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Post-collision interaction effects in the autoionization of atoms excited by ion impact
Beam tnteractions with Materials 8 Atoms
ELSEVIER
Nuclear
Instruments
and Methods
in Physics Research
B 132 (1997) 18%301
Post-collision interaction effects in the autoionization excited by ion impact R.O. Barrachina Centro
Atbmico
Brrrilochr.
Cotnisicitt
Nucionul
de Enrrgitr
Atcin?icu
/ CNEA
of atoms
’ I. 8400 S. C. de Bariloche.
Rio Negro.
Argmtinu
Abstract When an autoionizing state decays. the spectral line shape of the emitted electron is affected by the post-collision interaction (PCI) with other collision partners. This is the case in slow ion-atom collisions, where the autoionization line profile displays a strong enhancement in the forward direction. The aim of the present work is to describe the PC1 effects that occur in outer-shell autoionization by ion impact. Different theoretical descriptions and experimental results are reviewed. 0 1997 Elsevier Science B.V.
1. Introduction The ionization of atoms, ions and molecules by the impact of other particles can take place by a number of different mechanisms. In particular, it can proceed by the excitation to certain discrete states which decay spontaneously by the emission of electrons rather than photons. This radiationless process is called autoionization effect. The term “Auger effect” is also used in honour of Pierre Auger, who discovered it in 1925 [l]. However, this latter term is usually reserved for those electron emission processes that result from inner-shell vacancies. The electrons are emitted with a sharply defined energy and a characteristic angu-
’ Also member of the Consejo National de lnvestigaciones Cientificas y Tdcnicas (CONICET). Argentina. Tel.: 54 944 45234: fax: 54 944 45299: e-mail: [email protected]. 0168-583X/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PIIs0168-583x(97)00451-5
lar distribution. However, in the presence of other charged particles in the continuum, the spectral line shape of the emitted electron is strongly affected by the post-collision interaction (PCI) with the corresponding Coulomb fields. This is the case, for instance. of the Auger process where the emitted electron interacts with the low-energy photoelectron, or when the autoionizing state is formed by the impact of a charged projectile. PC1 effects were first observed in the outer-shell autoionization decay of atoms excited by the impact of slow ions 121. In inner-shell ionization processes, PC1 phenomena were recognized some years later [335]. Early theoretical descriptions [6,7] draw attention mainly to the distortion of the energy spectrum of the ejected or receding electrons. As was first shown on classical grounds by Barker and Berry in 1966 [6], the energy distribution of the emitted electrons is broadened and shifted. displaying an asymmetrical profile. These
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conclusions were validated by other semiclassical and quantum models (see e.g. the review by Kuchiev and Sheinerman [8]). Nevertheless, the interaction among the different post-collision partners can produce another effect on the electronic distribution which was not predicted by these early models. When the emitted particles move at velocities that are comparable in magnitude, the line profiles are not only broadened and shifted, but a sharp angle dependence is likely to occur. This effect, predicted by Dahl et al. in 1976 [9], was first observed by Swenson et al. [lo] in 2-15 keV He+ + He collisions as a sharp enhancement of the line shape in the direction of the receding ion. They explained this observation by modelling the classical deflection of the ejected electron in the Coulomb field of the projectile. The vanishing of the element of solid angle at zero degree leads to a singularity in the forward direction; an effect which has been named Coulomb fix-using [lo]. Including the interaction with the receding projectile in the post-collision electron state by means of a continuum distorted wave (CDW) approximation, Barrachina and Macek [l l] derived an analytical expression for the electronic emission amplitude. The corresponding cross section shows a similar sharp angular dependence that agrees with the experimental results of Swenson et al. [lo] both in shape and magnitude [12]. The general description of PC1 phenomena developed by Kuchiev and Sheinerman [13] in 1988 uses an alternative approximation for the final state which, for the particular case of heavy ion collisions, leads to results which are equivalent to those of the CDW model. The PC1 influence on the angular distribution of autoionization electrons is more easily investigated in ion-atom collisions than for the case of electron impact. where the projectile’s scattering might diffuse the strong effect observed by Swenson et al. [lo]. In this sense, the outer-shell autoionization decay induced by the impact of an ion represents an ideal situation where the final state is composed of one light particle moving in the continuum of two heavy ions. In this review we shall analyse mainly this latter case. In photoionization or electron collision processes, an experimental study of a PC1 angular dependence would
289
require a coincidence measurement of the autoionization and “primary” electrons [ 14,151. Other effects, besides the aforementioned PC1 phenomena, might distort the energy and angle dependence of the observed electron line shapes. For instance, as was first noticed by Rudd et al. in 1966 [ 161, an additional line broadening and energy shift can be produced by the recoil of the emitting atom [9,17,18]. Furthermore, the interference of the resonant and “direct” ionization amplitudes gives rise to strongly modified profiles [19,20], as was first discussed by Fano in 1961 [21]. Similarly, the interference between overlapping resonances might also affect the electron line shapes [l?]. Finally, let us note that the presence of another electron in inner-shell ionization processes might lead to some additional PC1 distortions of the Auger line shape [22]. The aim of the present work is to describe the PC1 effects that occur in autoionization processes induced by ion impact. We shall proceed by steps of increasing complexity, towards a full description of these phenomena. In Sections 2 and 3 we review and compare different classical and quantum-mechanical models for the PC1 distortion of the energy and angle distribution of the emitted electrons. In Section 4 we describe how the projectile influences the angular dependence of the emitted electron spectra. We show that the line shape may present a sharp enhancement in the forward direction when the charge of the projectile is positive. This effect occurs both in a quantum-mechanical description and in its classical approximation. and can be attributed to a forward glory effect. We also show that a rainbow effect might occur for negatively charged projectiles, characterized by a deep depletion of electron emission in the forward direction and an enhancement at intermediate emission angles [23]. In Sections 5-7 we discuss three different interference mechanisms that can modify the autoionization line shapes. These are a path interference mechanism proposed by Swenson et al. in 1991 [24]. and the interference with the direct ionization process and between overlapping resonances. Finally, we discuss in Section 8 the appearance of additional PC1 effects due to the presence of the other electron in inner-shell ionization processes.
290
R. 0. Barruchinu I Nd.
2. Phenomenological
Insrr. and Meih. in Ph~a. Rex B 132
model of Barker and Berry
We consider the collision of a heavy ion of charge Z, and velocity VP with an atom which, after being excited to an autoionizing state, decays by ejecting an electron of energy &. Let us assume that the electron emission probability decays exponentially with time as (atomic units are used throughout)
(I 997) -188-301
effect, the quantitative comparison of this phenomenological line shape with experimental data was not conclusive in Barker and Berry’s work [6]. Nevertheless, the distinctive features of the autoionization line shape in Eq. (2) were experimentally confirmed in later investigations on ionatom collision processes 126,271.
3. Quantum-mechanical
dP - = dt
r
eprr o(t),
where o(t) is Heaviside step function and l/r is the characteristic lifetime of the autoionizing target state. We are assuming that the decay occurs as a post-collision process, when the projectile is at a large distance R(t) M upt from the target. For the time being, we neglect any Stark mixing effect [25]. Thus, the action of the projectile on the target electrons reduces to a shift -Zp/R(t) of all their energies in an independent-electron description. The initial and final state energies at the time t of decay read E,(t) = E, - ZrZP/R(f) and Ef(t) = ef + E(Z, - l)Zp/R(t), where t, and cf are the energies of the corresponding localized orbitals, while E is the energy of the emitted electron. Therefore, in this first order approximation, the PC1 with the projectile modifies the energy of the emitted electron from E. = E, - cf to E M En - Zp/(~!pt). This equation provides a one-to-one relation between the energy of the ejected electron and the time t after the excitation stage. Changing variables from t to E in (l), we obtain the following emission probability: dP dE
ZPl-/UP =
(&
-
exp(-(E))
It is important to emphasize that the former model is of a phenomenological nature, and ignores important aspects of the autoionization process. For instance, the possible interference between different coherently excited states or the continuum is not taken into account. In order to consider these effects, it is necessary to resort to a quantum-mechanical treatment. In this section we shall review different quantum models in a unified description [23], although the original derivations were done independently. We consider that the emitted electron travels with velocity Y in the two-center Coulomb field of the recoil target and the receding projectile. The continuum state of the electron can be decomposed in the following way [l 11: Y,, = D(u,R)
(2) This energy distribution, obtained by Barker and Berry [6] in 1966, has a maximum at E = E. - Zpr/20p and a full width at half maxi1.07 Zpr/up, with a pronounced tail mumofA/?= towards the lower or higher energies, depending on the sign of the projectile’s charge. Due to an insufficient resolution and an unnoticed Doppler
G,(v).
where $, (v) is the continuum wave function of the electron moving with velocity v in the field of the residual target ion alone. o(r, R) incorporates the distortion due to the electron-projectile interaction. In a similar way, we separate the autoionization transition amplitude in the following way [11.13]: A = .f’(E, H)
E)’
theory
r-12 E - E. + ir/2
@,
0):
where E = v’/2 and H = arccos (i. GP) are the energy and angle of the emitted electron. The reduced amplitude k and the lorentzian term u-/2)/p - E. + ir/2) account for the excitation to the autoionizing state by the projectile-target collision and the natural decay which would take place if the projectile had no influence on the ejected electrons. In particular, the angular behavior of k resembles the symmetry of the autoioniz-
291
R. 0. Barrachina I Nucl. Insrr. and Meth. in Pkys. Res. B 132 (1997) 288-301
ing state. On the other hand, the multiplicative factor f(E, f3) is related to the dynamic of the final state, describing the PC1 between the projectile and the emitted electron. In a first-order description, this term can be approximated by [ 11,131 f(E, 0) = - i(E - E,) + iI/2) x j.o.(O,
“$) ( ,)izp/cp
0 x exp [i(E - E0+iF/2)t] dt.
