Semiclassical and quasiclassical calculation of reaction probabilities for collinear X + F2→XF + F (X = Mu,H, D,T)

Semiclassical calculations of reaction probabilities have been carried out for the collinear H + F₂ (n = 0, 1) reaction using the best extended LEPS surface No. II of Jonathan et al. Both real and complex valued classical trajectories have been included in the calculations for an energy range where the quasiclassical total reaction probability is unity. Comparison with quantum results shows the semiclassical reaction probabilities are accurate to about ± 0·05 provided only two real or complex stationary phase points make an important contribution to the S matrix element, so that the uniform Airy or integer Bessel approximations are valid. Real semiclassical calculations are also reported for the collinear Mu, D, T + F₂ (n = 0) reactions. For the D and T reactions, the semiclassical reaction probabilities are estimated to be accurate to ± 0·05, except close to the reaction threshold, but for the Mu reaction the estimated errors are much larger. In addition, quasiclassical calculations for the reaction probabilities have been carried out using half integer boxing and smooth sampling methods to quantize the product distributions. For the H + F₂ reaction, there are usually systematic deviations from the quantum reaction probabilities and the same is expected to be true for the Mu, D and T reactions.     

Analysis of data on proton-proton scattering in the energy range 100–1300 MeV

The results of a partial-wave analysis of data on proton-proton interaction in the energy range 100–1300 MeV are presented. The real parts of phase shifts were found for states of orbital angular momentum up to L = 9, while their imaginary parts were determined for states of orbital angular momentum up to L = 5. The sixth parameter of the S matrix was introduced in order to describe states of total angular momentum J = 2 and 4. The inelasticity thresholds were chosen individually for each state and were found to be substantially different from one another. The resulting solution was characterized by χ ² = 1.155 per point in the case where the number of experimental data was 12 841 and by a large imaginary part of the phase shift in the ³ P ₂-wave state at the edge of the energy range. Special features of the interaction in orbital states are discussed along with the energy dependence of integrated amplitudes and amplitudes of the scattering matrix at zero angle.     

Complete Analysis of the ηN S-Wave Scattering-Length Values and Its Natural Limitations in Any Single-Resonance Model

We show that the S-wave ηN scattering length can be extracted in a model-independent way only if the restricting assumption is adopted that one resonance describes the behaviour of the S-wave entirely. To obtain the ηN S-wave scattering length in a model-independent way one needs only two wellknown quantities; the πN elastic S-wave T-matrix at the η-production threshold, and the near threshold π ⁻ p →ηn total cross section slope. The results are independent of the particular parametrization of the elastic πN S-wave T-matrix and of the number of channels used. These assumptions are more general than the assumptions of the existing single-resonance models employed up to now for extracting the ηN S-wave scattering length. We show that the ηN S-wave scattering length value of other single-resonance models agrees with the model-independent estimate if the input data agree with the commonly accepted values. The existence of the upper limit of the real part and the almost fixed value of the imaginary part of the ηN S-wave scattering length are demonstrated, and the nature of limitations originating directly from first principles is explained as the reduction of the full model to the single resonance in the S-wave. The expected differences from the single-resonance estimate for models that are more general is shown. A simple criterion for recognizing the importance of parts of the model that are added in addition to the single resonance in the S-wave is given as well. Received May 15, 1995; revised September 14, 1995; accepted for publication October 11, 1995     

Quasiclassical Hamiltonians for large-spin systems

We extend and apply a previously developed method for a semiclassical treatment of a system with large spin S. A multisite Heisenberg Hamiltonian is transformed into an effective classical Hamilton function which can be treated by standard methods for classical systems. Quantum effects enter in form of multispin interactions in the Hamilton function. The latter is written in the form of an expansion in powers of J/(TS), where J is the coupling constant. Main ingredients of our method are spin coherent states and cumulants. Rules and diagrams are derived for computing cumulants of groups of operators entering the Hamiltonian. The theory is illustrated by calculating the quantum corrections to the free energy of a Heisenberg chain which were previously computed by a Wigner-Kirkwood expansion. Received 5 May 1999 and received in final form 24 September 1999     

Semiclassical approach to time-dependent tunnelling

We present an exact time-dependent semiclassical formulation of the dynamics of a -atom subjected to an oscillating electric field. Through a simple approximation the results of Ergenzinger's more intuitive analysis are obtained. We also comment on the important role played by the imaginary tunnelling time , which is quite distinct from the usual adiabatic tunnelling time. Received: 24 April 2001 / Revised version: 15 May 2001 / Published online: 3 August 2001     

