states in the ¹⁴N nucleus for the first five levels were obtained from an analysis of the experimental differential cross sections known from our measurements and from the literature for the reaction ¹³C(³He, d)¹⁴N in the projectile-energy range between about 15 and 40 MeV. The values obtained in this way were used to calculate the astrophysical S factors for the reaction ¹³C(p, γ)¹⁴N in the case of the population of the first five levels of the 14N nucleus. The calculations were based on the R-matrix approach. The calculated values are in good agreement with the experimental astrophysical S factor at energies below 1 MeV.     

Coordinate asymptotic behavior of the radial three-particle wave function for a bound state involving two charged particles

The asymptotic expression for the radial component of the wave function for a three-particle bound state involving two charged particles is derived in an explicit form. This expression contains a three-particle asymptotic normalization factor C(φ), where φ is a hyperangle in the six-dimensional space of intrinsic coordinates of the three-particle system. The resulting expressions are used to analyze the asymptotic behavior of the wave functions for the ⁹Be nucleus that were calculated within the α + α + n three-particle model for various forms of the an potential. A comparison of the asymptotic expression derived here and the asymptotic expressions for model wave functions makes it possible to extract C(φ) values, which turned out to be sensitive to the form of αn interaction. This permits deducing information about two-particle interaction from a comparison of the theoretical values of C(α) with their phenomenological counterparts found from an analysis of experimental differential cross sections for relevant nuclear reactions.     

High-frequency nonlinear response of thin superconducting films with a mixed <Emphasis Type="Italic">d</Emphasis> and <Emphasis Type="Italic">s</Emphasis> symmetry of the order parameter

Third-harmonic generation during reflection of electromagnetic radiation from a thin superconducting film with a mixed d-and s-order parameter is studied theoretically. The dependence of the third harmonic intensity on the temperature and amplitude of an incident wave is calculated in the framework of the Ginzburg-Landau theory for superconductors with a two-component order parameter, and its behavior in the vicinity of transitions between phases with different symmetries is analyzed. It is shown, in particular, that the third harmonic intensity in the vicinity of the temperature corresponding to the d ? d + s phase transition substantially increases and is a nonanalytic function of the amplitude of the incident wave, while no singularity in the nonlinear response is observed for the d ? d + is transition. The linear reflection coefficient is found to be virtually insensitive to these phase transitions.     

Radiative transition probabilities in ions of the silver isoelectronic sequence

The wave functions of one-electron states above the 4d ¹⁰ core have been calculated within the relativistic perturbation theory with a zero-order model potential. The wavelengths and probabilities of electric dipole transitions in an Ag-like ion have been calculated for the 5s-5p, 5p-5d, 5d-5f, and 4f-5d transitions. The data obtained are compared with the results of calculations by the relativistic Hartree-Fock method and within the relativistic many-body perturbation theory. The theoretical results are compared with experimental data on the lifetimes of energy levels in Ag-like ions.     

Photoabsorption cross section of the Thomas-Fermi atom

The photoabsorption cross section σ(ω) and the distribution of oscillator strengths df/dω [these values are related as σ=(2π²e²/mc)(df/dω)] were determined for an atom with a large Z value using the semiclassical approach. These values were found for low frequencies with the use of the Vlasov kinetic equations, which were numerically solved by the method of particles. The asymptotic behavior of the distribution of oscillator strengths at high frequencies was determined by semiclassical equations for the photoabsorption cross section of electron shells in a Coulomb potential. The asymptotic equations were used to suggest an interpolation equation for the distribution of oscillator strengths over the whole Thomas-Fermi frequency range 27 eV ? ?ω ? 27Z² eV. This equation was used to calculate the logarithmic mean excitation energy, which appears in problems of ionization loss of charged particles. The distribution of oscillator strengths in a neutral atom allows the radiative properties of dense matter to be determined.     