(4)
The great majority of the quantum-mechanical models [28,29] for autoionization processes in ionatom collisions allow for a factorization as given by Eq. (3). The basic advantage of this decomposition is that it predicts in a very simple way how the PC1 modifies the autoionization reaction. From Eq.(3), we obtain the following factorization:
dEdQ
(5)
for the autoionization cross section. This expression decouples the PC1 effects, as given by the distortion factor If(E, 0)]‘, from the actual process of autoionization. In principle, the various quantummechanical descriptions of PC1 effects are based on different approximations of the factor D, although the original derivations did not follow the same line of reasoning. Now let us briefly discuss these theoretical models.
qf Devdariani,
Ostrovskii and Sebayakin
The first quantum-mechanical calculation was developed by Devdariani et al. in 1977 [30]. They neglected the effect of the projectile ion on the electronic wave function by making D(r, R) = 1. This approximation implies that the projectile does modify the energy but not the trajectory of the emitted electron. The resulting PC1 amplitude jbos(E) = I(1
’
+iz)
exp(-2)
r/2
iZp/cp
E - E,, + iI-12 >
and the distortion factor
sinh(nZp/vp)
-2zarctan(F))
x exp
(7)
( are independent of the emission angle 0. They depend only on the dimensionless quantity vp = Zp/vp and the reduced energy E = 2(E - Eo)/r. We note that fDos(E) approaches unity for vp -+ 0. This means that the velocity spectrum approaches the natural linewidth distribution if the projectile exerts no influence upon the emitted electron. At the opposite limit ]vp] $1, Barker and Berry’s energy distribution is recovered [23]. We therefore see that the model of Devdariani et al. [30], which represents the most crude approximation of the electron’s continuum state (D = 1) leads to an energy distribution that is shifted from E = E0 and has the correct behavior at the limits vp -+ 0 and /vp( ---f+cc.
da
I-‘/4 (EYE~)~+F~/~
3.1. Model
nzp’vp
lfDos(E)12 =
3.2. Eikonal approximation
The model of Devdariani et al. is not consistent with the fact that, due to its infinite range, the electron-projectile Coulomb interaction does not vanish at large distances. Therefore, at least a logarithmic phase distortion must be kept in the lowest order approximation, namely D(r, R) M (v’lr - RI + V’ . (Y - R))izp’L”, with V’= v - vp the relative electron-projectile velocity. Substituting into Eq. (4), we obtain iZpJ7.p
YEA@,
4
=
r(~n$iv)
(E
_ Ey/:
E _ & + ir/2 X
(
ir,2) izdo’
u’up- v’ vp )
i
with v=---_.
ZP
ZP
up
0’
We note that the corresponding factor IDEA@,
e)l’
=z
PC1 distortion
sinh(nv)
(6) x exp (-2v
arctan (9))
(8)
is the same as the one obtained by van der Straten and Morgenstern [3 l] and differs from that of Devdariani et al., Eq. (7), only in that the dimensionless parameter vp = Zp/rp is now replaced by 1’. The discrepancies between both theories are negligible when U’ is much larger than tip. However, when 1)’< up, the corresponding energy shift of the intensity maximum AE = -l-r/2 changes its sign. In the case of a positively charged projectile, this leads to a high energy tail in the energy spectrum of the emitted electron, instead of the shift to lower energies predicted by the model by Devdariani et al. Furthermore. in the matching velocity region, a strong increase of AE is expected to occur. This enhancement agrees with the tendency of different experimental measurements [32.33]. Fig. 1 shows experimental results by Vikor et al. [33] for the energy shift of the Ne K-L?.3 L?.j (ID?) Auger line at 0” as a function of the velocity I’~ of a H+ projectile relative to the Auger electron velocity uo. We clearly see that the theoretical energy shift closely matches the experimental data.
Except eventually for the “reduced” cross section dii/dE dQ, the autoionization cross section in the eikonal approximation (Eqs. (5) and (8)) depends on the emission angle only through the parameter ~1. However, as was first observed by Swenson et al. [lo] in He++ He collisions. the PC1 with the projectile can strongly modify the angular distribution of the emitted electrons. In order to explain this result, Barrachina and Macek [l l] considered a CDW approximation [34,35] of the electron final state,
- i (011~- RI + v
f;-ow(E, 6) =
(Y - R))].
1 + i3 (
,
,
I
,
0.8
1.0
1.2
1.4
Projectile velocity ratio I 0.94
0.6
I / 1.48 2.13 Proton energy (MeV)
1.0
0.8
1.2
I 2.89
1.4
q”A Fig. I. Normalized
intensity and energy shift of the Ne K-L2,1 L2, (‘IL) Auger line at 0” as a function of the velocity Q of a H- projectile relative to the Auger electron velocity t’,. (0 l l). Experimental data by Vikor et al. [33] normalized to the total Auger-line intensity. (c 1a.1 ,L,).Experimental data normalized to the theoretical K-shell ionization cross section. ( ~~). CDW theory [l l] (From Ref. [33]).
L’P
)
x I(l-i%)exp($-2) r/2 X
E - E,, + iI/
““-”
’ vp
E - E,) + iF/2 )
in (4) we obtain
r
0.6
]F,[ _ i&/L’I: 1; e nZP’2r’ ’
D(u, R) =I-( 1 + iZp/rj)
Replacing
08iI
I&/i ,> ?F, i3: ) ( 1.’
(9).
The autoionization cross section is obtained, through Eq. (5). from the square modulus of this amplitude. It is clearly seen that it incorporates a dependence on the emission angle that is missed in previous theories (see Fig. 2). In particular, for a positively charged projectile, Z, > 0. it predicts a sharp enhancement in the forward direction when the projectile velocity rp is smaller than the
RO.
293
Barrachinu I Nucl. Instr. and Meth. itt Phv.7. Rrs. B 132 (1997) 288-301
3.4. Comparison of the diflerent theories The models of Devdariani et al. [30] and van der Straten and Morgenstern [31] can be obtained as particular cases of the CDW approximation in different regions of the electronic velocity space [I l] (see Fig. 3). For instance, in the forward direction, as given by V’VP- v’ . vp -cc l-12, we obtain
This amplitude differs from that of the model of Devdariani et al. only in the Coulomb normalization factor f( 1 - iZp/v’) enZp/“‘. This normalization factor generates the usual cusp-shaped peak in the electronic intensity when the electron velocity matches that of the projectile. On the other hand, far from the forward direction, i.e. for n’vp - v’ . VP> r/2, the CDW amplitude reduces to the eikonal approximation YEA. As it was previously explained, this amplitude approaches that of Devdariani et al. [30] when v’ is much larger than up. The same is true in the forward direction, where the CDW amplitude reduces to foes for large electron-projectile velocities (i.e. for v’ > Zp).
4. Glory and rainbow effects Fig. 2. The CDW approximation for the electron emission intensity in the autoionization of He excited to a (2~~)‘s state by the impact of a (a) proton (b) antiproton of velocity rp = 0. I au. (from Ref. [23]).
autoionization velocity ug. This behavior of the intensity profile was first observed by Swenson et al. [IO] in He+ - He collisions. Fig. 1 shows experimental data measured by Vikor et al. [33] for the intensity in the Ne K-L?.3 L2.3 (‘D2) Auger-line forward direction as a function of the velocity ZIP of a H+ projectile relative to that of the Auger electron t:“. The data are normalized to total intensity into 471 solid angle. We note that the agreement between experiment and theory (full curve) is reasonable. Furthermore, the enhancement of the intensity for up < L:~, as predicted by the CDW model, is clearly seen. Recently, different authors demonstrated that the [12,32,33,36,37] have CDW model is very successful in achieving a quantitative coincidence with the experimental data.
The strong dependence the CDW approximation,
on the emission angle in as shown in Fig. 2, can
DOS x CUSP
-1 -
Fig. 3. The model of Devdariani et al. (DOS) and the eikonal approximation of van der Straten and Morgenstern can be obtained as particular cases of the CDW approximation in different regions of the electronic velocity space. See text for details.
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I Nucl. Instr. urld hleth. in Ph_vs. Res. B 132 ( 1997) 28X-301
be better understood by means of a classical description of the autoionization process. In 1976, Dahl et al. [9] had pointed out that some kind of dependence of the autoionization cross section on the emission angle was likely to be produced by the deflection of the ejected electron in the Coulomb field of the outgoing projectile. Even though Dahl et al. [9], Arcuni [38] and van der Straten and Morgenstern [31] explicitly considered this postcollision distortion of the electron’s trajectory, a small deflection approximation prevented them from obtaining a significant dependence on the emission angle. Finally, in 1989, Swenson et al. [lo] developed a classical generalization of Barker and Berry’s theory which modelled the motion of the ejected electron in the field of a positively charged projectile. In a first order approximation. the angle dependence of the final velocity distribution for z)~ < u0 is shown to be dominated by an e$ f&title cross section [ 10,231
with 2’ and 0’ the projectile-frame emission angles before and after scattering, respectively. This quantity is operationally equivalent to the differential cross section for the elastic scattering of a beam of particles by a force center, except that here the trajectories are parametrized not by an impact parameter p but by the initial emission angle x’. which makes this CcL a dimensionless quantity [23]. In the approximation employed by Swenson et al. [lo], where the post-collision interaction of the emitted electron with the residual target is neglected, ,XcL can be analytically evaluated [23,39] .X‘ti_( cos 0’) 1 ’ =-[
1 - (1 fA)COSH (2(1 - cosH’) - (1 +.d)sin’H’)“~
and CI’is not one-to-one. There are two different initial emission angles c& which lead to the same deflection angle 0’. This means that. when calculating the autoionization cross section, both contributions have to be added up. The relation ~‘(0’) and the effective cross section, Eq. (lo), are shown in Fig. 4 for (a) a positively and (b) negatively charged projectile [23 . For Zr > 0, &_ diverges at 8’ = 0 as zf,_ % IAl’1’ /(3101). This divergence is characteristic of a for~~urcfglory eflect [40], and occurs whenever the dispersion function 0’(c(‘) passes smoothly through 0 as a function of the initial parameter a’ [23,40]. The electron-projectile potential deflects those trajectories with initial emission near angles ‘XL= arccos m into the forward direction, where the solid angle vanishes and the angular density of trajectories diverges. This effect has usually been described in the literature as a Coulomb f&wing mechanism [ 10,12,29,33]. However, this phenomenon, i.e. the deflection of one particular trajectory into the forward direction, can equally occur with practically any other attractive potential. Furthermore, the idea encompassing a ,fkusing mechanism, i.e. that “the deflection effectively compresses the emitter-frame solid angle, resulting in an enhancement, or focusing, of electron intensity in the direction of the ion” [lo], is misleading. In fact, those electrons emitted within the cone CL< XL are not focused in the forward direction, but scattered into the full (0,~) range. Hence. it seems reasonable to keep the standard definition introduced by Ford and Wheeler almost forty years ago in their classical paper on the semiclassical scattering theory [40], describing this divergence of the autoionization cross section as a “forward glory effect”. For the case of negatively charged projectiles (ZP < 0). the effective cross section diverges like
.?FJIs-A 1
(10) The dimensionless parameter A = -(ZP/R)/E;, represents the ratio between the potential and the kinetic energies (in the projectile’s frame) at the moment of emission. The two branches of CCL originate from the fact that the relation between H’
z;L
~
&Gmi 4cos(t&/2)
1 d_
@($’ - 0;)
at a characteristic angle 0; = 2 arctan strictly zero inside a shadow zone 0’ none of the trajectories can penetrate. Fig. 4 that this angle 0; represents of the dispersion relation 0’(~‘). Thus,
(11)
(a) and is < Hk, where We see in a minimum the effective
R.O. Burmchinu
I Nucl.Instr. and Meth. in Phys. Rex B 132 (1997) 288-301
60
120
C a)
395
118C
0’ (deg)
Fig. 4. Relation between the emission angle 2’ and the final angle 0’. and the corresponding effective cross sections for (a) a positively charged projectile with A = -7/3 and (b) a negative one with A = 2/3. There are two emission angles t$ for each 0’. For Z, > 0 there is a forward glory divergency. For 2, < 0 the effective cross section diverges at 0; due to a rainbow effect (from Ref. 1231).
