Eikonal Theory Research Articles

Second-order relativistic electron capture

Starting with the impulse approximation, we analyse second-order effects in relativistic electron capture. The relation of this model with relativistic distorted-wave approximations is clarified. In particular it is shown that the second-order spin-coupling terms in the RCDW theory are consistent with the correct form given by perturbation theory. In the semirelativistic limit, the RCDW results are shown to accord with the formulae of Moiseiwitsch for flip and nonflip transitions in the ultra-relativistic limit. This confirms that the continuum-distorted-wave model generalises to relativistic spinors, and highlights the defects of scalar models. We also present a new symmetric eikonal theory which gives reliable results for capture without change of spin, but leads to a divergent total cross section for spin-flip transitions in the second-order term. This effect, which is quite distinct from the spurious spin-flip amplitudes of the scalar symmetric eikonal theory, is taken as further evidence that the eikonal approximation is not valid for magnetic transitions.

Eikonal theory and computer algebra

I show how computer algebra can be used to turn the study of aberrations from a thorny branch of lens theory into a practical design tool. After a discussion of the underlying mathematics two applications are shown: the well-known problem of a lens used at different magnifications and the analysis of an unusual afocal element.

The multichannel eikonal theory program for electron-atom scattering

The operation of the code MET_cross for the solution of the semiclassical multichannel eikonal theory for electron-atom scattering is described. Also described is a second code, MET_states, which utilizes the results produced by MET_cross to generate a complete set of state multipoles and coherence and alignment parameters needed to characterize the polarization properties of the radiation emitted in the decay of the metastable atomic states excited in the collision. Included also is a discussion of the relationship between the present codes used to solve the semiclassical multichannel eikonal theory and codes used to implement other semiclassical and quantal scattering theories of electron-atom scattering.

Automatic generation of analytical matrix elements for electron-atom scattering

A code which can be used to automatically generate entire sets of analytical matrix elements of the instantaneous electrostatic interaction potential between the projectile electron and the target atom is described. The code can be used at present for both hydrogen and helium target atoms. In the case of hydrogen, the present code Vij generalizes and extends an earlier code of Jamison [1]; while in the case of helium, Hartree-Fock frozen-core wave-functions are used to represent the two-electron wavefunctions. When an entire set of matrix elements is generated, a complete set of FORTRAN subroutines is produced which can be directly incorporated into the code MET_cross used to solve the multichannel eikonal theory (see accompanying paper). A principal application of the code Vij is to aid the elucidation of the systematic trends observed in transitions among metastable states in electron-atom collisions by studying the functional form of the underlying interaction matrix elements.

In-plane polarisation correlation study of the 31 D state of helium excited by 40 eV electrons

Electron-photon polarisation correlations measured in the scattering plane defined by the incident and scattered electron momenta are reported for 40 eV electron impact excitation of the 31D state of helium and for electron scattering angles in the range 40°≦ϑ≦120°. When combined with the results of our earlier polarisation analysis on radiation emitted perpendicular to the scattering plane, detailed information on the shape and dynamics of the exicted state is obtained. In the small angular range of overlap (40–60°) there is excellent agreement with a pervious experimental study. The behaviour of the linear polarisation of the radiation emitted in the scattering plane is well reproduced by a multichannel eikonal theory for ϑ≦80°. Otherwise theories totally fail to described the excitation process.

Energy transfer between surface and internal waves in the North Pacific Ocean

The energy transfer between internal and surface waves is formulated as a scattering process of surface waves by packets of internal waves. For weak internal wave fields, the Born approximation may be used, and this is shown to give the same transfer rate as the “modulation model” of the resonant triad theory for weak nonlinear interactions. For stronger internal wave fields, the eikonal theory of ray paths is used to calculate surface wave scattering. The theory is applied to historical Väisälä frequency profiles and wind data for the North Pacific ocean. Energy transfer rates from the internal wave to the surface wave field in the range of 10−4 W/m2 and higher are found, except during winter months at higher latitudes, where the rates are somewhat lower. For the weaker internal wave fields, the resonant triad theory (or, equivalently, the Born theory) must be used. For the strongest fields observed, the weak interaction theory is not valid and the interaction is described as ray path scattering. There is a region of overlap for which the two theoretical models give comparable results.

Pion-nucleus scattering and the {spdf} interacting boson model

An approach in which the eikonal approximation theory is combined with the {spdf} interacting boson model is proposed in an attempt to include nucleon correlations in the description of pion-nucleus scattering. The channel-coupling effect in the excitation of the nuclear collective state is taken into account in the approach. The calculated cross sections for the 180 MeV pi +or--152Sm scattering agree with experimental data quite well.

