Electron-impact ionization cross sections and rates for ions of argon

Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 301 – 315

www.elsevier.com/locate/jqsrt

Electron-impact ionization cross sections and rates for ions of argon Y. Zhao, C.Y. Chen∗ , H.N. Xia, J.B. Qi, Y.S. Wang Institute of Modern Physics, Fudan University, Shanghai 200433, People’s Republic of China Received 5 April 2002; accepted 1 July 2002

Abstract A distorted-wave Born exchange (DWBE) approximation including relativistic correction is used to calculate the electron-impact ionization cross sections and rate coe8cients for the highly charged ions Ar 7+ ; : : : ; Ar 17+ . The comparison of the calculated results with the experimental data and other theoretical calculations shows that the DWBE method is valid for these ions of argon. The calculated results for direct ionization cross sections and excitation autoionization were :tted by empirical formulas to meet the requirements of applications. A set of improved empirical formulas are used for the fast and accurate calculations of rate coe8cients from the :t parameters of cross sections. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Highly charged ions; Ionization cross sections; Distorted wave approximation; Fit formula

1. Introduction The electron–ion collision is one of the fundamental processes in atomic physics. In astrophysical and laboratory plasma, and X-ray laser studies, large amounts of atomic data are required, including electron-impact ionization cross sections and rate coe8cients. These data are important for modeling the structure and dynamics of high-temperature plasmas occurring both naturally in space and arti:cially in fusion devices as discussed by Mark and Dunn [1] and Stephens and Botero [2]. In many situations, it is necessary to obtain quickly and accurately the data of cross sections and rate coe8cients over a wide range of incident electron energies. Thus, suitable empirical formulas for the ionization cross sections and rates become very useful.

∗

Corresponding author. E-mail address: [email protected] (C.Y. Chen).

0022-4073/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 2 ) 0 0 1 3 1 - 0

302

Y. Zhao et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 301 – 315

The electron-impact ionization cross sections and rate coe8cients of 11 highly charged ions, Ar 7+ ; : : : ; Ar 17+ , have been studied theoretically and experimentally in some papers. Salop [3] employed classical method to estimate the electron-impact ionization cross sections of Ar 7+ ; : : : ; Ar 12+ . Using distorted-wave Born exchange (DWBE) approximation, Younger [4–6] obtained the ionization cross sections and rate coe8cient of Ar 7+ , Ar 8+ and Ar 14+ . Laghdas et al. [7] used a method that combines the distorted-wave Born exchange approximation and the R-matrix theory to obtain the ionization cross sections of Ar 9+ . Chen and Reed [8,9] used the relativistic distortedwave and the multicon:guration Dirac–Fock method to get the ionization cross sections of Ar 7+ and Ar 15+ . Lotz [10] presented an empirical formula with three parameters for the single ionization cross sections of atoms and ions in the early years. Recently, Bernshtam et al. [11] analyzed published data and proposed an empirical formula for direct electron-impact ionization cross sections, which improves the Lotz formula. This formula predicts distorted-wave calculated results commonly to ±20%. Arnaud et al. [12] presented an evaluation of the ionization rates for argon ions. In Tawara et al.’s report [13], the experimental ionization cross sections and selected theoretical values for the ions of argon are compiled. Lennon et al. [14] gave the recommended cross sections and rate coe8cient of all the ions of argon. Recently, based on the recommended data by the Belfast group and the data derived from several other sources, Voronov [15] gave a practical :t formula for ionization rate coe8cients of all the ions of argon. Donets et al. [16] made some measurements of the electron-impact ionization cross sections of Ar 15+ , Ar 16+ and Ar 17+ . Recently Zhang et al. [17] measured the cross sections of Ar 8+ . Racha: et al. [18] gave some experimental data of the ionization cross sections of Ar 7+ and Ar 8+ . Though the ionization cross sections for the ions of argon have been widely discussed, there are not many available experimental data in a wide energy range of the incident electron for each ion of argon, especially, for highly charged ions of argon. It is necessary to calculate, systematically, the ionization cross sections and rate coe8cients of the highly charged ions of argon by using DWBE approximation. In this paper, (1) we use a semirelativistic DWBE approximation method to calculate, systematically, the total ionization cross sections and rate coe8cients of the highly charged ions of argon Ar 7+ ; : : : ; Ar 17+ . In the present calculations we consider inner-shell and outer-shell direct ionization for all the ions. For lithium-like (Ar 15+ ), beryllium-like (Ar 14+ ) and sodium-like (Ar 7+ ) ions, we also consider excitation autoionization with con:guration interaction. (2) We :t the calculated results of direct ionization and total excitation autoionization cross sections with corresponding empirical formulas to meet the practical applications. (3) We also give an outline about a set of improved empirical formulas developed by our group [19] to calculate rates easily and accurately by using these :t parameters of the cross sections. The remainder of the paper is arranged as follows. In Section 2 we describe the theoretical methods employed to determine the ionization cross sections and rate coe8cients, and the respective empirical formulas. In Section 3 the results and discussions are given. We :rst present the calculated ionization cross sections, and compare these results with the experimental measurements and other calculations. Then we present the theoretical rate coe8cients for all the 11 ions, systematically, and compare these rates with the values given by Voronov [15].

