The relationship between formal scattering theory and the density matrix

A study has been carried out on the relationship between formal scattering theory and the density matrix formalism. The density operator is developed in terms of the scattering operator S and the observable square modulus matrix elements of S are shown to be equivalent to elements of the density matrix. The Møller wave operators are similarly treated and subsequently used in obtaining the density matrix expression for the transition matrix. Finally, using the latter, it is shown that the hierarchy of approximations to the density matrix yields equivalent results to those obtained from the Born series.     

Transfer-matrix for the two-channel scattering problem

The transfer-matrix for the two-channel scattering problem is obtained. The elements of this matrix are expressed in terms of transmission T ₁, T ₂ and reflection R ₁, R ₂ amplitudes. The transfer-matrix for N localized and nonoverlapping scattering centers is presented. Recurrent equations for matrix elements are derived and initial conditions for them are defined.     

Derivation of extended boundary condition method from general definition of -matrix elements and Lippman–Schwinger equation for transition operator

Waterman's surface-integral expressions for the T-matrix elements are derived on the basis of the quantum-mechanical potential scattering approach in electromagnetic scattering problem. We use general definition of the elements of the T-matrix as the matrix elements of the dyadic transition operator and Lippman–Schwinger volume integral equation for the dyadic transition operator. The interrelations of the Q- and Waterman's T-matrix with the transition operator are shown.     

Allowing for Coulomb effects in the effective range expansion for two coupled channels

A Coulomb-modified matrix of scattering amplitudes (an [(F)\tilde]\tilde F matrix) is considered for the case of two coupled channels of elastic scattering of charged particles with different orbital angular momenta (l ₁ and l ₂ = l ₁ + 2). Matrix elements of the [(F)\tilde]\tilde F matrix are expressed in terms of the matrix elements of a [(K)\tilde] ^{- 1}\tilde K^{ - 1} matrix inverse to a modified reaction K matrix. The elements of the [(K)\tilde] ^{- 1}\tilde K^{ - 1} matrix are written as expansions that are generalizations of single-channel effective range expansion with allowance for the Coulomb interaction. If a system of colliding particles involves a bound state, the analytic continuation of these expansions into the region of negative energies makes it possible to obtain both the position of the pole corresponding to the bound state and the scattering amplitude residues in this pole, in terms of which the corresponding vertex constants and asymptotic normalization coefficients are expressed.     

Effective-range expansion for two coupled channels and properties of bound states

The S matrix and the scattering-amplitude matrix (F matrix) are considered for the case of two coupled elastic-scattering channels differing by the values of the orbital angular momentum (l ₁ and l ₂ = l ₁ + 2). The matrix elements of the S and F matrices in the absence of Coulomb interaction are expressed in terms of the matrix elements of the matrix K ⁻¹ inverse to the reaction K matrix. The elements of the K ⁻¹ matrix are written in the form of expansions that are generalizations of the single-channel effective-range expansion. If there is a bound state in the system of colliding particles, then an analytic continuation of these expansions to the region of negative energies makes it possible to obtain both the position of the pole corresponding to this bound state and the residues of scattering amplitudes at this pole, the respective vertex constants and asymptotic normalization coefficients being expressed in terms of these residues. By way of example, the developed formalism is applied to describing triplet neutron-proton scattering.     

强场中NO分子回归谱的长程散射矩阵的理论研究

采用半经典散射矩阵方法研究外磁场中高里德伯态双原子分子在能量范围为77010—77050cm^-1的回归谱.通过引进模型势简化强磁场中NO分子的高里德伯电子的势函数,找出其在核转动量子数分别为N=1,3,5的三个通道中的闭合轨道,重点分析了强磁场中NO分子的长程散射矩阵元实部的傅里叶变换谱与闭合轨道之间的一一对应关系.     

Experimental determination of scattering matrices of dust particles at visible wavelengths: The IAA light scattering apparatus

We present a new apparatus for measuring the complete scattering matrix as a function of the scattering angle of dust irregular particles. The design is based on the well-known apparatus located in Amsterdam, The Netherlands. In this improved version we have extended the scattering angle ranging from 3° to 177°. Moreover, the measurements are performed with a tunable argon–krypton laser that emit at a wavelength (λ) of 483, 488, 520, 568, or 647 nm. The apparatus has been developed at the Instituto de Astrofísica de Andalucía (IAA), Granada, Spain. To measure the scattering matrix elements we use a number of different optical components such as polarizers, a quarter-wave plate, and an electro-optic modulator. These components are used to manipulate the polarization state of light. By using eight different combinations for the orientation angles of the optical components, all scattering matrix elements are obtained as functions of the scattering angle. The accuracy of the instrument is tested by comparing the measured scattering matrices of water droplets at 488, 520, and 647 nm with Lorenz–Mie calculations for a distribution of homogeneous water droplets.     

