AIP CONFERENCE PROCEEDINGS 205

THE PHYSICS OF ELECTRONIC AND

ATOMIC COLLISIONS X V I I N T E R N A T I O N A L C O N F E R E N C E

EDITORS: A. DALGARNO HARVARD-SMITHSONIAN CENTER FOR ASTROPHYSICS R. S. FREUND AT&T BELL LABORATORIES P. M. KOCH STATE UNIVERSITY OF NEW YORK or STONY BROOK M. S. LUBELL CITY UNIVERSITY OF NEW YORK, CITY COLLEGE T. D. LUCATORTO NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

NEW YORK, NY 1989

A I P

American Institute of Physics New York

C O N T E N T S

Preface xi

I C P E A C Executive Committee 1 9 8 7 - 1 9 8 9 xiii I C P E A C General Committee 1 9 8 7 - 1 9 8 9 xiv

X V I I C P E A C Local Committee xv

K E Y N O T E L E C T U R E

O n the Utility and Ubiquity of Atomic Collision Physics 2 S. Datz

P L E N A R Y L E C T U R E S

Chaos in Atoms? 1 6 Κ. H. Welge

Transition State Spectroscopy of Hydrogen Transfer Reactions 3 3 D. M. Neumark

Low-Energy Molecular Collisions with Applications to Interstellar Cloud Problems 4 9 K. Takayanagi

R E V I E W T A L K S , P R O G R E S S R E P O R T S , A N D H O T T O P I C S

E L E C T R O N S

Electron-A torn Collision Theory 6 8 C. J. Joachain

Giant Resonances in Double Ionization of Atomic Ions 8 2 H. Suzuki, T. Hirayama, and T. Takayanagi

Determinations of the Products of Dissociative Recombination Reactions 9 0 N. G. Adams, C. R. Herd, and D. Smith

/^-Matrix Calculations of Electron Molecule Scattering 9 6 L. A. Morgan

Spin-Polarization, Orientation, and Alignment in Electron-Atomic Collisions 1 0 3 Μ. H. Kelley

Near-Threshold Studies of Atomic Hydrogen 1 1 5 J. F. Williams

Simultaneous Electron-Photon Excitation Experiments 1 2 2 W. R. Newell

Relativistic (e, 2e) Processes on Inner Shells of Heavy Atoms 1 3 0 J. Bonfert, H. Graf, and W. Nakel

The Theory of Electron-Ion Collisions 1 3 7 A. E . Kingston

Theoretical Calculations of Elastic and Inelastic Scattering of Electrons from Hydrogen 1 4 9 D. H. Madison

Electron Impact Ionization of U 8 8 + - U 9 , + 1 5 7 N. Claytor, B. Feinberg, Η. Gould, C. E . Bemis, Jr., J. Gomez del Campo, C. A. Ludemann, and C. R. Vane

P H O T O N S

Synchrotron Radiation Experiments on Atoms and Molecules 162 U. Becker

Synchrotron-Radiation Experiments with Recoil Ions 176 J. C. Levin

Atoms in Intense Radiation Fields 184 Μ. H. Mittleman

Photoionization of Positive Atomic Ions: A Review of Our Present Understanding 189 S. T. Manson

Polarizational Radiation in Electronic and Atomic Collisions ("Atomic" Bremsstrahlung) 201 Μ. Ya. Amusia

Correlation Effects in Electron Impact- and Photon-Induced Two-Electron Transitions in Rare Gases 215 K . - H . Schartner

Photoionization of Laser Excited Atoms by Synchrotron Radiation 224 D. Cubaynes, J. M. Bizau, B. Carre, and F. J. Wuilleumier

Photodetachment Collisions 233 D. J. Pegg

Double Photoionization near Threshold 241 V. Schmidt

IONS AND A T O M S

Theoretical Collision Physics of Highly Charged Ions 246 A. Bäräny

Optical Potentials in Ion-Atom Collisions 258 R. M. Dreizier, H. J. Ast, A. Henne, Η. J. Lüdde, and C. Stary

Distorted Wave Models for Ionization in Atomic Collisions 264 R. D. Rivarola, P. D. Fainstein, and H. Ponce

Electron Capture and Energy-Gain Spectroscopy 273 K. Taulbjerg

Correlation in Atomic Scattering 280 J. H. McGuire and J. C. Straton

Ion-Ion Collisions: Charge Transfer and Ionization 290 E . Salzborn

Multiply Differential Ionization Probabilities in Small Impact Parameter Ion-Atom Collisions 299 G. Schiwietz, B. Skogvall, N. Stolterfoht, D. Schneider, V. Montemayor

Energy Loss and Charge Exchange Processes of High Energy Heavy Ions Channeled in Crystals 309 J. C. Poizat, S. Andriamonje, R. Anne, Ν. V. de Castro Faria, M. Chevallier, C. Cohen, J. Dural, Β. Farizon-Mazuy, Μ. J. Gaillard, R. Genre, M. Hage-Ali, R. Kirsch, A. L'hoir, J. Mory, J. Moulin, Y . Quere, J. Remillieux, D. Schmaus, and M. Toulemonde

