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Absolute electron impact ionization cross sections for highly charged ions: Scaling law for hydrogen-like ions
2s
Nuclear Instruments
and Methods in Physics Research B 98 (1995) 135-137
Beam
Interactions with Materials 8 Atoms
ELSEVIER
Absolute electron impact ionization cross sections for highly charged ions: Scaling law for hydrogen-like ions H. Deutsch ‘, T.D. Miirk Institut
fir
lonenphysik,
Leopold Franzens Universitiit, Technikerstr.
25, A-6020 Innsbruck, Austria
Abstract Scaling laws are of great importance in the description of elementary processes in plasma physics and plasma chemistry. This is particularly true for electron impact induced ionization of atoms, molecules and their respective ions. The recently proposed semi-classical DM (Deutsch-MHrk) approach for the calculation of the single ionization cross section of atoms is applied here to the single ionization of hydrogen-like ions and the formula derived can be used to rationalize the empirically-based scaling law for the ionization cross section of hydrogen-like ions, i.e. the Z4 power dependence, where Z is the nuclear charge. The results obtained are compared with previous experimental and theoretical ionization cross section functions.
1. Introduction Scaling laws are of great interest in different fields of physics, for instance for the description of elementary processes in plasma physics and in plasma chemistry. As discussed in Ref. [l] there is a great need for scaling laws in describing electron impact ionization cross sections. Despite this need only a few empirical approaches are known today [1,2]. This paper presents a new scaling law for ionization cross sections for hydrogen-like ions, based on the DM (Deutsch-Msrk) approach reported recently [3]. The DM approach is based on a combination of the classical binary encounter approximation and the quantum mechanical Born-Bethe approximation (for a first formulation, see Ref. [3]). Although this DM formalism was originally devised for the single ionization of ground state atoms, it has been recently applied successfully in the case of excited state atoms, radicals, molecules and clusters. Following up these previous studies we are extending the DM approach in the present investigation for the first time to the ionization of ionized targets.
2. Theoretical
hydrogen-like targets. In this case the theory has to deal only with one target electron; therefore data on hydrogenic ions seem to be ideal for making comparisons between experiment and theory and to test, for instance, scaled impact ionization cross sections. Based on empirical considerations, Donets et al. [2] proposed for hydrogen-like ions that the electron impact ionization cross section scales with Z4, where Z is the nuclear charge of the atoms. As will be shown in what follows, it is possible to derive this scaling law also theoretically using a recently developed semiclassical treatment for electron impact ionization (DM approach 131). Starting from the DM formula for the integrated single ionization cross section of atoms:
(1) Il.1
where rzl is the mean square radius of the &shell (as calculated in Ref. [4]), tnnr the number of electrons in the nl-shell, g,, the weighting factor (for a definition see Refs.
[3,51), I/U-l\"
considerations
One of the most important classes of electron impact ionization processes is that concerning the ionization of
* Corresponding author. ’ Guest Professor at the Institut fiir Ionenphysik Universitlt Innsbruck. Permanent address: Fachbereich Physik, Ernst Moritz Amdt UniversitIt, D-17489 Greifswald, Germany.
(with n = f, b = 1, c = 1, for s-electrons [3,5]), and U = E/E,,, (where E is the energy of incident electrons and E,, is the binding energy of electrons in the nl-shell) one arrives at
0168-583X/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDZ 0168-583X(95)00090-9
(2)
3.1. COLLISIONS
WITH ELECTRONS
H. Deutsch, T.D. Miirk/ Nucl. Ins&. and Meth. in Phys. Res. B 98 (1995) 135-137
136 With g,,E,,
En,/r,:=
(T%/a;
= 50 eV for (Is)’ states [5] one then obtains:
3.23 x lO_‘“f(U),
(3)
where En is the ionization energy of hydrogen and a, is the Bohr-radius. Following Casnati et al. [6] we have developed here E,, and r,, in terms of E,, and a,,, respectively, i.e., E
nl
= 0 424 Z”.‘s”“E
.