High-energy pion-nucleus elastic scattering.

We investigate theoretical approaches to pion--nucleus elastic scattering at high energies (300 $\le T_\pi \le$ 1 GeV). A ``model--exact'' calculation of the lowest--order microscopic optical model, carried out in momentum space and including the full Fermi averaging integration, a realistic off--shell pion--nucleon scattering amplitude and fully covariant kinematics, is used to calibrate a much simpler theory. The simpler theory utilizes a local optical potential with an eikonal propagator and includes the Coulomb interaction and the first Wallace correction, both of which are found to be important. Comparisons of differential cross sections out to beyond the second minimum are made for light and heavy nuclei. Particularly for nuclei as heavy as $^{40}$Ca, the eikonal theory is found to be an excellent approximation to the full theory. (8 Figures available from authors)

Electron-metastable-helium differential and integral cross sections

The differential and integral cross sections for the excitation of the 23P and the 33L (L identical to S, P and D) states of He from the metastable 23S state are calculated using the semiclassical multichannel eikonal theory with a nine-channel basis set. Comparison is made with the recent experimental results of Muller-Fiedler et al. (1984) and Rall et al. (1989) for the differential and integral cross sections, respectively. The agreement between the present multichannel eikonal results and the experimental data for the 33S and 33D differential cross sections is satisfactory. However, significant differences are noted between the experimental data and the present multichannel eikonal theory for the 23S to 23P, 33P differential cross sections.

Cylindrical sound wave fronts in acousto-optics: multiple, local, strong interaction

The results of computer simulation experiments concerning strong interaction with strongly curved wave fronts of sound are presented. Such interaction is found to occur locally, at successive points in the interaction region. This is in agreement with the eikonal theory of acousto-optics and its associated diffracted ray diagrams. Based on the latter, measures are presented for the separation between interaction points, their relative independence, and the effective interaction length. These measures are shown to accurately describe the simulation results. Diffraction efficiency for the various orders shows close correspondence with the eikonal theory of uncoupled local interactions. Following the discussion of theory and simulation, implications for wideband acousto-optic devices are considered.

On the eikonal theory of potential scattering in the presence of a low frequency laser

The author investigates the eikonal theory of charged-particle potential scattering in the presence of a laser when the collision time is much smaller than the laser period while the translational velocity of the particle is much larger than its classical oscillatory velocity in the field. The author's detailed numerical calculations show that for small scattering angles, previous eikonal results of the Kroll-Watson type, based on a stationary phase approximation (SPA), are inaccurate for differential cross sections and become worse as one increases the range and/or the strength of the potential. The author replaces the SPA by a different approximation, ideally suited to small-angle scattering and obtain excellent agreement with the 'exact' numerical results.

Eikonal form of the dynamic polarization potential and its application to the scattering of exotic nuclei

The eikonal theory of the dynamic polarization potential (DDP) is developed. Application to the scattering of loosely bound exotic nuclei is made. In particular, the effect of our DPP on the scattering of 11 Li + 12 C at 85 A · MeV is discussed.

The multichannel eikonal theory of electron-helium collisions. I. Excitation of He(11S)

Calculations of the integral and differential cross sections for the electronic excitation of the 21L and 31L states of helium are reported. The multichannel eikonal theory employing a 10-channel basis set is used. The lambda and chi parameters for the 21P and 31P states at 80 eV are also reported. A number of misprints concerning the original multichannel eikonal theory results of Flannery and McCann (1975) have occurred in the literature. These misprints are corrected in the paper. A detailed description of both the theoretical and experimental difficulties involved in the determination of the integral cross sections sigma 11S to 31S and sigma 11S to 31D is provided. In addition, a critique of the assumptions made in previous experimental determinations of the differential cross sections (both composite and magnetic sublevel) for the 21L states of helium is given. A principal conclusion of the paper is that the first Born approximation underestimates experimental data in the intermediate energy region for 1S1D transitions, in contrast to the trend seen in 1S1P transitions. In the paper the importance of indirect mechanisms in the excitation of the 31S and 31D is shown, thereby confirming the early results of Chung and Lin (1968, 1969) and the tentative conclusions of Massey (1969, 1974).

