Andreas Schinner

Binary theory of electronic stopping

Abstract Binary stopping theory has been developed to characterize the electronic stopping of swift heavy ions in matter. It is an extension of Bohr’s classical theory of 1913 incorporating screening, higher-order-Z1 and shell corrections, high-speed quantum and relativity corrections as well as projectile excitation and ionization. The main numerical input comes from optical properties. The computation of shell corrections involves orbital velocity distributions of target and projectile electrons. Calculated stopping parameters depend on ion charge. Equilibrium stopping forces may be computed by adoption of a suitable model for the equilibrium charge state. This paper summarizes the current status of the theory, in particular the sensitivity of its predictions to pertinent input. Charge-dependent stopping forces have been calculated for selected systems and compared to experimental results. Equilibrium stopping forces calculated for a wide variety of ion–target combinations are compared with experimental data from the literature spanning over 6 decades in ion energy.

An empirical approach to the stopping power of solids and gases for ions from 3Li to 18Ar - Part II

A large collection of stopping power data for projectiles from 3Li to 18Ar is investigated as a possible basis for producing a table of stopping powers. We divide the experimental stopping powers for a particular projectile (nuclear charge Z1) by those for alpha particles in the same element, as given in ICRU Report 49. With proper normalization, we then obtain experimental stopping power ratios Srel that lie approximately on a single curve, provided we treat solid and gaseous targets separately, and provided we exclude H2 and He targets. For every projectile, this curve is then fitted by a 3-parameter sigmoid function Srel=Srel(a,b,c). We find that the three parameters a, b and c depend smoothly on Z1 and can themselves be fitted by suitable functions af,bf and cf of Z1, separately for solid and gaseous targets. The low energy limit (coefficient a) for solids agrees approximately with the prediction by Lindhard and Scharff. We find that agas<asol in almost all cases. Introducing the coefficients af , bf and cf in Srel, we can calculate the stopping power for any ion (3⩽Z1⩽18), and for any element (except H2 and He) and any mixture or compound contained in the ICRU table.

Judging the reliability of stopping power tables and programs for heavy ions

Using our large collection of electronic stopping power data for ions from 3Li to 36Kr taken from the literature (0.001–1000 MeV/nucleon), we compare these data to stopping power tables and computer codes by Steward, Northcliffe and Schilling, Ziegler et al. (SRIM), Hubert et al., Konac et al., Grande and Schiwietz, Sigmund and Schinner, Hiraoka and Bichsel, Paul and Schinner (MSTAR) and the Geant Collaboration. Using either representative Z1–Z2-combinations, or all the data from our collection, we determine the average relative difference between experiment and table, and its standard deviation. On the basis of these numbers, we estimate the reliability of the various tables and codes. We find that SRIM and MSTAR are best in the entire energy region, but MSTAR is better for gaseous targets.

Effective charge and related/unrelated quantities in heavy-ion stopping

Abstract Stopping cross sections for swift heavy ions are commonly parameterized in terms of a velocity-dependent effective charge defined with reference to proton or helium stopping. This assumes projectile screening by core electrons to be a dominating factor governing the velocity dependence of the stopping cross section. In this note we demonstrate, on the basis of improved theoretical estimates of stopping cross sections, that projectile screening only affects the quantitative details of the velocity dependence of the effective charge while the general behavior reflects the transition from the Born to the classical regime. This is consistent with the experience that effective charges derived from measured stopping cross sections show dependencies on projectile and target atomic number which differ distinctly from those of the equilibrium projectile charge. It follows that there is no theoretical basis for the effective-charge concept as it is commonly used in heavy-ion stopping.

Binary theory of light-ion stopping

Binary stopping theory, although developed to describe heavy-ion stopping, has a potential in light-ion stopping which is examined in this paper. We study antiprotons, hydrogen and helium ions penetrating through elemental materials and lithium fluoride. Several comparisons with experiment show good agreement over a wide velocity range both below, around and above the stopping maximum. Predictions are given for antiproton stopping in carbon and LiF, and comments are made on the Barkas effect in the stopping of swift light ions and frozen-charge stopping of helium in carbon.

