A three-parameter analytic phase function for multiple scattering calculations

A very simple procedure has been developed to fit the first three moments of an actual phase function with a three parameter analytic phase function. The exact Legendre Polynomial decomposition of this function is known which makes it quite suitable for multiple scattering calculations. The use of this function can be expected to yield excellent flux values at all depths within a medium. Since it is capable of reproducing the glory, it can be used in synthetic spectra computations from planetary atmospheres. Accurate asymptotic radiance values can also be achieved as long as the single scattering albedo ω₀ ?0.9.     

Regge pole analysis of large angle elastic scattering in chemically reactive systems

Complex angular momentum techniques are applied to the theory of elastic scattering in chemically reactive systems. The Watson-Sommerfeld transformation allows the scattering amplitude to be written as a background integral and a sum over Regge Poles in the right-hand complex plane. It is shown the Regge Poles are restricted to the first quadrant for physically meaningful optical potentials. Neglect of the background integral and all poles except for the one closest to the real axis allows the differential cross section for large scattering angles to be written in terms of a Legendre function of complex degree. The theory agrees with experimental results for the elastic angular distributions of direct and complex reactions. It predicts oscillations and a glory in the backward direction of the angular distributions. The oscillations arise from interference between orbiting or surface waves that propagate around the region of chemical reaction. Measurements of angular distributions at different collision energies can be used to determine a Regge trajectory.     

Eikonal description of quantal scattering

Elastic and inelastic quantal scattering is described by a theory in which the contribution of a range of impact parameters to the scattering amplitude is determined by a phase integral (“eikonal”) which is integrated along a real curved “quantal” trajectory. This amplitude reduces to the Glauber expression in the high-energy, forward-angle limit, and to the usual semiclassical amplitude in the classical limit. The formulation can be applied to the study of heavy-ion scattering. The quantal trajectories are investigated analytically for the case of Coulomb scattering. A numerical analysis of elastic ¹⁶O¹⁶O scattering is carried out. The results show appreciable improvement as compared with the Glauber approximation.     

Theory of multiple scattering

Consideration is given to problems of obtaining exact and approximate solutions of kinetic equations in the multiple scattering problem. For cross sections which are rational functions of χ² (χ = 2sin(δ/2), δ is the scattering angle) exact solutions are obtained as a series in terms of Legendre polynomials. The limits of validity of the kinetic equation for the distribution function in terms of the variable q = 2sin(?/2) are refined [1] and the solutions of this equation are compared with the exact solutions of the Rutherford and Mott cross sections. The problem of convergence of approximate solutions in the form of a series in terms of Legendre polynomials and a series in powers of 1/B is solved. These approximations are obtained and their limits of validity are determined.     

Light scattering,origin invariant multipole moments and molecular property tensors: the case of electric-field-gradient-induced birefringence

A comparison of semiclassical and quantum versions of molecular light scattering theory at finite temperatures is presented. A general formulation of the semiclassical radiation model is developed to the point where its relationship to the corresponding QED formalism can be established: the classical scattered electric field is proportional to the same R-matrix element as that obtained from QED for the photon scattering amplitude. The result is valid for non-resonant scattering at T = 0. The semiclassical theory conventionally also inherits aspects of a classical molecular model, principally origin-dependent molecular multipole moments. Origin independent multipoles, and corresponding response functions can be defined if the theory is cast in terms of centre-of-mass and translation invariant internal coordinates. Such a choice of coordinates brings molecular light scattering theory into line with the theory of the molecular Schrödinger equation. This is illustrated for the case of a diatomic molecule. A specific application of these results of current interest is electric-field-gradient induced birefringence (EFGB) for which there are four competing theories in the literature. In this paper we examine the treatment of finite temperature effects in two semiclassical accounts of EFGB in polar molecules and identify a likely source of the discrepancy between them revealed in a recent ab initio computational study.     

