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One and two electron transitions in ion-atom collisions
66
Nuclear
Instruments
and Methods
One and two electron transitions in ion-atom
in Physics
Research
B56/57
(1991) 66-69 North-Holland
collisions
R. Shingal Kansas State University,
Phpics Department, Cardwell Hall. Manhattan, KS 66.502-2601, US.4
One and two electron transitions in collisions between light bare ions and helium atoms have been studied for intermediate impact energies. The helium atom was treated as an effective one electron atom. The multichannel semiclassical impact parameter model with travelling atomic orbital expansion and rectilinear trajectories was used to calculate the transition probabilities. Cross sections for all one electron processes calculated in the independent electron appro~mation, employing a large basis set, are in good agreement with the experimental data. A two step mechanism is proposed to describe the two electron collision processes. Predicted total cross sections for various one and two electron processes are found to be in good agreement with the available experimental data.
1. introduction The collisions involving atomic species with one or more active electrons is interesting and has been widely studied experimentally due to the advent of accelerators. The basic understanding of the many electron phenomenon encountered in such collisions is essential not only for grasping the mysteries of the microscopic world but also, for example, for advancing the quality of our lives through the development of large machines (fusion reactors) capable of solving the energy problem of the living world. In principle, within the semiclassical impact parameter model, the theoretical study of the problem of ion-atom collisions reduces to the solution of the time dependent Schroedinger equation (H-i
a/at)qr,,
with the expansion system in terms of orbitals. For a two Hamiltonian, H, in H-
-fof-iv?+
P,, t) = 0
(1)
of the total wave function of the either the molecular or the atomic or pseudo-two electron system the eq. (1) takes the form V,(r,,)+
VA(rzA)
+ ~g(f-3d + ~B(pi~) + +c,,~
(2)
y( rij) being the Coulomb interaction of the i th electron (i = 1, 2) with the ionic core presented by the jth nucleus (j = A, B). Calculations along these lines have already been carried out for few representative processes and systems [l-8]. These ab initio calculations pose serious computational problems, in terms of processing time, due to the presence of coulombic interaction between the active electrons. Experience has shown that the electronelectron correlation plays a very important role in the 0168-583X/91/$03.50
0 1991 - Elsevier Science Publishers
collision processes involving both the electrons, and even in the cases where the end product of the reaction indicates that a single electron transition has taken place [5]. Furthermore, inclusion of extremely large basis sets, routinely used in one or pseudo-one active electron collision systems becomes computationally prohibitive in many electron systems. This problem becomes serious in the intermediate and high impact energy region where in addition to the one electron processes - viz., charge transfer, excitation and ionisation - two electron transitions such as double ionisation, double capture, transfer ionisation, transfer excitation and double excitation are important reaction channels (for a schematic representation of these processes, see for example ref. [9]). This complex situation has been tackled in certain circumstances by assuming that either of the active electrons moves in the average field provided by the remaining electron and the ionic core, in the form of the independent particle model (IPM) [lo]. In this work, the IPM in conjunction with the standard muiti~hannel impact parameter model [ll] has been used to study some observable quantities for the one and two electron processes in collisions involving light bare ions (H+Lise) and helium atoms at intermediate impact energies. A brief overview of the theoretical method is given in section 2.
2. Theoretical models The Hamiltonian (eq. (2)) can be written as a sum of two one particle Hamiltonians under the assumption that each electron of the helium atom moves in an effective central potential presented by the remaining electrons and the helium nucleus. The total wave func-
B.V. (North-Holland)
61
R. Shrngai / One and two electron transitions tion of the system can then be written as a product of two one particle wave functions each of which satisfies a one electron time dependent Schroedinger equation. This equation was solved in the standard coupled channel semiclassical impact parameter model with appropriate boundary conditions. The total wave function was expanded in travelling atomic orbitals placed on the scattering centres. The bases set contained exact and pseudostates accounting for higher bound, the direct ionisation and electron capture to the continuum channels [13]. The resulting scattering probabilities are combined,
transitions [14]. In this model ii) the collision is viewed as being two seperate events. In the first collision with the projectile the active electron (ionisation energy 0.9 a.u.) of the helium atom (target) is removed and assumed to be captured by the projectile at low and intermediate impact energies. The modified projectile then collides with the singly ionised helium ion for the same impact parameter. The calculated scattering amplitudes are then combined to compute cross sections for the two electron transitions.
