- Email: [email protected]
Ionization cross sections of helium and hydrogenic ions by antiproton impact
Nuclear Instruments and Methods in Physics Research B 214 (2004) 135–138 www.elsevier.com/locate/nimb
Ionization cross sections of helium and hydrogenic ions by antiproton impact A. Igarashi *, S. Nakazaki, A. Ohsaki Department of Applied Physics, Faculty of Engineering, Miyazaki University, Miyazaki 889-2192, Japan
Abstract The antiproton-impact cross sections are calculated for hydrogenic ions, hydrogen negative ion, and helium at lower collision energies than reported before. The electronic wave function is expanded in terms of the atomic orbitals centered on the target nucleus in a semiclassical common trajectory method, which includes the trajectory bending effect. The absorption potential is used to circumvent an unphysical reflection of electronic continuum wave function during the collision. At low energies, the absorption potential is effective for the neutral target and the trajectory bending effect is important for the ionized target. Ó 2003 Elsevier B.V. All rights reserved. PACS: 34.50.Fa; 31.15.Gy Keywords: Antiproton; Semiclassical method; Ionization; Excitation
1. Introduction Since antiproton-beam with energies lower than 1 keV is expected to be available in the near future owing to the ASACUSA collaborations at CERN AD, the collisional calculations of low-energy antiproton impact are required. In this work, we adopt the semiclassical treatment, called common trajectory (CT) method [1,2], which includes the effect of trajectory bending in contrast to the usual impact parameter (IP) method with straight-line trajectory. One-center atomic orbital close coupling (AOCC) expansion [3,4] is used to describe the electronic wave function. The cross sections are
*
Corresponding author. E-mail address: [email protected] (A. Igarashi).
calculated for H, D, H-like ions, H and He. Atomic units are used unless otherwise stated.
2. Theory The general theory of the CT method has been developed by Gaussorgues et al. [1]. To explain the present calculation briefly, we consider the colli and hydrogenic ion with nuclear sion between p charge Z as an example. The calculation is performed for each partial and electron wave J . The position vectors of p measured from the nucleus are expressed by R and r, respectively. In the CT method, the radius R is described by a classical equation of motion and the variables other than R are treated in quantum mechanics by a time dependent Shr€ odinger equation as
0168-583X/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0168-583X(03)01767-1
136
A. Igarashi et al. / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 135–138
o J ½J þ 1 € lR ¼ þ V0 ðRÞ ; oR 2lR2 o L2 ZT 1 i Wðr; RÞ ¼ HA ðrÞ þ R 2 þ ot 2lR R jR rj J ½J þ 1 V0 ðRÞ Wðr; RÞ: 2lR2
ð1Þ
Here, l is reduced mass, HA is the Hamiltonian of hydrogenic ion, and the potential V0 determines the trajectory. In the one-center AOCC expansion, the wave function is expanded P in terms of atomic basis fWi g b Þ, where i deas Wðr; R; tÞ ¼ i ai ðtÞeii t Wi ðr; R notes the atomic energy of channel i. The present one-center AOCC expansion describes the electronic wave function in the finite space around the target nucleus. A part of continuum wave function may reach the edge during collision, and may reflect inward. To circumvent such an unphysical reflection, a linear-ramp type absorption potential [5] is included in the Shr€ odinger equation (1). By substituting the wave function into the Shr€ odinger equation, we have coupled equations with respect to the expansion coefficients. The equation of motion for R and the coupled equations for fai g are solved under the initial condipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tions: Rðt ¼ 0Þ ¼ R0 , R_ ¼ k02 2lV0 ðR0 Þ=l, and ai ðt ¼ 0Þ ¼ di;0 , where R0 is a sufficiently large distance, k0 is the momentum of the initial channel 0. The partial ionization cross section is calculated by rJion ¼ kp2 ð2J þ 1Þð1 PB Þ, where PB is the sum 0
of the probabilities remaining in bound states.