cross section CCL( cos 0’)) being proportional to (d@/dcc’))‘, diverges. This kind of divergence, geometrically analogous to what happens to a beam of light rays scattered by a water droplet, is known as rainbow effect [40]. We see in Figs. 2 and 4 that the classical and quantum-mechanical cross sections show some important similarities. For instance, in both cases, the glory and rainbow effects are more pronounced, the greater the charge the slower the relative electron-projectile velocity. Furthermore, the shadow zone in the rainbow phenomenon becomes broader when the electron-projectile interaction increases. On the .other hand, the glory and rainbow divergences are replaced by finite maxima in the quantum-mechanical description, and the number of particles emitted into the forward dark zone for ZP < 0 is no longer zero. We
also note that the dependence of the CDW cross section on the parameters of the problem is more intricate than in the classical approach. In particular, it presents an oscillatory behavior which is not shown by CCL. These kinds of discrepancies between the quantum-mechanical treatment and its classical approximation are the same that occur between Airy’s undulatory description of the rainbow phenomenon and Descartes’ classical picture. In the following section we analyze this latter oscillatory structure of the autoionization line shape.
5. Path interference The CDW line profiles in Fig. 2 do not only show the expected glory and rainbow enhancement of the electron intensity but also an accom-
296
R. 0. Burruchintr I Nud
1nst11 und Meih
panying oscillatory structure at intermediate angles. These oscillations were predicted by Barrachina and Macek in 1989 [l l] for the case of a positively charged projectile. In 1991 Swenson et al. [24] reported to have found experimental evidence of this effect in an oscillation observed in the low-energy wing of the (2~‘)‘s autoionization line shape in a 10 keV He+ + He collision. This structure was reproduced by a direct extension of the CDW model by Macek and Cordrey [12] including contributions from adjacent lines (see Section 6). Swenson et al. explained this oscillation as due to a path interference mechanism between indistinguishable classical trajectories around opposite sides of the outgoing ion. Any given direction 0 can be reached by two distinct trajectories, giving rise to constructive or destructive interference, depending on the observation angle. A similar effect is observed for the case of a negatively charged projectile. In this latter case, the interference phenomenon has an optical counterpart in the luminous arcs which can sometimes be seen under the principal rainbow. The origin of these so-called supernurnerq~ WCS was explained by Thomas Young in 1803 in terms of this same path interference mechanism. The path interference mechanism was explicitly incorporated in a semi-classical calculation made by Swenson et al. [24]. In the quantum-mechanical calculation, the physical origin of these oscillations has been attributed to an interference process between the direct wave and the wave that scatters from the receding projectile [29,41,42]. In a recent paper [43], it has been shown that the interference structure of the CDW spectral line shape can be interpreted by means of the concept of nectrside+vside scattering (see also Ref. [24]), which was first used to describe elastic reactions of spin-zero nuclei [44]. In a nearside+farside decomposition of the autoionization amplitude, .f’ = ,f‘ + ,f + . each component [44]
it? Plzyx Rex B 132 (1997) 288-3-301
tive (nearside) or negative (farside) deflection [43]. In Fig. 5 we see that the nearside and farside contributions to the autoionization cross section for ZP > 0 show no oscillations when taken separately. The oscillations appear only in the interfer-
0
can be related in a semi-classical limit to trajectories which have traveled along one or the other side of the interaction potential, suffering a posi-
90
135
180
Angle (deg)
0
b) .f‘r(E, 0,J = lim g ’ .f‘(E, 4)sin4 d$ i.P-0~211 i’ cos0’ - cos4 *if .ll
45
I
I
I
45
90
135
180
angle (deg)
Fig. 5. Electron emission intensity at the peak position for the (3’)‘s autoionizing state of He excited by the collision of (a) a proton of velocity L’~ =0.35 a.u. and (b) an antiproton of velocity t’p =O.Ol a.“. The nearside (+), farside (-) and interference (i) contributions to the CDW intensity lAcowlZ are shown separately.
R.O. Barruchinu I Nucl. Instr. and Mrth. in Phys. Res. B 132 (19971 2X-301
ence term 2 Rev-J‘+.). For the case of a negatively charged projectile, Z, < 0, both indistiguishable trajectories travel along the same side of the projectile. Therefore, the oscillations are already present in the near term, while the far term, not representing any classical trajectory, is quite smooth. These results support the assumption by Swenson et al. [24] that the angle oscillations of an isolated lineshape are due to a path interference mechanism.
In an Eulerian description [39], the interference always occurs between trajectories that have and have not reached the glory (for ZP > 0) or rainbow (for Zp < 0) caustics. In this sense, this semi-classical explanation can be translated into quantummechanical terms as an interference between waves that are and are not scattered by the receding projectile [41,29]. This approach has recently been investigated by Kunikeev and Senashenko [42]. They separate the electron continuum wave function ‘l”, = D(Y, R) $,(v) in the CDW approximation into two components @(v, R) = eP’l”G(
‘4=
,,,2,. ci f,(E,0) 2fir J / E
where E,, l/I, and A; are the corresponding resonant energy, lifetime and inherent amplitudes. This interference is known to have a considerable influence on the emission lines, even if they seem to be well separated and no oscillatory structure is observed [31]. A direct extension of the CDW model by Macek and Cordrey [12] including contributions from adjacent lines, was found to be in good agreement with the experimental data of Swenson et al. [IO] for 5, 10 and 15 keV He+ + He collision. In particular, the results shown in Fig. 6 clearly demonstrate that the interference terms are essential for achieving a correct description of the angle dependence of the autoionization line shapes.
l
5keV
0 A
IOkeV expt. 15keV expt
1
fi,,’
I-( -iI+)
1.
expt.
-
-
__
1OkeV theory - 15keV theory
G(
(12)
_
- iv’, 1; -it),
0 (r. R) = - r(l + i”‘) e-rrr’/Ze-i;
291
5keV
theory
icr)
1_
with 1~’= Zr/D’ and i” = v’lr - R(+ v' (Y- R). Here G(a, c, z) is the irregular confluent hypergeometric function [45]. In the asymptotic region, these components represent the incoming (0) and outgoing (1) parts of the electron continuum wave function. These two terms lead to a separation of the autoionization amplitude that is related to the nearside-farside decomposition for the case of a positively charged projectile (Zr > 0) [44]. Within this separation, the oscillation observed by Swenson et al. is shown to be produced solely by the interference term.
I :I:::‘:I:I:‘:I:::~:I:I:‘:I:I:’ 4. n
0 b -. -
-
-
-
5keV expt. lOkeV expt. 15keV expt. 5keV talc. 1OkeV talc. 15keV cab.
(b)
6. Interference between overlapping lines The autoionization line shape can be significanly altered by the interference with neighboring overlapping lines. In this case, the autoionization amplitude, Eq. (3). has to be replaced by the coherent sum of the individual amplitudes corresponding to each state.
Fig. 6. Ratio of target to projectile autoionization total cross sections for 5. IO and 15 keV He+ + He collisions. Experimental data of Ref. [IO]. (a) CDW model calculations by Cordrey and Macek [12] for the (2~‘)‘s. (2p’)‘D and (Xp)‘P states. The interference term between overlapping lines has been omitted in the calculation shown in (b) (from Ref. [I?]),
198
R. 0. Burrutlzinu I Nucl. Insir. md Met/~. in Pizys. Res. B 132 (19Y7) 2X8-301
It is important to note that the relative intrinsic amplitudes Al in Eq. (12) can be modified by the post-collision Stark mixing of the close lying states due to the projectile’s electric field [25]. This effect was first observed by Stolterfoth, Brandt and Prost [47] in 1979, and included in the framework of the CDW model of Ref. [l 1] by Miraglia and Macek [25]. It was explicitly incorporated in the calculation shown in Fig. 6. This effect is known to produce a deviation from the backward-forward symmetry in the angular distribution of electron ejection, having its greatest effect in the tail of the line shape, which corresponds to small internuclear separations of target and projectile.
