Relativistic calculations of electron-impact ionization for highly charged hydrogen-like ions

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Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 161–171 www.elsevier.com/locate/jqsrt

Relativistic calculations of electron-impact ionization for highly charged hydrogen-like ions X.H. Shi, C.Y. Chen, Y. Zhao, Y.S. Wang Institute of Modern Physics, Fudan University, Shanghai 200433, People’s Republic of China Received 6 October 2003; accepted 21 May 2004

Abstract Electron-impact ionization cross-sections and rate coefficients of the 1s ground state for H-like C, O, Mg, Ar, Fe, Cu, As, Kr, Y, Mo ions with incident electron energies up to 15 times the ionization threshold energy have been systematically calculated by the relativistic distorted-wave Born exchange (DWBE) approximation. The comparison of the result with the experimental data, other theoretical calculations and recommended values shows the very good agreement. The influence from relativistic and the lowest order QED effect in the calculation is discussed. The calculated ionization cross-sections are fitted by empirical formulas. These fits can be readily integrated over a relativistic Maxwellian electron distribution function to obtain rate coefficient for plasma modeling. r 2004 Elsevier Ltd. All rights reserved. Keywords: Highly charged hydrogen-like ions; Ionization cross-section and rate coefficient; Relativistic distorted-wave method; 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, including electron-impact ionization (EI) cross-sections and rate coefficients, are required. Corresponding author.

E-mail address: [email protected] (C.Y. Chen). 0022-4073/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2004.05.054

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Although EI cross-sections have been measured for various ions and different incoming electron energies. The measurements for highly charged ions, especially for hydrogen-like ions, were performed quite recently. For low-Z hydrogen-like ions several species have been measured by the crossed-beams technique [1,2], and some intermediate- and high-Z hydrogen-like ions have been measured in an electron beam ion trap (EBIT) [3–6]. Some relativistic calculations including the Møller interaction [7,8] and generalized Breit interaction (GBI) [9,10] between bound and free electrons when Z is large have been published. GBI and Møller interaction are the first-order QED correction to the Coulomb interaction. Their calculations are in the range of incident electron energies up to about six times of the threshold energy. In Ref. [10] Fontes et al. also gave a formula for rate coefficients with a relativistic Maxwellian distribution. Besides, some scaling empirical formulas of EI for H-like ions have been proposed due to the importance for plasma modeling [2,5,10,11]. Recently, Bernshtam et al. [12] analyzed published data and proposed an empirical formula for direct electron-impact ionization cross-section, which improves the Lotz formula [11]. Based on the recommended data by the Belfast group and the data derived from several other sources, Voronov [13] gave a practical fit formula of ionization rate coefficients for the ions (Zp28). In previous work [14] we used a semi-relativistic DWBE approximation method to systematically calculate the electron-impact ionization cross-sections for several hydrogen-like ions (Zp30) and give an empirical formula to fit the calculated cross-sections. In this paper: (1) We use a relativistic DWBE approximation method to calculate the electron–ion collisional ionization cross-sections for H isoelectronic sequence (Zp42) in a wide range of incident electron energies up to 15 times the ionization threshold energy to meet the practical application. The calculation does not include the GBI (or Møller) interaction. We discussed the dependence of relativistic effect on the nuclear charge Z and incident electron energies, and the effects of QED on the EI for intermediate-Z H-like ions. The calculated results are compared with the experimental data and other theoretical calculations. (2) We fit the calculated cross-sections with empirical formulas and corresponding fit parameters. (3) The rate coefficients are calculated by a set of empirical formulas using relativistic Maxwellian electron distribution and those fit parameters of the cross-sections for plasma modeling. The calculated rate coefficients are compared with the recommended data [13]. The remainder of the paper is as follows. In Section 2, we describe the theoretical method employed to determine the ionization cross-sections and rate coefficients, and the respective empirical formulas. In Section 3, the results and discussions are given. The comparison of the calculated cross-sections and rate coefficients with the experimental measurements and other calculation shows the good agreement.

