2 electron emission in ion-atom collisions

Nuclear Instruments North-Holland,

and Methods Amsterdam

MECHANISMS

FOR u/2

in Physics

Research

ELECTRON

R40/41

(1989) 65-69

EMISSION

IN ION-ATOM

65

COLLISIONS

We have studied electron emission spectra in ionizing collisions in one dimension with interaction potentials of zero range, From low to intermediate collision energies the ionization mechanism is attributed to promotion of quasi-molecular orbitals formed between the eollkion partners into the continuum. At intermediate collision velocities we observe a hump in the ionization spectrum near ~1~= o/Z. The peak position follows approximately the motion of an improper saddle point as the charges of the projectile and of the target are varied. Double peak structures are observed at low collision energies in the forward emission spectra which can be traced to interferences between resonances in the projectile and target continuum.

1. Introduction

In recent years a newly discovered ionization phenomenon has gained considerable attention: the socalled u/2 electrons. i.e. electrons ejected with velocities of about half the velocity of the projectile, u. The traditional picture of the velocity distribution of ionized electrons was characterized by a two-center structure of electrons centered in velocity space either about the target (direct ionization, electron loss to continuum (ELC)) [1.2] or about the projectile (electron capture to continuum (ECC)) [3,4]. These structures are closely associated with the corresponding two-body final-state interactions for the ejected electrons with the target and the projectile. The binary encounter peak resulting from a head-on collisionof the projectile with the electrons plays a somewhat special role in that the final states are free electrons of velocity 20 well separated in velocity space from either the target or the projectile and are not subject to significant final-state interactions. Recent theoretical and experimental results have amended this global picture of the electron distribution by a new feature: electrons “stranded” in between the two Coulomb centers giving rise to a hump near o/2. “In between” refers here to both the coordinate space and the velocity space. Such a feature is clearly a true three-body effect caused by simultaneous interactions of comparable strengths of the electrons with both the target and the projectile. Several theoretical investigations [5-71 have found both direct and indirect evidence for the u/2 electron emission. Experimental evidence for u/2 electrons has recently been presented [8,9] and a shift of the rl,/2 hump to the lower charge of the collision partners has been reported [lo]. Invoking their unique location in phase space a qualitative interpretation of the e/2 0168-583X/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

electron distribution relies on the evolution of the electrons near the saddle point between the two Coulomb centers in the exit channel. Post-coll~sional evolution of a wave packet centered about the midpoint in phase space under the influence of a two-center [8] or singlecenter [ll] Coulomb distortion have been shown to lead to a formation of a ridge in the emission spectra. We have recently investigated the Q/Z electron emission in one-~mensional (1D) ion-atom collisions with zero-ranged (6 function) potentials [12]. We found direct evidence for the existence of the zr/2 electrons in an exact quantum-mechanical formulation. While being a drastic simplification, this 1D 6 problem has surprisingly many features in common with the 3D Coulomb problem, for exampie the well known Stiickelberg oscillations for quasi-resonant charge transfer 1131. This exactly solvable model may help to delineate the mechanisms of ionization in realistic three-body systems, in particular it allows more detailed studies of the mechanisms leading to 1:/Z electron emission which is the primary focus of this paper. The S potential problem is also of interest from a different point of view: it enables one to explore the ionization spectrum in the absence of any long-range forces. However as has been pointed out [12], there is an improper saddle point which is degenerated into a flat “plane”. We shall refer to it as the “degenerate saddle point”. As we will discuss below the degenerate saddle point plays an important role in the emission of u/2 electrons.

