Modified effective range analysis of low energy electron and positron scattering on CO 2

Transcription

1 15th Internationa Symposium on Eectron Moecue Coisions and Swarms IOP Pubishing Journa of Physics: Conference Series 115 (008) 0100 doi: / /115/1/0100 Modified effective range anaysis of ow energy eectron and positron scattering on CO Zbigniew Idziaszek 1, Grzegorz P. Karwasz, Roberto S. Brusa 3 1 Centrum Fizyki Teoretycznej, Poska Akademia auk, 0668 Warszawa, Poand Instytut Fizyki, Uniwersytet Mikołaja Kopernika, Toruń, Poand 3 Dipartimento di Fisica, Università degi Studi di Trento, Trento, Itay Abstract. Anaytica soutions for the modified effective range probem have been appied to positron and eectron scattering on carbon dioxide in the ow (beow 10 ev) energy range. For positrons, the soution with three partia waves reproduces very we experimenta resuts up to the positronium formation threshod; the s-wave contribution rises in the imit of zero energy and the p-wave contribution reaches a very broad maximum at about 0.5 ev. For eectron scattering, the present soution shows a sharp rise of the s-wave contribution in the imit of zero energy, expained by earier cacuations as a virtua negative ion state. The p- wave shows a resonant structure at about 5 ev corresponding to an experimentay we known Π u shape resonance. An additiona maximum in the p-wave contribution is observed at about 1- ev. The atter feature woud expain resonant-ike scattering observed recenty in high-resoution vibrationa excitation measurements [M. Aan, Phys. Rev. Lett. 87 (001) 03301]. PACS: x, d Correspondence author: zbyszek@cft.edu.p 1. Introduction experiments and theory Knowedge of cross sections for eectron scattering on CO is of basic importance for understanding operation of high power infrared asers; CO is aso the main factor in humaninduced greenhouse effect. The ow-energy eectron scattering tota cross section is characterized by a strong rise in the imit of zero energy, with the tota cross sections reaching as much as [1] 60x10 0 m at 0.1 ev and presence of a shape Π u resonance at about 3.8 ev, observed as a maximum in the tota cross section [,3], with a big part (about 1/3, see [4]) coming from the vibrationa excitation. Whie eary cacuations using a semiempirica poarization potentia [5,6] agreed pretty we with the eectron-scattering experiment in a arge ( ev) energy range, ab-initio theories covering this energy range have been deveoped ony recenty [7,8,9]. The ow energy rise has been attributed [10,11] to a virtua Σ g state. The recent measurements of the vibrationa excitation showed the presence of resonant enhancements aso in the 1-.5 ev energy range [1]. Positron scattering on CO was studied experimentay by few groups (see the discussion beow). For the theory, Gianturco and Paioetti [13] appied the density-functiona method for short-range interaction effects in order to obtain eastic and rotationa excitation cross sections in the ev energy range and Gianturco and Mukherjee [14] used the body-fixed vibrationa cose-couping scheme to obtain the eastic and the vibrationa excitation cross sections in the same energy range. The weak point of theoretica cacuations is that different c 008 IOP Pubishing Ltd 1

