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Quantum mechanical description of recoil ion production in C6+ +Ne collisions
Volume 145, number 4
PHYSICS LETTERS A
9 April 1990
QUANTUM MECHANICAL DESCRIPTION OF RECOIL ION PRODUCTION IN C6~+Ne COLLISIONS Hi. LUDDE a M. HORBATSCH b A. HENNE a and R.M. DREIZLER a b
a
Institutfür Theoretische Physik der Universität, Frankfurt, FRG Physics Department, York University, Toronto, Ontario, Canada M3J 1P3
Received 11 January 1990; revised manuscript received 31 January 1990; accepted for publication 7 February 1990 Communicated by B. Fricke
Total cross sections for recoil ion production in C6 + +Ne collisions are presented. The results are based on the solution of the time dependent effective single particle Schrodinger equation in terms of a target centred atomic basis. Jonisation channels are included globally via the Feshbach formalism adapted to time dependent systems.
1. Introduction
H(t)=H 0(t)+V(t)
The scattering of highly charged ions on rare gas atoms leads to very complex situations as a considerable number of target electrons are actively involved in the collision process. Experimental data have been published during recent years investigating charge transfer, ionisation and in particular the production of highly charged recoil ions [1—3]. From the theoretical side many electron capture and ionisation in collisions ofbare ions with rare gas atoms have been described so far only within a semiclassical independent particle model (1PM) [4—6] and a classical n-particle model [7] with neglect of the interelectronic potential. To our knowledge no nonperturbative quantum calculations have been reported in the literature, In this contribution we collisions present results for from multiple 6~+Ne obtained the electron loss in C solution of the effective time dependent single partide Schrodinger equation.
(1)
,
where we decompose the Hamiltonian into an effective single particle operator N ~
I-~0=
l~(i) I
_________
~ ( t)
=
— ~
s~—R(t)~— r~+ u1 ( t)
A~—
and the residual electron—electron interaction N 1 N V= ~, ,~ u~(t) ~,
—
.
We start from the many electron time dependent Schrodinger equation (/l=me=e= 1) i8,~1-’(r 1 rN,t)=H(t)W(rl rN,t),
(3)
Now we assume that (i) electron correlations can be neglected and (ii) the single particle potential is described in termsofthe of aNe frozen (HF) potential atomatomic and theHartree—Fock Coulomb potential of the incoming projectile, +
2. Theory
(2)
HF VNe.
(4)
This approach defines an independent particle model, where the many electron wave function is a time dependent single Slater determinant
0375-9601 /90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)
173
Volume 145, number 4
rN,
PHYSICS LETTERS A
9 April 1990
(5)
semiclassical independent particle model calculations. These results are included only for electron loss. In ref. [6] the individual capture and ionisation probabilities are presented giving the information in
evolving from the specified initial condition, that is the HF approximation of the Ne ground state. In order to solve the single particle Schrodinger equation
which impact parameter range electron loss is produced mainly by capture or ionisation. With increasing energy or impact parameter the loss probability is dominated by ionisation. The individual loss probabilities reflect the shell structure
det{~(r1,
=
~
I)
t)
~N(rN, t)},
(6)
t)=l~~(t)~(r~, I) ,
we use the time dependent projector formalism [81 defining the finite model space P in terms of the K and L shell HF orbitals of the Ne atom. Following the arguments given in ref. [8] the projected P space coupled channel equations take the form à= ~(1 —G
-
Gk~=
ifQ~/Ir—R ( t)
—
‘
—if~)a,
‘
(7)
where the vector a denotes the excitation amplitudes 2Po, 2Pi) and the diof the four Ne states (ls, 2s, agonal matrix ~ represents the corresponding HF energies respectively. f is a scaling factor which improves the impact parameter dependence ofthe local approximation to the optical potential,
f=
B+ 1 —~----.
