Lecture Notes in Physics Edited by H. Araki, Kyoto, J. Ehlers, München, K. Hepp, Zürich R. Kippenhahn, München, H. A. Weidenmüller, Heidelberg and J. Zittartz, Köln

2 1 3

Forward Electron Ejection in Ion Collisions Proceed ings of a Sympos ium He ld at the Phys i cs Institute, University of Aa rhus Aarhus, Denmark, June 2 9 - 3 0 , 1 9 8 4

Edi ted by K . O . Groeneve ld , W. M e c k b a c h and I.A. Se l l in

Springer-Verlag Berlin Heidelberg New York Tokyo 1984

Editors

K.O. Groeneveld Institut für Kernphysik, Johann-Wolfgang-Goethe Universität, Frankfurt/M. August-Euler-Straße 6, D-6000 Frankfurt/M. 90, FRG

W. Meckbach Centro Atömico Bariloche, Comisiön Nacional de Energia Atömica 8400 Bariloche, Argentina

I.A. Sellin Department of Physics, University of Tennessee Knoxville, TN 37916, USA

ISBN 3-540-13887-0 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13887-0 Springer-Verlag New York Heidelberg Berlin Tokyo

This work is subject to Copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2153/3140-543210 ' /

C o n t e n t s

I n t r o d u c t i o n

ELECTRON LOSS TO THE CONTINUUM FOR LIGHT IONS

M.W. L u c a * , / ( . F . Man a n d W. Stecfee^macfre-t

T h e o r y, S i n g l e C o l l i s i o n

THEORETICAL DESCRIPTION OF THE CUSP ELECTRONS

EJECTED IN ASYMMETRIC HEAVY-ION COLLISIONS

P.tf. Jakubaßa- A m u n d . 6 en 17

DOUBLE DIFFERENTIAL CROSS SECTION FOR ELECTRON

CAPTURE TO THE CONTINUUM WITH MOLECULAR PROJECTILES

C . E . G o n z a l e z LepeAa a n d l/.ff. P o n c e . 28

DENSITY MATRIX DESCRIPTION OF COLLISIONAL ELECTRON

TRANSFER INTO THE CONTINUUM OF IONIC PROJECTILES

J . ButLgdöKÜe.'i 32

T h e o r y, S o l i d T a r g e t

A TIME DEPENDENT SECONDARY ELECTRON TRANSPORT MODEL

J . V z v Q o g h t , J . C . V a k a n z , A. V u b i U a n d W. H o l l a * k y 52

E x p e r i m e n t , S i n g l e C o l l i s i o n

CONTINUUM-ELECTRON CAPTURE BY 25-250 keV PROTONS IN HELIUM

P. D a h l 6 2

THE INFLUENCE OF A DIFFUSE TARGET ON ELECTRON LOSS

INTO THE CONTINUUM DOUBLE DIFFERENTIAL DISTRIBUTIONS

G . C . B e . K n a n . d l , Z . B . Ütminov&ky, W. Mecfobac/i a n d C R . G a n i b o t t i 67

CUSP STUDIES FOR SIMPLE COLLISION SYSTEMS

V. B z n ' z n y l , L . Gulyä6, Ä. Kö've/i, E. Szmo£a, G(/. Szabö,

M. B u / i k k a / i d a n d K.O. G n o a n a v a l d 71

VI

DOUBLY DIFFERENTIAL EMISSION DISTRIBUTIONS FOR ELECTRON

LOSS TO THE CONTI NU UM FROM FAST HEAVY PROJECTILES IN GAS

TARGETS

S.B. E U t o n , S.V. B e i i y , M. B r e i n i g , U. V e . S e n . i o , C . E . G o n z a l e

L e . p e < x a , I .A . S e l l i n , K.O. G x o c n e v c i d , V. Ho&mann, P. K c s c h a n ,

