Ejected-energy differential cross sections for the electron-impact detachment of H-

Electron-impact detachment cross sections for H− are calculated using time-dependent close-coupling theory. The three-electron wavefunction is expanded in terms of a product of a frozen core 1s hydrogenic wavefunction and a correlated two-electron wavefunction which fully describes the ejected and scattered electrons at all times following the collision. Ejected-energy differential and total integrated cross sections are calculated at 10 eV and 20 eV incident electron energy. Equal energy anomalies in the differential cross section are avoided by direct projection of the time-dependent wavefunction onto lattice continuum eigenstates. The total cross sections are in excellent agreement with previous ion storage ring experiments, while the differential cross section results confirm Monte Carlo perturbation theory in predicting a zero energy signature which should be common to all negative ions. Renewed theoretical interest in electron-impact detachment of negative ions has been sparked by recent high precision experimental measurements using ion storage rings [1–4]. The final quantum state after detachment consists of two free electrons moving in the field of a neutral third body. This is qualitatively different from the final quantum state after ionization of an atom or positive ion, in which two free electrons move in the long range Coulomb field of a charged third body. However, common to both the detachment and ionization processes is the difficult task of representing the double-electron continuum. Classical [1] and semiclassical [5–7] approaches have been invoked to predict electron-impact detachment cross sections for negative ions. Fully quantal approaches have used non-perturbative R-matrix theory [8], standard firstorder perturbation theory [9], and non-standard first-order perturbation theory based on Monte Carlo integration of a scattering amplitude in a mixed coordinate system [10, 11]. In this letter we apply time-dependent close-coupling (TDCC) theory to the electronimpact detachment of H−. This fully quantal non-perturbative method has been used previously to calculate electron-impact ionization cross sections for a number of atoms [12,13] and positive ions [14–17]. To support experiment and to verify the predictions of Monte Carlo perturbation theory (MCPT), we calculate ejected-energy differential and total integrated cross sections for the detachment of H− at 10 eV and 20 eV incident energy. In the past, the calculation of ejected-energy differential ionization cross sections has presented problems for various non-perturbative quantal treatments, such as the R-matrix pseudo-state [18], converged closecoupling [19, 20], and hyperspherical close-coupling [21] methods. The problems appear to be connected with extraction of differential cross sections by boundary matching of the wavefunction [21, 22]. Non-perturbative methods that avoid asymptotic forms, such as the complex exterior scaling [23] and the time-dependent close-coupling methods, are also found to avoid these equal energy anomalies in the differential cross section. 0953-4075/00/120427+06$30.00 © 2000 IOP Publishing Ltd L427 L428 Letter to the Editor The time-dependent close-coupling calculations begin with a frozen core 1s hydrogenic wavefunction. A set of bound n̄l and continuum k̄l radial orbitals are then obtained by diagonalization of the single particle Hamiltonian given by: h(r) = − 2 ∂2 ∂r2 + l(l + 1) r2 − 1 r + VD(r) + VX(r) , (1) where the direct, VD(r), and local exchange, VX(r), potentials are calculated using the frozen core 1s orbital (atomic units are used throughout unless otherwise specified). Diagonalization is on a radial grid of 300 points with a uniform mesh spacing of r = 0.20. A parameter in the local exchange potential is adjusted so that the 1̄s orbital has a binding energy of −0.75 eV, in agreement with experiment. The 1̄s orbital is quite different from the 1s orbital, with a mean radius of 〈r〉 = 3.19, in contrast to the hydrogenic value of 〈r〉 = 1.50. The total three-electron wavefunction (e− + H− system) for a given 2L symmetry is expanded as a coupled product of a frozen core 1s wavefunction and a correlated two-electron wavefunction. Reduction of the time-dependent Schrodinger equation [12, 13] yields two uncoupled sets of close-coupled partial differential equations for each 2L symmetry given by: i ∂P l1l2 (r1, r2, t) ∂t = Tl1l2(r1, r2)P LS l1l2 (r1, r2, t) + ∑ l′ 1,l ′ 2 U l1l2,l′ 1l ′ 2 (r1, r2)P LS l′ 1l ′ 2 (r1, r2, t) , (2) where P l1l2 is a two-electron radial wavefunction, L = L, S = 0 or S = 1, and (l1, l2) are the angular momenta for the ejected and scattered electrons. The operator Tl1l2 contains kinetic energy, centrifugal barrier, nuclear, frozen core direct and frozen core exchange terms. The operator U l1l2,l ′ 1l ′ 2 couples the various (l1, l2) scattering channels. At a time t = 0 before the collision, the two-electron radial wavefunctions are taken to be S = 0 symmetric or S = 1 antisymmetric products of the 1̄s radial orbital and an incoming radial wavepacket for the l = L incident electron. Finite differencing methods are used to represent the close-coupled partial differential equations (equation (2)) on a 300 × 300 point numerical lattice with a uniform mesh spacing of r1 = r2 = 0.20. The lattice wavefunction is partitioned over the many processors of a distributed memory parallel computer. Each radial wavefunction is propagated in time using an explicit second-order differencing scheme. At a time t = T following the collision the two-electron radial wavefunctions may be projected onto the lone 1̄s bound radial orbital or onto the many k̄l continuum radial orbitals. By unitarity, both projection schemes will yield the same total integrated detachment cross section. For ejected-energy differential detachment cross sections the two-electron radial wavefunctions are projected onto the k̄l continuum radial orbitals to yield momentum space probabilities [13]. In the (k1, k2) plane the momentum space probabilities are peaked along a ridge of total energy E = k2 1/2 + k2 2/2 = E0 − Ip, where E0 is the incident energy and Ip is the ionization potential. The diagonalization of h(r) of equation (1) determines the density of states. To increase the density of states at low energies we extended the radial grid to 1500 points, while maintaining the same uniform mesh spacing of r = 0.20. Dividing the (k1, k2) plane into angular segments, θ , defined by the hyperspherical angle tan(θ) = k2 k1 , the differential cross section is given by: