Differential cross sections for fragmentation of positronium

NIM B Beam Interactions with Materials & Atoms

Nuclear Instruments and Methods in Physics Research B 247 (2006) 111–116 www.elsevier.com/locate/nimb

Differential cross sections for fragmentation of positronium H.R.J. Walters b

a,*

, C. Starrett a, Mary T. McAlinden

b

a Department of Applied Mathematics and Theoretical Physics, Queen’s University, Belfast BT7 1NN, United Kingdom School of Computing and Mathematical Sciences, Oxford Brookes University, Wheatley Campus, Oxford OX33 1HX, United Kingdom

Available online 10 March 2006

Abstract Cross sections differential with respect to energy and angle of ejected positrons and electrons for Ps(1s) fragmentation in collision with He, Ne, Ar, Kr and Xe targets are reported. For Ne, Ar, Kr and Xe, only the case where the target is not excited (target elastic collisions) is considered. For He, fragmentation with target excitation/ionization (target inelastic collisions) is also studied. The impulse approximation has been used for target elastic fragmentation, the first Born approximation for target inelastic processes. Ó 2006 Elsevier B.V. All rights reserved. PACS: 36.10.Dr; 95.30.Ky Keywords: Positronium; Positron; Electron; Fragmentation; Ionization; Noble gases; Differential cross sections; Impulse approximation; First Born approximation

1. Introduction In a paper of 2001 Ludlow and Walters [1] put forward some ideas, at the time seemingly impractical, for coincidence studies of positronium (Ps) fragmentation in Psatom collisions. In 2002 Armitage et al. [2] published the first experiment on differential fragmentation of Ps in collision with He. Although not yet able to explore fragmentation in the detail envisaged by Ludlow and Walters, this was the first major step in such a programme. In the experiment of Armitage et al. [2,3] it is the ejected positron that is observed. However, the experiment is constrained by the presence of a magnetic field which means that it is not possible to measure both the angle and the energy of the positron. Rather, what is measured is the kinetic energy of the positron in the direction of the incident Ps beam, the so-called longitudinal energy, Epl. The cross section that is extracted is the cross section differential with respect to Epl, i.e., dr/dEpl. This cross section gets contributions from all energies, Ep and all angles, hp, of the

*

Corresponding author. E-mail address: [email protected] (H.R.J. Walters).

0168-583X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.01.046

ejected positron consistent with a fixed value Epcos2 hp of the longitudinal energy Epl. The measurements show that, with increasing impact energy, dr/dEpl, as a function of Epl, acquires a peak at roughly one half of the allowed maximum value of Epl, see Fig. 1. This is due to the fact that the Ps tends to fragment with roughly equal energies for the dissociated electron and positron which also tend to continue to move in the same direction as their parent Ps [1,4]. Calculations by Sarkadi [5] in the classical trajectory Monte Carlo (CTMC) method gave good agreement with the shape of the measured dr/dEpl but failed to give the correct normalisation. Following on from the work of Ludlow and Walters, Starrett et al. [4] applied the impulse approximation to the problem, getting good agreement both in shape and normalisation with the data (Fig. 1). Besides examining the spectrum of the emitted positron, Sarkadi used the CTMC approximation to study the pattern of electron ejection. Although the spectra of the ejected electron and positron should be identical at asymptotic energies when the first Born approximation rules [4], at the impact energies of the Armitage et al. experiment, 633 eV, Sarkadi predicted a noticeable difference, a prediction that was later confirmed by Starrett et al. [4], see Fig. 1.

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H.R.J. Walters et al. / Nucl. Instr. and Meth. in Phys. Res. B 247 (2006) 111–116 0.5 IA Positron IA Electron FBA Expt

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Armitage et al. since the Ps must have at least 27 eV if it is to excite or ionize the He. However, at higher impact energies simultaneous excitation/ionization of the target must be considered (so-called ‘‘target inelastic’’ collisions [6]). One of the virtues of studying the ejected positron rather than the electron, is that we know definitely where it has come from, i.e., from the Ps. When the electron is observed, it may have come either from the Ps or from the target, if the impact energy is sufficiently high to ionize the target. In this paper, we announce some preliminary results on target inelastic fragmentation and combine them with the target elastic calculations of Starrett et al. to get a full picture, such as would be observed by experiment at higher impact energies. However, we first give some more information on target elastic processes. In the papers of Ludlow and Walters, Sarkadi, and Starrett et al. and with the exception of the total fragmentation cross section, only He and Xe targets have been studied. We think it is of interest to complete this picture by exhibiting some target elastic results which show the changes as we progress up the noble gas sequence from He, through Ne, Ar and Kr, to Xe. Unless otherwise stated, it is to be assumed that we are using atomic units in which h = me = e = 1. 2. Theory