2. Theor Using a classical constant velocity trajectory for the internuclear motion. R(t) = uz, the time dependent I. ATOMIC PHYSICS

J. Wang, J. Burgdiirfer / o/2 electron emission in ion-atom

66

“electronic” by (in a.u.) H=

Hamiltonian

for the 1D 8 probem

is given

-$-Zp8(x-it)-Z,S(x+;t),

(1)

In eq. (1) we have adopted the “center of velocity (CV)” frame denoted in the following by primed quantities. It should be noted that the solutions of the time-dependent Hamiltonian eq. (1) are Galilean invariant if translation factors for projectile (target) centered orbitals are included [14]. Accordingly the resulting transition probabilities are independent of the chosen frame. The projectile and the target with “nuclear charges” Z,., propagate with speed uk.r = f v/2 in the CV frame and the hump of c/2 electrons is localized around the origin with momentum k’ = 0. As usual, the internuclear potential has been neglected since it gives rise only to an irrelevant phase factor in the wavefunction within a constant-velocity approximation. The asymptotic initial and final states are defined with reference to the “atomic” Hamiltonian - 5 $

- ZT,P~(~)

GT.P = ~T,PGT.P.

i

(4

1

Eq. (2) possesses

only one bound

z T.P = - ?'G,P

state with (3)

and +T,P=

&

e-ZT.P+‘I,

(4)

which closely resembles a hydrogenic 1s state, and an infinite number of continuum states including resonances (“quasi-stationary states” [15]). The lack of an excited bound-state spectrum points to the absence of ELC or ECC peaks because of vanishing density of states near threshold. The time-dependent Schrodinger equation H$(x.

t) =i

a+(x.

at

t>

collisions

The numerical accuracy of p(k) which requires a numerical integration has been checked by calculating the total ionization probability P, =

m p(k) / -m

which, P &SIX

by -

dk.

(8)

unitarity

considerations,

1-

Due to scaling properties [13] the only independent parameters in the 8 model are Z,/v and Z,/u. Fig. 1 shows the ionization spectra for the symmetric collision system Z,/u = Z,/o at three different collision energies. The spectra at these energies have a common feature: ECC and ELC peaks are missing as expected. This is a combined effect of the short-ranged potential and the confinement of the motion to one dimension. At intermediate collision energy ( Zp/~l = Z,/v = 1) a broad peak near k/v = +( k’ = 0) is indeed clearly visible, which we identify as the v/2 peak. The occurrence of a o/2 emission feature independent of the range of the potential involved can be understood in terms of a quasi-molecular potential energy diagram fig. 2 (i.e. the static limit of the Schrodinger eq. (1)). The 6 molecule has one binding and one anti-binding state. The later is directly promoted to the continuum at small distances R, = a.u. in a non-adiabatic collision. At large distances the two quasi-molecular bound states are degenerate if Z, = Z,. The population of the quasi-molecular states during the collision process can be determined by projecting the exact scattering state onto the eigenstates of the quasi-molecule at fixed internuclear distances

(5)

:: “,

,(

,’

‘,

,;’

‘..

‘.

where U( t,, to ) is the time-evolution operator belonging to the Hamiltonian in eq. (l), and B( t ) is the operator of projection onto the continuum portion of the Hilbert space [12]. In eq. (6) we have uses the initial condition that the electron is in an atomic eigenstate at t = - co. The differential ionization probability of an electron with momentum k is given by

-1

0

1

3

2

k/v Fig.

1. Differential

momentum = z,/u

(7)

:’

lo-" - :: -2

I ‘,, :

,:’

I~(k>l’.

equal

3. Results and discussion

has been solved exactly for the Hamiltonian operator (eq. (1)) [13,15] to determine the elastic and the charge transfer probabilities. We are mainly interested in determining the differential ionization amplitude a (k ), defined by

p(k)=

must

Pcapture.

k (in units = 5: -z,/u

ionization probabilities of the projectile velocity = Z,/L’

0.1.

=l:

- --

z,/o

vs u). =

electron Z,/u z,/o

=

J. Wang, J. Burgdtirfer / u/2 electron emission in ion-atom

67

collisions

0.8

0.4 00

30

20

10

R (a.u

40

0

1

06

05

04

03

G cc 02

01

1

00 00

-10 R

3

4

5

Fig. 4. u/2 electron peak position k,,, (in units of D) vs the ratio of projectile and target charges v( = Z,/Z,). the calculated result: --- uqaddle + 0.1

