- Email: [email protected]
Cross sections by electron impact excitation and ionization of tungsten ions, W39+
Journal of Electron Spectroscopy and Related Phenomena 234 (2019) 86–90
Contents lists available at ScienceDirect
Journal of Electron Spectroscopy and Related Phenomena journal homepage: www.elsevier.com/locate/elspec
Cross sections by electron impact excitation and ionization of tungsten ions, W39+
T
⁎
A.A.A. Husseina, , Abdel-Haleem Abdel-Atyb, Udai Al-Jubooric, El Sayed Yousefd,e a
Physics Department, College of Science, Jouf University, P.O. Box 2014, Skaka, Saudi Arabia Department of Physics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia c College of Arts and Science, Applied Science University, Bahrain d Department of Physics, Faculty of Sciences, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia e Research Center for Advanced Materials Science (RCAMS), King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia b
A R T I C LE I N FO
A B S T R A C T
Keywords: Cross section R-matrix Excitation Ionization Tungsten
Excitation cross sections by electron impact of tungsten ions, W39+, have been calculated using R-matrix method for a configuration state list consists of [Ar]3d104s24p5, [Ar]3d104s24p44l, (l = d, f), and [Ar]3d104s24p45l′, l′ = s, p. A set of configuration state functions (CSFs) with 2-electron excitations up to 5l orbitals have been included within the MCDF-calculations of the used radial integrals of collisional excitations. R-matrix calculations of collision strength and cross section have been executed for the intercombination transitions between 4p5, 4p44d, and 4p44f at different angular momenta using DARC code. Resonance cross section have been observed at low energy of incident electron in the range of ∼5=-40 Ryd, however, the strong lines limited between 5 and 25 Ryd. The relativistic distorted-wave (DW) approximation, Coulomb–Born-Exchange (CBE) approximation, and the Binary-Encounter-Dipole (BED) theory have been used to calculate the electron impact ionization (EII) of the present ion. The accuracies of the present results are estimated by comparing R-matrix calculations with the available literature.
1. Introduction Tungsten is a favorable material for several magnetic confinement fusion facilities, due to its high melting point, excellent thermal conductivity, low sputtering rate and low retention of tritium. Tungsten has been used as a plasma-facing material, and is an important candidate material or wall material of the International Thermonuclear Experimental Reactor (ITER) tokamak due to its favorable properties [1–5]. The atomic structure and collisional processes whether it was excitation, ionization or recombination of different ionic species of tungsten have been studied experimentally and theoretically in several works, see for example, [6–14], but some focused on W39+ such as Refs. [15,16]. Electron impact excitation cross sections and linear polarization of W39+ have been performed by Priti et al. [15]. Atomic structure and spectroscopic properties of W39+ have been calculated using quasirelativistic Hartree–Fock method [17]. Experimental measurements of extreme ultraviolet (EUV) spectral lines of tungsten ions including W39+ using electron beam ion trap technique have been performed by Das et al. [18]. Short sequence of tungsten ions, W29+ through W43+
⁎
have been theoretically studied using multiconfiguration Dirac–Fock (MCDF) approach, the energy levels and radiative rates of electronic quadrupole (E2) and magnetic dipole (M1) transitions have been calculated [19]. The previous argument distinctly shows that, despite the efforts to fill the empty space of required accurate atomic data for tungsten, these valuable published data still limited. Most of previous works provided results of collision strengths or cross sections for just few low-lying transitions. In the present work, we performed calculations of electron impact excitation (EIE) and electron impact ionization (EII) of highly charged tungsten ion, W39+ using R-matrix method as implemented in the DARC code [20] and using FAC code [21]. 2. Theoretical approach and computations The multiconfiguration Dirac–Fock (MCDF) method as implemented in the computer package GRASP0 [22] has been used in parallel with Rmatrix method to construct the radial integrals used in the collisional calculations of W39+. The MCDF method depends on the Dirac–Coulomb Hamiltonian in the form
Corresponding author. E-mail address: [email protected] (A.A.A. Hussein).
https://doi.org/10.1016/j.elspec.2019.05.010 Received 9 March 2019; Received in revised form 22 May 2019; Accepted 24 May 2019 Available online 01 June 2019 0368-2048/ © 2019 Elsevier B.V. All rights reserved.
