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

ARTICLE IN PRESS

Journal of Quantitative Spectroscopy &

Radiative Transfer 91 (2005) 161–171

0022-4073/$ -

doi:10.1016/j.

�CorresponE-mail add

www.elsevier.com/locate/jqsrt

Relativistic calculations of electron-impact ionization forhighly 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 thresholdenergy 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 andrecommended values shows the very good agreement. The influence from relativistic and the lowest orderQED effect in the calculation is discussed. The calculated ionization cross-sections are fitted by empiricalformulas. These fits can be readily integrated over a relativistic Maxwellian electron distribution function toobtain 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. Inastrophysical 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.

see front matter r 2004 Elsevier Ltd. All rights reserved.

jqsrt.2004.05.054

ding author.

ress: [email protected] (C.Y. Chen).

www.elsevier.com/locate/jqsrt

ARTICLE IN PRESS

X.H. Shi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 161–171162

Although EI cross-sections have been measured for various ions and different incoming electronenergies. The measurements for highly charged ions, especially for hydrogen-like ions, wereperformed quite recently. For low-Z hydrogen-like ions several species have been measured by thecrossed-beams technique [1,2], and some intermediate- and high-Z hydrogen-like ions have beenmeasured 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. Theircalculations are in the range of incident electron energies up to about six times of the thresholdenergy. In Ref. [10] Fontes et al. also gave a formula for rate coefficients with a relativisticMaxwellian distribution. Besides, some scaling empirical formulas of EI for H-like ions have beenproposed 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 onthe 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-likeions (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 electronenergies up to 15 times the ionization threshold energy to meet the practical application. Thecalculation does not include the GBI (or Møller) interaction. We discussed the dependence ofrelativistic effect on the nuclear charge Z and incident electron energies, and the effects of QED onthe EI for intermediate-Z H-like ions. The calculated results are compared with the experimentaldata 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 plasmamodeling. 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 respectiveempirical formulas. In Section 3, the results and discussions are given. The comparison of thecalculated cross-sections and rate coefficients with the experimental measurements and othercalculation 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 theunit of pa20, a0 is Bohr radius) of the bound electron ðnblbjbÞ in the target ion can be written

ARTICLE IN PRESS

X.H. Shi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 91 (2005) 161–171 163

as [15]

QðnblbjbÞ ¼8

pð2jb þ 1Þk2i

Z ðEi�IÞ=2

0

dEe

Xl

QlðnblbjbÞ; ð1Þ

where I is the ionization potential of each ion, Ei and Ee are energies of the incident and ejectedelectron, respectively. ki is the relativistic wave number of the incident electron:

ki ¼ ðEi þ a2E2i =4Þ

1=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 incidentenergies. By using a partial wave expansion, the energy differential cross-section Ql can be dividedinto angular factor and Slater integrals.

QlðnblbjbÞ ¼X

li; le; lf

ji; je; jf

jPlðnblbjbkiliji; kelejekf lf jf Þj2; ð3Þ

where, k, l and j are the wave number, the angular momentum and total angular of free electron,respectively.

Pl ¼ ð2lþ 1Þ�1=2hjbkclkjeihjikclkjfi Dlðnblbjbkiliji; kelejekf lf jf Þ

þX

t

ð�1Þlþtð2lþ 1Þ1=2

jb jf t

ji je l

( ) hjbkctkjfihjikctkjei

Etðnblbjbkiliji; kelejekf lf jf Þ; (4)

in which Dl and Et are the direct and exchange radial Slater integral, respectively.

Dl ¼

Z 1

0

Z 1

0

½Pnblbjbðr1ÞPkelejeðr1Þ þ Qnblbjbðr1ÞQkeleje

ðr1Þ

rlo

rlþ14

½Pkilijiðr2ÞPkf lf jf ðr2Þ þ Qkilijiðr2ÞQkf lf jf

ðr2Þ dr1 dr2; (5)

Et ¼

Z 1

0

Z 1

0

½Pnblbjbðr1ÞPkf lf jf ðr1Þ þ Qnblbjbðr1ÞQkf lf jf

ðr1Þ

rto

rtþ14

½Pkilijiðr2ÞPkelejeðr2Þ þ Qkilijiðr2ÞQkeleje

ðr2Þ dr1 dr2; (6)

where P and Q are the large and small components of relativistic radial function, respectively.Pnblbjb 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 ownprogram. It should be pointed out that in Ref. [15] for the calculations of radial wave functions offree electrons, the exchange potential between free and bound electrons are taken in anapproximation of free electron gas, but in our calculation a local semiclassical approximation is