(8)
The P space coupled channel equations are solved for each of the initially occupied single particle states with a standard predictor corrector algorithm assuming straight line trajectories for the nuclear motion. 3. Results Fig. 1 shows the single particle loss probabilities for different impact energies. As the P space includes the important target excitation channels, electron loss can be interireted as the sum of the capture and ionisation probabilities. The curves represent the mdividual loss probabilities for the different initial states involved and the weighted average probability in order to allow comparison with results of the statistical 174
of the independent particle model: the more loosely bound 2p electrons are ionised with higher probability than the stronger bound 2s and is electrons, respectively. The situation, however, is not that obvious for collisions with small impact parameters, where capture is the dominant process. In this case the loss probability strongly depends on the energy difference of the initial target orbital and the final projectile state. The smaller the energy difference the larger the capture probability, which is in fact the condiction for resonance charge exchange. Thus the HF energy of the 2s level (ENe(2s)5 + (E~ 1.93043) corresponds to the M shellofenergy of C 5+(n~.3) while the levels Ne (ENe( 4) = lie2), between the 2p N 2p)=O.8504 and 0 shell energies of C5 + (E~ 5+(,, 4) = 1.125, Ec5+(,~s)= 0.72). Consequently capture of the 2s electrons becomes more likely than capture of the 2p electrons, although the latter are less strongly bound. The shell dependent results indicate that electron removal from all shells is possible for impact parameters larger than the corresponding shell radii. The ls-removal probability deviates most due to the significantly higher binding energy and smaller radius of the K shell. One can deduce from this observation that the previously employed statistical model [5,6] can describe properly at most the removal of 8 electrons beyond which it must overestimate the cross section. The difference between the statistical model calculations and the average optical potential results at lower energies, however, reflects the influence of a time dependent screening potential on the single partide loss probabilities. With increasing impact energy dynamic screening becomes less important. Within the framework of the independent particle model many electron loss probabilities can be obtamed from the single particle probabilities following the statistical approach,
Volume 145, number 4
PHYSICS LETTERS A
9 April 1990
I
I
I
I
I
Y0O~:65s789
_
9 (au.)
B (au.) I
I
I
I
CC
C
&
0.6
I
d
C.5~
i~
00O~3456789
B )a.u.)
B (au.)
Fig. 1. Single electron loss probability for (a) E= 100 keV/amu, (b) E= 250 keV/amu, (c) E= 400 keV/amu and (d) E= 1 MeV/amu. 2po; (——) 2p,. (0) CTMC, ref. [6]; (—) ls, weighted average; (— — —) 2s; (‘)
(N
a_s
(9)
07 0.6
Here Pq denotes the probability for the q-fold ionisation of Ne calculated from the average single electron loss probability p. Taking into account the individual shell-specific instead of the average single particle probabilities produces results for multiple electron loss, which differ only slightly for q< 6. Discrepancies are however observed for q ~ 6 which will be discussed in more detail in a subsequent publication. Fig. 2 shows the q-electron loss probabilities for the impact energy of 1 MeV/amu. The corresponding cross sections are given in fig. 3. Except for the
~
/
°~ °-“ 0.3 0 2
/
qH
/,~..,
q~3 ‘‘~.
f’-..
q8
“..
0 00
“
-
1 .
.
~-
‘I
2
3
B au.)
6
7
8
.
Fig. 2. Multiple electron loss probabilities for E= 1 MeV/amu. P 6 is obtainedfrom eq. (6), where q corresponds to the charge of the recoil ion.
175
Volume 145, number 4
1
I5~
‘
PHYSICS LETTERS A
~
1 .—
-~
~U2 I6L ~ F-----
I
— -
2)
1
I
21
21 --~
-
2
-~
-1.., 16
1/
~
-
~
.
-~
-..
-
IlL
6 L
j~
2
0
—
b
‘
‘
1 l’’’I’ I
I_li -
ing the information depopulation of individual shells given about in fig. the 1 can be improved by proper incorporation ofPauli blocking [10]. It is cxpected that this will have an effect on projectile scattering angle dependent cross sections where one is probing specific impact parameter regions [11].
I
Fig. 3. Total cross sections for recoil ion production. Theory: present results; (— — —) CTMC, ref [6]. Experiment: (•) ref. [1]; (A) ref. [3]; (•) ref. [2]. The numbers indicate the charge of the recoil ion.