0,
(4)
1.15 rHl
=-a
(5)
0’
z’.n5
By using these relations,
Eq. (3) results in reduced
,z4=
=
eleclron
energy U-E/E,,,-
py(U). (6)
Therefore the DM-approach is able to rationalize quantitatively the empirically-based Z” scaling law for the ionization cross sections of hydrogen-like ions [2]. Moreover, following up these considerations, the DMapproach allows also the development of a new and more generalized scaling law for the single ionization cross section of isoelectronic ions. Starting from Eq. (1) one gets:
~Elll = ~g,,E,,Lf(~). Taking into account that the product g,,E,, is constant for isoelectronic targets [3,5,7] (i.e., 50 eV for (Is)’ states) the reduced cross section given on the right-hand side of Eq. (7) is independent of Z for hydrogen-like ions. In order to use this formula for atoms with higher nuclear charges Z (i.e., involving higher ionization energies) it is necessary to take into account relativistic effects [7]. In this case it is necessary to start instead of Eq. (1) with the relativistic form of the DM-approach as given recently [7], i.e.
with
(1 + u)(u+
25)(1
+J)2
3’?
X
2J)
Ne9+ [2]; @ Ar17+ [2].
and m, the electron mass. Thus the scaling law [7] obtains the following form:
(9) with g,,E,,
4
i P(1+
Fig. 1. Scaled electron impact ionization cross section, oE,,, / ri,F(U), for hydrogen-like ions versus reduced electron energy, E/E,,,. Present theoretical results: solid line, previous experimental results: 0 Hef[8]; 0 C5’ [2]; 0 Nh’ [2]; A O’+ [2]; *
+ u(u+
2J)(l
+.Q2
I
’
= 50 eV for (Is)‘-states.
3. Results Using Eq. (9) we present in a scaled manner in Fig. 1 experimental cross sections for hydrogen-like ions from across the whole periodic system. It is interesting to note that there appears to be (within the experimental error bars) rather strong support for the scaling law as expressed by Eq. (9). First of all, the experimental cross section functions of various targets plotted in this reduced form versus E/E,, agree rather well with each other. Secondly, there is also a very satisfactory agreement between the calculated values using Eq. (9) (solid line in Fig. 1) and the experimental data points. All of these experimental points are in addition in very good agreement with theoretical predictions based on the DWE approximation by Younger [9] (e.g., see Fig. 8-6 in Ref. [l] giving a comparison between scaled experimental and theoretical data thereby confirming the Z4 scaling law) and thus confirm the validity of these measurements. Because hydrogen-like isoelectronic ions are the simplest atomic systems and the only ones without multielectron interactions, this ion sequence has been used for many years to study fundamental atomic structure, including experimental investigations on lamb shift, hyperfine splittings and most importantly in the present context on collision cross sections. Considerable attention has been
H. Deutsch, T.D. Miirk/Nucl.
Instr. and Meth. in Phys. Res. B 98 (1995) 135-137
given recently to the measurement of single ionization cross sections for few-electron high-Z ions, in particular the ions of uranium. Despite this increased interest, even the approximate magnitude of the high-Z ionization cross sections for 1s electrons has been uncertain in view of an accelerator stripping experiment [lo] that obtained values for an electron energy of 222 keV much larger than any theory and with error bars larger than a factor of 2. Production of hydrogen-like uranium ions is quite difficult, because many ionizing collisions with small cross sections are required, one of which must have at least a center-of-mass energy of 132 keV (ionization energy of the Is electron of uranium). Recently, Marrs et al. [ll] reported the first production of stationary hydrogen-like uranium ions in a upgraded electron beam ion trap (EBIT). From the observed equilibrium ionization balance in this trap they were able to determine the electron impact ionization cross section for U9’+ at 198 keV electron energy to give a value of (1.55 + 0.27) X lO_” m2. This value is much smaller than the previous experimental value of 3.9 x 1O-28 m2 and shghtly larger than previous theoretical results (DWE calculations by Younger [9] giving 0.8 X 1O-28 m2 [lo], QED calculations by Pindzola et al. [12] giving 1.1 X 10w2* m2, relativistic DW calculations by Zhang et al. [13] giving 0.93 X 1O-28 m2 [ll] and calculations using the Lotz formula [14] giving 0.7 X 10c2a m* [IO]) and the present result of 0.4 X 1O-28 m2. Some of these discrepancies could be due to the fact that at these low electron energies (Us 1.7) the cross section is a strong function of the electron energy and thus a small uncertainty in the electron energy results in a large change in the cross section (i.e., see Fig. 2 in Ref. [12]). Further more accurate data are needed to serve as a reliable frame of reference for a definitive comparison with theoretical predictions. Current calculations for higher isoelectronic sequences (e.g., helium- and lithium-like ions) using an extension of the present approach indicate the validity of an analogous scaling law [IS] for these other few-electron series.
