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Double-to-single ionization ratios of helium bombarded by low-to-intermediate velocity Cq+, Oq+ (q = 1 ∼ 3) ions
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 258 (2007) 340–344 www.elsevier.com/locate/nimb
Double-to-single ionization ratios of helium bombarded by low-to-intermediate velocity Cq+, Oq+ (q = 1 3) ions J.X. Shao, X.M. Chen *, B.W. Ding, H.B. Fu, Z.M. Gao, Y.W. Liu, J. Du, Y.X. Lu, Y. Cui Department of Modern Physics, Lanzhou University, Lanzhou 730000, China Received 19 July 2006; received in revised form 16 October 2006 Available online 26 March 2007
Abstract We measured the double-to-single ionization ratios R of Helium bombarded by Cq+ and Oq+ ions (q = 1 3). The velocity of the projectile varies from vBohr to 4 vBohr, the energy varies from 25 keV/amu to 500 keV/amu. The value of R increases rapidly with the collision velocity and reaches the maximum when the velocity is about 2 or 3 vBohr, then it decreases slowly for the higher velocity. A simple model is presented to estimate the value R varying with the collision velocity. The results calculated by the model are in agreement with the experimental data basically. Ó 2006 Published by Elsevier B.V. Keywords: Double ionization; Helium; Carbon ions; Oxygen ions
1. Introduction Ionization is one of the basic processes mediated by Coulomb forces during ion–atom collisions, and this process has been studied both theoretically and experimentally from the beginning of atomic and quantum physics. In high energy range (vion > 10 vBohr), the removal of a single-electron by the projectile is well described in the framework of the first Born approximation [1]. In the framework of first Born approximation, McGuire proposed that double ionization by ions at intermediate to high velocities (vion P 10 vBohr) can be understood in terms of two mechanisms [2,3]: (1) a two-step process (TS), vion 10 vBohr, in which both target electrons are removed in separate direct interactions with the projectile, the value of R increases with the projectile charge q and decreases with collision velocity v as q2/(v2 ln v); (2) a shake-off process (SO), vion 10 vBohr, in which the first electron is removed in a direct interaction with the projectile while the second electron is ejected when the *
Corresponding author. Tel.: +86 931 8913541. E-mail addresses: [email protected] (X.M. Chen), [email protected]. edu.cn (J.X. Shao). 0168-583X/$ - see front matter Ó 2006 Published by Elsevier B.V. doi:10.1016/j.nimb.2006.10.072
resulting ion ‘‘relaxes’’ to a continuum state and the ratio R is expected to be a constant. In the low energy range (vion < vBohr), single removal is understood well in the framework of Bohr’s ClassicalOver-Barrier-Model (COBM). COBM [4] indicates: that electron capture is the dominant process, direct ionization is neglected in this model. The cross section of capture is independent of the collision velocity v, as a result, the double-to-single ionization ratio R is also expected to be a constant. In the low-to-intermediate velocity range (vion P vBohr), many experiments had been made focusing on the ratios R [5–10]. No matter whether the projectile is bare ions or partially bare, the trends of R varying with the collision velocity have a similar character. That is, the value of R increases rapidly with the collision velocity and reaches the maximum when the velocity is about 2 or 3 vBohr then it decreases slowly for the higher velocity [1]. In the present work, Cq+ and Oq+ ions (q = 1 3) with the velocity varying from vBohr to 4 vBohr collide with He, the double-to-single ionization ratios R of Helium are measured using the time-to-flight and coincidence technique, the details of the experimental method are discussed
J.X. Shao et al. / Nucl. Instr. and Meth. in Phys. Res. B 258 (2007) 340–344
elsewhere [11]. When the velocity is about 2 or 3 vBohr, the value of R reaches the maximum, not only in the direct ionization process but also in the charge exchange process. The experimental data and the details of the model will be discussed in Sections 2 and 3. 2. A simple theoretical estimate The model we used to estimate the double-to-single ionization ratios R, is based on Bohr–Lindhard’s COBM [4]. They introduce two important ion–atom interaction distances in COBM. First, they introduced the release distance Rr, where the electron can be released from the target nucleus. The release distance Rr satisfies: q ðRr aÞ2
¼
Z ; a2
ð1Þ
where a is the orbital radius of the electron, q is the charge state of the projectile and Z is the atomic number of the target. Eq. (1) indicates that the electron can be released when the forces on the electron extracted by the projectile and the target atom are in balance. Second, the capture distance RC is introduced. RC is the ion–atom distance where the electron can be captured by the projectile. RC satisfies the following equation: q 1 ¼ V 2; RC 2 P