Y. Zhao et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 301 – 315

303

2. Theoretical and calculational methods 2.1. Ionization cross sections In the papers written by Chen et al. [20,21], we calculated successfully the electron-impact ionization cross sections of neon and sodium-like ions by using the DWBE approximation, and the calculated results agree well with the available experimental data. The DWBE method, which is described in more detail in our previous works given by Fang et al. [22] and Hu et al. [23], is a development of the Coulomb–Born (CB) approximation with the following four improvements: (1) we consider the distribution of target bound electrons rather than taking the eOect of these electrons as an equivalent point charge; (2) we consider the exchange eOect between the two free electrons in the :nal state; (3) we include the exchange eOect between the free electrons and bound electrons; and :nally (4) we take the relativistic correction of target electron wave functions into consideration which is shown in the book written by Cowan [24]. The direct ionization and excitation autoionization are assumed to be independent processes and the total ionization cross section is given by
Qi (Ei ) = Qd + Qea = Qo (Ei ) + Qi (Ei ) + Qej (Ei )Bja ; (1) j

where Qd is the total direct ionization cross section, Qea is total autoionization cross section. Qo (Ei ) and Qi (Ei ) are the outer-shell and inner-shell direct ionization, respectively, Qej (Ei ) is the excitation cross section of inner-shell electrons to the autoionization level j and Bja is the branching ratio for autoionization from the level j. Bja is given by  → m) a m Aa (j  ; (2) Bj =  m Aa (j → m) + k Ar (j → k) in which Aa (j → m) is the autoionization rate to channel m and Ar (j → k) is radiative transition rate to the bound level k. The direct ionization cross section (in units of a20 with a0 being the Bohr radius) in the DWBE approximation can be written as  E=2 Q(Ei ) = (Ee ; Ei ) dEe ; (3) 0

where Ei and Ee are the incident and ejected electron energies, respectively. E is the sum of the ejected and the scattered electron energies in the :nal state. (Ee ; Ei ) is the diOerential cross section as function of energy. In Eq. (3), we use 3 points Gauss integral when incident energy is less than 3 times ionization energy, and 5 points Gauss integral for the remaining incident energies. By using the partial wave expansion, the calculation of the scattering amplitude can be divided into the calculation of angular factor and Slater integrals. We can write 16
(Ee ; Ei ) = Ili le lf L (Ee ; Ei ); (4) Ei li le lf L

where li , le and lf are the angular momenta of incident, ejected and scattered electrons, respectively. L is the total angular momentum of the whole system, which is a conserved quantity. And Ili le lf L (Ee ; Ei ) = |f|2 + |g|2 − |fg|;

(5)

304

Y. Zhao et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 301 – 315

in which f and g are direct and exchange partial wave scattering amplitudes, respectively. Here, we use the maximum interference approximation ( = 1) discussed by Younger [4] and Gri8n [25] to treat the interaction between the direct and exchange terms, which can be written as
f (lb li le lf L)(Pb Pi |1=r12 |Pe Pf ) ; (6) f= g=




f (lb li lf le L)(Pb Pi |1=r12 |Pf Pe ) ;