(Born-Oppenheimer approximation). Plotted in fig. 3 are probabilities of the electron occupying the bound states as a function of relative coordinate R(negative (positive) R describes the incoming (outgoing) part of the trajectory of the projectile). At large negative R the two bound states are degenerate, therefore the electron has approximately equal probability (P = 4) of being in either of them. We have observed deviations from P = f even at large distances which is a direct measure of the influence of “translation factors” missing in a quasi-molecular state representation of the exact scattering state. As the incoming projectile approaches the critical distance R, the occupation of the anti-binding state drops sharply to zero, indicating the promotion into the continuum. At the same time the occupation probability of the binding state increases only slightly. This is another indication that electron emission occurs predominantly by direct promotion of the

-20

2 Y

Fig. 2. Molecular energy diagram at Z, = Z, =l. -the binding state: - - - the anti-binding state.

2 e 0

1

10

20

ia.u.1

Fig. 3. The occupation probabilities of the binding ) and the anti-binding state (- - -). (-

state

anti-binding state into continuum states. On the outgoing part of the trajectory. the population of the deeply bound state decreases at small distances R -c R,, i.e. before the anti-binding state is restored at R = R, enhancing thereby the ionization probability. The forward-backward asymmetry of the occupation probabilities illustrate the non-adiabaticity of the collision process. Subtracting from unity the sum of the two occupation numbers gives the total ionization probability which in this case is approximately 0.4, in agreement with the calculated result [13]. While in detail quite different, the quasimolecular orbital promotion mechanism for ionization is present both in the 1D S problem and the 3D Coulomb problem. This mechanism is effective at small distances irrespective of the long-range nature of the potential. The low-energy portion of the quasi-molecular ionization spectrum (in the CV frame) leads to the enhancement near v/2 in the labframe. This ionization process at small distances can be considered to be the likely candidate for the primary mechanism generating the wavepacket which is then subject to post-collisional distortion [8,11] in the 3D Coulomb problem. Fig. 4 shows the variation of the v/2 peak position k max (in units of u) as a function of the asymmetry ratio of the system y = Z,/Z, at a collision velocity of v = 1. Though the dependence on y is not monotonic there appears to be an overall trend in shifting the peak position towards lower velocities (from 0.7 to 0.48) as Z, increases, in surprising agreement with the shift of the classical saddle point in the 3D Coulomb problem [lo]. This observation has led us to an investigation of the motion of the saddle point for the S problem. As mentioned earlier the saddle point in this model is degenerated to a flat “plane”. However the motion of the saddle point can be studied by introducing a weaklimit representation of the S potentials in terms of I. ATOMIC PHYSICS

J. Wang, J. Burgdiirfer / u/2 electron emission in ion -atom

68

continuous functions. Choosing the target coordinate as .X= 0 and the projectile coordinate as x = R, the electronic potential is given by V(x)=

-Z,6(u)-Z,S(R-x)

(9) For finite c one can find the point of equiforce X saddle

R 1 +

y”j

to be

1 =

gz+y’/‘.

becomes the predominant feature at higher collision velocities while the emission cross section is still zero at threshold for the binary systems (electron-target, electron-projectile), i.e. at k = 0 and k = u. The spectrum displays now an enhancement near the projectile and target velocities which is similar but not identical to ELC and ECC peaks. In addition, structures at higher velocities in both forward and backward direction become visible which can be analyzed in terms of a perturbative multiple scattering series [17].

(10)

independent of 6. so that the saddle point remains well defined as E + 0. If the target ion is at rest and the projectile ion is travelling at a velocity ~1. then the velocity of the degenerate saddle point is given by L’saddle

collisions

(11)