3. Results and discussions 0; = 277 b dbP,(b), /
(3)
to obtain the cross section 0, for each one and two electron process i. In eq. (4), P,(b) is the probability for the corresponding process [9]. The one electron transitions were studied by assuming the active electron of the helium atom to move in an effective potential presented by the passive electron and the (Y particle. This potential was chosen to correctly represent the energies of the first few bound state of the helium atom. This model of the helium atom was unable to predict the cross sections for two electron transitions occuring in bare ion helium atom collisions. This result is not totally surprising as the above model gives 1.8 a.u. as the double ionisation energy in contrast to the actual value of 2.9 a.u. Two different physical models are proposed to find a way out of this predicament. i) Each electron of the helium atom was assumed to move in a pure Coulomb potential with the effective charge adjusted to give the binding energy of each electron as half of the double ionisation energy [9]. Alternatively, a mechanism in which the projectile undergoes two collisions is proposed to predict the cross sections for two electron
3.1. Hff
He
The passage of protons through a gas of helium atoms results in the production of coherently excited hydrogen atoms. The density matrices for the excited n = 3 manifold of the collisionally produced hydrogen atoms have recently been studied experimentally [15,16]. The density matrix can be constructed from the known values of the scattering amplitudes, at a given impact parameter, for the one electron charge transfer process. The diagonal elements of the integrated density matrix are related to the partial cross section for capture to an individual state. The real part of the integrated off-diagonal elements can be physically interpreted as the averaged z component of the dipole moment, (D,), while the imaginary part is related to the expectation value of the z component of the operator L x A constructed from the angular momentum vector L and the RungeLenz vector A. The predicted averaged dipole moment ( DZ) and the averaged velocity vector ((L X A)=) for the H(n = 3) atom are compared with the recent experimental data [15] in figs. la and lb, respectively. The results of an
2-
H+ + H&s*)->H(n=3) IL’S 10
20
3 30
c 40
Impact
8
50
Energy
*
80
c. 70
(keV/amu)
+ He+ 1 80
a 90
4.3~
‘, 100
3
IO
*’
a
I:pa:t
’
r
1
E&g;“(ke;o/a~u)
a
8
a’
+ I I
Iw
Fig. 1. (a) The average dipole moment for capture to the n = 3 state of the hydrogen atom in Hf + He collisions. Theory: ~ present; . . Slim et al. (1990) without exchange: - - - Slim et al. (1990) with exchange. Experiment: W Ashbum et al. (1990) (b) Average velocity vector for capture to the n = 3 state of the hydrogen atom in H+ + He collision. Symbols as m (a). I. ATOMIC/MOLECULAR
PHYSICS
R. Shrngal / One and two electron transitlons
Single Ionisation ~___L ~. ~. Y - -~- = - -. . * Single Cap *
_
- ~C_
.
.111
40 Impact
10% Energy (keV/amu)
300
Fig, 2. Cross section for single ionisation and single capture in single capture; He’+ + He collisions. Theory: present: -_single ionisation. Experiment: Shah et al. (1985): A smgle capture; 0 single ionisation.
explicit two electron calculation, without and with exchange, are also shown in these figures [17]. A positive value of the dipole moment is predicted throughout the energy region considered, indicating that the captured electron remains behind the proton. The computed quantities are in reasonable agreement with the symmetrised two electron results [17] and the experimental data [15]. The elements of the full density matrix, normalised to the diagonal 3s term, were also found to be in harmony with the two electron theory and the experimental data [12].