7. Interference with the direct ionization process The ionization of an atom by the impact of another particle can take place either directly or by the formation of an autoionizing unstable state. The amplitude for these two processes might interfere. In principle, interference with the continua is known to contribute significantly to outer-shell autoionization line shapes [46]. Generally, for a well isolated autoionizing line, the ionization amplitude is the coherent sum of a direct and a resonant contribution
A = Adir+ A,,, = &r
+
f(E, 0) E -
r/2 En t ir/2
L,
When the scattering of the projectile is negligible, the distortion factor can be taken outside the integral over the corresponding scattering angle and the following approximate expression for the ionization DDCS can be obtained [20]: da -= dE dSZ
account for the interference between the direct and resonant ionization amplitudes. With an appropriate definition of the parameters [17], this expression can also be written in the Fano form [21,48] da 1 (E-&+qr/2)’ dEdQ = Oa+ gh (E _ E0)2 + l-1/4 with (Th= -b/2 + J$ + /3’/2, ga = do/dE dQ],, -gh and q = c(/2ah. Early theoretical [49] and experimental studies [50,51] supposed that the Shore parameters, c( and p (or equivalently the Fano parameters q, CT~and ob), were weakly dependent on the electron energy E over the resonance width. However, it is now clear that this assumption is not always correct. In particular, the distortion factor ,f may produce a strong energy dependence of a and fi [52], mainly through a large variation of its phase [ll]. The first experimental evidence of this effect was published by Arcuni in 1986 [38,53]. An alternative parametrization of the autoionization line shape [54], which explicitly separates the distortion factor f’(E, 0) from the fitting parameters, has been recently employed by Moretto-Capelle et al. [46] for analysing the He(2jY) autoionization line shapes in 20-100 keV H ’ - He collisions. It reads da I do __ dE dL’ = dE da Id,, P/4 + If(E, @I’
(E - E$
do r/2 dEdL’ dir+ (E - Et))? + I-?/4 x (@(E-E”) + /K/2).
+ r2/4
do dE da
res (14)
(13)
where do/dE dQldi, = s IAdirl’ dQp is the cross section for the direct ionization process. The so-called Shore parameters.
where, now, the complex parameter cl = 2 s A& of the PC1 distorting factor f(E, 0). Recent measurements by Moretto-Capelle et al. [46] of the He(2L2C) autoionization line shapes in 20-100 keV Hi- He collisions are shown in Figs. 7 and 8. The direct ionization background (fitted to a second-order polinomial) has been sub-
A,, dQp is independent
R. 0. Barrachina
I Nucl. Insrr. and Merh. in PhFs. Res. B 132 (1997) 288-301
Fig. 7. Angle dependence of the He(2KX’) autoionization shapes in a 66 keV H+-He collision (from Ref. [46]).
line
stracted from all the spectra. All the line shapes have been fitted by Moretto-Capelle et al. using the previous parametrization, Eq. (14) (full curve). Fig. 7 displays the angle dependence of the autoionization spectra at 66 keV. We see that the well-delined lines observed at high emission angles change into typical Fano profiles at intermediate angles. At smaller emission angles, these profiles are completely obliterated by strong oscillating structures due to the interference with the direct ionization process. The projectile energy dependence of the autoionization line shapes is shown in Fig. 8 at a fixed emission angle (4”). The strong oscillatory structures already observed in Fig. 7 at 66 keV are also seen here for energies larger than 61 keV. These energies correspond to proton velocities up larger
Fig. 8. Projectile energy dependence zation line shapes at a fixed emission lision (from Ref. [46]).
of the He(2Ce’) autoioniangle (4”) in a H’ -He col-
than the velocity of the autoionizing electron ~0. The frequency of this interference increases for UP approaching uo. For up < ug these oscillations vanish and new structures appear in the low-energy wing of the autoionization spectra. Finally, below 29 keV these additional structures give place to rather simple line shapes. Only a broad satellite is seen in the low-energy side of each resonance line, resembling the one observed by Swenson et al. [24].
8. Post-collision the continuum
interaction with another electron in
In a typical Auger effect, an electron is removed from an inner-shell prior to the autoionization of
300
R. 0. Brtrruchirtcr
I Nucl. Instr.
trntl hleth.
the resulting ion. Through this mechanism, the final state includes an additional particle in the continuum. In principle, the post-collision interaction with other electrons (released during the collision stage through this or another mechanism) may lead to some additional distortion of the autoionization line shapes. Recently, T6th et al. [37] (see also Ref. [36]) have observed that the energy shift of the LMM spectra in 1.25, 1.5 and 2 MeV He- + Ar collisions shows a slight but systematic enhancement in the backward direction with respect to the CDW theory of Barrachina and Macek [l11. They explained this result as an experimental evidence of such an electron--electron PC1 effect originating from the electron loss process. Kuchiev and Sheinerman [29,55] have extended the eikonal approximation of Section 3.2 to the case when there is an additional electron in the final state. Assuming that the separation in velocity space of all the particles in the post-collision stage is large enough, the cross section for an Auger process induced by the impact of a charged projectile is modified by the same distortion factor given in Eq. (8), where now the parameter 1’ has to be replaced by ZP ~=----_+_ I’P
ZP
1
I:’
l’]
in Phvs. Rrs. B 132 (1997)
288-301
is due to the screening of the post-collision electron-projectile interaction by the primary electron.
9. Conclusions In this paper we have analyzed different theoretical descriptions and experimental results concerning the PC1 effects occurring in the autoionization of atoms by ion impact. For pedagogical reasons, we have proceeded from Barker and Berry’s phenomenological model for the line shape of a well isolated autoionization line, towards theoretical descriptions of increasing complexity. This analysis has led to the discussion of different manifestations of PC1 phenomena. Many of these effects have been theoretically and experimentally verified. This is the case of the energy shift enhancement in the matching velocity region or the glory enhanced emission in the forward direction. Other effects, like the rainbow phenomenon, are awaiting experimental verification. And still others, such as the influence of an additional electron in the final state, are in need of a better theoretical description.
1 Ir
-
Vi\
This factor incorporates the interaction of the products of the Auger decay ( i.e.. the residual target and the Auger electron), not only with the receding projectile, but also with the primary electron of velocity vI. The observation of this dependence on the velocities of all the particles in the final state would require a coincidence measurement. In order to compare with noncoincidence experiments, the cross section has to be integrated over the velocity of the primary electron [29,56,57]. This average intertwines the PC1 distortion factor with the reduced cross section for the emission of the primary electron [56]. A very interesting case occurs when the primary electron is captured into a bound state of the projectile. In 1991, Sarkadi et al. [32] observed that the ‘P-‘S Auger peak of Argon at 0” in a 700 keV H+ + Ar collision gets back to its nominal energy, showing no energy shift, when measured in coincidence with a neutral outgoing projectile. This effect
Acknowledgements I thank preparation
Mrs. Linda Yael for her help with the of the manuscript.
References [II P. Auger. J. Phys. Radium 6 (1925) 205. PI H.W. Berry. Phys. Rev. 117 (1961) 1634. H. Suzuki. K. Wakiya. Phys. PI S. Ohtani, H. Nishimura. Rev. Lett. 36 (1976) 863. 141 M.J. van der Wiel, G.R. Wight. R.R. Tol. J. Phys. B 9 (1976) L5. M.Y. Adam. F. 151 V. Schmidt, N. Sanders, N. Mehlhorn, Wuilleumier. Phys. Rev. Lett. 3X (1977) 63. 161 R.P. Barker. H.W. Berry. Phys. Rev. 1.51 (1966) 14. [71 A. Niehaus. J. Phys. B 10 (1977) 1845. Sov. Phys. -Usp. 32 PI M. Yu Kuchiev. S.A. Sheinerman. (1989) 569. PI P. Dahl. M. Radbro. B. Fastrup. M.E. Rudd. J. Phys. B 9 (1976) 1567. K. Sommer. IlO1 J.K. Swenson. C.C. Havener, N. Srolterfoht, F.W. Meyer. Phys. Rev. Lett. 63 (1989) 35.
R. 0. Barrachinu I Nucl. Ins@. and Meth in P/ITS. Rex B 132 ( 19971 2X8-301 [I I] R.O. Barrachina, J.H. Macek, J. Phys. B 22 (1989) 2151. [l?] 1.L. Cordrey, J.H. Macek, Phys. Rev. A 48 (1993) 1264. [13] M. Yu Kuchiev. S.A. Sheinerman, J. Phys. B 21 (1988) 2027. [l-l] B. Kammerling. B. Krassig. V. Schmidt, J. Phys. B 26 (1993) 261. [15] 0. Schwarzkopf. V. Schmidt, J. Phys. B 29 (1996) 3023. [16] M.E. Rudd. T. Jorgensen Jr.. D.J. Volz. Phys. Rev. Lett. 16 (1966) 929. [I 71 M.E. Rudd, J.H. Macek. Case Studies in Atomic Physics 3 (1972) 47. [1X] N. Stolterfoht. Phys. Rep. 146 (1987) 315. [19] B.W. Shore. Rev. Mod. Phys. 39 (1967) 439. [X] V.V. Balashov, S.S. Lipovetskii, V.S. Senashenko, Sov. Phys. -JETP 36 (1973) 858. [?I] I-1. Fano. Phys. Rev. 124 (1961) 1866. [X] M. Yu Kuchiev. S.A. Sheinerman. J. Phys. B 23 (1990) 2131. [23] I. Samengo, R.O. Barrachina. J. Phys. B 29 (1996) 4179. [23] J.K. Swenson. J. Burgdorfer, F.W. Meyer, C.C. Havener. D.C. Gregory. N. Stolterfoht, Phys. Rev. Lett. 66 (1991) 417. [X] J.E. Miraglia. J. Macek, Phys. Rev. A 43 (1990) 3971. [16] G. Gerber. R. Morgenstern. A. Niehaus. J. Phys. B 6 (1973) 493. [17] G. Gerber. A. Niehaus, J. Phys. B 9 (1976) 123. [28] S. Ritz. Nucl. Instr. and Meth. B 86 (1994) 119. [29] S.A. Sheinerman, Nucl. Instr. and Meth. B 86 (1994) 105. [30] A.Z. Devdariani. V.W. Ostrovskii, Yu. N. Sebayakin. Zh. Eksp. Teor. Fiz. 73 (1977) 412 [Sov. Phys. -JETP 46 (1977) 2151. [31] P. van der Straten. R. Morgenstern. J. Phys. B 19 (1986) 1361. [32] L. Sarkadi. T. Vajnai. J. Vtgh. A. Kovtr. J. Phys. B 24 (1991) 1655. [33] Gy. Vikor, S. Ritz, B. Sulik, L. Toth. A. Kover. J. Vlgh, J. Phys. B 29 (1996) L787. [ 341 1.M. Cheshire, Proc. Phys. Sot. 84 (1964) 89. [35] Dz. Belkic. R. Gayet, A. Salin, Phys. Rep. 56 (1979) 279. [36] E. Takacs. L. Sarkadi. S. Ritz, B. Sulik. L. Toth, J. Phys. B 24 (1991) L381.
