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Polarization effect in (e, 2e) collisions of argon
Physics Letters A 334 (2005) 192–196 www.elsevier.com/locate/pla
Polarization effect in (e, 2e) collisions of argon Xiao-Ying Hu ∗ , Ya-Jun Zhou, You-Qi Ke, Guang-Jun Nan Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, PR China Received 7 September 2004; received in revised form 8 November 2004; accepted 8 November 2004 Available online 13 November 2004 Communicated by B. Fricke
Abstract Calculations of the triple differential cross section (TDCS) for electron impact ionization of the Ar(2p) orbital in a highly asymmetric geometry, using modified distorted wave Born approximation (DWBA) methods, are reported. The role of the polarization effect in (e, 2e) collisions of Ar(2p) is considered in the calculations, and the calculated results shows that the polarization potential is particularly important. 2004 Elsevier B.V. All rights reserved. PACS: 34.80.Bm Keywords: (e, 2e) reaction; Polarization potential; Distorted wave Born approximation
1. Introduction In recent years, a number of new directions have been pursed in (e, 2e) investigations of electron-impact ionization processes. Much attention has been concentrated on atomic outer shells, particularly with simple targets, and the less bound intermediate shells of atoms and molecules. However, because of the smallness of the cross section the core region has been the subject of fewer studies, although the dynamical studies of inner-shell ionization in neon, argon [1–3] and xenon [4] revealed a number of interesting features of the * Corresponding author.
E-mail address: [email protected] (X.-Y. Hu). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.11.007
scattering which were distinctly different to outer-shell ionization. Experiment results [5,6] were reported for (e, 2e) ionization of the argon 2p orbital in a highly asymmetric geometry, which showed that the theoretical results obtained using the DWBA method had significant differences, compared with experimental data, in the overall shape and magnitude of the triple differential cross section. The recoil-to-binary ratio in relative TDCS for the Ar(2p) ionization was significantly underestimated by the DWBA calculations. The double peaks using the standard DWBA calculations were approximately symmetric about the direction of mo while the experimental results mentum transfer ±K, did not show this feature clearly. The calculated binary and recoil peaks showed no structure where the
X.-Y. Hu et al. / Physics Letters A 334 (2005) 192–196
experiments might possibly show some. All of these differences present a substantial challenge to theory. The discrepancy is probably due to neglecting to consider the higher-order electron–electron correlations in the DWBA method. Thus, a new dynamical mechanism and the modification of the DWBA method are needed to explain the experimental data properly. In order to test the significance of polarization, Whelan et al. [7] modified the DWBA calculations by introducing an ad hoc polarization potential Vpol in (e, 2e) collisions of He. In this Letter, we have extended the model proposed by Whelan et al. [7] and added another potential Vdft via density-functional theory (DFT) [8] for argon. We aim at showing the influence of the polarization potential [9] on the calculated TDCS of Ar(2p). In order to test the validity of our model, we have calculated the TDCS with the incident energy of 8256 and 5720 eV, respectively, in a highly asymmetric geometry. The calculated results have been compared with those obtained by the standard DWBA and the experimental data [5,6].
