ARTICLE IN PRESS
Radiation Physics and Chemistry 76 (2007) 542–545 www.elsevier.com/locate/radphyschem
Single and double K-shell ionization cross sections of silicon + ´ sib, M. Kavcˇicˇa, K. Toke a
b
J. Stefan Institute, P.O. Box 3000, SI-1001, Ljubljana, Slovenia Institute of Nuclear Research of the Hungarian Academy of Science (ATOMKI), H-4001 Debrecen, P.O. Box 51, Hungary Received 31 August 2005; accepted 10 December 2005
Abstract A four-body classical trajectory Monte Carlo (CTMC) method is applied in the study of single and double ionization cross-sections of the silicon K-shell. The calculations are based on the independent particle model. As projectiles, we consider protons with energies between 0.25 and 4.5 MeV. Our CTMC results are compared with the existing theoretical and experimental data. r 2006 Elsevier Ltd. All rights reserved. Keywords: Inner shell ionization cross-section; Classical trajectory Monte Carlo method; Two-step ionization
1. Introduction Double electron transitions in atomic collisions have received a growing interest from both experimental and + ´ si et al., theoretical points of view (McGuire, 1992; Toke 1999; Kavcˇicˇ et al., 2000; Vikor et al., 2000; Paripa´s, 2003; Kavcˇicˇ, 2003). The double ionization of K-shell electrons is a special case of double transitions. High-resolution charged particle induced X-ray spectra are often used to obtain the K-shell ionization cross-sections. The double K-shell ionization cross-sections can be obtained from the hyper-satellite line yields (Kobal et al., 2004; Kavcˇicˇ and + ´ si, 2005). In the previous works (Boschung et al., Toke 1995; Cindro et al., 1989), the theoretical predictions, applying the binary encounter and semi-classical approximations, overestimated the experimental values. The key point of the calculations is the proper description of the collisions. Most of the theoretical estimations are based on the independent particle approximation (IPA) and completely ignore the effect of the electron–electron interaction. The classical trajectory Monte Carlo (CTMC) method (Abrines and Percival, 1966; Olson and Salop, + ´ si and Hock, 1994) has the main advantage of 1977; Toke being a non-perturbative method. It was shown that the Corresponding author. Fax: +36 52 416 181.
E-mail addresses:
[email protected],
[email protected] + ´ si). (K. Toke 0969-806X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2005.12.058
CTMC method has been quite successful in dealing with + ´ si and ionization processes in ion–atom collisions (Toke Ko¨ve´r, 2000). In this work, we present single (Eq. (1)) and double ionization cross-sections (Eq. (2)) of silicon K-shell for proton impact within the framework of a four-body CTMC method. Hþ þ Sið1s2 Þ ! Sið1s1 Þ þ Hþ þ e ,
(1)
Hþ þ Sið1s2 Þ ! Sið1s0 Þ þ Hþ þ 2e .