3. Results þ H collision, many reliable calculaFor the p -impact energies ðEL Þ tions are available for the p above 1 keV [6], and they agree to within 10% level. We applied the present CT method for energies down to 0.1 keV. Fig. 1 shows the cross sections in three types of calculations: (a) the CT method with the absorption potential, (b) the CT method without the absorption potential and (c) the IP method with the absorption potential. The static potential is used in the CT method for the
Cross section (10-16 cm2)
1.5 Ion
1.0
n=2
0.5
n=3 0.0 10
-1
10
0
10
1
10
2
E L(keV) Fig. 1. The ionization cross section and excitation cross sec þ H (1s) tions into the n ¼ 2 and n ¼ 3 manifolds in the p collision. The CT method with absorption potential (solid line); the CT method without absorption potential (dashed line); the IP method with absorption potential (dotted line).
potential V0 in Eq. (1). The trajectory effect is included in the calculations (a) and (b). The absorption potential is included in the calculations (a) and (c). Good agreement is seen among the three calculations for the excitation cross sections into the n ¼ 2 and n ¼ 3 manifolds. The ionization cross section of (b) is smaller than those of (a) and (c) below 50 keV and decreases more rapidly below 1 keV. Therefore the absorption potential is very important below 1 keV in the present calculation. þ D using the The calculations are also done for p static or adiabatic potential as V0 (not shown in figure). These ionization cross sections for H and D are almost the same above 5 keV, and the differences appear for lower energies. Since the adiabatic potential is more attractive than the static potential, the ionization cross section using the adiabatic potential is larger than that of the static potential for the H or D target. When the same (static or adiabatic) potential is used, the ionization cross section of H is slightly larger than that of D, but the difference is not so significant, about 5% at EL ¼ 0:1 keV. The present ionization cross þH section obtained for EL P 0:1 keV in the p collision agrees well with that of the semiclassical calculation by Sakimoto [2] and agrees fairly well with that of the CTMC calculation by Shultz et al. [7].
A. Igarashi et al. / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 135–138
Cross section (10-16 cm2)
200
150
100
50
0 10
-2
10
-1
10
0
10
1
10
2
10
3
E L (keV) Fig. 3. Single electron detachment cross section of H ions by p impact. Present results within one-electron model in the CT method: including the Coulomb repulsion (dashed line); excluding the Coulomb repulsion (dotted line). Previous results in the IP method [4]: two-electron calculation (solid line); oneelectron model (triangle); Born approximation (plus).
We have reported the cross sections for the þ He collision at EL P 1 keV using the AOCC p expansion based on the IP method [4]. The similar 10
1
Ion 1
10
2 P
0
2
cm )
For the target of hydrogenic ion, the ionization is not the most dominant process at low energies, which differs from the case of H target. The excitation cross sections into the n ¼ 2 manifold exceeds the ionization cross section in the low-energy region. It does at Esc ¼ EL =Z 2 ¼ 0:5 keV in the Heþ target, where Z is the nuclear charge of hydrogenic ion. This occurs at larger Esc for hydrogenic ion with larger Z. As Z increases, the ionization is less probable and the effect of the absorption potential becomes less important, while the trajectory effect becomes more important to the contrary. In Fig. 2, the ionization cross sections are displayed for targets: H, Heþ , Be3þ and Ne9þ . For the collision between p and hydrogenic ion, the relative strength of interaction becomes weak in proportional to 1=Z [8]. Hence the obtained cross sections become closer to the Born cross section with Z increasing. The present cross sections are generally in good agreement with those of Tong et al. [8]. For the single electron detachment of H , the one-electron treatment works fairly well [4]. In the present work, the Coulomb repulsion between p and H is taken into account by use of the CT method within the one-electron calculation. The present results are shown in Fig. 3 with the previous results. It is seen that the Coulomb repulsion reduces the cross section significantly below 0.1 from approaching H . keV by preventing p
137
1
Cross section (10
-17
3 P
Z=1 1
1
10
2 S
-1
1
3 S
10
-2
4
Z × σ (10-16 cm2 )
2
Z=4
1
3 D
Z=2 10
Z=10 0 -1 10
10
0
10
1
10
2
10
3
2
E L / Z (keV) Fig. 2. Z-dependence of ionization cross section in the þ AðZ1Þþ (1s) collisions. AðZ1Þþ denotes hydrogenic ion with p nuclear charge Z. Present results in CT method with absorption potential (solid line); Born approximation (dashed line).