J . B . U e m i n o v & k y a n d L . I . L i i j e b y

PROJECTILE CONTINUUM ELECTRONS IN HIGHLY CHARGED

ION-ATOM COLLISIONS

L.H. A n d e n c n

L-SHELL VACANCY PRODUCTION BY ELECTRON CAPTURE TO P R O J E C T I L E -

CENTERED CONTINUUM STATES (ECC) IN PROTON-ARGON COLLISIOrtS

L . S a n . k a d i , J . B o y U e , i , R. H i p p i e . * a n d H.O. L a t z

ELECTRON CAPTURE INTO METASTABLE K r 8 + RECOIL IONS

K.M. CKamon a n d F. V o l k m a n n

E x p e r i m e n t , S o l i d T a r g e t

THREE DI MENS IONAL CONVOY ELECTRON VELOCITY DISTRIBUTIONS

PRODUCED BY 60-270 keV PROTON IMPACT ON CARBON FOILS

W. Meckbach, 1 . 8 . N e m i n . o v & k y a n d P.R. F o c k e

ANOMALOUS ME AN FREE PATHS FOR SCATTERING OF CONVOY ELECTRONS

GENERATED BY FAST , HIGHLY IONIZED IONS IN THIN SOLID TARGETS

I .A . S t i l i n , S.V. Be.nx.ij, M. B r e i n i g , C. ßottche.x, R. L a t z ,

M. B u i k h a i d , H. F o l g e n , H.J. F n i i c h k e i n . , K.O. G n o e . n e v c i d ,

V. Ho { ) m a n n a n d P. K o ^ c h a n

RYDBERG-STATE PRODUCTION IN COLLISIONS BETWEEN FAST IONS

AND CARBON TARGETS

H.-V. B e t z

CONVOY ELECTRONS FROM ATOMIC AND MOLECULAR HEAVY ION

COLLISIONS WITH SOLIDS

P. K o A c h a i , R. L a ^ z , J . Kümmle*, M. B u * k h a x d , H.J. F x i ^ c h k o x n

V. Ho&mann, J . S c h l a d e . * , R. S c h l a m m , K.O. G n o e n e v e l d ,

M. B r e i n i g , S. E l & t o n , I .A . S e l l i n a n d W. M e c k b a c h

VI!

ALIGNMENT OF HIGH RYDBFRG STATES IN HYDROGEN

H.G. B&tiMj, I . C . PeHaea , V . K . Weefe, L . P . Somme^u- t^e 150

AUTHOR INDEX 158

SUBJECT INDEX 160

THEORETICAL DESCRIPTION OF THE CUSP ELECTRONS

EJECTED IN ASYMMETRIC HEAVY-ION COLLISIONS

D.H.Jakubaßa-Amundsen

Physik-Department, Technische Un ivers i tä t München, 8046 Garching

and GSI Darmstadt, Germany

Abstract

S ta r t i ng from the Faddeev equations a ser ies expansion fo r the t r a n s i t i o n ampl i -

tude fo r e lect ron emission i s g i ven , which serves as a basis fo r the d iscuss ion of

approximations used in the l i t e r a t u r e and t h e i r ränge of v a l i d i t y fo r a given c o l l i -

s ion System and momentum of the ejected e lec t ron . Both target and p r o j e c t i l e e lec t ron

emission w i l l be considered. Emphasis i s l a i d on the asymmetry of the forward peak

and i t s Var ia t ion with System parameters, such as c o l l i s i o n ve loc i t y and Charge r a t i o

Z p / Z j . The theoret ica l r esu l t s w i l l be confirmed by a comparison with experimental

data.

