Charge-state-dependent energy loss of slow ions. II. Statistical atom model

A model for charge-dependent energy loss of slow ions is developed based on the Thomas-Fermi statistical model of atoms. Using a modified electrostatic potential which takes the ionic charge into account, nuclear and electronic energy transfers are calculated, the latter by an extension of the Firsov model. To evaluate the importance of multiple collisions even in nanometer-thick target materials we use the charge-state-dependent potentials in a Monte Carlo simulation in the binary collision approximation and compare the results to experiment. The Monte Carlo results reproduce the incident charge-state dependence of measured data well [see R. A. Wilhelm et al., Phys. Rev. A 93, 052708 (2016)], even though the experimentally observed charge exchange dependence is not included in the model.

Figure 1

Classical scattering of a charged ion with a neutral target atom. The ion, which is only partially screened, is split into a neutral atom with nuclear charge $Z_1-Q$ and point charge $Q$.

Figure 2

Deflection angle and nuclear energy transfer as a function of the impact parameter for Xe ions scattered by carbon atoms in different charge states and an initial kinetic energy of ${\varepsilon }_{\mathrm{kin}}=310\mathrm{eV}/\mathrm{amu}$ as calculated from Eqs. (10) and (12), respectively.

Figure 3

Nuclear stopping cross section for xenon on carbon and different xenon charge states from Eq. (13). For comparison the nuclear stopping cross section from SRIM2013 is plotted as the black line and coincides perfectly with the calculation for $Q=0$.

Figure 4

Nuclear stopping force as a function of the ion charge for 310 eV/amu xenon on a carbon membrane with a density of $\varrho =5.54\times {10}^{22}\mathrm{at}/{\mathrm{cm}}^3$. Data points are extracted from the calculations shown in Fig. 3 (vertical line) and fitted linearly.

Figure 5

Electrostatic potential ${\phi }_{\mathrm{total}}\left(\vec{r}\right)$ (blue tone) with contour lines. Scattering partners are at locations $(r_x,r_y)=(0,0)$ and $(10\mathrm{nm},0)$. (a) Two equal mass particles produce a planar surface $S$ (orange line). (b) A heavy neutral and light neutral particle produce a spherical surface $S$. (c) For a charged particle the potential ${\phi }_{\mathrm{ion}}\left(\vec{r}\right)$ is dominated by the unscreened point charge. (d) A zoom-in around the neutral carbon atom in (c) is shown, with a spherical surface $S$.

Figure 6

Electronically transferred energy as a function of the impact parameter $p$ for different xenon charge states on carbon. Energies are calculated for discrete impact parameters (as an example circles are shown for $Q=30)$ in the dipole approximation, Eq. (20), and fitted according to $T_{\mathrm{elec},\mathrm{ls}}\left(p\right)=a{(1+bp)}^{-5}$. As a comparison the result of Firsov is shown as the dashed line $\left(Q=0\right)$ [36].

Figure 7

Simulated energy distributions after transmission of 40-keV Xe ions at different charge states through a 1-nm-thick carbon membrane with a density of $\varrho =5.54\times {10}^{22}\mathrm{at}/{\mathrm{cm}}^3$ in the BCA under straightforward direction. The double-peak structure for each distribution results from three or four scattering events in the simulation. Dotted lines are Moyal fits to each peak. Dashed vertical lines are the means of each Moyal distribution.

Figure 8

Simulated energy distributions for 47-keV ${\mathrm{Xe}}^{25+}$ ions transmitted through a 1-nm-thick carbon membrane with a density of $\varrho =5.54\times {10}^{22}\mathrm{at}/{\mathrm{cm}}^3$ in the BCA under mean detection angles of $0^{\circ }$ (black), $2^{\circ }$ (blue), and $4^{\circ }$ (red). Results are fitted with a Moyal distribution (dotted lines) and the mean value of each distribution is shown as a vertical dashed line.

Figure 9

Calculated energy loss values as a function of the xenon charge state, at a fixed kinetic energy of 310 eV/amu. Nuclear losses are given according to Eq. (8) (dotted line) and with selected impact parameters (dashed-dotted line) (see text). Electronic losses are given with (dashed–multidotted line) and without (dashed line) impact parameter selection. Results of our BCA simulation are shown as blue circles and are fitted with the function $\alpha Q^2+\beta Q^{1.5}+\gamma$.

Figure 10

Calculated energy loss as a function of the mean exit angle of ${\mathrm{Xe}}^{25+}$ ions at 364 eV/amu specific kinetic energy in the BCA (see Fig. 8). Calculated data are fitted with a quadratic function.