one electron loss cross section, which is too large, the results for two- to six-fold ionisation of Ne are in qualitative agreement with experimental data [1—3] and the Vlasov—IPM calculation [6]. There is an apparent difficulty in establishing the correct (experimental) energy dependence of the recoil ion production cross sections as the various experiments seem to differ in the absolute normalisation of their data. It is remarkable to what extent the energy dependence of the present results agrees in shape with the semiclassical ones [6]. Improvements of the model presented here can be envisaged. The frozen core potential can be replaced by an effective time dependent one particle potential, obtained e.g. in a Vlasov~—lPMmodel calculation [6]. Such a potential corresponds to a mean field approximation with a time dependent electron density. The effect of a time dependent screening has been demonstrated in a previous calculation of resonant K—K transfer [9]. We expect a considerable reduction ofthe one electron loss cross section as the
176
time dependent potential will be moc~atiractive than the frozen atomic potential as ionisai.on sets in. This should be especially true for large impact parameters where the present calculation overestimates the singlc ionisation cross section. The extraction of many electron probabilities us-
‘-
-
1
9 April 1990
Acknowledgement We would like to thank the Deutsche Forschungsgemeinschaft for partial financial support of the Feshbach project, the NATO exchange programme for a cooperative travel grant and the EC for a twinning grant. References [1] S. Kelbch. CL. Cocke, S. Hagmann, M. Horbatsch. C. Kelbch. R. Koch, H. Schmidt-Bbcking and J. Ullrich, J. Phys. B (1989), in press [2]T.J. Gray, CL. Coeke and E. Justiniano. Phys. Rev. A 22 (1980) 849. [31A. MUller. B. Schuch. W. Groh, E. Salzborn, H.F. Beyer, PH. Moklerand R.E. Olson, Phys. Rev. A 33 (1986) 3010. [4]RE. Olson, J. Phys.B 12(1979)1843. [5]M. Horbatsch, Z. Phys. D 1 (1985) 337. [6] M. Horbatsch and R.M. Dreizler. Z. Phys. D 2 (1986) 183, [7]R.E. Olson. J. tJllrich and H. Schmidt-Bocking. Phys. Rex. A 39 (1989) 5572. [8] Hi. LUdde, A. Ast and R.M. Dreizier. J. Phys. B 21(1988) 4131. [91A. Toepfer, A. Henne, H.J. LUdde, M. Horbatsch and R.M. Dreizler. Phys. Lett. A 126 (1987) II. [10] H.J. LUdde, A. Henne and R.M. Dreizler, Z. Phys. D (1989), in press. Ii] M. Horbatsch. J. Phys. B 22 (1989) L639.
PHYSICS LETTERS A
9 April 1990
QUANTUM MECHANICAL DESCRIPTION OF RECOIL ION PRODUCTION IN C6~+Ne COLLISIONS Hi. LUDDE a M. HORBATSCH b A. HENNE a and R.M. DREIZLER a b
a
Institutfür Theoretische Physik der Universität, Frankfurt, FRG Physics Department, York University, Toronto, Ontario, Canada M3J 1P3
Received 11 January 1990; revised manuscript received 31 January 1990; accepted for publication 7 February 1990 Communicated by B. Fricke
Total cross sections for recoil ion production in C6 + +Ne collisions are presented. The results are based on the solution of the time dependent effective single particle Schrodinger equation in terms of a target centred atomic basis. Jonisation channels are included globally via the Feshbach formalism adapted to time dependent systems.
1. Introduction
H(t)=H 0(t)+V(t)
The scattering of highly charged ions on rare gas atoms leads to very complex situations as a considerable number of target electrons are actively involved in the collision process. Experimental data have been published during recent years investigating charge transfer, ionisation and in particular the production of highly charged recoil ions [1—3]. From the theoretical side many electron capture and ionisation in collisions ofbare ions with rare gas atoms have been described so far only within a semiclassical independent particle model (1PM) [4—6] and a classical n-particle model [7] with neglect of the interelectronic potential. To our knowledge no nonperturbative quantum calculations have been reported in the literature, In this contribution we collisions present results for from multiple 6~+Ne obtained the electron loss in C solution of the effective time dependent single partide Schrodinger equation.
(1)
,
where we decompose the Hamiltonian into an effective single particle operator N ~
I-~0=
l~(i) I
_________
~ ( t)
=
— ~
s~—R(t)~— r~+ u1 ( t)
A~—
and the residual electron—electron interaction N 1 N V= ~, ,~ u~(t) ~,
—
.
We start from the many electron time dependent Schrodinger equation (/l=me=e= 1) i8,~1-’(r 1 rN,t)=H(t)W(rl rN,t),
(3)
Now we assume that (i) electron correlations can be neglected and (ii) the single particle potential is described in termsofthe of aNe frozen (HF) potential atomatomic and theHartree—Fock Coulomb potential of the incoming projectile, +
2. Theory
(2)
HF VNe.