Dynamics of Inelastic Collisions of Electronically Excited Rare Gas Atoms 317 H. C. W. Beijerinck

S I F T and F A L P Determinations of Ionic Reactions Rate Coefficients 325 D. Smith and N. G. Adams

Position Sensitive Detection with Laser Induced Fluorescence 337 L . Hiiwel, A. M. Wodtke, P. Andresen, and H. Voges

Optical Spectroscopic Studies on Penning Ionization and Ion-Molecule Reactions at Thermal Energy by Using Flowing Afterglow 342

M. Tsuji Multi-Charged Ion/Slow Electron Collisions in Cold Plasmas 350

L . Shmaenok Spectral Distribution of Electrons Emitted into the Continuum of Fast Projectiles: Theoretical Approaches of Higher Order in Comparison with Experiment 358

D. H. Jakubassa-Amundsen

vi

Multiple Electron Capture in Close Ion-Atom Collisions 366 A. S. Schlachter, J. W. Stearns, Κ. H. Berkner, Ε. M. Bernstein, M. W. Clark, R. D. DuBois, W. G. Graham, T. J. Morgan, D. W. Mueller, Μ. P. Stockli, J. A. Tanis, and W. Τ. Woodland

Momentum Transfer between Projectile and Recoil Ion in Fast Ionizing Proton-Helium Collisions 372 J. Ullrich, R. E . Olson, R. Dörner, and H. Schmidt-Böcking

First Atomic Physics Experiments with Cooled Stored Ion Beams at the Heidelberg Heavy-Ion Ring T S R 378 A. Wolf, V. Balykin, W. Baumann, J. Berger, G. Bisoffi, P. Blatt, Μ. Blum, Α. Faulstich, Α. Friedrich, Μ. Gerhard, C. Geyer, Μ. Grieser, R. Grieser, D. Habs, H. W. Heyng, B. Hochadel, B. Holzer, G. Huber, E . Jaeschke, M. Jung, A. Karafillidis, G. Kilgus, R. Klein, D. Krämer, P. Krause, M. Krieg, T. Kühl, K. Matl, A. Müller, M. Music, R. Neumann, G. Neureither, W. Ott, W. Petrich, B. Povh, R. Repnow, S. Schröder, R. Schuch, D. Schwalm, P. Sigray, M. Steck, R. Stokstad, E . Szmola, M. Wagner, B. Wanner, K. Welti, and S. Zwickler

State-Selective Angular-Differential Single-Electron Capture in Very Slow A r 4 + - A r Collisions 384 C. Biedermann, Η. Cederquist, J. C. Levin, R. T. Short, L . Liljeby, L . R. Andersson, H. Rothard, K.-O. Groeneveld, C-S. O, C. R. Vane, J. P. Gibbons, S. B. Elston, and I. A. Sellin

/-State Selective Charge Exchange Cross Sections for Collisions of H e 2 + on Atomic and Molecular Hydrogen 390 R. Hoekstra, F . J. de Heer, and R. Morgenstern

M O L E C U L E S

Chaos in Molecular Rydberg States 398 M. Lombardi, P. Labastie, M. C. Bordas, and M. Broyer

Recent Developments in Molecular Ion Recombination Research 404 J. B. A. Mitchell

High-Resolution Electron-Molecule Scattering Using Synchrotron-Generated Electron Beams 410 D. Field, D. W. Knight, S. Lunt, G. Mrotzek, J. Randell, and J. P. Ziesel

G E N E R A L

Resonant Recombination and Autoionization in Electron-Ion Collisions 418 A. Müller

Collisions in, with and on Clusters 430 J. McCombie and G. Scoles

Quantum Mechanical Reactive Scattering Theory for Simple Chemical Reactions: Recent Developments in Methodology and Applications 442

W. H. Miller Harpooning and Chemistry at Surfaces 451

A. W. Kleyn Wavepacket Dynamics as a Tool for Calculation of Averaged Photoionization Cross Sections 458

W. P. Reinhardt and D. R. Kerner Collisions of Rydberg Atoms with Neutral Particles 466

V. S. Lebedev Collisionally Induced Stochastic Dynamics of Fast Ions in Solids 476

J. Bürgdorfer Observations of Excited H ° Atoms Produced by Relativistic Η Ions in Carbon Foils 487

A. H. Mohagheghi, P. G. Harris, C. Y . Tang, H. C. Bryant, J. B. Donahue, C. R. Quick, R. A. Reeder, Η. Sharifian, W. W. Smith, J. E . Stewart, H. Toutounchi, T. C. Altman, and D. C. Rislove

Classical Ghosts in Quantal Microwave Ionization 492 D. Richards and J. G. Leopold

Recent Progress in Above-Threshold Ionization 499 P. H. Bucksbaum

Generation of Very High Harmonics of Optical Radiation in Rare Gases 505 A. L'Huillier, L . A. Lompre, M. Ferray, and G. Mainfray

