- Email: [email protected]
Positron scattering from neon
Nuclear Instruments and Methods in Physics Research B 143 (1998) 37±40
Positron scattering from neon L.A. Parcell
a,*
, R.P. McEachran b, A.D. Stauer
b
a
b
Computing Department, School of MPCE, Macquarie University, NSW 2109, Australia Department of Physics and Astronomy, York University, Toronto, Ontario, Canada M3J 1P3 Received 31 October 1997
Abstract The cross sections for the excitation of the ®rst three 2p5 ns 1 P states of neon by positron impact have been determined within a distorted-wave framework for impact energies ranging from threshold to 200 eV. In addition, we have extended our previous calculations for the elastic cross section up to 200 eV. We analyze the relative contributions of these cross sections as well as the ionization cross section and compare our results with experimental measurements of the total cross section and the ionization cross section. Our results are generally in satisfactory agreement with experiment. Ó 1998 Elsevier Science B.V. All rights reserved. PACS: 34.90.+q
1. Introduction Although experimental techniques for the measurement of cross sections for positron scattering from atoms have improved greatly over the past few years, there are few experimental results for excitation of atoms by positron impact. The time-of-¯ight experiments (Refs. [1±3]) are the only experimental results for excitation of the rare gas atoms. In Refs. [4,5] we calculated cross sections for the excitation of helium by positron impact. We now present similar results for the excitation of the 2p5 ns 1 P, n 3; 4 and 5 states for neon in the energy range from 10 to 200 eV. We also ex* Corresponding author. Tel.: 61 2 9850 9577; fax: 61 2 9850 9551; e-mail: [email protected].
tend up to 200 eV our previous determination (Ref. [6]) of the elastic cross section by the polarized-orbital method. There are, however, no experimental results for excitation with which to compare beyond those referred to above although there are measurements of the total cross section up to 700 eV (Refs. [7,8]) and of ionization cross section up to 970 eV (Ref. [9]). In Section 2 we brie¯y outline our theoretical methods while in Section 3 we compare our results with the experimental data. Finally, in Section 4 we present our conclusions.
2. Theory Numerical Hartree±Fock wavefunctions were used for all the calculations presented here. The
0168-583X/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 7 ) 0 0 9 8 4 - 1
38
L.A. Parcell et al. / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 37±40
ground state Ne 2p6 1 S wavefunction was obtained by a regular fully varied Hartree±Fock procedure. However, the excited Ne 2p5 ns 1 Po states were obtained in a semi frozen-core manner, in which the 1s and 2s orbitals were held ®xed at their ground state values while the 2p and ns orbitals were varied. 2.1. Elastic scattering In Ref. [6] the elastic phase shifts for neon were obtained using the polarized-orbital method over the energy range from 0 to 45 eV. We now extend these calculations up to 200 eV. The basic procedure used here is identical to that which we employed previously and we shall therefore present only a brief outline of the theory (for details see Refs. [6,10]). In the polarized-orbital method the scattering phases shifts are determined from the solution of the dierential equation 2 d l l 1 2 ÿ 2V0 r V1 r k ulk r 0 dr2 r2 1 subject to the boundary conditions ulk 0 0
and
lp ulk rr!1 ! Al sin kr dl k : 2
2
Here the dl k are the partial-wave phase shifts and k is the magnitude of the momentum of the incident positron. Furthermore, V0 r is the usual static potential while V1 r is the polarization potential as determined by the polarized-orbital method (see Ref. [10]). This polarization potential can be expressed as X V1m r; 3 V1 r m
where asymptotically am V1m rr!1 ! 2m2 4 2r with the am being the multipole polarizabilities of the atom. For neon, all the atomic orbitals were polarized in the determination of the various contributions to V1m r and the m values from 1 to 14
were included in the above expansion. As with the other noble gases, we estimated the higher multipole contributions V1m r to the polarization potential by the approximate form v1 r m P 15; 5 mp where the function v1 r and the value of the parameter p were found from the calculated values of V1m r for m 13 and 14 (see Ref. [6] for details). As for all the other noble gases, the monopole contribution m 0 to the polarization potential was not included. V1m r
2.2. Excitation The optically allowed transitions from the 2p6 1 S ground state of Ne to the 2p5 ns 1 Po states give rise to the most important contributions to the inelastic excitation cross section. The cross sections for the excitation of Ne to these 3s, 4s and 5s excited states were calculated using a distortedwave approximation. From the magnitudes of these cross sections it is apparent that the excitations to higher states in this series contribute very little to the overall excitation cross section. The cross sections for excitation of optically forbidden states were not calculated since their magnitude would be substantially less than that for the ns 1 Po states. In general, the angle integrated cross section for excitation from the ground state of Ne to an excit1 ed 2p5 nl L state is the sum over the magnetic quantum number M of the cross section rM for excitation to the dierent M levels, where Z 2 M 2 j: 6 rM 4p kb =ka dXkb jtba Here ka and kb are the momenta of the positron in the ground and excited state channels respectively. The T matrix element is expressed as + * Z X M dr0 vÿ ÿ 1=ri0 Wa ; tba b r0 va r0 Wb i 7 vÿ b r0