Journal of Electron Spectroscopy and Related Phenomena 234 (2019) 86–90
A.A.A. Hussein, et al. N
HDC =
N
1
∑ (cαi·pi) + (βi − 1) c 2 + Vnuc (ri) + ∑ r i=1
i>j
ij
(1)
where c is the speed of light in atomic units, α and β are the 4 × 4 Dirac matrices and Vnuc(ri) is the potential due to the nucleus. R-matrix theory [23,24,20,25,26] has been used in the calculations of collision strengths and excitation cross sections of the present ion (W39+). The contribution of collision strength for a certain transition i → j (Ωij) to the cross section (σi→j) is given by:
σi → j =
πa02 IH Ωij ki2 gi
(2)
where gi = 2J + 1 and a0 is the Bohr radius. ki is the wave number of the incident electron and IH is the ionization potential of the hydrogen atom. In this method the Dirac Hamiltonian (Eq. (1)) for the (N + 1)electron atomic system is applied to construct bound state and target wave functions. The bound state (W39+) wave functions of the (N + 1)-electron atomic system are constructed using the multi-configurational DiracFock package (GRASP0) [22] in parallel with the collisional calculations using R-matrix method. Here, we considered the orbitals up to 3d10 as closed core, one electron (S) and two electrons (SD) excitations from the ground state (4s24p5) to 4l-orbitals plus the single excitations to 5l orbitals as a procedure to obtain the radial integrals using MCDF technique. Then, the 89 levels generated from the configurations 4p5, 4p44d, 4p44f, 4p45s, and 4p45p at different angular momenta have been included in the collisional calculations. Both configuration interaction (CI) and close-coupling (CC) expansions of the target have included the 89-level generated from the mentioned configurations above. All partial waves of odd and even states with 2J = 1–19 of the target expansion were included in the JΠ partial-wave expansion, where resonances may occur in the energy region of 15–25 Ryd. A fine mesh consists of 3200 mesh points by an energy mesh spacing of 0.02 Ryd in the resonance region for all partial waves has been performed. The size of Rmatrix boundary of W39+ is 2.2 au. In the second approach of collisional excitation calculations, the excitations of 2-electrons up to 4l levels are performed. Where, the 60-level generated from the configurations of 4p5, 4p44d, 4p44f, 4p34d2, and 4p34f2 have been considered as a 4l-DARC approach. In this case the R-matrix boundary is reduced to be 1.48 au. The electron impact ionization (EII) cross sections have been calculated using the flexible atomic code (FAC) [21]. FAC depends on diagonalizing the relativistic Hamiltonian H (in atomic units), this Hamiltonian is a combination of Dirac Hamiltonian of the multi-electronic systems (Eq. (1)) plus the relativistic correction. In this approximation, the configuration state functions (CSFs) of the ion under consideration are described by basis of states Φν of antisymmetric sums of the products of N one-electron Dirac spinor φnkm
φnkm =
1 ⎛ Pnk (r ) χkm (θ , ϕ, σ ) ⎞ ⎜ ⎟ r ⎝ iQ nk (r ) χ−km (θ , ϕ, σ ) ⎠
Fig. 1. Cross sections of the transition (4p5)3/2 → (4p5)1/2. Solid line for the obtained data from excitation up to 4l-orbitals while the dashed line for 5lexcitation.
have been applied in the calculations of direct electron impact ionization cross section. 3. Results and discussions 3.1. Electron impact excitation Cross sections of some transitions in W39+ (in units of cm2) are shown in Figs. 1–8. The present cross section calculations of these transitions have been performed based on R-matrix theory by using DARC code. For example, cross section of the transition 4p5(3/2)4p5(1/2), as a function of incident electron energy (in Ryd) are shown in Fig. 1. It is clear that, the resonance cross sections are observed at the region of 5–25 Ryd, while the rest of curve represents the direct excitation cross sections in the region of ∼25–40 Ryd. The cross sections of the transitions from ground state of W39+ to [(4p2)24d3/2]3/2, [(4p2)24d3/2]5/2, [(4p2)24d3/2]1/2, and [(4p2)04d5/2]5/2 are shown in Figs. 2–5. As shown in the figures, it is clear that the resonance background of cross sections dominate the incident electron energy in the range of ∼5–25 Ryd. As well as, there is a good agreement between the direct excitation cross sections produced from the 4l- and 5l-DARC approaches. The present calculations of cross section of the transitions 4p5(3/2)-[(4p2)24d5/2]5/2 have been compared with the calculated values using relativistic distorted wave approximation (RDW) [15] in Fig. 8. As it clear, there is an excellent agreement between the current
(3)
where χkm is the spin angular function, n is the principal quantum number, and k is the relativistic angular quantum number, which is written as:
k = (l − j )(2j + 1),
(4)
m is the z-component of the total angular momentum j and l is the orbital angular momentum. The atomic state functions (ASF) ψ are given by linear combination of the basis states Φν with same symmetries
ψ=
∑ bν Φν ν
(5)
where bν are the mixing coefficients obtained from diagonalizing the Hamiltonian matrix. The Distorted Wave (DW), Coulomb–Born-Exchange (CBE), and the Binary-Encounter-Dipole (BED) approximations
Fig. 2. The same as Fig. 1 for the transition (4p5)3/2 → [(4p2)24d3/2]3/2. 87
Journal of Electron Spectroscopy and Related Phenomena 234 (2019) 86–90
A.A.A. Hussein, et al.