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[37] L. Toth. Gy. Vikor. S. Ritz, P. Pelican. R. Miller. Nucl. Instr. and Meth. B 86 (1994) 151. [38] P.W. Arcuni. Phys. Rev. A 33 (1986) 105. [39] 1. Samengo. R.O. Barrachina. Eur. J. Phys. 15 (1994) 300. [40] K.W. Ford, J.A. Wheeler, Ann. Phys. (NY) 7 (1959) 259. 287. [41] J.H. Macek. private communication (1989). [42] Sh.D. Kunikeev. VS. Senashenko. Sot. Phys. -JETP. 82 (1996) 839. [43] 1. Samengo. R.O. Barrachina, J. Phys. B (1997) (submitted for publication). [44] R.C. Fuller. Phys. Rev. C 12 (1975) 1561. [45] M. Abramowitz, LA. Stegun. Handbook of Mathematical Functions. Dover. New York. 1970. p, 504. [46] P. Moretto-Capelle. M. Benhenni. D. Bordenave-Montesquieu. A. Bordenave-Montesquieu, J. Phys. B 29 ( 1996) 2007. [47] N. Stolterfoth. D. Brandt. M. Prost. Phys. Rev. Lett. 43 (1979) 1654. [48] U. Fano. J.W. Cooper, Phys. Rev. 137 (1965) A1364. [49] A.L. Godunov, V.N. Mileev, V.S. Senashenko, Sov. Phys. Tech. Phys. 28 (1983) 1170. [50] A. Bordenave-Montesquiey. P. Benoit-Cattin. M. Rodiere, A. Gleizes. H. Merchez. J. Phs. B 8 (1975) 874. [51] J.P. Giese, M. Schulz. J.K. Swenson. H. Schone. M. Benhenni. S.L. Varghese. C.R. Vane. P.F. Dittner. SM. Shafroth, S. Datz, Phys. Rev. A 42 (1990) 1231. [52] A.L. Godunov. Sh.D. Kunikeev. N.V. Novikov. V.S. Senashenko, Sov. Phys. -JETP 69 (1989) 927. [53] P.W. Arcuni. D. Schneider. Phys. Rev. A 36 (1987) 3059. [54] A.L. Godunov, V.A. Schipakov. Proc. 18th ICPEAC (Aarhus). Book of Abstracts 524 (1993). [55] M.Yu. Kuchiev. S.A. Sheinerman, Sov. Phys. Tech. Phys. 32 (1987) 879. [56] M.Yu. Kuchiev. S.A. Sheinerman, J. Phys. B 27 (1994) 2943. [57] Gy. Vikor. S. Ritz. B. Sulik. L. Gulyas. A. Kover. International Conference on the Physics of Electronic and Atomic Collisions. Vienna (1997). Abstracts of Contributed Papers.
ELSEVIER
Nuclear
Instruments
and Methods
in Physics Research
B 132 (1997) 18%301
Post-collision interaction effects in the autoionization excited by ion impact R.O. Barrachina Centro
Atbmico
Brrrilochr.
Cotnisicitt
Nucionul
de Enrrgitr
Atcin?icu
/ CNEA
of atoms
’ I. 8400 S. C. de Bariloche.
Rio Negro.
Argmtinu
Abstract When an autoionizing state decays. the spectral line shape of the emitted electron is affected by the post-collision interaction (PCI) with other collision partners. This is the case in slow ion-atom collisions, where the autoionization line profile displays a strong enhancement in the forward direction. The aim of the present work is to describe the PC1 effects that occur in outer-shell autoionization by ion impact. Different theoretical descriptions and experimental results are reviewed. 0 1997 Elsevier Science B.V.
1. Introduction The ionization of atoms, ions and molecules by the impact of other particles can take place by a number of different mechanisms. In particular, it can proceed by the excitation to certain discrete states which decay spontaneously by the emission of electrons rather than photons. This radiationless process is called autoionization effect. The term “Auger effect” is also used in honour of Pierre Auger, who discovered it in 1925 [l]. However, this latter term is usually reserved for those electron emission processes that result from inner-shell vacancies. The electrons are emitted with a sharply defined energy and a characteristic angu-
’ Also member of the Consejo National de lnvestigaciones Cientificas y Tdcnicas (CONICET). Argentina. Tel.: 54 944 45234: fax: 54 944 45299: e-mail: [email protected]. 0168-583X/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PIIs0168-583x(97)00451-5
lar distribution. However, in the presence of other charged particles in the continuum, the spectral line shape of the emitted electron is strongly affected by the post-collision interaction (PCI) with the corresponding Coulomb fields. This is the case, for instance. of the Auger process where the emitted electron interacts with the low-energy photoelectron, or when the autoionizing state is formed by the impact of a charged projectile. PC1 effects were first observed in the outer-shell autoionization decay of atoms excited by the impact of slow ions 121. In inner-shell ionization processes, PC1 phenomena were recognized some years later [335]. Early theoretical descriptions [6,7] draw attention mainly to the distortion of the energy spectrum of the ejected or receding electrons. As was first shown on classical grounds by Barker and Berry in 1966 [6], the energy distribution of the emitted electrons is broadened and shifted. displaying an asymmetrical profile. These
R. 0. Burrachiw
I Nucl. Instr. uncl Meth. in Phy.r. Rex B 132 (1997) 28&301
conclusions were validated by other semiclassical and quantum models (see e.g. the review by Kuchiev and Sheinerman [8]). Nevertheless, the interaction among the different post-collision partners can produce another effect on the electronic distribution which was not predicted by these early models. When the emitted particles move at velocities that are comparable in magnitude, the line profiles are not only broadened and shifted, but a sharp angle dependence is likely to occur. This effect, predicted by Dahl et al. in 1976 [9], was first observed by Swenson et al. [lo] in 2-15 keV He+ + He collisions as a sharp enhancement of the line shape in the direction of the receding ion. They explained this observation by modelling the classical deflection of the ejected electron in the Coulomb field of the projectile. The vanishing of the element of solid angle at zero degree leads to a singularity in the forward direction; an effect which has been named Coulomb fix-using [lo]. Including the interaction with the receding projectile in the post-collision electron state by means of a continuum distorted wave (CDW) approximation, Barrachina and Macek [l l] derived an analytical expression for the electronic emission amplitude. The corresponding cross section shows a similar sharp angular dependence that agrees with the experimental results of Swenson et al. [lo] both in shape and magnitude [12]. The general description of PC1 phenomena developed by Kuchiev and Sheinerman [13] in 1988 uses an alternative approximation for the final state which, for the particular case of heavy ion collisions, leads to results which are equivalent to those of the CDW model. The PC1 influence on the angular distribution of autoionization electrons is more easily investigated in ion-atom collisions than for the case of electron impact. where the projectile’s scattering might diffuse the strong effect observed by Swenson et al. [lo]. In this sense, the outer-shell autoionization decay induced by the impact of an ion represents an ideal situation where the final state is composed of one light particle moving in the continuum of two heavy ions. In this review we shall analyse mainly this latter case. In photoionization or electron collision processes, an experimental study of a PC1 angular dependence would
289
require a coincidence measurement of the autoionization and “primary” electrons [ 14,151. Other effects, besides the aforementioned PC1 phenomena, might distort the energy and angle dependence of the observed electron line shapes. For instance, as was first noticed by Rudd et al. in 1966 [ 161, an additional line broadening and energy shift can be produced by the recoil of the emitting atom [9,17,18]. Furthermore, the interference of the resonant and “direct” ionization amplitudes gives rise to strongly modified profiles [19,20], as was first discussed by Fano in 1961 [21]. Similarly, the interference between overlapping resonances might also affect the electron line shapes [l?]. Finally, let us note that the presence of another electron in inner-shell ionization processes might lead to some additional PC1 distortions of the Auger line shape [22]. The aim of the present work is to describe the PC1 effects that occur in autoionization processes induced by ion impact. We shall proceed by steps of increasing complexity, towards a full description of these phenomena. In Sections 2 and 3 we review and compare different classical and quantum-mechanical models for the PC1 distortion of the energy and angle distribution of the emitted electrons. In Section 4 we describe how the projectile influences the angular dependence of the emitted electron spectra. We show that the line shape may present a sharp enhancement in the forward direction when the charge of the projectile is positive. This effect occurs both in a quantum-mechanical description and in its classical approximation. and can be attributed to a forward glory effect. We also show that a rainbow effect might occur for negatively charged projectiles, characterized by a deep depletion of electron emission in the forward direction and an enhancement at intermediate emission angles [23]. In Sections 5-7 we discuss three different interference mechanisms that can modify the autoionization line shapes. These are a path interference mechanism proposed by Swenson et al. in 1991 [24]. and the interference with the direct ionization process and between overlapping resonances. Finally, we discuss in Section 8 the appearance of additional PC1 effects due to the presence of the other electron in inner-shell ionization processes.
290
R. 0. Barruchinu I Nd.
2. Phenomenological
Insrr. and Meih. in Ph~a. Rex B 132
model of Barker and Berry
We consider the collision of a heavy ion of charge Z, and velocity VP with an atom which, after being excited to an autoionizing state, decays by ejecting an electron of energy &. Let us assume that the electron emission probability decays exponentially with time as (atomic units are used throughout)
(I 997) -188-301
effect, the quantitative comparison of this phenomenological line shape with experimental data was not conclusive in Barker and Berry’s work [6]. Nevertheless, the distinctive features of the autoionization line shape in Eq. (2) were experimentally confirmed in later investigations on ionatom collision processes 126,271.