ARTICLE IN PRESS


used for the exchange interaction as in our previous work [17,18] and the calculations of Pindzolaet al. [19]. The Calculation of slowly converging Slater integrals has been described in detail byFang 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 isestimated to be less than 0.5%.We fit the calculation direct cross-sections with the empirical formula of Younger [21]

QR ¼ uIðRyÞ2Q ¼ A 1�1

u

� �þ B 1�

1

u

� �2

þ C ln u þ D ln u=u; ð7Þ

where IðRyÞ is ionization threshold energy in the unit of Rydberg energy, and u is the reducedincident energy defined as

u ¼ Ei=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 andD are four adjustable parameters (in unit of Ry2 pa20) obtained by an ‘‘Optimal calculationmethod’’ [22]. The average deviation from fit is defined as

Fð%Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N

XN

i¼1

QfitðuÞ � QcalðuÞ

QcalðuÞ

� �2

vuut � 100: ð9Þ

2.2. Rate coefficients and corresponding empirical formula

In many situations, ionization rate coefficients, rather than cross-sections, are needed. Forsufficiently high temperatures, at which 1s ionization of higher Z ions may become important, andhigh incident electron energy, a relativistic treatment of the free electron energy distribution isrequired. Using a relativistic Maxwellian distribution function, the rate coefficient formula for 1sionization by electron impact is given by [10]

að1S;TÞ ¼2e�m Ry

Neh

Z 1

1

e�ðI=kTÞuQR 1þa2

4IðRyÞu

� �du; ð10Þ

where Ne is the electron density, QR ¼ uIðRyÞ2Q (in unit of Ry2 pa20). em can be expressed (when

kTomc2) by

em ¼2

Neh3ð2pmkTÞ

3=2RðkTÞ; ð11Þ

RðkTÞ ¼ 1þ15

8

kT

mc2þ

105

108

kT

mc2

� �2

�315

1024

kT

mc2

� �2

þ : ð12Þ

Using Eqs. (10) and (11), we have

að1S;TÞ ¼1

RðkTÞ

pe4ffiffiffiffiffiffiffiffiffi2pm

pðkTÞ

3=2

Z 1

1

e�ðI=kTÞuQR 1þa2

4IðRyÞu

� �du; ð13Þ

ARTICLE IN PRESS


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

að1S;TÞ

¼1:090� 10�6 cm3=s

RðkTÞðkTÞ3=2

A þ B �a2IðRyÞ

4B

� �e�x

xþ

a2IðRyÞ

4ðA þ B þ CÞ

e�x

x2

�

� A þ 2B �a2IðRyÞ

4B

� �e�xf 1ðxÞ þ Be�x � Be�xf 1ðxÞ þ

a2IðRyÞ

4C

e�x

x2f 1ðxÞ

þ C þa2IðRyÞ

4D

� �e�x

xf 1ðxÞ þ De�xf 2ðxÞ

�; (14)

where kT is in eV and x is defined as

x ¼ I=kT ð15Þ

and

f 1ðxÞ ¼ ex

Z 1

1

e�ux

udu; ð16Þ

f 2ðxÞ ¼ ex

Z 1

1

ln u

udu: ð17Þ

Given the four fit parameters of the scaled cross-sections uI2Q, the calculation of the ratecoefficients is reduced to the calculation of function f 1ðxÞ and f 2ðxÞ which can be easily obtainedwith high accuracy by two empirical formulas, respectively (see our previous work [23]). WhenkT�mc2 and ða2IðRyÞ=4Þu�1, 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 highlycharged hydrogen-like ions: C5þ, O7þ, Mg11þ, Ar17þ, Fe25þ, Cu28þ, As32þ, Kr35þ, Y38þ andMo41þ. Figs. 1–3 show the cross-sections for Ar17þ, Fe25þ and Mo41þ, and as a comparison, theexperimental 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 semi-relativistic 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 calculateionization cross sections using relativistic method for hydrogen-like ions with ZX18.

ARTICLE IN PRESS

0

2

4

6

8

10

Ar17+

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

Cro

ss S

ecti

on (

10-2

2 cm2 )

Energy (keV)

10 20 30 40 50 60 70

Fig. 1. Ionization cross-section for Ar17þ.

0 20 40 600

1

2

3

4 Fe25+

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

Cro

ss S

ecti

on (

10- 2

2 cm2 )

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 about5–10% for Mo41þ in wide energy range.