v a r0
and are the distorted waves for where the two channels normalized to unit ¯ux, Wa and
L.A. Parcell et al. / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 37±40
Wb are the ground and excited state wave functions for the atom and ri0 is the distance from the positron to the ith electron (Ref. [11]). The partial-wave contributions to the distorted waves were calculated in the ®eld of the static and polarization potentials in each channel by solving Eq. (1). The static potentials were derived from the corresponding Hartree±Fock atomic wave functions. For the ground state the polarization potential given by Eq. (3) was used, so that the partial-wave functions were the same as used for elastic scattering. The polarization potentials for each of the excited states was calculated by using an extension to Stone's method (Ref. [12]). Converged values of the T -matrix were determined by summing typically 20±100 partial waves and then using a Born subtraction technique to estimate the remainder. 3. Results In Fig. 1 we present our individual cross sections for positron excitation to the 2p5 ns 1 P, n 3; 4 and 5 states of Ne together with the experimental data of Refs. [1,2]. Clearly the n 3
Fig. 1. Excitation cross sections for the 2p5 ns 1 P states of neon. Theoretical results: ,±±±, present cross section for n 3; ± ± , for n 4; - - - - -, for n 5. Experimental data: , Ref. [1]; , Ref. [2], both for excitation to all bound levels.
39
cross section dominates the excitation cross section and the contributions from states with n > 5 are relatively insigni®cant. The experimental data of Refs. [1,2] for excitation to all bound levels lie above our results and rise much more rapidly from the excitation threshold than that predicted by our theoretical results. Nonetheless, the overall magnitude of the experimental data is quite comparable to that predicted by our theory near the maximum of our n 3 cross section. In Fig. 2 we present our elastic and total excitation cross sections (the sum of the n 3, 4 and 5 contributions), together with our previous determination of the ionization cross section (Ref. [13]) and the sum of these three separate cross sections in the energy range from 1 to 200 eV. To this sum we have added the smoothed experimental data of Ref. [14] for positronium formation to allow a comparison with the experimental total cross section. Also given are the experimental total cross sections of Refs. [7,8] and the ionization cross section of Ref. [9]. Our elastic cross section is in good agreement with, although slightly smaller than, the experimental total cross of Ref. [7] below the pos-
Fig. 2. Cross sections: ±±± ±±±, present excitation cross section; ±±± ±±± ±±±, present elastic cross section; ± ± ±, ionization cross section of Ref. [13]; ± ±, present total cross section minus positronium formation; - - - -, total cross section including positronium cross section of Ref. [14]. Experimental data: , total cross section of Refs. [7,8], , ionization cross section of Ref. [9].
40
L.A. Parcell et al. / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 37±40
itronium threshold (14.759 eV) where elastic scattering is the only channel which is open. At approximately 16.76 eV the excitation channels also become accessible while at 21.559 eV ionization is also possible. In the energy region between the positronium threshold and 21 eV the dominant scattering processes will be elastic scattering and positronium formation with excitation playing only a minor role. Clearly, the experimental results indicate the importance of positronium formation in this energy region. There is quite good agreement between the experimental ionization cross section of Ref. [9] and the theoretical predictions of Ref. [13]. Our results show that the excitation cross sections for Ne appear to be much less important than the ionization channel. This is the reverse of the situation in Mg (Ref. [15]) where the excitation channels contribute far more to the total cross section than ionization. 4. Conclusions We have investigated the various contributions to the total cross section for the scattering of positrons from Ne up to 200 eV. Below the positronium formation threshold where only elastic scattering is possible our results are in satisfactory agreement with experiment. The experimental results indicate that positronium formation is very important immediately above threshold. However, the energy range over which positronium formation yields a signi®cant contribution to the total cross section is as yet unknown and more experimental data over a wide energy range is needed. The excitation process in Ne would appear to be not nearly as signi®cant as the ionization channel in the determination of the total cross section. Further theoretical and experimental investigations, particularly in the energy range below 50 eV would be very desirable.