2. Theoretical and calculation method 2.1. Ionization cross-sections and corresponding empirical formula On the relativistic DWBE approximation, the direct ionization cross-section (in the unit of pa20 , a0 is Bohr radius) of the bound electron ðnb l b j b Þ in the target ion can be written

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163

as [15] 8 pð2j b þ 1Þk2i

Qðnb l b j b Þ ¼

Z

ðE i IÞ=2

dE e 0

X

Ql ðnb l b j b Þ;

ð1Þ

l

where I is the ionization potential of each ion, E i and E e are energies of the incident and ejected electron, respectively. ki is the relativistic wave number of the incident electron: ki ¼ ðE i þ a2 E 2i =4Þ1=2 ;

ð2Þ

where a is the fine structure constant. In Eq. (1), we use a three-point Gaussian integral when the incident energy is less than three times ionization threshold energy, and a five-point Gaussian integral for the remaining incident energies. By using a partial wave expansion, the energy differential cross-section Ql can be divided into angular factor and Slater integrals. X jPl ðnb l b j b ki l i j i ; ke l e j e kf l f j f Þj2 ; ð3Þ Ql ðnb l b j b Þ ¼ li; le; lf ji; je; jf where, k, l and j are the wave number, the angular momentum and total angular of free electron, respectively. Pl ¼ ð2l þ 1Þ1=2 hj b kcl kj e ihj i kcl kj f i Dl ðnb l b j b ki l i j i ; ke l e j e kf l f j f Þ ( ) X jb jf t lþt 1=2

hj b kct kj f ihj i kct kj e i ð1Þ ð2l þ 1Þ þ j l j t i e

E t ðnb l b j b ki l i j i ; ke l e j e kf l f j f Þ;

(4)

in which Dl and E t are the direct and exchange radial Slater integral, respectively. Z 1Z 1 l D ¼ ½Pnb l b j b ðr1 ÞPke l e je ðr1 Þ þ Qnb l b jb ðr1 ÞQke l e je ðr1 Þ 0

Et ¼

Z

0 l ro rlþ1 4

1

0

Z 0



½Pki l i ji ðr2 ÞPkf l f jf ðr2 Þ þ Qki l i ji ðr2 ÞQkf l f jf ðr2 Þ dr1 dr2 ;

(5)

1

½Pnb l b jb ðr1 ÞPkf l f jf ðr1 Þ þ Qnb l b jb ðr1 ÞQkf l f jf ðr1 Þ

rto ½Pki l i ji ðr2 ÞPke l e je ðr2 Þ þ Qki l i ji ðr2 ÞQke l e je ðr2 Þ dr1 dr2 ; rtþ1 4

(6)

where P and Q are the large and small components of relativistic radial function, respectively. Pnb l b jb for bound electron is obtained from the relativistic atomic structure program of Grant [16]. The radial wave functions for incident, ejected and scattered electron are calculated in our own program. It should be pointed out that in Ref. [15] for the calculations of radial wave functions of free electrons, the exchange potential between free and bound electrons are taken in an approximation of free electron gas, but in our calculation a local semiclassical approximation is

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used for the exchange interaction as in our previous work [17,18] and the calculations of Pindzola et al. [19]. The Calculation of slowly converging Slater integrals has been described in detail by Fang and Wang [20]. In the present calculation, the summation of each partial wave was truncated when the increments were less than 0.2%. The total error introduced by the numerical calculation is estimated to be less than 0.5%. We fit the calculation direct cross-sections with the empirical formula of Younger [21]     1 1 2 2 þ C ln u þ D ln u=u; ð7Þ þB 1 QR ¼ uIðRyÞ Q ¼ A 1  u u where IðRyÞ is ionization threshold energy in the unit of Rydberg energy, and u is the reduced incident energy defined as u ¼ E i =I:

ð8Þ

The energy range in our calculation is quite large, from u ¼ 1:125 to 15.0. In Eq. (7), A, B, C and D are four adjustable parameters (in unit of Ry2 pa20 ) obtained by an ‘‘Optimal calculation method’’ [22]. The average deviation from fit is defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N  u 1 X Q ðuÞ  Q ðuÞ2 fit cal  100: ð9Þ Fð%Þ ¼ t N i¼1 Qcal ðuÞ

2.2. Rate coefficients and corresponding empirical formula In many situations, ionization rate coefficients, rather than cross-sections, are needed. For sufficiently high temperatures, at which 1s ionization of higher Z ions may become important, and high incident electron energy, a relativistic treatment of the free electron energy distribution is required. Using a relativistic Maxwellian distribution function, the rate coefficient formula for 1s ionization by electron impact is given by [10]  Z 2em Ry 1 ðI=kTÞu a2 að1S; TÞ ¼ e QR 1 þ IðRyÞu du; ð10Þ N eh 4 1 where N e is the electron density, QR ¼ uIðRyÞ2 Q (in unit of Ry2 pa20 ). em can be expressed (when kTomc2 ) by 2 ð2pmkTÞ3=2 RðkTÞ; N e h3     15 kT 105 kT 2 315 kT 2 RðkTÞ ¼ 1 þ þ  þ

: 8 mc2 108 mc2 1024 mc2

em ¼

Using Eqs. (10) and (11), we have 1 pe4 að1S; TÞ ¼ pffiffiffiffiffiffiffiffiffi RðkTÞ 2pmðkTÞ3=2

Z

1 ðI=kTÞu

e 1

 a2 QR 1 þ IðRyÞu du; 4

ð11Þ

ð12Þ

ð13Þ

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Using Eq. (7), we can easily perform the integration over energy in Eq. (13) and obtain að1S; TÞ

 x  a2 IðRyÞ e a2 IðRyÞ ex B ðA þ B þ CÞ þ A þ B  4 4 x x2 RðkTÞðkTÞ3=2   a2 IðRyÞ a2 IðRyÞ ex  A þ 2B  B ex f 1 ðxÞ þ Bex  Bex f 1 ðxÞ þ C 2 f 1 ðxÞ 4 4 x   a2 IðRyÞ ex D þ Cþ f ðxÞ þ Dex f 2 ðxÞ ; 4 x 1

¼

1:090  106 cm3 =s

(14)

where kT is in eV and x is defined as x ¼ I=kT and f 1 ðxÞ ¼ e

x

ð15Þ Z

1 ux

e

f 2 ðxÞ ¼ ex

Z

1 1

du;

ð16Þ

ln u du: u

ð17Þ

u

1

Given the four fit parameters of the scaled cross-sections uI 2 Q, the calculation of the rate coefficients is reduced to the calculation of function f 1 ðxÞ and f 2 ðxÞ which can be easily obtained with high accuracy by two empirical formulas, respectively (see our previous work [23]). When kTmc2 and ða2 IðRyÞ=4Þu1, Eq. (13) will reduce to the nonrelativistic rate coefficient formula (Eq. (6) in Ref. [23]).

3. Results and discussions 3.1. Ionization cross-sections We calculate electron-impact ionization cross-sections and rate coefficients for 10 highly charged hydrogen-like ions: C5þ , O7þ , Mg11þ , Ar17þ , Fe25þ , Cu28þ , As32þ , Kr35þ , Y38þ and Mo41þ . Figs. 1–3 show the cross-sections for Ar17þ , Fe25þ and Mo41þ , and as a comparison, the experimental data [5,6,24], our own semi-relativistic calculation, other relativistic calculations, and the results from improved Lotz formula [12], are also plotted in Figs. 1–3. From Figs. 1–3, we can see that: (1) Our relativistic calculations are in good agreement with experiments and other theoretical calculation without Breit interaction. The results from improved Lotz formulas is near our semirelativistic calculations. (2) The relativistic cross-section is larger than semi-relativistic calculation by 6% when u ¼ 2:5 and by 12% when u ¼ 10 for Ar17þ ; by 9% when u ¼ 2:5 and by 21% when u ¼ 10 for Fe25þ ; by14% when u ¼ 2:5 and by 48% when u ¼ 10 for Mo41þ . So it is necessary to calculate ionization cross sections using relativistic method for hydrogen-like ions with ZX18.