Fig. 3. The same as Fig. 1 for the transition (4p5)3/2 → [(4p2)24d3/2]5/2.
Fig. 6. The same as Fig. 1 for the transition (4p5)3/2 → [(4p1/24p33/2)24d3/2]3/2.
Fig. 4. The same as Fig. 1 for the transition (4p5)3/2 → [(4p2)24d3/2]1/2.
Fig. 7. The same as Fig. 1 for the transition (4p5)3/2 → (4p3/2)3/2(4d3/2)3(4d5/ 2)9/2.
Fig. 5. The same as Fig. 1 for the transition (4p5)3/2 → [(4p2)04d5/2]5/2.
Fig. 8. Comparison between the present calculations of cross sections (solid line) and those of Priti et al. [15] (black squares) for the transition (4p5)3/2 → [(4p2)24d5/2]5/2.
cross sections an those in Ref. [15]. For high excitations such as the transition to [(4p1/24p33/2)2 4d3/2]3/ 2, the agreement between the 4l- and 5l-excitations show a noticeable discrepancy, where a difference of 0.05×10−20 cm2 is observed in Fig. 6. The differences between the background collisional excitation cross sections depend on many factors, such as, the bound state wave functions calculated using GRASP code are accurate to at least one part
in 106 while the solution of the free particle Dirac–Fock equation converges to one part in 105, or because the partial wave number of continuum electron, where the radial integration of the continuum functions does not converge to the function on the surface boundary. As well as, the radial integration involving these functions is affected by
88
Journal of Electron Spectroscopy and Related Phenomena 234 (2019) 86–90
A.A.A. Hussein, et al.
the accuracy of integrands and the integration method [15]. The cross sections of the excitation to higher levels such as the transition from ground state to (4p3/2)3/2(4d3/2)3(4d5/2)9/2 did not show resonance in the energy range of incident electrons, see Fig. 7, in such cases this is familiar because the probability of electron capture by higher levels will be more difficult comparable to lower levels. 3.2. Electron impact ionization If tungsten ion (W39+) is hit by an electron has sufficiently high collision energy this process may remove one or more (n) target electrons from the ion.
[Ar]3d10 4s 2 4p5
+
e−
10 2 4 − ⎧3d 4s 4p + 2e 10 2 3 → [Ar] 3d 4s 4p 4d + 2e− ⎨ 10 2 3 − ⎩3d 4s 4p 4f + 2e
Fig. 10. Electron impact ionization cross section from the ground state of W39+ to 4d target states with J = 2, 3. Solid lines for BED-calculations and dash lines formats for CB approximation. The dot lines for DW approximation.
In the direct ionization, a bound electron of the target ion may obtain energy by the collision with the incident electron more than its binding energy, so that the target electron is released. This technique is known as single ionization of the parent Wq+ and is known as direct ionization. But if the previous process being done in two stages, one of them was mentioned above already, the excitation of an inner-shell electron with subsequent autoionization. This type of ionization is usually known as excitation-autoionization (EA), the first stage of excitation-autoionization is carried out through the formation of an intermediate resonance state [W q +]* . In this case, an inner-shell electron will be excited, and the ejected electron will be captured to a bound state. The resulting short lived recombined ion state is highly excited so that two electrons can be expelled in the relaxation process. Finally, one electron will be lost in the whole process [27]. The ionization cross sections of 4p, 4d and 4f electrons for the ground state of W39+ calculated using FAC code have been computed in the incident electron energy range of 50–600 Ryd. For example, the cross sections of 4p4 with J = 0,1, and 2 are shown in Fig. 9. As well as, the EII cross sections of 4p34dJ=2,3 and for 4p34fJ=2,3,4 are presented in Figs. 10 and 11, respectively. The produced results from the Relativistic Distorted-Wave (DW), Coulomb–Born-Exchange (CBE), and the BinaryEncounter-Dipole (BED) approximations are plotted together to estimate the differences and then the accuracy of the EII cross section. In Fig. 9 the ionization cross sections are shown side by side for the three used methods, i.e. BED, CB, and DW. It is clear that a reasonable agreement between BED and CB approaches is noticeable, while the DW-values show large differences from the other calculations. The reason for why the DW-cross sections are much smaller may be due to
Fig. 11. Electron impact ionization cross section from the ground state (4p5 J = 1/2) of W39+ to 4f target states with J = 2, 3, and 4. Solid lines for BEDcalculations and dash lines formats for CB approximation. The dot lines for DW approximation.
