Charge State Distribution of Recoil 16O from 4He(12C, 16O) g

in Astrophysical interest　　劉盛進 A 、相良建至 B 、寺西高 B 、 藤田訓裕 B 、山口祐幸 B 、　松田沙矢香 B 、三鼓達輝 B 、岩崎諒 B 、 Maria T. RosaryB 、櫻井

誠 A

神户大学理学研究科 A 九州大学理学府物理学専攻 B

2011-12-03

introduction

Total S-factor of 4He+12C 16O+g

Ruhr U.Kyushu U.

extrapolation

Astrophysics.4He(12C, 16O)g Ecm=2.4 MeV,1.5MeV Succeeded !! 　

For the recoils 16Oq+, to determine 4He(12C, 16O)g total cross section, fraction of 16Oq+ should be precisely known.

Investigation: charge exchange processes and charge state distribution of Oxygen though Helium gas.

16O1+,2+,….8+

The fraction of 16Oq+??

16Oq+ detector

Features of ion-atom collision

physical processes:Capture, Ionization, Excitation,…..

q-1

q-2

q+1

Chargedistribution

Charge exchange

e- e-(Auger)

Ionization of Projectile, Aq+ A(q+1)+

For a collision, from an initial state i to a final state f, ionization cross section is given:

PWBA: plan wave Born Approximation

• Projectile in initial and final states is represented by plane wave, (~exp(iqr)).

• Perturbation of the projectile orbits is neglected.• Screening effect is taken into account.• projectile assumed to linearly without deviation of trajectory.

(initial) (final)

min

2 2

12 30

8( , , ) ( )if nlm nlm

nlm nlm q

dqd d f q Z q

v q

2 2 2( , , ) exp( ) ( ) ( )nlm f nlmf q iRq r r dRdr Scattering amplitudeEffective charge of target(He):

2

1( )Z q Ejection electron energy:

Projectile velocity: n

Minimum momentum transfer: qmin=（ Ip+e） /n Ip is binding energy of the projectile electron, e is energy of the ejected electron,

2

2

2 ( 1)2 0

dR dR Z l lE R

dr r dr r r

The final state: to solve radial

equation in coulomb field

21 1( )

2

n

aa aba ab

ZH

r r nlm nlmH E

For the initial state(nlm) of ionized electron, the Schrodinger equation at multi-electrons system :

,

Capture of Projectile, Aq+ A(q-1)+

Impact parameter treatment 2

0

2 ( )ij ijb d

• electron exchange is neglected.• The corresponding trajectory may be uniquely

distinguished by an impact parameter and a velocity. • Nuclear and nuclear interaction is neglected.

The H-like wave functions of an optical electron in the initial and final states:

1/ 2( ) ( )z Hnl nlP r Z P Zr

( ) exp( ( ) )A Bij i jb i i t dt

*

2 2 1( ) ( ) ( ) exp( )B A Aj i rr V r r ir dv

1( )Ai r : electron initial state

2( )Bj r : electron final state

22

( , ) A B AZ Z ZV r t

R r

,A Bi j : corresponding eigen-energies

(initial) (final)

100 1000 10000

1E-18

1E-17

1E-16

1E-15

1E-14

1E-13

1E-12

Cro

ss s

ect

ion

cm

2

Energy (keV)

Arsenv Our CalculationCDW

1000 10000

1E-20

1E-19

1E-18

1E-17

1E-16

1E-15

cm

2

E , KeV

Experimental data Theory calculation

16O6+ 16O7+

16O5+ 16O4+

Arseny: adiabatic approximation, low energyCDW: Continuum distorted wave approximation, high energy

F.M.Martine et al. PRA,1965(1971)

Possibility>1

Ionization cross sectionCapture cross section

Equilibrium Charge State Distributions

Variation of the charge state distribution:

Equilibrium distribution:

All fractions reach a certain value and keep constant.

, / 0x dF dx

' ', , '', '

( ) 0q q q q q qq q q

F F

' ', , '', '

( )qq q q q q q

q q q

dFF F

dx

1qq

F Non-equili.

equili.

