Variational Matrix Padé Approximants in Two Body Scattering

The convergence and bounding properties of the variational matrix Padé approximants are investigated for non relativistic two body interactions. Selecting L – 1 discrete values qi, i = 1, …, L – 1 and the physical momentum q₀ the off shell scattering amplitudes are L X L matrices. The [N/N] Padé approximants to the Born series of these matrices are the variational solution of the Schwinger principle and the corresponding physical amplitude has variational properties in the off shell momenta. For positive interactions the best approximants to the phase shift is an absolute minimum on the q_i and monotonic convergence to the exact result for N → ∞ or L → ∞ ca be proved. Similar properties are shown for the bound states using the Ritz variational principle. The required mathematical background is extensively worked out, the extensions to non positive, singular and long range potentials are considered and some numerical examples are presented.     

自发破缺规范理论中的光子-光子散射

本文在SU(2)_L×U(1)模型里研究中间态的W介子对光子-光子散射的贡献。由于自发破缺规范理论的可重整化性以及自发破缺以后电磁规范不变性仍成立,在单迴路近似下,光子-光子散射幅中的无穷大必然互相抵消。本文用具体计算说明这一点,并且在低能近似下给出电磁场的等效非线性相互作用拉格朗日量和散射截面。在低能近似下,轻子中间态对散射的贡献为主,然而在高能极限下,W介子中间态的贡献将变得可以和轻子中间态的贡献相比较。     

Photon interaction cross-sections and anomalous scattering factors of Cu and Ag

Total attenuation cross sections of copper and silver have been measured in the energy range 5 to 85 keV in a narrow beam good geometry set up using X- andγ-rays emitted from radio isotopes, by employing a high resolution hyper pure germanium detector. From the measured values, the photoeffect cross-sections have been derived by subtracting a small contribution of the sum of the theoretical coherent and incoherent scattering cross sections. The photoeffect cross-sections so obtained are found to be in better agreement with the unrenormalized values of Scofield [10]. These photoeffect cross-sections have been used to evaluate the dispersion corrections (also called anomalous scattering factors)f ⁺ andf″ for the forward Rayleigh scattering amplitude by a numerical evaluation of the dispersion integral that relates them at the energies at which the cross-sections have been measured. To thef ⁺ values so obtained, the relativistic corrections proposed by different investigators are included separately and the valuesf′ so obtained are compared with the available data and discussed. Possible conclusions are drawn from the present study.     

Quasi-elastic and elastic charge-exchange differential cross sections (critical review)

More than 50 years ago the charge-exchange of one nucleon in the quasi-elastic nucleon deuteron scattering was proposed to determine the spin independent, or the spin dependent parts of the elastic backward neutron proton differential cross section. For this purpose two measurements are to be performed using unpolarized particles only. Such a suggestion was very attractive for experimentalists, since at that time polarized beams were rare and polarized targets were nonexistent. One experiment consists in the inclusive measurement of the charge-exchange quasi-elastic nucleon deuteron differential cross section. The energy of the fast outgoing nucleon is to be approximately the same as that of the incident nucleon. The intrinsic momenta of the remaining two identical nucleons should be small. The impulse approximation has been assumed, i.e. in the final state of the two nucleons their interaction can be neglected. The quasi-elastic result is to be compared with the elastic np differential cross section in the backward direction at the same energy. The theory related the ratio R _QE (π) of the quasi-elastic to the elastic np charge-exchange differential cross sections to the spin dependent part of the np backward elastic scattering amplitude. Several experiments in the nucleon kinetic energy interval below 1 GeV were carried out, where the R _QE (π) values were determined. Recently the energy interval was extended up to 2 GeV by successful experiments at the JINR VBLHE Nuclotron. The quasi-elastic results can be compared with the values of the analogous quantity R _np (π) for the np elastic scattering, calculated from the phase shift analyses below 1.3 GeV. The present paper is a critical review checking expressions for R _np (π) in different amplitude representations, listing numerical values of elastic np quantities and results of existing quasi-elastic experiments. Conclusions and statements of some authors and the validity of the relevant theory is discussed.     