the radiative damping effects in the direct ionizations are significantly considered [28,29]. As shown in Figs. 9–11, the ionization cross sections calculated using BED-method are smaller at near threshold energies as compared using CB-calculations. This is due to that, when the calculation of the ionization radial integrals using BED approach is applied, the total ionization cross sections are scaled by a factor of E/ (E + I), where E is the energy of incident electron, and I is the ionization threshold energy [14]. 4. Conclusion The current study provides theoretical calculations of electron impact excitation and ionization cross sections of tungsten ions, W39+. Excitations up to 5l-orbitals are taken into account in the calculations of collision strengths, hence, cross sections of the transitions under consideration. R-matrix calculations of cross sections have been performed for the transitions between the first 89-low-lying levels at electron energy range of 5–70 Ryd, and compared with the available literature. The electron impact ionization cross sections were computed for the present ion. The EII cross sections have been performed in the energy range of 50–600 Ryd using the CB, BED, and DW approximations. The comparisons show good agreement between different calculations using Coulomb–Born-Exchange (CBE) and the Binary-Encounter-Dipole
Fig. 9. Electron impact ionization cross section from the ground state of W39+ (N + 1) to the next ionization stage (N). Solid lines for BED-calculations and dash lines formats for CB approximation. The dot lines for DW approximation. 89
Journal of Electron Spectroscopy and Related Phenomena 234 (2019) 86–90
A.A.A. Hussein, et al.
(BED), while noticeable discrepancies of the calculated ionization cross sections using Distorted-Wave (DW) approximation are recorded.
551–570. [13] X. Guo, J. Grumer, T. Brage, R. Si, C. Chen, P. Jönsson, K. Wang, J. Yan, R. Hutton, Y. Zou, Energy levels and radiative data for Kr-like W38+ from MCDHF and RMBPT calculations, J. Phys. B 49 (2016) 135003. [14] A. El-Maaref, M.A. Halaka, M. Tammam, E. Shaaban, E.S. Yousef, Atomic structure and excitation cross section by electron impact of tungsten ion, W38+, J. Phys. B 52 (2019) 065202. [15] Priti, L. Sharma, R. Srivastava, Study of electron excitation of Rb-like to Br-like tungsten ions and polarization of their photon emission, Eur. Phys. J. D 71 (2017) 100. [16] K. Wang, C. Zhang, R. Si, S. Li, Z. Chen, X. Zhao, C. Chen, J. Yan, Energy levels, lifetimes and radiative rates for transitions in the bromine isoelectronic sequence La XXIII-Dy XXXII, W XL, At. Data Nucl. Data Tables 123–124 (2018) 114–167. [17] P. Bogdanovich, R. Karpuškienė, R. Kisielius, Energy spectra of the tungsten ion 4s24pN, 4s24pN−14d and 4s4pN+1 configurations, Lith. J. Phys. 55 (2015) 162–173. [18] T. Das, Y.A. Podpaly, J. Reader, J.D. Gillaspy, Y. Ralchenko, Analysis of EUV spectra from N-shell tungsten ions observed with an electron beam ion trap, Eur. Phys. J. D 72 (2018) 124. [19] P. Quinet, A theoretical survey of atomic structure and forbidden transitions in the 4p4k and 4dk ground configurations of tungsten ions W29+ through W43+, J. Phys. B: At. Mol. Opt. Phys. 45 (2012) 025003. [20] K.A. Berrington, W.B. Eissner, P.H. Norrington, RMATRX1: Belfast atomic R-matrix codes, Comput. Phys. Commun. 92 (1995) 290–420. [21] M.F. Gu, The flexible atomic code, Can. J. Phys. 86 (2008) 675–689. [22] I. Grant, B. McKenzie, P. Norrington, D. Mayers, N. Pyper, An atomic multiconfigurational Dirac–Fock package, Comput. Phys. Commun. 21 (1980) 207–231. [23] P.G. Burke, A. Hibbert, W.D. Robb, Electron scattering by complex atoms, J. Phys. B 4 (1971) 153. [24] K. Berrington, P. Burke, J. Chang, A. Chivers, W. Robb, K. Taylor, A general program to calculate atomic continuum processes using the R-matrix method, Comput. Phys. Commun. 8 (1974) 149–198. [25] P.G. Burke, K.T. Taylor, R-matrix theory of photoionization. Application to neon and argon, J. Phys. B 8 (1975) 2620. [26] P.G. Burke, R-Matrix Theory of Atomic Collisions, Springer-Verlag Berlin Heidelberg, 2011. [27] A. Müller, Electron-Ion Collisions: Fundamental Processes in the Focus of Applied Research, vol. 55 of Advances in Atomic, Molecular, and Optical Physics, Academic Press, 2008, pp. 293–417. [28] S. Pakalka, S. Kučas, Š. Masys, A. Kynienė, A. Momkauskaitė, V. Jonauskas, Electron-impact single ionization of the se3+ ion, Phys. Rev. A 97 (2018) 012708. [29] L. Liu, P. Liu, J. Zeng, Electron impact ionization cross sections of the ground and excited levels of se3+, J. Electron Spectrosc. Relat. Phenom. 209 (2016) 9–13.