Photodetachment in Strong Oscillating Fields 513 D. J. Larson, P. S. Armstrong, M. C. Baruch, T. F . Gallagher, and T. Olsson

vii

Atomic Physics in Surface Studies: An Overview 519 F. B. Dunning

Hydrogen-Surface Electron Transition Rates 529 P. Nordlander and J. C. Tully

S Y M P O S I U M : C O R R E L A T E D T R A N S F E R / E X C I T A T I O N A N D A U T O I O N I Z A T I O N

General Considerations 538 J. A. Tanis

Transfer and Excitation with Heavy Projectiles and Targets 544 W. G. Graham

Resonant Processes in Atomic Collisions and a Unified View 550 Y . Hahn

Recombination between Free Electrons and Multiply Charged Ions 556 P. Hvelplund

R T E of Hydrogen-like and Lithium-like Ions 562 R. Schuch, E . Justiniano, M. Schulz, P. H. Mokier, S. Reusch, S. Datz, P. F . Dittner, J. Giese, P. D. Miller, H. Schoene, T. Kambara, A. Müller, Ζ. Stachura, R. Vane, A. Warzcak, and G. Wintermeyer

Observation of Electron-Electron Interaction in Collisions of O s + Ions with H 2 Targets 568 T. J. M. Zouros, D. H. Lee, and P. Richard

S Y M P O S I U M : C O L L I S I O N S W I T H C O L D P A R T I C L E S

Measurements on Very Low-Energy Ion/Atom-Molecule Collisions 574 G. H. Dunn, Μ. M. Schauer, and S. R. Jefferts

Theory of Ultracold Atomic Collisions in Optical Traps 580 P. S. Julienne

Theoretical Treatment of the Associative Ionization Reaction between Laser Excited Sodium Atoms: Energy Dependence and Anisotropy Effects 586

A. Henriet and F. Masnou-Seeuws Collisional Loss Mechanisms in Light-Force Atom Traps 593

T. G. Walker, D. W. Sesko, C. Monroe, and C. Wieman Atomic Hydrogen: Gas and Surface Collisions for Γ-+ 0 599

J. Τ. M. Walraven Experiments in Cold and Ultracold Collisions 607

J. Weiner

S Y M P O S I U M : C O L L I S I O N S I N V O L V I N G P O S I T R O N S

Introduction 614 J. W. Humberston

The Workshop on Annihilation in Gases and Galaxies 616 R. J. Drachman

Low Energy Positron Hydrogen Atom Scattering Using C C A 622 A. S. Ghosh, M. Mukherjee, and M. Basu

Measurements of Positron and Electron Total and Elastic Scattering by Atoms 627 W. E . Kauppila and T. S. Stein

Positron-Impact Ionization of Atomic Hydrogen 633 W. Raith, B. Olsson, G. Sinapius, W. Sperber, and G. Spicher

Slowing Down of Positrons in Solids 639 P. J. Schultz, L. R. Logan, W. N. Lennard, and G. R. Massoumi

Theoretical Calculations of Positron Collisions with Atoms 645 A. D. Stauffer, K. Bartschat, R. I. Campeanu, M. Horbatsch, R. P. McEachran, L . A. Parcell, and S. J. Ward

viii

S Y M P O S I U M : S U P E R C O M P U T A T I O N A L C O L L I S I O N P H Y S I C S

Supercomputers and the Future of Computational Atomic Scattering Physics 652 S. M. Younger

Electron Correlation in the Continuum 658 C. Böttcher and M. R. Strayer

Ion-Metal and Ion-Atom Collisions: Instant Replays and Mean-Field Theories 663 J. D. Garcia, Ν. H. Kwong, and K. J. Schäfer

Large Scale Calculations of Electron-Atom/Ion Processes 668 Κ. T. Taylor

Author Index 675

ix

SPECTRAL DISTRIBUTION OF ELECTRONS EMITTED INTO THE CONTINUUM OF FAST PROJECTILES: THEORETICAL APPROACHES OF HIGHER ORDER IN COMPARISON WITH EXPERIMENT

D.H.Jakubaßa-Amundsen

Physics Section, University of Munich, 8046 Garching, Germany

A compilation of recent experimental data on the forward peak r e s u l t i n g from the capture of target electrons i n t o the continuum of bare, p a r t l y stripped and neutral p r o j e c t i l e s i s presented. The impact-parameter dependence and the dependence of the peak shape on the p r o j e c t i l e charge state as well as on the angular acceptance i s considered. An i n t e r p r e t a t i o n i s attempted w i t h i n the second-order Born theory and the impulse approximation. Results from Monte Carlo calculations at lower impact energies are also included.

