M.J. CONDENSED MATTER VOUME 3, NUMBER JUY Electron impact ionization o Ar + A. Riahi, K. aghdas, S. Rachai Département de Physique, Faculté des Sciences, Université Chouaib Douali, B.P., 4 ElJadida, Morocco. Total and single dierential cross sections have been calculated or electron impact ionization o Ar + (s s p 4 3 P e using a method that combines the distorted-wave Born approximation and R-matrix theory. In this method, the incident/scattered electron is described by the distorted-wave Born approximation, while both the initial bound state and the inal continuum states are expanded in terms o an R-matrix basis. Eight states o the inal Ar + ion are considered, namely s s p 3 4 S o D o P o and s sp 4 4 P e D e S e P e P o. Up to the 4 - pole components o the interaction with ionizing electron were included, giving ten distinct Ar + continuum symmetries. We calculated single dierential cross sections summed over inal ionic states, as a unction o the energy loss. These cross sections exhibit considerable structure due to autoionizing resonances. Total cross sections or production o Ar + in each o the eight states are presented or impact energies rom threshold energy at 9.8 au to au. Our theoretical values or the total cross section are in good agreement with recent experimental results. I. INTRODUCTION In usion plasma research, as in astrophysics, a good nowledge o physical properties and collisional atomic data is needed in order to interpret the observations made o various plasma parameters. One o the undamental processes in this ield is the electron-impact ionization o atoms or ions [], because it governs the relative proportions o the various ionization stages o each ion charge present in the plasma. The energetic radiation rom highly ionized impurities in high-temperature plasmas is a major cause o energy loss which has so ar prevented such devices rom reaching the required ignition temperature (KT ev and coninement condition (nô 4cm3.s. The eect o impurity ions on plasma coninement is also poorly understood. For many years there has been considerable theoretical interest in electron impact ionization processes, motivated on the one hand by the need to develop reliable methods o calculating (or example the ion charge state distribution in the plasma and the corresponding ionization cross sections, and on the other by the urgent practical need to obtain accurate data or plasma and usion research []. For single ionization, theoretical results have been obtained using various applications o the Coulomb- Born approximations where both the incident and outgoing electrons are described by pure or distorted Coulomb or plane waves [3]. The undamental problem here is the act that in this type o calculation all correlations between the incident and outgoing electrons with the initial and inal ionic states are completely neglected. On the experimental side, the inormation available regarding the electron impact ionization process has been analysed in detail [4] or a large number o targets o thermonuclear interest. Accurate models or plasmas require reliable cross section data or electron-impact excitation, ionization and recombination o these impurity ions. In the present study, we have used the distortedwave Born approximation (DWBA-R-matrix method o Bartschat and Bure [5] to calculate total and single dierential cross sections or electron-impact ionization. This method incorporates the indirect process o excitation-autoionization (EA by treating the inal continuum states (consisting o the inal ion plus the ejected electron by an accurate R-matrix ormulation. The initial bound state o the target ion is also calculated by the R-matrix method and is thus consistent with the inal continuum states. In the R-matrix method the close-coupling [6] problem is solved or the inner region, r a, subject to a ixed boundary condition at ra to give the basis unctions. Wave unctions or a system containing (N+ electrons or any system energy can be expanded in terms o products o N-electrons residual ion unctions multiplied by ully optimised unctions or an ejected electron and matched to unctions or the outer region r > a. The advantage o the R-matrix method compared with the usual close-coupling approach [6] is the act that a broad range o energies o the ejected electron can be studied by a single diagonalisation o the Hamiltonian matrix, instead o solving a system o coupled integrodierential equations at each energy. Because the ast ionizing electron is described by the distortedwave Born approximation we neglect all the correlations between this electron and the electrons o target. Exchange eects are approximated by modiying the range o integration over the energy o the ejected electron hal range. The plan o this paper is as ollows. In section we present a broad general outline o the theory o the R- matrix method. More details can ound in the paper o Bartschat and Bure [5]. Computational details o the method have been described by Bartschat [7]. Section 3 gives details o the speciic calculation or Ar +. The results or single dierential cross sections and total cross sections are presented, and compared with experiment, in section 4. 3 8 The Moroccan Statistical Physical Society