137
Acknowledgements Work partially supported by the osterreichischer Fonds zur Fiirderung der wissenschaftlichen Forschung und Bundesministerium fiir Wissenschaft und Forschung, Wien, Austria.
References 111T.D. Mlrk and G.H. Dunn, eds., Electron Impact Ionization (Springer, Wien, 1985). [2] E.D. Donets and V.P. Gvsyannikov, Sov. Phys. JETP 53 (1981) 466. [3] H. Deutsch and T.D. Mark, Int. J. Mass Spectrom. Ion Proc. 79 (1987) Rl. [4] J.P. Desclaux, At. Nucl. Data Tables 12 (1973) 325. [5] D. Margreiter, Thesis, Universitlt Innsbruck, 1993; H. Deutsch and T.D. Mark, Contrib. Plasma Phys. 34 (1994) 19; D. Margreiter, H. Deutsch and T.D. Mark, Int. J. Mass Spectrom. Ion Proc. 139 (1994) 127. [6] E. Casnati, A. Tartari and C. Baraldi, J. Phys. B 15 (1982) 155; J. Phys. B 16 (1983) 505. [7] H. Deutsch, D. Margreiter and T.D. M&k, Z. Phys. D 29 (1994) 31. [s] K.T. Dolder, M.F.A. Harrison and P.C. Thonemann, Proc. R. Sot. London A 264 (1961) 367. [9] S.M. Younger, Phys. Rev. A 22 (1980) 111. [lo] N. Claytor, B. Feinberg and H. Gould, Phys. Rev. Lett. 61 (1988) 2081. [ll] R.E. Marrs, S.R. Elliott and D.A. Knapp, Phys. Rev. Lett. 72 (1994) 4082. [12] M.S. Pindzola, D.L. Moores and D.C. Griffin, Phys. Rev. A 40 (1989) 4941. [13] H.L. Zhang and D.H. Sampson, Phys. Rev. A 42 (1990) 5378. (141 W. L&z, Z. Phys. 216 (1968) 241. [15] H. Deutsch, K. Becker and T.D. Msrk, Int. J. Mass Spectrom. Ion Proc., in press (1995).
3.1. COLLISIONS WITH ELECTRONS
Nuclear Instruments
and Methods in Physics Research B 98 (1995) 135-137
Beam
Interactions with Materials 8 Atoms
ELSEVIER
Absolute electron impact ionization cross sections for highly charged ions: Scaling law for hydrogen-like ions H. Deutsch ‘, T.D. Miirk Institut
fir
lonenphysik,
Leopold Franzens Universitiit, Technikerstr.
25, A-6020 Innsbruck, Austria
Abstract Scaling laws are of great importance in the description of elementary processes in plasma physics and plasma chemistry. This is particularly true for electron impact induced ionization of atoms, molecules and their respective ions. The recently proposed semi-classical DM (Deutsch-MHrk) approach for the calculation of the single ionization cross section of atoms is applied here to the single ionization of hydrogen-like ions and the formula derived can be used to rationalize the empirically-based scaling law for the ionization cross section of hydrogen-like ions, i.e. the Z4 power dependence, where Z is the nuclear charge. The results obtained are compared with previous experimental and theoretical ionization cross section functions.
1. Introduction Scaling laws are of great interest in different fields of physics, for instance for the description of elementary processes in plasma physics and in plasma chemistry. As discussed in Ref. [l] there is a great need for scaling laws in describing electron impact ionization cross sections. Despite this need only a few empirical approaches are known today [1,2]. This paper presents a new scaling law for ionization cross sections for hydrogen-like ions, based on the DM (Deutsch-Msrk) approach reported recently [3]. The DM approach is based on a combination of the classical binary encounter approximation and the quantum mechanical Born-Bethe approximation (for a first formulation, see Ref. [3]). Although this DM formalism was originally devised for the single ionization of ground state atoms, it has been recently applied successfully in the case of excited state atoms, radicals, molecules and clusters. Following up these previous studies we are extending the DM approach in the present investigation for the first time to the ionization of ionized targets.