2. Theory The triple differential cross section in the distortedwave Born approximation is given by [10] 2 kf ks d 3σ kf ks |T |α k0 , = (2π)4 df ds dE0 k0 av (1) where the subscript f and s represent the fast and slow electrons, respectively. E is kinetic energy, k momentum, E0 the kinetic energy of the incident electron. The incident electron with momentum k0 incidents on an uncharged system consisting of electron s bound to the core in the state α with separation energy εα . The notation av represents a sum over the final state and average over the initial magnetic and spin degeneracy state. In the calculation, the atomic units (h¯ = me = e = 1) are used. The DWBA T-matrix element is kf ks |T |α k0 = χ (−) (kf )χ (−) (ks )vf s αχ (+) (k0 ) , (2) where vf s is the interaction potential between the fast and slow electron. |χ (−) (kf ), |χ (−) (ks ) and
193
|χ (+) (k0 ) are the distorted waves at different asymptotic conditions. The superscripts refer to the outgoing and incident waves, respectively. The partial-wave expansions of the distorted waves can be written as (−) r χ (k) 1/2 2 1 ∗ ˆ LM = i −L eiσL uL (k, r)YLM (k)Y (ˆr ), π kr LM (3) (+) r χ (k) 1/2 1 2 ∗ ˆ LM (ˆr ), (4) i L uL (k, r)YLM (k)Y = π kr LM
where σL is the Coulomb phase shift. The coordinate representation of the orbital |α is r |α = r −1 unl YLM (ˆr ),
(5)
and the partial wave µL (k, r) can be obtained by solving the differential equation as follows: 2 L(L + 1) η d − − dρ 2 ρ2 ρ v(r) + VP (r) + 1 uL (k, r) = 0, − (6) E where the potential v(r) includes the direct distorting potentials VD (r) and the spin-averaged staticexchange potential VE (r), ρ = kr, η = −Z/k, E = 1 2 2 k , VP (r) is the polarization potential which we introduce in our calculations. The asymptotic form of the adiabatic polarization potential is simply given by the well-know secondorder perturbation expansion formula [11,12]. VpT (r) ∼ − r→∞
∞ l=1
αl , 2r 2l+2
for r > r0 ,
(7)
αl is the multipolar polarizability of the target atom and r0 separates the outer region from the inner shortrange region. In the near-target region, the main drawback of the expansion (7) is that it grossly overestimates polarization and incorrectly diverges near the origin of the force field. According to Whelan et al. [7], we only consider the dipole term. Hence the polarization po-
194
X.-Y. Hu et al. / Physics Letters A 334 (2005) 192–196
tential Vpol becomes VP (r) = Vpol =
α0 − 2r 4,
− α04 , 2r0
3. Results and discussion r > r0 , r r0 ,
(8)
where α0 is the dipole polarizability of neutral Ar for the incident channel, and the polarizability of Ar+ for the outgoing channels. In both cases we take the cutoff radius r0 = 3.16a0 according to Refs. [7,8]. In the near-target region, Gianturco et al. [8] took the nonadiabatic corrections into account. On basis of the model suggested by Lee, Young and Parr (LYP) [13], they obtained the polarization potential via the density-functional theory in near-target region as follows, 8 DFT = −a(F1 ρ + F1 ) − abCF ρ 5/3 G1 ρ + G1 V(r) 3
ab − G1 ρ|∇ρ|2 + G1 3|∇ρ|2 + 2ρ∇ 2 ρ 4 + 4G1 ∇ 2 ρ −
ab 3G1 ρ|∇ρ|2 + G1 5|∇ρ|2 + 6ρ∇ 2 ρ 72 + 4G1 ∇ 2 ρ , for r < r0 , (9)
where −1 F1 ρ(r) = 1 + dρ −1/3 ,
G1 ρ(r) = F1 (ρ)−5/3 exp cρ −1/3 , CF =
3 2 2/3 3π , 10
(10a) (10b) (10c)
ρ is the electron density. The three parameters (a)–(c) are the same as in [14]. The formula of Eq. (9) for the short-range polarization potential is a pure density functional and can be easily obtained. The final total polarization potential Vdft via DFT approach is VP (r) = Vdft =
DFT , V(r)
when r r0 ,
VpT (r), when r > r0 .
(11)
In Eq. (11) r0 = 5.39a0 according to Ref. [8,14]. Thus, DFT can be nicely connected the potential function of V(r) with VpT (r).