1. Introduction

Since i t s discovery, the forward peak (cusp) i n the secondary electron spectrum has attrac t e d great i n t e r e s t both experimentally and t h e o r e t i ­c a l l y . I t consists of target or p r o j e c t i l e electrons which are emitted i n t o low-lying con­tinuum states of the p r o j e c t i l e , and hence ap­pears i n the laboratory frame at forward elec­tron angles 3f & 0 and comprises electron momen­ta kf i n the v i c i n i t y of the c o l l i s i o n v e l o c i t y v. Recent coincidence experiments^ have offered the p o s s i b i l i t y to separate the contributions from target electrons (CTC) and from p r o j e c t i l e electrons (ELC). Although a very recent compila-t i o n of ELC data-3 c a l l s f o r an improvement of the t h e o r e t i c a l approaches beyond the customary f i r s t - o r d e r Born theory, I s h a l l r e s t r i c t myself to the CTC process since i t i s much more sensi­t i v e to higher-order e f f e c t s than the electron loss process. St a r t i n g with the derivation of the second Born theory and the impulse approxi­mation (IA) f o r structured p r o j e c t i l e s (section 2), I s h a l l consider CTC by bare p r o j e c t i l e s and show how the dependence of the peak shape on the angular acceptance i s related to the nonanalytic behaviour of the doubly d i f f e r e n t i a l CTC cross section at kf = ν (section 3). I t i s f u r t h e r shown how the v a r i a t i o n of the peak shape with p r o j e c t i l e charge Zp depends on the c o l l i s i o n v e l o c i t y . The influence of the c o l l i s i o n dyna­mics on the peak formation i s displayed with the

help of a c l a s s i c a l t r a j e c t o r y Monte Carlo (CTMC) ca l c u l a t i o n ^ , and the impact-parameter dependence of the forward electrons i s studied w i t h i n the impulse approximation. In section 4, I consider electron capture by p a r t l y stripped p r o j e c t i l e s and show that a description of the p r o j e c t i l e i n terms of a p o i n t l i k e i o n i c charge i s incorrect. The l a s t section i s devoted to CTC by neutral p r o j e c t i l e s where the observation of a cusp-like peak^ i s a great challenge to theory.

2. Theory

For a t h e o r e t i c a l description of charge transfer w i t h i n a perturbative approach, one has to r e s t r i c t oneself to energetic c o l l i s i o n s where the v e l o c i t y ν exceeds the s h e l l v e l o c i t y of the electron i n i t s i n i t i a l target bound state, or where the r a t i o between target charge and p r o j e c t i l e charge Zp deviates larg e l y from u n i t y . I n the following I s h a l l t r e a t the target as a quasi one-electron system, but w i l l take the p r o j e c t i l e electrons (as f a r as they e x i s t ) ex­p l i c i t l y i n t o account. The neglect of multiple target excitations i s especially j u s t i f i e d f or the case Zp/Z-p « 1.

In the semiclassical picture where the i n t e r -nuclear motion i s described by a c l a s s i c a l t r a ­j e c t o r y , the capture amplitude i s given by ( i n atomic uni t s h = e = m = l )

3 5 8 © 1990 American Institute of Physics

D. Η. Jakubassa-Amundsen 359

a f i f 1 ι Τ i P v (2.1)

where i s the wavefunction of the bound target • Ρ

electron and describes the electronic ground state of the p r o j e c t i l e . The i n i t i a l perturbation

i s composed of the i n t e r a c t i o n Vpp> of the pro­j e c t i l e electrons with the target electron, the

Ν i n t e r a c t i o n Vprp of the p r o j e c t i l e electrons with the target nucleus, and the i n t e r a c t i o n Vp be­tween the p r o j e c t i l e nucleus and the target elec­t r o n . I n the ( p r i o r ) impulse approximation, v a l i d for Zp « Ζρ·, the exact scattering function ^ i s approximated by

l y ^ } > = (ι + G ( ~ ) v f ) | f > # (ι + 4^v f)|f>

A

(2.2)

λϋΐ

where Vp = Vp-, the p o t e n t i a l between the target electron and the target nucleus, and consequently I Ο eigenstate to Hp = Η - V^ where Η i s the f u l l e l ectronic Hamiltonian (H Hp + T e + V p +

e Ν Vrp + Vp-p + Vp^; Hp describes the p r o j e c t i l e elec­trons, T e the one-electron k i n e t i c energy). The state | f > can be represented i n terms of eigen-

P+l states y to the p r o j e c t i l e plus one electron ( i n absence of the target nucleus) which again can be w r i t t e n as a superposition of electronic Φ Ρ ^P

m of the p r o j e c t i l e alone, where <jp^m i s a one-electron scattering state. The f u l l Greens function G (-) (iäf - Η - i f ) * of the exact formulation i s i n the IA replaced by Gp^' corres-

N ponding to Hp^ = Hp + T g + + ^ρτ· The second-order Born approximation would r e s u l t i f i n öS \ Η were replaced by Hp + Τ .

Introducing a complete set of eigenstates χΡΤ -χ o. |U) q y where |q> i s a one-electron plane wave

τΡΤ of momentum q and (p n eigenstate to Hpp, = Hp + Vprp, and going on-shell, one obtains IA

± Jdt: J d q ^ I <^ΙΦ» 3><Φ^· νΐ'ΫΪ Φΐ> η Η

(2.3)

where (ρ- i s an electronic continuum target state. xPT "\— Ρ Expanding φ η = / ^ a ( j ^ and making use of

the orthogonality properties of the eigenstates, one arrives at

IA ä f i

(2.4)

- K i ^ i v ^ +

^<^\^?Ul>] The expansion c o e f f i c i e n t s a n^ and bp^ describe p r o j e c t i l e excitations due to the i n t e r a c t i o n Ν

Vp^ with the target nucleus:

b f A - < % P + 1 | ( i + G f V ^ T ) i T

(2.5) P+l.

where Gpp* corresponds to Hpp and Gp corresponds to Hf.

