18 July 2002, Metz (France) Omar FOJON (CONICET-UNR) Argentina 1 Molecular structure Two-center continuum wavefunction for the ejected electron Not oriented

18 July 2002, Metz (France)

Omar FOJON (CONICET-UNR) Argentina

1

• Molecular structure• Two-center continuum

wavefunction for the ejected electron

• Not oriented target nor particular rovibrational initial state in experiments

• Vibrational nor Rotational states resolved

Introduction

Ionization of molecules compared to atoms

Theory Experiment



2

• DWVA (Zurales et al, 1988)• DWO (Monzani et al, 1999)• Influence of vibrational states, electronic vertical

transitions (Weck et al, 2000)

• e+/H2+ (Fojón et al, 1996)• Effective Center Models for ion impact (Rivarola et al,

1991-2002)

Theoretical work on H2 TDCS



3

• Absolute Triple Differential Cross Sections (TDCS) at very high impact energy in coplanar asymmetric geometry (Chérid et al, 1989)

• Relative TDCS at high impact energy (Jung et al, 1975 in coplanar asymmetric geometry

Experimental work on H2 (TDCS)



4

1 22 22g ge H e H

• Modelization– Electronic vertical transitions – Fixed molecular orientation during the collision– Neglect of exchange effects

The (e,2e) process of interest is

-Intermediate and high impact energies

-Asymmetric coplanar geometries



5

Cross Sections

• Fivefold differential cross section for coplanar asymmetric geometries

– Averaging over all molecular orientations, triple differential cross sections are obtained

5

24(5)02 ( )ee s

fie s s i

k kdt

d d d dE k

3

2430

12 ( )

4ee sfi

e s s i

k kdd t

d d dE k



6

T-Matrix Amplitude• The prior version of the T-matrix element reads

1 2

1 1 A Bi

P P A B

Z ZV

r r R R

1 2 1 2( ) 2 ( , , ; ) ( , , ; )efi f i it R r r V R r r

.

1 2 1 23/ 2( , , ; ) ( , ; )2

iik R

i i

eR r r r r



7

Initial bound wavefunctions

1 2( , ; ) 1 2i ir r N

1 2 2 1jA jB jA jBr r r rj e e e e Berencz Wavefunction

* * * *1 2 1 2

1 2( , ; ) A B B Br r r ri HLr r N e e e e

Heitler-London by Wang

0 1 2 00.002712 0.0711 1.1202 1.4354N

*01.166 1.406 . .a u



8

Final wavefunction

2 22 0( ; ) A Br r

f fr N e e

Final Bound wavefunction

01.3918 1.406 . . ( )a u vertical transitions

.

1 2 13/ 2( , , ; ) ( ; )2

sik R

f f C

eR r r r r

C r Continuum wavefunction TEC, TCC, BBK

approximations



9

Two Effective Center (TEC) Model

ks

ke

HH

ee

e

• Ionization near nucleus• Complete screening by passive electron• One continuum

r1



10

TEC final wavefunction

13/ 2 .

1 1 1, , 2 , ,eik rC e C j e e j jk r k r e C k r Z

*1exp( )jr

C j is chosen

appears in the t-matrix element

with j a or b

If

Weck, Fojón, Hanssen, Joulakian, and Rivarola, PRA 63 (2001) 042709

/ 21 1, , 1 / / ;1; ( . )z kC k r Z e iZ k F iZ k i kr k r



11

Two-Center Continuum (TCC) Model

ks

ke

HH

e

e

• H2+ targets

• Two-continuum wave function

Joulakian, Hanssen, Motassim and Rivarola PRA 54 (1996) 1473 for H2+

r1



12

TCC wavefunction for H2

13/ 2 .