3.2. Fit of ionization cross-sections

Fig. 4 shows the dependence of uI2Q on u. Table 1 lists the ionization threshold energies and thefit parameters in Eq. (7) for individual ions, respectively. The average deviations are also given inTable 1. It can be seen that most average deviations are less than 0.5%.In Fig. 5, the uI2Q=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 tothe form

uI2Q=Z ¼ aðuÞ þ bðuÞ=Z; ð18Þ

ARTICLE IN PRESS

0 50 100 150 2000

10

20

30

40

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]

Cro

ss S

ecti

on (

10-2

2 cm2 )

Energy (keV)

Fig. 3. Ionization cross-section for Mo41þ.

0 2 4 6 8 10 12 14 16 18

0

2

4

6

8

10

12

14

Z=6

Z=29

Z=36

Z=18Z=26

Z=33

Z=39Z=42

uI2 Q

u

Fig. 4. The variation of uI2Q 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 uI2Q=Z with 1=Z in two regions separately, Zp20 and 20oZp42. Aformula like Eq. (7) can in turn be used to fit the variation of a and b with u. So that

aðuÞ ¼ A1 1�1

u

� �þ B1 1�

1

u

� �2

þ C1 ln u þ D1ln u

u; ð19Þ

bðuÞ ¼ A2 1�1

u

� �þ B2 1�

1

u

� �2

þ C2 ln u þ D2ln u

u; ð20Þ

where A1, B1, C1, D1, A2, B2, C2 and D2 are adjustable parameters. These fit parameters are givenin Table 2. One can use these parameters, the threshold energies in Table 1, and Eqs. (18)–(20) toquickly obtain the ionization cross-sections of the other H-like ions with Zp42 which have notbeen tabulated. For still higher Z ions, the extrapolation is inappropriate because of the largerQED effect.

ARTICLE IN PRESS

Table 1

Ionization threshold energies IðRyÞ, fit parameters ðRy2 pa20Þ and fit errors for each ion

H-like ions IðRyÞ A B C D F (%)

C5þ 3.603E+1 1.339E+1 �4.926 2.280E�1 �1.061E+1 0.4206

O7þ 6.408E+1 1.271E+1 �4.742 3.979E�1 �1.014E+1 0.3730

Mg11þ 1.443E+2 9.861 �3.739 9.941E�1 �7.982 0.3448

Ar17þ 3.255E+2 4.766 �2.169 2.189 �4.144 0.2434

Fe25þ 6.825E+2 �4.671 6.751E�1 4.442 2.983 0.2672

Cu28þ 8.507E+2 �8.993 1.919 5.504 6.236 0.3243

As32þ 1.106E+3 �1.514E+1 3.604 7.057 1.083E+1 0.4096

Kr35þ 1.320E+3 �1.997E+1 4.955 8.273 1.445E+1 0.5391

Y38þ 1.554E+3 �2.584E+1 6.684 9.693 1.889E+1 0.6289

Mo41þ 1.808E+3 �3.097E+1 8.073 1.101E+1 2.270E+1 0.7562

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.180.0

0.2

0.4

0.6

0.8

1.0

1.2

u=1.5

u=15u=12u=10

u=8

u=6u=5

u=4

u=3

u=2

uI2 Q

/Z

1/Z

Fig. 5. The variation of uI2Q=Z with 1=Z.


As a test, we compared the ionization cross-sections from the relativistic DWBE calculationwith the ones from Eq. (18) using the fit parameters in Table 2, and found that the deviationsbetween 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,O7þ, Mg11þ, Ar17þ and Fe25þ given by Voronov [13]. It can be seen that the present relativisticresults 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, andrelativistic rates are larger than semi-relativistic results by 10% at kT ¼ 25 keV and by 15% atkT ¼ 50keV. The calculation also shows the same conclusion for Fe25þ: the relativisticcalculation rate coefficients are larger than the semi-relativistic calculation by 12% at kT ¼

25 keV and by 18% at kT ¼ 50 keV.

ARTICLE IN PRESS

Table 2

Fit parameters of aðuÞ and bðuÞ for H-like ions

a(u) A1 B1 C1 D1 F ð%Þ

Zp20 �6.408E�1 2.096E�1 1.451E�1 4.788E�1 7.820

20oZp42 �1.133 2.803E�1 3.107E�1 8.247E�1 7.346

b(u) A2 B2 C2 D2 F ð%Þ

Zp20 1.823E+1 �6.488 �8.666E�1 �1.427E+1 0.5003

20oZp42 4.272E+1 �1.333E+1 �6.935 �3.295E+1 1.515

1 10 100

1

10

100

Fe25+

Ar17+

Mg11+

O7+

C5+

Rat

e C

oeff

icie

nt (1

0-12 cm

3 /s)

T (keV)

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].