Acknowledgement The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University (KKU) for funding this research project number (R.G.P1/81/40). References [1] D.-H. Kwon, W. Lee, Dielectronic recombination of Ni-, Cu-, and Zn-like tungsten ions, J. Quant. Spectrosc. Radiat. Transf. 179 (2016) 98–104. [2] D.-H. Kwon, W. Lee, Dielectronic recombination of Cu-like W45+, J. Quant. Spectrosc. Radiat. Transf. 170 (2016) 182–188. [3] X. Ding, I. Murakami, D. Kato, H.A. Sakaue, F. Koike, C. Dong, Collisional-radiative modeling of W27+, Plasma Fusion Res. 7 (2012) 2403128 (4 pp.). [4] Z. Wu, Y. Fu, X. Ma, M. Li, L. Xie, J. Jiang, C. Dong, Electron impact excitation and dielectronic recombination of highly charged tungsten ions, Atoms 3 (2015) 474–494. [5] B. Naranjo, J. Gimzewski, S. Putterman, Observation of nuclear fusion driven by a pyroelectric crystal, Nature 434 (2005) 1115–1117. [6] L. Xie, X. Ma, C. Dong, Z. Wu, Y. Shi, J. Jiang, Polarization of the nf → 3d (n = 4, 5, 6) X-rays from tungsten ions following electron-impact excitation and dielectronic recombination processes, J. Quant. Spectrosc. Radiat. Transf. 141 (2014) 31–39. [7] M.M. Bluteau, M.G. O’Mullane, N.R. Badnell, Dirac R-matrix and Breit–Pauli distorted wave calculations of the electron-impact excitation of W44+, J. Phys. B 48 (2015) 195701. [8] A. El-Maaref, M.A. Halaka, M. Tammam, E. Shaaban, E.S. Yousef, Electron impact excitation and ionization cross section of tungsten ions, W44+, J. Quant. Spectrosc. Radiat. Transf. 224 (2019) 147–153. [9] H.A. Sakaue, D. Kato, N. Yamamoto, N. Nakamura, I. Murakami, Spectra of W19+–W32+ observed in the EUV region between 15 and 55 Å with an electronbeam ion trap, Phys. Rev. A 92 (2015) 012504. [10] I. Murakami, H. Sakaue, C. Suzuki, D. Kato, M. Goto, N. Tamura, S. Sudo, S. Morita, Development of quantitative atomic modeling for tungsten transport study using LHD plasma with tungsten pellet injection, Nucl. Fusion 55 (2015) 093016. [11] A. Kramida, T. Shirai, Energy levels and spectral lines of tungsten, W III through W LXXIV, At. Data Nucl. Data Tables 95 (2009) 305–474. [12] A. Kramida, Recent progress in spectroscopy of tungsten, Can. J. Phys. 89 (2011)
90
Contents lists available at ScienceDirect
Journal of Electron Spectroscopy and Related Phenomena journal homepage: www.elsevier.com/locate/elspec
Cross sections by electron impact excitation and ionization of tungsten ions, W39+
T
⁎
A.A.A. Husseina, , Abdel-Haleem Abdel-Atyb, Udai Al-Jubooric, El Sayed Yousefd,e a
Physics Department, College of Science, Jouf University, P.O. Box 2014, Skaka, Saudi Arabia Department of Physics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia c College of Arts and Science, Applied Science University, Bahrain d Department of Physics, Faculty of Sciences, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia e Research Center for Advanced Materials Science (RCAMS), King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia b
A R T I C LE I N FO
A B S T R A C T
Keywords: Cross section R-matrix Excitation Ionization Tungsten
Excitation cross sections by electron impact of tungsten ions, W39+, have been calculated using R-matrix method for a configuration state list consists of [Ar]3d104s24p5, [Ar]3d104s24p44l, (l = d, f), and [Ar]3d104s24p45l′, l′ = s, p. A set of configuration state functions (CSFs) with 2-electron excitations up to 5l orbitals have been included within the MCDF-calculations of the used radial integrals of collisional excitations. R-matrix calculations of collision strength and cross section have been executed for the intercombination transitions between 4p5, 4p44d, and 4p44f at different angular momenta using DARC code. Resonance cross section have been observed at low energy of incident electron in the range of ∼5=-40 Ryd, however, the strong lines limited between 5 and 25 Ryd. The relativistic distorted-wave (DW) approximation, Coulomb–Born-Exchange (CBE) approximation, and the Binary-Encounter-Dipole (BED) theory have been used to calculate the electron impact ionization (EII) of the present ion. The accuracies of the present results are estimated by comparing R-matrix calculations with the available literature.