(7)



where f is an angular factor in which lb is the angular momentum of the bound electron, and (Pb Pi |1=r12 |Pe Pf ) , (Pb Pi |1=r12 |Pf Pe ) are Slater integrals in which Pb , Pi , Pe and Pf are the radial wave functions of bound, incident, ejected and scattered electrons, respectively. Pb is obtained from the atomic structure program of Cowan [24] in an Hartree–Fock approximation with relativistic correction (HFR) model. In the calculations of Pi , Pe and Pf , the consideration for the potentials is the same as Younger’s [4–6]. The calculation of slowly converging Slater integral has been described in detail by Fang and Wang [26]. In present calculation, the summation of each partial wave was truncated when the increments were less than 0.2%. We :t the calculated direct cross sections with the empirical formula of Younger [5,6]: uId2 Q = A(1 − 1=u) + B(1 − 1=u)2 + C ln u + D ln u=u;

(8)

where Q is outer-shell or inner-shell direct ionization cross section, and Id is outer-shell or inner-shell ionization energy (see Table 1). u is the reduced incident energy de:ned as u = Ei =Id ;

(9)

in which Ei is the incident electron energy. The energy range in our calculation is quite large, from u = 1:125 to u = 15:00. In Eq. (8), A, B, C and D are four adjustable parameters (in unit of Ry2 a20 , Ry is the Rydberg energy) obtained by the least-squares method. In our previous papers written by Fang [22] and Hu [27], we have compared the Younger’s formula with Lotz’s [10] and Sampson’s [28] empirical formula, and found that the Younger’s formula gives a smaller average deviation which is de:ned as    N  1
Q:t (u) − Qcal (u) 2  F(%) = × 100; (10) N i=1 Qcal (u) i where Qcal and Q:t are calculated and :tted values, respectively, and N is the number of calculated points used in making the :t. In the DWBE approximation, the inner-shell excitation cross section for the transition from initial level i to :nal level j under LS coupling scheme, that is Qej (Ei ) in Eq. (1), can be written as
2a20
(2J + 1) × |R(i Li Si Ji lsjJ ; f Lf Sf Jf l
s
j
J )|2 ; (11) Qej = gi Ei (Ry) J

ll jj

where gi = (2Li + 1)(2Si + 1) is the statistical weight of initial level, Ei is the incident energy in unit of Ry, and J is the total angular momentum of the system. lsj and l
s
j
are the orbital, spin and total angular momenta of the incident (unprimed) and scattered (primed) electron, respectively. Li , Si and Ji are the orbital, spin and total angular momenta of the target ion, i represents the other

Y. Zhao et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 301 – 315

305

Table 1 Ionization energies (Ry), :t parameters (Ry2 a20 ) and :t errors of direct ionization cross sections for individual ions Charge states

Id

A

B

C

D

F (%)

Ar 17+ (1s) Ar 16+ (1s) Ar 15+ (1s) Ar 15+ (2s) Ar 14+ (2s) Ar 13+ (2s) Ar 13+ (2p) Ar 12+ (2s) Ar 12+ (2p) Ar 11+ (2s) Ar 11+ (2p) Ar 10+ (2s) Ar 10+ (2p) Ar 9+ (2s) Ar 9+ (2p) Ar 8+ (2s) Ar 8+ (2p) Ar 7+ (2s) Ar 7+ (2p) Ar 7+ (3s)

325.55 303.20 296.41 67.586 62.581 57.870 55.519 53.318 50.205 48.926 45.103 44.694 40.214 40.626 35.539 36.722 31.081 34.784 29.077 10.564

12.692 25.226 24.107 7.4654 15.221 15.252 13.030 16.173 25.943 16.890 38.689 17.386 51.916 17.641 66.680 17.887 79.954 19.303 81.406 5.8635

−4.6586 −9.3834 −9.1621 −2.5313 −5.1928 −5.2596 −4.7489 −5.6857 −9.5411 −6.0319 −14.406 −6.2536 −19.763 −6.3772 −26.165 −6.5548 −32.477 −7.2573 −34.060 −2.2683