3.2. He”+
He
The calculated cross section for single electron transfer and single ionisation in collisions of 01particles with helium atoms are compared with experimental data [18] in fig. 2. The predicted cross sections are in agreement with the measurements in the entire energy region considered. Similar agreement between the predicted cross sections and the experimental data have been found for other bare projectiles with a charge less than ‘or equal to 8 [9]. However, some discrepancy was observed for one electron transfer in F9++ He collisions, which is most likely to be attributed to the limited size of the bases set. The two electron phenomenon was studied using both the effective charge and the relaxation models explained earlier. The predicted cross sections for transfer ionisation and double ionisation, calculated in the two approximations, are compared with the experimental data [18] in fig. 3. The cross sections calculated in the two models predict similar qualitative behaviour in reasonable agreement with the experiment. However, a better agreement between the theoretical results obtained m the model which takes relaxation of the target atom into account and the experimental data is evident.
4--‘o
Impact
Energy
(keV/amu)
Fig. 3. Cross section for double and transfer ionisation in He*+ + He collisions. Theory: present: double ionisation - - model i) see text; double ionisation . model ii) see text; transfer ionisation - - - model i), model ii). Experiment: Shah et al. (1985): A double ionisatlon; W transfer ionisation.
3.3. Li3 + + He Finally, the collision between the bare lithium ion and the helium atom has been studied for impact energies lying in the range of 50-500 keV/u. The calculated cross sections for single electron capture and single ionisation are again found to be in harmony with the experimental data. The predicted cross section, calculated employing the relaxation model, for total double capture, transfer ionisation and double ionisation are compared with the experimental data in fig. 4. The computed double capture and transfer ionisation cross section are found to be in excellent agreement with the experimental data [18]. Reasonable agreement is also found between the experimental data and the calculated double ionisation cross section.
.‘.I 50
103
Impact Energy
501
(keV/amu)
Fig. 4. Cross section for double and transfer iomsation and double capture in He2+ + He collisions. Theory: present (model ii), see text); double ionisation; - - - double capture; transfer ionisation. Experiment: Shah et. al. (1985): A double ionisation; 0 double capture; v transfer iomsation.
R. Shingal / One and two electron transitrons
Acknowledgements Useful conversations with Professors B.H. Bransden, D.R. Flower and C.D. Lin and J H McGuire are gratefully acknowledged. This work is partially supported by the Division of Chemical Sciences, Office of the Basic Energy Sciences, US Department of Energy.
References PI B.H. Bransden,
A.M. Ermolaev and R. Shingal, J. Phys. B17 (1984) 4515. PI W. Fritsch and C.D. Lin, J. Phys. B19 (1986) 2683. [31 J.F. Reading and A.L. Ford, Phys. Rev. Lett. 58 (1987) 543. [41 J.F. Reading and A.L. Ford, J. Phys. B20 (1987) 3859. 151 R. Shingal and B.H. Bransden, Nucl. Instr. and Meth. B40/41 (1989) 242.
69
[61R. Shingal, B.H. Bransden
and D.R. Flower, J. Phys. B22 (1989) 855. [71 W. Fritsch and CD. Lin, Phys. Rev. Lett. 61 (1988) 690. PI W. Fritsch and CD. Lin, Phys. Rev. A41 (1990) 4776. 191 R. Shingal and C.D. Lin, J. Phys. B, to be published. [IO1 J H McGuire and L.O. Weaver, Phys. Rev. Al6 (1977) 41. [III See for example: B.H. Bransden, Atomic Collision Theory (Benjamin-Cummings, Reading, MA, 1983). WI R. Shmgal and C.D. Lin, submitted to J. Phys. B. [I31 R. Shingal J. Phys. B21 (1988) 2065. [I41 R. Shingal, Kansas State University Preprint, unpublished (1988). [I51 J.R. Ashburn, R.A. Cline, P.J.M. van der Burgt, W.B. Westerveld and J.S. Risley, Phys. Rev. A41 (1990) 2407. M.C. Brower and F.M. Pipkin, Phys. Rev. A39 (1989) 3323. H.A. Slim, E.L. Heck, B.H. Bransden and D.R. Flower, J. Phys. B, in press. M.B. Shah and H.B. Gildbody, J. Phys. B18 (1985) 899.