1. Introduction Tungsten is a favorable material for several magnetic confinement fusion facilities, due to its high melting point, excellent thermal conductivity, low sputtering rate and low retention of tritium. Tungsten has been used as a plasma-facing material, and is an important candidate material or wall material of the International Thermonuclear Experimental Reactor (ITER) tokamak due to its favorable properties [1–5]. The atomic structure and collisional processes whether it was excitation, ionization or recombination of different ionic species of tungsten have been studied experimentally and theoretically in several works, see for example, [6–14], but some focused on W39+ such as Refs. [15,16]. Electron impact excitation cross sections and linear polarization of W39+ have been performed by Priti et al. [15]. Atomic structure and spectroscopic properties of W39+ have been calculated using quasirelativistic Hartree–Fock method [17]. Experimental measurements of extreme ultraviolet (EUV) spectral lines of tungsten ions including W39+ using electron beam ion trap technique have been performed by Das et al. [18]. Short sequence of tungsten ions, W29+ through W43+
⁎
have been theoretically studied using multiconfiguration Dirac–Fock (MCDF) approach, the energy levels and radiative rates of electronic quadrupole (E2) and magnetic dipole (M1) transitions have been calculated [19]. The previous argument distinctly shows that, despite the efforts to fill the empty space of required accurate atomic data for tungsten, these valuable published data still limited. Most of previous works provided results of collision strengths or cross sections for just few low-lying transitions. In the present work, we performed calculations of electron impact excitation (EIE) and electron impact ionization (EII) of highly charged tungsten ion, W39+ using R-matrix method as implemented in the DARC code [20] and using FAC code [21]. 2. Theoretical approach and computations The multiconfiguration Dirac–Fock (MCDF) method as implemented in the computer package GRASP0 [22] has been used in parallel with Rmatrix method to construct the radial integrals used in the collisional calculations of W39+. The MCDF method depends on the Dirac–Coulomb Hamiltonian in the form
Corresponding author. E-mail address: [email protected] (A.A.A. Hussein).
https://doi.org/10.1016/j.elspec.2019.05.010 Received 9 March 2019; Received in revised form 22 May 2019; Accepted 24 May 2019 Available online 01 June 2019 0368-2048/ © 2019 Elsevier B.V. All rights reserved.
Journal of Electron Spectroscopy and Related Phenomena 234 (2019) 86–90
A.A.A. Hussein, et al. N
HDC =
N
1
∑ (cαi·pi) + (βi − 1) c 2 + Vnuc (ri) + ∑ r i=1
i>j
ij
(1)
where c is the speed of light in atomic units, α and β are the 4 × 4 Dirac matrices and Vnuc(ri) is the potential due to the nucleus. R-matrix theory [23,24,20,25,26] has been used in the calculations of collision strengths and excitation cross sections of the present ion (W39+). The contribution of collision strength for a certain transition i → j (Ωij) to the cross section (σi→j) is given by:
σi → j =
πa02 IH Ωij ki2 gi
(2)
where gi = 2J + 1 and a0 is the Bohr radius. ki is the wave number of the incident electron and IH is the ionization potential of the hydrogen atom. In this method the Dirac Hamiltonian (Eq. (1)) for the (N + 1)electron atomic system is applied to construct bound state and target wave functions. The bound state (W39+) wave functions of the (N + 1)-electron atomic system are constructed using the multi-configurational DiracFock package (GRASP0) [22] in parallel with the collisional calculations using R-matrix method. Here, we considered the orbitals up to 3d10 as closed core, one electron (S) and two electrons (SD) excitations from the ground state (4s24p5) to 4l-orbitals plus the single excitations to 5l orbitals as a procedure to obtain the radial integrals using MCDF technique. Then, the 89 levels generated from the configurations 4p5, 4p44d, 4p44f, 4p45s, and 4p45p at different angular momenta have been included in the collisional calculations. Both configuration