0.3349 0.7513 1.1510 0.6398 1.2474 1.3156 0.0625 1.2382 0.2875 1.1992 0.7159 1.1910 1.3219 1.2301 1.9959 1.3053 3.3097 1.1847 3.9002 0.72095

−10.393 −20.858 −3.6554 −5.1929 −10.771 −10.966 −9.7209 −11.907 −19.622 −12.696 −29.726 −13.330 −40.669 −13.797 −53.434 −14.303 −65.702 −15.736 −68.213 −3.8872

0.22 0.21 0.50 0.25 0.25 0.22 0.32 0.21 0.32 0.21 0.32 0.23 0.31 0.25 0.29 0.20 0.30 0.22 0.28 0.41

quantum numbers of the target ion. Lf , Sf , Jf and f are the corresponding quantum numbers of the residual ion. The reactance matrix R has a direct and an exchange terms, which can be written as R = Rd − Re ;

(12)

where the exchange scattering matrix element Re represents the exchange eOect between the scattered electron and the electron in the excited state. In our calculation, the eOect of con:guration interaction (CI) on the autoionization cross section is also considered. Then the excitation cross section should be 2

2a20
j


Qe = (2J + 1) c(i)c(f)R( L S J lsjJ ;  L S J l s j J ) (13) ; i i i i f f f f gi Ei (Ry)

J

ll jj

 i L i S i f L f Sf

where the con:guration mixing coe8cients c(i) is the abbreviation for c(i Li Si ; Ji ). We can :t the reduced cross sections uIe2 Qej , with the formula [25] uIe2 Qej = A + B=u + C=u2 + D ln u;

(14)

where u is the reduced incident energy de:ned as u = Ei =Ie ;

(15)

306

Y. Zhao et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 301 – 315

where Ei is the electron incident energy, and Ie is the excitation energy of the calculated transitions. In Eq. (14), A, B, C and D are also four adjustable parameters (in unit of Ry2 a20 ). 2.2. Ionization rate coe7cients In many situations, ionization rate coe8cients, rather than cross sections, are needed. The rate coe8cient is de:ned as  ∞ = vf(v)Q(v) dv; (16) 0

where f(v) is the velocity distribution of electrons and Q(v) is the total electron-impact ionization cross section as a function of electron velocity. If we assume a Maxwell distribution of velocities, then the rate coe8cient as a function of electron temperature is given by  ∞ 1 4 (kT ) = d + ea = √ × e−E=kT Qi (E)E dE 2m (kT )3=2 0

  ∞
 ∞ 1:090 × 10−6 (cm3 =s) = e−uId =kT uId2 Qd du + Bj e−uIe =kT uIe2 Qej du ; (17) (kT )3=2 1 1 j where d and ea are contributions from direct ionization and excitation autoionization, respectively, kT and I are in eV. For the direct ionization cross section, inserting Eq. (8) into Eq. (17), we obtain  ∞ e− x {A[1 − xf1 (x)] + B[1 + x − x(x + 2)f1 (x)] e−uId =kT uId2 Q du = x 1 + Cf1 (x) + Dxf2 (x)};

(18)

where reduced ionization energy x is de:ned as x = Id =kT

(19)

and f1 (x) = e

x

f2 (x) = e

x



∞

e−ux du; u

(20)

∞

ln u −ux e du: u

(21)

1



1

For excitation autoionization cross section, inserting Eq. (14) into Eq. (17), we can obtain  ∞ e− x {A + Bxf1 (x) + Cx[1 − xf1 (x)] + Df1 (x)}: e−uIe =kT uIe2 Qe du = (22) x 1 It can be seen that the integral in Eq. (22) is only depended on f1 (x).