Finite-size scaling from the self-consistent theory of localization

Accepting the validity of Vollhardt and Wölfle’s self-consistent theory of localization, we derive the finite-size scaling procedure used for studying the critical behavior in the d-dimensional case and based on the consideration of auxiliary quasi-1D systems. The obtained scaling functions for d = 2 and d = 3 are in good agreement with numerical results: it signifies the absence of substantial contradictions with the Vollhardt and Wölfle theory on the level of raw data. The results ν = 1.3–1.6, usually obtained at d = 3 for the critical exponentν of the correlation length, are explained by the fact that dependence L + L ₀ with L ₀ > 0 (L is the transversal size of the system) is interpreted as L ^1/ν with ν > 1. The modified scaling relations are derived for dimensions d ≥ 4; this demonstrates the incorrectness of the conventional treatment of data for d = 4 and d = 5, but establishes the constructive procedure for such a treatment. The consequences for other finite-size scaling variants are discussed.     

Dimension of an optimum impurity cluster and multiple-occupancy corrections in the theory of scattering of vibrational excitations in a solid solution

A relationship is derived for the correlation length L determining the size of the region in a solid solution in which excitations are scattered coherently. The correlation length depends on the fraction of impurity atoms x in the solid solution and the lattice dimension d. In the physical analysis of single-particle scattering processes in the solid solution and calculations, it is sufficient to take into account clusters with the number of cells n corresponding to the correlation volume L ^d . A theoretical analysis is illustrated by calculations of the spectral functions of the solid solution at different values of x and n. The multiple-occupancy corrections (polynomials in powers of x) to scattering diagrams are calculated using the method of sequential breaking apart of the interaction lines in the diagrams for the self-energy part. The method used was previously applied to the case of scattering by a single impurity. In this paper, the efficiency of the method is checked for scattering by multi-impurity clusters. It is demonstrated that the method can be useful in analyzing and estimating the contributions of scattering diagrams.     

Radiative Lifetimes of the Energy Levels of Palladium-like Ions Pr XIV and Nd XV

Radiative lifetimes for the 4d⁹4f, 4d⁹5p, 4d⁹5d, and 4d⁹6s levels in the spectra of palladium-like ions Pr XIV and Nd XV are calculated in the electric-dipole approximation by the intermediate coupling semiempirical method with the use of experimental energy levels available in the literature. The radial integrals of transitions necessary for calculating absolute values of transition probabilities are obtained in the length form with Hartree–Fock functions.     

Conductance distribution near the Anderson transition

Using a modification of the Shapiro approach, we introduce the two-parameter family of conductance distributions W(g), defined by simple differential equations, which are in the one-to-one correspondence with conductance distributions for quasi-one-dimensional systems of size L ^d–1 × L _z , characterizing by parameters L/ξ and L _z /L (ξ is the correlation length, d is the dimension of space). This family contains the Gaussian and log-normal distributions, typical for the metallic and localized phases. For a certain choice of parameters, we reproduce the results for the cumulants of conductance in the space dimension d = 2 + ? obtained in the framework of the σ-model approach. The universal property of distributions is existence of two asymptotic regimes, log-normal for small g and exponential for large g. In the metallic phase they refer to remote tails, in the critical region they determine practically all distribution, in the localized phase the former asymptotics forces out the latter. A singularity at g = 1, discovered in numerical experiments, is admissible in the framework of their calculational scheme, but related with a deficient definition of conductance. Apart of this singularity, the critical distribution for d = 3 is well described by the present theory. One-parameter scaling for the whole distribution takes place under condition, that two independent parameters characterizing this distribution are functions of the ratio L/ξ.     