3. Quantum-mechanical
dP - = dt
r
eprr o(t),
where o(t) is Heaviside step function and l/r is the characteristic lifetime of the autoionizing target state. We are assuming that the decay occurs as a post-collision process, when the projectile is at a large distance R(t) M upt from the target. For the time being, we neglect any Stark mixing effect [25]. Thus, the action of the projectile on the target electrons reduces to a shift -Zp/R(t) of all their energies in an independent-electron description. The initial and final state energies at the time t of decay read E,(t) = E, - ZrZP/R(f) and Ef(t) = ef + E(Z, - l)Zp/R(t), where t, and cf are the energies of the corresponding localized orbitals, while E is the energy of the emitted electron. Therefore, in this first order approximation, the PC1 with the projectile modifies the energy of the emitted electron from E. = E, - cf to E M En - Zp/(~!pt). This equation provides a one-to-one relation between the energy of the ejected electron and the time t after the excitation stage. Changing variables from t to E in (l), we obtain the following emission probability: dP dE
ZPl-/UP =
(&
-
exp(-(E))
It is important to emphasize that the former model is of a phenomenological nature, and ignores important aspects of the autoionization process. For instance, the possible interference between different coherently excited states or the continuum is not taken into account. In order to consider these effects, it is necessary to resort to a quantum-mechanical treatment. In this section we shall review different quantum models in a unified description [23], although the original derivations were done independently. We consider that the emitted electron travels with velocity Y in the two-center Coulomb field of the recoil target and the receding projectile. The continuum state of the electron can be decomposed in the following way [l 11: Y,, = D(u,R)
(2) This energy distribution, obtained by Barker and Berry [6] in 1966, has a maximum at E = E. - Zpr/20p and a full width at half maxi1.07 Zpr/up, with a pronounced tail mumofA/?= towards the lower or higher energies, depending on the sign of the projectile’s charge. Due to an insufficient resolution and an unnoticed Doppler
G,(v).
where $, (v) is the continuum wave function of the electron moving with velocity v in the field of the residual target ion alone. o(r, R) incorporates the distortion due to the electron-projectile interaction. In a similar way, we separate the autoionization transition amplitude in the following way [11.13]: A = .f’(E, H)
E)’
theory
r-12 E - E. + ir/2
@,
0):
where E = v’/2 and H = arccos (i. GP) are the energy and angle of the emitted electron. The reduced amplitude k and the lorentzian term u-/2)/p - E. + ir/2) account for the excitation to the autoionizing state by the projectile-target collision and the natural decay which would take place if the projectile had no influence on the ejected electrons. In particular, the angular behavior of k resembles the symmetry of the autoioniz-
291
R. 0. Barrachina I Nucl. Insrr. and Meth. in Pkys. Res. B 132 (1997) 288-301
ing state. On the other hand, the multiplicative factor f(E, f3) is related to the dynamic of the final state, describing the PC1 between the projectile and the emitted electron. In a first-order description, this term can be approximated by [ 11,131 f(E, 0) = - i(E - E,) + iI/2) x j.o.(O,
“$) ( ,)izp/cp
0 x exp [i(E - E0+iF/2)t] dt.
(4)
The great majority of the quantum-mechanical models [28,29] for autoionization processes in ionatom collisions allow for a factorization as given by Eq. (3). The basic advantage of this decomposition is that it predicts in a very simple way how the PC1 modifies the autoionization reaction. From Eq.(3), we obtain the following factorization:
dEdQ
(5)
for the autoionization cross section. This expression decouples the PC1 effects, as given by the distortion factor If(E, 0)]‘, from the actual process of autoionization. In principle, the various quantummechanical descriptions of PC1 effects are based on different approximations of the factor D, although the original derivations did not follow the same line of reasoning. Now let us briefly discuss these theoretical models.
qf Devdariani,
Ostrovskii and Sebayakin
The first quantum-mechanical calculation was developed by Devdariani et al. in 1977 [30]. They neglected the effect of the projectile ion on the electronic wave function by making D(r, R) = 1. This approximation implies that the projectile does modify the energy but not the trajectory of the emitted electron. The resulting PC1 amplitude jbos(E) = I(1
’
+iz)
exp(-2)
r/2
iZp/cp
E - E,, + iI-12 >
and the distortion factor
sinh(nZp/vp)
-2zarctan(F))
x exp
(7)
( are independent of the emission angle 0. They depend only on the dimensionless quantity vp = Zp/vp and the reduced energy E = 2(E - Eo)/r. We note that fDos(E) approaches unity for vp -+ 0. This means that the velocity spectrum approaches the natural linewidth distribution if the projectile exerts no influence upon the emitted electron. At the opposite limit ]vp] $1, Barker and Berry’s energy distribution is recovered [23]. We therefore see that the model of Devdariani et al. [30], which represents the most crude approximation of the electron’s continuum state (D = 1) leads to an energy distribution that is shifted from E = E0 and has the correct behavior at the limits vp -+ 0 and /vp( ---f+cc.
da
I-‘/4 (EYE~)~+F~/~
3.1. Model
nzp’vp
lfDos(E)12 =
3.2. Eikonal approximation
The model of Devdariani et al. is not consistent with the fact that, due to its infinite range, the electron-projectile Coulomb interaction does not vanish at large distances. Therefore, at least a logarithmic phase distortion must be kept in the lowest order approximation, namely D(r, R) M (v’lr - RI + V’ . (Y - R))izp’L”, with V’= v - vp the relative electron-projectile velocity. Substituting into Eq. (4), we obtain iZpJ7.p
YEA@,
4
=
r(~n$iv)
(E
_ Ey/:
E _ & + ir/2 X
(
ir,2) izdo’
u’up- v’ vp )
i
with v=---_.
ZP
ZP
up
0’
We note that the corresponding factor IDEA@,
e)l’
=z
PC1 distortion
sinh(nv)
(6) x exp (-2v
arctan (9))
(8)
is the same as the one obtained by van der Straten and Morgenstern [3 l] and differs from that of Devdariani et al., Eq. (7), only in that the dimensionless parameter vp = Zp/rp is now replaced by 1’. The discrepancies between both theories are negligible when U’ is much larger than tip. However, when 1)’< up, the corresponding energy shift of the intensity maximum AE = -l-r/2 changes its sign. In the case of a positively charged projectile, this leads to a high energy tail in the energy spectrum of the emitted electron, instead of the shift to lower energies predicted by the model by Devdariani et al. Furthermore. in the matching velocity region, a strong increase of AE is expected to occur. This enhancement agrees with the tendency of different experimental measurements [32.33]. Fig. 1 shows experimental results by Vikor et al. [33] for the energy shift of the Ne K-L?.3 L?.j (ID?) Auger line at 0” as a function of the velocity I’~ of a H+ projectile relative to the Auger electron velocity uo. We clearly see that the theoretical energy shift closely matches the experimental data.
Except eventually for the “reduced” cross section dii/dE dQ, the autoionization cross section in the eikonal approximation (Eqs. (5) and (8)) depends on the emission angle only through the parameter ~1. However, as was first observed by Swenson et al. [lo] in He++ He collisions. the PC1 with the projectile can strongly modify the angular distribution of the emitted electrons. In order to explain this result, Barrachina and Macek [l l] considered a CDW approximation [34,35] of the electron final state,
- i (011~- RI + v
f;-ow(E, 6) =
(Y - R))].
1 + i3 (
,
,
I
,
0.8
1.0
1.2
1.4
Projectile velocity ratio I 0.94
0.6
I / 1.48 2.13 Proton energy (MeV)
1.0
0.8
1.2
I 2.89
1.4
q”A Fig. I. Normalized
intensity and energy shift of the Ne K-L2,1 L2, (‘IL) Auger line at 0” as a function of the velocity Q of a H- projectile relative to the Auger electron velocity t’,. (0 l l). Experimental data by Vikor et al. [33] normalized to the total Auger-line intensity. (c 1a.1 ,L,).Experimental data normalized to the theoretical K-shell ionization cross section. ( ~~). CDW theory [l l] (From Ref. [33]).
L’P
)
x I(l-i%)exp($-2) r/2 X
E - E,, + iI/
““-”
’ vp
E - E,) + iF/2 )
in (4) we obtain
r
0.6
]F,[ _ i&/L’I: 1; e nZP’2r’ ’
D(u, R) =I-( 1 + iZp/rj)
Replacing
08iI
I&/i ,> ?F, i3: ) ( 1.’
(9).
The autoionization cross section is obtained, through Eq. (5). from the square modulus of this amplitude. It is clearly seen that it incorporates a dependence on the emission angle that is missed in previous theories (see Fig. 2). In particular, for a positively charged projectile, Z, > 0. it predicts a sharp enhancement in the forward direction when the projectile velocity rp is smaller than the
RO.
293
Barrachinu I Nucl. Instr. and Meth. itt Phv.7. Rrs. B 132 (1997) 288-301
3.4. Comparison of the diflerent theories The models of Devdariani et al. [30] and van der Straten and Morgenstern [31] can be obtained as particular cases of the CDW approximation in different regions of the electronic velocity space [I l] (see Fig. 3). For instance, in the forward direction, as given by V’VP- v’ . vp -cc l-12, we obtain
This amplitude differs from that of the model of Devdariani et al. only in the Coulomb normalization factor f( 1 - iZp/v’) enZp/“‘. This normalization factor generates the usual cusp-shaped peak in the electronic intensity when the electron velocity matches that of the projectile. On the other hand, far from the forward direction, i.e. for n’vp - v’ . VP> r/2, the CDW amplitude reduces to the eikonal approximation YEA. As it was previously explained, this amplitude approaches that of Devdariani et al. [30] when v’ is much larger than up. The same is true in the forward direction, where the CDW amplitude reduces to foes for large electron-projectile velocities (i.e. for v’ > Zp).
