ions by electron impact are suggested for ejection three or more electrons n 2 3. ... The problem of the multiple ionization (MI) arising in electron-atom and ...
J. Phys. B: At. Mot. Opt. F'hys. 28 (1995) L589-L5d4. Plinted in the UK
LETTER TO THE EDITOR
Semiempirical formulae for multiple ionization of neutral atoms and positive ions by electron impact V P Shevelkot and H Tawara t P N Lebedev physics Institute, Russian Academy of Sciences,
117924 Moscow, Russia
$ National Institute for Fusion Science. Nagoya 464-01, Japan Received 26 lune 1995
Abstract. Semiempirical formulae for multiple-ionization (MI) cross seaions 0 . of atoms and ions by electron impact are suggested for ejection three or more electrons n 2 3. The formulae have been deduced on the basis of available experimental data on c,,and the assumption of the Born-Bethe dependence of cn on the incident electron energy E. The formulae suggested can be used for estimation of MI cross sections an, n 3, from threshold up lo high electmn energies E w IO5 eV for an arbitmy atomic or ionic target.
>
The problem of the multiple ionization (MI) arising in electron-atom and ion-atom collisions is of great interest both for our understanding of the many-electron processes (multielectron transitions, electron correlation effects)and for different physical applications such as plasma kinetics problem, charge-state evaluation of atoms exposed to an electron beam. contribution of Auger and shake-off processes and others (see McGuire 1992). In the case of MI of atoms and positive ions by electron impact the available experimental data on the cross sections U" are often not consistent and complete and large discrepancies exist among MI cross sections, in particular for the large numbers of the ejected electrons n (see a compilation by Tawara and Kat0 (1987)). The quantum mechanical calculations of MI cross sections for n > 2 are still unknown, therefore analytical semiempirical formulae constitute a special interest. Deutsch et al (1995) apply the semiempirical formalism to predict double- and triple ionization cross sections in the vicinity of ionization threshold of some specific atomic targets. Very recently, a scaling of the MI cross sections and semiempirical formulae for un have been considered by Fisher et al (1995). Our aim in this work is to investigate the MI process in electron-atom and electron-ion collisions e
+ A'+
+ e + A'z+"'+
+ ne
nz 1
(1) and to obtain a semiempirical formula for MI cross section un which could describe its average behaviour over a wide range of incident electron energies. The measured threshold energy E& for MI cross sections U,, corresponds to the minimal ionization energy I,, required to remove n outmost electrons, i.e.
0953-4075/95/180589+06$19.50
@ 1995 IOP Publishing Ltd
L589
L590
Letter to the Editor
+
where /,,i+, is the one-electron ionization energy from the charge i to i 1 (see, e.g., Wetzel et a1 (1987) and Tinschert et al (1987)). The minimal ionization potentials In can be obtained from the tables in Lotz (1968, 1970) and Carlson et nl (1970). For example, the minimal energy 16 required for the ionization of six electrons in the Kr atom I(%+) /(K?+) I(@+) /(Kr4+) I ( @ + ) = is estimated to be: /6 = I ( % ) 14.0+27.89+41.78+55.67+70.31+84.52=294.17 eV. Each target atom or ion is characterized by its own set of minimal ionization energies I.. so it is natural to choose /" as a scaling parameter for the incident electron energy E :
+
+
+
+
+