2. Theoretical
hydrogen-like targets. In this case the theory has to deal only with one target electron; therefore data on hydrogenic ions seem to be ideal for making comparisons between experiment and theory and to test, for instance, scaled impact ionization cross sections. Based on empirical considerations, Donets et al. [2] proposed for hydrogen-like ions that the electron impact ionization cross section scales with Z4, where Z is the nuclear charge of the atoms. As will be shown in what follows, it is possible to derive this scaling law also theoretically using a recently developed semiclassical treatment for electron impact ionization (DM approach 131). Starting from the DM formula for the integrated single ionization cross section of atoms:
(1) Il.1
where rzl is the mean square radius of the &shell (as calculated in Ref. [4]), tnnr the number of electrons in the nl-shell, g,, the weighting factor (for a definition see Refs.
[3,51), I/U-l\"
considerations
One of the most important classes of electron impact ionization processes is that concerning the ionization of
* Corresponding author. ’ Guest Professor at the Institut fiir Ionenphysik Universitlt Innsbruck. Permanent address: Fachbereich Physik, Ernst Moritz Amdt UniversitIt, D-17489 Greifswald, Germany.
(with n = f, b = 1, c = 1, for s-electrons [3,5]), and U = E/E,,, (where E is the energy of incident electrons and E,, is the binding energy of electrons in the nl-shell) one arrives at
0168-583X/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDZ 0168-583X(95)00090-9
(2)
3.1. COLLISIONS
WITH ELECTRONS
H. Deutsch, T.D. Miirk/ Nucl. Ins&. and Meth. in Phys. Res. B 98 (1995) 135-137
136 With g,,E,,
En,/r,:=
(T%/a;
= 50 eV for (Is)’ states [5] one then obtains:
3.23 x lO_‘“f(U),
(3)
where En is the ionization energy of hydrogen and a, is the Bohr-radius. Following Casnati et al. [6] we have developed here E,, and r,, in terms of E,, and a,,, respectively, i.e., E
nl
= 0 424 Z”.‘s”“E
.
0,
(4)
1.15 rHl
=-a
(5)
0’
z’.n5
By using these relations,
Eq. (3) results in reduced
,z4=
=
eleclron
energy U-E/E,,,-
py(U). (6)
Therefore the DM-approach is able to rationalize quantitatively the empirically-based Z” scaling law for the ionization cross sections of hydrogen-like ions [2]. Moreover, following up these considerations, the DMapproach allows also the development of a new and more generalized scaling law for the single ionization cross section of isoelectronic ions. Starting from Eq. (1) one gets:
~Elll = ~g,,E,,Lf(~). Taking into account that the product g,,E,, is constant for isoelectronic targets [3,5,7] (i.e., 50 eV for (Is)’ states) the reduced cross section given on the right-hand side of Eq. (7) is independent of Z for hydrogen-like ions. In order to use this formula for atoms with higher nuclear charges Z (i.e., involving higher ionization energies) it is necessary to take into account relativistic effects [7]. In this case it is necessary to start instead of Eq. (1) with the relativistic form of the DM-approach as given recently [7], i.e.
with
(1 + u)(u+
25)(1
+J)2
3’?
X
2J)
Ne9+ [2]; @ Ar17+ [2].
and m, the electron mass. Thus the scaling law [7] obtains the following form:
(9) with g,,E,,
4
i P(1+
Fig. 1. Scaled electron impact ionization cross section, oE,,, / ri,F(U), for hydrogen-like ions versus reduced electron energy, E/E,,,. Present theoretical results: solid line, previous experimental results: 0 Hef[8]; 0 C5’ [2]; 0 Nh’ [2]; A O’+ [2]; *
+ u(u+
2J)(l
+.Q2
I
’
= 50 eV for (Is)‘-states.
3. Results Using Eq. (9) we present in a scaled manner in Fig. 1 experimental cross sections for hydrogen-like ions from across the whole periodic system. It is interesting to note that there appears to be (within the experimental error bars) rather strong support for the scaling law as expressed by Eq. (9). First of all, the experimental cross section functions of various targets plotted in this reduced form versus E/E,, agree rather well with each other. Secondly, there is also a very satisfactory agreement between the calculated values using Eq. (9) (solid line in Fig. 1) and the experimental data points. All of these experimental points are in addition in very good agreement with theoretical predictions based on the DWE approximation by Younger [9] (e.g., see Fig. 8-6 in Ref. [l] giving a comparison between scaled experimental and theoretical data thereby confirming the Z4 scaling law) and thus confirm the validity of these measurements. Because hydrogen-like isoelectronic ions are the simplest atomic systems and the only ones without multielectron interactions, this ion sequence has been used for many years to study fundamental atomic structure, including experimental investigations on lamb shift, hyperfine splittings and most importantly in the present context on collision cross sections. Considerable attention has been