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Here we briefly sketch the necessary theory, a fuller account of the theory for target elastic (TE) processes is given in [4] and that for target inelastic (TI) processes will be elaborated elsewhere. For TE fragmentation we use the impulse approximation (IA) with peaking as described by Starrett et al. [4]. In the peaking approximation the IA amplitude takes the particularly simple form: f IA;Peak ðPs : a ! j; At : b ! b0 Þ   þ iqt=2 0 ¼ 2 / j/a ðtÞ fbb 0 ðvþ ; vþ Þ j ðtÞje     0 þ 2 /j ðtÞjeiqt=2 j/a ðtÞ fbb 0 ðv ; v Þ; 

ð1Þ

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Longitudinal energy (eV) Fig. 1. dr/dEpl (solid curve) and dr/dEel (dashed curve), calculated in the impulse approximation (IA) [4], for Ps(1s) + He(11S) collisions at impact energies from 13 to 33 eV. The dotted curve gives the first Born cross section which is the same for dr/dEpl and dr/dEel. Experimental data for dr/dEpl are from [2,3]. The cross gives the average value of the IA cross section dr/dEpl over the first experimental bin from 0 to 1 eV.

In the calculations of Ludlow and Walters, Sarkadi and Starrett et al. it was assumed that the target atom remained in its initial state after the collision, a so-called ‘‘target elastic’’ collision [6]. This is adequate for the experiment of

v0 ¼ v  q; q ¼ 2v0  vp  ve ; 1 j ¼ ðvp  ve Þ. 2

ð2Þ

Here v0 is the velocity of the incident Ps, vp (ve) the velocity of the ejected positron (electron), /a the initial state of the Ps, / j the final Coulomb continuum state (with ingoing scattered wave boundary conditions) representing the ionþ  ized Ps, and fbb 0 (f 0 ) is the amplitude for free positron bb (electron) scattering by the atom resulting in a transition of the atom from state b to b 0 . For TE collisions b = b 0 . Starrett et al. use the static and static-exchange approximaþ tions respectively to calculate the elastic amplitudes fbb and

H.R.J. Walters et al. / Nucl. Instr. and Meth. in Phys. Res. B 247 (2006) 111–116  fbb , these amplitudes are also placed on-energy-shell according to a prescription of Hartley and Walters [7]. In the present state of the art, experiment can observe either an ejected positron or an ejected electron [8]. If the positron is observed, then we know that it must have come from the Ps but we do not know what the final state of the atom is unless the impact energy is so low that the atom cannot be excited. Accordingly, in general we must sum theoretical cross sections over all energetically allowed final states of the atom before making comparison with experiment, i.e., we must include both TE and TI processes. If an ejected electron is observed, then it may have come either from the Ps or, if the impact energy is high enough, from the atom. If it comes from the Ps, then we do not know the final state of the atom and, again, we must sum over all final states of the atom as before. If it comes from the atom, then we do not know the final state of the Ps and we must sum over all permitted Ps final states. Both sources need to be taken into account in order to compare with observations of electron ejection. The summing over final states of the atom and the Ps is a major task in itself even in the simplest approximation, the first Born approximation (FBA). We therefore restrict ourselves to the FBA for all cases except TE fragmentation, for which we use the IA. The FBA amplitude for a general collision between the Ps and the atom takes the form [4]:

f B1 ðPs : a ! a0 ; At : b ! b0 Þ  4 ¼ 2 /a0 ðtÞjeiqt=2  eiqt=2 j/a ðtÞ q Z X eiqri ÞjWb ðXÞi;  hWb0 ðXÞjðZ þ

ð3Þ

i¼1

where /a/Wb (/a0 /Wb0 ) is the initial (final) state of the Ps/ atom, q is the momentum transfer in the collision, and where we assume that the neutral atom contains Z electrons with coordinates ri relative to the nucleus. The Ps form factor can usually be calculated analytically [6,9], the difficulty lies with the atomic matrix element. Amongst other things, for the situations considered here we need to know the atomic matrix element in (3) for the case where Wb is the atomic ground state and Wb0 is an ionized state. For this case we adopt an independent electron approximation in which we consider ionization separately from each spin orbital in Wb. Then, for an electron ionized with momentum k from the orbital wnlm(r), the atomic matrix element in (3) becomes 

 iqr w jwnlm ðrÞ . k ðrÞje

ð4Þ

The ionized wave function w k is calculated in the spherically averaged static field of the ion and is Lagrange orthogonalized to all the bound orbitals in the atomic ground state Wb, this orthogonalization gives an approximate representation of electron exchange between the ionized electron and the remaining electrons in the ion [10].