1. Introduct ion

Elect ron spectroscopy has received a great deal of a t ten t ion l a t e l y , since the

production of electrons by fas t ion impact i s of p r inc ipa l i n te res t in many branches

of phys ics . In p a r t i c u l a r , the so -ca l l ed cusp e lectrons which are emitted with

nearly zero ve loc i ty r e l a t i v e to the p r o j e c t i l e , i . e . appear in the target frame in

forward d i rec t i on with a momentum that equals the c o l l i s i o n ve loc i t y v have been

focused in a var iety of experimental and theore t i ca l inves t iga t ions (see i . e . Meck­

bach and Burgdörfer, proceedings of th i s Conference). The spectra l d i s t r i b u t i o n of

the cusp electrons provides a sens i t i ve tes t fo r the a p p l i c a b i l i t y of f i r s t - o r d e r

theo r ies , and I w i l l show in the fo l lowing that these theor ies are i n v a l i d not only

fo r e lectrons i n i t i a l l y bound to the ta rge t , but a lso for e lect ron loss from the

p r o j e c t i l e in asymmetric c o l l i s i o n Systems, even at very high impact v e l o c i t y .

2. Faddeev Theory

The basic formalism for e lect ron emission i s most e a s i l y described in the three-

pa r t i c l e p i c tu re , where the e f fec t of a l l e lect rons of the p ro j ec t i l e - t a rge t System

except the act ive one i s incorporated into the po ten t i a l s . Let ^ be the i n i t i a l

e lec t ron ic bound State (with A denoting e i t he r target T or p r o j e c t i l e P ) , the

in te rac t ion f i e l d between the e lect ron and A, and V ß the e lec t ron i c perturbat ion in

the i n i t i a l Channel. In the semic lass ica l theory the t r a n s i t i o n amplitude i s given

by ( in atomic uni ts)

18

8 F I = - 1 f dt < Y ; | v B | Y ? > (2-D

where y ^ i s the exact Solut ion to the e lec t ron ic two-center probleni with the boun-

dary condi t ion of a f ree e lect ron in the f i n a l State. For fas t c o l l i s i o n s , which I

sha l l concentrate on, the in ternuc lear motion can be described by a c l a s s i c a l

s t ra igh t -1 ine path, thus neglect ing the in ternuc lear p o t e n t i a l .

i

Accordmg to Faddeev , y ^ can be wr i t ten as a sum of wavefunctions which are

determined by a set of two coupled equations

i V f > • | y 0 . f > + l v i > + i v 2 > ( 2 - 2 >

l Y l > = GB VB I V o , f > + GB VB | V 2 >

I V 2 > = GA VA | V o , f > + 6 A V Ä | Y I >

where y Q ^ i s an e lec t ron i c plane wave, and G^ g is a Green's funct ion defined by

Gß g = ( i ä / 3 t + A / 2 - g - i€)~^ with S —>0. In order to avoid convergence Prob­

lems from the long-range Coulomb f i e l d s i t i s assumed here and in the fo l lowing that

Vß, V ß are screened Coulomb potent ia l s , with the wel l -def ined l i m i t of Screening—>

0 taken in the f i n a l T matr ices. From (2 .2 ) , ser ies expansions fo r can be

der ived. Using the d e f i n i t i o n of an eigenstate to nucleus A, y ^ = (1 + G/\ v/\) V o f

the ser ies can be wr i t ten in three d i f f e ren t ways

| V " > = (1 • G ß V B + G A V A + G ß V B G A V A + G A V A G ß V B + . . . ) | Y ( ) ) f > (2.3a)

= ( 1 + GA VA + G B W A + • • •> l r ? > ( 2 - 3 b )

= (1 • G B V ß + G A V A G B V ß + . . . ) | y } > (2.3c)

where the form (2.3a) i s a Symmetrie expansion a l lowing fo r both a p r o j e c t i l e or

target e lec t ron ic f i n a l S ta te .