I f e x c i t a t i o n of the p r o j e c t i l e i s neglected (5 nj <.) the conventional form i s ob­

tained ( a n k ~ bnk

IA i U t T d q <<j>*± I q > (2.6)

where Cppp i s a one-electron scattering state to ι Ρ

the p r o j e c t i l e which i s i n the ground state (pp. In complete analogy, the second Born appro­

ximation reads B2 a . £v f ι f d t < ^ . | ( l + V T ^ | q > 1—γ- <q|)

+ i f

« ( v p + < Φ ι I v J T i > )\<?l> (2- 7>

In accordance with the omission of the i n t e r -nuclear p o t e n t i a l i n the t r a n s i t i o n matrix ele­ment, also the ground-state expectation value of Ν

Vprp (which only depends on the internuclear co­ordinate) has been omitted i n (2.7).

360 Spectral Distribution of Electrons

The impulse approximation i n i t s post form, suitable f o r Zp/ZT >7 1, can be derived 5 i n a s i ­milar way as (2.6) and contains an intermediate p r o j e c t i l e eigenstate instead of the target state

3. CTC by bare p r o j e c t i l e s

For bare p r o j e c t i l e s , Vprj. i n (2.6) and (2.7) ~P Ρ i s zero and reduces to a Coulomb wave <f£ to

the charge Zp. I n order to investigate the pro­perties of the forward electrons i t i s i n s t r u c -

p t i v e to look at the Fourier transform of <f£ at

ο (*\ ν i s the electron momentum i n the p r o j e c t i l e reference frame) which appears e x p l i c i t l y i n the t r a n s i t i o n amplitude (2.6)

IT* e Γ ( 1

*exp(-2iZp[ cosn3scos0£ + sinJ ssin9^cos^ s J/s),

(3.1)

where 0£ i s the electron emission angle i n the p r o j e c t i l e frame, and 7 ^ = Zp/*£.

Upon i n s e r t i o n i n t o (2.6) i t follows that the doubly d i f f e r e n t i a l cross section diverges l i k e Κ£^ due to the normalisation factor of the Cou­lomb wave. Furthermore, the dependence on θ| i s nonanalytical because the i n t e g r a t i o n variable s = q - ν can a t t a i n the value zero^. Since the phase i n (3.1) switches sign i f one considers forward electrons with momentum below the peak

f ο ( k f <^ v, 6 f = 180 ) and above the peak ( k f >> v» t ο

= 0 ) , the i n t e n s i t y of the emitted electrons i s d i f f e r e n t on the two sides of the peak, lead­ing to an asymmetric peak shape. This behaviour i s , however, only v i s i b l e i n higher-order theo­r i e s , because i n the f i r s t - o r d e r OBK theory the phase information from (3.1) drops out when c a l -

2 culating |afχ I · he asymmetry of the forward peak i s c l e a r l y seen i n the experiments as shown i n F i g . l i n the case of 0 + Ne c o l l i s i o n s . ^

u) 1 c D _ö -*—> (7) C Φ

0 8 +-Ne θ0= 0,4°

B=X2C

kf[a.u] 7 8-1-F i g . l . Cusp electron spectrum 7 from 40 MeV 00i" +

Ne c o l l i s i o n s at an energy resolution of 0.5% and 0O = 0.4° (a) and 1.2° ( b ) . I n (a) , the f i t by a 6-term expansion (eq. (3.2)) i s indistinguishable from the data. (b) - - - - constructed " f i t " with use of the same B^m as f o r 0.4°

In order to compare theory with experiment, the d i f f e r e n t i a l cross section has to be averaged over the energy resolution A E q and the angular resolution θ 0 of the detector. Conventionally, an expansion i n terms of Legendre polynomials P-j and powers of i s made^

Λ ι E f + 4 E ° / 2

< d E ^ > © 0 , ^ 0 = Σ > η ΐ Ζ ^ J N I Ε £-ΔΕ 0/2

k f d E f

(3.2)

s0o

^ sinrS^ da^ Ρ (cos9 f )

i n order to extract c o e f f i c i e n t s Β ηχ which are independent of the detector r e s o l u t i o n . However, since the cross section i s nonanalytic i n &f a truncation of the sum over 1 i n (3.2) i s i n gene­r a l not possible. From Table 1 i t i s seen that

D. Η. Jakubassa-Amundsen 361

9f N l N2 N3

0° 1 1 1 30° 1.02 1.01 1.05 60° 1.06 1.03 1.11 90° 1.15 1.09 1.28

120° 1.29 1.18 1.53 150° 1.42 1.27 1.79 180° 1.47 1.30 1.89

Table 1. Cross section ( i n the p r o j e c t i l e frame) for cusp electron emission at Zf = 0 w i t h i n the transverse peaked p r i o r IA. For ρ + He c o l l i s i o n s with ν = 4.4745, a26/aetaXt = Ν^2.02·10 5 barn/keV-sr, with ν = 6.328, d 2tf/de faM^ = N2*6.43-103 barn/keV-sr, fo r He 2 + + He c o l l i s i o n s with ν = 6.328, d 2 6-/de fd^ = N3*4.57-104 barn/keV-sr the d i f f e r e n t i a l cross section i n the p r o j e c t i l e frame (which r e s u l t s upon m u l t i p l i c a t i o n by *f/kp) increases weakly with near 0° and 180°, but rather strongly around 90°. A s i m i l a r r e s u l t as i n the IA i s also found with the second Born theory. The slope i s the larger, the smaller the v e l o c i t y ν and the larger the p r o j e c t i l e charge. Tentative considerations indicate that the nonanalyticity reveals i t s e l f i n a singular behaviour of the h i g ­her derivatives, especially around 90°.