Cross sections of ionization and capture are required

,Gain Loss

Results

0 300 600 900 1200 1500 1800 2100 24000.0

0.2

0.4

0.6

0.8

1.0

1+5+

4+2+

Fra

ctio

n

Thickness ( x 1014 atom/cm2)

3+

4.5 MeV 16O3+ + He

0 300 600 900 1200 1500 1800 2100 24000.0

0.2

0.4

0.6

0.8

1.0

2+6+

5+

4+

Fra

ctio

n

Thickness x ( 1014 atoms/cm2)

7.2MeV 16O3+ + He

3+

1+,7+,8+

The thickness evolution of fraction at 7.2 MeV and 4.5 MeV

Equilibrium distribution can be obtained

Evolution of fraction against thickness

Equili. Distri.Equili. Distri.

3+， 100% 3+， 100%

HeHe

Charge state distribution

0 2 4 6 8

0.0

0.1

0.2

0.3

0.4

0.5

Fra

ctio

n

Charge state

Theory Experiment

4.5MeV16O1+ + He

0 2 4 6 8

0.0

0.1

0.2

0.3

0.4

0.5

Fra

ctio

n

Chareg state

Theory experiment7.2 MeV

16O1+ + He

The results of calculation deviated from the experimental results and shift to left.Reflecting the shape of experimental data well but mean charge state is smaller.

Theory Experiment

4.5 MeV 2.88 3.50

7.2 MeV 4.15 4.38

Mean charge state:

0 2 4 6 8

0

10

20

30

40

50

Fra

ctio

n

Charge state

Theory experiment

2.2 MeV

0 2 4 6 8

0

10

20

30

40

50

Fra

ctio

n

Charge state

theory exp.

3.2MeV

0 2 4 6 8-5

0

5

10

15

20

25

30

35

40

45

Fra

ctio

n

Charge state

Theory Experiment

5.2MeV

Data from TRIUMF Lab. (energy: 2.2MeV --- 14 MeV)

Shift to left.!!

0 2 4 6 8-5

0

5

10

15

20

25

30

35

40

45

50

Fra

ctio

n

Charge state

Theory Exp.

9.4MeV

0 2 4 6 8-5

0

5

10

15

20

25

30

35

40

45

50

Fra

ctio

n

Charge state

Theory Exp.

12 MeV

0 2 4 6 8

0

10

20

30

40

50

60

Fra

ctio

n

Charge state

Theory Exp.

14MeV

1 2 3 4 5 6 7-5

0

5

10

15

20

25

30

35

40

Fra

ctio

n

Charge state

Theory(Correction) Experiment Theory(Original)

4.5MeV

1 2 3 4 5 6 7

0

10

20

30

40

Fra

ctio

n

Charge state

Theory(modification)Experiment Theory(Original)

7.2MeV

Enhanced ionization cross section

If ionization cross section is enhanced 2 times, calculation agrees with experimental data

• Blue curve: Calculation (σion. ×2)• Red curve: Experiment• Black curve: Calculation (σion. ×1)

Mean Charge state

Theory(σion. ×1)

Experiment

Theory(σion. ×2)

4.5 MeV 2.88 3.50 3.60

7.2 MeV 4.15 4.38 4.54

Enhanced ionization cross section

Ionization cross section is enhanced 2 times

0 2 4 6 8

0

10

20

30

40

50

Fra

ctio

n

Charge state

Theory(Modification) experiment Theory(original)

2.2 MeV

0 2 4 6 8-5

0

5

10

15

20

25

30

35

40

45

50

Fra

ctio

n

Charge state

Theory(Correction)Experiment Theory(original)

3.2MeV

-1 0 1 2 3 4 5 6 7 8 9

0

10

20

30

40

50

Fra

ctio

n

Charge state

Theory(Correction) Experiment Theory(Original)

5.2MeV

0 2 4 6 8

0

10

20

30

40

50

Fra

ctio

n

Charge state

Theory(correction) Experiment Theory(original)

9.4MeV

-1 0 1 2 3 4 5 6 7 8 9

0

10

20

30

40

50

60

Fra

ctio

n

Charge state

Theory(Correction) Experiment Theor(Original)

12MeV

0 2 4 6 8-10

0

10

20

30

40

50

60

70

Fra

ctio

n

Charge state

Theory (Correction) Experiment Theory(Original)

14MeV

Date from TRIUNF Lab.

