Search

Electron-impact ionization of the hydrogen atom is the prototypical three-body Coulomb breakup problem in quantum mechanics. The combination of subtle correlation effects and the difficult boundary conditions required to describe two electrons in the continuum have made this one of the outstanding challenges of atomic physics. A complete solution of this problem in the form of a 'reduction to computation' of all aspects of the physics is given by the application of exterior complex scaling, a modern variant of the mathematical tool of analytic continuation of the electronic coordinates into the complex plane that was used historically to establish the formal analytic properties of the scattering matrix. This review first discusses the essential difficulties of the three-body Coulomb breakup problem in quantum mechanics. It then describes the formal basis of exterior complex scaling of electronic coordinates as well as the details of its numerical implementation using a variety of methods including finite difference, finite elements, discrete variable representations and B-splines. Given these numerical implementations of exterior complex scaling, the scattering wavefunction can be generated with arbitrary accuracy on any finite volume in the space of electronic coordinates, but there remains the fundamental problem of extracting the breakup amplitudes from it. Methods are described for evaluating these amplitudes. The question of the volume-dependent overall phase that appears in the formal theory of ionization is resolved. A summary is presented of accurate results that have been obtained for the case of electron-impact ionization of hydrogen as well as a discussion of applications to the double photoionization of helium.

Since the invention of quantum mechanics, even the simplest example of the collisional breakup of a system of charged particles, e − + H → H + + e − + e − (where e − is an electron and H is hydrogen), has resisted solution and is now one of the last unsolved fundamental problems in atomic physics. A complete solution requires calculation of the energies and directions for a final state in which all three particles are moving away from each other. Even with supercomputers, the correct mathematical description of this state has proved difficult to apply. A framework for solving ionization problems in many areas of chemistry and physics is finally provided by a mathematical transformation of the Schrödinger equation that makes the final state tractable, providing the key to a numerical solution of this problem that reveals its full dynamics.

We analyze the consequences of discretizing one of the two continua in three-body breakup to reduce it to a two-body close-coupling problem. We identify the origin of oscillations in the singly differential cross section in those {open_quotes}convergent close-coupling{close_quotes} calculations as lying only in the way the cross section is calculated from the wave function and not in the wave function itself. The anomalous {open_quotes}step-function{close_quotes} behavior of those calculations is derived from a stationary-phase argument. Calculations are presented on the Temkin-Poet model for electron-impact ionization of hydrogen, a breakup problem with exponential potentials, and an analytically solvable model. The anomalies associated with two-body close-coupling calculations are demonstrated using wave functions from complex exterior scaling calculations that otherwise give converged results without any anomalies. {copyright} {ital 1999} {ital The American Physical Society}

Exterior complex scaling enables one to compute the outgoing wave portion of the wave function for three charged particles without explicitly imposing the asymptotic boundary condition for three-body breakup. This technique is used in connection with a high-order finite difference scheme to provide numerically accurate single-differential ionization cross sections for the Temkin-Poet (s-wave) model of $e\ensuremath{-}\mathrm{H}$ scattering. These benchmark values are compared with results obtained from several recent close-coupling approaches that employ pseudostates to discretize the ionization continuum, but use a strictly two-body scattering formalism.

We examine finite basis set implementations of complex scaling procedures for computing scattering amplitudes and cross sections. While ordinary complex scaling, i.e., the technique of multiplying all interparticle distances in the Hamiltonian by a complex phase factor, is known to provide convergent cross-section expressions only for exponentially bounded potentials, we propose a generalization, based on Simon's exterior complex scaling technique, that works for long-range potentials as well. We establish an equivalence between a class of complex scaling transformations carried out on the time-independent Schr\"odinger equation and a procedure commonly referred to as the method of complex basis functions. The procedure is illustrated with a numerical example.

Absolute doubly differential cross sections for the electron-impact ionization of atomic hydrogen have been measured from near threshold to intermediate energies. The measurements are calibrated to the well-established, accurate differential cross section for electron-impact excitation of the atomic hydrogen transition H(12S22S+22P). In these experiments background secondary electrons are suppressed by moving the atomic hydrogen target source to and from the collision region. Measurements cover the incident electron energy range of 14.6–40 eV, for scattering angles of 10°–120° and are found to be in very good agreement with the results of the most advanced theoretical models—the convergent close-coupling model and the exterior complex scaling model.

The formal theory of atomic ionization by electron impact relates the breakup amplitude to an integral expression involving the exact wave function and an unperturbed final state that contains two Coulomb functions with effective, noninteger charges. This integral expression has an associated phase factor that diverges logarithmically on an infinite volume unless the effective charges are chosen to satisfy a kinematic relationship, the so-called "Peterkop condition. "We derive the explicit form of the Peterkop phase for two commonly used models of electron-hydrogen ionization, the Temkin-Poet model and the collinear model. We show that the formal theory can be used to identify and remove this physically insignificant, volume-dependent phase from amplitudes computed using numerically stable integral expressions that do not satisfy the Peterkop condition.

We report triply differential measurements of atomic hydrogen ionization by 17.6-eV electrons, with the outgoing electrons both having 2 eV energy. These measurements supersede some of the existing data. The complete set is critically analyzed and is found to be much more internally consistent than before, thereby providing one of the most stringent tests for theory to date. Comparison with the calculations from the exterior complex scaling and convergent close-coupling theories shows excellent overall agreement in both shapes and magnitude.