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Ar17+ Relativistic Semi-Relativistic improved Lotz formula [12] Experiment [24]

Cross Section (10-22cm2)

8

6

4

2 10

0

20

30

40

50

60

70

Energy (keV)

Fig. 1. Ionization cross-section for Ar17þ .

Cross Section (10-22cm2)

4 Fe25+ Relativistic Semi-Relativistic improved Lotz formula [12] O'Rourke distorted-wave[5] Sampson with QED [10] EBIT[5]

3

2

1

0 0

20

40 60 Energy (keV)

80

100

Fig. 2. Ionization cross-section for Fe25þ .

(3) The contribution from QED effect are about 1–2% for hydrogen-like Fe25þ and about 5–10% for Mo41þ in wide energy range. 3.2. Fit of ionization cross-sections Fig. 4 shows the dependence of uI 2 Q on u. Table 1 lists the ionization threshold energies and the fit parameters in Eq. (7) for individual ions, respectively. The average deviations are also given in Table 1. It can be seen that most average deviations are less than 0.5%. In Fig. 5, the uI 2 Q=Z as a function of 1=Z for different values of u are plotted. From the figure, it can be seen that the variation is nearly a straight line. We can fit the scaled cross-sections to the form uI 2 Q=Z ¼ aðuÞ þ bðuÞ=Z;

ð18Þ

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Cross Section (10-22cm2)

40

30

Mo41+ Relativistic Semi-Relativistic improved Lotz formula [12] Sampson without QED [10] Sampson with QED [10] EBIT (Mars et al. 1997) [4] EBIT (Watanabe et al. 2002) [6]

20

10

0 0

50

100

150

200

Energy (keV)

Fig. 3. Ionization cross-section for Mo41þ .

14 Z=42 Z=39 Z=36 Z=33 Z=29 Z=26 Z=18

12 10

uI2Q

8

Z=6 6 4 2 0 0

2

4

6

8

10

12

14

16

18

u

Fig. 4. The variation of uI 2 Q with u.

where aðuÞ and bðuÞ are two adjustable parameters. Due to the need of the fit with high accuracy, we fit the variation of uI 2 Q=Z with 1=Z in two regions separately, Zp20 and 20oZp42. A formula like Eq. (7) can in turn be used to fit the variation of a and b with u. So that     1 1 2 ln u þ C 1 ln u þ D1 þ B1 1  ; ð19Þ aðuÞ ¼ A1 1  u u u  bðuÞ ¼ A2

   1 1 2 ln u ; 1 þ C 2 ln u þ D2 þ B2 1  u u u

ð20Þ

where A1 , B1 , C 1 , D1 , A2 , B2 , C 2 and D2 are adjustable parameters. These fit parameters are given in Table 2. One can use these parameters, the threshold energies in Table 1, and Eqs. (18)–(20) to quickly obtain the ionization cross-sections of the other H-like ions with Zp42 which have not been tabulated. For still higher Z ions, the extrapolation is inappropriate because of the larger QED effect.