ð2Þ
where VP is the velocity of the ion. Eqs. (1) and (2) means: it is only that the process of release occurred within the distance of capture the electron can be captured, otherwise, the release is temporary. In the low energy range, RC P Rr, the released electrons are all captured, the capture cross section is given by rC ¼ pR2r :
ð3Þ
For higher energy, RC < Rr. Only the electrons released within the distance of RC can be captured, as a result, the capture cross section rC ¼ pR2C f :
ð4Þ
f is the probability of release process occurring within the . distance of RC Where f ¼ vae VRCP , means the ratio of the duration of collision to the orbital period of the bound electron. ve is the orbital velocity of this electron. In Bohr’s theory of COBM [4], the electrons which are released but not captured will go back to the target atom after collision. The energy level of the released electron had already been estimated by Bohr et al. [4] and Niehaus [12], it is I n þ Rqr , the ionization energy of the target electron added by the stark-energy by the ion. We consider that the released but not captured electrons will be accelerated by the approaching ion. At some dis-
341
tance RI, the electrons will get enough kinetic energy from the ion to escape from the target atom, meanwhile, these electrons will not be captured, they will be ionized. The ionization distance RI satisfies: q q P In þ : ð5Þ RI Rr It means when the stark-energy transferred to the kinetic energy of the electron is larger than the ionization energy of the quasi-molecular states, the electron will escape from the target atom. If the released electron has enough energy to escape from the target atom and can not form the stable bound states of the ion, ionization occurs. According to Bohr’s theory, electrons have not been released during the collision are in the perturbative bound states of the target atom all the time, the kinetic energy of these electrons are almost the initial value, are not large enough to escape, thus, the ionization of these electrons is neglected in our model. That is, in our model, the electrons released within the distance RC will be captured, those released between Rr and RC will not be ionized until the ion entered the distance RI. Before we calculate the cross sections about two-electron system, it is necessary to get the ionization or capture cross section of the single-electron system. Then, those cross sections about the two electrons will be calculated by the independent electron model (IED) easily. Several approximations are made: (1) These projectiles move along the linear trajectories with the collision parameter b and the velocity VP. (2) The orbital period of the electron is T ¼ 2pa , neglectv ing the influence of the ions. For the given collision parameter b and velocity VP, the probability pffiffiffiffiffiffiffiffiffithat electron will be released is 2 R2r b2 1 P r ðbÞ ¼ V P T , the ratio of the duration of collision (duration of the release process) to the period of the bound pffiffiffiffiffiffiffiffiffiffi electron, for b 6 Rr. 2 R2 b2 The capture probability is P C ðbÞ ¼ V CP T1 , the ratio of the duration of the capture process to the period of the bound electron, for b 6 RC. Then the ionization probability is PI(b) = Pr(b) PC(b), for b 6 RI. When we construct the cross sections of twoelectron system with these single-electron probabilities, the only thing is to distinguish these two electrons with subscript 1 and 2, e.g. Rr1, RC1, RI1, the distances for the first electron; Rr2, RC2, RI2, the distances for the second electron. They satisfy the following equations: qi 2
¼
Zi ; a2
ðRri aÞ qi 1 ¼ V 2P ; RCi 2 qi q ¼ Ii þ i RIi Rri
ð6Þ ði ¼ 1; 2Þ;
J.X. Shao et al. / Nucl. Instr. and Meth. in Phys. Res. B 258 (2007) 340–344
where qi is the ion charge state to the ith electron of the target, Z1 = 1, Z2 = 2; I1 = 0.93, I2 = 2 of the helium system. The reason for Z2 5 Z1 is that the first released electron loses the screening ability to both nuclei. The value of qi will not change for the direct ionization process, q2 = q1; one will be subtracted from qi if the first electron is captured, that is, q2 = q1 1. The values of PC1(b), PI1(b) and PC2(b), PI2(b) are calculated from Rr1, RC1, RI1 and Rr2, RC2, RI2. After integrating PC1(b), PI1(b) and PC2(b), PI2(b) for all collision parameter b, we get the cross sections of the two-electron system. The direct single-ionization cross section is Z rSI ¼ 2p P I1 ðbÞ ð1 P I2 ðbÞ P C2 ðbÞÞ b db Z þ 2p P I2 ðbÞ ð1 P I1 ðbÞ P C1 ðbÞÞ b db: The direct double ionization cross section is Z rDI ¼ 2p P I1 ðbÞ P I2 ðbÞ b db:
0.14 0.12 0.10 0.08
DI/SI
342
0.06 0.04 0.02 0.00 0
50
100
150
200
250
300
350
400
Beam energy (keV/amu)
Fig. 2. Experimental data of direct double-to-single ionization ratios for ion charge q = 2 colliding with He and the theoretical estimate (solid line). The data were obtained by Sanders (10 in 5) (m: Li2+), present data (j: C2+; s: O2+).