interaction (CI) and close-coupling (CC) expansions of the target have included the 89-level generated from the mentioned configurations above. All partial waves of odd and even states with 2J = 1–19 of the target expansion were included in the JΠ partial-wave expansion, where resonances may occur in the energy region of 15–25 Ryd. A fine mesh consists of 3200 mesh points by an energy mesh spacing of 0.02 Ryd in the resonance region for all partial waves has been performed. The size of Rmatrix boundary of W39+ is 2.2 au. In the second approach of collisional excitation calculations, the excitations of 2-electrons up to 4l levels are performed. Where, the 60-level generated from the configurations of 4p5, 4p44d, 4p44f, 4p34d2, and 4p34f2 have been considered as a 4l-DARC approach. In this case the R-matrix boundary is reduced to be 1.48 au. The electron impact ionization (EII) cross sections have been calculated using the flexible atomic code (FAC) [21]. FAC depends on diagonalizing the relativistic Hamiltonian H (in atomic units), this Hamiltonian is a combination of Dirac Hamiltonian of the multi-electronic systems (Eq. (1)) plus the relativistic correction. In this approximation, the configuration state functions (CSFs) of the ion under consideration are described by basis of states Φν of antisymmetric sums of the products of N one-electron Dirac spinor φnkm
φnkm =
1 ⎛ Pnk (r ) χkm (θ , ϕ, σ ) ⎞ ⎜ ⎟ r ⎝ iQ nk (r ) χ−km (θ , ϕ, σ ) ⎠
Fig. 1. Cross sections of the transition (4p5)3/2 → (4p5)1/2. Solid line for the obtained data from excitation up to 4l-orbitals while the dashed line for 5lexcitation.
have been applied in the calculations of direct electron impact ionization cross section. 3. Results and discussions 3.1. Electron impact excitation Cross sections of some transitions in W39+ (in units of cm2) are shown in Figs. 1–8. The present cross section calculations of these transitions have been performed based on R-matrix theory by using DARC code. For example, cross section of the transition 4p5(3/2)4p5(1/2), as a function of incident electron energy (in Ryd) are shown in Fig. 1. It is clear that, the resonance cross sections are observed at the region of 5–25 Ryd, while the rest of curve represents the direct excitation cross sections in the region of ∼25–40 Ryd. The cross sections of the transitions from ground state of W39+ to [(4p2)24d3/2]3/2, [(4p2)24d3/2]5/2, [(4p2)24d3/2]1/2, and [(4p2)04d5/2]5/2 are shown in Figs. 2–5. As shown in the figures, it is clear that the resonance background of cross sections dominate the incident electron energy in the range of ∼5–25 Ryd. As well as, there is a good agreement between the direct excitation cross sections produced from the 4l- and 5l-DARC approaches. The present calculations of cross section of the transitions 4p5(3/2)-[(4p2)24d5/2]5/2 have been compared with the calculated values using relativistic distorted wave approximation (RDW) [15] in Fig. 8. As it clear, there is an excellent agreement between the current
(3)
where χkm is the spin angular function, n is the principal quantum number, and k is the relativistic angular quantum number, which is written as:
k = (l − j )(2j + 1),
(4)
m is the z-component of the total angular momentum j and l is the orbital angular momentum. The atomic state functions (ASF) ψ are given by linear combination of the basis states Φν with same symmetries
ψ=
∑ bν Φν ν
(5)
where bν are the mixing coefficients obtained from diagonalizing the Hamiltonian matrix. The Distorted Wave (DW), Coulomb–Born-Exchange (CBE), and the Binary-Encounter-Dipole (BED) approximations
Fig. 2. The same as Fig. 1 for the transition (4p5)3/2 → [(4p2)24d3/2]3/2. 87
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Fig. 3. The same as Fig. 1 for the transition (4p5)3/2 → [(4p2)24d3/2]5/2.
Fig. 6. The same as Fig. 1 for the transition (4p5)3/2 → [(4p1/24p33/2)24d3/2]3/2.
Fig. 4. The same as Fig. 1 for the transition (4p5)3/2 → [(4p2)24d3/2]1/2.
Fig. 7. The same as Fig. 1 for the transition (4p5)3/2 → (4p3/2)3/2(4d3/2)3(4d5/ 2)9/2.
Fig. 5. The same as Fig. 1 for the transition (4p5)3/2 → [(4p2)04d5/2]5/2.