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Table 1 Ionization threshold energies IðRyÞ, fit parameters ðRy2 pa20 Þ and fit errors for each ion H-like ions

IðRyÞ

A

B

C

D

F (%)

C5þ O7þ Mg11þ Ar17þ Fe25þ Cu28þ As32þ Kr35þ Y38þ Mo41þ

3.603E+1 6.408E+1 1.443E+2 3.255E+2 6.825E+2 8.507E+2 1.106E+3 1.320E+3 1.554E+3 1.808E+3

1.339E+1 1.271E+1 9.861 4.766 4.671 8.993 1.514E+1 1.997E+1 2.584E+1 3.097E+1

4.926 4.742 3.739 2.169 6.751E1 1.919 3.604 4.955 6.684 8.073

2.280E1 3.979E1 9.941E1 2.189 4.442 5.504 7.057 8.273 9.693 1.101E+1

1.061E+1 1.014E+1 7.982 4.144 2.983 6.236 1.083E+1 1.445E+1 1.889E+1 2.270E+1

0.4206 0.3730 0.3448 0.2434 0.2672 0.3243 0.4096 0.5391 0.6289 0.7562

1.2

u=15 u=12 u=10 u=8

1.0

u=6 u=5

uI2Q/Z

0.8

u=4

0.6 u=3

0.4 u=2

0.2

u=1.5

0.0 0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

1/Z

Fig. 5. The variation of uI 2 Q=Z with 1=Z.

As a test, we compared the ionization cross-sections from the relativistic DWBE calculation with the ones from Eq. (18) using the fit parameters in Table 2, and found that the deviations between them are less than about 5% in wide energy region. 3.3. Ionization rates Fig. 6 shows the present relativistic results from Eq. (14) and the recommended rates of C5þ , O , Mg11þ , Ar17þ and Fe25þ given by Voronov [13]. It can be seen that the present relativistic results are larger than Voronov’s rates by 5–15% in wide temperature region. Such as for Ar17þ , from Fig. 7 it can be seen that Voronov’s rates are near the semi-relativistic calculated values, and relativistic rates are larger than semi-relativistic results by 10% at kT ¼ 25 keV and by 15% at kT ¼ 50 keV. The calculation also shows the same conclusion for Fe25þ : the relativistic calculation rate coefficients are larger than the semi-relativistic calculation by 12% at kT ¼ 25 keV and by 18% at kT ¼ 50 keV. 7þ

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Table 2 Fit parameters of aðuÞ and bðuÞ for H-like ions a(u)

A1

B1

C1

D1

F ð%Þ

Zp20 20oZp42

6.408E1 1.133

2.096E1 2.803E1

1.451E1 3.107E1

4.788E1 8.247E1

7.820 7.346

b(u)

A2

B2

C2

D2

F ð%Þ

Zp20 20oZp42

1.823E+1 4.272E+1

6.488 1.333E+1

8.666E1 6.935

1.427E+1 3.295E+1

0.5003 1.515

Rate Coefficient (10-12cm3/s)

100

C5+ O7+

10

Mg11+

Ar17+ 1

Fe25+

1

10

100

T (keV)

Rate Coefficient (10-12cm3/s)

Fig. 6. Ionization rate coefficients for H-like ions. The curves are the rates for five ions: C5þ , O7þ , Mg11þ , Ar17þ , Fe25þ , from the highest to the lowest curves, respectively. The open circles are recommended data by Voronov [13].

6

4

2 relativistic rate coefficient semi-relativistic rate coefficient recommanded rates of Voronov [13] 0 0

20

40

60

80

100

T (keV)

Fig. 7. Ionization rate coefficient for Ar17þ .

120

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4. Conclusion In this paper, electron-impact ionization cross-sections and rate coefficients for highly charged H-like ions are systematically calculated by using the relativistic DWBE approximation. Our calculation agrees well with the experimental data and other relativistic calculations without QED effect. Also, we present the empirical formulas and corresponding fit parameters to calculate the ionization cross-sections and rate coefficients for ions with Zp42 quickly and precisely. The calculation shows relativistic method is necessary for H-like ions with ZX18.

Acknowledgements This work is supported by the National Natural Science Foundation of China Project 10104005, National High-tech ICF Committee in China, Chinese Association of Atomic and Molecular Data and the research foundation of Zhonglu Corporation.

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