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Positron- and electron-impact double ionisation of helium at low and intermediate energies
a‘Es -li!iJ
9 September 1996 PHYSICS
LETTERS
A
Physics Leners (1996) 237-241
EiSEVIER
Positron- and electron-impact double ionisation of helium at low and intermediate energies Jamal Berakdar Atomic
and Molecular
Physics
Laboratories,
Research School of Physical Sciences and Engineering, Canberra, ACT 0200, Australia
Australian
National
University,
Received 2 May 1996; revised manuscript received 20 June 1996: accepted for publication 24 June 1996 Communicated by B. Fricke
fl\bstract
IJsinga correlatedfour-body final-statewavefunctionthe angulardistributionof fragmentsfollowing doubleionisationof heliumby positronand electronimpactis studied.Distortion of angular patterns due to projectile-electron and projectilenucleus interactions is investigated. Further, the influence of two-body Coulomb-density of states in the four-body continuum is envisaged. It is shown that these interactions, previously neglected in Born-type treatments, play a major role in determining
the crosssectionat low and intermediateenergies. t’ACS:
34.50.H; 34.80.D; 35.80.fS
There has been an increased activity in the study of double ionisation prompted by recent advancesin coincidence-detection techniques [ l-41 (see Ref. [ 51 ‘or a comprehensive review). A major aim of these ‘neasurementsis to simultaneously determinethe monentum distributions of the double-ionisation fragnents and to study the influence of various interac:ions between particles involved in the collisions on :hesedistributions. The first experiments of this kind ssing high-energy electrons (5 keV) as projectiles, :he so-called (e, 3e) measurements,have been performed using argon and krypton targets under condi;.ionswhere littlemomentum is being transferredto the larget atoms [ I ,2]. In this casea first-order Born-type approximation (FBA) in the projectile-target potenI:ial is justified. Basically, the problem reducesthen to I:he description of a three-body continuum final state and has been treated by numerous authors [ 6-131.
3375.9601/96/$12.00 Copyright 0 1996 Elsevier Science B.V. PLI SO375-9601(96)00537-3
All
Within the FBA the samecrosssection is obtained for both particle and anti-particle collisions. In addition, it has been shown in Ref. [ 111 that ISA-cross sections for double ionisation by particles of arbitrary masses can be obtained from the (e, 3e) crosssection using a scaling formula which hasbeen derived from pure geometrical arguments.In other words, within the FBA, the properties of the projectile have no dynamical effects on the cross sections. In this Letter we presentthe first calculations of the fully differential crosssection for double-ionisation of helium in its ground stateby positron and electron impact where all particles in the final statesare treated on equal footing. Thus, double ionisation at low and intermediate impact energiescan then be treated. This is of particular interest since the magnitude of the double-ionisation cross section is known to peak at an excessenergy a few times the ionisation-potential
rights reserved.
238
J. Berukdur/Physic.s
Letters
energy. In addition, the first measurements at intermediate incident energies have been performed using the cold-target recoil-ion momentum spectroscopy [ 3,141. Hence it appears timely to develop a theoretical model which goes beyond the FBA and to consider the influence of the projectile-nucleus and the projectile-electron interactions on the cross sections. The extension of the model presented here to the treatment of double ionisation of helium by an arbitrary, structureless projectile will be published elsewhere. Atomic units are used throughout, the nucleus is assumed infinitely heavy with respect to the electron mass. The multiply differential cross section for the simultaneous detection of the scattered projectile and the two ionised electrons emerging with momenta (with respect to the nucleus) k,, kb, k,, respectively, is given by