Acknowledgements We wish to express our appreciation to Dr. G. Laricchia for sending us their experimental ionization data prior to publication. Two of us (RPM and ADS) wish to thank the National Sciences and Engineering Research Council of Canada for ®nancial assistance. References [1] P.G. Coleman, J.T. Hutton, D.R. Cook, C.A. Chandler, Can. J. Phys. 60 (1982) 584. [2] S. Mori, O. Sueoka, J. Phys. 27 (1994) 4349. [3] O. Sueoka, S. Mori, J. Phys. 27 (1994) 5038. [4] L.A. Parcell, R.P. McEachran, A.D. Stauer, J. Phys. 16 (1983) 4249. [5] L.A. Parcell, R.P. McEachran, A.D. Stauer, J. Phys. 20 (1987) 2307. [6] R.P. McEachran, A.G. Ryman, A.D. Stauer, J. Phys. 11 (1978) 551. [7] T.S. Stein, W.E. Kauppila, V. Pol, J.H. Smart, G. Jesion, Phys. Rev. 17 (1978) 1600. [8] W.E. Kauppila, T.S. Stein, J.H. Smart, M.S. Dababneh, Y.K. Ho, J.P. Downing, V. Pol, Phys. Rev. A 24 (1981) 725. [9] V. Kara, K. Paludan, J. Moxom, P. Ashley, G. Laricchia, J. Phys. B 30 (1997) in press. [10] R.P. McEachran, D.L. Morgan, A.G. Ryman, A.D. Stauer, J. Phys. B 10 (1977) 663. [11] J.R. Taylor, Scattering Theory, Wiley, New York, 1972. [12] R.P. McEachran, L.A. Parcell, A.D. Stauer, J. Phys. B 28 (1995) 2487. [13] R.I. Campeanu, R.P. McEachran, A.D. Stauer, Can. J. Phys. 74 (1996) 544. [14] L.M. Diana, S.C. Sharma, L.S. Fornari, P.G. Coleman, P.K. Pendleton, D.L. Brooks, B.E. Seay, in: P.C. Jain, R.M. Singru, K.P. Gopinathan (Eds.), Seventh International Conference on Positron Annihilation, World Scienti®c, Singapore, 1985, p. 428. [15] R.I. Campeanu, R.P. McEachran, L.A. Parcell, A.D. Stauer, this issue, Nucl. Instr. and Meth. B 143 (1998) 21.
Positron scattering from neon L.A. Parcell
a,*
, R.P. McEachran b, A.D. Stauer
b
a
b
Computing Department, School of MPCE, Macquarie University, NSW 2109, Australia Department of Physics and Astronomy, York University, Toronto, Ontario, Canada M3J 1P3 Received 31 October 1997
Abstract The cross sections for the excitation of the ®rst three 2p5 ns 1 P states of neon by positron impact have been determined within a distorted-wave framework for impact energies ranging from threshold to 200 eV. In addition, we have extended our previous calculations for the elastic cross section up to 200 eV. We analyze the relative contributions of these cross sections as well as the ionization cross section and compare our results with experimental measurements of the total cross section and the ionization cross section. Our results are generally in satisfactory agreement with experiment. Ó 1998 Elsevier Science B.V. All rights reserved. PACS: 34.90.+q
1. Introduction Although experimental techniques for the measurement of cross sections for positron scattering from atoms have improved greatly over the past few years, there are few experimental results for excitation of atoms by positron impact. The time-of-¯ight experiments (Refs. [1±3]) are the only experimental results for excitation of the rare gas atoms. In Refs. [4,5] we calculated cross sections for the excitation of helium by positron impact. We now present similar results for the excitation of the 2p5 ns 1 P, n 3; 4 and 5 states for neon in the energy range from 10 to 200 eV. We also ex* Corresponding author. Tel.: 61 2 9850 9577; fax: 61 2 9850 9551; e-mail: [email protected].
tend up to 200 eV our previous determination (Ref. [6]) of the elastic cross section by the polarized-orbital method. There are, however, no experimental results for excitation with which to compare beyond those referred to above although there are measurements of the total cross section up to 700 eV (Refs. [7,8]) and of ionization cross section up to 970 eV (Ref. [9]). In Section 2 we brie¯y outline our theoretical methods while in Section 3 we compare our results with the experimental data. Finally, in Section 4 we present our conclusions.