Elastic scattering of 11B from 209Bi in the energy range 49.8 ⩽ E ⩽ 84.1 MeV

The elastic scattering of ¹¹B from ²⁰⁹Bi has been measured at laboratory energies of 49.8, 51.3, 52.2, 52.8, 54.3, 55.8, 59.8, 64.8, 69.8, 74.8 and 84.1 MeV. These data have been analyzed using a microscopic optical model and the energy dependence of the real and the imaginary parts of the optical potential at near barrier energies has been determined. The “threshold anomaly” observed in the real part of the potential is found to be consistent with the dispersion relation which connects the real and the imaginary parts of the potential. Inelastic scattering and transfer reactions have also been measured at energies of 51.3, 55.8, 59.8 and 74.8 MeV. DWBA calculations for the 3⁻ state in ²⁰⁹Bi are made. From the measured transfer probabilities, using a semiclassical approach the strength of the form factors have been obtained. The fusion cross sections have been derived at these energies from the corresponding quasi-elastic scattering angular distribution data. The fusion cross sections calculated using the energy dependent barriers extracted from the energy dependent real parts of the potential compare well with, that determined from quasi-elastic scattering data and are also in good agreement with simplified coupled channels calculation for fusion incorporating important inelastic and transfer channels.     

High frequency properties of resonant tunneling diode

The small signal analysis for the resonant tunneling diode (RTD) is carried out by using a semiclassical transport theory. Multiple scattering effects are accounted for in an optical approximation by using a complex mean free path. An analytical expression for the conduction current is given. The results show that the negative differential conductance prevails up to the frequency f₀ limited by the quantum well transit time. The imaginary part of the admittance can be presented by a series inductance as has been recently found experimentally. In addition, the equivalent circuit has a capacitor in parallel with the conductance-inductance branch. Above f₀ the admittance shows an oscillatory behaviour. The oscillations are associated with the quantum well transit time resonances.     

Pion-nucleus optical potential: The pion-absorption contribution

The real and imaginary parts of pion-nucleus optical potential arising from pion absorption channel have been computed. A two-nucleon model of pion absorption which includesπ andρ rescattering andS-wave interaction has been used. The effects of short-range nucleon-nucleon correlations, Pauli blocking and formfactors have been included. The threshold values of imaginary absorption potential are reasonably close to density-squared terms of phenomenological potentials. The real part ofP-wave potential is attractive and that ofS-wave potential is weakly attractive at lower pion energies and changes sign as pion energy is increased. The calculation shows that the real part of absorption is significantly affected by short-range correlations and Pauli-blocking.     

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.     

Calculation of accurate imaginary frequencies and tunnelling coefficients for hydrogen abstraction reactions using IRCmax

The effect of level of theory on the imaginary frequency and corresponding tunnelling coefficients has been studied for a test set of hydrogen abstraction reactions: ?CH₂X + CH₃Y → CH₃X + ?CH₂Y for (X,Y) = (H,H), (H,CN), (H,F), (H,Li) and (F,Li). It is found that the imaginary frequency is very sensitive to the level of theory used, with Hartree-Fock (HF) methods severely overestimating the imaginary frequency compared with high-level CCSD(T)/6-311G(d,p) calculations. The errors for the other methods are smaller but nonetheless significant, with MP2 methods overestimating the imaginary frequency and density functional theory (DFT) methods underestimating it. In the case of the HF methods, this leads to errors in the tunnelling coefficient of several orders of magnitude, while for the better DFT and MP2 methods errors of a factor of 2–3 are observed. To address this problem, an IRCmax procedure for estimating the imaginary frequency has been developed and it is found that IRCmax imaginary frequencies calculated with CCSD(T)/6-311G(d,p) single points along a low-level HF/6-31G(d) minimum energy path provide excellent approximations to the high-level values, at a fraction of the computational cost.     

Degeneracy of Resonances: Branch Point and Branch Cuts in Parameter Space

The rich phenomenology of crossings and anticrossings of energies and widths, observed in an isolated doublet of resonances when one control parameter is varied, is fully explained in terms of the topological properties of the energy hypersurfaces close to the degeneracy point. The hypersurface representing the complex resonance eigenvalues, as functions of the control parameters, has an algebraic branch point of rank one, and branch cuts in its real and imaginary parts, in parameter space. Associated with this singularity in parameter space, the scattering matrix, S _ℓ (E), and the Green’s function, G _ℓ ⁽⁺⁾(k; r,r'), have one double pole in the unphysical sheet of the complex energy plane. We characterize the universal unfolding or deformation of any degeneracy point of two unbound states in parameter space by means of a universal 2-parameter family of functions which is contact equivalent to the pole position function of the isolated doublet of resonances at the exceptional point and includes all small perturbations of the degeneracy condition up to contact equivalence.     