Y. Zhao et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 301 – 315

307

Given the four :t parameters of the scaled cross sections uI 2 Q, the calculation of the rate coe8cients is then reduced to the calculation of functions f1 (x) and f2 (x). The values of the exponential integral f1 (x) can be obtained from Abramowitz and Stegun [29]. Due to the rapid decrease of f2 (x), we have to :t f2 (x) in three diOerent ranges to ensure the accuracy of the :tting [30]. These ranges are 0 ¡ x ¡ 0:2; 0:2 6 x ¡ 1, and 1 6 x ¡ ∞. In the empirical formula f2 (x) =

1 x 3 + a1 x 2 + a2 x + a 3 x 2 x 3 + a4 x 2 + a5 x + a 6

(23)

the :t parameters i are diOerent in the diOerent ranges; their values are given in Table A in Ref. [30]. The relative deviation of this :t from the actual value for f2 (x) is mostly in the range from 10−3 to 10−5 , and the maximum deviation is below 1%. Based on the high precision of the empirical formulas for f1 (x) and f2 (x), the relative error between the rates calculated from the :tted values and the rates obtained with accurate values of the two functions is in the range 10−3 –10−5 for most cases and the maximum error is below 2%.

3. Results and discussions 3.1. Total ionization cross sections We calculate electron-impact ionization cross sections and rate coe8cients of the highly charged ions of argon, Ar 7+ ; : : : ; Ar 17+ , and the calculated results are :tted by the empirical formulas given above. Table 1 lists the ionization energies and the :t parameters in Eq. (8) for the outer-shell and inner-shell direct ionization cross sections for individual ions, respectively, the average deviations are also given in Table 1. It can be seen that the average deviations are less than 0.5% for all. In Eqs. (1) and (17), Qea and ea for Ar 7+ , Ar 14+ and Ar 15+ are the summation over the contribution from each level. In order to calculate Qea and ea conveniently, Qea can also be :tted to the following formula: 2 Qea = A + B=u + C=u2 + D ln u; uIea

(24)

where the reduced incident energy u is de:ned as u = Ei =Iea :

(25)

Iea is the lowest excitation energy for each ion. Using only four :t parameters in Eqs. (24) and (22) (Ie and Qe in Eq. (22) should be replaced by Iea and Qea , respectively.) ea can be directly obtained. The calculation shows that the excitation autoionization cross section is twice the direct ionization cross section for Ar 7+ , and the contribution of excitation autoionization cross sections to the total ionization cross sections are about 5% and 2% for Ar 15+ and Ar 14+ , respectively. Table 2 shows the four :t parameters and the lowest excitation energy in Eq. (24) for Ar 7+ , Ar 14+ and Ar 15+ . As an example of a :t, in Fig. 1, the total excitation autoionization cross section

308

Y. Zhao et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 301 – 315

Table 2 Lowest excitation energy (Ry) and :t parameters (Ry2 a20 ) for excitation autoionization cross sections Ions

Iea

A

B

C

D

Ar 7+ Ar 14+ Ar 15+

17.90 225.8 226.4

−4.84856 −1.00539 −0.43180

22.5059 2.98309 1.45923

−17.2034 −1.06137 −0.52683

10.5368 0.42131 0.66792

16 14

10

-19

2

Qea(10 cm )

12

8 6 4 2 0 0

2

4

6

8

10

12

14

16

u(E/243.9eV) Fig. 1. The solid curve is the total excitation autoionization cross section obtained from the summation over the contribution from each level and the dotted curve is from the :t parameters of Qea for Ar 7+ .

of Ar 7+ as a function of u is plotted. The solid curve is the total excitations–autoionization cross section obtained from the summation over the contribution of each level and the dotted curve is from the :t. It can be seen that the :tting is quite successful. The calculated results indicate that the deviation between the rates from :t parameters of Qea and from the summation over the contribution of each level for Ar 7+ , Ar 14+ and Ar 15+ , is mostly less than 1%. Fig. 2 shows the comparison of the present results of Ar 17+ with the results of Lotz [10] and Lennon et al. [14] and experimental data of Donets [16]. Our DWBE calculations are in good agreement with experimental results, about 5% lower than Lotz’s results and 10% higher than Lennon’s data. Fig. 3 is the comparison of our calculated cross sections of Ar 15+ with the experimental and other theoretical values. The calculations are about 15% lower than the measurements of Donets [16], and about 7% higher than the recommendation of Lennon et al. [14], and about 10% lower than the calculated results of Reed and Chen [9] with relativistic distorted wave. The results from Lotz [10] are much higher than the others. Fig. 4 is the calculated result of Ar 14+ . As Younger’s calculation

Y. Zhao et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 301 – 315

309

10 17+

Ar Cross section (10-22cm2)

8

6

4

2

0

0

10

30

20

40

50

60

Electron energy (keV) Fig. 2. Total ionization cross section for Ar 17+ . The solid curve represents the present calculation. The dashed curve shows the calculated total cross section by Lennon et al. [14]. The dotted curve denotes the theoretical result by Lotz [10]. 4 denotes the experiment data of Donets et al. [16].