2. Theory Numerical Hartree±Fock wavefunctions were used for all the calculations presented here. The
0168-583X/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 7 ) 0 0 9 8 4 - 1
38
L.A. Parcell et al. / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 37±40
ground state Ne 2p6 1 S wavefunction was obtained by a regular fully varied Hartree±Fock procedure. However, the excited Ne 2p5 ns 1 Po states were obtained in a semi frozen-core manner, in which the 1s and 2s orbitals were held ®xed at their ground state values while the 2p and ns orbitals were varied. 2.1. Elastic scattering In Ref. [6] the elastic phase shifts for neon were obtained using the polarized-orbital method over the energy range from 0 to 45 eV. We now extend these calculations up to 200 eV. The basic procedure used here is identical to that which we employed previously and we shall therefore present only a brief outline of the theory (for details see Refs. [6,10]). In the polarized-orbital method the scattering phases shifts are determined from the solution of the dierential equation 2 d l l 1 2 ÿ 2V0 r V1 r k ulk r 0 dr2 r2 1 subject to the boundary conditions ulk 0 0
and
lp ulk rr!1 ! Al sin kr dl k : 2
2
Here the dl k are the partial-wave phase shifts and k is the magnitude of the momentum of the incident positron. Furthermore, V0 r is the usual static potential while V1 r is the polarization potential as determined by the polarized-orbital method (see Ref. [10]). This polarization potential can be expressed as X V1m r; 3 V1 r m
where asymptotically am V1m rr!1 ! 2m2 4 2r with the am being the multipole polarizabilities of the atom. For neon, all the atomic orbitals were polarized in the determination of the various contributions to V1m r and the m values from 1 to 14
were included in the above expansion. As with the other noble gases, we estimated the higher multipole contributions V1m r to the polarization potential by the approximate form v1 r m P 15; 5 mp where the function v1 r and the value of the parameter p were found from the calculated values of V1m r for m 13 and 14 (see Ref. [6] for details). As for all the other noble gases, the monopole contribution m 0 to the polarization potential was not included. V1m r
2.2. Excitation The optically allowed transitions from the 2p6 1 S ground state of Ne to the 2p5 ns 1 Po states give rise to the most important contributions to the inelastic excitation cross section. The cross sections for the excitation of Ne to these 3s, 4s and 5s excited states were calculated using a distortedwave approximation. From the magnitudes of these cross sections it is apparent that the excitations to higher states in this series contribute very little to the overall excitation cross section. The cross sections for excitation of optically forbidden states were not calculated since their magnitude would be substantially less than that for the ns 1 Po states. In general, the angle integrated cross section for excitation from the ground state of Ne to an excit1 ed 2p5 nl L state is the sum over the magnetic quantum number M of the cross section rM for excitation to the dierent M levels, where Z 2 M 2 j: 6 rM 4p kb =ka dXkb jtba Here ka and kb are the momenta of the positron in the ground and excited state channels respectively. The T matrix element is expressed as + * Z X M dr0 vÿ ÿ 1=ri0 Wa ; tba b r0 va r0 Wb i 7 vÿ b r0
v a r0
and are the distorted waves for where the two channels normalized to unit ¯ux, Wa and
L.A. Parcell et al. / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 37±40
Wb are the ground and excited state wave functions for the atom and ri0 is the distance from the positron to the ith electron (Ref. [11]). The partial-wave contributions to the distorted waves were calculated in the ®eld of the static and polarization potentials in each channel by solving Eq. (1). The static potentials were derived from the corresponding Hartree±Fock atomic wave functions. For the ground state the polarization potential given by Eq. (3) was used, so that the partial-wave functions were the same as used for elastic scattering. The polarization potentials for each of the excited states was calculated by using an extension to Stone's method (Ref. [12]). Converged values of the T -matrix were determined by summing typically 20±100 partial waves and then using a Born subtraction technique to estimate the remainder. 3. Results In Fig. 1 we present our individual cross sections for positron excitation to the 2p5 ns 1 P, n 3; 4 and 5 states of Ne together with the experimental data of Refs. [1,2]. Clearly the n 3
Fig. 1. Excitation cross sections for the 2p5 ns 1 P states of neon. Theoretical results: ,±±±, present cross section for n 3; ± ± , for n 4; - - - - -, for n 5. Experimental data: , Ref. [1]; , Ref. [2], both for excitation to all bound levels.