• Blue curve: Calculation (σion. ×2)• Red curve: Experiment• Black curve: Calculation (σion. ×1)

Discussion Theory Correction PWBA: Polarization effect• Perturbation of projectile orbits.• Result: a reduction in electron binding energy and increase

the ionization probability

And excited state effects may play an important role in ionization of projectile. (Probability: Pexcited > Pdirect)

Ionization cross section:

1s 1s

2s

Ground state 16O5+ Excited state 16O5+

total direct subsequent

Polarization effect

Proj. IonProj. Ion Targ. atom

ground--excited--ionized

Ioni.Ioni.

2s

Conclusion Charge exchange and charge distribution have been calculated

Ionization cross section: PWBA Capture cross section: Impact Parameter Model

Comparison between theory and experiment. Theory reflected experiment data but mean charge state was smaller Try to correct theory (σion. ×2) and almost agreed well with experimental data.

Future: Theory Correction (polarization effect and excited effect) Prediction of non-equilibrium distribution at various energyMeasurement of 4He(12C, 16O) g at low energy :1.15MeV, 1.0MeV,……

THANK YOU FOR ATTENDANCE

r2

r’1

r’’1

R

Z1

Z2

Projectile Target

Hamiltonian:

2 2 2 21 1 1 1 2 2

' '' ' ''1 1 2 1 1

' ' ' 1

2 2 2 2M Z Z Z

H VM r r r r r

1 2 2 2 1' '' ' '

1 1 2 2 1 2 2

1 1Z Z Z Z ZV

R R r R r R r R r r R r r

Interaction potential:

Proj. electron

TargetElectron 1

TargetElectron 2

Projectile as a plane wave: Initial state:exp(-ikR) Final state: exp(-ik’R) q=k’-k (momentum

transfer)

Ionized electron state Initial state: fi final state: ff

The state of atom electron j1s1s(r’1, r’’2) , jn’l’(r’1, r’’2)

According the quantum transition, the cross section

22

2 2

'( )

4if

M vd f q d

v

2 ' ' 1 1 2 1 1 1 1 1 2( ) exp( ) ( ) ( ' '') ( ) ( ' '')f n l i s sf q iRq r r r V r r r dRdr dr

' 'i f

n l

d d

2 1nl nlm

nlm

N

l

M: reduced mass, v initial velocity, v’ the final velocity

Summation of the states of target atom :

Summation of the states of projectile:

'

2

1 4'

'

iR qi Rq irq irqe

e dR e dR eR r R q

2 ' ' 1 1 2 1 1 1 1 1 2( , , ) exp( ) ( ) ( ' '') ( ) ( ' '')f n l i s sf q iRq r r r V r r r dRdr dr 2

(1 ) 1 1 1nl nl

s nl s nl nl s 2 2

1

(1 ) 1 (1 1 )nl

s nl s s

2 sind d 2 2 2' 2 'cosq k k kk 22 sin

'

qdqd d

kk

min

2 2

12 30

8( , , ) ( )

2 1 2 1nl nl

nlmnl nl q

N N dqd d f q Z q

l l v q

Minimum momentum transfer: qmin=Ip+e+ E△ T

Ip is binding energy of the projectile electron, e is energy of the ejected electron,△ET is the excitation energy of the target electron

The closure condition:

22

2( , , ) exp( )f q f iqr nlm

( ) ( , )nlm nl lmR r Y 21 1( )

2

n

aa aba ab

ZH

r r

( )! 2 1( 1) (cos )

( )! 4m m im

lm l

l m lY P e

l m

Scattering amplitude: F factor

For the nlm initial state, the Hamiltonian at multi-electorns system :

Spherical function:

2

2

2 ( 1)2 0

dR dR Z l lE R

dr r dr r r

The final state: to solve radial equation at the coulomb central field

Capture Impact parameter treatment Assume: 1. electron exchange may be neglected. 2. effects of the identity of the nuclei may be

neglected 3. the relative motion of the nuclei takes

place at such velocities v, that the scattering is confined to negligibly small angles.

4. the corresponding trajectory for the incident nucleus may be uniquely distinguished by an impact parameter and a velocity.