Nearside—farside theory of elastic angular scattering for strongly absorptive collisions

A systematic investigation has been made of ways for combining the method of Yennie, Ravenhall and Wilson (YRW) for the resummation of a partial wave Legendre series with the nearside—farside (NF) angular decompositions of Fuller and Hatchell. Of the resulting four procedures, the most useful one first applies the YRW resummation method m times (m = 0,1,2,…) to the partial wave scattering amplitude, followed by the Fuller NF decomposition. This NF procedure performs best for the physical interpretation of structure in the angular scattering of strongly absorptive elastic collisions. Numerical results from all four NF procedures are reported for a partial wave series possessing a simple parametrized scattering matrix element. The theory and calculations are relevant to atom—atom and atom—molecule elastic scattering in the presence of other open channels, for example, chemical reaction.     

Light atom scattering from adsorbates at low coverage

A multiple scattering theory is developed for the scattering of light atoms from a disordered adsorbate on a smooth surface. Using gas phase potentials for adatom potentials, excellent agreement is found with data for He scattering from adsorbed Xe and surprisingly good agreement with the extensive data for He scattering from adsorbed CO. The attractive adatom potential must be included in the calculation if quantitative comparison with experimental data is to be made. An important contribution to the total cross section of an adatom are quantum mechanical oscillations similar to glory oscillations in gas phase scattering. The dependence of the total cross section on the incident angle is strongly influenced by the varying number of adatoms seen by the probe.     

The scattering of quantized solitons in the non-linear Schrödinger theory

The scattering of quantized solitons in non-linear Schrödinger theory is treated using the collective coordinate method of Gervais, Jevicki and Sakita. The phase shift for soliton-soliton scattering is calculated up to the one-loop level. We find that the quantum correction vanishes. This result coincides in the first two terms of an expansion in $\text{h}\text{?}$ with the exact amplitude calculated from a quantum mechanical N-body problem.     

Dirac-delta function approximations to the scattering phase function

Dirac-delta function approximations are used to represent the single scattering phase function of large spherical particles or voids. The phase function for a spherical particle or void can be represented by a series of Legendre polynomials; however, as the diameter is increased, forward scattering becomes dominant and the number of terms in the series becomes very large. A Dirac-delta function approximation consists of a Dirac-delta function in the forward direction plus a finite series of Legendre polynomials. The Dirac-delta function accounts for strong forward scattering. Particular attention is given to large ice spheres and spherical voids in ice. The Dirac-delta function is shown effective in reducing the number of terms needed to describe the phase function.     

Doubly uniform semiclassical quantization formula for resonances

The radial Schrödinger equation with an effective potential containing a single well and a single barrier is treated with an improved uniform semiclassical method. The improved quantization formula for complex energies (or resonances) contains a correction term that originates from a uniform treatment of the classically forbidden region near the origin in addition to the more familiar uniform treatment of the barrier region. In the present case the origin has a second-order pole, due to the centrifugal barrier potential term, and/or a Coulomb-type singularity, and these terms dominate the region inside the innermost classical turning point.Numerical results for first-order and third-order approximate complex resonance energies are compared with those of a standard (first- and third-order) barrier-uniform semiclassical method and also with those of ‘exact’ numerical computations.The improved quantization formula provides results in significantly better agreement with the exact results as the angular momentum quantum number l approaches zero.     

Quantum-chaotic scattering effects in semiconductor microstructures

We show that classical chaotic scattering has experimentally measurable consequences for the quantum conductance of semiconductor microstructures. These include the existence of conductance fluctuations-a sensitivity of the conductance to either Fermi energy or magnetic field-and weak-localization-a change in the average conductance upon applying a magnetic field. We develop a semiclassical theory and present numerical results for these two effects in which we model the microstructures by billiards attached to leads. We find that the difference between chaotic and regular classical scattering produces a qualitative difference in the fluctuation spectrum and weak-localization lineshape of chaotic and nonchaotic structures. While the semiclassical theory within the diagonal approximation accounts well for the weak-localization lineshape and for the spectrum of the fluctuations, we uncover a surprising failure of the semiclassical diagonal-approximation theory in describing the magnitude of these quantum transport effects.     