2 15th Internationa Symposium on Eectron Moecue Coisions and Swarms IOP Pubishing Journa of Physics: Conference Series 115 (008) 0100 doi: / /115/1/0100 modes were separatey appied to positron and eectron projecties and for the very ow and resonance energy regions in eectron scattering. The present work gives an anaysis of positron and eectron scattering on CO in the framework of the modified effective-range theory (MERT) [15]. It is based on the anaytica soution for scattering from a poarization potentia with short-range effects introduced through parametrization of the phase shifts [16]. The anaysis uses experimenta tota cross sections at energies beow the threshod for eectronic excitation (or positronium formation for positrons). In the this energy range (apart from resonances) the tota cross section, measured with a few per cent error, approximates we the integra eastic cross section (usuay measured with much bigger uncertainty and by normaized methods).. Choice of experimenta data For eectron scattering the time-of-fight experiments in the very-ow-energy range [1, 3] merge we with the eectrostatic-beam experiment in the region of a few ev []. These three sets of data, weighted by their uncertainties form recommended vaues given in [4], are used for the present anaysis. Positron-scattering tota cross sections were measured down to 0.5 ev by Hoffman et a. [17] with an apparatus using a ong (109 cm) scattering ce, with rather sma apertures (1. mm radius at the entrance and.4 mm at the exit) and a ongitudina guiding magnetic fied (16 G at ev coision energy) [18]. First measurements by Sueoka and coaborators [19] were performed down to 1 ev using 67.5 mm ong scattering ce (we quote the geometrica ength and not the effective 79.7 mm ength) with 4 mm radius apertures and 9 G guiding fied. The more recent data [0] (quoted aso in [1]) were performed down to 0.3 ev with 1.8 G guiding fied and 3 mm radius apertures; at 1 ev those data are by 10% ower than the vaue of Kwan et a. [] and 0% higher than the eary set [19]. The new apparatus constructed at Trento aboratory [3] uses a 10 mm ong scattering ce with 0.75 mm apertures radii and 9 G guiding fied. The positron scattering data for CO extending down to 0.1 ev [4] rise quicky in the ow-energy imit and form the highest set. As we discussed in detai for N measurements [5], the use of arge apertures in the scattering ce combined with high magnetic fieds can ead to a strong underestimation of the tota cross section in the ow energy imit. At sufficienty ow energies, when the cycotronic radius of positrons in the magnetic fied equas or is smaer than the scattering ce exit apertures, a positrons scattered into the forward cone (up to 90º scattering ange) are recaptured by the magnetic fied and guided to the detector. Assuming an isotropic differentia cross section and no ineastic processes present, the measured tota cross section woud then amount to 50% of the rea vaue. Such a haf-vaue coision energy equas to 0.8 ev in the experiment of [17], 1.1 ev in experiment of Sueoka and Mori [19], 0.06 ev in their repeated experiment [6], and as itte as ev in the Trento experiments [5]. A second possibe source of error in the absoute magnitude of tota cross sections in the ow energy imit can be the energy scae shift. In the Trento apparatus, detaied procedures both with a retarding fied anayzer [7] and using the N and Ar positronium threshods [5] were appied, so a possibe error in the energy scae is within ±0.1 ev. Note aso that the energy resoution of the Trento apparatus is about 150 mev, as proved by observation of sharp resonant structures in He at ev [8]. For a these arguments we beieve that the cross sections from Trento aboratory can be considered presenty the most reiabe set for positron scattering.

3 15th Internationa Symposium on Eectron Moecue Coisions and Swarms IOP Pubishing Journa of Physics: Conference Series 115 (008) 0100 doi: / /115/1/ Anaytica soution for modified effective range approximation As aready stated by O Maey et a. [9] the zero energy imit in eectron and positron scattering shoud be we approximated by the modified effective range theory. In this theory scattering is assumed to be due to poarization potentia soey and the short-range effects are introduced by an effective range R 0. Eectron scattering on nobe gases has been extensivey evauated in the framework of MERT by Buckman and Mitroy [30] who used five-parameter expansion in series of momentum k for the s and p-wave phase shifts. Eectron scattering on non-poar moecues (H, N, CO ) was studied, among others by Fabrikant [31]. In a these studies an upper imit for MERT anaysis was confined to energies beow 1 ev (0. ev for CO ). In a recent work [16] we have appied anaytica soutions for scattering on the poarization potentia to extend the appicabiity of MERT to higher energies. We obtained MERT expansion by deveoping the parameter characterizing the short range interaction into series of k. We tested this scheme for positron scattering on Ar and N up to ev - the parameters of the short range potentia were obtained by fitting recent experimenta tota cross sections [5]. We resume here the main notions. The radia Schrödinger equation for a partice moving in 4 the poarization potentia V ( r) = αe r is h ( + 1) αe µ r r r 4 E Ψ ( r) = 0, (1) where µ is the reduced mass, α is the static dipoe poarizabiity, E is the reative energy of the partices, and Ψ (r) denotes the radia wave function for the partia wave. With appropriate change of variabes, the Schrödinger equation (1) can be transformed into Mathieu's differentia equation of the imaginary argument [3], and soved anayticay in terms of the continued fractions (see e.g. [16]). At sma distances where the poarization potentia dominates over centrifuga potentia and the constant energy term, the asymptotic behaviour of Ψ (r) is described by * where αµ Ψ ( r) r 0 r sin ( R * r+ φ ) R = e / h describes a typica ength reated to the poarization interaction, and φ is some short-range phase, that depends on the short-range part of the potentia. For =0 the short range phase determines the s-wave scattering ength: a = - R*cot φ 0. At arge distances Ψ (r) must take the form of the scattered wave where Ψ ( kr π + η ) r ( r) sin (3) k = µ E / h and phase-shifts η can be expressed in terms of the short-range phases: 1) tan( φ + π ) tanδ ( φ + π ) (1 m tan δ ) tan m tan δ + ( m η = (4) (1 m ) tanδ + tan Here, δ = π/(ν--1/), and m and ν are some parameters characterizing Mathieu functions of the imaginary argument (see [16] for detais). The effective range is introduced by expanding of tan(φ + π/) in powers of k 1 ( + π ) = A + R * R k + K tanφ, (5) 3 ()