2.2 2.0

15+

Ar

Cross section (10-20cm2)

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

2

4

6

8

10

12

14

Electron energy (keV) Fig. 3. Total ionization cross section for Ar 15+ . The solid curve represents the present calculation. The dashed curve shows the calculated total cross section by Lennon et al. [14], the dotted curve denotes the theoretical result by Lotz [10].  is the calculated result of Chen [9]. 4 denotes the experiment data of Donets et al. [16].

310

Y. Zhao et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 301 – 315 5.0 14+

4.5

Ar

Cross section (10-20cm2)

4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0

2

4

6

8

10

12

14

Electron energy (keV) Fig. 4. Total ionization cross section for Ar 14+ . The full curve represents the present calculation. The dashed curve shows the calculated total cross section by Lennon [14]. The dotted curve denotes the theoretical result by Lotz [10]. ◦ denotes theoretical calculation of Younger [6].

[6], the contribution from inner shell (1s) is not included in the direct ionization cross sections, because the 1s2s2 is autoionization state. From Fig. 4 we can see that our DWBE calculations of Ar 14+ are in good agreement with Younger’s [6] and Lennon’s data. And about 15% lower than Lotz’s results [10]. Fig. 5 shows that our calculation of Ar 10+ are about 10% lower than Lotz’s [10] and about 10% higher than Lennon’s recommendation [14]. But the classical calculations of salop [3] are much higher than the others. In fact, the classical calculations of salop [3] are higher from Ar 7+ to Ar 17+ . In Fig. 6 we compared our calculated cross of Ar 9+ section with the experimental and other theoretical values. The calculations agree well with the measurements of Racha: [18], and the data given by Lennon [14] and Lotz [10], and are about 25% lower than the calculation of Laghdas et al. [7]. From Fig. 7 we can see our calculated cross sections of Ar 8+ agree well with the measurements of Zhang et al. [17], and a little lower than Lotz’s, and are about 8% lower than the data of Younger [6] and Lennon et al. [14]. From Fig. 8, it can be seen that our result is between the two experimental results of Racha: [18] and Zhang et al. [17], and our theoretical direct ionization cross section is about 20% lower than Lotz’s. We also :nd that the results from the empirical formula of Bernshtam [11] are near to our calculation for s-subshell, and same as Lotz’s results for p-subshell. The calculated sum of the direct ionization and the excitation autoionization cross sections is about 20% lower than the results of Reed and Chen [8]. We can also see the calculation from the excitation autoionization cross section very important and it is about double to the direct ionization cross section.

Y. Zhao et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 301 – 315

311

4.5 10+

Ar 4.0

Cross section (10-19cm2)

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0

1

2

3

4

5

6

7

8

Electron energy (keV) Fig. 5. Total ionization cross section for Ar 10+ . The full curve represents the present calculation. The dashed curve shows the calculated total cross section by Lennon [14], the dotted curve denotes the theoretical result by Lotz [10]. ♦ denotes theoretical calculation of Salop et al. [3].

6

Cross section (10-19cm2)

Ar

9+

5

4

3

2

1

0 0

1

2

3

4

5

6

7

Electron energy (keV) Fig. 6. Total ionization cross section for Ar 9+ . The full curve represents the present calculation. The dashed curve shows the calculated total cross section by Lennon et al. [14], the dotted curve denotes the theoretical result by Lotz [10]. denote theoretical calculation of Laghdas et al. [7]. 4 denotes the experiment data of Racha: et al. [18].