(4)
This approach defines an independent particle model, where the many electron wave function is a time dependent single Slater determinant
0375-9601 /90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)
173
Volume 145, number 4
rN,
PHYSICS LETTERS A
9 April 1990
(5)
semiclassical independent particle model calculations. These results are included only for electron loss. In ref. [6] the individual capture and ionisation probabilities are presented giving the information in
evolving from the specified initial condition, that is the HF approximation of the Ne ground state. In order to solve the single particle Schrodinger equation
which impact parameter range electron loss is produced mainly by capture or ionisation. With increasing energy or impact parameter the loss probability is dominated by ionisation. The individual loss probabilities reflect the shell structure
det{~(r1,
=
~
I)
t)
~N(rN, t)},
(6)
t)=l~~(t)~(r~, I) ,
we use the time dependent projector formalism [81 defining the finite model space P in terms of the K and L shell HF orbitals of the Ne atom. Following the arguments given in ref. [8] the projected P space coupled channel equations take the form à= ~(1 —G
-
Gk~=
ifQ~/Ir—R ( t)
—
‘
—if~)a,
‘
(7)
where the vector a denotes the excitation amplitudes 2Po, 2Pi) and the diof the four Ne states (ls, 2s, agonal matrix ~ represents the corresponding HF energies respectively. f is a scaling factor which improves the impact parameter dependence ofthe local approximation to the optical potential,
f=
B+ 1 —~----.
(8)
The P space coupled channel equations are solved for each of the initially occupied single particle states with a standard predictor corrector algorithm assuming straight line trajectories for the nuclear motion. 3. Results Fig. 1 shows the single particle loss probabilities for different impact energies. As the P space includes the important target excitation channels, electron loss can be interireted as the sum of the capture and ionisation probabilities. The curves represent the mdividual loss probabilities for the different initial states involved and the weighted average probability in order to allow comparison with results of the statistical 174
of the independent particle model: the more loosely bound 2p electrons are ionised with higher probability than the stronger bound 2s and is electrons, respectively. The situation, however, is not that obvious for collisions with small impact parameters, where capture is the dominant process. In this case the loss probability strongly depends on the energy difference of the initial target orbital and the final projectile state. The smaller the energy difference the larger the capture probability, which is in fact the condiction for resonance charge exchange. Thus the HF energy of the 2s level (ENe(2s)5 + (E~ 1.93043) corresponds to the M shellofenergy of C 5+(n~.3) while the levels Ne (ENe( 4) = lie2), between the 2p N 2p)=O.8504 and 0 shell energies of C5 + (E~ 5+(,, 4) = 1.125, Ec5+(,~s)= 0.72). Consequently capture of the 2s electrons becomes more likely than capture of the 2p electrons, although the latter are less strongly bound. The shell dependent results indicate that electron removal from all shells is possible for impact parameters larger than the corresponding shell radii. The ls-removal probability deviates most due to the significantly higher binding energy and smaller radius of the K shell. One can deduce from this observation that the previously employed statistical model [5,6] can describe properly at most the removal of 8 electrons beyond which it must overestimate the cross section. The difference between the statistical model calculations and the average optical potential results at lower energies, however, reflects the influence of a time dependent screening potential on the single partide loss probabilities. With increasing impact energy dynamic screening becomes less important. Within the framework of the independent particle model many electron loss probabilities can be obtamed from the single particle probabilities following the statistical approach,
Volume 145, number 4
PHYSICS LETTERS A
9 April 1990
I
I
I
I
I
Y0O~:65s789
_
9 (au.)
B (au.) I
I
I
I
CC
C
&
0.6
I
d
C.5~
i~
00O~3456789
B )a.u.)
B (au.)