4 15th Internationa Symposium on Eectron Moecue Coisions and Swarms IOP Pubishing Journa of Physics: Conference Series 115 (008) 0100 doi: / /115/1/0100 A = tan φ + π The owest order correction in k is quadratic, and can be k= interpreted as effective range R for partia wave [33]. where ( ) 0 4. Appication for positron and eectron scattering on CO. Using for the (spherica) poarizabiity of CO the vaue obtained from eectron-scattering eectronic excitation experiments α=16.9 a 0 3 [34] we obtain R* = a 0. For MERT fit for positrons we used a experimenta data from Trento ab from 0.1 ev up to the positronium formation threshod (i.e. 6.8 ev). The experimenta data can be approximated quite we assuming contributions just from two partia waves s and p. If MERT expansion is performed aso for d wave, the agreement remains within the statistica error bar for the whoe energy range considered, see figure 1. The s-wave partia wave cross sections rises in the imit of ow energies and dominates over the p-wave contribution for energies beow 0. ev; the s-wave contribution passes through zero at ev. Parameters of the potentia (zero-energy contributions A and the effective ranges R ) for the three partia waves are given in tabe 1. Note the zero vaue for the p-wave zero-energy parameter A 1 and a negative vaue for the s- wave scattering ength. Fig. 1. MERT anaysis for positron scattering on CO. Experimenta data are: fu circes Zecca et a. [4], open circes Hoffman et a. [17], trianges Sueoka and Hamada [0]. Theoretica curves are: MERT fit for the tota cross-section (soid ine), and contributions from s wave (dash ine), p wave (dot ine) and d wave (dash-dot ine). 4

5 15th Internationa Symposium on Eectron Moecue Coisions and Swarms IOP Pubishing Journa of Physics: Conference Series 115 (008) 0100 doi: / /115/1/0100 a/r*=-1/a 0 A 1 A R 0 /R* R 1 /R* R /R* E + CO e CO Tabe 1. Positron and eectron scattering parameters obtained by fitting MERT mode to the experimenta data. For eectron scattering we used the recommended data [4] from 0.1 to.0 ev: the fit with MERT expansion for two partia waves reproduced the set within experimenta uncertainties, see fig.. The obtained scattering ength a =- 6.61a 0 agrees we with the vaue a=- 6.17a 0 predicted by Morisson [35] and the imits given by Fabrikant [31]: -7. a 0 < a < -6.8a 0. Fig.. MERT anaysis for eectron scattering on CO. Depicted are: recommended experimenta data from review [4] (squares), mutistate cacuation of Morgan [11] (dash ine), and theoretica fits with MERT expansion for two (soid ine) and three (dot ine) partia waves. For the atter cacuation we assumed p-wave resonance at the experimentay observed peak. The inset shows in addition the s-wave and p-wave contributions to the MERT fit with two partia waves. Differenty from previous MERT approaches and in agreement with experiments, our MERT mode predicts the existence of a shape resonance in p-wave channe at 5 ev. This is sighty higher than the peak in the experimenta tota cross section but aso the cacuation by Morgan [11] overestimates the energy of the shape resonance, see fig. 1. The shape resonance appears in the regime where the d-wave contribution can be significant, therefore we have 5