On the analytic extrapolation of scattering amplitudes in L2 norm

The problem of the analytic extrapolation of scattering amplitudes to interior points from a least-squares error channel is studied and found to be stable if some L² norm of the amplitude on the remainder of the cuts obeys a certain bound. The minimal possible value of this bound is of relevance to a model-independent location of the various singularities of scattering amplitudes. A simple and rigorous method is given for its construction and the numerical results of a singularity search are presented. The method is shown to be capable of accomodating unequal errors on the real and imaginary parts of the amplitude and is applicable to both simply and doubly connected domains (energy and cosine extrapolations).     

Regge Models and Dispersion Relations for Partial Waves

The asymptotic behaviour of partial wave amplitudes is calculated supposing various Regge models for the total scattering amplitude A(s, t, u). The high energy partial wave behaviour obtained is combined with the validity of partial wave dispersion relations. It is shown that consistency of these assumptions can only be achieved by demanding. 1) a definite asymptotic behaviour of the discontinuity of the left hand cut of partial wave amplitudes. 2) the validity of partial wave sum rules of similar kind as the well-known finite energy sum rules for the total amplitude. All steps of the derivation shall first be demonstrated for elastic scattering of identical scalar particles. Then within the helicity formalism the results are generalized for particles with arbitrary spin and different masses. Finally the question is studied whether the sum rules can be employed to determine unknown CDD-pole parameters in an N/D approach for the I = J = 1/2 state in πN scattering. It is shown that the sum rules of highest order are able to do that.     

Scattering approach to the von Klitzing effect

Earlier results of the scattering theoretical approach to the quantum Hall effect are simplified and generalized. Finite size corrections to the plateau values are found to be not of order (l/L _x ) but (l ² /L _x L _y )²,l,L _x ,L _y being the magnetic length, and the sample dimensions inx andy-direction respectively. An expression for the current parallel to the electric field in terms of a scattering matrix is derived. In the weak scattering regime this expression leads to a vanishing diagonal conductivity _xx .     

Rosette motion and string scattering of 20 MeV electrons in MgO single crystals

Analytical estimates and computer simulations were undertaken to perceive the motion of negative particles through a lattice structure, the interaction being classical binary scattering. Three distinct modes of particle motion along atomic strings were found depending on the magnitude of the transverse energy and the angular momentum L of the particle with regard to the string axis. At small and large L increased scattering on the strings as compared with random penetration dominates. At medium L and negative transverse energy (bound state particles in the attractive potential) a rosette motion along the string occurs. In this case small impact parameters to the string atoms are avoided and thus an increased penetrability of the negative particles results. The influence of thermal lattice vibrations on these motions was studied.

Experimentally, the negative particle motion modes manifested themselves in the penetration profiles of 20 MeV electrons through an 8 μm MgO single crystal.     

A new method for the calculation of Sommerfeld screening parameter σ1 in x-ray spectra

The paper describes a new method for the calculation of the Sommerfeld screening parameter σ₁. It requires neither the knowledge of the energy separations of spin doublet levels nor is it based on the application of the Hertz law. The only data required for the calculation are the experimental energy values of the level concerned for the series of elements belonging to the same subshell in which the element in the question is situated. As an illustration the values of σ₁ are calculated for the L ₁, L ₂ and L ₃ levels for elements belonging to the 4f subshell and these are found to be in excellent agreement with those published earlier by Gokhale and Misra. The method brings out the constancy of σ₁ (L ₂ L ₃)−σ₁(L ₁) in a natural way and may thus be regarded as providing theoretical explanation of the Hertz law.     

