Electron ionization rate coefficient for highly ionized iron and scandium

J. Quant. Spectros¢. Radiat. Trans[er Vol. 29, No. I, pp. 61-66, 1983

0022--40731531010061.-06503.0010

Printed in Great Britain.

Pergamon Press Ltd.

ELECTRON IONIZATION RATE COEFFICIENTS HIGHLY IONIZED IRON AND SCANDIUM

FOR

S. M. YOUNGER Atomic and Plasma Radiation Division, Center for Radiation Research, National Bureau of Standards, Washington, DC 20234, U.S.A.

(Received 5 February 1982) Abstract---Cross sections and rate coefficients for the electron ionization of Fe(IX)-(XV) and Sc(IV)-(X) have been computed in a distorted wave Born exchange approximation. The scaled cross section for ejection of a 3p electron was found to be roughly linear in the number of 3p electrons in the ion. Analytic fits to the distorted wave cross sections and rate coefficients are included. 1. I N T R O D U C T I O N

Cross sections and rate coefficients describing the electron impact ionization of the positive ions of iron and scandium are required for modeling the structure and dynamics of high temperature, laboratory and astrophysical, plasmas. In tokamak plasma machines, for example, iron enters the plasma via scrape-off from stainless steel vacuum vessel walls, limiters, etc., while scandium is deliberately injected into the plasma as a diagnostic probe. RecentlyY we presented results of distorted wave calculations of the electron ionization cross sections and rate coefficients of Fe(XVI)--(XXVI). Here, we present results for Fe(IX)-(XV), as well as for the isoelectronic scandium ions Sc(IV)-(XI). Only direct ionization from the 3p and 3s subshells will be considered. Estimates of inner shell cross sections, which are much smaller than those for ejection of a 3p or 3s subshell electron, are given, as are estimates for the indirect ionization mechanism when ionization proceeds by way of autoionizing resonances in the ejected electron continuum. 2. DISTORTED WAVE CROSS S E C T I O N S

The distorted wave Born exchange approximation has been described in detail elsewhere, 3-4 Briefly, a triple partial wave expansion describes the incident, scattered, and ejected electrons, with a Hartree-Fock ground state wavefunction5 representing the target. Scattering exchange is handled in the maximum interference approximation. Experimental ionization energies6'7 were employed. The 3s ionization energy of the configuration 3s23p q was computed from the configuration-averaged energy of the 3s3pq configuration. Figure 1 presents the scaled electron ionization cross section, uI2Q, for ejection of a 3p electron from Fe(IX)-(XIV)vs the number of 3p electrons in the target; u is the reduced incident energy (u = El1) where E is the incident electron energy and I is the ionization energy. Each curve in Fig. 1 corresponds to a fixed reduced incident energy. It is significant that the quantity uI2Q varies nearly linearly with the number of 3p electrons in the target, although the cross section itself, Q, does not scale in such manner. Figure 2 is a similar plot for scandium ions. Figures 3 and 4 present the scaled electron ionization cross sections, ul2Q, for ejection of a 3s electron from (Fe(IX)-(XV) and Sc(IV)-(X) respectively vs the number of 3p electrons in the target. If there were no changes in the atomic structure of the ion as 3p electrons are removed, then the curves in Figs. 3 and 4 would be horizontal lines. Deviation of the scaled 3s cross section from a straight line reflects the effect of the emptying of the 3p subshell on the 3s orbital. As one would expect, the deviation from straight lines in Figs. 1-4 is more pronounced for scandium ions than for iron ions since the former are less dominated by the nuclear potential than the latter. In the limit of infinite nuclear charge, where the interelectronic interaction vanishes compared to the nuclear potential, the 3p cross section would scale linearly with the number of 3p electrons and the 3s cross section would be constant vs the number of 3p electrons. In order to facilitate the calculation of electron ionization rate coefficients from the cross 61

62

S.M. YOUNOER I

40.0

,

l

I

I

] s.o

3.5

Iron IX-XV Scaled Ionization Cross Section vs. Number of ~p Electrons 3s23pn~3s23pn-I + e"

30J

-

g

@ o

cb

2.25 20.{

1.5 10.0

u = 1.25

0.0

0

1

2

3

4

5

6

nO

Fig. 1. Scaled electron impact ionization cross sections, ~13pI3pQ3p, 2 for ejection of a 3p electron from M-shell iron ions vs n3p,the number of 3p electrons in the ion: u3p is the incident electron energy measured in units of I3p, the 3p ionization energy. Each curve corresponds to a fixed reduced incident energy. Table 1, Parameters describing the electron impact ionization cross sections and rate coefficients of Fe(IX)-(XV) (in 10-14 cm3-eV2). Ion Fe IX