0.12
The single-capture cross section is Z rSC ¼ 2p P C1 ðbÞ ð1 P I2 ðbÞ P C2 ðbÞÞ b db Z þ 2p P C2 ðbÞ ð1 P I1 ðbÞ P C1 ðbÞÞ b db:
0.10
And the transfer-ionization cross section is Z Z rTI ¼ 2p P I1 ðbÞ P C2 ðbÞ b db þ 2p P I2 ðbÞ P C1 ðbÞ b db: The ratios of direct double-to-single ionization is RðDIÞ ¼ rrDI , the ratios of transfer-ionization-to-single-capSI TI ture is RðTIÞ ¼ rrSC .
DI/SI
0.08
0.06
0.04
0.02
0.00 50
100
150
200
250
300
350
400
450
500
Beam energy (keV/amu)
Fig. 3. Experimental data of direct double-to-single ionization ratios for ion charge q = 3 colliding with He and the theoretical estimate (solid line). The data were obtained by Shah and Gilbody [5] (n: Li3+), present data (j: C3+; s: O3+).
0.10
DI/SI
0.08
The calculation results are plotted in Figs. 1–6 together with the results of our experiment and other experiments in the same energy range. The experimental results in Figs. 1–6 indicate that with varying ion charges, the double-to-single ionization ratio R reaches the maximum when the velocity is about 2 or 3 vBohr.
0.06
0.04
0.02
3. Discussion
0.00 0
100
200
300
400
Beam energy (keV/amu)
Fig. 1. Experimental data of direct double-to-single ionization ratios for ion charge q = 1 colliding with He and the theoretical estimate (solid line). The data were obtained by Dubois [10] (n: He+), present data (j: C+; s: O+).
Figs. 1–6 indicate that (1) The ratios R have the similar trend for the various systems. The trend is, R increases rapidly with the collision velocity and reaches the maximum at
J.X. Shao et al. / Nucl. Instr. and Meth. in Phys. Res. B 258 (2007) 340–344 0.22
0.50
0.20
0.45
0.18
0.40
0.16
0.35 0.30
0.12
TI/SC
TI/SC
0.14
0.10
0.25 0.20
0.08 0.06
0.15
0.04
0.10
0.02
0.05
0.00
0.00 -50
0
50
100
150
200
250
300
350
0
Fig. 4. Experimental data of transfer-ionization-to-single-capture ratios for ion charge q = 1 colliding with He and the theoretical estimate (solid line). The data were obtained by Dubois [10] (n: He+), present data (h: C+; d: O+).
0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0
50
100 150 200 250 300 350 400 450 500 550
Beam energy (keV/amu)
Fig. 5. Experimental data of transfer-ionization-to-single-capture ratios for ion charge q = 2 colliding with He and the theoretical estimate (solid line). The data were obtained by Sanders (10 in [5]) (n: Li2+), present data (j: C2+; s: O2+).