For p rac t i ca ! purposes, these ser ies have to be t runcated, such that i t i s no

longer i r r e l e v a n t , which of them i s used. I t i s thus c ruc ia l to determine from the

momentum k and the d i r ec t i on td* of the ejected e lec t ron r e l a t i v e to the parent nuc le­

us , as well as from the Charge r a t i o Z A / Z ß , whether V"A or V ß w i l l be the dominating

p o t e n t i a l , in order to truncate the appropriate s e r i e s . An estimate fo r the poten­

t i a l act ing on the e lec t ron a time T a f te r the c o l l i s i o n can be found by using the

Coulomb formula

19

V. « — , V R ~ 5 (2.4) k X T [ ( V - k c o s * ) 2

+ k 2 s i n 2 , 9 - ] 1 / 2

R e s t r i c t i n g the d iscuss ion to the cusp e lec t rons , one has to use (2.3c) fo r k 0

i f Z A >> Z ß (because V A i s dominant), and (2.3b) fo r k—*v and/d*—*0 i f Z ß » Z^

(as Vg i s dominant), but in the cases k—>0, Z^ « Z ß as well as k—>v,/d*-^0,

Z ß ä Z A , the Symmetrie se r ies (2.3a) should be pre fer red.

3.Approximations f o r the Ca lcu la t ion of the Forward Peak: Target Ion isat ion

Let me f i r s t consider the case where bare p ro j ec t i l e s are impinging on the target

atom, such that only target e lect rons are cont r ibu t ing to the forward peak. S o l i d

State e f f e c t s , as fo r example capture in to h igh- l y ing bound p r o j e c t i l e s tates which 2

are eventua l ly pa r t i e i pa t i ng in the convoy e lect ron produetion , w i l l not be d i scuss -

ed here. In t h i s case, A i s i d e n t i f i e d with the target T, B with the p r o j e c t i l e P.

a) F i r s t - o r d e r Bern Approximation

The f i r s t Born approximation fo r target i on i sa t i on i s obtained by re ta in ing the

f i r s t term in the ser ies (2 .3c ) , such that the t r a n s i t i o n matrix element reads

M};B- = <H>] I v p l y T > (3.D

Pioneer experiments on the forward peak in c o l l i s i o n s of H + on He by Rudd and cowor-3

kers have c l e a r l y demonstrated that (3.1) s t rongly underestimates the number of

e lectrons e jected with beam ve loc i t y near 0 ° .

b) F i r s t - o r d e r Faddeev Approximation

From the estimate (2.4) i t i s obvious that in th i s case the p r o j e c t i l e f i e l d must

not be neglected in the expansion of y ^ . Using the Symmetrie ser ies (2.3a) and re ­

ta in ing the f i r s t three terms 1 + G p V p + GyVy, i . e .

M}: f a = ^ y ? | v p | y { > + < v j | v p | y { > - < f y 0 , f I v p | y { > (3.2)

4 Macek could exp la in the enhanced i n tens i t y of the cusp e lec t rons . I t o r ig ina tes

p from the normal isat ion of the p r o j e c t i l e eigenstate Y f > a n c ' diverges l i k e F Q =

2trZp/lk^ - v | as the momentum k^ of the e lec t ron approaches v , unless the f i n i t e

detector reso lu t ion i s taken into aecount. However, a care fu l ana lys is of the peak 5

shape reveals an enhanced i n tens i t y of e lect rons with k^ < v as compared to those

with > v , a feature which can not be reproduced by the Symmetrie fac to r F Q con-

20

ta ined in (3 .2) .

c) Second-order Born Approximation

I nc i den ta l l y , the i nsu f f i c i ency of a

f i r s t - o r d e r theory f o r the descr ip t ion of

Charge t rans fer to the continuum (CTC)

which i s the bas ic process leading to the

cusp e lec t rons , i s not s u r p r i s i n g . It i s

known from bound-state Charge t rans fe r

that the Brinkman-Kramers theory provides

the wrong h igh-ve loc i t y l i m i t which has

only recent ly been v e r i f i e d experimental-

l y ^ s and also gives a wrong r e l a t i v e

occupation p robab i l i t y of the f i n a l

s u b s h e l l s . Therefore, Dettmann 7, Shake-Q

shaf t and coworkers have appl ied the second Born approximation, which cons is ts in

r e ta i n i ng the f i e l d s up to f i r s t order in the ser ies (2 .3b) :

F i g . 1 Momenta of the ejected e lec t ron in the p r o j e c t i l e or target frame