FeX

Fe XI

Fe XII

Fe X I I I

Fe XIV

Fe XV

Subshell

ZCeV)

A

B

3p

233.6

71.0

-23.9

9. 47

-51.9

3s

275.9

19.1

-5.55

2.32

-12.7

3p

262.1

57.0

-18.6

7.64

-39.7

3s

297.4

20.8

-7.40

2.28

-14.1

3p

290.3

44.7

-14.4

5.88

-30.1

3s

325.2

22.3

-8.65

2.22

-15.2

3p

330.8

32.7

-9.73

4.40

-20.9

3s

368.4

24.7

-10.2

2.27

-17.0

3p

361.0

20.5

-5.65

2.82

-12.3

3s

403.0

26.2

-I 1.3

2.25

-18.2

3p

392.2

9.77

-2.58

1.35

-5.57

3s

426.0

27.5

-12.0

1.91

-19.3

3s

457.0

38.8

-16.7

1.87

-28.8

C

D

Electron ionization rate coefficients for highly ionized iron and scandium 40.0

I

I

I

I

I

63

1

Scandium IV-X Scaled Ionization Cross Section vs. Number of 3p Electrons 3s23pn -3s23p n-1 + e-

30.0

o o

~

20.0

2.25

10.0 1.5

1.25

0.0 t g ~ ' " 0

1

I

I

I

I

I

1

2

3

4

5

6

n3p

Fig. 2. Scaled electron impact ionization cross sections, u3pI3pQ3p,2 for ejection of a 3p electron from M-shell scandium ions vs n3p, the number of 3p electrons in the ion: u3p is the incident electron energy measured in units of I3p, the 3p ionization energy. Each curve corresponds to a fixed reduced incident energy. Table 2. Parameters describing the electron impact ionization cross sections and rate coefficients of Sc(IV)-(IX) (in lO -t4 cm2-eV2). Ion Sc IV

Subshell 3p 3s

Sc V

Sc VI

l(eV)

A

73.49

74.3

-24,2

6.98

-63.0

17.6

-3.82

1.95

-13.8

I01 .l

B

C

D

3p

91.9

55.8

-15,8

6.41

-44.5

3s

114.0

16.2

-3.18

1.77

-11.6

3p

llO.7

47.1

-14,5

4.75

-35.5

3s

133.1

18.9

-5.13

1.62

-13.2

Sc VII

3p

138.0

40.9

-13.6

3.44

-30.1

3s

164.2

20.4

-6.33

2.13

-13.8

Sc V l l l

3p

158.1

22.9

-7.37

2.84

-15.9

3s

181.7

21.9

-7.70

1.90

-14.9

Sc IX

Sc X

30

180.0

II .I

-3.35

1.26

-7.29

3s

202.3

22.7

-8.65

1.89

-15.5

3s

225.1

24.2

-9.69

1.75

-16.7

S. M. YOUNGER

64 I

1

t

i

I

I

Iron IX-XV Scaled Ionization Cross Section vs. Number of 3p Electrons 3s23pn- 3s3p n + e12.0 ~f

~

~.--.~

5.0

~...-.---

3.5

10.0 '

~

o

o

8.0

6

E 2.25

6.0

4.0 1.5

2.0

0.0

u=1.25

I

1

1

I

I

i

1

2

3

4

5

6

n3p

--

Fig. 3. Scaled electron impact ionization cross sections, u3sI]sQ3~for ejection of a 3s electron from M-shell iron ions vs n~p, the number of 3p electrons in the ion: u3~ is the incident electron energy measured in units of I3,, the 3s ionization energy. Each curve corresponds to a fixed reduced incident energy.

sections, we have fit the distorted wave data by the analytic function

uI:Q=A(I-

+B 1-

+Clnu+Dlnu'u

(1)

where A, B, and D are free parameters chosen by a least squares tit to the data. C is a Bethe coefficients computed from the extensive photoionization cross section tabulation of Manson. 9 Details of the fitting procedures are given in Ref. 2. The results of the fits are given in Table 1 for the iron ions and Table 2 for the scandium ions. For a Maxwellian distribution of incident electron velocities in a plasma, the distorted wave exchange electron ionization rate coefficient, SDWE, a function of the plasma electron temperature kT, may be written in the form of a correction to the commonly used Seaton formula, 8 viz. SDW E =

F(x)SSeat°n(x),

(2)

where the Seaton approximation is t° S se~t°" = 2.2 × 10-6 1-312"k/X e

1/xcm3 eV3/2; s

(3)

X is the reduced electron temperature x = kTlI

(4)

Electron ionization rate coettcients for highly ionized iron and scandium I

I

I

I

I

65

I

Scandium IV-X Scaled Ionization Cross Section vs. Number of 3p Electrons

3s23pn- 3s3pn+ e12.0

~

10.0

5

.