-3
10
-1
10
0
10
1
10
2
10
3
E L (keV) Fig. 4. The single ionization cross section and excitation cross þ He (11 S) collision. Present results in the CT sections in the p method: including absorption potential (solid line); excluding absorption potential (dashed line). Previous results in the IP method [4]; (dot–dashed line).
138
A. Igarashi et al. / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 135–138
AOCC calculation in the CT method is applied to estimate the cross section for lower energies. The atomic wave function of He is expanded in terms of {1s,2s,2p} of Heþ [4]. The static potential is used for V0 . The cross sections in the present calculation are shown in Fig. 4 with the previous result in the similar basis set. The cross sections in the present and previous works agree well around 1 keV. The calculations with and without the absorption potential give similar excitation cross sections. The single ionization cross section without the absorption potential is smaller than that with the absorption potential below 1 keV. These situations are similar to the case of H target. The present ionization cross section agrees well with that of Kirchner et al. [9] for energies above 1 keV. As a summary, good agreement among calcu þ H collision lations has been obtained for the p even for energies below 1 keV [10]. More calculations are required for lower collision energies and
for other systems in accordance with the experimental progress.
References [1] C. Gaussorgues et al., J. Phys. B 8 (1975) 239. [2] K. Sakimoto, J. Phys. B 33 (2000) 3149; J. Phys. B 33 (2000) 5165; J. Phys. B 34 (2001) 1769. [3] A. Igarashi et al., Phys. Rev. A 61 (2000) 062712. [4] A. Igarashi et al., Phys. Rev. A 62 (2000) 052722; Phys. Rev. A 64 (2001) 042717. [5] N.D. Balakrishnan et al., Phys. Rep. 280 (1997) 7. [6] X.M. Tong et al., Phys. Rev. A 64 (2001) 022711; N. Toshima, Phys. Rev. A 64 (2001) 024701; J. Azuma et al., Phys. Rev. A 64 (2001) 062704, and references therein. [7] D.R. Schultz et al., Phys. Rev. Lett. 76 (1996) 2882. [8] X.M. Tong et al., Phys. Rev. A 66 (2002) 032708. [9] T. Kirchner et al., J. Phys. B 35 (2002) 925. [10] K. Sakimoto, Phys. Rev. A 65 (2001) 012706.
Ionization cross sections of helium and hydrogenic ions by antiproton impact A. Igarashi *, S. Nakazaki, A. Ohsaki Department of Applied Physics, Faculty of Engineering, Miyazaki University, Miyazaki 889-2192, Japan
Abstract The antiproton-impact cross sections are calculated for hydrogenic ions, hydrogen negative ion, and helium at lower collision energies than reported before. The electronic wave function is expanded in terms of the atomic orbitals centered on the target nucleus in a semiclassical common trajectory method, which includes the trajectory bending effect. The absorption potential is used to circumvent an unphysical reflection of electronic continuum wave function during the collision. At low energies, the absorption potential is effective for the neutral target and the trajectory bending effect is important for the ionized target. Ó 2003 Elsevier B.V. All rights reserved. PACS: 34.50.Fa; 31.15.Gy Keywords: Antiproton; Semiclassical method; Ionization; Excitation
1. Introduction Since antiproton-beam with energies lower than 1 keV is expected to be available in the near future owing to the ASACUSA collaborations at CERN AD, the collisional calculations of low-energy antiproton impact are required. In this work, we adopt the semiclassical treatment, called common trajectory (CT) method [1,2], which includes the effect of trajectory bending in contrast to the usual impact parameter (IP) method with straight-line trajectory. One-center atomic orbital close coupling (AOCC) expansion [3,4] is used to describe the electronic wave function. The cross sections are
*
Corresponding author. E-mail address: [email protected] (A. Igarashi).
calculated for H, D, H-like ions, H and He. Atomic units are used unless otherwise stated.