M2

F:B- = <"Yf ! v

p + v T G 0 v p | y { > (3.3)

where G Q i s the f ree propagator. The asymmetry property of the cusp e lect rons i s

r e a d i l y displayed by wr i t i ng the doubly d i f f e r e n t i a l cross sect ion in terms of a

p a r t i a l wave expansion

~ .» 1 ^ y a . P,(cos©) (3 .4) dE fdJe f | k f - v | £ — 1 1

o where = k^/2, P-j a Legendre polynomial and 9 the e lect ron emission angle in the

p r o j e c t i l e frame with respect to v . I t has been shown9 that the assumption of c o n t i -

nu i ty of the capture amplitude across the ion i sa t i on threshold leads to f i n i t e a^

( i nc lud ing odd values of 1) in the l i m i t k f—>v. This i s s a t i s f i e d for the second o i n

Born theory ' , whereas a-j (1 > 0) vanishes in the f i r s t - o r d e r theory. For zero

emission angle td^ in the target frame, e lectrons with k f > v are emitted at 9 = 0,

such that P-j = 1, whi le e lect rons with k^ < v appear in the backward d i r ec t i on

& =7t, where P-| = ( - 1 ) \ resu l t i ng in a d i scon t inu i t y of the cross sect ion at k^ = v.

The increased in tens i t y at 0= 0 may be explained by the fac t that these e lect rons

are e jected in the same d i rec t i on as the target motion (when viewed from the pro­

j e c t i l e frame). Another d i s t i n c t i o n between the Brinkman-Kramers and the second Born approxima-

21

t i on i s found f o r the pos i t i on E of the forward peak as a funct ion of emission peaK .g

angle tf^. From (3.4) i t can be v e r i f i e d that the f i r s t - o r d e r theory shows a quadra-

t i c dependence on ̂ f o r small ang les , whi le the second Born theory d isp lays a l i n e a r

decrease of E p e a ^ wi th «3^. The l a t t e r dependence i s c l e a r l y v e r i f i e d by experiment in

H e + + on Ar ( r e f s . 10,11) and H + on He c o l l i s i o n s 1 2 .

d) Impulse Approximation (IA)

I f the p r o j e c t i l e i s much heavier than the ta rge t , a f i r s t - o r d e r treatment of the

potent ia l Vp in the T matrix i s not s u f f i c i e n t . Instead, a consis tent f i r s t - o r d e r

expansion of y ^ in the weak target f i e l d Vj leads with the help of (2.3b) to

| y - > = 0 + GOV t + GpV p G 0 v T ) | y j >

M}} = <yP

f\ V T(1 + GpVp) I ^ { > (3.5)

where use has been made of the re la t ions G p = G Q + G p V p G o and <yf | V p I ^ ! > = ^ Vj/£ | V T [yT > . An Inser t ion of a complete set of plane waves y Q * behind the

Operator (1 + G pVp) in (3.5) al lows fo r the descr ip t ion of the CTC process in terms of e jec t ion of a target e lec t ron in to an intermediate p r o j e c t i l e State with momentum q + v , with a subsequent sca t te r ing by the target f i e l d in to the f i n a l State 4>£.

F ig .2 Ratio of peak energy to v /2 as a funct ion of fi^ f o r c o l l i s i o n s

of Ar 1 8 + (v=18.1 a .u . ) and N e 1 0 +

(v=17.62 a .u . ) with He

F ig .3 Cross sect ion ra t i o between im-pulse approximation and f i r s t ^ Born theory times F Q =2»Zp/ Ik f -v l

a t / f r f =0° and 1.7° fo r 155 MeV

N e 1 0 + on He

22

Of f - she l l e f f e c t s fo r CTC are expected to be small and neglected in the f o l l ow ing .

