Stopping power calculations

We have shown in Section 5 that the non-linear approach yields a good representation of the energy loss in the high-energy region, showing a connection with the BBB models for bare ions, and extending the possibilities of calculations to all types of ions. 

In the case of light ions, this approach has been applied to calculate the energy loss for the following systems (i) helium ions in aluminum targets [37], and (ii) protons and antiprotons in various solids [38] (with particular evaluation of the Barkas effect). In all these cases the agreement with experimental results was very good. 

Now we will show more extensive calculations following this approach, including light and heavy ions, and the corresponding comparisons with experimental results. 

As it was mentioned, one remarkable feature of the low-energy stopping phenomenon is the oscillatory dependence on the ion atomic number Zj. The question of representing this structure constitutes an important test for the theoretical descriptions. 

In Fig. 5 we show, with solid line, the non-linear calculations of the stopping power of carbon for all ions with atomic numbers in the range 1 Zj 40, and with a fixed velocity v = 0.8 a.u., together with experimental results from various authors [10,46,47]. We also show the theoretical results obtained from the DFT [32] (which, for the electron density of carbon targets, are available only in the range Zj 17), and the calculations according to the Brandt-Kitagawa model (BK) [15]. This model is based on linear theory and includes a statistical model for the ion stmcture as well as 

---

Fig. 2. Stopping power calculations for bare ions, H and according to the linear (DF) and non-linear (NL) formulations described in the text, for a medium with Tg = 1.6. The asymptotic Bethe and Bloch limits are also indicated.

The results of Fig. 8(a) show that the non-linear stopping power calculations are consistent with the first assumption, and in clear discrepancy with the second. 

Figure 1. Depth profile of trans-vinylene absorbance, 2 months from the irradiation. Dotted line represents the depth profile of stopping power calculated with SR1M-2003.

Equations 4.2-4.4 give the result of the stopping power calculation if the particle moves in a pure element. If the particle travels in a compound or a mixture of several elements, the stopping power is given by... 

Ema data can be quantitated to provide elemental concentrations, but several corrections are necessary to account for matrix effects adequately. One weU-known method for matrix correction is the 2af method (7,31). This approach is based on calculated corrections for major matrix-dependent effects which alter the intensity of x-rays observed at a particular energy after being emitted from the corresponding atoms. The 2af method corrects for differences between elements in electron stopping power and backscattering (the correction), self-absorption of x-rays by the matrix (the a correction), and the excitation of x-rays from one element by x-rays emitted from a different element, or in other words, secondary fluorescence (the f correction). 

P. Duncamb, S. j. B. Reed in K. F. J. Heinrich (ed.) The Calculation of Stopping Power and Backscattcr Effects in Electron Probe Microanalysis, NBS Special Publ. 298, Washington, 1968. 

Our particular interest lies in the calculation of the linear energy deposition, or stopping power, of swift ions in materials, 5o(v). In the first Born approximation, and for a fully stripped projectile, this quantity can be written [2-4]... 

Here Z is the charge of the projectile with velocity v. In order to calculate stopping powers for atomic and molecular targets with reliability, however, one must choose a one-electron basis set appropriate for calculation of the generalized oscillator strength distribution (GOSD). The development of reasonable criteria for the choice of a reliable basis for such calculations is the concern of this paper. 

Figure 5 Stopping power for protons on He calculated with the standard basis (basis A), with two consistent bases (B and C), and in the Bethe approximation using the kinetic theory [17, 18],...

Another procedure for calculating the W value has been developed by La Verne and Mozumder (1992) and applied to electron and proton irradiation of gaseous water. Considering a small section Ax of an electron track, the energy loss of the primary electron is S(E) Ax, where S(E) is the stopping power at electron energy E. The average number of primary ionizations produced over Ax is No. Ax where o. is the total ionization cross section and N is the number density of molecules. Thus, the W value for primary ionization is 0)p = S(E)/No.(E). If the differential ionization cross section for the production... 

---