4. Glory and rainbow effects Fig. 2. The CDW approximation for the electron emission intensity in the autoionization of He excited to a (2~~)‘s state by the impact of a (a) proton (b) antiproton of velocity rp = 0. I au. (from Ref. [23]).
autoionization velocity ug. This behavior of the intensity profile was first observed by Swenson et al. [IO] in He+ - He collisions. Fig. 1 shows experimental data measured by Vikor et al. [33] for the intensity in the Ne K-L?.3 L2.3 (‘D2) Auger-line forward direction as a function of the velocity ZIP of a H+ projectile relative to that of the Auger electron t:“. The data are normalized to total intensity into 471 solid angle. We note that the agreement between experiment and theory (full curve) is reasonable. Furthermore, the enhancement of the intensity for up < L:~, as predicted by the CDW model, is clearly seen. Recently, different authors demonstrated that the [12,32,33,36,37] have CDW model is very successful in achieving a quantitative coincidence with the experimental data.
The strong dependence the CDW approximation,
on the emission angle in as shown in Fig. 2, can
DOS x CUSP
-1 -
Fig. 3. The model of Devdariani et al. (DOS) and the eikonal approximation of van der Straten and Morgenstern can be obtained as particular cases of the CDW approximation in different regions of the electronic velocity space. See text for details.
194
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be better understood by means of a classical description of the autoionization process. In 1976, Dahl et al. [9] had pointed out that some kind of dependence of the autoionization cross section on the emission angle was likely to be produced by the deflection of the ejected electron in the Coulomb field of the outgoing projectile. Even though Dahl et al. [9], Arcuni [38] and van der Straten and Morgenstern [31] explicitly considered this postcollision distortion of the electron’s trajectory, a small deflection approximation prevented them from obtaining a significant dependence on the emission angle. Finally, in 1989, Swenson et al. [lo] developed a classical generalization of Barker and Berry’s theory which modelled the motion of the ejected electron in the field of a positively charged projectile. In a first order approximation. the angle dependence of the final velocity distribution for z)~ < u0 is shown to be dominated by an e$ f&title cross section [ 10,231
with 2’ and 0’ the projectile-frame emission angles before and after scattering, respectively. This quantity is operationally equivalent to the differential cross section for the elastic scattering of a beam of particles by a force center, except that here the trajectories are parametrized not by an impact parameter p but by the initial emission angle x’. which makes this CcL a dimensionless quantity [23]. In the approximation employed by Swenson et al. [lo], where the post-collision interaction of the emitted electron with the residual target is neglected, ,XcL can be analytically evaluated [23,39] .X‘ti_( cos 0’) 1 ’ =-[
1 - (1 fA)COSH (2(1 - cosH’) - (1 +.d)sin’H’)“~
and CI’is not one-to-one. There are two different initial emission angles c& which lead to the same deflection angle 0’. This means that. when calculating the autoionization cross section, both contributions have to be added up. The relation ~‘(0’) and the effective cross section, Eq. (lo), are shown in Fig. 4 for (a) a positively and (b) negatively charged projectile [23 . For Zr > 0, &_ diverges at 8’ = 0 as zf,_ % IAl’1’ /(3101). This divergence is characteristic of a for~~urcfglory eflect [40], and occurs whenever the dispersion function 0’(c(‘) passes smoothly through 0 as a function of the initial parameter a’ [23,40]. The electron-projectile potential deflects those trajectories with initial emission near angles ‘XL= arccos m into the forward direction, where the solid angle vanishes and the angular density of trajectories diverges. This effect has usually been described in the literature as a Coulomb f&wing mechanism [ 10,12,29,33]. However, this phenomenon, i.e. the deflection of one particular trajectory into the forward direction, can equally occur with practically any other attractive potential. Furthermore, the idea encompassing a ,fkusing mechanism, i.e. that “the deflection effectively compresses the emitter-frame solid angle, resulting in an enhancement, or focusing, of electron intensity in the direction of the ion” [lo], is misleading. In fact, those electrons emitted within the cone CL< XL are not focused in the forward direction, but scattered into the full (0,~) range. Hence. it seems reasonable to keep the standard definition introduced by Ford and Wheeler almost forty years ago in their classical paper on the semiclassical scattering theory [40], describing this divergence of the autoionization cross section as a “forward glory effect”. For the case of negatively charged projectiles (ZP < 0). the effective cross section diverges like
.?FJIs-A 1
(10) The dimensionless parameter A = -(ZP/R)/E;, represents the ratio between the potential and the kinetic energies (in the projectile’s frame) at the moment of emission. The two branches of CCL originate from the fact that the relation between H’
z;L
~
&Gmi 4cos(t&/2)
1 d_
@($’ - 0;)
at a characteristic angle 0; = 2 arctan strictly zero inside a shadow zone 0’ none of the trajectories can penetrate. Fig. 4 that this angle 0; represents of the dispersion relation 0’(~‘). Thus,
(11)
(a) and is < Hk, where We see in a minimum the effective
R.O. Burmchinu
I Nucl.Instr. and Meth. in Phys. Rex B 132 (1997) 288-301
60
120
C a)
395
118C
0’ (deg)
Fig. 4. Relation between the emission angle 2’ and the final angle 0’. and the corresponding effective cross sections for (a) a positively charged projectile with A = -7/3 and (b) a negative one with A = 2/3. There are two emission angles t$ for each 0’. For Z, > 0 there is a forward glory divergency. For 2, < 0 the effective cross section diverges at 0; due to a rainbow effect (from Ref. 1231).
cross section CCL( cos 0’)) being proportional to (d@/dcc’))‘, diverges. This kind of divergence, geometrically analogous to what happens to a beam of light rays scattered by a water droplet, is known as rainbow effect [40]. We see in Figs. 2 and 4 that the classical and quantum-mechanical cross sections show some important similarities. For instance, in both cases, the glory and rainbow effects are more pronounced, the greater the charge the slower the relative electron-projectile velocity. Furthermore, the shadow zone in the rainbow phenomenon becomes broader when the electron-projectile interaction increases. On the .other hand, the glory and rainbow divergences are replaced by finite maxima in the quantum-mechanical description, and the number of particles emitted into the forward dark zone for ZP < 0 is no longer zero. We
also note that the dependence of the CDW cross section on the parameters of the problem is more intricate than in the classical approach. In particular, it presents an oscillatory behavior which is not shown by CCL. These kinds of discrepancies between the quantum-mechanical treatment and its classical approximation are the same that occur between Airy’s undulatory description of the rainbow phenomenon and Descartes’ classical picture. In the following section we analyze this latter oscillatory structure of the autoionization line shape.
5. Path interference The CDW line profiles in Fig. 2 do not only show the expected glory and rainbow enhancement of the electron intensity but also an accom-
296
R. 0. Burruchintr I Nud
1nst11 und Meih
panying oscillatory structure at intermediate angles. These oscillations were predicted by Barrachina and Macek in 1989 [l l] for the case of a positively charged projectile. In 1991 Swenson et al. [24] reported to have found experimental evidence of this effect in an oscillation observed in the low-energy wing of the (2~‘)‘s autoionization line shape in a 10 keV He+ + He collision. This structure was reproduced by a direct extension of the CDW model by Macek and Cordrey [12] including contributions from adjacent lines (see Section 6). Swenson et al. explained this oscillation as due to a path interference mechanism between indistinguishable classical trajectories around opposite sides of the outgoing ion. Any given direction 0 can be reached by two distinct trajectories, giving rise to constructive or destructive interference, depending on the observation angle. A similar effect is observed for the case of a negatively charged projectile. In this latter case, the interference phenomenon has an optical counterpart in the luminous arcs which can sometimes be seen under the principal rainbow. The origin of these so-called supernurnerq~ WCS was explained by Thomas Young in 1803 in terms of this same path interference mechanism. The path interference mechanism was explicitly incorporated in a semi-classical calculation made by Swenson et al. [24]. In the quantum-mechanical calculation, the physical origin of these oscillations has been attributed to an interference process between the direct wave and the wave that scatters from the receding projectile [29,41,42]. In a recent paper [43], it has been shown that the interference structure of the CDW spectral line shape can be interpreted by means of the concept of nectrside+vside scattering (see also Ref. [24]), which was first used to describe elastic reactions of spin-zero nuclei [44]. In a nearside+farside decomposition of the autoionization amplitude, .f’ = ,f‘ + ,f + . each component [44]
it? Plzyx Rex B 132 (1997) 288-3-301
tive (nearside) or negative (farside) deflection [43]. In Fig. 5 we see that the nearside and farside contributions to the autoionization cross section for ZP > 0 show no oscillations when taken separately. The oscillations appear only in the interfer-
0
can be related in a semi-classical limit to trajectories which have traveled along one or the other side of the interaction potential, suffering a posi-
90
135
180
Angle (deg)
0
b) .f‘r(E, 0,J = lim g ’ .f‘(E, 4)sin4 d$ i.P-0~211 i’ cos0’ - cos4 *if .ll
45
I
I
I
45
90
135
180
angle (deg)
Fig. 5. Electron emission intensity at the peak position for the (3’)‘s autoionizing state of He excited by the collision of (a) a proton of velocity L’~ =0.35 a.u. and (b) an antiproton of velocity t’p =O.Ol a.“. The nearside (+), farside (-) and interference (i) contributions to the CDW intensity lAcowlZ are shown separately.
R.O. Barruchinu I Nucl. Instr. and Mrth. in Phys. Res. B 132 (19971 2X-301
ence term 2 Rev-J‘+.). For the case of a negatively charged projectile, Z, < 0, both indistiguishable trajectories travel along the same side of the projectile. Therefore, the oscillations are already present in the near term, while the far term, not representing any classical trajectory, is quite smooth. These results support the assumption by Swenson et al. [24] that the angle oscillations of an isolated lineshape are due to a path interference mechanism.