As the impulse approximation reduces to the second Born theory fo r asymptotic

c o l l i s i o n v e l o c i t i e s , i t i s expected to show s im i l a r features c h a r a c t e r i s t i c f o r a

higher-order theory. F ig .2 d isp lays the l i nea r dependence of E M / . on * k fo r small 18+ 10+ P angles in c o l l i s i o n s of Ar and Ne with He at nearly equal impact v e l o c i t i e s . In

order to make the d i scon t inu i t y of the d i f f e r e n t i a l cross sect ion at k f = v v i s i b l e ,

F ig .3 shows the r a t i o d 2 < 5 I A / ( F Q d V * B *) which i s f i n i t e at E f = v 2 / 2 . At nonzero

angles the d i scon t i nu i t y i s replaced by a rather steep f a l l - o f f of the r a t i o when E f

i s increased.

In order to study the dependence of the d i scon t inu i t y on the 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 , l e t me def ine the r a t i o

d ^ / d E , d J L ( k * &=0) s = t t t t — ( 3 a 6 )

d ^ / d E f d J ^ ( k * ,J f=0)

where the e lec t ron momenta k^ and k^ are chosen such that tf[ = Zp / |k^ - v j i s

equal (and » 1, but f i n i t e fo r numerical reasons) fo r both, with k* £ v and k ^ ^ v .

As the e lec t ron i s in a f i n a l p r o j e c t i l e S ta te , the f i e l d strength Zp i s taken as

H +-> He

Zp/v •2 .h >i

Fig .4 D iscont inu i ty S at ^ = 0 andij=30

as a funct ion of inverse ve lo ­

c i t y f o r c o l l i s i o n s of C 6 + , Ne 1 0"*

and A r 1 8 + with He

F ig .5 D iscont inu i ty S at «^3:0 as a

funct ion of Z p / v in H —*He co l ­

l i s i o n s . Shown are resu l ts f o r the IA ( ) and the asymptotic second Born theory ( — ) . Data are from Dahl 12

23

reference parameter a lso with respect to the ve loc i t y v. F ig .4 shows S in the IA fo r

C 6 + , N e 1 0 + and A r 1 8 + impinging on a He ta rget . For large c o l l i s i o n v e l o c i t i e s , v Z D

13 a peaking approximation reveals that in the l i m i t of k^—> v, ^ — * 0 the matr ix

element in (3.5) becomes proport ional to a conf luent hypergeometric f unc t i on , the a r -

gument of which i s x » 16iZp/3v fo r k^ ^ v and x s* 0 fo r k^ ^ v. Thus the

d i scon t i nu i t y vanishes (S — ^ 1) fo r Z p / v —> 0. At f i xed Z p / v which may a l so be con-

sidered as parameter of the i n i t i a l perturbing f i e l d , S increases when Z p/Z-j- becomes

sma l le r , i . e . when the r e l a t i v e inf luence of the target becomes st ronger . In the case of H+—> He c o l l i s i o n s , the systematics of the d i scon t i nu i t y has re -

12 " cent ly been invest igated experimental ly . In F ig .5 the re lated quant i ty S =

k> FQ(k>) d ^ ( k < , 0 ) / [ k < FQ(k<) d 2 «-(k>,0)] i s shown, where ( k j - v ) 2 = (k* - v ) 2 =

2*10~^ eV. However, the data are taken at r e l a t i v e l y low v where the IA i s no longer

v a l i d . Never the less, the trend shown in F ig .4 i s in good agreement with the e x p e r i -

ment. That the theory extrapolated to H + p ro j ec t i l e s l i e s above the data resu l t s from

the average over the angular acceptance in the experiment, which reduces the d i scon­

t i n u i t y (c f . F i g . 3 ) . For comparison, the second Born resu l ts are a lso shown fo r H + —*

He at = 0, applying the h igh-ve loc i t y formula from r e f . 8 . I t i s seen that even at

rather high v e l o c i t y , t h i s formula breaks down, whereas at the highest v e l o c i t y i n ­

ves t iga ted , agreement i s found with the IA wi th in the numerical accuracy of the

l a t t e r .