Simulation of carrier capture in quantum well lasers due to strong inelastic scattering

The effects of strong inelastic scattering on carrier transport over and capture into the quantum wells of quantum well lasers are simulated. In contrast to most semiconductor devices, strong scattering is beneficial to the operation of quantum well lasers. However, such strong inelastic scattering in nanostructures can be expected to produce intermediate degrees of phase coherence, limiting the applicability of both classical models, such as Bethe thermionic emission theory, and commonly used quantum mechanical treatments, such as Fermi's Golden Rule. Two computational approaches are demonstrated for simulating such transport with intermediate degrees of phase coherence. First, absorbing potentials are used within Schrödinger's equation to represent inelastic scattering. This simple approach both exhibits much of the essential physics of such transport and is computationally efficient. Then a more rigorous approach, Schrödinger equation (based) Monte Carlo (SEMC), is demonstrated. While SEMC is rigorously quantum mechanical, the numerical algorithm has more in common with semiclassical Monte Carlo methods than path integral-based quantum Monte Carlo methods. Both of these methods demonstrate nonlinear variations in carrier capture with variations in scattering, and the destruction of quantum resonances for transmission over the quantum well.     

Quantum chaotic scattering with a mixed phase space: the three-disk billiard in a magnetic field

We study the classical and semiclassical scattering behavior of electrons in an open three-disk billard in the presence of a homogeneous magnetic field, which is confined to the inner part of the scattering region. As the magnetic field is increased the phase space of the invariant set of the classical scattering trajectories changes from hyperbolic (fully chaotic) to a mixed situation, where KAM tori are present. The "stickiness" of the stable trajectories leads to a much slower decay of the survival probability of trajectories as compared to the hyperbolic case. We show that this effect influences strongly the quantum fluctuations of the scattering amplitude and cross sections.     

Semiclassical expansion of quantum characteristics for many‐body potential scattering problem

In quantum mechanics, systems can be described in phase space in terms of the Wigner function and the star‐product operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve the evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many‐body potential scattering problem simplifies to a statistical‐mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to the initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in ?. We present semiclassical expansion of quantum characteristics for many‐body scattering problem and provide tools for calculation of average values of time‐dependent physical observables and cross sections. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as non‐locality and coherence in propagation of particles, into the semiclassical transport models. We formulate the principle of stationary action for quantum Hamilton's equations and give quantum‐mechanical extensions of the Liouville theorem on conservation of the phase‐space volume and the Poincaré theorem on conservation of 2p‐forms. The lowest order quantum corrections to the Kepler periodic orbits are constructed. These corrections show the resonance behavior.     

Evaluation of multidimensional canonical integrals in semiclassical collision theory

The evaluation of the multidimensional canonical integrals that occur in the uniform asymptotic representations of the S matrix in the semiclassical theory of inelastic and reactive molecular collisions is considered. For the non-separable two-dimensional canonical integral considered earlier, an exact series expansion is obtained with the help of convergence factors. This method avoids the complex variable techniques used previously. The uniform asymptotic formulae derived by Miller and Marcus are discussed, and compared with the approach adopted in the present paper.     

The Weyl representation in classical and quantum mechanics

The position representation of the evolution operator in quantum mechanics is analogous to the generating function formalism of classical mechanics. Similarly, the Weyl representation is connected to new generating functions described by chords and centres in phase space. Both classical and quantal theories relie on the group of translations and reflections through a point in phase space. The composition of small time evolutions leads to new versions of the classical variational principle and to path integrals in quantum mechanics. The strong resemblance between the two theories allows a clear derivation of the semiclassical limit in which observables evolve classically in the Weyl representation. The restriction of the motion to the energy shell in classical mechanics is the basis for a full review of the semiclassical Wigner function and the theory of scars of periodic orbits. By embedding the theory of scars in a fully uniform approximation, it is shown that the region in which the scar contribution is oscillatory is separated from a decaying region by a caustic that touches the shell along the periodic orbit and widens quadratically within the energy shell.