The pion mass difference in threshold pion reactions on light nuclei

We investigate the effect of the pion mass difference in scattering and photoproduction of pions on light nuclei near threshold. The effect is found to be most easily described in a nuclear K-matrix formalism where the nuclear K-matrix elements are shown to be independent of the mass difference.     

K −-Proton atomic transitions

The relation between theK ^}-P scattering length and the X-ray spectrum for the 2p → 1s electromagnetic transition inK ^?-P atoms is examined. A coupled-channel potential model is used to explicitly calculate the energy of theS-matrix pole in the 1s channel, which is then compared with the energy obtained from the scattering lengths via the standard equation. The X-ray spectrum is calculated and compared with the Lorentzian shape associated with the complex energy of theS-matrix pole. In addition, theK ^?p branching ratios are compared at threshold and at the complexS-matrix pole energy.     

The Gell-Mann-Goldberger formula for long-range potential scattering

A derivation of the Gell-Mann-Goldberger (GG) formula and cut-off versions of this formula for the T-matrix involving long-range potentials is given. The derivation is based on the time-dependent and recently developed stationary formalisms for scattering via long-range potentials. A stationary S-operator expression for two-body Coulomb-like scattering is derived. Using the well-known expression for the off-energy-shell “T-matrix” for a pure Coulomb potential the energy-shell limit of this stationary expression is shown to yield the pure Coulomb scattering amplitude. A proof of the convergence of the perturbation series corresponding to the Gell-Mann-Goldberger formula for the two-body Coulomb-like T-matrix is given.     

Quantum scattering from a class of anisotropic potentials

The nonrelativistic quantum scattering problem for a class of 3-dimensional anisotropic, or explicitly angle-dependent potentials, is evaluated. The S-matrix is expressed in terms of the time-dependent propagator and evaluated using earlier results of a path integral solution. The scattering amplitude is then obtained from the S-matrix.     

Summation of partial waves for large-angle scattering

For large-angle elastic scattering different methods of summing partial wave amplitudes are investigated for their accuracy and simplicity of computation. It is found that among the approximations considered, the method of expanding the T-matrix in terms of the weighted orthogonal polynomials proposed by Brysk is the most accurate way of calculating the scattering amplitude in the backward direction. If the two-particle interaction is assumed to be a Yukawa potential, then the lth partial sum of the T-matrix with the weighted polynomials can be expressed as the lth partial sum with the Legendre polynomials and a correction term which depends on the phase shift for the lth partial wave.     

Elastic unitarity and the K-matrix in a Lorentz covariant representation of the NN interaction

A Lorentz covariant representation of the NN t-matrix has been obtaained over the energy range from 150 to 500 MeV by solving the integral equation that connects the t-matrix with the K-matrix. The K-matrix is expanded in a complete set of on-shell Lorentz invariant amplitudes represented phenomenologically by isoscalar and isovector “meson” exchanges. The real part of the K-matrix is fit over the energy range from 150 to 500 MeV using coupling strengths that are allowed to vary quadratically with energy. Above the pion production threshold at T_lab = 280 MeV, the real K-matrix is supplemented by an imaginary part with linear energy dependence. The K-matrix parameters are fit to thesmost recent (January 1999) Arndt amplitudes [R.A. Arndt, D. Roper, VPI and SU Scattering Analysis Interactive Dial-in Program and Data Base]. Direct and exchange contributions to the K-matrix are handled explicitly in the formalism. The resulting t-matrix satisfies elastic unitary below the pion production threshold and contains non-local terms that are not present in direct Love-Franey parameterizations of the t-matrix. Results are given for the NN amplitudes and compared with both the Arndt amplitudes and amplitudes obtained from a direct fit of the t-matrix [O.V. Maxwell, Nucl. Phys. A 600 (1996) 509]. Results are also given for a selected set of np and pp observables.     

On the relation between resonances andK-matrix poles

The relation between resonances andK-matrix poles for the potential scattering problem is investigated. The analysis is carried out in two stages. We first discuss the analytic properties of aK-matrix related to a model phase-shift which embodies the relevant features of a potential scattering resonance; we then consider the specific case of a square-well potential. The result emerging from the analysis is that to each resonance two poles of theK-matrix are associated; they appear either as a real resonance-echo pair or as a complex conjugate doublet, according to the value of a suitable background parameter. The relevance of this result in connection with the Huby theory is briefly discussed.     

Impact parameter representation from the Watson-Sommerfeld transform

Using the Watson-Sommerfeld transform the elastic scattering amplitude of two spinless particles is shown to have an exact and unique impact parameter, or Fourier-Bessel (FB) representation. The representation is valid for all physical energies and scattering angles. Wallace's recent work is found to be an asymptotic expansion of the FB amplitude obtained from the partial-wave expansion. The way singularities of the partial-wave amplitude in the l-plane enter in the FB amplitude is also explicitly shown.     