H. Deutsch, T.D. Miirk/Nucl.
Instr. and Meth. in Phys. Res. B 98 (1995) 135-137
given recently to the measurement of single ionization cross sections for few-electron high-Z ions, in particular the ions of uranium. Despite this increased interest, even the approximate magnitude of the high-Z ionization cross sections for 1s electrons has been uncertain in view of an accelerator stripping experiment [lo] that obtained values for an electron energy of 222 keV much larger than any theory and with error bars larger than a factor of 2. Production of hydrogen-like uranium ions is quite difficult, because many ionizing collisions with small cross sections are required, one of which must have at least a center-of-mass energy of 132 keV (ionization energy of the Is electron of uranium). Recently, Marrs et al. [ll] reported the first production of stationary hydrogen-like uranium ions in a upgraded electron beam ion trap (EBIT). From the observed equilibrium ionization balance in this trap they were able to determine the electron impact ionization cross section for U9’+ at 198 keV electron energy to give a value of (1.55 + 0.27) X lO_” m2. This value is much smaller than the previous experimental value of 3.9 x 1O-28 m2 and shghtly larger than previous theoretical results (DWE calculations by Younger [9] giving 0.8 X 1O-28 m2 [lo], QED calculations by Pindzola et al. [12] giving 1.1 X 10w2* m2, relativistic DW calculations by Zhang et al. [13] giving 0.93 X 1O-28 m2 [ll] and calculations using the Lotz formula [14] giving 0.7 X 10c2a m* [IO]) and the present result of 0.4 X 1O-28 m2. Some of these discrepancies could be due to the fact that at these low electron energies (Us 1.7) the cross section is a strong function of the electron energy and thus a small uncertainty in the electron energy results in a large change in the cross section (i.e., see Fig. 2 in Ref. [12]). Further more accurate data are needed to serve as a reliable frame of reference for a definitive comparison with theoretical predictions. Current calculations for higher isoelectronic sequences (e.g., helium- and lithium-like ions) using an extension of the present approach indicate the validity of an analogous scaling law [IS] for these other few-electron series.
137
Acknowledgements Work partially supported by the osterreichischer Fonds zur Fiirderung der wissenschaftlichen Forschung und Bundesministerium fiir Wissenschaft und Forschung, Wien, Austria.
References 111T.D. Mlrk and G.H. Dunn, eds., Electron Impact Ionization (Springer, Wien, 1985). [2] E.D. Donets and V.P. Gvsyannikov, Sov. Phys. JETP 53 (1981) 466. [3] H. Deutsch and T.D. Mark, Int. J. Mass Spectrom. Ion Proc. 79 (1987) Rl. [4] J.P. Desclaux, At. Nucl. Data Tables 12 (1973) 325. [5] D. Margreiter, Thesis, Universitlt Innsbruck, 1993; H. Deutsch and T.D. Mark, Contrib. Plasma Phys. 34 (1994) 19; D. Margreiter, H. Deutsch and T.D. Mark, Int. J. Mass Spectrom. Ion Proc. 139 (1994) 127. [6] E. Casnati, A. Tartari and C. Baraldi, J. Phys. B 15 (1982) 155; J. Phys. B 16 (1983) 505. [7] H. Deutsch, D. Margreiter and T.D. M&k, Z. Phys. D 29 (1994) 31. [s] K.T. Dolder, M.F.A. Harrison and P.C. Thonemann, Proc. R. Sot. London A 264 (1961) 367. [9] S.M. Younger, Phys. Rev. A 22 (1980) 111. [lo] N. Claytor, B. Feinberg and H. Gould, Phys. Rev. Lett. 61 (1988) 2081. [ll] R.E. Marrs, S.R. Elliott and D.A. Knapp, Phys. Rev. Lett. 72 (1994) 4082. [12] M.S. Pindzola, D.L. Moores and D.C. Griffin, Phys. Rev. A 40 (1989) 4941. [13] H.L. Zhang and D.H. Sampson, Phys. Rev. A 42 (1990) 5378. (141 W. L&z, Z. Phys. 216 (1968) 241. [15] H. Deutsch, K. Becker and T.D. Msrk, Int. J. Mass Spectrom. Ion Proc., in press (1995).
3.1. COLLISIONS WITH ELECTRONS