100 keV/amu or 200 keV/amu then it begins to decrease slowly. (2) The values of R bombarded by C, O, He and Li ions, which have the same charge state q, are still different from each other. The reason is, even for the same q, the effective charge states are different for various systems. Ionization and capture cross sections of these systems, which have the same q value, are not exactly the same. So the values of R of these same q systems are different in some degree. It is difficult to estimate the effective charge of various systems in collision. In our model, the influences by the difference of the effective charge states are neglected. Two parameters are introduced: q and E, the charge state and
50 100 150 200 250 300 350 400 450 500 550
Beam energy (keV/amu)
Beam energy (keV/amu)
TI/SC
343
Fig. 6. Experimental data of transfer-ionization-to-single-capture ratios for ion charge q = 3 colliding with He and the theoretical estimate (solid line). The data were obtained by Shah and Gilbody [5] (n: Li3+), present data (j: C3+; s: O3+).
the energy of the ions. The aim of our model is trying to understand a little about the trends of R varying with the q and E. The model well describes the following fact. The fact is, if q = 1, R(DI) and R(TI) increases with the beam energy and reach the maximum when the energy is about 100 keV/amu then decrease slowly; if q = 2 or 3, R(DI) and R(TI) reach the maximum when the energy is about 200 keV/amu. In order to get the conclusion conveniently, we discuss the ionization probability when the collision parameter b = 0. 2 2Rr 1 Rr RC P I ð0Þ ¼ ðRr RC Þ ¼ ; T VP VP T Rr C r we let h ¼ 2R 1 ; g ¼ Rr R , h is the ratio of the duration of VP T Rr 2Rr collision, V P , to the period of the bound electrons, T. g is the un-captured fraction of the released electrons, g ¼ 1 2q 1. Rr V 2P P I ð0Þ ¼ h g ¼ V1P 2q 1 , has the maximum value at a Rr V 3 P
certain VP. This will cause the single-, double- and transferionization cross sections to have the maximum too. So the ratios R also have the maximum value. Figs. 1–6 also indicate that the curves of R(DI) have a shoulder below the maximum, but R(TI) generally does not. For the two-electron system of Helium 2 ðRr1 RC1 Þ; T VP 2 ðRr2 RC2 Þ; P I2 ð0Þ ¼ T VP P I1 ð0Þ reaches the maximum when V P ¼ V 1 ;
P I1 ð0Þ ¼
P I2 ð0Þ reaches the maximum when V P ¼ V 2 ;
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Z1 = 1, Z2 = 2, RC1 = RC2 and Rr1 > Rr2, so the value of V2 must be larger than V1. That is, for the given collision system, rSI reaches the maximum before rDI does, rSI has already decreased when rDI reaches the maximum. This is why the curves of R(DI) have the shoulder below the maximum. Generally, the curves of R(TI) have no shoulder, because PI has the maximum value at a certain VP. PC always decreases with VP. rTI has the maximum value while rSC decreases with VP monotonously, thus the curves of R(TI) only have the maximum value but no shoulder. 4. Conclusion In the energy range from 25 keV/amu to 500 keV/amu, the ratios of double-to-single ionization of helium bombarded by Cq+ and Oq+ (q = 1, 2, 3) ions are measured. A simple model developed from Bohr’s theory of COBM is presented to estimate the ratios. Both the experimental results and the model indicate that the ratios reach the maximum when the energy is about 100 keV/amu or 200 keV/amu.
Acknowledgement We thank Prof. Z.Y. Liu for the helpful discussions. References [1] M. Inokuti, Rev. Mod. Phys. 43 (1971) 297; M. Inokuti, Rev. Mod. Phys. 50 (1978) 23. [2] J.H. McGuire, in: Proceeding of the Second US – Mexico Symposium of Atomic and Molecular Physics: Two Electron Phenomena, Cocyoc Mexico, 1986. [3] J.H. McGuire, L. Weaver, Phys. Rev. A 16 (1977) 41. [4] N. Bohr, J. Lindhard, K. Dan, Vidensk. Selsk. Mat. Fys. Medd. 28 (7) (1954). [5] M.B. Shah, H.B. Gilbody, J. Phys. B 18 (1985) 899; M.B. Shah, H.B. Gilbody, J. Phys. B 15 (1982) 413; M.B. Shah, H.B. Gilbody, J. Phys. B 14 (1981) 2361. [6] R.M. Wood, A.K. Edwards, R.L. Ezell, Phys. Rev. A 34 (1986) 4415. [7] A.K. Edwards, R.M. Wood, R.L. Ezell, Phys. Rev. A 32 (1985) 1346. [8] R.D. Dubois, L.H. Toburen, M.E. Rudd, Phys. Rev. A 29 (1984) 70. [9] R.D. Dubois, Phys. Rev. A 36 (1987) 2585. [10] R.D. Dubois, Phys. Rev. A 39 (1989) 4440. [11] B.W. Ding et al., unpublished. [12] A. Niehaus, J. Phys. B 19 (1986) 2925.