On the symmetries between neutrino and antineutrino-nucleon elastic scattering

Various symmetry relations developed between neutrino-neutron and antineutrino-proton elastic scattering cross sections are surveyed and an identity between scattering amplitudes and a symmetry between cross sections of these processes established by consideringCPT andG conjugation invariance of current matrix elements. A symmetry is obtained giving rise to a theorem on the nature of contribution of form factors to terms in the cross sections.     

Photon emission induced by elastic exciton-carrier scattering in semiconductor quantum wells

We present a study of the elastic exciton-electron (X-e⁻) and exciton-hole (X-h) scattering processes in semiconductor quantum wells, including fermion exchange effects. The balance between the exciton and the free carrier populations within the electron-hole plasma is discussed in terms of ionization degree in the nondegenerate regime. Assuming a two-dimensional Coulomb potential statically screened by the free carrier gas, we apply the variable phase method to obtain the excitonic wavefunctions, which we use to calculate the 1s exciton-free carrier matrix elements that describe the scattering of excitons into the light cone where they can radiatively recombine. The photon emission rates due to the carrierassisted exciton recombination in semiconductor quantum-wells (QWs) at room temperature and in a low density regime are obtained from Fermi’s golden rule, and studied for mid-gap and wide-gap materials. The quantitative comparison of the direct and exchange terms of the scattering matrix elements shows that fermion exchange is the dominant mechanism of the exciton-carrier scattering process. This is confirmed by our analysis of the rates of photon emission induced by electron-assisted and hole-assisted exciton recombinations.     

Calculation of the minimum scattering field of an arbitrary antenna by its directivity pattern

A rigorous statistically closed expression has been obtained to calculate the elements of the polarization scattering matrix S _p which describes the quadratically minimum scattering field of an arbitrary passive matched antenna. The elements S _p are analytically expressed only through the complex vector pattern of the antenna.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 39, No. 8, pp. 1026–1035, August, 1996.     

Electronic transmission through a ladder with a single side-attached impurity

We study the electronic transmission of a model system composed by two coupled chains with an impurity attached to one of them. Analytical espressions for the transmittivity and for the diagonal and the off-diagonal Green’s function matrix elements are derived. Green’s function behaviour as function of the charge carrier energy is exploited to interpret the system transmittivity calculated by the scattering matrix formalism. We find that while a single substitutional impurity in the ladder may generate a Fano resonance in the transmittivity in the lower or in the higher energy part of the spectrum, in the case of a single side-attached impurity to the ladder, a resonance is found in each energy region. We interpret such resonances in terms of local density of states and off-diagonal Green’s function matrix elements.     

Studies on the vibrational excitation differential cross sections of non-resonant e-N2 scattering using augmented polarization potentials

Distributed spherical Gaussian (DSG) correlation-polarization model potentials with higher-order terms and exact exchange effects in single-configuration Slater determinant are taken into account for low-energy vibrational excitation e-N₂ scattering system. The integrodifferential coupled channel equations are solved using a combination of linear-algebraic and R-matrix-propagator algorithms. Analytic Born completion is used to calculate high-order scattering matrix elements in order to obtain convergent differential cross sections. The energy range is set to 4–15 eV which is not tested by the present theoretical method before. The overall agreement of theoretical results with the latest experiments emphasizes the importance of higher-order correlation-polarization potentials and rigorous exchange effects in vibrational excitation scattering.     

Analysis of Millimeter Wave Scattering by the Infinite Plane Metallic Grating Using PCG-FFT Technique

In this paper, both fast Fourier transformation (FFT) and preconditioned CG technique are introduced into method of lines (MOL) to further enhance the computational efficiency of this semi-analytic method. Electromagnetic wave scattering by an infinite plane metallic grating is used as the examples to describe its implementation. For arbitrary incident wave, Helmholz equation and boundary condition are first transformed into new ones so that the impedance matrix elements are calculated by FFT technique. As a result, this Topelitz impedance matrix only requires O(N) memory storage for the conjugate gradient FFT method to solve the current distribution with the computational complexity O(N log N) . Our numerical results show that circulate matrix preconditioner can speed up CG-FFT method to converge in much smaller CPU time than the banded matrix preconditioner.     