The multichannel eikonal theory of electron-hydrogen collisions. I. Excitation of H (1s)

The integral and differential cross sections for the electronic excitation of the 2l and 3l states of hydrogen obtained by the multichannel eikonal theory in a 10-channel basis are reported. The lambda , R and I parameters for the 2p state of hydrogen at 54.40 and 350 eV are also provided. Good agreement with the experimental data of Mahan et al. (1976) for the integral cross sections sigma 1s to 3l is achieved by the updated and extended multichannel eikonal theory. Two general conclusions follow from the results reported in the paper: (a) the first Born results underestimates experiment in s to d transitions and (b) indirect coupling mechanisms for the excitation of the 3d state of hydrogen are important and cannot be neglected. These conclusions indicate that the pseudostate basis used in a number of optical potential model calculations needs further refinement in the intermediate energy region.

Schwinger action principle, eikonal theory and two-time rotation-translation of coordinates

Via the Schwinger action principle, we derive a formula for the transformation function (or propagator) of time-dependent systems that can be expressed into a quadratic form. We present also a new type of two-time rotation-translation of coordinates which makes it possible to evaluate the exact transformation function of the three-dimensional time-dependent charged harmonic oscillator in cross time-varying, arbitrary magnetic and laser fields. We demonstrate that such a transformation function can be obtained from that for a free-particle in the new two-time and space coordinate system, whereby the scalar and vector (quadratic) potentials are transformed away from the quantum-mechanical motion. We also show how the quantum-mechanical eikonal theory of this problem arises in a natural way from the Schwinger action principle.

PCI influence on the atomic Auger spectrum induced by fast electron impact

A theoretical consideration of PCI in the processes of inner-atomic shell ionisation by fast electron impact is given. The shift of the maximum of the Auger line is calculated for a number of Auger transitions in the atoms of noble gases and lanthanum. The consideration is based on the eikonal theory.

Charge equilibrium of relativistic uranium ions in matter

Calculations are made for the electron capture and loss cross sections and the charge state distributions of uranium ions at energies of 200 MeV/nucleon, 437 MeV/nucleon and 962 MeV/nucleon colliding with low- Z and high- Z targets. The charge states of the projectile ions are determined from the equilibrium condition between electron loss from the ion and electron capture from a target atom into the projectile ion. The electron loss cross sections are formulated using the relativistic Born approximation. The electron capture process is considered to be non-radiative electron capture (non-REC) and radiative electron capture (REC) in the relativistic impact velocity range. The non-REC cross sections are calculated by Eichler's formula based on the eikonal theory, while for the evaluation of the relativistic REC process the contributions of internal e +e − pair creation and successive e +e − pair annihilation processes are taken into account in addition to the usual REC photon emission process.

Numerical techniques in eikonal function theory

Many problems in eikonal function theory can be formulated as an optimization problem in one or more sets of two variables. We show that such problems can be attacked quite successfully by purely numerical methods instead of the usual series developments. As an example, we determine the angle eikonal of a lens designed to form a perfect one-to-one image of a curved object plane onto a flat image plane and then calculate the aberrations when the lens is used backward.

Effect of large-amplitude perpendicularly propagating radio frequency waves on the interchange instability

Results are presented from hybrid 2-D quasineutral Darwin simulations of the interchange instability in the presence of a large rf wave in the ion-cyclotron frequency range. The simulation models the plane perpendicular to the background magnetic field using cold particle ions and a cold E×B electron fluid. Related theory is also discussed. Fluid equations appropriate to the simulation model are derived and their properties demonstrated and compared to simulation. A method for solving for the rf-modified growth rates from the fluid equations is described. It is generally expected that the current component associated with the mean, rf-induced ion drift is capable of influencing the stability of the interchange mode; however, no modification of the mean ion drift is observed in simulations in which rf is present. Instead, in both the theory and simulation, an electron rf-field oscillation current dominates the modification to the gravitational current. As a result, even in the presence of large rf fields (Brf/B0=15%), only modest corrections to the interchange growth rates are observed. The effect is stabilizing for kLn≲0.8–0.9, apparently for both signs of the square-electric-field gradient, and is destabilizing for larger values of kLn, although the credibility of the simulation begins to become suspect here. Fractional reduction of the interchange growth rate is observed to be quadratically dependent on the rf wave amplitude, independent of ion-cyclotron resonant effects, and proportional to ∇B2rf/∇B20, consistent with an eikonal theory developed for the study of stabilizing effects on perpendicularly propagating fast Alfvén waves. The results also suggest that additional gradient-independent stabilizing effects may be operative when kLn∼1. Finally, it is also observed that, while the rf wave has little effect on the interchange instability, the interchange mode strongly affects the rf wave, damping it significantly as the mode saturates.

Extension of "Four-dimensional eikonal theory of linear mode conversion" to parametrically unstable media.

Received 20 October 1987DOI:https://doi.org/10.1103/PhysRevLett.59.2709©1987 American Physical Society