Nuclear Instruments and Methods in Physics Research B 258 (2007) 340–344 www.elsevier.com/locate/nimb
Double-to-single ionization ratios of helium bombarded by low-to-intermediate velocity Cq+, Oq+ (q = 1 3) ions J.X. Shao, X.M. Chen *, B.W. Ding, H.B. Fu, Z.M. Gao, Y.W. Liu, J. Du, Y.X. Lu, Y. Cui Department of Modern Physics, Lanzhou University, Lanzhou 730000, China Received 19 July 2006; received in revised form 16 October 2006 Available online 26 March 2007
Abstract We measured the double-to-single ionization ratios R of Helium bombarded by Cq+ and Oq+ ions (q = 1 3). The velocity of the projectile varies from vBohr to 4 vBohr, the energy varies from 25 keV/amu to 500 keV/amu. The value of R increases rapidly with the collision velocity and reaches the maximum when the velocity is about 2 or 3 vBohr, then it decreases slowly for the higher velocity. A simple model is presented to estimate the value R varying with the collision velocity. The results calculated by the model are in agreement with the experimental data basically. Ó 2006 Published by Elsevier B.V. Keywords: Double ionization; Helium; Carbon ions; Oxygen ions
1. Introduction Ionization is one of the basic processes mediated by Coulomb forces during ion–atom collisions, and this process has been studied both theoretically and experimentally from the beginning of atomic and quantum physics. In high energy range (vion > 10 vBohr), the removal of a single-electron by the projectile is well described in the framework of the first Born approximation [1]. In the framework of first Born approximation, McGuire proposed that double ionization by ions at intermediate to high velocities (vion P 10 vBohr) can be understood in terms of two mechanisms [2,3]: (1) a two-step process (TS), vion 10 vBohr, in which both target electrons are removed in separate direct interactions with the projectile, the value of R increases with the projectile charge q and decreases with collision velocity v as q2/(v2 ln v); (2) a shake-off process (SO), vion 10 vBohr, in which the first electron is removed in a direct interaction with the projectile while the second electron is ejected when the *
Corresponding author. Tel.: +86 931 8913541. E-mail addresses: [email protected] (X.M. Chen), [email protected]. edu.cn (J.X. Shao). 0168-583X/$ - see front matter Ó 2006 Published by Elsevier B.V. doi:10.1016/j.nimb.2006.10.072
resulting ion ‘‘relaxes’’ to a continuum state and the ratio R is expected to be a constant. In the low energy range (vion < vBohr), single removal is understood well in the framework of Bohr’s ClassicalOver-Barrier-Model (COBM). COBM [4] indicates: that electron capture is the dominant process, direct ionization is neglected in this model. The cross section of capture is independent of the collision velocity v, as a result, the double-to-single ionization ratio R is also expected to be a constant. In the low-to-intermediate velocity range (vion P vBohr), many experiments had been made focusing on the ratios R [5–10]. No matter whether the projectile is bare ions or partially bare, the trends of R varying with the collision velocity have a similar character. That is, the value of R increases rapidly with the collision velocity and reaches the maximum when the velocity is about 2 or 3 vBohr then it decreases slowly for the higher velocity [1]. In the present work, Cq+ and Oq+ ions (q = 1 3) with the velocity varying from vBohr to 4 vBohr collide with He, the double-to-single ionization ratios R of Helium are measured using the time-to-flight and coincidence technique, the details of the experimental method are discussed
J.X. Shao et al. / Nucl. Instr. and Meth. in Phys. Res. B 258 (2007) 340–344
elsewhere [11]. When the velocity is about 2 or 3 vBohr, the value of R reaches the maximum, not only in the direct ionization process but also in the charge exchange process. The experimental data and the details of the model will be discussed in Sections 2 and 3. 2. A simple theoretical estimate The model we used to estimate the double-to-single ionization ratios R, is based on Bohr–Lindhard’s COBM [4]. They introduce two important ion–atom interaction distances in COBM. First, they introduced the release distance Rr, where the electron can be released from the target nucleus. The release distance Rr satisfies: q ðRr aÞ2
¼
Z ; a2
ð1Þ
where a is the orbital radius of the electron, q is the charge state of the projectile and Z is the atomic number of the target. Eq. (1) indicates that the electron can be released when the forces on the electron extracted by the projectile and the target atom are in balance. Second, the capture distance RC is introduced. RC is the ion–atom distance where the electron can be captured by the projectile. RC satisfies the following equation: q 1 ¼ V 2; RC 2 P
ð2Þ