Generator coordinate amplitude for scattering

In the generator coordinate method for scattering the proper boundary condition is accomplished by requiring the GC amplitude to satisfy an integral equation of the first kind. Attempts to solve this problem are first reviewed and then an improved approximation is proposed which is applicable to a wider class of scattering problems in addition to the Coulomb scattering.A better approximation is obtained in the asymptotic region, where the generator coordinate, i.e., the distance between two shell-model wells of the fragments, is larger than the touching distance of the colliding nuclei, by deriving partial differential equations of first order for the terms of an asymptotic series in $\text{1}\text{E}$, where E is the scattering energy.Extracting the information on the GC amplitude for small values of the generator parameter from the integral equation of the first kind is an ill-posed problem. It is shown that the method of statistical regularization offers a powerful and controllable procedure to uncover the GC amplitude. The unknown GC amplitude is treated as a random function with an a priori distribution of probability which is based on the assumption that the amplitude is bounded and that the errors in the input are random with zero expectation value. A useful procedure is found for fixing parameters of the a priori distribution. The solution for small values of the GC parameter is expressed in the form of a Dini series.The method is applied to the calculation of the GC amplitude for scattering of two α-particles at 15 MeV c.m. energy. The measure of the accuracy is the difference between the input wave function of relative motion and the result of folding of the GC amplitude with the kernel of the integral equation. The prescribed accuracy is reached with this method on a much larger interval than with any previously proposed method.     

Strong suppression of Coulomb corrections to the <Emphasis Type="Italic">e</Emphasis><Superscript>+</Superscript><Emphasis Type="Italic">e</Emphasis><Superscript>−</Superscript> pair production cross section in ultrarelativistic nuclear collisions

The semiclassical correction to Molière’s formula for multiple scattering is derived. The consideration is based on the scattering amplitude obtained with the first semiclassical correction taken into account for an arbitrary localized but not spherically symmetric potential. Unlike the leading term, the correction to Molière’s formula contains the target density n and thickness L not only in the combination nL (areal density). Therefore, this correction can be referred to as the bulk density correction. It turns out that the bulk density correction is small even for high density. This result explains the wide range of applicability of Molière’s formula. The article is published in the original.     

Correction to Molière’s formula for multiple scattering

The semiclassical correction to Molière’s formula for multiple scattering is derived. The consideration is based on the scattering amplitude obtained with the first semiclassical correction taken into account for an arbitrary localized but not spherically symmetric potential. Unlike the leading term, the correction to Molière’s formula contains the target density n and thickness L not only in the combination nL (areal density). Therefore, this correction can be referred to as the bulk density correction. It turns out that the bulk density correction is small even for high density. This result explains the wide range of applicability of Molière’s formula.     

Atomic Rayleigh scattering cross-sections and the associated anomalous dispersion in the X-ray regions

Elastic scattering cross-sections for Pd, Ag, Cd, In, Sn, Sb, Pt, Au and Pb are measured at an angle of 90 in the X-ray region 5.41 keV. These energies fall between the high-energy side of the L- and M-shell absorption edges of the atoms considered. The present atomic region is significant for solid X-rays to assess the contribution of resonance and solid-state environmental effects. Also it is the anomalous scattering region for many of the atoms of the periodic table. Experimental results are compared with theoretical calculations based on form factor formalisms including the anomalous corrections and available recent S-matrix values. Based on the experimental evidence, the present results indicate the influence of solid-state environmental effects, the importance of anomalous corrections nearer to absorption edges, the correctness of revised high-energy limit values, the superiority of S-matrix predictions over form factor values on measured elastic scattering cross-sections in the X-ray regime and also show the resonance behavior around K, L and M absorption edges. Received: 27 January 1998 / Received in final form: 4 January 1999     

Asymptotics of the S-matrix and unitarisation

The manner in which the elastic scattering amplitude obeys unitarity, how it enters the circle of unitarity, and what its asymptotic limit is, remains a problem for models which include terms that rise fast with s. We have checked that the features of cross sections which come from unitarisation are present for most unitarisation schemes, e.g. those that saturate the profile function or those that describe multiple exchanges via an analytic formula. We have also obtained a scheme which interpolates between different classes of the unitarisation and found corresponding non-linear equations. Considering different forms of energy dependence of the scattering amplitude, and a variety of unitarisation schemes, we show that, in order to reproduce the data, the fits choose an amplitude that corresponds to an asymptotic value S = 0.     