Z 4-symmetric factorizedS-matrix in two space-time dimensions

The factorizedS-matrix with internal symmetryZ ₄ is constructed in two space-time dimensions. The two-particle amplitudes are obtained by means of solving the factorization, unitarity and analyticity equations. The solution of factorization equations can be expressed in terms of elliptic functions. TheS-matrix contains the resonance poles naturally. The simple formal relation between the general factorizedS-matrices and the Baxter-type lattice transfer matrices is found. In the sense of this relation theZ ₄-symmetricS-matrix corresponds to the Baxter transfer matrix itself.     

Semiclassical theory of integrable and rough Andreev billiards

We study the effect on the density of states in mesoscopic ballistic billiards to which a superconducting lead is attached. The expression for the density of states is derived in the semiclassical S-matrix formalism shedding light onto the origin of the differences between the semiclassical theory and the corresponding result derived from random matrix models. Applications to a square billiard geometry and billiards with boundary roughness are discussed. The saturation of the quasiparticle excitation spectrum is related to the classical dynamics of the billiard. The influence of weak magnetic fields on the proximity effect in rough Andreev billiards is discussed and an analytical formula is derived. The semiclassical theory provides an interpretation for the suppression of the proximity effect in the presence of magnetic fields as a coherence effect of time reversed trajectories. It is shown to be in good agreement with quantum mechanical calculations. Received 21 August 1999 and Received in final form 21 March 2001     

Phase-function method for complex potentials

We consider the scattering problem for absorptive interactions within the framework of phase-function method. A Green’s function approach is used to derive the phase equation. As a case study we apply the algorithm presented on a shallow α-α potential, the real and imaginary parts of which have been deduced from experimental data. The real and imaginary parts of theS-wave phase shift are found to vary smoothly with energy while those forD andG waves show some fluctuations in the low-energy region. It is shown that studies in spatial behaviour of the phase function provide a plausible explanation for the dynamical origin of these fluctuations.     

Generalised ABCD matrix treatment for laser resonators and beam propagation

A generalised ABCD matrix treatment for laser resonators and beam propagation is developed for computer programming. In this treatment, imaginary parts are introduced into the matrixes for all optical elements, and the beam quality factor M² and the index of the medium are also taken into account. When the imaginary parts of the complex matrix are zero and the beam quality factor M² and medium index are unity, the stated method is transformed back into the more classic ABCD matrix format in which the fundamental-mode Gaussian beam transmits through real elements in the vacuum. Based on this method, laser resonator software is realised by the VB programme language. The software can be used to analyse and design stable/unstable standing/travelling cavities, phase-conjugate cavities and beam transformations.     

Associative ionization in slow collisions of atoms

An analysis of the modern theory of associative ionization (AI) is performed, and ways of its further development are discussed. The threshold behavior of the cross section of endothermic AI reactions is considered and it is shown that, in the quantum case, its dependence on the above-threshold energy E is strictly linear, i.e., significantly different from the E ^3/2 law ensuing from the semiclassical theory. This has a simple explanation, since the matrix elements of the scattering operator are finite at E = 0 due to the tunneling effect. The possibility of describing the dynamics of an elementary AI event within the framework of the diffusion approximation is substantiated, and the conditions of applicability of the theory of “quantum chaos” to treating the spectrum of highly excited Rydberg molecules are examined. The quantum properties of the exothermic processes of AI and Penning ionization in the case of the states of the interacting atoms being autoionizing are discussed in detail. A comparison with the semiclassical theory is presented.     

Spin exchange during collisions of potassium atoms: Complex cross sections

Singlet (X ¹Σ⁺) and triplet (a ³Σ⁺) potentials of interaction of two potassium atoms residing in the ground state (4s ² S _1/2) are presented. Based on the given interaction potentials, the complex cross sections of spin exchange q = [`(q)] + i[`([`(q)])]q = \bar q + i\overline{\overline q} for the system under investigation are calculated. Obtained dependences of the real and imaginary parts of the spin-exchange cross section on temperature allow one to obtain information both on the broadening of a magnetic resonance line of K atoms and on the frequency shift of the magnetic resonance during collision.     

Giant Gamow-Teller resonance: 40 Years after the prediction

A brief survey of the physics of the giant Gamow-Teller resonance, from the history of its prediction and the first attempts at theoretically describing it to the present-day state of this realmof nuclear physics, is given. The structure of Gamow-Teller resonances is analyzed within the theory of finite Fermi systems. Simple expressions for the energy of the Gamow-Teller resonance and for the matrix elements of its excitation are derived via approximately solving the equations of this theory by the semiclassicalmethod. The calculated values of the energy difference between the Gamow-Teller resonance and the analogous resonance are found to be in good agreement with their experimental counterparts. Strength functions for the beta decay of neutron-rich nuclei, S _β (E), are analyzed, and the calculated values of S _β (E) are contrasted against relevant experimental data.