6 15th Internationa Symposium on Eectron Moecue Coisions and Swarms IOP Pubishing Journa of Physics: Conference Series 115 (008) 0100 doi: / /115/1/0100 aso tried a fit with MERT for three partia waves fixing additionay the position of the resonance at the experimentay observed vaue. Parameters of such a fit are given for sake of comparison with positron parameters in tabe I. This demonstrates the compatibiity of MERT expansion for sma energies and the existence of the p-wave shape resonance. Our MERT anaysis shows aso a pecuiar behaviour of the p-wave shift just beow the shape resonance. The p-wave cross section changes quicky just beow the resonance, reaching another maximum at 1 ev. We hypothesize that this maximum is responsibe for a resonance-ike enhancement of the vibrationa excitation observed between 1 and.5 ev by Aan [1]. 6. Concusions We show that the recenty deveoped scheme based on anaytica soutions and MERT expansion gives very good agreement with experimenta data both for positron and eectrons. For positrons the agreement extends up to the threshod for positronium formation. For eectrons, the fit performed in the ow energy range (beow ev) predicts the existence of a shape resonance at 5 ev, and an additiona enhancement in the p-wave channe at about 1 ev. The obtained position of the resonance is sighty higher than the experimentay observed peak at 3.8 ev, and is cose to the theoretica vaue of [11]. The agreement with the experiment can be improved, as far as the position of the resonance is concerned, using three partia waves in MERT expansion. Summarizing, the present MERT cacuations reproduces we the observed cross section for both type projecties. In the case of positron scattering MERT shows a sharp rise of the tota cross section in the imit of zero energy; in the case of eectrons it shows a simiar rise in the zero-energy imit pus a shape resonance at about 5 ev. References [1] J. Ferch, C. Masche, and W. Raith, J. Phys. B 14 (1981) L97. [] Cz. Szmytkowski, A. Zecca, G. Karwasz, S. Oss, K. Maciąg, B. Marinković, R.S. Brusa, and R. Grisenti J. Phys. B 0 (1987) 581. [3] S. J. Buckman, M. T. Efordt and D. S. Newman, J. Phys. B 0 (1987) [4] G. P. Karwasz, A. Zecca and R.S. Brusa, Eectron Scattering with Moecues, in: Photon and Eectron Interaction, with Atoms, Moecues and Ions, Landot-Börstein New Series vo I/17, Springer-Verag, Berin, Heideberg, 003, pp [5] M. A. Morrison, N. F. Lane, and L. A. Coins, Phys. Rev. A 15 (1977) 186. [6] D. Di, J. Wech, J. L. Dehmer, and J. Siege, Phys. Rev. Lett. 43 (1979) 136. [7] T. N. Recigno, D.A. Byrum, W. A. Isaacs, and C. W. McCurdy, Phys. Rev. A 60 (1999) 186. [8] C.-H. Lee, C. Winstead, and V. McKoy, J. Chem. Phys. (1999) [9] T. N. Rescigno, W. A. Isaacs, A.E. Ora, H.-D. Meyer, and C. W. McCurdy, Phys. Rev. A 65 (00) [10] H. Estrada and W. Domcke, J. Phys. B 18 (1985) [11] L. A. Morgan, Phys. Rev. Lett. 80 (1998) [1] M. Aan, Phys. Rev Lett. 87 (001)

7 15th Internationa Symposium on Eectron Moecue Coisions and Swarms IOP Pubishing Journa of Physics: Conference Series 115 (008) 0100 doi: / /115/1/0100 [13] F. A. Gianturco and P. Paioetti, Phys. Rev. A 55 (1997) [14] F. A. Gianturco and T. Mukherjee, J. Phys. B 30 (1997) [15] L. Spruch, T. F. O Maey, and L. Rosenberg, Phys. Rev. Lett. 5 (1960) 347. [16] Z. Idziaszek and G. P. Karwasz, Phys. Rev. A 73 (006) [17] K. R. Hoffman, M. S. Dababneh, Y.-F. Hsieh, W. E. Kauppia, V. Po, J. H. Smart, and T. S. Stein, Phys. Rev. A 5 (198) [18] W. E. Kauppia, T. S. Stein, G. Jesion, M. S. Dababneh, and V. Po, Rev. Sci. Instrum. 48 (1977) 8. [19] O. Sueoka and S. Mori, J. Phys. Soc. Japan 53 (1984) 491. [0] O. Sueoka and A. Hamada, J. Phys. Soc. Japan 6 (1993) 669. [1] M. Kimura, O. Sueoka, A. Hamada, M. Takekawa, Y. Itikawa, H. Tanaka, and L. Boesten, J. Chem. Phys. 107 (1997) [] C. K. Kwan, W. E. Kauppia, S. Nazaran, D. Przybya, N. Scahi, T. S. Stein, Nuc. Instr. Meth. B 143 (1998) 61. [3] G. Karwasz, R.S. Brusa, M. Barozzi, and A. Zecca, Nuc. Instr. Meth. B 171 (000) 178. [4] A. Zecca, C. Perazzoi, N. Moser, D. Sanya, M. Chakrabarti, and M. J. Brunger, Phys. Rev. A 74 (006) [5] G. P. Karwasz, D. Piszka, and R. S. Brusa, Nuc. Instr. Meth. B 47 (006) 68. [6] O. Sueoka and S. Mori, J. Phys. B 7 (1994) [7] G.P. Karwasz, R.S. Brusa, and D. Piszka, Nuc. Instr. Meth. B 51 (006) 50. [8] G. P. Karwasz, D. Piszka, A. Zecca, and R. S. Brusa, Nuc. Instr. Meth. B 40 (005) 666. [9] T. F. O'Maey, L. Rosenberg, and L. Spruch, Phys. Rev. 15 (196) [30] S. J. Buckman and J. Mitroy, J. Phys. B (1989) [31] I. I. Fabrikant, J. Phys. B 17 (1984) 43. [3] E. Vogt and G. H. Wannier, Phys. Rev. 95 (1954) [33] T. F. O Maey, L. Spruch, and L. Rosenberg, J. Math. Phys. (1961) 491. [34] T. N. Oney, N. M. Cann, G. Cooper, and C. E. Brion, Chem. Phys 3 (1997) 59. [35] M. A. Morrison, Phys. Rev. A 5 (198)