where VP is the velocity of the ion. Eqs. (1) and (2) means: it is only that the process of release occurred within the distance of capture the electron can be captured, otherwise, the release is temporary. In the low energy range, RC P Rr, the released electrons are all captured, the capture cross section is given by rC ¼ pR2r :
ð3Þ
For higher energy, RC < Rr. Only the electrons released within the distance of RC can be captured, as a result, the capture cross section rC ¼ pR2C f :
ð4Þ
f is the probability of release process occurring within the . distance of RC Where f ¼ vae VRCP , means the ratio of the duration of collision to the orbital period of the bound electron. ve is the orbital velocity of this electron. In Bohr’s theory of COBM [4], the electrons which are released but not captured will go back to the target atom after collision. The energy level of the released electron had already been estimated by Bohr et al. [4] and Niehaus [12], it is I n þ Rqr , the ionization energy of the target electron added by the stark-energy by the ion. We consider that the released but not captured electrons will be accelerated by the approaching ion. At some dis-
341
tance RI, the electrons will get enough kinetic energy from the ion to escape from the target atom, meanwhile, these electrons will not be captured, they will be ionized. The ionization distance RI satisfies: q q P In þ : ð5Þ RI Rr It means when the stark-energy transferred to the kinetic energy of the electron is larger than the ionization energy of the quasi-molecular states, the electron will escape from the target atom. If the released electron has enough energy to escape from the target atom and can not form the stable bound states of the ion, ionization occurs. According to Bohr’s theory, electrons have not been released during the collision are in the perturbative bound states of the target atom all the time, the kinetic energy of these electrons are almost the initial value, are not large enough to escape, thus, the ionization of these electrons is neglected in our model. That is, in our model, the electrons released within the distance RC will be captured, those released between Rr and RC will not be ionized until the ion entered the distance RI. Before we calculate the cross sections about two-electron system, it is necessary to get the ionization or capture cross section of the single-electron system. Then, those cross sections about the two electrons will be calculated by the independent electron model (IED) easily. Several approximations are made: (1) These projectiles move along the linear trajectories with the collision parameter b and the velocity VP. (2) The orbital period of the electron is T ¼ 2pa , neglectv ing the influence of the ions. For the given collision parameter b and velocity VP, the probability pffiffiffiffiffiffiffiffiffithat electron will be released is 2 R2r b2 1 P r ðbÞ ¼ V P T , the ratio of the duration of collision (duration of the release process) to the period of the bound pffiffiffiffiffiffiffiffiffiffi electron, for b 6 Rr. 2 R2 b2 The capture probability is P C ðbÞ ¼ V CP T1 , the ratio of the duration of the capture process to the period of the bound electron, for b 6 RC. Then the ionization probability is PI(b) = Pr(b) PC(b), for b 6 RI. When we construct the cross sections of twoelectron system with these single-electron probabilities, the only thing is to distinguish these two electrons with subscript 1 and 2, e.g. Rr1, RC1, RI1, the distances for the first electron; Rr2, RC2, RI2, the distances for the second electron. They satisfy the following equations: qi 2
¼
Zi ; a2
ðRri aÞ qi 1 ¼ V 2P ; RCi 2 qi q ¼ Ii þ i RIi Rri
ð6Þ ði ¼ 1; 2Þ;
J.X. Shao et al. / Nucl. Instr. and Meth. in Phys. Res. B 258 (2007) 340–344
where qi is the ion charge state to the ith electron of the target, Z1 = 1, Z2 = 2; I1 = 0.93, I2 = 2 of the helium system. The reason for Z2 5 Z1 is that the first released electron loses the screening ability to both nuclei. The value of qi will not change for the direct ionization process, q2 = q1; one will be subtracted from qi if the first electron is captured, that is, q2 = q1 1. The values of PC1(b), PI1(b) and PC2(b), PI2(b) are calculated from Rr1, RC1, RI1 and Rr2, RC2, RI2. After integrating PC1(b), PI1(b) and PC2(b), PI2(b) for all collision parameter b, we get the cross sections of the two-electron system. The direct single-ionization cross section is Z rSI ¼ 2p P I1 ðbÞ ð1 P I2 ðbÞ P C2 ðbÞÞ b db Z þ 2p P I2 ðbÞ ð1 P I1 ðbÞ P C1 ðbÞÞ b db: The direct double ionization cross section is Z rDI ¼ 2p P I1 ðbÞ P I2 ðbÞ b db:
0.14 0.12 0.10 0.08
DI/SI
342
0.06 0.04 0.02 0.00 0
50
100
150
200
250
300
350
400
Beam energy (keV/amu)
Fig. 2. Experimental data of direct double-to-single ionization ratios for ion charge q = 2 colliding with He and the theoretical estimate (solid line). The data were obtained by Sanders (10 in 5) (m: Li2+), present data (j: C2+; s: O2+).