The d i scon t i nu i t y S reveals i t s e l f as a dec is ive parameter fo r the shape of the

forward peak. Considering only pa r t i a l waves with 1 £ 1 in the expansion ( 3 . 4 ) , the

d i f f e r e n t i a l cross sect ion may be approximated by

= , J * Z f L , a f ( S + 1) - (S - 1) cosöl (3.7) dEfd<Äf | k f - v | L J

k . c o s ^ - v cos© = 4

| k f - v i

which has the property that at ^ = 0, d^>(k* , 0 ) / dV (k * ,0) = S, whi le d2cT i s c o n t i -

nuous for ^ f 0. In order to compare with experimental peak shapes, (3.7) has to be

integrated over the angular a c c e p t a n c e / ^ , g iv ing a simple ana ly t i c formula, and over

the energy reso lu t ion 4 E r . The constant a in (3.7) accounts for the absolute value 10+

which h i ther to has not been determined exper imental ly . In the case of 155 MeV Ne c o l l i d i n g with He (ah = 1 . 4 ° , 4 E . = 2.2%) the peak shape i s compared with experimen-

14 ta l data in F i g . 6 . The dashed l i n e i s the formula (3.7) with S from F i g . 4 , and

gives a reasonable f i t to the peak shape. The agreement between the IA and (3.7) in

the t a i l s might be improved by int roducing the energy dependence of the f i r s t Born 8+

theory into the constant a , as suggested from F i g . 3 . For the c o l l i s i o n System 0 — *

24

F ig .6 D i f f e ren t i a l cross sect ion for cusp e lec t ron emission in

Ne 1 0 +—»He c o l l i s i o n s at v=17.62 a . u . Fül l l i n e , IA, dashed l i n e , (3.7) normalised at the peak.

14 Data are from Berry et al

F ig .7 D i f f e ren t i a l cross sect ion for 8+

cusp e lec t ron emission in 0 —^ He c o l l i s i o n s at v=16.64 a . u . The dashed l i n e i s the formula (3 .7 ) ,

14 the data are from Berry et al (a rb i t ra ry un i ts )

He (<{h = 1.4°, 4 E f = 1.4%) the formula (3.7) with S = 3.49 a lso compares well with 14

the experimental data . Note that apart from the normal isat ion of the peak height

there are no f ree Parameters in t h i s theory.

4 . P r o j e c t i l e Ion isat ion

When the p r o j e c t i l e i s not f u l l y s t r i pped , e lec t ron loss w i l l i n general provide

the dominant cont r ibu t ion to the forward peak at high c o l l i s i o n v e l o c i t i e s . Then in

the formulas (2.1) - (2 .3 ) , A has to be i den t i f i ed with the p r o j e c t i l e P, B with the

ta rge t T.

a) F i r s t - o r d e r Born Approximation

The f i r s t Born approximation fo r p r o j e c t i l e i o n i s a t i o n , which i s obtained from

the f i r s t term in the ser ies (2 .3c ) , y i e l d i ng

MJ:b

- = ^ y j | vT| > ( 4 . 1 )

has been frequent ly appl ied ' fo r the descr ip t ion of the forward peak in Systems

wi th Zp j£ lj and large v. In a s i m i l a r way as for target i o n i s a t i o n , the d i f f e r e n -

25

t i a l cross sec t ion near = v can be decomposed into pa r t i a l waves

2n

dE f dÄ f Ik f " v1 r r r 1=0

^> ' a ] P ^ - cos©) (4.2)

where in the f i r s t Born theory, 1 can only take even values ' such that the cross

sect ion does not exh ib i t a d i scon t inu i t y when© i s switched from 0° to 180°. Expe r i -19

ments show that the e lec t ron loss peak i s much more Symmetrie than the correspon-

ding CTC peak which would support the a p p l i c a b i l i t y of the f i r s t Born theory ; however,

the dependence of the peak width on ve loc i t y in c o l l i s i o n s of e . g . Si with Ne and 19 20 0 with Ar show deviat ions from f i r s t Born p red ic t i ons , and a de ta i led study of

the angular d i s t r i b u t i o n for H —^ He c o l l i s i o n s in the cusp region ind ica tes that

more p a r t i a l waves than present in (4.2) should cont r ibu te .