2. Theory The general theory of the CT method has been developed by Gaussorgues et al. [1]. To explain the present calculation briefly, we consider the colli and hydrogenic ion with nuclear sion between p charge Z as an example. The calculation is performed for each partial and electron wave J . The position vectors of p measured from the nucleus are expressed by R and r, respectively. In the CT method, the radius R is described by a classical equation of motion and the variables other than R are treated in quantum mechanics by a time dependent Shr€ odinger equation as
0168-583X/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0168-583X(03)01767-1
136
A. Igarashi et al. / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 135–138
o J ½J þ 1 € lR ¼ þ V0 ðRÞ ; oR 2lR2 o L2 ZT 1 i Wðr; RÞ ¼ HA ðrÞ þ R 2 þ ot 2lR R jR rj J ½J þ 1 V0 ðRÞ Wðr; RÞ: 2lR2
ð1Þ
Here, l is reduced mass, HA is the Hamiltonian of hydrogenic ion, and the potential V0 determines the trajectory. In the one-center AOCC expansion, the wave function is expanded P in terms of atomic basis fWi g b Þ, where i deas Wðr; R; tÞ ¼ i ai ðtÞeii t Wi ðr; R notes the atomic energy of channel i. The present one-center AOCC expansion describes the electronic wave function in the finite space around the target nucleus. A part of continuum wave function may reach the edge during collision, and may reflect inward. To circumvent such an unphysical reflection, a linear-ramp type absorption potential [5] is included in the Shr€ odinger equation (1). By substituting the wave function into the Shr€ odinger equation, we have coupled equations with respect to the expansion coefficients. The equation of motion for R and the coupled equations for fai g are solved under the initial condipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tions: Rðt ¼ 0Þ ¼ R0 , R_ ¼ k02 2lV0 ðR0 Þ=l, and ai ðt ¼ 0Þ ¼ di;0 , where R0 is a sufficiently large distance, k0 is the momentum of the initial channel 0. The partial ionization cross section is calculated by rJion ¼ kp2 ð2J þ 1Þð1 PB Þ, where PB is the sum 0
of the probabilities remaining in bound states.
3. Results þ H collision, many reliable calculaFor the p -impact energies ðEL Þ tions are available for the p above 1 keV [6], and they agree to within 10% level. We applied the present CT method for energies down to 0.1 keV. Fig. 1 shows the cross sections in three types of calculations: (a) the CT method with the absorption potential, (b) the CT method without the absorption potential and (c) the IP method with the absorption potential. The static potential is used in the CT method for the
Cross section (10-16 cm2)
1.5 Ion
1.0
n=2
0.5
n=3 0.0 10
-1
10
0
10
1
10
2
E L(keV) Fig. 1. The ionization cross section and excitation cross sec þ H (1s) tions into the n ¼ 2 and n ¼ 3 manifolds in the p collision. The CT method with absorption potential (solid line); the CT method without absorption potential (dashed line); the IP method with absorption potential (dotted line).
potential V0 in Eq. (1). The trajectory effect is included in the calculations (a) and (b). The absorption potential is included in the calculations (a) and (c). Good agreement is seen among the three calculations for the excitation cross sections into the n ¼ 2 and n ¼ 3 manifolds. The ionization cross section of (b) is smaller than those of (a) and (c) below 50 keV and decreases more rapidly below 1 keV. Therefore the absorption potential is very important below 1 keV in the present calculation. þ D using the The calculations are also done for p static or adiabatic potential as V0 (not shown in figure). These ionization cross sections for H and D are almost the same above 5 keV, and the differences appear for lower energies. Since the adiabatic potential is more attractive than the static potential, the ionization cross section using the adiabatic potential is larger than that of the static potential for the H or D target. When the same (static or adiabatic) potential is used, the ionization cross section of H is slightly larger than that of D, but the difference is not so significant, about 5% at EL ¼ 0:1 keV. The present ionization cross þH section obtained for EL P 0:1 keV in the p collision agrees well with that of the semiclassical calculation by Sakimoto [2] and agrees fairly well with that of the CTMC calculation by Shultz et al. [7].
A. Igarashi et al. / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 135–138
Cross section (10-16 cm2)
200
150
100
50
0 10
-2
10
-1
10
0
10
1
10
2
10
3
E L (keV) Fig. 3. Single electron detachment cross section of H ions by p impact. Present results within one-electron model in the CT method: including the Coulomb repulsion (dashed line); excluding the Coulomb repulsion (dotted line). Previous results in the IP method [4]: two-electron calculation (solid line); oneelectron model (triangle); Born approximation (plus).