0.12
The single-capture cross section is Z rSC ¼ 2p P C1 ðbÞ ð1 P I2 ðbÞ P C2 ðbÞÞ b db Z þ 2p P C2 ðbÞ ð1 P I1 ðbÞ P C1 ðbÞÞ b db:
0.10
And the transfer-ionization cross section is Z Z rTI ¼ 2p P I1 ðbÞ P C2 ðbÞ b db þ 2p P I2 ðbÞ P C1 ðbÞ b db: The ratios of direct double-to-single ionization is RðDIÞ ¼ rrDI , the ratios of transfer-ionization-to-single-capSI TI ture is RðTIÞ ¼ rrSC .
DI/SI
0.08
0.06
0.04
0.02
0.00 50
100
150
200
250
300
350
400
450
500
Beam energy (keV/amu)
Fig. 3. Experimental data of direct double-to-single ionization ratios for ion charge q = 3 colliding with He and the theoretical estimate (solid line). The data were obtained by Shah and Gilbody [5] (n: Li3+), present data (j: C3+; s: O3+).
0.10
DI/SI
0.08
The calculation results are plotted in Figs. 1–6 together with the results of our experiment and other experiments in the same energy range. The experimental results in Figs. 1–6 indicate that with varying ion charges, the double-to-single ionization ratio R reaches the maximum when the velocity is about 2 or 3 vBohr.
0.06
0.04
0.02
3. Discussion
0.00 0
100
200
300
400
Beam energy (keV/amu)
Fig. 1. Experimental data of direct double-to-single ionization ratios for ion charge q = 1 colliding with He and the theoretical estimate (solid line). The data were obtained by Dubois [10] (n: He+), present data (j: C+; s: O+).
Figs. 1–6 indicate that (1) The ratios R have the similar trend for the various systems. The trend is, R increases rapidly with the collision velocity and reaches the maximum at
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0.50
0.20
0.45
0.18
0.40
0.16
0.35 0.30
0.12
TI/SC
TI/SC
0.14
0.10
0.25 0.20
0.08 0.06
0.15
0.04
0.10
0.02
0.05
0.00
0.00 -50
0
50
100
150
200
250
300
350
0
Fig. 4. Experimental data of transfer-ionization-to-single-capture ratios for ion charge q = 1 colliding with He and the theoretical estimate (solid line). The data were obtained by Dubois [10] (n: He+), present data (h: C+; d: O+).
0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0
50
100 150 200 250 300 350 400 450 500 550
Beam energy (keV/amu)
Fig. 5. Experimental data of transfer-ionization-to-single-capture ratios for ion charge q = 2 colliding with He and the theoretical estimate (solid line). The data were obtained by Sanders (10 in [5]) (n: Li2+), present data (j: C2+; s: O2+).
100 keV/amu or 200 keV/amu then it begins to decrease slowly. (2) The values of R bombarded by C, O, He and Li ions, which have the same charge state q, are still different from each other. The reason is, even for the same q, the effective charge states are different for various systems. Ionization and capture cross sections of these systems, which have the same q value, are not exactly the same. So the values of R of these same q systems are different in some degree. It is difficult to estimate the effective charge of various systems in collision. In our model, the influences by the difference of the effective charge states are neglected. Two parameters are introduced: q and E, the charge state and
50 100 150 200 250 300 350 400 450 500 550
Beam energy (keV/amu)
Beam energy (keV/amu)
TI/SC
343
Fig. 6. Experimental data of transfer-ionization-to-single-capture ratios for ion charge q = 3 colliding with He and the theoretical estimate (solid line). The data were obtained by Shah and Gilbody [5] (n: Li3+), present data (j: C3+; s: O3+).