3 EECTRON IMPACT IONIZATION OF Ar + 8 II. THE R-MATRIX METHOD OF IMPACT IONIZATION A. GENERA THEORY Consider the problem o calculating the cross section or the electron-impact ionization o complex positive ions : q * [ ] ( q+ + [ + N + ( ] q+ + e ( + X X + e ( α X e Here X q + and X q ( + + ( are the initial and inal states o the ionic target, the asteris on the [ X q+ ] * indicating ormation o a continuum state. This continuum state represents the intermediate states o a (N+-electrons system or which N electrons are bound in an atomic ion represented by the inal-ion state X ( q + + and one electron e N + can escape to ininity with l th partial wave. The ionization process can be resolved into partial waves. In the present wor we describe this process using the ollowing approximations : The ionizing electron e ( with initial and inal momenta and is described by the distorted-wave Born approximation. e N + is a ejected electron with momenta and the initial and inal continuum (N+ electron states Ψ i ( q+ + and { X + e ( } Ψ N + are calculated using the R-matrix approach. The process is treated in S coupling where only Coulomb interaction between the ionizing electron and the target electrons is taen into account. B. THE R-MATRIX EXPANSION In this subsection we discuss the use o an R-matrix basis to represent the initial and inal (N+-electron states. We start with a brie summary o the basic equations o the general R-matrix theory [8,9]. In the R-matrix method, the coniguration space, describing the scattered or ejected electron and the target, can be separated into two distinct regions. The inner region is deined as that region enclosed by a sphere o radius a centred on the origin o the atomic system, and is chosen to be large enough to just envelop the charge distribution o the target states o interest. The outer region consists o all other space and we can assume that in this region the target eigenstate is zero. In this method, since the initial and inal (N+ electron states Ψ i and Ψ are obtained by diagonalising the same target Hamiltonian H ion [,,], they are orthogonal. In the inner region, the waveunction o the target can be expanded in terms o energy- independent R-matrix basis unctions or any energy E : Ψ E A E ψ ( The R-matrix basis unctions are deined by ψ + Α Φ (\ r r u ( r c j d j ij i N+ N+ ij N+ ij ϕj( XN+ (3 (\ r Here Φ i N + are channel unctions ormed by coupling a target state with the spin and angular components o the scattered electron, and are channel eigenstates o and S respectively and o π o the system. \ r N + means the inclusion o all coordinates except the radial coordinate o the (N+ th electron, or writing generally, { } \ r X,..., X, X,..., X,, r i i i+ N+ i i σ (4 Α is the antisymmetrisation operator which ensures that the Pauli principe is satisied. The ϕ j ( X N + are quadratically integrable correlation unctions, which are short range and consist totally o bound conigurations. Finally, u ij are the R-matrix continuum orbitals which satisy the dierential equation : d li( li + + Vi( r + i uij( r dr r p ( r n ij nl i (5 or each angular momentum, l i, subject to the R-martix boundary conditions u ij ( a duij u ( a dr ij r a b (6 The logarithmic derivative b is usually taen to be zero and the R-matrix boundary a is chosen in such a way that all the bound orbitals Pnl i (used r in the description o the channel and correlation unctions essentially vanish or r > a. The ij are agrange multipliers, and are chosen to ensure that the continuum orbitals are orthogonal within a particular symmetry to the bound orbitals Pnl i ( r In order to eep the notation compact the equation (3 can be write as : φ ' (7 ψ ω ' '