The f a i l u r e of the conventional truncation of the series to 6 terms ( 1 < 2, η <. 1) i s r e a d i l y seen i f QQ i s varied i n the experiment. I n Fi g . l a where © Q = 0.4°, the Bp m are f i t t e d to the expe­rimental spectrum. I f these Bp m are used to con­s t r u c t the spectrum f o r e.g. 0 O = 1.2° (Fig. l b ) a clear discrepancy with the experimental data i s found.

As a measure of the peak asymmetry, the half width at half maximum to the l e f t (Pp) and to the r i g h t (TR) of the l i n e kf = ν can be used. In Fig.2 the r a t i o ^τ/Γβ f°ll°wing from experiment and from the second Born theory i s shown. The ex­perimental decrease of the peak asymmetry for θ0—> 0 (which i s another argument against a t r u n ­cated series expansion) i s q u a l i t a t i v e l y reprodu­ced by theory. Note that although eq. (2.7) has

3

2 L ?

-

1

-

.—'

/ /

τ / / /

I /

-

0 0.5 10 1 eoiDeg]

Fig.2. Ratio T L / r R f o r 40 MeV 0 8+ + Ne as a func­t i o n of detector resolution'. Solid l i n e , eye-guide to the data, dashed l i n e , 2.Born

been evaluated without resorting to the asympto­t i c approximation of Shakeshaft^, the second Born theory i s not expected to give qu a n t i t a t i v e agreement with the data f o r systems with Zp ~ Z^ ^ v.

The shape of the forward peak depends strong­l y on the system parameters l i k e p r o j e c t i l e char­ge and c o l l i s i o n v e l o c i t y . Fig.3 displays the cross section r a t i o for He 2 + impact to proton im­pact on He at a rather low c o l l i s i o n v e l o c i t y of 2 au, and also the cross section r a t i o for 0^+

impact to hydrogen impact on Ne at ν = 10 au. The low-velocity data, which are well reproduced by a Monte Carlo c a l c u l a t i o n ^ , scale approximately

2 with Zp at small momenta of the ejected electron, but show an increase when kf > v. This increase r e f l e c t s a decreasing peak asymmetry with Zp. On the other hand, the high-velocity data scale ap-

2 3 proximately with Zp* but decrease when kf i n ­creases beyond v. This behaviour i s q u a l i t a t i v e l y explained by the (post) impulse approximation^ ( f o r a He target i n order to obey the IA v a l i d i t y

3 c r i t e r i o n s ) which scales l i k e Zp and which leads to an asymmetry which increases according to Zp/v. A s i m i l a r difference i n the ν dependence of the asymmetry (an increase with ν at small ν but a

362 Spectral Distribution of Electrons

500

10

5

d ^ ( Q 8 U H e ) d 2 a(H + — He)

v=10

d 2 a ( H e 2 - H e ) d 2 a(H* —He)

v=2

-1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2

kf— vla.u.)

Fig.3. (Top) Ratio of the doubly d i f f e r e n t i a l cross sec­t i o n f o r 2.5 MeV/N 0 8 + + Ne to H+ + Ne c o l l i s i ­ons (exp: histogramm; 9 Q ~ 0.8°, AEQ/E^ = 0.5%) as a function of electron momentum r e l a t i v e to v; so l i d l i n e , cross section r a t i o f o r 2.5 MeV/N 0 8 +.+ He to H + + He at α% = 1° wi t h i n the post IA. (Bottom) Cross section r a t i o f o r 100 keV/N He^+ + He to H + + He. Data points, Bernardi et a l , dashed l i n e , CTMC cal c u l a t i o n at = 5° (taken from Ref. 10) decrease at large v) has been observed by Da h l 1 2 .

A supplementary information about the cusp electrons can be extracted from an inv e s t i g a t i o n of the c o l l i s i o n dynamics. Especially suited i s the Monte Carlo method, where the c l a s s i c a l t r a ­j e c t o r i e s of the p r o j e c t i l e can be followed. I n Fig.4 i s shown how the forward peak develops as the internuclear distance between the proton and the He target i s increased a f t e r the encounter. Distances of the order of 105 au are necessary i n order to give the correct energy spectrum which compares well with the experimental data of Gib-son-LJ and Dahl , pointing to strong p o s t - c o l l i -sional e f f e c t s i n medium-energy c o l l i s i o n s . I n

^ ( c V / e V - s r )

10' •17

10" •18

10'

p-*He

/ X · : j

I ι k ι Γ

Γ · ι ;