(1) where ki is the momentum of the incident projectile and E; and Ef are the total energies in the initial and the final channels, respectively. The two-particle transition matrix element 7” has the form Tf, = Wk,,kb.k,
(r,,
rb, rc> lvl@(rQ,
rb. rc)),
(2)
where the asymptotic initial state of the system is described by @(r,,rb,rc)
= (2r)-3’2exp(iki
‘r,)
$D(Tb,Tc). (3)
In Eqs. (2), (3) the vectors rG, rb, rc denote the positions, with respect to the nucleus, of the projectile and the two target electrons, respectively. The twoelectron bound-state wavefunction of He( ‘Se) is denoted by (D( rb, r,) . The transition operator v is taken as the projectile-target potential in the initial channel (4) In Eq. (4) the projectile and the nuclear charges are designated by Z, and Zr, respectively. The four-body wavefunction Pk;,., k, ( r,, rb, rc) describes three charged particles in the continuum of He2+ with outgoing boundary conditions. In the context of electronimpact single ionisation followed by Auger decay,
A 220 (19961237-241
some aspects of the four-body Coulomb continuum problem have been studied within the eikonal approximation [ 15,161. In this work we adopt a procedure similar to that employed in Ref. [ 171 in which the four-body system is broken down into six spatially decoupled two-body subsystems which results in the approximate four-body wavefunction w‘k;,kl, k, (r,, rb, rc) % (2%-) p9’2n x
IF~ [iaj, 1, -i(kjrj
+ kj - rj)]
y;eik’.’ ,
(5)
where j E {a, b, c, ab, ac, be}. The momenta kab, kac, kbc are respectively conjugate to the relative coordinates rob := ra - rb, rat := ra - r,., rbc := rb - rc. The Sommerfeld parameter LY~Lwhich describes the strength of interaction between particle “k” and “I” is given by Lvkl := ZkZpkl/kkl, where Zk and Zl are the charges of the particles “k” and “I” respectively and ,+l is their reduced mass (thus LY, = ZpZT/ka, ab = -.&/kb, ac = -&/kc, %zb = -&&k,b, %c = -&/ 2k,, abc = 1/2kbC). In Eq. (5) the normalisation constants Nj are given by Ni = exp( -ajr/2)
I’( 1 - iaj).
(6)
It is straightforward to show that the wavefunction (5) satisfies the Redmond-boundary conditions (see Ref. [ 181) for large interparticle separations. An important feature of the wavefunction (5) is that it exactly diagonalises the total potential and neglects parts of the kinetic energy operator. Calculations of scattering amplitudes using Eq. (5) as final-state representation require, however, the evaluation of a ninedimensional integral (over rat rb. r,) which is not feasible yet. Thus, we are obliged to make a further approximation. If we assume the total potential to be exactly diagonalised by the wavefunction (5) only along the paths of free particles, which is reasonable for large interparticle separations, we can employ a procedure similar to that introduced in Ref. [ 191 and the wavefunction, given by Eq. (5), reduces to the expression
X
IF~ [ia,(k,,ka,k,),l,-i(k~ir,i+kj.rj)], (7)
J. Berddar/Physics
Letters
where j E {a, b, c}. The wavefunction (7) treats all particles on equal footing in the sensethat all interactions of the continuum particles with the nucleus are assumedto be of the sameorder. This in contrast to FBA treatments where the projectile-atom finalstate interactions are disregarded. The momentumcependent Sommerfeld parameters have the form z?j := zL(k,, kb, k,)/kj and the dynamical product charges 2, read [20] tk +zjkb+ n L1
-‘p
k2,,
( k, + kc) k,, (8)
;?,, =-& + 2
-ZP (ka + kb)k,b
1 + (kb + k,)kbc
$3
(9) 1
tkb+k,)kbck;.