I. ATOMIC/MOLECULAR
PHYSICS
Nuclear
Instruments
and Methods
One and two electron transitions in ion-atom
in Physics
Research
B56/57
(1991) 66-69 North-Holland
collisions
R. Shingal Kansas State University,
Phpics Department, Cardwell Hall. Manhattan, KS 66.502-2601, US.4
One and two electron transitions in collisions between light bare ions and helium atoms have been studied for intermediate impact energies. The helium atom was treated as an effective one electron atom. The multichannel semiclassical impact parameter model with travelling atomic orbital expansion and rectilinear trajectories was used to calculate the transition probabilities. Cross sections for all one electron processes calculated in the independent electron appro~mation, employing a large basis set, are in good agreement with the experimental data. A two step mechanism is proposed to describe the two electron collision processes. Predicted total cross sections for various one and two electron processes are found to be in good agreement with the available experimental data.
1. introduction The collisions involving atomic species with one or more active electrons is interesting and has been widely studied experimentally due to the advent of accelerators. The basic understanding of the many electron phenomenon encountered in such collisions is essential not only for grasping the mysteries of the microscopic world but also, for example, for advancing the quality of our lives through the development of large machines (fusion reactors) capable of solving the energy problem of the living world. In principle, within the semiclassical impact parameter model, the theoretical study of the problem of ion-atom collisions reduces to the solution of the time dependent Schroedinger equation (H-i
a/at)qr,,
with the expansion system in terms of orbitals. For a two Hamiltonian, H, in H-
-fof-iv?+
P,, t) = 0
(1)
of the total wave function of the either the molecular or the atomic or pseudo-two electron system the eq. (1) takes the form V,(r,,)+
VA(rzA)
+ ~g(f-3d + ~B(pi~) + +c,,~
(2)
y( rij) being the Coulomb interaction of the i th electron (i = 1, 2) with the ionic core presented by the jth nucleus (j = A, B). Calculations along these lines have already been carried out for few representative processes and systems [l-8]. These ab initio calculations pose serious computational problems, in terms of processing time, due to the presence of coulombic interaction between the active electrons. Experience has shown that the electronelectron correlation plays a very important role in the 0168-583X/91/$03.50
0 1991 - Elsevier Science Publishers
collision processes involving both the electrons, and even in the cases where the end product of the reaction indicates that a single electron transition has taken place [5]. Furthermore, inclusion of extremely large basis sets, routinely used in one or pseudo-one active electron collision systems becomes computationally prohibitive in many electron systems. This problem becomes serious in the intermediate and high impact energy region where in addition to the one electron processes - viz., charge transfer, excitation and ionisation - two electron transitions such as double ionisation, double capture, transfer ionisation, transfer excitation and double excitation are important reaction channels (for a schematic representation of these processes, see for example ref. [9]). This complex situation has been tackled in certain circumstances by assuming that either of the active electrons moves in the average field provided by the remaining electron and the ionic core, in the form of the independent particle model (IPM) [lo]. In this work, the IPM in conjunction with the standard muiti~hannel impact parameter model [ll] has been used to study some observable quantities for the one and two electron processes in collisions involving light bare ions (H+Lise) and helium atoms at intermediate impact energies. A brief overview of the theoretical method is given in section 2.
2. Theoretical models The Hamiltonian (eq. (2)) can be written as a sum of two one particle Hamiltonians under the assumption that each electron of the helium atom moves in an effective central potential presented by the remaining electrons and the helium nucleus. The total wave func-
B.V. (North-Holland)
61
R. Shrngai / One and two electron transitions tion of the system can then be written as a product of two one particle wave functions each of which satisfies a one electron time dependent Schroedinger equation. This equation was solved in the standard coupled channel semiclassical impact parameter model with appropriate boundary conditions. The total wave function was expanded in travelling atomic orbitals placed on the scattering centres. The bases set contained exact and pseudostates accounting for higher bound, the direct ionisation and electron capture to the continuum channels [13]. The resulting scattering probabilities are combined,
transitions [14]. In this model ii) the collision is viewed as being two seperate events. In the first collision with the projectile the active electron (ionisation energy 0.9 a.u.) of the helium atom (target) is removed and assumed to be captured by the projectile at low and intermediate impact energies. The modified projectile then collides with the singly ionised helium ion for the same impact parameter. The calculated scattering amplitudes are then combined to compute cross sections for the two electron transitions.