similar to a single ionization. Our analysis of the experimental data available on MI cross sections us for atoms and ions by electron impact has shown that the majority of the cross sections has a similar shape for all targets and all cases with n > 3 and the electron-impact energy dependence is described by the universal Born-Bethe type formula (Shevelko and Tawara 1995a):
where 1 Ryd = 13.6 eV, U is defined in equation (3). The constant c = 1 for neutral targets ( z = 0) and c = 0.75 for positive ions (z 1). Unfortunately, the energy dependence of doubleionization cross sections.(n = 2) cannot be described properly by equation (4) and will be considered separately. The coefficient C(n,N ) in equation (4) depends only on two parameters: the number of the ejected electrons n and the total number of the target electrons N . The dnalysis of the available experimental data with the fixed n number and different N has shown that the constant C ( n , N ) can be written in the form:
>
~ ( nN ,) = a(n)Nb(") (5) where a and b are the approximation parameters. They were obtained by fitting equations (4)-(5) to the experimental data at low as well as at high electron energies. As the references for experimental data for electron-atom and electron-ion collisions we used the results from Lebius et nl (1989), Freund et ol (1990). McCallion et d (1992a. b), Shah et al (1993), Bolorizadeh era! (1994) and for electron-ion collisions those of Muller and Frodl (1980), Miiller et al (1983, 1984, 1985), Pindzola et al (1984), Howald et al (1986). Tinschert et a! (1987) and Stenke et nl (1994). Most of these cross sections were measured using the crossed beam technique. Table 1. Fitting parameters o(n) and b(n) in equation (6)for removal of 3 < n from atom and ions
40
b(4
5
6.30 0.50 0.140
1.20 1.73 1.85
6 I
0.049 0.021
8 9
0.0096
2.00 2.00
0.0049
2.00
10
0.0027
2.00
n
3 4
1.96
~. .
< 10 elecvons
Letter to the Editor
L591
.. X
U
0
A
103
2
3
4 5 6
Schram. 1966 Lcbius et d.,1989 Syagc.1992 prcrcnt
104
I
2
3
4
5 6
105
Electron energy, eV Figure 1. Typical MI MSS sections of neutral atoms by electron impact. (a) Ionization of atom, n = 6 4 , Experiment: x. Schram (1966). n = 6-9.0, Lebiw etnl (1989), n = 6.7 and 8; A, Syage (1992), n = 6. ( b ) Ioniration of Xe atoms. Experiment: S c h m (1966). n = 11 a d 13. Theory: full curve equation (6).
Finally, the expression for the MI cross section o;, for electron-atom and electron-ion collisions can be written in the form:
where the constant c is defined in equation (4). The analysis has shown that it is possible to describe MI cross sections of atoms and ions by the same set of fitting parameters a(n) and b(n). For ejection of 3 n 10 electrons the parameters are listed in table 1; for
<
e
+
9+
Xe
+
3e
4 -
n
> 10 one can use the asymptotic values: b(n) =constant = 2.00
(7) We note that the parameters a(n) and b ( n ) given in table 1 are the smooth functions of n. One can see that the ~1 cross section on falls off approximately as a(n) % 1350n-'.'
n > 10.
-
on n-6. According to equation (6) the cross section E:= % 4.21,,, i.e. onm
%
(8) U,
reaches its maximum at
0.27a(n)Nb(")(l,/Ryd)-2[10-'8cm2].
U,
%
3.2,
(9)
Leffer to the Editor
E93
Equation (9) gives quite a good estimate for the cross section maximum unm and the corresponding electron energy Enm. For example, for the experimental triple-ionization cross section of Bi atoms we have from the data by Freund etal (1990): U;""
= 2.7 x
cm2
E,-
= 170 eV
meanwhile equation (9) gives, respectively ( I 3 = 51.3 eV): u3-
= 0.27 x 6.3 x (83)'.20(51.3/13.6)-210-'8 cmz = 2.40 x
cmz