(10)
,\ccording to Eqs. (8)-( IO) the correlations between the continuum particles arecompletely subsumedin an effective interaction of each continuum particle with Ithe nucleus. With the product charges (8)-(10) the xvavefunction given by Eq. (7) exactly diagonalises ;.hetotal potential along the trajectories of free particles. This is readily concluded from the fact that the :?roductcharges,given by Eqs. (8) -( lo), conservethe :otal potential along the paths of free particles since along these trajectories the relation C, Z,/r,i = V; ,j; E {a, b, c} applies where V is the total potential. In addition, the wavefunction given by Eq. (7) hasa regJlar asymptotic behaviour in subspacesof the asymp:otic regions. Details of the derivations and the physical interpretations will be given elsewhere [20]. We note here that disregarding any correlation between the continuum particles, i.e. in a model of three independent particles moving in the field of He*+, the product charges,given by Eq. (8) -( lo), reduce to the first term of the respective equations.In this casedouble ionisation is prohibited provided initial and final statesare eigenfunctions of the sameHamiltonian (the Initial-state wavefunction then contains no electronic correlations) ~A vanishing z, leads to the FBA treatment. For a radially correlated helium-ground state wavefunction [ 211 andusing the wavefunction (5) for the description of the final state, the nine-dimensional
A 220 (1996) 237-241
239
integral, given by Eq. ( 1)) can be analytically reduced to a one-dimensionalintegral on the real axis [ 201. To consider the symmetry properties of the finalstate wavefunction we remark that for the caseconsidered here (Zr = 2) the spin-orbit interaction is negligibly small so that L and S are good quantum numbers and an L-S-coupling schemecan be employed. Thus, for positron impact the final-state wavefunction, given by Eq. (7)) must be symmetric with respect to exchangeof the ionisedelectrons sincethe spatial part of the helium ground state and the transition operator (Eq. (4)) are both symmetric under this operation. In contrast, for electron-impact double-ionisation the cross section is a statistical average of doubletand quartet-scatteringcrosssections[ 201 respectively, correspondingto a symmetric and antisymmetric spatial wavefunction in the final state. For the caseof large impact velocity and little momentumbeing transferred to the target atom, cross sectionsresulting from electron and positron impacts converge to each other (see below) which meansthat in this limit electron-impact double ionisation into the doublet channel dominates. To assess the influence of the projectile-nucleus and projectile-electron interactions on the cross sections, we consider, at first, the caseof moderateincident energy (Fig. 1) . In this situation all particles in the final channel are detected with quite different energies. Neglecting the projectile-target final-state interactions (as done in an FBA treatment or in a plane-wave description of the continuum particles) results in equivalent cross sectionsfor positron and electron impact. The present model, however, predicts in the case of Fig. 1 a larger cross section for electron impact than for positron impact. The interpretation of the angular distribution (Fig. 1) of the secondaryelectron is quite obvious as the fast electron (hereafter electron “P) is predominantly emitted into the momentum transfer direction q := ki - k,. This allows the conclusion that the projectile directly collides with electron “b” which is then ejected into the direction q. The slower electron (electron “c”) is then ionisedby meansof initialstate correlations (shake-off). Since the ejection of electron “6” is mainly due to two-body interactions, its angular correlation pattern resemblesthe angular distribution of slow secondary electrons produced by singleionisation of atomsby fast electrons 1221. This strongly supportsthe suggestionthat the peak in Fig. 1 which emergesin the backward direction with respect
240
J. Berakd~~r / Physics
Letters A 220 t 19961237-241
e
Fig. I. The angular distribution of a secondary electron (electron “B”) escaping with an energy of 100 eV after electron- or positron-impact double ionisation of He. The other electron (electron “c”) is detected under a fixed angle of 130’ with respect to the incoming projectile direction and with a fixed energy of I eV. The emission angles of the electrons are measured with respect to ki. The incident energy is chosen as 800 eV and the projectile is scattered through a Y-angle with respect to the incident direction. All momenta ki, kn, kb, k, lie in the same plane. Results are shown for double ionisation by positron (solid line) and electron (dotted line) impact. A Slater-type wavefunction (o(rb, Tc) = N[exp( -C,,f-,, - W-c) + eXp( --c&, - C/,l-,)] has been used to describe the ground state of helium, where N is a normalisation constant and cl, = 2.17, cc = 1.21 are variational parameters. For electron impact, the magnitude of the cross section for the emission of electron “b” under an angle of 40 = 90’. where cos 4b = k, . pi, is equal to I .96 x 10W5 a.u.
to q is a recoil peak, i.e. after an encounter with the projectile the atomic electron “8’ scatters off the nucleus to emerge in a direction opposite to q. The latter interaction with the nucleus is provided by the initialbinding wavefunction [ 231. We remark here that the well-known q-cylindrical symmetry of cross sections obtained by the FBA for electron-impact single ionisation [ 241 is, in general, not reproduced by the present model due to final-state interactions. If the projectile and the secondary electrons are well separated in momentum space the difference between cross sections resulting from positron and electron impacts diminishes with increasing incident energies (Fig. 2), as final-state interactions of the projectile with the target atom become marginal in this case. Nevertheless, at incident energies as high as 5 keV there are still small differences between double ionisation by electron and positron impact which signifies a slow convergence to the FBA limit. In the case of Fig. 2 the angular
Fig. 2. The same collision geometry as in Fig. 1, but the incident energy is increased to 5000 eV. For electron impact, the cross section at Dot, = 90” is equai to 2.66 x 10-a a.u.