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To sum over all final excited/ionized states of the atom we adopt a procedure of Hartley and Walters [11]. This procedure is suitable when ionization dominates discrete excitation, which should be the case for noble gas targets. In the Hartley–Walters approximation we calculate only the ionization contribution to the sum over final atom states, this contribution is then multiplied by the factor SðqÞ=SðqÞ where S(q) is the incoherent scattering function which takes into account excitation/ionization to all final states and for which tabulations exist [11,12], and where SðqÞ is its counterpart calculated from the ionization component evaluated using the approximation (4). In this way a correction is applied for all omitted discrete excitations and for defects in modelling the ionization using the approximation (4). The reader is referred to Hartley and Walters [11] for a more complete discussion. 3. Results We first look at further results for TE fragmentation of Ps in the IA, these supplement the calculations of Starrett et al. [4] by extending the picture to Ne, Ar and Kr. Fig. 2 shows the TE cross section differential in the longitudinal energy of the ejected positron (dr/dEpl) and of the ejected electron (dr/dEel) for Ps(1s) impacting with an energy of 33 eV and for all five noble gases, He through Xe [13]. We see that dr/dEpl has a peak at roughly one half of the maximum value of Epl (i.e. near 13.1 eV) for all five targets. By contrast, dr/dEel only shows peaks for Ar, Kr and Xe. Generally speaking, the cross section steadily increases on going from He to Xe but the close similarity of Ar and Kr is noteworthy. Unfortunately, the single differential cross sections do not show the real structural differences between the targets. These first appear at the double differential level. Fig. 3(a) and (b) compares the double differential cross sections d2r/ dE dXp [14] and d2r/dE dXe for all five targets at an impact energy of 50 eV and with equal energy sharing, i.e. Ep = Ee = 21.6 eV, between the outgoing positron and electron. Notice the smooth variation in all of the positron cross sections d2r/dE dXp and in the helium electron cross section d2r/dE dXe as compared with the structures that appear in the electron cross section d2r/dE dXe for Ne, Ar, Kr and Xe. As shown by Starrett et al. [4], these structures reflect those in free electron scattering by the target,  they enter through the fbb amplitude in the IA (1). The structures for the Ne and Ar targets are not fully developed until an impact energy of about 100 eV, those for Kr and Xe not until 300 eV, see Fig. 3(c). Let us now turn to the main import of this paper, target inelastic processes. In Fig. 4 we show the TE total Ps fragmentation cross section for He as calculated by Starrett et al. [4] in the IA together with our present FBA result for the corresponding TI cross section. It is seen that the TI component becomes dominant above 180 eV but is negligible below 40 eV. The sum of the two gives the total Ps fragmentation cross section. This has a shoulder at around

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80 eV resulting from the decline in the TE cross section combined with the rise in the TI contribution. The single differential cross sections dr/dE [14] and dr/ dEpl for Ps(1s) impact on He at 200 eV are shown in Fig. 5(a) and (b), respectively. At 200 eV Fig. 4 tells us that TE and TI total fragmentation are about equal. From Fig. 5(a) we see that the TE contribution to dr/dE has the higher peak and is more or less symmetrical while the TI contribution has a lower peak but a long low energy tail. This tail is easily understood. As Starrett et al. point out [4], the FBA to drbb0 =dE, for a given atomic transition b ! b 0 , is symmetrical about Em/2 where Em is the maximum possible ejected energy for that transition. If we imagine a symmetrical such cross section for each final atomic state with decreasing peak height for increasing atomic excitation energy, smeared out since the atomic transitions are largely ionizations (in the model used here they are all treated as ionizations), then the origin of the tail is clear. The total dr/dE cross section, as Fig. 5(a) shows, has a noticeably larger peak than the TE cross section, positioned at a slightly lower energy Ep, and with a low energy tail. The cross section differential with respect to the longi-

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tudinal energy of the ejected positron, dr/dEpl, shown in Fig. 5(b), follows a similar pattern. Finally, in Fig. 6 we plot the cross section dr/dE that would be obtained if we observed the energy Ee of the outgoing electrons. Again, the target is He and the impact energy, E0, is 200 eV. Like Fig. 5(a), this cross section contains a TE component in which the Ps is ionized but the target is not excited, a TI component in which the target is excited or ionized as well as the Ps but only the electron from the ionized Ps is observed, and a component of electrons coming from ionization of the target, the final state of

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the Ps not being observed. Because of preferred ejection from the Ps with equal energies and in the forward direction, the TE and TI peaks appear at just below E0/2 and will move as the Ps energy is changed. By contrast, the electrons ejected from the target tend to remain under the influence of the target and so give rise to a sharp peak at Ee = 0. The overall cross section dr/dE therefore exhibits a sharp