(2)
The energies of the projectiles investigated were in the range of 0.25 and 5 MeV. The total cross-section as a function of the impact energy is calculated and compared with the available experimental and theoretical values. As an example, certain characteristic deviations, observed in the double electron ionization at high and low impact energies, are demonstrated with the help of individual trajectory calculations. 2. Theoretical framework In the present CTMC approach, Newton’s classical nonrelativistic equations of motion for a four-body system are solved numerically for a statistically large number of trajectories. The silicon atom is imitated and represented as three particles. Then the four particles in this CTMC model were chosen as follows: the projectile (Hþ ), two active
ARTICLE IN PRESS + ´si / Radiation Physics and Chemistry 76 (2007) 542–545 M. Kavcˇicˇ, K. Toke
2pbmax Sbi . (3) N The statistical uncertainty for a cross-section is given by N N i 1=2 Dsk ¼ sk . (4) NN i
sk ¼
3.5e-20 3.0e-20 2.5e-20 σK [cm2]
target K-shell electrons (e ), and the target nucleus. The forces acting among the four bodies are taken to be pure coulombic. The interaction between the two active electrons of the silicon atom is neglected. The binding energies of the K-shell electrons in the Si atom are chosen as 68.81 and 75.37 a.u., respectively. These values are related to the first and second ionization energies calculated with the help of the Hartee Fock method. To distinguish between the various final states, the exit channels are tested at large distances (5000 a.u.) from the collision center. The total cross-sections for a specific event k are calculated by
We have calculated the single and double K-shell ionization cross-sections of silicon as a function of impact energies, lying between 0.25 and 4.5 MeV using a four-body CTMC approach. A very large number of classical trajectories were computed to calculate the cross-sections. The total number of primary histories was 2 000 000 for each impact energy. The large numbers of trials were required because the total cross-sections for deeply bound electrons were very small, especially for the case of double ionization. Fig. 1 shows the total cross-sections for single ionizations of silicon K-shell. Below 1.5 MeV our results are in good agreement with the data of Paul and Sacher (1989), calculations based on the first-order semi-classical approximation (SCA) (Trautmann and Ro¨sel, 1980a,b), and calculations based on SCA by Sˇmit (1996). Above 1.5 MeV our predictions overestimate the other results. In the following discussion, our attention will focus on the double ionization channel. Fig. 2 shows the total crosssections for double ionizations of silicon K-shell in comparison with recent data obtained experimentally + ´ si, 2005) and other calculations using (Kavcˇicˇ and Toke independent electron approximations. The errors of the experimental data are smaller or comparable to the size of the symbols. At small collision energies, the agreement between the calculations and the experimental data are satisfactory. The SCA results are close to the experimental data for the whole range of impact energy. The shape of the theoretical curves is the same. However, the CTMC
2.0e-20 1.5e-20 1.0e-20 5.0e-21
0
1
2 3 Energy [MeV]
4
5
Fig. 1. Total K-shell single-ionization cross-sections of silicon by proton impact. Solid line: present CTMC approach; open circle: Paul and Sacher (1989); dashed line: Trautmann and Ro¨sel (1980a,b); dash-dotted line: Sˇmit (1996).
N is the total number of trajectories calculated for the impact parameters less than bmax , N i is the number of trajectories that satisfy the criteria for the process under consideration, and bi is the actual impact parameter for the event i specified by a set of collision product criteria.
5e-23
4e-23
σK [cm2]
3. Results and discussion
543
X0.5
3e-23
2e-23
1e-23
0 0
1
2 3 Energy [MeV]
4
5
Fig. 2. Total K-shell double-ionization cross-sections of silicon by proton impact. Solid line: present CTMC approach; open circle: Kavcˇicˇ and + ´ si (2005); dashed line: Trautmann and Ro¨sel (1980a,b); dash-dotted Toke line: Sˇmit (1996).
cross-sections are much larger than the experimental ones and, for some cases, the discrepancy is almost an order of magnitude. The possible source of the discrepancies can be ascribed partly to the double ionization process being more sensitive for static nuclear charge used in the calculations thereby indicating the significant role of the shielding effect of the outer electrons and partly to neglecting the electron–electron correlation between the two K-shell electrons during the calculation. Figs. 3 and 4 show two typical examples for the double ionizations of silicon K-shell at 3 and 30 MeV impact energy, respectively. Contrary to the rather different appearances of the two electron states in Fig. 3, they fulfill the requirement of the ionization of 1s2 state. In Fig. 3, at low impact energy (3 MeV) compared to the binding energy of the electrons in the K-shell of silicon, the two-step mechanism can be observed. In the first step, the projectile
ARTICLE IN PRESS + ´si / Radiation Physics and Chemistry 76 (2007) 542–545 M. Kavcˇicˇ, K. Toke
544 0.10
0.10 p scattering
P scatterings 0.05
0.05
0.00 y [a.u.]
y [a.u.]
0.00
-0.05
-0.05
Target core
-0.10 -0.10
-0.15 (a) -0.20 -0.3
p scattering
-0.2
0.0
0.1
0.2
0.3
z [a.u.]