We have calculated the TDCS of Ar(2p) in a highly asymmetric geometry. The impact energy of 8256 and 5720 eV is chosen, respectively, and correspondingly the polar angle of the fast electron is settled at θf = 1.5◦ and θf = 0.5◦ , respectively. The only variable parameter is the polar angle θs of the ejected electron, which is scanned from 0◦ to 360◦ . This is the experimental condition adopted by Stefani et al. and Taouil et al. [5,6], which is also used in our calculations so that our theoretical results can be compared with the experimental data. The experimental data are not absolute, so only shapes can be compared. Hence we normalize the present calculated results using Vdft to the experimental binary peak height. The results of the standard DWBA and using Vpol are relative to the calculations of using Vdft . The calculated TDCS of Ar(2p) with the impact energy of 8256 eV is shown in Fig. 1. We show the TDCS in coplanar asymmetric geometry with three different the DWBA approximations. (a) Standard DWBA without polarization potential (dash line). (b) DWBA with polarization potential Vpol (dotted line). (c) DWBA with polarization potential Vdft via density-functional theory (solid line). In Fig. 1, the calculated results using standard DWBA do not agree with the experimental data, but a significant improvement is obtained with the modified DWBA by the introduction of a polarization potential. The recoil-to-binary ratio from the experiment data [5] is about 1.4 : 1, whereas the ratio of the two modified DWBA is about 1.3 : 1. In contrast, the ratio is only 1.2 : 1 for using the standard DWBA. The present results obtained using the modified DWBA by introducing Vpol and Vdft have significantly improved those from the standard DWBA, which indicates that the polarization potential is very important. Our theoretical calculations also exhibit that these two peaks locate the same positions as those reported by Stefani et al. [5] experimentally. It worth mentioning that there is no significant difference for our calculated results using Vpol and Vdft , respectively. The calculated TDCS of Ar(2p) with the impact energy of 5720 eV is shown in Fig. 2, in which it can easily be seen that the calculated results show a similar trend as in Fig. 1. The ratio of the two peaks obtained from the calculation taking into account polarization
X.-Y. Hu et al. / Physics Letters A 334 (2005) 192–196
195
Fig. 1. Relative TDCS for the Ar(2p) ionization, E0 = 8256 eV, Es = 7 eV, θf = 1.5◦ : (a) open triangles: experiment by Stefani et al. [5]. The error bars are one standard deviation statistical error. (b) The standard DWBA calculation: dash line; (c) polarization potential Vpol : dotted line; (d) Vdft via DFT: solid line.
Fig. 2. Same as in Fig. 1 but for E0 = 5720 eV, Es = 10 eV, θf = 0.5◦ . Full circles: experiment by Taouil et al. [6]. The meanings of the symbols are the same as in Fig. 1.
is in better agreement with the experiment than that from the DWBA, which indicates the importance of the polarization potential again. However, there is still a deviation between the theoretical calculations and the experimental results in Fig. 2. For example, compared to the experimental data, the calculated recoil
peak seems to be shifted forward with respect to the incident direction by some ∼ 20◦ , whereas the binary peak is shifted backwards by ∼ 30–40◦. Perhaps these deviations come from a small momentum transfer K of only 0.5 au, which is to a regime where Avaldi et al. [4] reported a rapid deterioration in accord be-
196
X.-Y. Hu et al. / Physics Letters A 334 (2005) 192–196
tween DWBA theory and experiment as the momentum transfer is reduced. From Figs. 1 and 2, we can see that the use of Vpol or Vdft appears to significantly improve the DWBA. The results of TDCS show a slight difference between the modified models using the two polarization potential Vpol or Vdft , which may be due to the different treatments of the short-range polarization. The potential Vpol is derived from the model of He empirically, while Vdft , in the near-target region, is obtained by taking into account the nonadiabatic correction via density-functional theory. The advantage of using Vdft is that every term in the expression of Vdft can be written as a functional of the electron density and therefore an even more transparent way of evaluating the polarization potential required in the scattering studies is provided. In our calculations we also can conclude that there is a very strong interaction between the slow electron and the core because of the very large ratio (25 : 1, 35 : 1) of Ar(2p) binding energy to ejected energy. Several numerical tests have been carried out to verify our calculations are sensitive to the choices of various r0 . The value of r0 adopted by Gianturco et al. [8, 14] gives a better description of short-range polarization potential and provides better agreement with the experiment.
4. Conclusions Calculations of the triple differential cross section for electron impact ionization of the Ar(2p) orbital in a highly asymmetric geometry, using modified distorted wave Born approximation methods, are reported. In the present calculations, the ratio of the two peaks obtained from the calculation taking into account polarization is in better agreement with the experiment than that from the DWBA, which shows that the polarization effects are very important for the 2p shell ionization of argon. Consequently, polarization effects must
be incorporated into the DWBA. Although the present work dramatically improves the results of DWBA calculation, deviations between the theoretical calculations and the experimental results still exist. Some higher-order electron–electron interactions have been neglected in the modified approximation. In our future work, we will center on the higher-order electron– electron correlations to improve the calculations.