distribution of the ionised electron is sharply peaked around q. For even higher impact energies the cross section depicted in Fig. 2 tends to a S-function distribution centred around q. This is due to the fact that the initial-momentum distribution of the ionised electron becomes less significant with increasing projectile velocity in the initial channel. The projectile-electron encounter then resembles more and more a classical projectile scattering from a stationary free particle. The product charges (8)-( 10) are suitable to describe strong correlations between collision partners. Whenever two final-state particles approach each other in momentum space the corresponding dynamical product charges change in such a way that strong repulsion or attraction between these two particles is simulated depending on whether the collision partners are of the same or different (physical) charge states. A striking example of this situation is when an incident positron and one of the ionised electrons emerge with comparable velocities (Fig. 3). In this case the angular distribution of this electron (and in fact of the e. projectile) reveals a sharp peak at the direction kb-k, M 1. In the projectile-electron relative-velocity, upe, spectrum the latter peak develops to a pole of the first order which occurs at upe = 0. This singular structure is due to a diverging Coulomb-density of states of the projectile-electron two-body subsystem. The analogous process in single ionisation of atomic systems by positively charged particles is the electron capture to the projectile continuum (ECC) 125,261. In the case of negatively charged projectiles an expo-
.I. Berakdar
/Physics
Letters
A 220 (1996)
237-241
Foundation and the Australian National University der contract V.3-FLF.
241
un-
References
Fi:;. 3. The double-ionisation cross section of He( ‘S”) by positron impact. The incident energy is equal to 900 eV. The ionised electron “c” is detected under a fixed angle of 130’ with respect to the incident direction and with an energy of 10 eV. The momenta k,. k,, ko. k, lie in the same plane. The angular distribution of the otl:er secondary electron (electron “b”), which emerges with an energy of 360 eV, is depicted. The projectile is scattered through an angle of 5’ with respect to the positron beam. The magnitude of the cross section at 40 = 0’ is equal to 3.5 x IO-* au.
nential dip instead of the divergence appears in the cross section. We note here that in case of soft collisions, i.e. of very slow escaping electrons, the crosssection singularities arising from diverging Coulombdensity of states of the nucleus-ionised electron subsystems are exactly removed by the kinematical flux factors contained in Eq. (1). In contrast, due to the projectile-nucleus repulsion there is an exponentially vanishing probability for a projectile positron to lose its whole momentum during the collision and emerge with vanishing velocity with respect to the nucleus. To summarize, the first calculations for positronand electron-impact double ionisation of a helium target in its ground state have been performed using a four-body correlated final-state wavefunction. It has been shown how the projectile-target final-state interactions modify the angular distribution of the ionised electrons. In addition, the influence of the Coulomb&nsity of states in a four-body system in the continulrn has been pointed out. I would like to thank Steve Buckman and Erich Weigold for helpful comments and suggestions. This u ork was supported by the Alexander von Humboldt
[20] 1211 [223 [231 [24] 1251 1261