3. Results and discussions 0; = 277 b dbP,(b), /
(3)
to obtain the cross section 0, for each one and two electron process i. In eq. (4), P,(b) is the probability for the corresponding process [9]. The one electron transitions were studied by assuming the active electron of the helium atom to move in an effective potential presented by the passive electron and the (Y particle. This potential was chosen to correctly represent the energies of the first few bound state of the helium atom. This model of the helium atom was unable to predict the cross sections for two electron transitions occuring in bare ion helium atom collisions. This result is not totally surprising as the above model gives 1.8 a.u. as the double ionisation energy in contrast to the actual value of 2.9 a.u. Two different physical models are proposed to find a way out of this predicament. i) Each electron of the helium atom was assumed to move in a pure Coulomb potential with the effective charge adjusted to give the binding energy of each electron as half of the double ionisation energy [9]. Alternatively, a mechanism in which the projectile undergoes two collisions is proposed to predict the cross sections for two electron
3.1. Hff
He
The passage of protons through a gas of helium atoms results in the production of coherently excited hydrogen atoms. The density matrices for the excited n = 3 manifold of the collisionally produced hydrogen atoms have recently been studied experimentally [15,16]. The density matrix can be constructed from the known values of the scattering amplitudes, at a given impact parameter, for the one electron charge transfer process. The diagonal elements of the integrated density matrix are related to the partial cross section for capture to an individual state. The real part of the integrated off-diagonal elements can be physically interpreted as the averaged z component of the dipole moment, (D,), while the imaginary part is related to the expectation value of the z component of the operator L x A constructed from the angular momentum vector L and the RungeLenz vector A. The predicted averaged dipole moment ( DZ) and the averaged velocity vector ((L X A)=) for the H(n = 3) atom are compared with the recent experimental data [15] in figs. la and lb, respectively. The results of an
2-
H+ + H&s*)->H(n=3) IL’S 10
20
3 30
c 40
Impact
8
50
Energy
*
80
c. 70
(keV/amu)
+ He+ 1 80
a 90
4.3~
‘, 100
3
IO
*’
a
I:pa:t
’
r
1
E&g;“(ke;o/a~u)
a
8
a’
+ I I
Iw
Fig. 1. (a) The average dipole moment for capture to the n = 3 state of the hydrogen atom in Hf + He collisions. Theory: ~ present; . . Slim et al. (1990) without exchange: - - - Slim et al. (1990) with exchange. Experiment: W Ashbum et al. (1990) (b) Average velocity vector for capture to the n = 3 state of the hydrogen atom in H+ + He collision. Symbols as m (a). I. ATOMIC/MOLECULAR
PHYSICS
R. Shrngal / One and two electron transitlons
Single Ionisation ~___L ~. ~. Y - -~- = - -. . * Single Cap *
_
- ~C_
.
.111
40 Impact
10% Energy (keV/amu)
300
Fig, 2. Cross section for single ionisation and single capture in single capture; He’+ + He collisions. Theory: present: -_single ionisation. Experiment: Shah et al. (1985): A smgle capture; 0 single ionisation.
explicit two electron calculation, without and with exchange, are also shown in these figures [17]. A positive value of the dipole moment is predicted throughout the energy region considered, indicating that the captured electron remains behind the proton. The computed quantities are in reasonable agreement with the symmetrised two electron results [17] and the experimental data [15]. The elements of the full density matrix, normalised to the diagonal 3s term, were also found to be in harmony with the two electron theory and the experimental data [12].