1 1 1, 2 , , , ,eik rC e e A sc e B sck r e C k r Z C k r Z

/ 21 1, , 1 / / ;1; ( . )z kC k r Z e iZ k F iZ k i kr k r

TCC for H2 molecules

2 1scZ Asymptotic Charge 1

Weck, Fojón, Hanssen, Joulakian, and Rivarola, PRA 63 (2001) 042709

Weck, Fojón, Joulakian, Stia, Hanssen and Rivarola, PRA 66 (2002) in press



13

• Boundary conditions satisfied

• Passive electron included

• Only one continuum: easy to handle

• Boundary conditions not satisfied (correction 2Zsc=1)

• Passive electron plays no role (correction 2Zsc=1)

• Two continua

TEC vs. TCC

TEC TCC



14

0 50 100 150 200 250 300 3500

1

2

3

4 Exp. from Chérid et al (1989) CWLA from Zurales et al (1988) TCC C-TCC TEC

s= 1 deg

3DC

S (

a.u.

)

e (deg)



15

• TEC treatment

• BBK correlated continuum wavefunction in the final channel

Molecular BBK Model

ks

ke

HH

e

e

e



16

13/ 2 .

1 1 1 1, 2 , , , , , ,eik rC e e j e s j s P P ePk r e C k r C k R C k r

ee

Z

k

ss

Z

k

1

1

2ePPk

Molecular BBK Final Continuum Wf.

17

Direct and Indirect terms

ks

ke

HH

e

e

e

, ,, ( )

, ,e dir indfi j l l j

j l j l

t t t j l a b

* *2 11 2 2. .

, 1 11

1 1, , j je l lr rik r r rdir iK R

j l C j j pp j

t e r R r e e e e er R

r1pR1

Direct Term

18

Direct and Indirect terms

ks

ke

HH

e

e

e

, ,, ( )

, ,e dir indfi j l l j

j l j l

t t t j l a b

* *2 11 2 2. .

, 1 12

1 1, , j je l lr rik r r rind iK R

j l C j j pp l

t e r R r e e e e er R

r2p

R2Indirect Term



19

MBBK Cross Sections• Neglecting indirect terms and averaging over

all molecular orientations

ek K

3 2253 *

0

0

4 2 , ; ,

sin( )1

e se

e s e i

k kdN I K k

d d dE k

i sK k k

20

210 240 270 300 3300

1

290 180 270

1

2

390 180 270

0

1

2

3

4

(c) s= 3°

Ejection angle e (deg)

(b) s= 1.5°

TD

CS

(a.u

.)

Ei= 4087 eV

Ee= 20 eV (a)

s= 1°

MBBK TEC Chérid M et al (1989) 2 x BBK

21

0.00

0.09

0.18

0.270.00

0.09

0.18

0.27

210 240 270 300 3300.00

0.09

0.18

(b) s=8.9°


TDCS

(a.u

.)

(a) s=8.2°E

i=4168eV

Ee=100eV

MBBK TEC Chérid et al (1989) 2x BBK (H)

(c) s=9.6°

22

0 90 180 270 3600

3

6

0

4

8

12

16

20

(b) s=8°

T

DC

S (

a.u

.)


(a) s=4°E

i=250eV

Ee=4.5eV

MBBK TEC Jung et al (1975) 2xBBK(H) DWVA

230 90 180 270 360

0

1

2

3

0

2

4

6

(c) s=12°


MBBK TEC Jung et al (1975) Monzani et al (1999) 2 x BBK (H) (b)

s=8°

Ei=250eV

Ee=9eV

TD

CS

(a

.u.)

0

3

6

9

12(a)

s=4°

240 90 180 270 360

0

2

4

0

3

6

9

12E

i= 100 eV

Ee= 4.5 eV

(b) s=15°


(a) s=7°

TD

CS

(a

.u.)

MBBK TEC Jung et al (1975) 2 x BBK (H)



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• MBBK: Good agreement between experiments at intermediate and high impact energy

• TEC and TCC: describe well experiments at high impact energy

• Inclusion of e-e repulsion crucial to describe the binary and recoil peak region

• Extension of TEC treatment to other diatomic molecules such as N2

Conclusions and Perspectives

Documents

18 July 2002, Metz (France) Omar FOJON (CONICET-UNR) Argentina 1 Molecular structure Two-center continuum wavefunction for the ejected electron Not oriented