The description of reflection coefficients of the scattering problems for finding solutions of the Korteweg–de Vries equations

The results of inverse scattering problem associated with the initial-boundary value problem (IBVP) for the Korteweg–de Vries (KdV) equation with dominant surface tension are formulated. The necessary and sufficient conditions for given functions to be the left- and right-reflection coefficients of the scattering problem are established. The time-dependence t, t > 0 of each element of the scattering matrix s(k,t) is found in respective sector of the k-spectral plane by expansion formulas which are constructed from the known initial and boundary conditions of the IBVP. Knowing the right-reflection coefficient calculated from the elements of s(k,t), we solve the Gelfand–Levitan–Marchenko (GLM) equation in the inverse problem. Then the solution of the IBVP is expressible through the solution of the GLM equation. The asymptotic behavior at infinity of time of the solution of the IBVP is shown     

Elastic scattering of 81 keVγ rays by aluminium,nickel, tantalum,gold and lead

A recently reported study [Phys. Rev. A49, 3664 (1994)] of elastic scattering of 81 keVγ rays in the angular range from 60° to 133° has been extended to smaller and larger angles. Previously reportedS matrix calculations of atomic Rayleigh scattering have been shown to require a subtraction of contributions from spurious resonances. Most of the experimental data are in agreement with the calculations. Calculations (MF + ASF) based on a combination of relativistic modified form factors (MF’s) and angle independent anomalous scattering factors (ASF’s) are found to be inadequate for an explanation of experimental cross-sections in the case of highZ elements at angles larger than about 120°.     

Positron-hydrogen charge-exchange scattering

The positron hydrogen charge-exchange problem has been investigated using the three-particle scattering formulation. 1s and 2s intermediate states of hydrogen atom have been taken into account for calculation of second order matrix elements. The effect of 2s term is to decrease total cross-section as compared to the results involving 1s state only. The total cross-sections results are compared with results of other calculations in the energy region 1.5 to 10 Ry.     

Reaction matrix for nonlocal potentials. I. Magic nuclei

The binding energies of⁴He,¹⁶O, and⁴⁰Ca are computed applying the Brueckner theory of the reaction matrix for the Tabakin and the Mongan nonlocal separable potentials. Binding energies (10 MeV/nucleon) obtained are substantially greater than similar values for the local Hamada-Johnston potential (approx. 4÷5 MeV/nucleon). Detailed comparison of reference and exact reaction matrices for both types of potentials (local and nonlocal) in the coupled³ S-³ D channel confirms the existence of a strong correlation between the magnitude of the matrix elements of the reaction matrix and the intensity of a tensor force. A similar correlation also exists in the¹ S ₀ channel between the magnitude of matrix elements of the reaction matrix and the singlet scattering length. A different off-shell behaviour of individual potentials plays also a certain role.     

Analytical Solution of Partial-Wave Faddeev Equations with Application to Scattering and Statistical Mechanical Properties

In this work, the Faddeev equations for three-body scattering at arbitrary angular momentum are exactly solved and the transition matrices for some transition processes, including scattering and rearrangement channels are formulated in terms of free-particle resolvent matrix. A generalized Yamaguchi rank-two nonlocal separable potential has been used to obtain the analytical expressions for partial wave scattering properties of a three-particle system. The partial-wave analysis for some transition processes in a three-particle system is suggested. The partial-wave three-particle transition matrix elements have been constructed via knowledge of the matrix elements of the free motion resolvent.The calculation of a number of scattering properties of interest of the system such as transition matrix and its poles(bound states and resonances) and consequently other related quantities like scattering amplitudes, scattering length,phase shifts and cross sections are feasible in a straightforward manner. Moreover, we obtain a new analytical expression for the third virial coefficient in terms of three-body transition matrix.     

Analytical Solution of Partial-Wave Faddeev Equations with Application to Scattering and Statistical Mechanical Properties

In this work, the Faddeev equations for three-body scattering at arbitrary angular momentum are exactly solved and the transition matrices for some transition processes, including scattering and rearrangement channels are formulated in terms of free-particle resolvent matrix. A generalized Yamaguchi rank-two nonlocal separable potential has been used to obtain the analytical expressions for partial wave scattering properties of a three-particle system. The partial-wave analysis for some transition processes in a three-particle system is suggested. The partial-wave three-particle transition matrix elements have been constructed via knowledge of the matrix elements of the free motion resolvent. The calculation of a number of scattering properties of interest of the system such as transition matrix and its poles (bound states and resonances) and consequently other related quantities like scattering amplitudes, scattering length, phase shifts and cross sections are feasible in a straightforward manner. Moreover, we obtain a new analytical expression for the third virial coefficient in terms of three-body transition matrix.     

Influence of the scattering matrix elements on the gain of Stokes and anti-Stokes components in the nonstationary regime

The influence of the scattering matrix elements that are responsible for the amplification of the Stokes and anti-Stokes components on the process of stimulated Raman scattering in the essentially nonstationary regime is analyzed. It is demonstrated that the optimum proportion between the scattering matrix elements, which maximizes the anti-Stokes component, is determined by the ratio between the optical frequencies of the components and the initial proportion between the intensities of the pump and initiating pulses. In the case of the optimum proportion between the scattering matrix elements, the variation in the excitation of a medium compensates the self-induced modulation of gain, and the efficiency of converting the radiation energy into the Stokes and anti-Stokes components increases substantially.