A. Lahmam-Bennani, C. Dupfe and A. Duguet, Phys. Rev. Len. 63 (1989) 1582. A. Lahmam-Bennani, A. Duguet, A.M. Grisogono and M. Lecas, J. Phys. B 25 (1992) 2873. R. DBmer, V. Mergel, L. Spielberger, 0. Jagutzki, S. Nuttgens, M. Unverzagt, H. Schmidt-Backing, J. Ulhich, R.E. Olson, K. Tokesi, WE. Meyerhof, W. Wu and CL. Cocke, Nucl. Instr. Meth. B 99 ( 1995) 111. B. El Marji, A. Duguet, A. Lahmam-Bennani, M. Lecas and HF. Wellenstein, J. Phys. B 26 (1995) L733. A. Lahmam-Bennani. I. Phys. B 24 (1991) 2401. B. Joulakian, C. Dal Cappello and M. Brauner, J. Phys. B 25 (1992) 2863. B. Joulakian and C. Dal Cappello, Phys. Rev. A 47 ( 1993) 788. C. Dal Cappello and H. Le Rouzo, Phys. Rev. A 43 (1991) 1395. R.J. Tweed, J. Phys. B 6 (1973) 270. Yu.F. Smimov, A.V. Pavlitchenkov, V.G. Levin and G. Neudatchin, J. Phys. B I1 ( 1978) 20. I. Berakdar, J. Phys. (Paris) IV 3 (1993) C6-135. J. Berakdar and H. Klar, J. Phys. B 26 (1993) 4219. I. Berakdar. Phys. Rev. A 53 (1996) 2281. 0. Jagutzki, L. Spielberger, R. Diimer, S. Niittgens, V. Mergel, H. Schmidt-B&king, J. Ullrich, R.E. Olson and U. Buck, Z. Phys. D 36 (1996) 5. M.Yu. Kuchiev and S.A. Scheinerman, Sov. Phys. Usp. 32 ( 1989) 569. L. Avaldi, P Belotti, P Bolognesi, R. Camilloni and G. Stefani, Phys. Rev. Lett. 75 ( 1995) 1915. I. Berakdar, Phys. Rev. A 53 (1996) 2314. L. Rosenberg, Phys. Rev. D 8 ( 1973) 1833. S. Jetzke and F.H.M. Faisal, J. Phys. B 25 ( 1992) 1543. J. Berakdar ( 1996) to be published. A. Hylleraas, Z. Phys. 54 (1929) 347. I.E. McCarthy and E. Weigold, Rep. Prog. Phys. 54 ( 1991) 781. J.S. Briggs, Comments At. Mol. Phys. 23 (1989) 155. H. Ehrhardt, M. Schulz. T. Tekaat and K. Willmann, Phys. Lett. A 110 (1985) 92. A. Salin, J. Phys. B 2 (1969) 631. J.H. Macek, Phys. Rev. A 1 ( 1970) 235.
9 September 1996 PHYSICS
LETTERS
A
Physics Leners (1996) 237-241
EiSEVIER
Positron- and electron-impact double ionisation of helium at low and intermediate energies Jamal Berakdar Atomic
and Molecular
Physics
Laboratories,
Research School of Physical Sciences and Engineering, Canberra, ACT 0200, Australia
Australian
National
University,
Received 2 May 1996; revised manuscript received 20 June 1996: accepted for publication 24 June 1996 Communicated by B. Fricke
fl\bstract
IJsinga correlatedfour-body final-statewavefunctionthe angulardistributionof fragmentsfollowing doubleionisationof heliumby positronand electronimpactis studied.Distortion of angular patterns due to projectile-electron and projectilenucleus interactions is investigated. Further, the influence of two-body Coulomb-density of states in the four-body continuum is envisaged. It is shown that these interactions, previously neglected in Born-type treatments, play a major role in determining
the crosssectionat low and intermediateenergies. t’ACS:
34.50.H; 34.80.D; 35.80.fS
There has been an increased activity in the study of double ionisation prompted by recent advancesin coincidence-detection techniques [ l-41 (see Ref. [ 51 ‘or a comprehensive review). A major aim of these ‘neasurementsis to simultaneously determinethe monentum distributions of the double-ionisation fragnents and to study the influence of various interac:ions between particles involved in the collisions on :hesedistributions. The first experiments of this kind ssing high-energy electrons (5 keV) as projectiles, :he so-called (e, 3e) measurements,have been performed using argon and krypton targets under condi;.ionswhere littlemomentum is being transferredto the larget atoms [ I ,2]. In this casea first-order Born-type approximation (FBA) in the projectile-target potenI:ial is justified. Basically, the problem reducesthen to I:he description of a three-body continuum final state and has been treated by numerous authors [ 6-131.
3375.9601/96/$12.00 Copyright 0 1996 Elsevier Science B.V. PLI SO375-9601(96)00537-3
All
Within the FBA the samecrosssection is obtained for both particle and anti-particle collisions. In addition, it has been shown in Ref. [ 111 that ISA-cross sections for double ionisation by particles of arbitrary masses can be obtained from the (e, 3e) crosssection using a scaling formula which hasbeen derived from pure geometrical arguments.In other words, within the FBA, the properties of the projectile have no dynamical effects on the cross sections. In this Letter we presentthe first calculations of the fully differential crosssection for double-ionisation of helium in its ground stateby positron and electron impact where all particles in the final statesare treated on equal footing. Thus, double ionisation at low and intermediate impact energiescan then be treated. This is of particular interest since the magnitude of the double-ionisation cross section is known to peak at an excessenergy a few times the ionisation-potential
rights reserved.