3.2. He”+
He
The calculated cross section for single electron transfer and single ionisation in collisions of 01particles with helium atoms are compared with experimental data [18] in fig. 2. The predicted cross sections are in agreement with the measurements in the entire energy region considered. Similar agreement between the predicted cross sections and the experimental data have been found for other bare projectiles with a charge less than ‘or equal to 8 [9]. However, some discrepancy was observed for one electron transfer in F9++ He collisions, which is most likely to be attributed to the limited size of the bases set. The two electron phenomenon was studied using both the effective charge and the relaxation models explained earlier. The predicted cross sections for transfer ionisation and double ionisation, calculated in the two approximations, are compared with the experimental data [18] in fig. 3. The cross sections calculated in the two models predict similar qualitative behaviour in reasonable agreement with the experiment. However, a better agreement between the theoretical results obtained m the model which takes relaxation of the target atom into account and the experimental data is evident.
4--‘o
Impact
Energy
(keV/amu)
Fig. 3. Cross section for double and transfer ionisation in He*+ + He collisions. Theory: present: double ionisation - - model i) see text; double ionisation . model ii) see text; transfer ionisation - - - model i), model ii). Experiment: Shah et al. (1985): A double ionisatlon; W transfer ionisation.
3.3. Li3 + + He Finally, the collision between the bare lithium ion and the helium atom has been studied for impact energies lying in the range of 50-500 keV/u. The calculated cross sections for single electron capture and single ionisation are again found to be in harmony with the experimental data. The predicted cross section, calculated employing the relaxation model, for total double capture, transfer ionisation and double ionisation are compared with the experimental data in fig. 4. The computed double capture and transfer ionisation cross section are found to be in excellent agreement with the experimental data [18]. Reasonable agreement is also found between the experimental data and the calculated double ionisation cross section.
.‘.I 50
103
Impact Energy
501
(keV/amu)
Fig. 4. Cross section for double and transfer iomsation and double capture in He2+ + He collisions. Theory: present (model ii), see text); double ionisation; - - - double capture; transfer ionisation. Experiment: Shah et. al. (1985): A double ionisation; 0 double capture; v transfer iomsation.
R. Shingal / One and two electron transitrons
Acknowledgements Useful conversations with Professors B.H. Bransden, D.R. Flower and C.D. Lin and J H McGuire are gratefully acknowledged. This work is partially supported by the Division of Chemical Sciences, Office of the Basic Energy Sciences, US Department of Energy.
References PI B.H. Bransden,
A.M. Ermolaev and R. Shingal, J. Phys. B17 (1984) 4515. PI W. Fritsch and C.D. Lin, J. Phys. B19 (1986) 2683. [31 J.F. Reading and A.L. Ford, Phys. Rev. Lett. 58 (1987) 543. [41 J.F. Reading and A.L. Ford, J. Phys. B20 (1987) 3859. 151 R. Shingal and B.H. Bransden, Nucl. Instr. and Meth. B40/41 (1989) 242.
69
[61R. Shingal, B.H. Bransden
and D.R. Flower, J. Phys. B22 (1989) 855. [71 W. Fritsch and CD. Lin, Phys. Rev. Lett. 61 (1988) 690. PI W. Fritsch and CD. Lin, Phys. Rev. A41 (1990) 4776. 191 R. Shingal and C.D. Lin, J. Phys. B, to be published. [IO1 J H McGuire and L.O. Weaver, Phys. Rev. Al6 (1977) 41. [III See for example: B.H. Bransden, Atomic Collision Theory (Benjamin-Cummings, Reading, MA, 1983). WI R. Shmgal and C.D. Lin, submitted to J. Phys. B. [I31 R. Shingal J. Phys. B21 (1988) 2065. [I41 R. Shingal, Kansas State University Preprint, unpublished (1988). [I51 J.R. Ashburn, R.A. Cline, P.J.M. van der Burgt, W.B. Westerveld and J.S. Risley, Phys. Rev. A41 (1990) 2407. M.C. Brower and F.M. Pipkin, Phys. Rev. A39 (1989) 3323. H.A. Slim, E.L. Heck, B.H. Bransden and D.R. Flower, J. Phys. B, in press. M.B. Shah and H.B. Gildbody, J. Phys. B18 (1985) 899.
I. ATOMIC/MOLECULAR
PHYSICS