I I

: i ι I

I i i i i

ι ι ιί

^ Λ θ ·

\ · _

ι ί ii ^ < \

10 30 90 50 70 E f(eV)

Fig.4. Forward peak and i t s formation as a function of the internuclear distance R (where the CTMC cal­culation i s stopped) for 100 keV ρ + He c o l l i s i -

1°. ons at ojjr i t RQ = 500 au,

~ 10^ au. Experiments:φ, Gib ι, D a h l 1 2 (taken from Ref.4)

, RQ = 100 au, 3000 au, bson and Reid Ϊ3* R °

O, quantum mechanical calculations, the information on the relevant p r o j e c t i l e - t a r g e t distances i s contained i n the impact parameter d i s t r i b u t i o n . Fig.5 gives a comparison of the experimental data of Jagutzki et a l 1 ^ with the ( p r i o r ) peaked im­pulse approximation 6 (which i s scaled by the r a ­t i o between the unpeaked IA and the peaked IA for l s - l s capture i n the same c o l l i s i o n at b = 0 ) . For H + + Ne, theory reproduces experiment w i t h i n the experimental uncertainty of the normalisation (/v50%). A measurement of the b - d i s t r i b u t i o n of έ-electrons emitted under zero degrees gives w i ­t h i n the error bars the same shape as the b-dis­t r i b u t i o n of the cusp electrons. This confirms the v a l i d i t y of the peaked IA which fa c t o r i s e s i n t o the i o n i s a t i o n p r o b a b i l i t y times the squared normalisation constant of the f i n a l - s t a t e Cou­lomb wave. 6' 1 5 For the more symmetric 3He 2 + + Ne c o l l i s i o n , the IA gives poor re s u l t s at an impact v e l o c i t y as low as the electronic v e l o c i t y of the

D. Η. Jakubassa-Amundsen 363

P(b)

1Ö3

ΙΟ"4

1.5 MeV 3 H e * * ^ N e \ \

1 1 ' · » · \

0.5 MeV H*-~Ne

0.01 b (au.)

Fig.5. Impact-parameter d i s t r i b u t i o n of the cusp elec­trons w i t h i n J f ί 3° i n 0.5 MeV/Ν H + + Ne and % e 2 + + Ne c o l l i s i o n s . The experimental data-^ are obtained by in t e g r a t i n g the electron spectrum over the peak region. The calculations are perfor­med with the f u l l y peaked pr i o r IA (integrated over the region kf = ν + O.lv) and scaled down by a factor of 0Λ ( s o l i d l i n e ) and 0.23 (dashed l i n e ) , respectively (see t e x t ) target L-shell which yields the dominant c o n t r i ­bution.

4. CTC by p a r t l y stripped p r o j e c t i l e s

To s i m p l i f y the t h e o r e t i c a l description of the cusp electrons when the p r o j e c t i l e carries elec­trons i t i s often assumed that the p r o j e c t i l e acts l i k e a p o i n t l i k e p a r t i c l e of ionic charge. From the general theory (e.g. eq. (2.6)) i t follows that the p r o j e c t i l e f i e l d enters twofold, f i r s t d i r e c t l y as a t r a n s i t i o n operator i n the matrix element, and second, i m p l i c i t l y i n the f i n a l - s t a t e electronic wavefunction <^fi- I n Fig.6 i s shown the doubly d i f f e r e n t i a l cross section r a t i o f o r He + + He r e l a t i v e to the H + + He c o l l i s i o n system at an intermediate c o l l i s i o n v e l o c i t y of 2 au from a CTMC cal c u l a t i o n when a s t a t i c screened p r o j e c t i l e f i e l d i s used-^. For small electron momenta, kf < v, the r a t i o i s approximately one which i s con­form with the picture of an ionic point charge. For k f > v, however, He+ i s much more e f f i c i e n t i n

dV(He+) dV(H +)

10

Ne. v= 6.328

He. ν = 2.0

0 -15 -0,5 -0,5 15

kf-v (au.)

Fig.6. Ratio of the doubly d i f f e r e n t i a l cross section fo r 100 keV/N He+ + He to H + + He at f3~f = 5° wi­t h i n the Monte Carlo method^ (dashed l i n e ) and for 1 MeV/N He+ + Ne to H + + Ne for θ 0 = 1 ° wi­t h i n the f u l l y peaked p r i o r IA ( s o l i d l i n e ) as a function of electron momentum r e l a t i v e to ν producing CTC cusp electrons than H+. This i s r e ­lated to the f a c t that f a s t electrons require a large momentum transfer q, which i s easier pro­vided by the s t a t i c screened He + p o t e n t i a l which i n the l i m i t of q —> oo coincides with the He2+ f i e l d . Also shown i n Fig.6 i s the r a t i o for He + + Ne r e l a t i v e to H + + Ne i n a fast c o l l i s i o n (v = 6.328 au) from the ( p r i o r ) peaked impulse appro­ximation. A r a t i o of 4 i s expected since the pro­j e c t i l e f i e l d i n the i o n i s a t i o n matrix element i n (2.6) acts very much l i k e a He 2 + f i e l d f o r the large momenta required. The deviation of the r a ­t i o from 4 resu l t s from the consideration of the i o n i c f i e l d (including p o l a r i s a t i o n and exchange) i n the c a l c u l a t i o n of the f i n a l - s t a t e electronic wavefunction.