0

2

4

6

relativistic rate coefficient

semi-relativistic rate coefficient

recommanded rates of Voronov [13]

Rat

e C

oeff

icie

nt (

10-1

2 cm3 /s

)

T (keV)20 40 60 80 100 1200

Fig. 7. Ionization rate coefficient for Ar17þ.


ARTICLE IN PRESS


4. Conclusion

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

Acknowledgements

This work is supported by the National Natural Science Foundation of China Project10104005, National High-tech ICF Committee in China, Chinese Association of Atomic andMolecular Data and the research foundation of Zhonglu Corporation.

References

[1] Tinschert K, Muller A, et al. Experimental cross-sections for electron impact ionization of hydrogen-like Li2þ ions.

J Phys B 1989;22:531.

[2] Aichele K, Hartenfeller U, et al. Electron impact ionization of the hydrogen-like ions B4þ, C5þ, N6þ and O7þ.

J Phys B 1998;31:2369.

[3] Marrs RE, Elliott SR, Knapp DA. Production and trapping of hydrogenlike and bare uranium ions in an electron

beam ion trap. Phys Rev Lett 1994;72:4082.

[4] Marrs RE, Elliott SR, Scofield JH. Measurement of electron-impact ionization cross sections for hydrogen-like

high-Z ions. Phys Rev A 1997;56:1338.

[5] O’Rourke B, Currell FJ, Kuramoto H, Li YM, Ohtani S, Tong XM, et al. Electron-impact ionization of hydrogen-

like iron ions. J Phys B 2001;34:4003.

[6] Watanabe H, Currell FJ, Kuramoto H, Ohtani S, O’Rourke BE, Tong XM. Electron impact ionization of

hydrogen-like molybdenum ions. J Phys B 2002;35:5095.

[7] Moores DL, Reed KJ. Effect of the Møller interaction on electron-impact ionization of high-Z hydrogenlike ions.

Phys Rev A 1995;51:R9.

[8] Moores DL, Reed KJ. Electron collisions with very highly charged ions—relativistic calculations. Nucl Instrum

Methods B 1995;98:122.

[9] Fontes CJ, Sampson DH, Zhang HL. Relativistic calculation of cross sections for ionization of U90þ and U91þ ions

by electron impact. Phys Rev A 1995;51:R12.

[10] Fontes CJ, Sampson DH, Zhang HL. Fully relativistic calculations of and fits to 1s ionization cross sections. Phys

Rev A 1999;59:1329.

[11] Lotz W. Electron-impact ionization cross-sections and ionization rate coefficients for atoms and ions from

hydrogen to calcium. Z Phys 1968;216:241.

[12] Bernshtam VA, Ralchenko YV, Maron Y. Empirical formula for cross section of direct electron-impact ionization

of ions. J Phys B 2000;33:5025.

[13] Voronov GS. A practical fit formula for ionization rate coefficients of atoms and ions by electron-impact:

Z ¼ 1–28. At Data Nucl Data Tables 1997;65:1.

[14] Fang D, Hu W, Chen C, Wang Y, et al. Electron–ion collisional ionization cross sections for the H and He

isoelectronic sequences. At Data Nucl Data Tables 1995;61:91.

ARTICLE IN PRESS


[15] Sampson DH, Zhang HL. Rapid-ionization approach based on the factorization method. Phys Rev A

1992;45:1657.

[16] Parpia FA, Fischer CF, Grant IP. Grasp 92: a package for large-scale relativistic atomic structure calculations.

Comput Phys Commun 1996;94:249.

[17] Fang D, Hu W, Tang J, Wang Y, Yang F. Energy distribution of secondary electrons in electron-impact ionization

of hydrogenlike and heliumlike ions. Phys Rev A 1993;47:1861.

[18] Chen CY, Qi JB, Wang YS, et al. Electron-impact collision ionization cross sections and rates for the Na

isoelectronic sequence. At Data Nucl Data Tables 2001;79:65.

[19] Pindzola MS, Griffin DC, Macek JH. Electron-impact ionization of the Fe atom. Phys Rev A 1995;51:2186.

[20] Fang DF, Wang YS. Calculation of slowly converging integrals in electron–ion collision problems. J Phys B

1991;24:1749.

[21] Younger SM. Cross sections an rate for direct electron-impact ionization of sodiumlike ions. Phys Rev A

1981;24:1271.

[22] Shaolin Xi. The optimal calculation method. Shanghai: Shanghai Sci. Technol. Press; 1980.

[23] Teng ZX, Chen CY, Yan SX, Wang YS. Rate coefficients of electron-impact ionization for highly ionized ions.

JQSRT 1999;61:123.

[24] Donets ED, et al. Investigation of ionization of positive ions by electron-impact. Sov Phys JETP 1981;53:466.

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

Documents