Fig. 1. Single electron loss probability for (a) E= 100 keV/amu, (b) E= 250 keV/amu, (c) E= 400 keV/amu and (d) E= 1 MeV/amu. 2po; (——) 2p,. (0) CTMC, ref. [6]; (—) ls, weighted average; (— — —) 2s; (‘)
(N
a_s
(9)
07 0.6
Here Pq denotes the probability for the q-fold ionisation of Ne calculated from the average single electron loss probability p. Taking into account the individual shell-specific instead of the average single particle probabilities produces results for multiple electron loss, which differ only slightly for q< 6. Discrepancies are however observed for q ~ 6 which will be discussed in more detail in a subsequent publication. Fig. 2 shows the q-electron loss probabilities for the impact energy of 1 MeV/amu. The corresponding cross sections are given in fig. 3. Except for the
~
/
°~ °-“ 0.3 0 2
/
qH
/,~..,
q~3 ‘‘~.
f’-..
q8
“..
0 00
“
-
1 .
.
~-
‘I
2
3
B au.)
6
7
8
.
Fig. 2. Multiple electron loss probabilities for E= 1 MeV/amu. P 6 is obtainedfrom eq. (6), where q corresponds to the charge of the recoil ion.
175
Volume 145, number 4
1
I5~
‘
PHYSICS LETTERS A
~
1 .—
-~
~U2 I6L ~ F-----
I
— -
2)
1
I
21
21 --~
-
2
-~
-1.., 16
1/
~
-
~
.
-~
-..
-
IlL
6 L
j~
2
0
—
b
‘
‘
1 l’’’I’ I
I_li -
ing the information depopulation of individual shells given about in fig. the 1 can be improved by proper incorporation ofPauli blocking [10]. It is cxpected that this will have an effect on projectile scattering angle dependent cross sections where one is probing specific impact parameter regions [11].
I
Fig. 3. Total cross sections for recoil ion production. Theory: present results; (— — —) CTMC, ref [6]. Experiment: (•) ref. [1]; (A) ref. [3]; (•) ref. [2]. The numbers indicate the charge of the recoil ion.
one electron loss cross section, which is too large, the results for two- to six-fold ionisation of Ne are in qualitative agreement with experimental data [1—3] and the Vlasov—IPM calculation [6]. There is an apparent difficulty in establishing the correct (experimental) energy dependence of the recoil ion production cross sections as the various experiments seem to differ in the absolute normalisation of their data. It is remarkable to what extent the energy dependence of the present results agrees in shape with the semiclassical ones [6]. Improvements of the model presented here can be envisaged. The frozen core potential can be replaced by an effective time dependent one particle potential, obtained e.g. in a Vlasov~—lPMmodel calculation [6]. Such a potential corresponds to a mean field approximation with a time dependent electron density. The effect of a time dependent screening has been demonstrated in a previous calculation of resonant K—K transfer [9]. We expect a considerable reduction ofthe one electron loss cross section as the
176
time dependent potential will be moc~atiractive than the frozen atomic potential as ionisai.on sets in. This should be especially true for large impact parameters where the present calculation overestimates the singlc ionisation cross section. The extraction of many electron probabilities us-
‘-
-
1
9 April 1990
Acknowledgement We would like to thank the Deutsche Forschungsgemeinschaft for partial financial support of the Feshbach project, the NATO exchange programme for a cooperative travel grant and the EC for a twinning grant. References [1] S. Kelbch. CL. Cocke, S. Hagmann, M. Horbatsch. C. Kelbch. R. Koch, H. Schmidt-Bbcking and J. Ullrich, J. Phys. B (1989), in press [2]T.J. Gray, CL. Coeke and E. Justiniano. Phys. Rev. A 22 (1980) 849. [31A. MUller. B. Schuch. W. Groh, E. Salzborn, H.F. Beyer, PH. Moklerand R.E. Olson, Phys. Rev. A 33 (1986) 3010. [4]RE. Olson, J. Phys.B 12(1979)1843. [5]M. Horbatsch, Z. Phys. D 1 (1985) 337. [6] M. Horbatsch and R.M. Dreizler. Z. Phys. D 2 (1986) 183, [7]R.E. Olson. J. tJllrich and H. Schmidt-Bocking. Phys. Rex. A 39 (1989) 5572. [8] Hi. LUdde, A. Ast and R.M. Dreizier. J. Phys. B 21(1988) 4131. [91A. Toepfer, A. Henne, H.J. LUdde, M. Horbatsch and R.M. Dreizler. Phys. Lett. A 126 (1987) II. [10] H.J. LUdde, A. Henne and R.M. Dreizler, Z. Phys. D (1989), in press. Ii] M. Horbatsch. J. Phys. B 22 (1989) L639.