Scattering length and effective range in closed form for the Coulomb plus Yamaguchi potential

From the known expression for the off-shell T-matrix corresponding to the potential consisting of the sum of the Coulomb potential and the Yamaguchi potential, the physical scattering amplitude can be derived in a satisfactory way. We derive simple exact closed formulae for the scattering length a_cs and the effective range r_cs from this amplitude. These are compared with approximate formulae derived by Harrington. Also a few numerical calculations are reported and compared with results obtained by Harrington and by Ali et al.     

Investigation on electromagnetic scattering from rough soil surface of layered medium using the small perturbation method

Electromagnetic scattering from a rough surface of layered medium is investigated, and the formulae of the scattering coefficients for different polarizations are derived using the small perturbation method. A rough surface with exponential correlation function is presented for describing a rough soil surface of layered medium, the formula of its scattering coefficient is derived by considering the spectrum of the rough surface with exponential correlation function; the curves of the bistatic scattering coefficient of HH polarization with variation of the scattering angle are obtained by numerical calculation. The influence of the permittivity of layered medium, the mean layer thickness of intermediate medium, the roughness surface parameters and the frequency of the incident wave on the blstatic scattering coefficient is discussed. Numerical results show that the influence of the permittivity of layered medium, the mean layer thickness of intermediate medium, the rms and the correlation length of the rough surface, and the frequency of the incident wave on the bistatic scattering coefficient is very complex.     

Two-dimensional friction oscillator: group-preserving scheme and handy formulae

With vibration isolation of buildings and turbomachinery blades in mind, we study the dynamic behaviour of a single-mass two-degree-of-freedom oscillator with dry friction damper, viscous damper and elastic spring connected in parallel. The mass is mounted on an elastic supporting plate allowing movement in two directions on a plane. We formulate a multi-dimensional friction model, from which the sliding conditions and the sticking conditions of the mass are derived. For calculations we develop a group-preserving scheme, which preserves the projective proper orthochronous Lorentz group PSO_o(2,1) symmetry of the model in the sliding phase so as to satisfy automatically the sliding conditions at each time step without iteration at all. The oscillator is then subjected to simple harmonic excitations, and the responses are displayed. According to the simple harmonic balance method together with a circular orbit assumption on displacements, we derive closed-form formulae for handily estimating the steady state responses, which are then compared with the results calculated by the group-preserving scheme to confirm the applicability of the formulae. We also derive formulae specifically for a two-dimensional friction oscillator with rigid base support, which include an exact formula of the magnification factor and a simple formula for estimating the minimum driving force amplitude (or the maximum friction force bound) to avoid sticking.     

Approximate formulae for the overlap integral of two harmonic oscillator wave functions

The use of second-order perturbation theory to derive approximate formulae for the overlap integral of two harmonic oscillator wave functions is discussed, and the results applied to the theory of intensity distributions in vibrational progressions in electronic spectra. For the vibrational progression m←0 an approximate formula is given which, when the vibrational frequencies of the initial and final states differ by less than 10%, reproduces to an accuracy of 1% or less the intensity profile calculated using the exact formulae for the overlap integrals.     

Semiclassical S-matrix theory for atomic fragmentation

A semiclassical scattering approach is developed which can handle long-range (Coulomb) forces without the knowledge of the asymptotic wave function for multiple charged fragments in the continuum. The classical cross section for potential and inelastic scattering including fragmentation (ionization) is derived from first principles in a form which allows for a simple extension to semiclassical scattering amplitudes as a sum over classical orbits and their associated actions. The object of primary importance is the classical deflection function which can show regular and chaotic behavior. Applications to electron impact ionization of hydrogen and electron–atom scattering in general are discussed in a reduced phase space, motivated by partial fixed points of the respective scattering systems. Special emphasis, also in connection with chaotic scattering, is put on threshold ionization. Finally, motivated by the reflection principle for molecules, a semiclassical hybrid approach is introduced for photoabsorption cross sections of atoms where the time-dependent propagator is approximated semiclassically in a short-time limit with the Baker–Hausdorff formula. Applications to one- and two-electron atoms are followed by a presentation of double photoionization of helium, treated in combination with the semiclassical S-matrix for scattering.     