We compare calculations of the fully differential cross section for ionization of atomic hydrogen by electron impact using two different theories--the perturbative CDW-EIS (continuum distorted wave with eikonal initial state) approximation and the nonperturbative ECS (exterior complex scaling) method. For this comparison, we chose an impact energy of 54.4 eV, since this is near the lowest energy that our perturbative approach would be applicable and near the highest energy that can be tackled by the ECS method with our present computational resources. For the case of equal-energy outgoing electrons investigated here, the two theories predict nearly identical results except that CDW-EIS underestimates the ECS values nearly uniformly by about 30%. Interestingly, when initial-state projectile-target interactions are neglected by replacing the eikonal initial state with the unperturbed initial state (the approximation of Brauner, Briggs, and Klar [J. Phys. B 22, 2265 (1989)]), the cross section oscillates by {+-}50% about the ECS values.

The formulation of the time-dependent close-coupling method is extended to allow the calculation of electron-impact triple-differential cross sections for atoms. The fully quantal method is applied to the electron-impact ionization of hydrogen at an incident energy of 54.4 eV for various scattering geometries. The time-dependent close-coupling results are found to be in very good agreement with those obtained by a time-independent exterior complex-scaling method. On the other hand even though the incident energy is relatively large, significant differences are found between the two nonperturbative cross-section results and those obtained using perturbative distorted-wave methods. Large differences are found for those geometries that require an accurate knowledge of the correlation between two outgoing continuum electrons in the presence of a Coulomb nuclear field.

We demonstrate experimentally how the time-dependent phase modulation induced by molecular rotational wave packets can manipulate the phase and spectral content of ultrashort light pulses. Using impulsively excited rotational wave packets in ${\mathrm{CO}}_{2}$, we increase the bandwidth of a probe pulse by a factor of 9, while inducing a negative chirp. This chirp is removed by propagation through a fused silica window, without the use of a pulse compressor. This is a very general technique for optical phase modulation that can be applied over a broad spectral region from the IR to the UV.

We show that the ``two-potential'' formalism of conventional distorted-wave rearrangement theory, which is formally valid only for short-range interactions, can be used to evaluate amplitudes for the ionization of atomic hydrogen by electron impact. The triply differential cross sections calculated using this method validate earlier results obtained by extrapolating the quantum mechanical flux. Although it uses the same time-independent wave functions, this method offers significant advantages over flux extrapolation. It is more accurate, can be applied in practical calculations over a broader range of collision energies, and unlike the flux-extrapolation method, can be applied for arbitrary values of energy sharing between the ejected electrons. Since this rearrangement formalism provides the complete scattering amplitude for ionization, it can be used to calculate any differential cross section.

We report the first completely ab initio calculations of doubly differential cross sections for the electron impact ionization of hydrogen at incident energies of 17.6 eV, 25 eV, and 30 eV. These cross sections have been extracted from wave functions obtained by directly solving a finite difference discretized Schr\"odinger equation without explicit reference to any assumed asymptotic form. Outgoing wave boundary conditions are assured by the use of the exterior complex scaling method. Our calculations suggest the need for additional experiments to augment the one available measurement.

Since the invention of quantum mechanics, even the simplest example of collisional breakup in a system of charged particles, e{sup {minus}} + H {r_arrow} H{sup +} + e{sup {minus}} + e{sup {minus}}, has stood as one of the last unsolved fundamental problems in atomic physics. A complete solution requires calculating the energies and directions for a final state in which three charged particles are moving apart. Advances in the formal description of three-body breakup have yet to lead to a viable computational method. Traditional approaches, based on two-body formalisms, have been unable to produce differential cross sections for the three-body final state. Now, by using a mathematical transformation of the Schrodinger equation that makes the final state tractable, a complete solution has finally been achieved, Under this transformation, the scattering wave function can be calculated without imposing explicit scattering boundary conditions. This approach has produced the first triple differential cross sections that agree on an absolute scale with experiment as well as the first ab initio calculations of the single differential cross section.

We have calculated, independently, the {sup 1}S singly differential cross section for the electron-impact ionization of hydrogen at an incident energy of 4.0 eV above threshold using two different methods: time-dependent close coupling and exterior complex scaling. The absolute value of the {sup 1}S cross section for equal energy sharing is critical in assessing recent theoretical and experimental results for coplanar triply differential cross sections at low energies. Convergence of the cross section is studied as a function of the number of coupled channels. We find that at this energy the singly differential cross section is relatively flat over the entire ejected-energy range as predicted by semiclassical calculations.

Absolute doubly differential cross sections for the ionization of atomic hydrogen by electron impact have been measured at energies ranging from near threshold to intermediate values. The measurements are normalized to the accurate differential cross section for the electron-impact excitation of the H 12S22S+22P transition. These measurements were made possible through the use of a moveable target source which enables the collection of hydrogen energy loss spectra free of all backgrounds. The measurements cover the incident electron energy range of 14.6–40 eV and scattering angles from 12° to 127°, and are in very good agreement with the results of the latest theoretical models—the convergent close-coupling model and the exterior complex scaling model.