= 4.2 x 51.3 = 215 eV. A comparison of the experimental MI data with the semiempirical formula (6) is given in figures 1 and 2 where typical examples of the cross sections are shown for atoms (Kr, n = &9; Xe, n = 11 and 13) and ions @b+, n = 3 and 4; Xe", n = 3), respectively. It is seen that the agreement between experimental data and OUI predictions is quite good. In general, a comparison of the cross section described by the semiempirical formula (6) with the experimental data available for neutral atomic targets from Ne up to U and ejection up to 13 electrons and with those for positive ions from AI+ up to W4+ with ejection up to four electrons has shown that in most cases considered the accuracy of the present formula is within a factor of 2 or even better. Of course, the present empirical formula (6) based on the Born-Bethe approximation cannot describe properly the energy dependence near the cross section maximum where the indirect processes are known to play a significant role. However, the present formula is very simple, depends on three main atomic parameters (the minimal ionization potential, the number of the target electrons, the number of the ejected electrons) and can be used for estimating the MI cross section for an arbitrary atomic or ionic target in a wide range of incident elecWon energies. and
The research described in this publication was made possible in part by Grant MKlOOO from the International Science Foundation (ISF). One of us W S ) was supported by the International Atomic Energy Agency (Vienna), contract 8552/RB. References Bolorizadeh M A, Panon C J, Shah M B and Gilbody H B 1994 3. Phys. B: At. Mol. Opt, Phys. 27 175 Cadson T A, Nestor C W Jr, Wassernwn N and McDowell I D 1970 Atomic Data Tabks 2 63 Deutsch H. Becker K and Miirk T D 1995 Calculoted Cross Sections for Double ond Triple Ionization o j A t o m ondlonr by Electron Impact, P l m m Physics (in print) Fisher V, Ralchenko Yu, Goldgirsh A, Fisher D and Mamn Y 1995 J. Phys. B: At. Mol. Opl. Phys. 28 3027 Freund R S, Wetzel R C, Shull R J and Hayes T R I990 Phys. Rev. A 41 3575 Hawald A M,Gregory D C Phaneuf R A. Cnndall D Hand Pindzola M S 1986 Phys. Rev. Letf. 56 1675 Hughes D W and Feeney R K 1981 Phys. Rev. A 23 2241 Lebius H, Binder J, Koslowski H R. Wiesemmn K and Huber B A 1989 J, Phys. B: At. Mol. Opt. Phys. 22 83 Lotz W 1968 3, Opt. Soc. Am 58 915 -1970 3. Opt. Soc. Am 60 206 McCallion P, Shah M B and Gilbody H B 1992a J, Phys. B: At. Mol. Opt. Phys. 25 1051 McGuire I H 1992 Ad". At. Mal. Opt. Phys. 29 217 -1992b 3. Phys. B: At. Mol. Phys. 25 1061 Miiller A, Achenbach C, Sdzbom E and Becker R 1984 J. Phys. B: At. Mol. Phys. 17 1427 Miiller A and Fmdl R 1980 Phys. Rev. Lett. 44 29 Miiller A, Groh W. Kneisel Y. Heil R, Stroher H and Salzbom E 1983 3. Phys B: At. Mol. Phys. 16 2039 Miiller A, 'Enschert K. Achenbach C, Becker R and Salzbom E 1985 3. Phys. B: At. Mol. Phys. 18 3011 Pindzola M S.Giftin D C, Bottcher C, Crandall D If, Phaneuf R A and Gregory D C 1984 Phys. Rev. A 29 1749 Schmm B L 1966 Physica 32 197 Shah M E, Mccallion P, Okuno K and Gilbody H B 1993 J, Phys. B: At, Mol. Opt. Phys. 26 2393
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Letter to the Editor
Shevelko V P and Tawara H 1995a Semiempirical formuhe for multipbioniution cmss sections of atom and ions by electron impact. Absrracts of the 11th Coloquium on W o n d X - r a y Spectroscopy of Asimphydml md Laboratmy P l m m , NaEnya Universiw, Iapm, May 29-Ime 2, 1995 p 109 -1995b Semiempirical formula for multiple-ionization cmss sections of atoms by electron impact Phys. Scr. (accepted) Syage J A 1992 Phys. Rev. A 46 5666 Tawara H and Kat0 T I981 At. Data and NucL Dam Tables 36 167 linschert K, Milller A, Becker R.and S&om E 1981 I. Phys. E: At. Mal. Phys. 20 1823 WetA R C, Baiocchi P A, Hayes T R and Freuod R S 1987 Phys. Rev. A 35 559