Acknowledgements The authors acknowledge financial support from the National Natural Science Foundation of China with Grant No. 10274024.
References [1] P. Bickert, Thesis, University of Würzburg, 1991. [2] X. Zhang, C.T. Whelan, H.R.J. Walter, R.J. Allan, P. Bickert, W. Hink, S. Schönberger, J. Phys. B: At. Mol. Opt. Phys. 25 (1992) 4325. [3] A. Lahmam-Bennani, H.F. Wellenstein, A. Duguet, A. Daoud, Phys. Rev. A 30 (1984) 1511. [4] L. Avaldi, R. Camilloni, R. Multari, G. Stefani, X. Zhang, H.R.J. Walters, C.T. Whelan, Phys. Rev. A 48 (1993) 1195. [5] G. Stefani, L. Avaldi, A. Lahmam-Bennani, A. Duguet, J. Phys. B: At. Mol. Opt. Phys. 19 (1986) 3787. [6] L. Taouil, A. Duguet, A. Lahmam-Bennani, B. Lohmann, J. Rasch, C.T. Whelan, H.R.J. Walters, J. Phys. B: At. Mol. Opt. Phys. 32 (1999) L5–L11. [7] C.T. Whelan, R.J. Allan, J. Rasch, H.R.J. Walters, X. Zhang, J. Röder, K. Jung, H. Ehrhardt, Phys. Rev. A 50 (1994) 4394. [8] F.A. Gianturco, J.A. Rodriguez-Ruiz, Phys. Rev. A 47 (1993) 1075. [9] F. Rouet, R.J. Tweed, J. Langlois, J. Phys. B: At. Mol. Opt. Phys. 29 (1996) 1767. [10] I.E. McCarthy, Aust. J. Phys. 48 (1995) 1. [11] F.A. Gianturco, K.T. Tang, J.P. Toennies, D. De Fazio, J.A. Rodriguez-Ruiz, Z. Phys. D 33 (1995) 27. [12] K.T. Tang, J.P. Toennies, J. Chem. Phys. 80 (1984) 3726. [13] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [14] F.A. Gianturco, J.A. Rodriguez-Ruiz, J. Mol. Struct. (Theochem) 260 (1992) 99.
Polarization effect in (e, 2e) collisions of argon Xiao-Ying Hu ∗ , Ya-Jun Zhou, You-Qi Ke, Guang-Jun Nan Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, PR China Received 7 September 2004; received in revised form 8 November 2004; accepted 8 November 2004 Available online 13 November 2004 Communicated by B. Fricke
Abstract Calculations of the triple differential cross section (TDCS) for electron impact ionization of the Ar(2p) orbital in a highly asymmetric geometry, using modified distorted wave Born approximation (DWBA) methods, are reported. The role of the polarization effect in (e, 2e) collisions of Ar(2p) is considered in the calculations, and the calculated results shows that the polarization potential is particularly important. 2004 Elsevier B.V. All rights reserved. PACS: 34.80.Bm Keywords: (e, 2e) reaction; Polarization potential; Distorted wave Born approximation
1. Introduction In recent years, a number of new directions have been pursed in (e, 2e) investigations of electron-impact ionization processes. Much attention has been concentrated on atomic outer shells, particularly with simple targets, and the less bound intermediate shells of atoms and molecules. However, because of the smallness of the cross section the core region has been the subject of fewer studies, although the dynamical studies of inner-shell ionization in neon, argon [1–3] and xenon [4] revealed a number of interesting features of the * Corresponding author.