In an Eulerian description [39], the interference always occurs between trajectories that have and have not reached the glory (for ZP > 0) or rainbow (for Zp < 0) caustics. In this sense, this semi-classical explanation can be translated into quantummechanical terms as an interference between waves that are and are not scattered by the receding projectile [41,29]. This approach has recently been investigated by Kunikeev and Senashenko [42]. They separate the electron continuum wave function ‘l”, = D(Y, R) $,(v) in the CDW approximation into two components @(v, R) = eP’l”G(
‘4=
,,,2,. ci f,(E,0) 2fir J / E
where E,, l/I, and A; are the corresponding resonant energy, lifetime and inherent amplitudes. This interference is known to have a considerable influence on the emission lines, even if they seem to be well separated and no oscillatory structure is observed [31]. A direct extension of the CDW model by Macek and Cordrey [12] including contributions from adjacent lines, was found to be in good agreement with the experimental data of Swenson et al. [IO] for 5, 10 and 15 keV He+ + He collision. In particular, the results shown in Fig. 6 clearly demonstrate that the interference terms are essential for achieving a correct description of the angle dependence of the autoionization line shapes.
l
5keV
0 A
IOkeV expt. 15keV expt
1
fi,,’
I-( -iI+)
1.
expt.
-
-
__
1OkeV theory - 15keV theory
G(
(12)
_
- iv’, 1; -it),
0 (r. R) = - r(l + i”‘) e-rrr’/Ze-i;
291
5keV
theory
icr)
1_
with 1~’= Zr/D’ and i” = v’lr - R(+ v' (Y- R). Here G(a, c, z) is the irregular confluent hypergeometric function [45]. In the asymptotic region, these components represent the incoming (0) and outgoing (1) parts of the electron continuum wave function. These two terms lead to a separation of the autoionization amplitude that is related to the nearside-farside decomposition for the case of a positively charged projectile (Zr > 0) [44]. Within this separation, the oscillation observed by Swenson et al. is shown to be produced solely by the interference term.
I :I:::‘:I:I:‘:I:::~:I:I:‘:I:I:’ 4. n
0 b -. -
-
-
-
5keV expt. lOkeV expt. 15keV expt. 5keV talc. 1OkeV talc. 15keV cab.
(b)
6. Interference between overlapping lines The autoionization line shape can be significanly altered by the interference with neighboring overlapping lines. In this case, the autoionization amplitude, Eq. (3). has to be replaced by the coherent sum of the individual amplitudes corresponding to each state.
Fig. 6. Ratio of target to projectile autoionization total cross sections for 5. IO and 15 keV He+ + He collisions. Experimental data of Ref. [IO]. (a) CDW model calculations by Cordrey and Macek [12] for the (2~‘)‘s. (2p’)‘D and (Xp)‘P states. The interference term between overlapping lines has been omitted in the calculation shown in (b) (from Ref. [I?]),
198
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It is important to note that the relative intrinsic amplitudes Al in Eq. (12) can be modified by the post-collision Stark mixing of the close lying states due to the projectile’s electric field [25]. This effect was first observed by Stolterfoth, Brandt and Prost [47] in 1979, and included in the framework of the CDW model of Ref. [l 1] by Miraglia and Macek [25]. It was explicitly incorporated in the calculation shown in Fig. 6. This effect is known to produce a deviation from the backward-forward symmetry in the angular distribution of electron ejection, having its greatest effect in the tail of the line shape, which corresponds to small internuclear separations of target and projectile.
7. Interference with the direct ionization process The ionization of an atom by the impact of another particle can take place either directly or by the formation of an autoionizing unstable state. The amplitude for these two processes might interfere. In principle, interference with the continua is known to contribute significantly to outer-shell autoionization line shapes [46]. Generally, for a well isolated autoionizing line, the ionization amplitude is the coherent sum of a direct and a resonant contribution
A = Adir+ A,,, = &r
+
f(E, 0) E -
r/2 En t ir/2
L,
When the scattering of the projectile is negligible, the distortion factor can be taken outside the integral over the corresponding scattering angle and the following approximate expression for the ionization DDCS can be obtained [20]: da -= dE dSZ
account for the interference between the direct and resonant ionization amplitudes. With an appropriate definition of the parameters [17], this expression can also be written in the Fano form [21,48] da 1 (E-&+qr/2)’ dEdQ = Oa+ gh (E _ E0)2 + l-1/4 with (Th= -b/2 + J$ + /3’/2, ga = do/dE dQ],, -gh and q = c(/2ah. Early theoretical [49] and experimental studies [50,51] supposed that the Shore parameters, c( and p (or equivalently the Fano parameters q, CT~and ob), were weakly dependent on the electron energy E over the resonance width. However, it is now clear that this assumption is not always correct. In particular, the distortion factor ,f may produce a strong energy dependence of a and fi [52], mainly through a large variation of its phase [ll]. The first experimental evidence of this effect was published by Arcuni in 1986 [38,53]. An alternative parametrization of the autoionization line shape [54], which explicitly separates the distortion factor f’(E, 0) from the fitting parameters, has been recently employed by Moretto-Capelle et al. [46] for analysing the He(2jY) autoionization line shapes in 20-100 keV H ’ - He collisions. It reads da I do __ dE dL’ = dE da Id,, P/4 + If(E, @I’
(E - E$
do r/2 dEdL’ dir+ (E - Et))? + I-?/4 x (@(E-E”) + /K/2).
+ r2/4
do dE da
res (14)
(13)
where do/dE dQldi, = s IAdirl’ dQp is the cross section for the direct ionization process. The so-called Shore parameters.
where, now, the complex parameter cl = 2 s A& of the PC1 distorting factor f(E, 0). Recent measurements by Moretto-Capelle et al. [46] of the He(2L2C) autoionization line shapes in 20-100 keV Hi- He collisions are shown in Figs. 7 and 8. The direct ionization background (fitted to a second-order polinomial) has been sub-
A,, dQp is independent
R. 0. Barrachina
I Nucl. Insrr. and Merh. in PhFs. Res. B 132 (1997) 288-301
Fig. 7. Angle dependence of the He(2KX’) autoionization shapes in a 66 keV H+-He collision (from Ref. [46]).
line
stracted from all the spectra. All the line shapes have been fitted by Moretto-Capelle et al. using the previous parametrization, Eq. (14) (full curve). Fig. 7 displays the angle dependence of the autoionization spectra at 66 keV. We see that the well-delined lines observed at high emission angles change into typical Fano profiles at intermediate angles. At smaller emission angles, these profiles are completely obliterated by strong oscillating structures due to the interference with the direct ionization process. The projectile energy dependence of the autoionization line shapes is shown in Fig. 8 at a fixed emission angle (4”). The strong oscillatory structures already observed in Fig. 7 at 66 keV are also seen here for energies larger than 61 keV. These energies correspond to proton velocities up larger
Fig. 8. Projectile energy dependence zation line shapes at a fixed emission lision (from Ref. [46]).
of the He(2Ce’) autoioniangle (4”) in a H’ -He col-
than the velocity of the autoionizing electron ~0. The frequency of this interference increases for UP approaching uo. For up < ug these oscillations vanish and new structures appear in the low-energy wing of the autoionization spectra. Finally, below 29 keV these additional structures give place to rather simple line shapes. Only a broad satellite is seen in the low-energy side of each resonance line, resembling the one observed by Swenson et al. [24].
8. Post-collision the continuum
interaction with another electron in
In a typical Auger effect, an electron is removed from an inner-shell prior to the autoionization of
300
R. 0. Brtrruchirtcr
I Nucl. Instr.
trntl hleth.
the resulting ion. Through this mechanism, the final state includes an additional particle in the continuum. In principle, the post-collision interaction with other electrons (released during the collision stage through this or another mechanism) may lead to some additional distortion of the autoionization line shapes. Recently, T6th et al. [37] (see also Ref. [36]) have observed that the energy shift of the LMM spectra in 1.25, 1.5 and 2 MeV He- + Ar collisions shows a slight but systematic enhancement in the backward direction with respect to the CDW theory of Barrachina and Macek [l11. They explained this result as an experimental evidence of such an electron--electron PC1 effect originating from the electron loss process. Kuchiev and Sheinerman [29,55] have extended the eikonal approximation of Section 3.2 to the case when there is an additional electron in the final state. Assuming that the separation in velocity space of all the particles in the post-collision stage is large enough, the cross section for an Auger process induced by the impact of a charged projectile is modified by the same distortion factor given in Eq. (8), where now the parameter 1’ has to be replaced by ZP ~=----_+_ I’P
ZP
1
I:’
l’]
in Phvs. Rrs. B 132 (1997)
288-301
is due to the screening of the post-collision electron-projectile interaction by the primary electron.
9. Conclusions In this paper we have analyzed different theoretical descriptions and experimental results concerning the PC1 effects occurring in the autoionization of atoms by ion impact. For pedagogical reasons, we have proceeded from Barker and Berry’s phenomenological model for the line shape of a well isolated autoionization line, towards theoretical descriptions of increasing complexity. This analysis has led to the discussion of different manifestations of PC1 phenomena. Many of these effects have been theoretically and experimentally verified. This is the case of the energy shift enhancement in the matching velocity region or the glory enhanced emission in the forward direction. Other effects, like the rainbow phenomenon, are awaiting experimental verification. And still others, such as the influence of an additional electron in the final state, are in need of a better theoretical description.
1 Ir
-
Vi\
This factor incorporates the interaction of the products of the Auger decay ( i.e.. the residual target and the Auger electron), not only with the receding projectile, but also with the primary electron of velocity vI. The observation of this dependence on the velocities of all the particles in the final state would require a coincidence measurement. In order to compare with noncoincidence experiments, the cross section has to be integrated over the velocity of the primary electron [29,56,57]. This average intertwines the PC1 distortion factor with the reduced cross section for the emission of the primary electron [56]. A very interesting case occurs when the primary electron is captured into a bound state of the projectile. In 1991, Sarkadi et al. [32] observed that the ‘P-‘S Auger peak of Argon at 0” in a 700 keV H+ + Ar collision gets back to its nominal energy, showing no energy shift, when measured in coincidence with a neutral outgoing projectile. This effect
Acknowledgements I thank preparation
Mrs. Linda Yael for her help with the of the manuscript.
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