Pion-nucleon scattering: The P11 channel

A simple model is presented for the πN scattering in the P₁₁ channel from the threshold to the Roper resonance N(1470). The model starts with the πBB′ coupling of the Yukawa type where B (and B') stands for N, R or Δ. A unitarized scattering amplitude is constructed from the K-matrix which is obtained perturbatively. With appropriate coupling constants and form factors for the πBB′ interactions, which conform to the quark model, the P₁₁ scattering can be well reproduced. It is emphasized that the process πN → ππΔ → πN, which has been ignored hitherto, plays a crucial role in the energy region considered.     

Perturbation theories in low energy electron diffraction: t-matrix and τ-matrix approaches

Two perturbation approaches to low energy electron diffraction (LEED) are discussed within the inelastic collision model: t-matrix approach recently proposed by Tong, Rhodin and Tait (all scatterings treated to third order), and τ-matrix approach (layer scattering is treated exactly while all interlayer scatterings are treated to third order). Both approaches are applied to the calculation of rotation diagrams for the (001) surface of Al. In the case of a single layer third-order perturbation results for the magnitude and phase of the scattering amplitude are compared to the exact results and found to be in fairly good agreement over a large range of elastic scattering cross-sections. When all scatterings (up to third order) are considered for an infinite crystal, the differences between the exact and third-order t-matrix perturbation resutls become quite substantial ; marked improvement is achieved if the layer scattering is treated exactly, but the remaining difference may be significant enough (depending on the application) to suggest that interlayer scattering may not always be treated by perturbation, and thus renormalization of the type proposed by Pendry may be necessary.     

Influence of corrugation on light-scattering properties of capsule and finite-cylinder particles: Analytic solution using Sh-matrices

We use the Sh-matrices to derive an analytic T-matrix solution for the light scattering from capsule and finite-cylinder particles with corrugated surfaces. The solution is tested by comparison with discrete-dipole-approximation (DDA) calculations. We present results from finite circular cylinders and capsules whose diameter/length ratio is 1/2. Such particles have interest as simulants of aerosolized spores. We analyze and compare their two-dimensional scattering patterns. The effect of corrugated surfaces of relatively small amplitude is quite significant on the resulting scattering patterns.     

Multiphoton detachment from a zero-range potential revisited

Direct detachment of an electron bound by a regularized zero-range potential is reinvestigated. For the definition of the S-matrix amplitude for detachment, the wave function of the electron after the laser pulse has passed through is projected on an exact scattering state of the zero-range potential rather than a plane-wave state, which is normally used. The exact scattering state generates an additional contribution to the amplitude, which leads to an isotropic term in the angle-resolved energy spectrum.     

A new approach to potential scattering

An approximate integral-representation of theS-matrix in partial-wave expansion is derived for a scalar Schrödinger particle in a central field. The method consists of linearizingCalogero's Riccati equation for the interpolatingS-matrix in such a way that the solution of the linearized equation deviates as little as possible from the exact one. TheS-matrix thus obtained exhibits exact crossing-symmetry and uniform convergence independent of the coupling constant of the scattering potential. In the weak coupling limit it is especially shown thatour method is more accurate than the second Born approximation. In the second part of the paper we specialize ourS-matrix to low and large energies. At low energies, a general integral for the scattering length is obtained and at large energies the summation over all angular momenta is carried out yielding an expression for the scattering amplitude.     

Solving the relativistic inverse scattering problem with allowance for inelasticity effects on the basis of <Emphasis Type="Italic">N/D</Emphasis> equations and application of the resulting solution to an analysis of nucleon-nucleon interaction

A manifestly Poincaré-invariant approach to solving the inverse scattering problem is developed with allowance for inelasticity effects. The equations of the N/D method are used as dynamical equations in this approach. Two versions of the approach are considered. In the first version (method A), the required equations are constructed on the basis of the maximal-analyticity principle, which constitutes the basis of dynamical S-matrix theory. In formulating the second version of equations (method B), it is assumed that a partial-wave scattering amplitude may develop dynamical singularities that violate the requirement of maximal analyticity. The dynamics of interaction components that violate maximal analyticity is described within the model of a nonlocal separable potential. The method is used to analyze nucleon-nucleon interaction in the ¹S₀ and ³S₁ states. The results obtained by solving the inverse scattering problem for potential functions are compared with the predictions of the one-boson-exchange model.     

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.     

Scattering theory of three-body systems

The Faddeev amplitude is expressed in the N/D form in terms of the real reciprocal matrix K. The S-matrix is written in the unitary form (1 + iπK)S = 1 ?iπK. The Breit-Wigner formula for the three-body system including the break-up channel is derived. In the present method, the three-body problem is reduced to solve the eigenvalue problem for the real symmetric kernel.     

Comprehensive T-matrix reference database: A 2009–2011 update

The T-matrix method is one of the most versatile and efficient theoretical techniques widely used for the computation of electromagnetic scattering by single and composite particles, discrete random media, and particles in the vicinity of an interface separating two half-spaces with different refractive indices. This paper presents an update to the comprehensive database of peer-reviewed T-matrix publications compiled by us previously and includes the publications that appeared since 2009. It also lists several earlier publications not included in the original database.