5. CTC by neutral p r o j e c t i l e s

For a short-range p o t e n t i a l no cusp behaviour for the CTC electrons i s expected as long as the p r o j e c t i l e remains i n i t s ground st a t e . I f the f i n a l - s t a t e electronic wavefunction i s taken as a scattering eigenstate of the p r o j e c t i l e (calcu-

364 Spectral Distribution of Electrons

k f(o.u.)

Fig.7. Forward electron spectrum i n 2.5 MeV Η + He c o l ­l i s i o n s f o r θ 0 = 0.5° w i t h i n the f u l l y peaked pri o r IA f o r d i f f e r e n t charge states of the pro­j e c t i l e . , H+, , H°( 3S), , H°(1S) lated from a Schrödinger-type equation including p o l a r i s a t i o n and exchange^) a forward peak i s indeed obtained, although with a much larger width than i n case of an io n i c p o t e n t i a l . Fig.7 displays the forward peak fo r neutral hydrogen p r o j e c t i l e s i n the spin s i n g l e t and spin t r i p l e t state i n comparison with H + on a He tar g e t . The calculations are performed w i t h i n the ( p r i o r ) peaked impulse approximation where from the f i ­nal-state wavefunction, only the 1 = 0 p a r t i a l wave normalisation constant i s r e q u i r e d ^ . The existence of a peak f o r /£ —» 0 re s u l t s from the a t t r a c t i v e p o l a r i s a t i o n p o t e n t i a l , and the en­hancement of the peak f o r the s i n g l e t state i s due to the exchange i n t e r a c t i o n .

Experiments, where the p r o j e c t i l e charge sta­te i s measured i n coincidence with the electrons i n order to i s o l a t e the CTC cont r i b u t i o n show, however, a cusp-like structure even for neutral p r o j e c t i l e s . 2 Fig.8 displays the forward peak i n 75 keV/N H° + Ar and He° + Ar c o l l i s i o n s separa-

kr(a-u.)

Fig.8. Forward electron spectrum i n 75 keV/N H° + Ar ( l e f t ) and He 0 + Ar ( r i g h t ) c o l l i s i o n s . Shown are the r e l a t i v e i n t e n s i t i e s f o r the electron loss co n t r i b u t i o n (ELC) and capture to continuum con­t r i b u t i o n (ECC) fo r θ 0 =3.5°. The s o l i d l i n e s are eye-guides to the data (taken from Ref.18) ted i n t o electron loss and electron capture con­t r i b u t i o n s 1 8 . The fa c t that f o r the He 0 projec­t i l e , the cusp i s even narrower than f o r a He +

p r o j e c t i l e has been t e n t a t i v e l y explained i n terms of a low-lying shape resonance 1^. I n order to observe such a resonance which introduces a s i n g u l a r i t y i n t o the f i n a l - s t a t e e l e c t r o n i c wave-function i t i s , however, necessary that the pro­j e c t i l e i s excited to a sp e c i f i c state p r i o r to the CTC process. For He0, the metastable 2*S s t a ­te (which may be present i n the beam) can account for the occurrence of a resonance 2^. For neu­t r a l hydrogen on the other hand, i t i s d i f f i c u l t to imagine i n which way a sp e c i f i c excited reso­nance-supporting p r o j e c t i l e state can be s u f f i ­c i e n t l y strong populated: From Fig.8 i t follows that the peak i n t e n s i t y f o r CTC i s about 20% of that f o r ELC. Assuming ( o p t i m i s t i c a l l y ) an equal t r a n s i t i o n p r o b a b i l i t y f o r electron loss and e l ­ectron capture by a p r o j e c t i l e with f i x e d elec­t r o n i c configuration, the e x c i t a t i o n p r o b a b i l i t y of the p r o j e c t i l e has to be as large as 0.2 !

D. Η. Jakubassa-Amundsen 365

I n conclusion, i t has been shown that the im­pulse approximation as well as the second Born theory are able to reproduce the features of the forward peak for energetic c o l l i s i o n s with char­ged p r o j e c t i l e s , such as the peak asymmetry and i t s dependence on v e l o c i t y , p r o j e c t i l e charge and angular acceptance. For neutral p r o j e c t i l e s , the question of existence and i n t e r p r e t a t i o n of a cusp-like peak c a l l s f o r further i n v e s t i g a t i ­ons.

Acknowledgments

The c o l l a b o r a t i o n with W.Oswald (who provided the second Born r e s u l t s ) and with the other mem­bers of the experimental group, H.-D.Betz and R.Schramm, i s greatly acknowledged. I would also l i k e to thank R.Koch and H.Schmidt-Böcking f o r providing the data on the b - d i s t r i b u t i o n , and CO.Reinhold f o r communicating the CTMC res u l t s p r i o r t o pub l i c a t i o n . Especially I would l i k e to thank D.Berenyi and his group for discussions and f o r t h e i r great h o s p i t a l i t y during my v i s i t i n Debrecen.

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