8 A. RIAHI, K. AGHDAS, AND S. RACHAFI 3 The coeicients ω ', which correspond collectively to the c ij and d j in equation (3, are determined by diagonalising the (N+-electrons Hamiltonian as : ω" < φ H φ > ω δ E (8 ' ' ' "' " " "' Both the initial and inal (N+-electrons states are now expanded in terms o the R-matrix basis unctions as ollows : and Ar Ψ i A ψ i Ψ A ψ ( In order to determine the coeicients A i and A K it is necessary to calculate the radial waveunctions o the continuum electron in the outer region. The complete solution can be obtained by matching the inner region and outer region waveunctions at the boundary. III. COMPUTATIONA DETAIS We have considered the ionization o Ar + in its ground state (s s p 4 3 P e (9 (8 + 4 3 e + 3 π e ( E, + Ar ( s s p P e ( E, + Ar ( ε ( Equation ( shows the excitation o Ar + to continuum state o symmetry ( ε 3 π caused by the 4 -pole components o the interaction with the incident electron, e ( E, lwhich has incident energy E and angular momentum l. The electrostatic interaction between e ( E, l and the N+ electrons o Ar + causes e ( E, l to lose energy E E E and excites Ar + π ( ε + 3 rom the ground state to a continuum state o symmetry ( ε 3 π. With an initial 3 P e state (, the inal state symmetry is identiied by - + andπ (. In practice, we have included only the interaction components with all < 4, and so ten continuum states are accessible, with symmetries: 3 P e, 3 S o, 3 P o, 3 D o, 3 D e, 3 F e, 3 F o, 3 G o and 3 H e. We too an R-matrix radius o.984a o and constructed radial continuum orbitals unl ( r(n, or each angular momentum l,..., 7. Equation ( shows that, in the outer region, the continuum states consist o + 3 4 o s+ Ar ( s s p S + en + ( E(, l + 3 o s+ Ar ( s s p D + en + ( E(, l + 3 o s+ Ar ( s s p P + e ( E (, l N + + 4 4 e s+ Ar ( s sp P + en + ( E(, l ( + 4 e s+ Ar ( s sp D + e ( E (, l N + + 4 e s+ Ar ( s sp S + en + ( E(, l + 4 e s+ Ar ( s sp P + e ( E (, l N + + 5 o s+ Ar ( s p P + e ( E (, l Ar + and an ejected N + s+ electron, e ( E, l, with energy E ( N + and angular momentum l. We have included the eight states 3 4 with conigurationss s p, s sp and s p 5. The bound s, s and p radial orbitals rom which the waveunctions are constructed were obtained rom those given by Clementi and Roetti [3] or 3 4 ( s s p S o reoptimized to the sum o the 3 4 energies o the ( s s p S o, D o, P o states using the CIV3 code [4]. Our calculated energy o the ground state o Ar + is 453.835 au, which compares avourably with the value o 453.84 au given by Clementi and Roetti [3] and is indicative o the quality o the waveunction. The energy o the inal continuum state o the Ar + system is characterised by the energy-loss E : s+ E E E E + I( (3 s where I( + is the ionization potential relative to the initial bound state or production o the inal ion in
3 EECTRON IMPACT IONIZATION OF Ar + 83 each Ar + core. Since I( s+ is ixed, the ionization is characterised by E and E. IV. RESUTS A. SINGE DIFFERENTIA CROSS SECTION The single dierential cross section or ionizing 4 3 ( s s p P e state o Ar + with production o the Ar + ion in the inal state via continuum symmetry is given by : dσ de E E 6 (, ( < Ψ ϑ ( ( Ψi > E( + l l l (4 with where N + ( 4π ϑ ( R (, ; ri Y ( r i + i ( + + R + ( (, ; r γ ( r, r F (, r F (, r dr ' ' ' ' i i (6 (5 ( r, r r / r ' + whereγ i < > are the lesser and greater o r and r i. The unctions F(, r are radial waveunctions which are solutions o with d dr ( + + V ( r F (, r (7 r F(, (8 r π + ( F (, r r sin Z lnr + σ+ η where σ arg Γ + iz In equation (4, the operator ϑ ( ( (9 is proportional to the -pole components o the interaction between e and the Ar + system overlapped between the th and th partial waves o e at energies o E and E, respectively [5,6]. These partial waves were calculated using the static potential o Ar + ( s s p 4 3 S e obtained rom the waveunction given by Clementi and Roetti [3]. The maximum value o l and l that was suicient to ensure convergence o the single dierential cross section, ranged rom l ' 4 or E au to l '' 66 or E 8 au. In igure l we present the single dierential cross section or electron-impact ionization o Ar + ( s s p 4 3 P e or impact energy E 5 au as a unction o energy loss. Each contribution is rom a particular continuum symmetry o Ar +, shown at the right margin, and is summed over the inal states o Ar +. The cross section demonstrates the regular Rydberg series o resonances. The resonances are due to quasibound autoionizing states consisting o an electron bound to an excited Ar + core such that the eective principal quantum number n * is given by n δ with n an integer and the quantum deect δ virtually constant or each series. We have identiied 43 distinct series o resonances, and table shows the resonance energy ε r and eective principal quantum number n * or resonances with (n < 5. B. TOTA CROSS SECTIONS The total cross section or ionization that leaves the inal ion in state is : where σ E I( l dσ ( E ( de E, E de ( σ dde E E dσ (, ( de E, E (