Approximate,non-relativistic scattering phase shifts,bound state energies,and wave function normalization factors for a screened Coulomb potential of the Hulthén type

Non-relativistic scattering phase shifts, bound state energies, and wave function normalization factors for a screened Coulomb potential of the Hulthén type are presented in the form of relatively simple analytic expressions. These formulae have been obtained by a suitable renormalization procedure applied to the quantities derived from an approximate Schrödinger equation which contains the exact Hulthén potential together with an approximate angular momentum term. When the screening exponent vanishes, our formulae reduce to the exact Coulomb expressions. The interrelation between our formulae and Pratt's ‘analytic perturbation theory for screened Coulomb potentials’ is discussed.     

Geometric singularities and spectra of Landau-Ginzburg models

Some mathematical and physical aspects of superconformal string compactification in weighted projective space are discussed. In particular, we recast the path integral argument establishing the connection between Landau-Ginzburg conformal theories and Calabi-Yau string compactification in a geometric framework. We then prove that the naive expression for the vanishing of the first Chern class for a complete intersection (adopted from the smooth case) is sufficient to ensure that the resulting variety, which is generically singular, can be resolved to a smooth Calabi-Yau space. This justifies much analysis which has recently been expended on the study of Landau-Ginzburg models. Furthermore, we derive some simple formulae for the determination of the Witten index in these theories which are complimentary to those derived using semiclassical reasoning by Vafa. Finally, we also comment on the possible geometrical significance ofunorbifolded Landau-Ginzburg theories.     

Solitons in stimulated Raman scattering and resonant two-photon propagation

We are starting from equations of motion describing both stimulated Raman scattering and resonant two-photon absorption and emission processes. Application of the Estabrook-Wahlquist method leads to a system of differential equations whose integrability conditions are the original equations. This system is used as the starting point for applying the inverse scattering method. Implicit N-soliton formulae and explicit one-soliton formulae are derived. Possible applications under specified experimental conditions are discussed.     

Zur Theorie der nichtstationären stimulierten Raman-Streuung

The theory of transient stimulated Raman scattering is developed taking into account the deplation of the pump pulse. Two coupled differential equations describing the conversion of a laser pulse in a Stokes wave are derived with the aid of a semiclassical nonstationary perturbation calculation without the phonon concept. The equations are solved both analytically by a successive approximation and by computer calculations. Estimations are given of the delay of the Stokes pulse, the gain and a characteristic length concerning the conversion.     

Long range absorption and other optical-model effects from strong inelastic coupling

An optical potential component is constructed to represent the effect of a strongly coupled inelastic excitation upon elastic scattering. In the particular case of quadrupole Coulomb excitation a long range imaginary potential component is derived in closed form. The effects of long range absorption upon the elastic scattering are considered in a general way by inserting this potential into a weak absorption model and deriving an elastic scattering cross section in closed form. Below the Coulomb barrier the formula takes a simple form which may be related to the semiclassical theory of Coulomb excitation. The potential component arising from nuclear excitation of an inelastic state may be evaluated numerically on a computer. Two examples computed (50 MeV α-scattering on ¹⁵⁴Sm and 60 MeV ¹⁶O scattering on ⁴⁰Ca) exhibit strong l-dependence in the potential component.     

Phosphorescence from the two triplet states of p-benzoquinone and toluquinone vapour

Molecular formulae for the isotropic and anisotropic scattering are derived by considering the internal field acting upon a molecule within a dense medium and its fluctuation caused by density fluctuations. The anisotropic scattering formula of classical theory is retained but modified by the factors of the internal field. The new formula for the isotropic scattering depends sensitively on the principal values of the optical polarizability tensor and on the parameters describing the anisotropic internal field. Assuming the internal field to be given by the semimacroscopic approach of the Onsager-Scholte model with an ellipsoidal cavity, comparison of the calculated isotropic Rayleigh factor with the experimental value allows a prediction of the degree of the cavity anisotropy for different liquid densities.     

Adiabatic anholonomy and canonical transformations

Biswas and Soni [4] have surmised a semiclassical formula for Berry’s phase in terms of a generating function. We derive this formula apart from showing that it is not true in general and investigate its domain of validity. We also derive transformation formulae for Berry’s phase (Hannay’s angle) under general canonical transformations. A simpler proof for total angle invariance than hitherto available, is given.     