39
cross section dominates the excitation cross section and the contributions from states with n > 5 are relatively insigni®cant. The experimental data of Refs. [1,2] for excitation to all bound levels lie above our results and rise much more rapidly from the excitation threshold than that predicted by our theoretical results. Nonetheless, the overall magnitude of the experimental data is quite comparable to that predicted by our theory near the maximum of our n 3 cross section. In Fig. 2 we present our elastic and total excitation cross sections (the sum of the n 3, 4 and 5 contributions), together with our previous determination of the ionization cross section (Ref. [13]) and the sum of these three separate cross sections in the energy range from 1 to 200 eV. To this sum we have added the smoothed experimental data of Ref. [14] for positronium formation to allow a comparison with the experimental total cross section. Also given are the experimental total cross sections of Refs. [7,8] and the ionization cross section of Ref. [9]. Our elastic cross section is in good agreement with, although slightly smaller than, the experimental total cross of Ref. [7] below the pos-
Fig. 2. Cross sections: ±±± ±±±, present excitation cross section; ±±± ±±± ±±±, present elastic cross section; ± ± ±, ionization cross section of Ref. [13]; ± ±, present total cross section minus positronium formation; - - - -, total cross section including positronium cross section of Ref. [14]. Experimental data: , total cross section of Refs. [7,8], , ionization cross section of Ref. [9].
40
L.A. Parcell et al. / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 37±40
itronium threshold (14.759 eV) where elastic scattering is the only channel which is open. At approximately 16.76 eV the excitation channels also become accessible while at 21.559 eV ionization is also possible. In the energy region between the positronium threshold and 21 eV the dominant scattering processes will be elastic scattering and positronium formation with excitation playing only a minor role. Clearly, the experimental results indicate the importance of positronium formation in this energy region. There is quite good agreement between the experimental ionization cross section of Ref. [9] and the theoretical predictions of Ref. [13]. Our results show that the excitation cross sections for Ne appear to be much less important than the ionization channel. This is the reverse of the situation in Mg (Ref. [15]) where the excitation channels contribute far more to the total cross section than ionization. 4. Conclusions We have investigated the various contributions to the total cross section for the scattering of positrons from Ne up to 200 eV. Below the positronium formation threshold where only elastic scattering is possible our results are in satisfactory agreement with experiment. The experimental results indicate that positronium formation is very important immediately above threshold. However, the energy range over which positronium formation yields a signi®cant contribution to the total cross section is as yet unknown and more experimental data over a wide energy range is needed. The excitation process in Ne would appear to be not nearly as signi®cant as the ionization channel in the determination of the total cross section. Further theoretical and experimental investigations, particularly in the energy range below 50 eV would be very desirable.
Acknowledgements We wish to express our appreciation to Dr. G. Laricchia for sending us their experimental ionization data prior to publication. Two of us (RPM and ADS) wish to thank the National Sciences and Engineering Research Council of Canada for ®nancial assistance. References [1] P.G. Coleman, J.T. Hutton, D.R. Cook, C.A. Chandler, Can. J. Phys. 60 (1982) 584. [2] S. Mori, O. Sueoka, J. Phys. 27 (1994) 4349. [3] O. Sueoka, S. Mori, J. Phys. 27 (1994) 5038. [4] L.A. Parcell, R.P. McEachran, A.D. Stauer, J. Phys. 16 (1983) 4249. [5] L.A. Parcell, R.P. McEachran, A.D. Stauer, J. Phys. 20 (1987) 2307. [6] R.P. McEachran, A.G. Ryman, A.D. Stauer, J. Phys. 11 (1978) 551. [7] T.S. Stein, W.E. Kauppila, V. Pol, J.H. Smart, G. Jesion, Phys. Rev. 17 (1978) 1600. [8] W.E. Kauppila, T.S. Stein, J.H. Smart, M.S. Dababneh, Y.K. Ho, J.P. Downing, V. Pol, Phys. Rev. A 24 (1981) 725. [9] V. Kara, K. Paludan, J. Moxom, P. Ashley, G. Laricchia, J. Phys. B 30 (1997) in press. [10] R.P. McEachran, D.L. Morgan, A.G. Ryman, A.D. Stauer, J. Phys. B 10 (1977) 663. [11] J.R. Taylor, Scattering Theory, Wiley, New York, 1972. [12] R.P. McEachran, L.A. Parcell, A.D. Stauer, J. Phys. B 28 (1995) 2487. [13] R.I. Campeanu, R.P. McEachran, A.D. Stauer, Can. J. Phys. 74 (1996) 544. [14] L.M. Diana, S.C. Sharma, L.S. Fornari, P.G. Coleman, P.K. Pendleton, D.L. Brooks, B.E. Seay, in: P.C. Jain, R.M. Singru, K.P. Gopinathan (Eds.), Seventh International Conference on Positron Annihilation, World Scienti®c, Singapore, 1985, p. 428. [15] R.I. Campeanu, R.P. McEachran, L.A. Parcell, A.D. Stauer, this issue, Nucl. Instr. and Meth. B 143 (1998) 21.