238
J. Berukdur/Physic.s
Letters
energy. In addition, the first measurements at intermediate incident energies have been performed using the cold-target recoil-ion momentum spectroscopy [ 3,141. Hence it appears timely to develop a theoretical model which goes beyond the FBA and to consider the influence of the projectile-nucleus and the projectile-electron interactions on the cross sections. The extension of the model presented here to the treatment of double ionisation of helium by an arbitrary, structureless projectile will be published elsewhere. Atomic units are used throughout, the nucleus is assumed infinitely heavy with respect to the electron mass. The multiply differential cross section for the simultaneous detection of the scattered projectile and the two ionised electrons emerging with momenta (with respect to the nucleus) k,, kb, k,, respectively, is given by
(1) where ki is the momentum of the incident projectile and E; and Ef are the total energies in the initial and the final channels, respectively. The two-particle transition matrix element 7” has the form Tf, = Wk,,kb.k,
(r,,
rb, rc> lvl@(rQ,
rb. rc)),
(2)
where the asymptotic initial state of the system is described by @(r,,rb,rc)
= (2r)-3’2exp(iki
‘r,)
$D(Tb,Tc). (3)
In Eqs. (2), (3) the vectors rG, rb, rc denote the positions, with respect to the nucleus, of the projectile and the two target electrons, respectively. The twoelectron bound-state wavefunction of He( ‘Se) is denoted by (D( rb, r,) . The transition operator v is taken as the projectile-target potential in the initial channel (4) In Eq. (4) the projectile and the nuclear charges are designated by Z, and Zr, respectively. The four-body wavefunction Pk;,., k, ( r,, rb, rc) describes three charged particles in the continuum of He2+ with outgoing boundary conditions. In the context of electronimpact single ionisation followed by Auger decay,
A 220 (19961237-241
some aspects of the four-body Coulomb continuum problem have been studied within the eikonal approximation [ 15,161. In this work we adopt a procedure similar to that employed in Ref. [ 171 in which the four-body system is broken down into six spatially decoupled two-body subsystems which results in the approximate four-body wavefunction w‘k;,kl, k, (r,, rb, rc) % (2%-) p9’2n x
IF~ [iaj, 1, -i(kjrj
+ kj - rj)]
y;eik’.’ ,
(5)
where j E {a, b, c, ab, ac, be}. The momenta kab, kac, kbc are respectively conjugate to the relative coordinates rob := ra - rb, rat := ra - r,., rbc := rb - rc. The Sommerfeld parameter LY~Lwhich describes the strength of interaction between particle “k” and “I” is given by Lvkl := ZkZpkl/kkl, where Zk and Zl are the charges of the particles “k” and “I” respectively and ,+l is their reduced mass (thus LY, = ZpZT/ka, ab = -.&/kb, ac = -&/kc, %zb = -&&k,b, %c = -&/ 2k,, abc = 1/2kbC). In Eq. (5) the normalisation constants Nj are given by Ni = exp( -ajr/2)
I’( 1 - iaj).
(6)
It is straightforward to show that the wavefunction (5) satisfies the Redmond-boundary conditions (see Ref. [ 181) for large interparticle separations. An important feature of the wavefunction (5) is that it exactly diagonalises the total potential and neglects parts of the kinetic energy operator. Calculations of scattering amplitudes using Eq. (5) as final-state representation require, however, the evaluation of a ninedimensional integral (over rat rb. r,) which is not feasible yet. Thus, we are obliged to make a further approximation. If we assume the total potential to be exactly diagonalised by the wavefunction (5) only along the paths of free particles, which is reasonable for large interparticle separations, we can employ a procedure similar to that introduced in Ref. [ 191 and the wavefunction, given by Eq. (5), reduces to the expression
X
IF~ [ia,(k,,ka,k,),l,-i(k~ir,i+kj.rj)], (7)
J. Berddar/Physics
Letters
where j E {a, b, c}. The wavefunction (7) treats all particles on equal footing in the sensethat all interactions of the continuum particles with the nucleus are assumedto be of the sameorder. This in contrast to FBA treatments where the projectile-atom finalstate interactions are disregarded. The momentumcependent Sommerfeld parameters have the form z?j := zL(k,, kb, k,)/kj and the dynamical product charges 2, read [20] tk +zjkb+ n L1
-‘p
k2,,
( k, + kc) k,, (8)
;?,, =-& + 2
-ZP (ka + kb)k,b
1 + (kb + k,)kbc
$3
(9) 1
tkb+k,)kbck;.
