Lotten Hägg

Interaction of highly charged ions with surfaces

An overview is given of recent advances in the theoretical description of the interaction of multiply charged ions with surfaces. Simulations are presented displaying the formation of hollow atoms when slow highly charged ions approach the surface. It is shown that above-surface neutralisation proceeds via hollow-atom formation. Relaxation of the multiply excited states to the ground state occurs only subsequent to the close encounter with the topmost atomic layer.

Maximum entropy method and equilibrium charge state distributions

The maximum entropy method is implemented in order to describe equilibrium distributions arising in beam-foil spectroscopy. Since there are very few charge states involved, the usual moment conditions, based on simple powers xi, give rise to severe numerical difficulties already for three moments and cannot be applied in these systems. Earlier devised methods, based on Lagrange interpolation polynomials Qi(x) with abscissae chosen as zeros of Chebyshev polynomials in the interval being studied, are adapted and implemented for the present problems of charge state distributions. A reduced variable x=(q-(q))/(q), where q is the charge and (q) the mean charge, is chosen. Using the method described above calculations of equilibrium charge state distributions for copper ions (exit energy range 0.559-2.304 MeV u-1) colliding with carbon foils are carried out in order to exhibit the usability of the method. The new moment conditions associated to the Qi(x) provide a framework for a systematic analysis of equilibrium distributions.

Curve-crossing analysis for potential sputtering of insulators

We develop a theoretical model for the recently observed threshold for potential sputtering of LiF by slow singly and doubly charged ions. The threshold coincides with the potential energy to create a cold hole in the valence band of LiF by resonant neutralization. We calculate the level shift of the incident ion and the deformation of the valence band under the influence of the projectile. Resonant neutralization becomes possible for ions with recombination energies larger than 10 eV in agreement with the experimental findings.

Energy gain of highly charged ions in front of LiF

Abstract We present estimates of the energy gain of highly charged ions approaching a LiF surface, based on a modified classical-over-barrier model for insulators. The analysis includes the energy gain by image acceleration as well as the deceleration due to charge-up of the surface in a staircase sequence. The resulting velocity-dependent total energy gain is studied in detail and the results are compared with experimental data.

Gaussian approximations to equilibrium charge-state distributions with the maximum entropy method

Charge-state distributions for ions in beam-foil experiments are discussed in detail and analysed using the maximum entropy method (MEM). The symmetric distributions can be well described by one Gaussian in the whole charge-state interval. It is found that asymmetric distributions, exhibiting the shell effect can be treated using two disjoint Gaussians, in two separate intervals. The relative weights of these Gaussians are interpreted by using Fermi Dirac statistics. Physically this corresponds to varying occupancy of some of the shells in the ions. The fact that no more than two moments, in one interval or two, is sufficient is of significant importance in the analysis of charge-state distributions in general. An interpolation procedure to obtain the charge-state distribution for an energy is developed and tested. The procedure, which is made possible by the Gaussian character, uses MEM and experimental information. It is only possible to predict distributions in energy regions where experimentally measured values for other energies exist. The method is tested on asymmetric distributions of copper ions in the energy range 0.902-1.505 MeV u-1 and symmetric distributions for calcium, titanium, yttrium, rhodium, holmium and bismuth ions for energies from 0.191 to 3.184 MeV u-1.

The maximum entropy method and relaxation for multiple collisions involving highly charged ions

Advantages and disadvantages of the maximum entropy method (MEM) in application to the theory of relaxation are studied. The time evolution of distributions and of associated moments must obey stringent conditions for both finite and infinite intervals. The theoretical considerations are illustrated with examples from charge-state distributions arising in beam-foil spectroscopy. The examples indicate that the possibility to include more than two moments (extension to non-Gaussian case) is severely limited (though feasible) in the static case due to nonpositive definiteness as well as stiffness of the Hessian matrices appearing in the computations. This takes place already for the finite charge-state distribution intervals. For infinite intervals, this is a severe problem as required by the Marcinkiewicz theorem, affecting characteristic functions and, hence, the description of the time evolution of distributions.

General Computational Strategies

The poor convergence of the partial sums of the 1/D expansion for atomic and molecular energies is due to the fact that the energy function E(δ), δ ≡ 1/D, is not a polynomial. The large-D limit appears to be an excellent qualitative model, but in order to accurately continue the solution from δ = 0 to δ = 1/3 it is necessary to take into account the functional form of E(δ) in that general region of the comples plane. The most important feature in this functional form is a second-order pole at δ = 1. Incorporating this pole into the form of the summation approximates, using either of three methods that we describe, accounts for over 99% of the solution at δ = 1/3 for ground-state energies of 2 + and helium. Almost all the remaining error can be accounted for by including a rather complicated singularity at δ = 0. Through an analysis of the large-order behavior of the δ expansion, we construct a functional form for this singularity. The functional form suggests that the energy expansion can be summed most effectively using Borel summation, with the integrand of the Borel sum approximated by Pade approximants or, better yet, by approximants in the form of Darboux functions. We present numerical results that support this conclusion.

Dimensional scaling of local pseudopotentials

Abstract The integral equations of Gelfand and Levitan were previously used by Abraham and Moses (Phys. Rev. A, 22 (1980) 1333) to construct a modified local potential Vpert(r), which in this work is added to a coulombic potential in order to remove the lowest s state from the spectrum. The total modified potential Vnew(itr), for excited states is used in conjunction with the shifted 1/D expansions of dimensional scaling. It is verified that it is feasible to find the lowest energies and the mean radius for Vnew, treating it now as a ground state problem. This indicates that a potential for excited states can be generated by the procedure, with the possibility of using it in the study of excited states and valence states. The potentials applied can be considered as local Pseudopotentials.