The relativistic bound and scattering states of the Manning-Rosen potential with an improved new approximate scheme to the centrifugal term

The approximately analytical bound and scattering state solutions of the arbitrary l-wave Klein-Gordon equation for the mixed Manning-Rosen potentials are carried out by an improved new approximation to the centrifugal term. The normalized analytical radial wave functions of the l-wave Klein-Gordon equation with the mixed Manning-Rosen potentials are presented and the corresponding energy equations for bound states and phase shifts for scattering states are derived. It is shown that the energy levels of the continuum states, reduce to the bound states of those at the poles of the scattering amplitude. Some useful figures are plotted to show the improved accuracy of our results and the special case for wave is studied briefly.      

Energy filtration of charge carriers in a nanostructured material based on bismuth telluride

The dependences of the electrical conductivity and thermopower on the size of grains in a nanocrystalline material based on Bi₂Te₃-Sb₂Te₃ solid solutions of the p type have been investigated theoretically and experimentally. The relaxation time in the case of hole scattering by nanograin boundaries in an isotropic polycrystal has been calculated taking into account the energy dependence of the probability of tunneling of charge carriers and the dependence of the scattering intensity on the nanograin size L _n . A decrease in the probability of boundary scattering with an increase in the energy of charge carriers leads to an increase in the thermopower. The dependences of the thermopower and electrical conductivity on the nanograin size, which have been obtained taking into account the boundary scattering and scattering by acoustic phonons, are in good agreement with experimental data. For the material under consideration, the thermopower coefficient increases by 10–20% compared to the initial solid solution at L _n = 20–30 nm. This can lead to an increase in the thermoelectric figure of merit by 20–40%, provided that the decrease in the electrical conductivity and the decrease in the lattice thermal conductivity compensate each other. Despite the absence of a complete compensation, it has been possible to increase the thermoelectric figure of merit for the samples under investigation to ZT = 1.10–1.12.     

Phase shift analysis of pion-4He scattering by optimal polynomial expansion

A novel method of phase shift analysis for the scattering of charged pion from⁴He nuclei is proposed. The nuclear amplitudef _N (ϑ) has been parametrized in terms of a polynomial expansion in a conformally mapped variable, which is obtained by optimally exploiting the analytic property off _N . The method exhibits a significant reduction in the number of free parameters required for the fixed energy phase shift analysis of the differential cross-section data. The nuclear amplitude thus constructed is then used to obtain phase shifts and inelastic parameters of all possible orders. Reliable values of the real and imaginary parts of the forward amplitude are also obtained.     

Structural interpretation of endurance of polymeric fibers

Endurance results in cyclic tension of several polymeric fibers are reviewed and analyzed in terms of Prevorsek-Lyons theory. It is shown that the plots of log (cycles to failure) against strain amplitude can be divided into regions of short and long term endurance. An expression is derived correlating the inflection point on the S-N curve with the structural parameters: fracture surface energy p, tensile modulus E, activation energy associated with the process of crack growth F*, and stress concentration factor, q. Manson-Coffin endurance plots expressed in terms of log (strain amplitude), log (stress amplitude), and log (strain energy amplitude) are examined. From the log Nf—log (strain energy amplitude) are derived the relative values of fracture surface energy of these fibers. Experimental data are shown which reflect the structural changes which occur in fibers during endurance experiments and an analysis is carried out leading to an estimate of the corresponding changes in p, E, and q.     

Peculiarities of the amplitude of elastic subbarrier impurity scattering

A formula for the amplitude of electron elastic subbarrier (in the domain of imaginary momenta) scattering by an impurity modeled as a spherical potential well is derived. An equation is obtained to find the features (poles) of this amplitude in a complex energy plane in the close vicinity of which the energy spectrum of quantum resonant percolation trajectories in M-I-M contacts (M = N (normal metal) or S (superconductor) and I is an insulator) with weak (low impurity concentration) structural disorder in the I layer is concentrated [1–4].