312

Y. Zhao et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 301 – 315 8

Ar

7

8+

Cross section (10-19cm2)

6 5 4 3 2 1 0 0

1

2

3

4

5

6

Electron energy (keV) Fig. 7. Total ionization cross section for Ar 8+ . The full curve represents the present calculation. The dashed curve shows the calculated total cross section by Lennon et al. [14]. The dotted curve denotes the theoretical result by Lotz [10]. ◦ denotes theoretical calculation of Younger [4]. 4 denote the experiment data of Zhang et al. [17].

3.5

7+

Ar

Cross section (10-18cm2)

3.0

2.5 2.0 1.5 1.0 0.5 0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Electron energy (keV) Fig. 8. Total ionization cross section for Ar 7+ . The short dashed curve are our calculated values without branch ratio, and the solid curve are values with branch ratio. The dashed curve represents our direct ionization cross section. The dotted curve denotes direct ionization cross sections by Lotz [10] and the dash dotted curve is from Bernshtam [11].  is the calculated result of Reed and Chen [9]. 4 and • denote experimental data of Racha: et al. [18] and Zhang et al. [17], respectively.

Rate coefficient (cm3/s)

Y. Zhao et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 301 – 315 10

-7

10

-9

10

-11

10

-13

10

-15

10

-17

10

-19

10

-21

10

0

10

1

2

3

4

10 10 10 Electron Temperature (eV)

10

313

5

Fig. 9. Ionization rate coe8cients for ions of argon. The solid curve shows the present calculation. o denotes data from Voronov [15]. The curves are the rates for 11 ions: Ar 7+ ; : : : ; Ar 17+ from the highest to lowest curve, respectively. Table 3 Ar 15+ rate coe8cient (cm3 =s) kT(eV)

Calculated

Voronov

F (%)

100 300 1000 3000 10,000

3.10E-15 2.07E-12 2.29E-11 4.52E-11 5.06E-11

3.31E-15 2.19E-12 2.35E-11 4.59E-11 4.95E-11

6.02 2.36 0.56 0.47 5.59

3.2. Ionization rates In Fig. 9, Table 3 and Table 4, we compare the present results with the recommended rates given by Voronov [15]. It can be seen that our results agree well with the rates given by Voronov [15] except for the relative lower charged ion Ar 7+ . We also compare our calculation with Arnaud’s result [12]. The comparison shows that the agreement is also good.

314

Y. Zhao et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 301 – 315

Table 4 Ar 8+ rate coe8cient (cm3 =s) kT(eV)

Calculated

Voronov

F (%)

100 300 1000 3000 10,000

1.346E-11 3.310E-10 1.091E-9 1.425E-9 1.255E-9

1.56E-11 3.68E-10 1.18E-9 1.55E-9 1.47E-9

13.7 10.0 7.5 8.0 14.6

4. Conclusions In this paper, electron-impact ionization cross sections and rate coe8cients for highly charged ions Ar 7+ ; : : : ; Ar 17+ are calculated by using the distorted-wave Born exchange (DWBE) approximation including relativistic correction. Our calculation agree well with the experimental measurements and other distorted-wave calculations, and much better than the calculations of classical theory. The systematically calculated results above show that the DWBE approximation is quite valid to calculate the ionization cross sections for highly charged ions. Also, we present some empirical formulas and corresponding :t parameters to calculate the direct ionization and excitation autoionization cross sections and rate coe8cients for argon ions quickly and precisely. Acknowledgements This work was supported by the National Natural Science Foundation of China Project 10104005, Chinese Research Association of Atomic and Molecular Data, National High-Tech ICF Committee in China and the research foundation of Zhonglu Corporation. References [1] Mark TD, Dunn GH. Electron-impact Ionization. Berlin: Springer, 1986. [2] Stephens JA, Botero J. International bulletin on atomic and molecular data for fusion. Vilenna: IAEA, 1995 –2000. p. 50 –1. [3] Salop A. Electron-impact ionization of multicharged ions. Phys Rev A 1976;14:2095–102. [4] Younger SM. Distorted-wave electron-impact-ionization cross sections for highly ionized neonlike atoms. Phys Rev A 1981;23:1138–46. [5] Younger SM. Cross sections and rate for direct electron-impact ionization of sodiumlike ions. Phys Rev A 1981;24:1272–7. [6] Younger SM. Electron-impact ionization cross sections and rates for highly ionized berylliumlike ions. Phys Rev A 1981;24:1278–85. [7] Laghdas K, Reid RHG, Joachain CJ, Burke PG. Electron-impact ionization of Ar 9+ . J Phys B 1995;28:4811–22. [8] Reed KJ, Chen MH. Distorted-wave cross sections for electron-impact ionization of Ar 7+ . Phys Rev A 1996;54: 2967–72. [9] Chen MH, Reed KJ. Electron-impact ionization of charged lithiumlike ions. Phys Rev A 1992;45:4525–9. [10] Lotz W. Electron-impact ionization cross-sections and ionization rate coe8cients for atoms and ions from hydrogen to calcium. Z Phys 1968;216:241–7.