A first-principle study on the phase transition,electronic structure,and mechanical properties of three-phase ZrTi<Subscript>2</Subscript> alloy under high pressure*

We employed density-functional theory (DFT) within the generalized gradient approximation(GGA) to investigate the ZrTi₂ alloy, and obtained its structural phase transition,mechanical behavior, Gibbs free energy as a function of pressure, P-V equation of state,electronic and Mulliken population analysis results. The lattice parameters andP-V EOS for α, β and ω phases revealed by ourcalculations are consistent with other experimental and computational values. The elasticconstants obtained suggest that ω-ZrTi₂ and α-ZrTi₂ are mechanically stable, and that β-ZrTi₂ is mechanically unstableat 0 GPa, but becomes more stable with increasing pressure. Our calculated resultsindicate a phase transition sequence of α → ω → β forZrTi₂. Both thebulk modulus B and shear modulus G increase linearly withincreasing pressure for three phases. The G/B values illustrated goodductility of ZrTi₂alloy for three phases, with ω<α<β at0 GPa. The Mulliken population analysis showed that the increment of d electron occupancystabilized the β phase. A low value for B '₀ is the feature of EOS for ZrTi₂ and this softness in the EOS isrepresentative of pressure induced s-d electron transfer.     

Process <Emphasis Type="Italic">γ</Emphasis>*<Emphasis Type="Italic">γ</Emphasis> → <Emphasis Type="Italic">σ</Emphasis> at large virtuality of <Emphasis Type="Italic">γ</Emphasis>*

The process γ*γ → σ is investigated in the framework of the SU(2)×SU(2) chiral NJL model. The form factor of the process is derived for arbitrary virtuality of γ* in the Euclidean kinematic domain. The asymptotic behavior of this form factor resembles the asymptotic behavior of the γ*γ → π form factor.     

Discussion of isomeric ratios in (<Emphasis Type="Italic">p,n</Emphasis>) and (<Emphasis Type="Italic">d</Emphasis>, 2<Emphasis Type="Italic">n</Emphasis>) reaction

Isomeric ratios (IR) in the (p, n) and (d, 2n) reactions are considered. The dependence of IR values on the projectile type and energy, the target- and product-nucleus spin, the spin difference between the isomeric and ground states of products, and the product mass number is discussed. The isomeric ratios for 46 product nuclei (from ^44m,gSc to ^127m,gXe) obtained in reactions where target and product nuclei have identical mass numbers were calculated at energies from the reaction threshold to 50 MeV (with a step of ΔE = 1 MeV). The calculations in question were performed with the aid of the TALYS 1.4 code package. The calculated IR values were compared with their experimental counterparts available from the literature (EXFOR database). In the majority of cases, the calculated IR values agree well with the experimental data in question. It is noteworthy that the IR values obtained in (d, 2n) reactions are substantially greater than those in (p, n) reactions.     

Contribution of meson-exchange currents to the electromagnetic structure of the deuteron at high momentum transfers

The leading term of the asymptotic expansion of the contributions of the πNN and ρπγ processes to the electromagnetic form factors of the deuteron at high momentum transfers is calculated. The resulting asymptotic behavior is compared with experimental data obtained at the Thomas Jefferson Laboratory. Constraints on the functional dependence of meson-nucleon vertex functions that ensure a correct asymptotic behavior of deuteron form factors are discussed.     

Phase change on transmission of microwaves through metal disc delay dielectrics

Expressions for the phase changeΦ suffered by microwaves when transmitted through an artificial dielectric composed of metallic discs arranged in a three-dimensional array have been derived with different approaches as follows (i) molecular theory, (ii) electromagnetic theory and (iii) transmission line theory. The phase change depends on the distancet that the wave traverses inside the dielectric and also the spacingd between centre to centre of any two adjacent discs in the three principal directions. Molecular theory indicatesΦ as an increasing function oft, whereas, the other two theories indicateΦ as an oscillatory function oft. The transmission line theory also exhibitsΦ to be real or imaginary depending ont. Experimental values ofΦ as a function oft have been obtained with the help of a microwave (3·2 cms wavelength) interferometer for two dielectrics havingd as 1·91 cms and 2·22 cms respectively.     