X

Z

Y

Velocity : n

Rimpact parameter: r

B(ion) A(target)

n

r2

r

r1

r2=r-R/2 r1=r+R/2

R O (mid-poit)

2

1 1 1 1( , )

( ) ( ) / 2V r t

R r R t r R t

R vt

e

:t :t

For the atom electron:

( ) ( , ) 0el nH i r tt

0 ( , )elH H V r t

20

1

1 1( )2 rH

r

Omit nuclear interaction to get a rectilinear orbits from the equation of motion for the nuclei

2

1( , )V r t

r

2

1 1 1 1( , )

( ) ( ) / 2V r t

R r R t r R t

Defining initial unperturbed eigen function of the electron on A

0 1 1( ) ( )Ai i iH r r

To make an expansion of y(r,t)in the state f(s) of A

1 1 1( , ) ( , ) ( ) ( )Ai

i t

i i i itr t r t r e r

1( , ) ( ) ( ) exp( )n nk k pk

r t a t r i t

z r v

21 1

2 8k

Ap vz v

( ) ( )B Bf k k kH x x

After reaction:

( , )el f fH H V r t

21 1( )2f rH

x

1

1 1fV

R r

1 1 1( , ) ( , ) ( ) ( )Bfi tB B B

i f f ftr t r t r e r

To make an expansion of y(r,t)in the state f(s) of B

( , ) ( ) ( ) exp( )n nk k qk

r t b t x i t 21 1

2 8k

Bq vz v

As it is the cross section describing the capture of an electron from state i of A to state jof B is given by

2

0

2 ( )ij jb d

The initial conditions being ( ) 1, ( ) 0( )i pa a p i

jdbi

dt 2

2

1 1( ) ( , ) exp( ) exp[ ( ) ]

2 8B A Bj r jr V r t ivz d i v t

21

1 1( , ) ( ) exp( ( ) )

2 8A A

i ir t r ivz i v t *

2 1 1

exp( ( ) )

( ) ( ) ( ) exp( )

A Bji i j

B A Aii j i r

g i t

g r V r r ivz d

z r v

*2 1 1( ) exp( ( ) ) ( ) ( ) ( ) exp( )A B B A A

j i j j i rb i i t dt r V r r ivz d

Finally:

* *( ) ( ) exp( ) , ( ) ( ) exp( )i

A A Bi j jp V r ipr dr q r iqr dr

22

1( , ) ( , ) exp( )

(2 ) p

b v f q v iq d qv

*( , ) ( ) ( )A B

i jf q v p q 2 2, 2q p v q p w

So the radial part of the exchange amplitude for the n0l0m0-n1l1m1 transition has the form

0 1

0 0 1 1 0 1

0 0 1 1

2 2 2 2

4( , ) (cos )

( ) ( ( / / 2) ) ( ( / / 2) )

m ml m l m l l

n l n l

b v pdpC C P Pv

J p F k w v v F k w v v

Using the Fourier transform

w :The difference between the binding energies of capture electron in the initial and final state

Function F for the initial and final states are defined by the radial integrals

0 0 0 0 0

1 1 1 1 1

2 2

0

( ) ( ) ( ) ,

( ) ( ) ( ) ( ) , 2

n l n l l

n l n l l

F y P r j ry rdr

F x P r j rx V r rdr x y w

The H-like wave functions of an optical electron in the initial and final states:

1/ 2( ) ( )z Hnl nlP r Z P Zr

PnlH is the radial wave function of a hydrogen atom and Coulomb interaction

potential has the form :

1( ) /V r Z r

min

2 2

12 30

8( , , ) ( )if nlm nlm

nlm nlm q

dqd d f q Z q

v q

2 2 2( , , ) exp( ) ( ) ( )nlm f nlmf q iRq r r dRdr Scattering amplitude

Effective charge of target(He):2

1( )Z q Ejection electron energy:

Projectile velocity: n

Minimum momentum transfer: qmin=（ Ip+e） /n Ip is binding energy of the projectile electron, e is energy of the ejected electron,

2

2

2 ( 1)2 0

dR dR Z l lE R

dr r dr r r

The final state: to solve radial equation in coulomb field

21 1( )

2

n

aa aba ab

ZH

r r nlm nlmH E

For the initial state(nlm) of ionized electron, the Schrodinger equation at multi-electrons system :

,

Capture cross section

Impact parameter treatment Assumption: 1. electron exchange may be neglected. 2. the relative motion of the nuclei takes

place at such velocities v, that the scattering is confined to negligibly small angles.

3. the corresponding trajectory for the incident nucleus may be uniquely distinguished by an impact parameter and a velocity.

Proj. Ion Targ. atom

Ioni.

Aq+ A(q-1)+