Fig. 8. Comparison between the present calculations of cross sections (solid line) and those of Priti et al. [15] (black squares) for the transition (4p5)3/2 → [(4p2)24d5/2]5/2.
cross sections an those in Ref. [15]. For high excitations such as the transition to [(4p1/24p33/2)2 4d3/2]3/ 2, the agreement between the 4l- and 5l-excitations show a noticeable discrepancy, where a difference of 0.05×10−20 cm2 is observed in Fig. 6. The differences between the background collisional excitation cross sections depend on many factors, such as, the bound state wave functions calculated using GRASP code are accurate to at least one part
in 106 while the solution of the free particle Dirac–Fock equation converges to one part in 105, or because the partial wave number of continuum electron, where the radial integration of the continuum functions does not converge to the function on the surface boundary. As well as, the radial integration involving these functions is affected by
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the accuracy of integrands and the integration method [15]. The cross sections of the excitation to higher levels such as the transition from ground state to (4p3/2)3/2(4d3/2)3(4d5/2)9/2 did not show resonance in the energy range of incident electrons, see Fig. 7, in such cases this is familiar because the probability of electron capture by higher levels will be more difficult comparable to lower levels. 3.2. Electron impact ionization If tungsten ion (W39+) is hit by an electron has sufficiently high collision energy this process may remove one or more (n) target electrons from the ion.
[Ar]3d10 4s 2 4p5
+
e−
10 2 4 − ⎧3d 4s 4p + 2e 10 2 3 → [Ar] 3d 4s 4p 4d + 2e− ⎨ 10 2 3 − ⎩3d 4s 4p 4f + 2e
Fig. 10. Electron impact ionization cross section from the ground state of W39+ to 4d target states with J = 2, 3. Solid lines for BED-calculations and dash lines formats for CB approximation. The dot lines for DW approximation.
In the direct ionization, a bound electron of the target ion may obtain energy by the collision with the incident electron more than its binding energy, so that the target electron is released. This technique is known as single ionization of the parent Wq+ and is known as direct ionization. But if the previous process being done in two stages, one of them was mentioned above already, the excitation of an inner-shell electron with subsequent autoionization. This type of ionization is usually known as excitation-autoionization (EA), the first stage of excitation-autoionization is carried out through the formation of an intermediate resonance state [W q +]* . In this case, an inner-shell electron will be excited, and the ejected electron will be captured to a bound state. The resulting short lived recombined ion state is highly excited so that two electrons can be expelled in the relaxation process. Finally, one electron will be lost in the whole process [27]. The ionization cross sections of 4p, 4d and 4f electrons for the ground state of W39+ calculated using FAC code have been computed in the incident electron energy range of 50–600 Ryd. For example, the cross sections of 4p4 with J = 0,1, and 2 are shown in Fig. 9. As well as, the EII cross sections of 4p34dJ=2,3 and for 4p34fJ=2,3,4 are presented in Figs. 10 and 11, respectively. The produced results from the Relativistic Distorted-Wave (DW), Coulomb–Born-Exchange (CBE), and the BinaryEncounter-Dipole (BED) approximations are plotted together to estimate the differences and then the accuracy of the EII cross section. In Fig. 9 the ionization cross sections are shown side by side for the three used methods, i.e. BED, CB, and DW. It is clear that a reasonable agreement between BED and CB approaches is noticeable, while the DW-values show large differences from the other calculations. The reason for why the DW-cross sections are much smaller may be due to
Fig. 11. Electron impact ionization cross section from the ground state (4p5 J = 1/2) of W39+ to 4f target states with J = 2, 3, and 4. Solid lines for BEDcalculations and dash lines formats for CB approximation. The dot lines for DW approximation.
the radiative damping effects in the direct ionizations are significantly considered [28,29]. As shown in Figs. 9–11, the ionization cross sections calculated using BED-method are smaller at near threshold energies as compared using CB-calculations. This is due to that, when the calculation of the ionization radial integrals using BED approach is applied, the total ionization cross sections are scaled by a factor of E/ (E + I), where E is the energy of incident electron, and I is the ionization threshold energy [14]. 4. Conclusion The current study provides theoretical calculations of electron impact excitation and ionization cross sections of tungsten ions, W39+. Excitations up to 5l-orbitals are taken into account in the calculations of collision strengths, hence, cross sections of the transitions under consideration. R-matrix calculations of cross sections have been performed for the transitions between the first 89-low-lying levels at electron energy range of 5–70 Ryd, and compared with the available literature. The electron impact ionization cross sections were computed for the present ion. The EII cross sections have been performed in the energy range of 50–600 Ryd using the CB, BED, and DW approximations. The comparisons show good agreement between different calculations using Coulomb–Born-Exchange (CBE) and the Binary-Encounter-Dipole
Fig. 9. Electron impact ionization cross section from the ground state of W39+ (N + 1) to the next ionization stage (N). Solid lines for BED-calculations and dash lines formats for CB approximation. The dot lines for DW approximation. 89
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(BED), while noticeable discrepancies of the calculated ionization cross sections using Distorted-Wave (DW) approximation are recorded.
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