84 A. RIAHI, K. AGHDAS, AND S. RACHAFI 3 Here, the hal-range approximation is used in the integration over E to tae account o the indistinguishability o the two inal, continuum electrons. The upper limit o ejected-electron energy E is taen to be hal o the maximum possible value E I( [5]. Our calculated total cross sections are also presented in table 3 or impact energies rom au to au. This table shows the total cross sections or electron-impact ionization o Ar + with production o Ar + in various states (columns 3-9. Column σ Total is the cross section summed over the states o Ar +. Figure shows cross sections plotted or impact energies rom threshold energy at 9.8 au to au. In igure 3, we present the total cross section compared with the recent experimental results [6]. Our results are in overall good agreement with the semi-empirical otz ormula [7] with inclusion o the s and p contributions. No theoretical results can be compared to the present results..4.3.. 4 4 4 43 Total.4.3...3.. 4 4 43 4 4 43 FIG. : Contribution rom a particular continuum symmetry to the single dierential cross section or ionization o Ar + by an electron with energy 5 au as a unction o energy loss. The graph shows a region or energy loss ÄE between the thresholds or producing Ar + 3 ( s s p 4 S o and Ar + 3 ( s s p P o Cross section (a o 6 8 3 F e 3 F o σ[p 5 P o ] σ[p 4 P e ] σ[p 4 S e ] σ[p 4 D e ] σ[p 4 4 P e ] σ[p 3 P o ] σ[p 3 D o ] σ[p 3 4 S o ] σ Total. 3 P e 4.5.3.. 4 4 43 3 S o 5 5 Impact energy (au FIG. : Cross sections or electron impact ionization o Ar +, producing eight states o inal Ar +. 4 4 43 5 dσ/de(a o /au....5..5 3 o P 4 4 43 3 o D 4 4 43 Total cross section ( -9 cm 4 3 Our results The experimental results otz 4 6 Impact energy (ev 3 D e.5.5 4 4 43 FIG. 3 : Total Cross sections or electron impact ionization o Ar +, summed over the inal states o Ar +. The results o Derance and al are also shown (+. The ull curve is the semi empirical otz ormula.
3 EECTRON IMPACT IONIZATION OF Ar + 85 V. CONCUSIONS We have calculated total and single dierential cross sections or electron impact ionization o Ar + or a wide range o impact energies, rom au (which is just above the threshold energy at 9.8 au to au. Our calculations has demonstrated the ability o the Bartschat-Bure R-matrix ormulation to describe the initial and inal state o the ion. The total cross section has a maximum value o 3.86-3 a at an impact energy o 56 au. Our total cross section, summed over the inal states, is in good agreement with the predictions o the semi-empirical otz ormula and with experimental results [6]. We also calculated total cross section relative to production o each inal state o the residual ion, but, we can not compare these results with experimental results. The contribution o autoionization to the total cross section is not signiicant. These resonances are incorporated automatically by the accurate treatment o the inner region in the R-matrix method. The exchange eects can be included exactly in this ormulation and results may be improved signiicantly in medium impact energy. Table : The energies used in our ionization calculation, I( s+ are relative to the ground state o Ar +. Adopted states Energy I( s+ (au s s p 3 4 S o 9.85896 s s p 3 D o.64746 s s p 3 P o.57656 s sp 4 4 P e.7736 s sp 4 D e.55796 s sp 4 S e 3.6836 s sp 4 P e 3.6 s p 5 P o 5.4896 Table : Details o autoionization states o Ar +. The energies are relative to the ground state Ar + (s s p 4 3 P e. They are given as I in Table Energy symmetry (Ryd 3 P e 39.665 39.6889 39.796 39.8787 4.35 3 S o 39.685 39.977 39.9877 3 P o 4.5765 4.5968 4.965 4.547 4.598 4.7383 3 D o 39.6439 39.679 39.7778 39.95 39.9674 4.884 4.66 4.998 3 D e 4.5859 4.63 4.534 4.6895 3 F e 4.63 4.5568 4.653 4.758 4.476 3 F o 39.6476 39.666 39.666 4.88 3 G o 36.39 37.344 37.344 37.7856 3 G e 36.39 37.344 37.344 3 H e 4.6566 Ar (+ Ar (+ cores n* min P o 9p D o p D o 4 P e 6s 4 P e 6d D o d 4 P e 6p P o p P o 5s P o 5d 4 P e 7p D e 6p D e 6 P o 5p D o g D o s D o 3d P o p 4 P e 6p 4 P e 6 D e 5p D e 5 P o 5p P o 5 4 P e 7d D e 6s P o 5 4 P e 8d 4 P e 8g D e 6d D e 7g D o 9d D o d P o d P o g D o d P o g 4 P e 7 D e 6 4 P e 9g D e 6d S e 6g 9.456.37.337 5.7439 5.9337.959 5.86.45 4.6367 4.97 6.847 5.7965 5.8596 5.375.6895.6895.6873 9.99 5.846 6.9 4.9949 5. 4.7587 4.9969 6.967 5.946 4.969 7.7988 7.9945 5.9983 6.775 9.3.83.9969.55 4.999 4.9993 4.999 4 P e 8g 8.