the energy of the ions. The aim of our model is trying to understand a little about the trends of R varying with the q and E. The model well describes the following fact. The fact is, if q = 1, R(DI) and R(TI) increases with the beam energy and reach the maximum when the energy is about 100 keV/amu then decrease slowly; if q = 2 or 3, R(DI) and R(TI) reach the maximum when the energy is about 200 keV/amu. In order to get the conclusion conveniently, we discuss the ionization probability when the collision parameter b = 0. 2 2Rr 1 Rr RC P I ð0Þ ¼ ðRr RC Þ ¼ ; T VP VP T Rr C r we let h ¼ 2R 1 ; g ¼ Rr R , h is the ratio of the duration of VP T Rr 2Rr collision, V P , to the period of the bound electrons, T. g is the un-captured fraction of the released electrons, g ¼ 1 2q 1. Rr V 2P P I ð0Þ ¼ h g ¼ V1P 2q 1 , has the maximum value at a Rr V 3 P
certain VP. This will cause the single-, double- and transferionization cross sections to have the maximum too. So the ratios R also have the maximum value. Figs. 1–6 also indicate that the curves of R(DI) have a shoulder below the maximum, but R(TI) generally does not. For the two-electron system of Helium 2 ðRr1 RC1 Þ; T VP 2 ðRr2 RC2 Þ; P I2 ð0Þ ¼ T VP P I1 ð0Þ reaches the maximum when V P ¼ V 1 ;
P I1 ð0Þ ¼
P I2 ð0Þ reaches the maximum when V P ¼ V 2 ;
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Z1 = 1, Z2 = 2, RC1 = RC2 and Rr1 > Rr2, so the value of V2 must be larger than V1. That is, for the given collision system, rSI reaches the maximum before rDI does, rSI has already decreased when rDI reaches the maximum. This is why the curves of R(DI) have the shoulder below the maximum. Generally, the curves of R(TI) have no shoulder, because PI has the maximum value at a certain VP. PC always decreases with VP. rTI has the maximum value while rSC decreases with VP monotonously, thus the curves of R(TI) only have the maximum value but no shoulder. 4. Conclusion In the energy range from 25 keV/amu to 500 keV/amu, the ratios of double-to-single ionization of helium bombarded by Cq+ and Oq+ (q = 1, 2, 3) ions are measured. A simple model developed from Bohr’s theory of COBM is presented to estimate the ratios. Both the experimental results and the model indicate that the ratios reach the maximum when the energy is about 100 keV/amu or 200 keV/amu.
Acknowledgement We thank Prof. Z.Y. Liu for the helpful discussions. References [1] M. Inokuti, Rev. Mod. Phys. 43 (1971) 297; M. Inokuti, Rev. Mod. Phys. 50 (1978) 23. [2] J.H. McGuire, in: Proceeding of the Second US – Mexico Symposium of Atomic and Molecular Physics: Two Electron Phenomena, Cocyoc Mexico, 1986. [3] J.H. McGuire, L. Weaver, Phys. Rev. A 16 (1977) 41. [4] N. Bohr, J. Lindhard, K. Dan, Vidensk. Selsk. Mat. Fys. Medd. 28 (7) (1954). [5] M.B. Shah, H.B. Gilbody, J. Phys. B 18 (1985) 899; M.B. Shah, H.B. Gilbody, J. Phys. B 15 (1982) 413; M.B. Shah, H.B. Gilbody, J. Phys. B 14 (1981) 2361. [6] R.M. Wood, A.K. Edwards, R.L. Ezell, Phys. Rev. A 34 (1986) 4415. [7] A.K. Edwards, R.M. Wood, R.L. Ezell, Phys. Rev. A 32 (1985) 1346. [8] R.D. Dubois, L.H. Toburen, M.E. Rudd, Phys. Rev. A 29 (1984) 70. [9] R.D. Dubois, Phys. Rev. A 36 (1987) 2585. [10] R.D. Dubois, Phys. Rev. A 39 (1989) 4440. [11] B.W. Ding et al., unpublished. [12] A. Niehaus, J. Phys. B 19 (1986) 2925.