(10)
,\ccording to Eqs. (8)-( IO) the correlations between the continuum particles arecompletely subsumedin an effective interaction of each continuum particle with Ithe nucleus. With the product charges (8)-(10) the xvavefunction given by Eq. (7) exactly diagonalises ;.hetotal potential along the trajectories of free particles. This is readily concluded from the fact that the :?roductcharges,given by Eqs. (8) -( lo), conservethe :otal potential along the paths of free particles since along these trajectories the relation C, Z,/r,i = V; ,j; E {a, b, c} applies where V is the total potential. In addition, the wavefunction given by Eq. (7) hasa regJlar asymptotic behaviour in subspacesof the asymp:otic regions. Details of the derivations and the physical interpretations will be given elsewhere [20]. We note here that disregarding any correlation between the continuum particles, i.e. in a model of three independent particles moving in the field of He*+, the product charges,given by Eq. (8) -( lo), reduce to the first term of the respective equations.In this casedouble ionisation is prohibited provided initial and final statesare eigenfunctions of the sameHamiltonian (the Initial-state wavefunction then contains no electronic correlations) ~A vanishing z, leads to the FBA treatment. For a radially correlated helium-ground state wavefunction [ 211 andusing the wavefunction (5) for the description of the final state, the nine-dimensional
A 220 (1996) 237-241
239
integral, given by Eq. ( 1)) can be analytically reduced to a one-dimensionalintegral on the real axis [ 201. To consider the symmetry properties of the finalstate wavefunction we remark that for the caseconsidered here (Zr = 2) the spin-orbit interaction is negligibly small so that L and S are good quantum numbers and an L-S-coupling schemecan be employed. Thus, for positron impact the final-state wavefunction, given by Eq. (7)) must be symmetric with respect to exchangeof the ionisedelectrons sincethe spatial part of the helium ground state and the transition operator (Eq. (4)) are both symmetric under this operation. In contrast, for electron-impact double-ionisation the cross section is a statistical average of doubletand quartet-scatteringcrosssections[ 201 respectively, correspondingto a symmetric and antisymmetric spatial wavefunction in the final state. For the caseof large impact velocity and little momentumbeing transferred to the target atom, cross sectionsresulting from electron and positron impacts converge to each other (see below) which meansthat in this limit electron-impact double ionisation into the doublet channel dominates. To assess the influence of the projectile-nucleus and projectile-electron interactions on the cross sections, we consider, at first, the caseof moderateincident energy (Fig. 1) . In this situation all particles in the final channel are detected with quite different energies. Neglecting the projectile-target final-state interactions (as done in an FBA treatment or in a plane-wave description of the continuum particles) results in equivalent cross sectionsfor positron and electron impact. The present model, however, predicts in the case of Fig. 1 a larger cross section for electron impact than for positron impact. The interpretation of the angular distribution (Fig. 1) of the secondaryelectron is quite obvious as the fast electron (hereafter electron “P) is predominantly emitted into the momentum transfer direction q := ki - k,. This allows the conclusion that the projectile directly collides with electron “b” which is then ejected into the direction q. The slower electron (electron “c”) is then ionisedby meansof initialstate correlations (shake-off). Since the ejection of electron “6” is mainly due to two-body interactions, its angular correlation pattern resemblesthe angular distribution of slow secondary electrons produced by singleionisation of atomsby fast electrons 1221. This strongly supportsthe suggestionthat the peak in Fig. 1 which emergesin the backward direction with respect
240
J. Berakd~~r / Physics
Letters A 220 t 19961237-241
e
Fig. I. The angular distribution of a secondary electron (electron “B”) escaping with an energy of 100 eV after electron- or positron-impact double ionisation of He. The other electron (electron “c”) is detected under a fixed angle of 130’ with respect to the incoming projectile direction and with a fixed energy of I eV. The emission angles of the electrons are measured with respect to ki. The incident energy is chosen as 800 eV and the projectile is scattered through a Y-angle with respect to the incident direction. All momenta ki, kn, kb, k, lie in the same plane. Results are shown for double ionisation by positron (solid line) and electron (dotted line) impact. A Slater-type wavefunction (o(rb, Tc) = N[exp( -C,,f-,, - W-c) + eXp( --c&, - C/,l-,)] has been used to describe the ground state of helium, where N is a normalisation constant and cl, = 2.17, cc = 1.21 are variational parameters. For electron impact, the magnitude of the cross section for the emission of electron “b” under an angle of 40 = 90’. where cos 4b = k, . pi, is equal to I .96 x 10W5 a.u.