In this paper, we have extended the TE differential fragmentation cross sections of Starrett et al. [4] in the peaking IA to Ne, Ar and Kr targets. This has enabled us to give a feel for the variation in these cross sections from target to target as we go up the noble gas sequence from He to Xe, and, through the double differential cross sections, Fig. 3, to show how structures, related to free electron scattering, develop in the sequence. We hope that this will be an inspiration to further experimental work and in this we are greatly encouraged by the agreement obtained between theory and experiment for a He target, Fig. 1. We have also addressed the question of TI fragmentation which becomes important as the impact energy is increased and have shown in the case of He that it dominates TE fragmentation above about 180 eV. It would be nice to see experiments in this domain. We have also explored the spectrum of ejected electrons, showing that, at sufficiently high impact energy, there are two distinct groups, one group moving with roughly half the energy of the incident Ps and coming from TE and TI ionization of the Ps, the other group lingering around the target with relatively low velocities in the laboratory and coming from ionization of the target. Again, an experimental investigation of this prediction would be interesting. A comment on our approximations is in order. For TE collisions we have used the IA of Starrett et al. [4], for TI processes the FBA. Experience of electron, atom, and Ps collisions [6,15,12,16] with ‘‘heavy’’ atoms such as, e.g. Ar, Kr and Xe, indicates that the FBA does not become viable for TE collisions until quite high impact energies. The problem lies with the increasing static potential

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presented to the projectile by the atom as the atom grows in size, this eventually ceases to be treatable in a first-order pertubation approximation except at very high energies [15]. The IA, which is non-pertubative in the projectile–target interaction, gets over the problem. For TI scattering the situation is different. Here the FBA cross section does not grow strongly with the atom size as it does for TE transitions. Firstly, the FBA matrix element for a TI collision, see Eq. (3), does not contain the nuclear charge Z because of orthogonality of the initial and final atom states, and so cannot increase directly with Z. Secondly, in an independent electron approximation to the atomic wave functions the exponential terms eiqri (i = 1 to Z) in the atomic matrix element, see (3), reduce to a single matrix element connecting the excited orbital to the orbital being excited, see (4), i.e. there is no growth with Z here either. Of course, the strong static potential of the atom will enter in second and higher orders of pertubation theory for TI collisions but unitarity keeps a lid on its effect. As a first guess therefore, we would suggest that the FBA sets the right scale for TI processes even for heavy targets, experience with other collision systems [7,11,15,12] is supportive of this suggestion. For the light He target, the only case considered here for TI scattering, the FBA for TI processes should be quite good. There is, at this stage, also a major practical consideration in not going beyond the FBA for TI collisions and that is because, already in this simple approximation, the sum over excited/ionized atom states is a substantial task. Acknowledgments This work was supported by EPSRC grants GR/ N07424, GR/R83118/01 and GR/R62557/01 and by the

Department for Employment and Learning Northern Ireland (DEL). References [1] J. Ludlow, H.R.J. Walters, in: J. Berakdar, J. Kirschner (Eds.), Many-Particle Spectroscopy of Atoms, Molecules, Clusters, and Surfaces, Kluwer/Plenum, New York, 2001, p. 319. [2] S. Armitage, D.E. Leslie, A.J. Garner, G. Laricchia, Phys. Rev. Lett. 89 (2002) 173402. [3] G. Laricchia, S. Armitage, D.E. Leslie, Nucl. Instr. and Meth. B 221 (2004) 60. [4] C. Starrett, M.T. McAlinden, H.R.J. Walters, Phys. Rev. A 72 (2005) 012508. [5] L. Sarkadi, Phys. Rev. A 68 (2003) 032706. [6] M.T. McAlinden, F.G.R.S. MacDonald, H.R.J. Walters, Can. J. Phys. 74 (1996) 434. [7] H.M. Hartley, H.R.J. Walters, J. Phys. B 20 (1987) 3811. [8] See article by S. Armitage, D.E. Leslie, J. Beale, G. Laricchia, Nucl. Instr. and Meth. B, these Proceedings, doi:10.1016/j.nimb.2006.01. 044. [9] J.P. Coleman, in: E.W. McDaniel, M.R.C. McDowell (Eds.), Case Studies in Atomic Physics I, North-Holland, Amsterdam, 1969, p. 101. [10] P.G. Burke, W.D. Robb, Adv. At. Mol. Phys. 11 (1975) 143. [11] H.M. Hartley, H.R.J. Walters, J. Phys. B 20 (1987) 1983. [12] D.P. Dewangan, H.R.J. Walters, J. Phys. B 11 (1978) 3983. [13] By dr/dEpl (dr/dEel) we mean the cross section corresponding to the collection of both forward and backward moving positrons (electrons), see Eq. (8) of [4]. [14] Note that dE is a generic notation for a differential change in the energy of the observed particle, electron or positron. In ionization of Ps the ejected electron energy, Ee, and the positron energy, Ep, are linked by conservation of energy so that dEe =  dEp. We use dE for the generic magnitude of the change. [15] H.R.J. Walters, J. Phys. B 8 (1975) L54. [16] H.M. Hartley, H.R.J. Walters, J. Phys. B 21 (1988) L43.