We have reported the cross sections for the þ He collision at EL P 1 keV using the AOCC p expansion based on the IP method [4]. The similar 10
1
Ion 1
10
2 P
0
2
cm )
For the target of hydrogenic ion, the ionization is not the most dominant process at low energies, which differs from the case of H target. The excitation cross sections into the n ¼ 2 manifold exceeds the ionization cross section in the low-energy region. It does at Esc ¼ EL =Z 2 ¼ 0:5 keV in the Heþ target, where Z is the nuclear charge of hydrogenic ion. This occurs at larger Esc for hydrogenic ion with larger Z. As Z increases, the ionization is less probable and the effect of the absorption potential becomes less important, while the trajectory effect becomes more important to the contrary. In Fig. 2, the ionization cross sections are displayed for targets: H, Heþ , Be3þ and Ne9þ . For the collision between p and hydrogenic ion, the relative strength of interaction becomes weak in proportional to 1=Z [8]. Hence the obtained cross sections become closer to the Born cross section with Z increasing. The present cross sections are generally in good agreement with those of Tong et al. [8]. For the single electron detachment of H , the one-electron treatment works fairly well [4]. In the present work, the Coulomb repulsion between p and H is taken into account by use of the CT method within the one-electron calculation. The present results are shown in Fig. 3 with the previous results. It is seen that the Coulomb repulsion reduces the cross section significantly below 0.1 from approaching H . keV by preventing p
137
1
Cross section (10
-17
3 P
Z=1 1
1
10
2 S
-1
1
3 S
10
-2
4
Z × σ (10-16 cm2 )
2
Z=4
1
3 D
Z=2 10
Z=10 0 -1 10
10
0
10
1
10
2
10
3
2
E L / Z (keV) Fig. 2. Z-dependence of ionization cross section in the þ AðZ1Þþ (1s) collisions. AðZ1Þþ denotes hydrogenic ion with p nuclear charge Z. Present results in CT method with absorption potential (solid line); Born approximation (dashed line).
-3
10
-1
10
0
10
1
10
2
10
3
E L (keV) Fig. 4. The single ionization cross section and excitation cross þ He (11 S) collision. Present results in the CT sections in the p method: including absorption potential (solid line); excluding absorption potential (dashed line). Previous results in the IP method [4]; (dot–dashed line).
138
A. Igarashi et al. / Nucl. Instr. and Meth. in Phys. Res. B 214 (2004) 135–138
AOCC calculation in the CT method is applied to estimate the cross section for lower energies. The atomic wave function of He is expanded in terms of {1s,2s,2p} of Heþ [4]. The static potential is used for V0 . The cross sections in the present calculation are shown in Fig. 4 with the previous result in the similar basis set. The cross sections in the present and previous works agree well around 1 keV. The calculations with and without the absorption potential give similar excitation cross sections. The single ionization cross section without the absorption potential is smaller than that with the absorption potential below 1 keV. These situations are similar to the case of H target. The present ionization cross section agrees well with that of Kirchner et al. [9] for energies above 1 keV. As a summary, good agreement among calcu þ H collision lations has been obtained for the p even for energies below 1 keV [10]. More calculations are required for lower collision energies and
for other systems in accordance with the experimental progress.
References [1] C. Gaussorgues et al., J. Phys. B 8 (1975) 239. [2] K. Sakimoto, J. Phys. B 33 (2000) 3149; J. Phys. B 33 (2000) 5165; J. Phys. B 34 (2001) 1769. [3] A. Igarashi et al., Phys. Rev. A 61 (2000) 062712. [4] A. Igarashi et al., Phys. Rev. A 62 (2000) 052722; Phys. Rev. A 64 (2001) 042717. [5] N.D. Balakrishnan et al., Phys. Rep. 280 (1997) 7. [6] X.M. Tong et al., Phys. Rev. A 64 (2001) 022711; N. Toshima, Phys. Rev. A 64 (2001) 024701; J. Azuma et al., Phys. Rev. A 64 (2001) 062704, and references therein. [7] D.R. Schultz et al., Phys. Rev. Lett. 76 (1996) 2882. [8] X.M. Tong et al., Phys. Rev. A 66 (2002) 032708. [9] T. Kirchner et al., J. Phys. B 35 (2002) 925. [10] K. Sakimoto, Phys. Rev. A 65 (2001) 012706.