E-mail address: [email protected] (X.-Y. Hu). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.11.007
scattering which were distinctly different to outer-shell ionization. Experiment results [5,6] were reported for (e, 2e) ionization of the argon 2p orbital in a highly asymmetric geometry, which showed that the theoretical results obtained using the DWBA method had significant differences, compared with experimental data, in the overall shape and magnitude of the triple differential cross section. The recoil-to-binary ratio in relative TDCS for the Ar(2p) ionization was significantly underestimated by the DWBA calculations. The double peaks using the standard DWBA calculations were approximately symmetric about the direction of mo while the experimental results mentum transfer ±K, did not show this feature clearly. The calculated binary and recoil peaks showed no structure where the
X.-Y. Hu et al. / Physics Letters A 334 (2005) 192–196
experiments might possibly show some. All of these differences present a substantial challenge to theory. The discrepancy is probably due to neglecting to consider the higher-order electron–electron correlations in the DWBA method. Thus, a new dynamical mechanism and the modification of the DWBA method are needed to explain the experimental data properly. In order to test the significance of polarization, Whelan et al. [7] modified the DWBA calculations by introducing an ad hoc polarization potential Vpol in (e, 2e) collisions of He. In this Letter, we have extended the model proposed by Whelan et al. [7] and added another potential Vdft via density-functional theory (DFT) [8] for argon. We aim at showing the influence of the polarization potential [9] on the calculated TDCS of Ar(2p). In order to test the validity of our model, we have calculated the TDCS with the incident energy of 8256 and 5720 eV, respectively, in a highly asymmetric geometry. The calculated results have been compared with those obtained by the standard DWBA and the experimental data [5,6].
2. Theory The triple differential cross section in the distortedwave Born approximation is given by [10] 2 kf ks d 3σ kf ks |T |α k0 , = (2π)4 df ds dE0 k0 av (1) where the subscript f and s represent the fast and slow electrons, respectively. E is kinetic energy, k momentum, E0 the kinetic energy of the incident electron. The incident electron with momentum k0 incidents on an uncharged system consisting of electron s bound to the core in the state α with separation energy εα . The notation av represents a sum over the final state and average over the initial magnetic and spin degeneracy state. In the calculation, the atomic units (h¯ = me = e = 1) are used. The DWBA T-matrix element is kf ks |T |α k0 = χ (−) (kf )χ (−) (ks )vf s αχ (+) (k0 ) , (2) where vf s is the interaction potential between the fast and slow electron. |χ (−) (kf ), |χ (−) (ks ) and
193
|χ (+) (k0 ) are the distorted waves at different asymptotic conditions. The superscripts refer to the outgoing and incident waves, respectively. The partial-wave expansions of the distorted waves can be written as (−) r χ (k) 1/2 2 1 ∗ ˆ LM = i −L eiσL uL (k, r)YLM (k)Y (ˆr ), π kr LM (3) (+) r χ (k) 1/2 1 2 ∗ ˆ LM (ˆr ), (4) i L uL (k, r)YLM (k)Y = π kr LM
where σL is the Coulomb phase shift. The coordinate representation of the orbital |α is r |α = r −1 unl YLM (ˆr ),
(5)
and the partial wave µL (k, r) can be obtained by solving the differential equation as follows: 2 L(L + 1) η d − − dρ 2 ρ2 ρ v(r) + VP (r) + 1 uL (k, r) = 0, − (6) E where the potential v(r) includes the direct distorting potentials VD (r) and the spin-averaged staticexchange potential VE (r), ρ = kr, η = −Z/k, E = 1 2 2 k , VP (r) is the polarization potential which we introduce in our calculations. The asymptotic form of the adiabatic polarization potential is simply given by the well-know secondorder perturbation expansion formula [11,12]. VpT (r) ∼ − r→∞
∞ l=1
αl , 2r 2l+2
for r > r0 ,
(7)
αl is the multipolar polarizability of the target atom and r0 separates the outer region from the inner shortrange region. In the near-target region, the main drawback of the expansion (7) is that it grossly overestimates polarization and incorrectly diverges near the origin of the force field. According to Whelan et al. [7], we only consider the dipole term. Hence the polarization po-
194
X.-Y. Hu et al. / Physics Letters A 334 (2005) 192–196
tential Vpol becomes VP (r) = Vpol =
α0 − 2r 4,
− α04 , 2r0
3. Results and discussion r > r0 , r r0 ,
(8)
where α0 is the dipole polarizability of neutral Ar for the incident channel, and the polarizability of Ar+ for the outgoing channels. In both cases we take the cutoff radius r0 = 3.16a0 according to Refs. [7,8]. In the near-target region, Gianturco et al. [8] took the nonadiabatic corrections into account. On basis of the model suggested by Lee, Young and Parr (LYP) [13], they obtained the polarization potential via the density-functional theory in near-target region as follows, 8 DFT = −a(F1 ρ + F1 ) − abCF ρ 5/3 G1 ρ + G1 V(r) 3
ab − G1 ρ|∇ρ|2 + G1 3|∇ρ|2 + 2ρ∇ 2 ρ 4 + 4G1 ∇ 2 ρ −
ab 3G1 ρ|∇ρ|2 + G1 5|∇ρ|2 + 6ρ∇ 2 ρ 72 + 4G1 ∇ 2 ρ , for r < r0 , (9)
where −1 F1 ρ(r) = 1 + dρ −1/3 ,
G1 ρ(r) = F1 (ρ)−5/3 exp cρ −1/3 , CF =
3 2 2/3 3π , 10
(10a) (10b) (10c)
ρ is the electron density. The three parameters (a)–(c) are the same as in [14]. The formula of Eq. (9) for the short-range polarization potential is a pure density functional and can be easily obtained. The final total polarization potential Vdft via DFT approach is VP (r) = Vdft =
DFT , V(r)
when r r0 ,
VpT (r), when r > r0 .