to q is a recoil peak, i.e. after an encounter with the projectile the atomic electron “8’ scatters off the nucleus to emerge in a direction opposite to q. The latter interaction with the nucleus is provided by the initialbinding wavefunction [ 231. We remark here that the well-known q-cylindrical symmetry of cross sections obtained by the FBA for electron-impact single ionisation [ 241 is, in general, not reproduced by the present model due to final-state interactions. If the projectile and the secondary electrons are well separated in momentum space the difference between cross sections resulting from positron and electron impacts diminishes with increasing incident energies (Fig. 2), as final-state interactions of the projectile with the target atom become marginal in this case. Nevertheless, at incident energies as high as 5 keV there are still small differences between double ionisation by electron and positron impact which signifies a slow convergence to the FBA limit. In the case of Fig. 2 the angular
Fig. 2. The same collision geometry as in Fig. 1, but the incident energy is increased to 5000 eV. For electron impact, the cross section at Dot, = 90” is equai to 2.66 x 10-a a.u.
distribution of the ionised electron is sharply peaked around q. For even higher impact energies the cross section depicted in Fig. 2 tends to a S-function distribution centred around q. This is due to the fact that the initial-momentum distribution of the ionised electron becomes less significant with increasing projectile velocity in the initial channel. The projectile-electron encounter then resembles more and more a classical projectile scattering from a stationary free particle. The product charges (8)-( 10) are suitable to describe strong correlations between collision partners. Whenever two final-state particles approach each other in momentum space the corresponding dynamical product charges change in such a way that strong repulsion or attraction between these two particles is simulated depending on whether the collision partners are of the same or different (physical) charge states. A striking example of this situation is when an incident positron and one of the ionised electrons emerge with comparable velocities (Fig. 3). In this case the angular distribution of this electron (and in fact of the e. projectile) reveals a sharp peak at the direction kb-k, M 1. In the projectile-electron relative-velocity, upe, spectrum the latter peak develops to a pole of the first order which occurs at upe = 0. This singular structure is due to a diverging Coulomb-density of states of the projectile-electron two-body subsystem. The analogous process in single ionisation of atomic systems by positively charged particles is the electron capture to the projectile continuum (ECC) 125,261. In the case of negatively charged projectiles an expo-
.I. Berakdar
/Physics
Letters
A 220 (1996)
237-241
Foundation and the Australian National University der contract V.3-FLF.
241
un-
References
Fi:;. 3. The double-ionisation cross section of He( ‘S”) by positron impact. The incident energy is equal to 900 eV. The ionised electron “c” is detected under a fixed angle of 130’ with respect to the incident direction and with an energy of 10 eV. The momenta k,. k,, ko. k, lie in the same plane. The angular distribution of the otl:er secondary electron (electron “b”), which emerges with an energy of 360 eV, is depicted. The projectile is scattered through an angle of 5’ with respect to the positron beam. The magnitude of the cross section at 40 = 0’ is equal to 3.5 x IO-* au.
nential dip instead of the divergence appears in the cross section. We note here that in case of soft collisions, i.e. of very slow escaping electrons, the crosssection singularities arising from diverging Coulombdensity of states of the nucleus-ionised electron subsystems are exactly removed by the kinematical flux factors contained in Eq. (1). In contrast, due to the projectile-nucleus repulsion there is an exponentially vanishing probability for a projectile positron to lose its whole momentum during the collision and emerge with vanishing velocity with respect to the nucleus. To summarize, the first calculations for positronand electron-impact double ionisation of a helium target in its ground state have been performed using a four-body correlated final-state wavefunction. It has been shown how the projectile-target final-state interactions modify the angular distribution of the ionised electrons. In addition, the influence of the Coulomb&nsity of states in a four-body system in the continulrn has been pointed out. I would like to thank Steve Buckman and Erich Weigold for helpful comments and suggestions. This u ork was supported by the Alexander von Humboldt
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