Electronic stopping of swift lithium and carbon ions

An attempt is made to estimate stopping powers for swift nonrelativistic heavy ions in the velocity range where projectile screening is significant. In contrast to existing tabulations and computer codes, our calculations take account of the fact that Bohrs classical approach is superior in this velocity regime to the Bethe theory. A theoretical scheme for estimating stopping powers at a fixed projectile charge, developed recently by one of us, has been expanded so as to account for charge equilibrium. Both target and projectile excitation/ionization are taken into account. The agreement with available experimental data appears promising in the considered velocity range (v0<v<2Z1v0) considering the fact that shell and polarization corrections still need to be incorporated.

Polarization effect in stopping of swift partially screened heavy ions: Perturbative theory

We report a calculation of the polarization correction – also called Barkas or Z13 effect – to the electronic energy loss of screened heavy ions above the Bohr velocity. For distant collisions we extend the classical calculation of Ashley et al. (Phys. Rev. B 5 (1972) 2393) to a screened projectile, while for close collisions we make use of an argument by Lindhard. The variation of energy loss with impact parameter is illustrated for Li on C for different charge fractions and projectile velocities v. Stopping numbers are shown at fixed charge fraction as a function of velocity, and for a velocity-dependent equilibrium charge. The polarization correction is found to be sizable over most of the velocity range covered by Bohr theory (the classical regime), but for the three ions for which results are reported, Li, C and Ar, very similar curves are found when stopping numbers are plotted against the Bohr parameter mv3/Z1e2ω, where ω is a characteristic resonance frequency of the target. The polarization correction is similar in magnitude to the leading term over a wide velocity range below mv3/Z1e2ω≃3. Therefore the perturbative approach presented here, which includes only one order beyond the leading term, cannot be expected to deliver very accurate results. This work will therefore be followed up by a nonperturbative evaluation of the stopping cross-section.

Material dependence of electronic stopping

Abstract The dependence of the electronic stopping cross-section on the atomic number of the target material has been studied on the basis of binary stopping theory over a wide range of beam energies. A distinct dependence on ion type and energy is found for the predicted Z2-structure which becomes significantly less pronounced for screened heavy ions than for protons and antiprotons. This behavior is caused primarily by the interplay between projectile screening and closing of inner target shells as a function of the beam velocity and introduces a pronounced Z2-structure in the effective-charge ratio. Related effects are found for stopping in three different phases of carbon and in lithium fluoride.

Anatomy of the Barkas effect

Abstract The stopping force on a swift charged particle penetrating through matter differs from that on its anti-particle. This difference, commonly called Barkas effect, has been studied theoretically as a function of velocity and projectile-target combination. In Bohr’s harmonic-oscillator model the Barkas ratio L+/L− is governed by one single variable. An explicit estimate of this dependency has been found from binary stopping theory. With the addition of shell and quantum effects, more complex scaling properties emerge, but for a given material, essentially the same Barkas ratio has been found for hydrogen, lithium and argon ions when plotted as a function of the Bohr parameter mv3/|Z1|e2ω. Projectile screening reduces the Barkas effect and may even invert its sign. We conclude that the effect is larger for protons than for heavier ions. This finding is shown not to contradict recently reported Barkas corrections under channeling conditions.

Electron emission yield of Al, Cu and Au targets induced by fast hydrogen and helium ions

Abstract The electron emission yield γ is measured for impact of H + , He + , He 2+ ions and of electrons on Al, Cu, and Au targets. The ion energy is between 0.5 MeV (H + ) and 4.8 MeV (He 2+ ), the electron energies are around 1 keV. The bare ion yield is compared with predictions of a model, which assumes the yield to be proportional to the stopping power and takes collective electric fields generated by the projectiles into account. The He + yield is explained by investigating the additional electron production of stripped electrons. It is found that the mean depth He + ions penetrate the surface layer without losing their bound electron and is about 60% of the mean escape length of the emitted electrons.