0.15
Fig. 4. Typical trajectories for double-ionization of K-shell electrons from silicon by proton impact at 30 MeV impact energy in the laboratory frame and in an arbitrary plane (y–z) projection. The arrays show the positions of the electron and projectile at the instant of the ionization. The impact parameter is b ¼ 0:05 a:u.
0.10 y [a.u.]
-0.1
0.05 0.00
Target core
rearrangement as expected to dominate at large impact energies.
-0.05
(b)
4. Conclusion p scattering
0.15
y [a.u.]
0.10 0.05 0.00 -0.05
Target core
(c)
-0.10 -0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
z [a.u.]
Fig. 3. Typical trajectories for double-ionization of K-shell electrons from silicon by proton impact at 3 MeV impact energy in the laboratory frame and in an arbitrary plane (y–z) projection. The arrays show the positions of the electron and projectile at the instant of the ionization: (a) The impact parameter, b is 0 a.u.; (b) the impact parameter, b is 0.05 a.u.; (c) the impact parameter, b is 0.1 a.u.
ionizes one of the target electrons. The arrows show the positions of the projectile and an electron, where the distinct ionization occurs. The second target electron is ionized at a well-separated distance with respect to the position of ionization of the first electron. This two-step mechanism is observable classically during a slow collision. Fig. 4 shows the trajectory of the particles at much higher impact energy (30 MeV). In contradiction to the previous case, here the two electrons of the target atom are ionized at the same instance of time by the projectile. This simultaneous ionization occurs as a sudden (resonant)
In the present work, we have calculated cross-sections for single and double K-shell ionization as a function of the impact energy for proton impact. The cross-sections were obtained with the CTMC method using the independent electron model, i.e. with the complete neglect of the electron–electron interaction during the collision. Our present CTMC data for single ionization cross-sections are in general agreement with the experimental values and with the SCA calculations. For the case of double ionization cross-sections, the CTMC calculations overestimate the results of experimental and SCA data. The discrepancies can be partly attributed to the fact that we completely neglect the screening of the outer electrons and the electron–electron interaction of the two active electrons during the collisions. Acknowledgments This work was supported by the Slovenian Ministry of Education, Science and Sport of Slovenia through the research program Low Energy Physics (PO-0521-0106-02), the Hungarian Scientific Research Found OTKA nos. T046095, T046454, the grant ‘‘Bolyai’’ from the Hungarian Academy of Sciences, and Te´T Grant no. SLO-15/05. References Abrines, R., Percival, I.C., 1966. Proc. Phys. Soc. (London) 88, 861. Boschung, B., Dousse, J.-Cl., Galley, B., Herren, Ch., Hoszowska, J., Kern, J., Rheeˆme, Ch., Halabuka, Z., Ludziejewski, T., Rymuza, P., Sujkowski, Z., Polasik, M., 1995. Phys. Rev. A 51, 3650.
ARTICLE IN PRESS + ´si / Radiation Physics and Chemistry 76 (2007) 542–545 M. Kavcˇicˇ, K. Toke Cindro, V., Budnar, M., Kregar, M., Ramsˇ ik, V., Sˇmit, Zˇ., 1989. J. Phys. B 22, 2161. Kavcˇicˇ, M., 2003. Phys. Rev. A 68, 022713. + ´ si, K., 2005. Phys. Rev. A 72, 062704. Kavcˇicˇ, M., Toke Kavcˇicˇ, M., Budnar, M., Mu¨hleisen, A., Pelicon, P., Sˇmit, Zˇ., Zˇitnik, M., Castella, D., Corminboeuf, D., Dousse, J.-Cl., Hoszowska, J., + ´ si, K., 2000. Phys. Rev. A 61, 052711. Raboud, P.A., Toke Kobal, M., Kavcˇicˇ, M., Budnar, M., Dousse, J.Cl., Maillard, Y.-P., Mauron, O., Raboud, P.A., To˜ke´si, K., 2004. Phys. Rev. A 70, 2720. McGuire, J.H., 1992. Adv. At. Mol. Opt. Phys. 29, 217. Olson, R.E., Salop, A., 1977. Phys. Rev. A 16, 531.
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