86 A. RIAHI, K. AGHDAS, AND S. RACHAFI 3 Table 3 : Total cross section or electron-impact ionization o Ar + with production o Ar + in various states (columns 3-9. Column (Σ is the cross section summed over the states o Ar + Impact Cross section (a energy σ Total σ[p 3 4 S o ] σ[p 3 D o ] σ[p 3 P o ] σ[p 4 4 P e ] σ[p 4 D e ] σ[p 4 S e ] σ[p 4 P e ] σ[p 5 P o ] (au -3-3 -3-3 -3-4 -5-3 -3..........564.8.88.397.336.... 6 7.3538.534.457.346.776.95.633..7 3.63 3.453 3.38.7966.558.74.83.4533.65 36.538 3.64 3.8644.899.78.655.53.673.977 4.8955 3.7874 4.8.9.8856.79.659.7634.99 46 3596 3.979 4.3.366.553.75.754.8496.5 5 3.86 3.944 4.3543.3535.9.763.776.884.57 56 3.8675 3.958 4.3559.3557.656.78.768.99.75 6 3.875 3.893 4.3353.346.756.795.749.9.73 7 3.45 3.759 4.5.794.488.79.673.955.43 8 5439 3.575 3.943.44.9676.859.534.8364.5 9.987 3.359 3.7637.487.889.836.445.844.98.486 3.96 3.69.963.878.377.343.776.45.458.96 3.77.7963.658.67.4.75.948 4 9566.6545 3.56.646.558.53.85.6445.864 6 8.754.4336.747.4886.396.97..599.788 8 8.86.56.5573.3988.86.3.95.5466.75 7.58.4.3533.747.995.38.863.589.676 [] Mar TD 99 Plasma Phys.Control.Fusion 34 83 [] M.F.A. Harrison, `` The plasma boundary region and the role o Atomic and Molecular Processes in Atomic and Molecular Physics o Control Thermonuclear Fusion, ed. C.J. Joachain and D.E. Post, NATO Advanced Study Institue Series B : Physics, vol. (Plenum, new Yor and ondon, 983, pp. 73-474. [3] Rudge MRH 968 Rev.Mod.Phys.4 564 [4] Derance P,Duponcelle M and Moores D 995 Atomic and Molecular Processes in usion Edge Plasmas ed RK Janev (New yor:plenum p 53 [5] Bartschat K and Bure PG 987 J.Phys.B.At.Mol.Phys. 39 [6] Jaubowicz H and Moores D 98 J.Phys.B At.Mol.Phys.4 3733 [7] Bartschat K 993 Compt.Phys.Commun 7 9 [8] Bure PG, Hibbert A and Robb WD 97 J.Phys. B :At. Mol. Phys. 4 53 [9] Bure PG and Robb WD 975 Adv.At.Mol.Phys. 43 [] Raeer A, Bartschat K and Reid R H G 994 J. Phys. B : At. Mol.Opt. Phys. 7 39 [] Reid R H G, bartschat K and Bure P G 99 J. Phys. B :At. Mol. Opt. Pys. 5 375 [] aghdas K, Reid R H G, Joachain C J and Bure P G 995 J ; Phys. B : At. Mol. Opt. Phys. 8 48 [3] Clementi E and Roetti C974 At.Data.tables 4 7 [4] Hibbert A 975 Comput.Phys.Commun.9.4 [5] Moores D and reed K J 994 Adv ; At. Mol. Phys. 34 3 [6] Derance P, and al 999 Private communication. [7] otz W 967 Astrophys. J Suppl. 4 7