As indicated in fig. 4, the curve ( usaddle+ 0.1) reproduces the dependence of the position of the u/2 hump qualitatively quite well despite noticeable deviations. The source of the constant shift by 0.1 is not yet known. It is also remarkable that eq. (11) possesses a striking similarity to the motion of the saddle point in 3D Coulomb problem with an exponent l/2 rather than l/3. Obviously the coupling of quasi-molecular orbitals to the continuum is related to the dynamics of the saddle point. Another interesting feature appears at lower collision velocities (Z,/u = Z,/v = 5) where the spectrum displays a double-peak structure in the forward direction in the range 0 I k/u I 1. Numerical analysis shows that it is an interference effect between two groups of “quasi-stationary” resonances in the continuum which are formed during the collision process. The first group consists of states that are centered around the target. The second group is composed of states that are centered around the projectile. These two groups can be identified by their respective Galilean translation factors resulting from transformation of wavefunctions to a uniformly moving reference frame. The two groups strongly interfere with each other at low collision energies, giving rise to the double-peak structure. The interference is strongest for slow, symmetric systems. This feature has a similarity to the interferences observed in the autoionization spectrum in symmetric He+ + He collisions where electron emission of doubly excited projectile and target states can not be distinguished [16]. Note also that a separate binary peak is not visible at low collision energies because the widths of the internal momentum distribution of the atomic states are large compared to the momentum of the projectile. In the limit of high collision velocities (Z,/v = Z,/u = 0.1) the u/2 electron peak disappears (fig. 1). The binary encounter peak in the forward direction

4. Conclusions A detailed study has been performed to investigate the ionziation mechanisms in ion-atom collisions using 1D 8 potentials. At low collision energies a double-peak structure appears in forward direction due to the interference effects. At intermediate energies the mechanism for v/2 electron production is attributed to promotion of quasi-molecular states into the continuum. The shift of peak position of u/2 electrons is qualitatively explained by considering the shift of the classical saddle point. We should like to thank Anders Barany for illuminating discussions on the subject. This work was supported in part by the National Science Foundation and by the US Department of Energy under contract No. DE AC05-840R21400 with Martin Marietta Energy Systems, Inc.

References to the VI M.R.C. McDowell and J.P. Coleman, Introduction Theory of Ion-Atom Collisions (North-Holland, Amsterdam. 1970). PI F. Drepper and J.S. Briggs, J. Phys. B9 (1976) 2063. [31 C.B. Crooks and M.E. Rudd. Phys. Rev. Lett. 25 (1970) 1599. [41 J. Macek, Phys. Rev. Al (1970) 235: A. Salin. J. Phys. B2 (1969) 631. [51 R.E. Olson. Phys. Rev. A27 (1983) 1871. [61 T.G. Winter and C.D. Lin. Phys. Rev. A29 (1984) 3071. [71 R.E. Olson. Phys. Rev. A33 (1986) 4397. F31 W. Meckbach. P.J. Focke, A.R. Goni. S. Suarez, J. Macek and M.G. Menendez, Phys. Rev. Lett. 57 (1986) 1587. [91 R.E. Olson. T.J. Gay. H.G. Berry. E.B. Hale and V.D. Irby, Phys. Rev. Lett. 59 (1987) 36. DOI V.D. Irby. T.J. Gay, J. Wm. Edwards. E.B. Hale. M.L. McKenzie and R.E. Olson, Phys. Rev. A37 (1988) 3612. and A. Niehaus. 15th P11 P. van der Straten. R. Morgenstern Int. Conf. Physics of Electronic and Atomic Collisions. Abstracts of contributed papers, eds. J. Geddes, H.B. Gilbody. A.E. Kingston, C.J. Latimer and H.J.R. Walters (Brighton. UK. 1987) p. 605.

J. Wang, J. Btrrg&rjer

/ v/2

electron emMon

[12] J. Burgdarfcr, J, Wang and A. B&&y, Phys. Rev. A38 (1988) 4919. [13] W. Dtippen, J. Phys. BlO (1977) 2399: H. Danared. J. Phys. B17 (1984) 2619. [14] D.A. Park, Introduction to the Quantum Theory (McGraw-Hill, New York, 1974) p. 624.

in ion -atom

collrsions

69

[15] S.K. Zhdanov and A.S. Chikhachev. Sov. Phys. Dokl. 19 (1975) 696: E.A. Solovev, Teor. Mat. Fiz. 28 (1976) 240. 1161 E. Boskamp. R. Morgenstern and A. Niehaus. J. Phys. BlS (1982) 4577. [17] J. Wang and J. Burgdiirfer, to be published.

I. ATOMIC

PHYSICS