Y. Zhao et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 301 – 315

315

[11] Bernshtam VA, Ralchenko YV, Maron Y. Empirical formula for cross section of direct electron-impact ionization of ions. J Phys B 2000;33:5025–32. [12] Arnaud M, RothenTug R. An updated evaluation of recombination and ionisation rates. Astron Astrophys Suppl 1985;60:425–45. [13] Tawara H, Kato T. Ionization by electron-impact. At Data Nucl Data Tables 1987;36:167–353. [14] Lennon MA, et al. Recommended data on the electron-impact ionization of atoms and ions Tuorine to nickel. J Phys Chem Ref Data 1988;17(3):1285–363. [15] Voronov GS. A practical :t formula for ionization rate coe8cients of atoms and ions by electron-impact: Z = 1–28. At Data Nucl Data Tables 1997;65:1. [16] Donets ED, et al. Investigation of ionization of positive ions by electron-impact. Sov Phys-JETP 1981;53:466. [17] Zhang Y, Reddy CB, Smith RS, Golden DE, Muller DW. Measurement of electron-impact single-ionization cross sections of Ar 8+ . Phys Rev A 1991;44:4368–71. [18] Racha: S. Absolute cross section measurements for electron-impact ionisation of Ar 7+ . J Phys B 1996;24:1037–47. [19] Teng ZX, Chen CY, Yan SX, Wang YS. Rate coe8cients of electron-impact ionization for highly ionized ions. JQSRT 1999;61:123–9. [20] Chen CY, Yan SQ, Wang YS, Yang FJ, Sun YS. Electron-impact ionization cross sections and rates for ions of neon. J Phys B 1998;31:2667–79. [21] Chen CY, Qi JB, Wang YS, Yang FJ. Electron-ion collisional ionization cross sections and rates for the Na isoelectronic sequence. At Data Nucl Data Tables 2001;79:65–107. [22] Fang DF, Hu W, Tang J, Wang YS, Yang FJ. Energy distribution of secondary electrons in electron-impact ionization of hydrogenic and heliumlike ions. Phys Rev A 1993;47:1861–5. [23] Hu W, Fang DF, Wang YS, Yang FJ. Electron-impact-ionization cross section for the hydrogen atom. Phys Rev A 1994;49:989–91. [24] Cowan RD. The theory of atomic structure and spectra. Berkeley: University of California Press, 1981. [25] Gri8n DC, Bottcher C, Pindzola MS. Distorted-wave calculations of the electron-impact ionization of highly ionized Na-like ions. Phys Rev A 1987;36:3642–53. [26] Fang DF, Wang YS. Calculation of slowly converging integrals in electron–ion collision problems. J Phys B 1991;24:1749–56. [27] Hu W, Chen CY. Systematic study on ionization cross sections of electron-ion collisions for the Li-like isoelectronic sequence. J Phys B 1996;29:2887–95. [28] Golden IB, Sampson DH. Ionisation from 4s, 4p, 4d and 4f sublevels of highly charged ions. J Phys B 1980;13: 2645–52. [29] Abramowitz M, Stegun IA. Handbook of mathematical functions with formulas, graphs and mathematical tables. New York: Dover Pub., Inc., 1972. [30] Teng ZX, Chen CY, Yan SX, Wang YS, Yang FJ, Sun YS. Rate coe8cients of electron-impact ionization for highly ionized ions. JQSRT 1999;61:123–9.