(11)
In Eq. (11) r0 = 5.39a0 according to Ref. [8,14]. Thus, DFT can be nicely connected the potential function of V(r) with VpT (r).
We have calculated the TDCS of Ar(2p) in a highly asymmetric geometry. The impact energy of 8256 and 5720 eV is chosen, respectively, and correspondingly the polar angle of the fast electron is settled at θf = 1.5◦ and θf = 0.5◦ , respectively. The only variable parameter is the polar angle θs of the ejected electron, which is scanned from 0◦ to 360◦ . This is the experimental condition adopted by Stefani et al. and Taouil et al. [5,6], which is also used in our calculations so that our theoretical results can be compared with the experimental data. The experimental data are not absolute, so only shapes can be compared. Hence we normalize the present calculated results using Vdft to the experimental binary peak height. The results of the standard DWBA and using Vpol are relative to the calculations of using Vdft . The calculated TDCS of Ar(2p) with the impact energy of 8256 eV is shown in Fig. 1. We show the TDCS in coplanar asymmetric geometry with three different the DWBA approximations. (a) Standard DWBA without polarization potential (dash line). (b) DWBA with polarization potential Vpol (dotted line). (c) DWBA with polarization potential Vdft via density-functional theory (solid line). In Fig. 1, the calculated results using standard DWBA do not agree with the experimental data, but a significant improvement is obtained with the modified DWBA by the introduction of a polarization potential. The recoil-to-binary ratio from the experiment data [5] is about 1.4 : 1, whereas the ratio of the two modified DWBA is about 1.3 : 1. In contrast, the ratio is only 1.2 : 1 for using the standard DWBA. The present results obtained using the modified DWBA by introducing Vpol and Vdft have significantly improved those from the standard DWBA, which indicates that the polarization potential is very important. Our theoretical calculations also exhibit that these two peaks locate the same positions as those reported by Stefani et al. [5] experimentally. It worth mentioning that there is no significant difference for our calculated results using Vpol and Vdft , respectively. The calculated TDCS of Ar(2p) with the impact energy of 5720 eV is shown in Fig. 2, in which it can easily be seen that the calculated results show a similar trend as in Fig. 1. The ratio of the two peaks obtained from the calculation taking into account polarization
X.-Y. Hu et al. / Physics Letters A 334 (2005) 192–196
195
Fig. 1. Relative TDCS for the Ar(2p) ionization, E0 = 8256 eV, Es = 7 eV, θf = 1.5◦ : (a) open triangles: experiment by Stefani et al. [5]. The error bars are one standard deviation statistical error. (b) The standard DWBA calculation: dash line; (c) polarization potential Vpol : dotted line; (d) Vdft via DFT: solid line.