0

o

o

6

3.5

0.0

e,J 6.0

2.25 4.0

1.5 2.0 u=1.25

o.0 0

I

I

I

I

I

I

1

2

3

4 nap

5

6

Fig. 4. Scaled electron impact ionization cross sections, u3sI],Q3s for ejection of a 3s electron from M-shell scandium ions vs n3p, the number of 3p electrons in the ion: u3~ is the incident electron energy measured in units of I3s, the 3s ionization energy. Each curve corresponds to a fixed reduced incident energy.

and

F(X)is a distorted wave factor approximated by

10'3tt.[[A+ B (1 + 1 ) ] + [C-I(A+x\2B + B'~]X/Ja(x)+DB(X)}

F(X) = 3.0x.X

(5)

with 0.001193 + 0.9764X + 0.6604X2+ 0.02590X3 1.0 + 1.488X + 0.2972X2+ 0.004925X3

(6)

-0.0005725 + 0.01345X + 0.8691X2+ 0.03404X3 1.0 + 2.197x + 0.2454X2 + 0.002053X3

(7)

a (X) = and

B(X) =

3. DISCUSSION In the present work, we have considered only direct ionization of M-shell electrons in iron and scandium ions by electron impact, i.e., the process whereby a 3s or 3p electron is promoted directly into its continuum by means of a single electron collision. For ions containing only a few M-shell electrons, inner shell ionization may make a significant contribution to the total electron ionization cross section. Direct ionization of an inner shell electron may be approximated fairly accurately by means of the simple Lotz formula 1'

uI2Q= (4.5 x 10-~4) n In u QSRTVoL29,No. l--E

(cm2 eV2),

(8)

66

S.N. YOUNGER

where n is the number of electrons in the subshell under consideration. An important indirect mechanism for electron ionization is the electron impact excitation of the target into a core-excited autoionizing resonance which is effectively a bound-type configuration above the ionization limit. Such a state may then either decay radiatively to a bound state or autoionize. An example for sodium-like ions is the core-excited configuration ls~2s*'2p53s 2 which lies above the ionization limit of the initial ground state for all values of the nuclear charge. Cowan and Mann ~2have demonstrated that, for sodium-like iron, the cross section for indirect ionization is comparable to direct ionization. Since the M-shell ionization cross section is roughly proportional to the number of electrons in the shell, a constant inner shell contribution will constitute a decreasing fraction of the total cross section as the occupation of the M-shell increases. The inner shell indirect ionization cross section for non-sodium-like ions should be smaller than for the sodium-like cases, since the important 3s subshell is closed in such cases. For argon-like ions, where the 3p subshell is complete, inner shell effects are expected to account for only a few percent of the total electron ionization cross section. There are presently no experimental measurements available with which the present results may be compared. For singly ionized ions isoelectronic with the M-shell cases considered here, agreement with experiment is generally within 20-100%. Since complications due to target electron correlation and other complex electron-atom interactions are expected to decrease in importance with increasing nuclear charge, we anticipate a similar level of accuracy for the present results. REFERENCES 1. S. M. Younger, JQSRT 27, 541 (1982). 2. S. M. Younger, Phys. Rev. A 24, 1272(1981). 3. S. M. Younger, Phys. Rer. A 22, Ill (1980). 4. S. M. Younger, Phys. Rev. A 23, 1138(1981). 5. E. Clementi and C. Roetti, At. Data Nucl. Data Tables 14, 177 (1974). 6. J. Sugar and C. Corliss, J. Phys. Chem. Ref. Data 9, 473 (1980). 7. J. Reader and J. Sugar, J. Phys. Chem. Ref,. Data 4, 353 0975). 8. S. M. Younger, JQSRT 26, 329 (1981). 9. S. T. Manson, private communication. 10. M. J. Seaton, Planet. Space Sci. 12, 55 (1964). ll. W. Lotz, Z. Phys. 216, 241 (1968). 12. R. D. Cowan and J. B. Mann, Astrophys. J. 232, 940 (1979).