Infrared problems in quantum electrodynamics

We present a self-contained treatment of the infrared problem in Quantum Electrodynamics. Our program includes a derivation and proof of finiteness of modified reduction formulae for scattering in Coulomb potentials and unitary extensions of the relativistic Coulomb amplitudes in the forward direction. The renormalization structure of the theory is discussed in connection with the infrared problem and the renormalization group is reconsidered and shown to be inadequate for the “improvement” of perturbation theoretic results. However, simple forms of the renormalization group equations are easily established, which allow for a simple discussion of the renormalization structure and the extraction of physical quantities out of Green functions normalized at an arbitrary mass μ < m (m is the fermion mass). As an example of such a quantity we consider the construction of a renormalized and infrared finite mass-operator in presence of external fields. Scattering theory in Quantum Electrodynamics is elaborated in the context of the coherent state formulation of the asymptotic condition. Dimensional regularization techniques are systematically used for the reduction of coherent states and the construction of S-matrix elements and the cross-section formulae. The latter are obtained in a relatively simple form, which allows for a direct comparison with the exact cross-section formulae derived in the traditional context. This establishes the equivalence of the two approaches at the cross-section level. Various applications illustrate the techniques presented here and relative topics are discussed.     

A Soliton Perturbation Theory for the MNLS Equation with Corrections in terms of an Auxiliary Spectral Parameter

Based on the lakharov-Shabat equation of the inverse scattering transform for the MNLS equation, a perturbation theory for the MNLS equation with corrections is developed in terms of an auxiliary spectral parameter ζ. All necessary formulae for calculating the scattering data are derived without any assumption beyond the usual conditions for the inverse scattering transform. Based upon these formulae, the effects due to the corrections can be calculated.     

Improved analytical formulae for correcting elastic-scattering effects in X-ray photoelectron spectroscopy

We present new predictive formulae for the correction parameters Q_x and β_eff that are used to account for the effects of elastic scattering in quantitative X-ray photoelectron spectroscopy. These formulae were derived from an analysis of published calculations of Q_x and β_eff from extensive Monte Carlo simulations for a group of elemental solids. Two formulae are given for each parameter. One formula is a function of the single-scattering albedo and the photoelectron emission angle, and is useful for emission angles between 0° and 80°. The other formula is a function only of the single-scattering albedo and is useful for emission angles between 0° and 50°. The single-scattering albedo is in turn a simple function of two material parameters, the inelastic mean free path and the transport mean free path. The latter parameters can be determined readily from available predictive formulae for any material or from databases. The root-mean-square and mean deviations of the Monte Carlo values of Q_x and β_eff from those found with the new formulae are comparable to or smaller than those found with formulae published by Seah and Gilmore.     

A SOLITON PERTURBATION THEORY FOR THE UNSTABLE NONLINEAR SCHR?DINGER EQUATION WITH CORRECTIONS

Based on the Zakharov-Shabat equation of the inverse scattering transform for the unstable nonlinear Schr?dinger equation, for which a perturbation theory with corrections is developed in this paper. All necessary formulae for calculating the scattering data are derived. Based upon these formulae, the effect due to the corrections can be studied. As an example, the correction due to the damping is calculated.     

World-line approach to the Bern-Kosower formalism in two-loop Yang-Mills theory

Based on the world-line formalism with a sewing method, we derive the Yang-Mills effective action in a form useful to generate the Bern-Kosower-type master formulae for gluon scattering amplitudes at the two-loop level. It is shown that four-gluon (φ⁴ type sewing) contributions can be encapsulated in the action with three-gluon (φ³ type) vertices only, the total action thus becoming a simple expression. We then derive a general formula for a two-loop Euler-Heisenberg type action in a pseudo-abelian su(2) also studied.     

Higher-order <Emphasis Type="Bold">?</Emphasis> corrections in the semiclassical quantization of chaotic billiards

In the periodic orbit quantization of physical systems, usually only the leading-order ? contribution to the density of states is considered. Therefore, by construction, the eigenvalues following from semiclassical trace formulae generally agree with the exact quantum ones only to lowest order of ?. In different theoretical work the trace formulae have been extended to higher orders of ?. The problem remains, however, how to actually calculate eigenvalues from the extended trace formulae since, even with ? corrections included, the periodic orbit sums still do not converge in the physical domain. For lowest-order semiclassical trace formulae the convergence problem can be elegantly, and universally, circumvented by application of the technique of harmonic inversion. In this paper we show how, for general scaling chaotic systems, also higher-order ? corrections to the Gutzwiller formula can be included in the harmonic inversion scheme, and demonstrate that corrected semiclassical eigenvalues can be calculated despite the convergence problem. The method is applied to the open three-disk scattering system, as a prototype of a chaotic system. Received 10 September 2001 and Received in final form 3 January 2002