Fig. 2. Same as in Fig. 1 but for E0 = 5720 eV, Es = 10 eV, θf = 0.5◦ . Full circles: experiment by Taouil et al. [6]. The meanings of the symbols are the same as in Fig. 1.
is in better agreement with the experiment than that from the DWBA, which indicates the importance of the polarization potential again. However, there is still a deviation between the theoretical calculations and the experimental results in Fig. 2. For example, compared to the experimental data, the calculated recoil
peak seems to be shifted forward with respect to the incident direction by some ∼ 20◦ , whereas the binary peak is shifted backwards by ∼ 30–40◦. Perhaps these deviations come from a small momentum transfer K of only 0.5 au, which is to a regime where Avaldi et al. [4] reported a rapid deterioration in accord be-
196
X.-Y. Hu et al. / Physics Letters A 334 (2005) 192–196
tween DWBA theory and experiment as the momentum transfer is reduced. From Figs. 1 and 2, we can see that the use of Vpol or Vdft appears to significantly improve the DWBA. The results of TDCS show a slight difference between the modified models using the two polarization potential Vpol or Vdft , which may be due to the different treatments of the short-range polarization. The potential Vpol is derived from the model of He empirically, while Vdft , in the near-target region, is obtained by taking into account the nonadiabatic correction via density-functional theory. The advantage of using Vdft is that every term in the expression of Vdft can be written as a functional of the electron density and therefore an even more transparent way of evaluating the polarization potential required in the scattering studies is provided. In our calculations we also can conclude that there is a very strong interaction between the slow electron and the core because of the very large ratio (25 : 1, 35 : 1) of Ar(2p) binding energy to ejected energy. Several numerical tests have been carried out to verify our calculations are sensitive to the choices of various r0 . The value of r0 adopted by Gianturco et al. [8, 14] gives a better description of short-range polarization potential and provides better agreement with the experiment.
4. Conclusions Calculations of the triple differential cross section for electron impact ionization of the Ar(2p) orbital in a highly asymmetric geometry, using modified distorted wave Born approximation methods, are reported. In the present calculations, the ratio of the two peaks obtained from the calculation taking into account polarization is in better agreement with the experiment than that from the DWBA, which shows that the polarization effects are very important for the 2p shell ionization of argon. Consequently, polarization effects must
be incorporated into the DWBA. Although the present work dramatically improves the results of DWBA calculation, deviations between the theoretical calculations and the experimental results still exist. Some higher-order electron–electron interactions have been neglected in the modified approximation. In our future work, we will center on the higher-order electron– electron correlations to improve the calculations.
Acknowledgements The authors acknowledge financial support from the National Natural Science Foundation of China with Grant No. 10274024.
References [1] P. Bickert, Thesis, University of Würzburg, 1991. [2] X. Zhang, C.T. Whelan, H.R.J. Walter, R.J. Allan, P. Bickert, W. Hink, S. Schönberger, J. Phys. B: At. Mol. Opt. Phys. 25 (1992) 4325. [3] A. Lahmam-Bennani, H.F. Wellenstein, A. Duguet, A. Daoud, Phys. Rev. A 30 (1984) 1511. [4] L. Avaldi, R. Camilloni, R. Multari, G. Stefani, X. Zhang, H.R.J. Walters, C.T. Whelan, Phys. Rev. A 48 (1993) 1195. [5] G. Stefani, L. Avaldi, A. Lahmam-Bennani, A. Duguet, J. Phys. B: At. Mol. Opt. Phys. 19 (1986) 3787. [6] L. Taouil, A. Duguet, A. Lahmam-Bennani, B. Lohmann, J. Rasch, C.T. Whelan, H.R.J. Walters, J. Phys. B: At. Mol. Opt. Phys. 32 (1999) L5–L11. [7] C.T. Whelan, R.J. Allan, J. Rasch, H.R.J. Walters, X. Zhang, J. Röder, K. Jung, H. Ehrhardt, Phys. Rev. A 50 (1994) 4394. [8] F.A. Gianturco, J.A. Rodriguez-Ruiz, Phys. Rev. A 47 (1993) 1075. [9] F. Rouet, R.J. Tweed, J. Langlois, J. Phys. B: At. Mol. Opt. Phys. 29 (1996) 1767. [10] I.E. McCarthy, Aust. J. Phys. 48 (1995) 1. [11] F.A. Gianturco, K.T. Tang, J.P. Toennies, D. De Fazio, J.A. Rodriguez-Ruiz, Z. Phys. D 33 (1995) 27. [12] K.T. Tang, J.P. Toennies, J. Chem. Phys. 80 (1984) 3726. [13] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [14] F.A. Gianturco, J.A. Rodriguez-Ruiz, J. Mol. Struct. (Theochem) 260 (1992) 99.