Asymptotic behavior of the Gell-Mann-Low function in quantum field theory

The reconstruction of the Gell-Mann-Low function in quantum field theory from its asymptotic series, whose first terms are calculated using perturbation theory, is discussed. This mathematical problem cannot be solved uniquely. Nevertheless, the desired function can be reconstructed in a certain finite range of coupling constant g under reasonable assumptions about this function. However, attemps to determine the behavior of the function for g→∞ are, in our opinion, groundless. Conditions under which the sum of the divergent perturbation series can rapidly decrease at infinity are determined.     

<Emphasis Type="Italic">K</Emphasis><Superscript>±</Superscript><Emphasis Type="Italic">p</Emphasis> elastic scattering in the QCD inspired eikonalized model

Based on our previous study of the QCD inspired eikonalized model for describing vector meson photoproduction, pp, and \(\bar p\) p elastic scattering at high energies, we apply the mode to high energy K ^± p elastic scattering. The total cross section σ _tot(s), differential cross section dσ/dt, the ratio of the real part to imaginary part of the forward scattering amplitude ρ(s), and nuclear slope parameter function β(s) are calculated in the model. Our results show that the theoretical prediction for σ _tot(s) is in a good agreement with the experimental data within error bars of the data. For the other theoretical predictions there are no data to test the predictive power of the model. We need the corresponding experimental data to examinate the validity of our QCD inspired eikonalized model. However, our calculations clearly show that the Odderon exchange in the process makes a significant contribution to the observable of ρ(s) and β(s). Therefore, we may conclude that there is a good opportunity to find the QCD Odderon in the K ^± p elastic scattering at high energies.     

On the reactions <Emphasis Type="Italic">A</Emphasis>+<Emphasis Type="Italic">A</Emphasis>+‖+<Emphasis Type="Italic">A</Emphasis>→0 at a one-dimensional periodic lattice of catalytic centers: Exact solution

The kinetics of the diffusion-controlled chemical reactions A+A+‖+A→0 that occur at catalytic centers periodically arranged along a straight line is considered. Modes of the behavior of reaction probability W(t) were studied at different times and different densities of the catalyst. Within the Smoluchowski approximation, it was rigorously proved that at large times the function W(t) is independent of the lattice period. This means that, in the given asymptotic mode, the probability of the reaction on a lattice with a catalyst placed in each lattice site is the same as on a lattice with a catalyst placed in sparse sites.     

Elastic Deuteron Scattering on the <Superscript>12</Superscript>C Nucleus within a Three-Body Model

The formalism developed earlier for elastic pd scattering on the basis of Glauber theory with allowance for a total spin dependence is modified by replacing pN amplitudes by amplitudes for N¹²C scattering and is applied to elastic deuteron scattering on the ¹²C nucleus. The amplitudes for elastic N¹²C scattering are obtained within the optical model. Respective numerical calculations performed at the kinetic deuteron-beam energy of 270 MeV lead to results that agree well with data on the differential cross section for d¹²C scattering into the forward hemisphere, but the calculated spin observable A _y ^d agrees with experimental data only qualitatively.     

Bestimmung des Kompoundkern- und Stripping-Anteils der (d,p)- und (d,n)-Reaktionen an Zr94 und Zr96

An experiment has been accomplished which gives information on the reaction mechanism of the (Zr^94,96+d)-reactions. Total cross sections and excitation functions up to 11,8 MeV have been measured for the reactions Zr⁹⁴ (d, p), Zr⁹⁴ (d, n), Zr⁹⁶(d, p), Zr⁹⁶(d, n) and Zr⁹⁶(d, 2n) by the activation method. The results are compared with cross sections calculated according to the statistical theory of nuclear reactions. It is shown that the (d, p) -reactions proceed almost completely by the stripping-mechanism. The (d, n)- and (d, 2n)-excitation functions on the other hand are in good agreement with the predictions of the statistical theory and one can conclude that the compound nucleus mechanism is the dominating process.