b) Second-order Faddeev Approximation

For asymmetric Systems with Zp Zy a treatment of the target f i e l d Vy in f i r s t

order i s not s u f f i c i e n t , as long as the c o l l i s i o n ve loc i t y i s not very much greater

than Zy. As in the cusp region both potent ia ls w i l l be important, the Symmetrie

ser ies (2.3a) fo r ify should be used. The second-order Faddeev approximation i n c l u d -

es a l l terms e x p l i c i t l y wr i t ten in (2.3a)

such that the t r a n s i t i o n matrix element

becomes

M 2: F A = + V p G p ) v T | v ; >

- < y J l v T l v f > - < y ? l vTl Y ? >

where the l a s t three terms correct fo r

double counting and are nothing but the

f i r s t - o r d e r Faddeev approximation. The

f i r s t term, , descr ibes the Charge

t ransfer to the target continuum and

inf luences the t a i l s of the forward

F ig .8 Cross sect ion d iv ided by F Q f o r

e lec t ron emission in 2.5 MeV H—> Ne c o l l i s i o n s at«J)^=0. Only the

second term in (4.3) i s inc luded

26

peak , whereas the second term, M 9 , accounts fo r the p r o j e c t i l e i on i sa t i on v ia an intermediate target continuum State and i s dominant fo r k ^ « v. At = 0, the ma-

21 t r i x element corresponding to t h i s second term exh ib i t s a phase proport ional to ZpCOSÖ which gives r i s e to a d i scon t inu i t y as k^ t raverses v. Unfor tunate ly , th is phase depends a lso on in tegra t ion var iab les which cannot be f i xed by a simple peaking approximation. In F ig .8 the d iscon t inu i t y of the cross sect ion ca lcu la ted only from i s c l e a r l y v i s i b l e . In contrast to the target i o n i s a t i o n , the h igh-energy side i s enhanced, because the electrons which are ejected with 8 = 0 move in the same d i rec t ion as the p r o j e c t i l e from which they o r i g i na te . Note that the wings of the peak w i l l be inf luenced by M-j which is not included in the c a l c u l a t i o n .

Again, one may def ine the r a t i o S according to (3.6) as a measure fo r the d iscon­t i n u i t y . F ig .9 shows S as a funct ion of Z p / v fo r H on C, Ne and Ar ta rge ts . While the f i r s t Born theory always gives a ra t i o c lose to 1, the dev ia t ion from unity in the Faddeev approximation i s the 1 arger , the heavier the target atom and the lower the c o l l i s i o n v e l o c i t y , i nd ica t ing the importance of the target f i e l d . In Fig.10 the ab­solute values of the d i f f e r e n t i a l cross sect ion at ^ v and = 0 are compared to the f i r s t Born r e s u l t s . Considerable deviat ions are found even fo r a C ta rge t , and the Born theory becomes only v a l i d at v Zy.

In conc lus ion , i t has been shown fo r asymmetric c o l l i s i o n Systems, using the Im­pulse approximation for target i on i sa t i on and the second Faddeev approximation for e lec t ron loss that the forward peak d isp lays a d i scon t inu i t y for zero emission angle

F ig .9 Discont inu i ty S at 5*̂ =0 and 4f=10 as funct ion of inverse ve loc i t y fo r H c o l l i d i n g with C, Ne and Ar

F ig.10 Cross sect ion r a t i o between the Faddeev and the Born approxima­t ion fo r H c o l l i d i n g with C, Ne and Ar at ^ = 0 and ^=10

27

at k.p = v i r r espec t i ve whether the e lect rons o r ig ina te from the target or the pro jec­

t i l e . The magnitude of th i s d i scon t i nu i t y can be c lose l y re la ted to the experimental

peak shapes which provide strong evidence fo r the necessi ty of a h igher-order theory.

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