Charge State Distribution of Recoil 16 O from 4 He( 12 C, 16 O) 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. Rosary B 、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

introduction Total S-factor of 4 He+ 12 C  16 O+  Ruhr U. Kyushu U. extrapolation Astrophysics. 4 He( 12 C, 16 O)g E cm =2.4 MeV,1.5MeV Succeeded !!  For the recoils 16 O q+, to determine 4 He( 12 C, 16 O)g total cross section, fraction of 16 O q+ should be precisely known. Investigation: charge exchange processes and charge state distribution of Oxygen though Helium gas. 16 O 1+,2+,….8+ The fraction of 16 O q+ ?? 16 O q+ detector

Features of ion-atom collision physical processes: Capture, Ionization, Excitation,….. q-1 q-2 q+1 Charge distribution Charge exchange e-e- e-e- (Auger)

Ionization of Projectile, A q+ 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)  Scattering amplitude  Effective charge of target(He):Ejection electron energy:  Projectile velocity: n  Minimum momentum transfer: q min = （ I p + e ） / n I p is binding energy of the projectile electron, e is energy of the ejected electron,  The final state: to solve radial equation in coulomb field  For the initial state(nlm) of ionized electron, the Schrodinger equation at multi- electrons system :,

Capture of Projectile, A q+ A (q-1)+ Impact parameter treatment 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: : electron initial state : electron final state : corresponding eigen-energies (initial) (final)

16 O 6+16 O O O 4+ Arseny: adiabatic approximation, low energy CDW: Continuum distorted wave approximation, high energy F.M.Martine et al. PRA,1965(1971) Possibility>1 Ionization cross section Capture cross section

Equilibrium Charge State Distributions Variation of the charge state distribution: Equilibrium distribution: All fractions reach a certain value and keep constant. Non- equili. equili. Cross sections of ionization and capture are required, GainLoss

Results 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. 3+ ， 100% He

Charge state distribution 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. TheoryExperiment 4.5 MeV MeV Mean charge state:

Data from TRIUMF Lab. (energy: 2.2MeV MeV) Shift to left.!!

Enhanced ionization cross section If ionization cross section is enhanced 2 times, calculation agrees with experimental data Blue curve: Calculation ( σ ion. ×2 ) Red curve: Experiment Black curve: Calculation ( σ ion. ×1 ) Mean Charge state Theory ( σ ion. ×1 ) ExperimentTheory ( σ ion. ×2 ) 4.5 MeV MeV

Enhanced ionization cross section Ionization cross section is enhanced 2 times Date from TRIUNF Lab. Blue curve: Calculation ( σ ion. ×2 ) Red curve: Experiment Black curve: Calculation ( σ ion. ×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: P excited > P direct ) Ionization cross section: 1s 2s Ground state 16 O 5+ Excited state 16 O 5+ Polarization effect Proj. Ion Targ. atom ground--excited --ionized 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 ( σ ion. ×2 ) and almost agreed well with experimental data.  Future:  Theory Correction (polarization effect and excited effect)  Prediction of non-equilibrium distribution at various energy  Measurement of 4 He( 12 C, 16 O) g at low energy :1.15MeV, 1.0MeV,……

r2r2 r’ 1 r’’ 1 R Z1Z1 Z2Z2 Projectile Target Hamiltonian: Interaction potential: Proj. electron Target Electron 1 Target Electron 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:  i final state: f f  The state of atom electron j 1s1s ( r’ 1, r’’ 2 ), j n’l’ ( r’ 1, r’’ 2 )

 According the quantum transition, the cross section M: reduced mass, v initial velocity, v’ the final velocity Summation of the states of target atom : Summation of the states of projectile:

Minimum momentum transfer: q min =I p +e+ △ E T I p is binding energy of the projectile electron, e is energy of the ejected electron, △ E T is the excitation energy of the target electron The closure condition:

Scattering amplitude: F factor For the nlm initial state, the Hamiltonian at multi-electorns system : Spherical function: 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 : R impact parameter: r B(ion) A (target) r2r2 r r1r1 r 2 =r-R/2r 1 =r+R/2 R O (mid-poit) e

 For the atom electron: Omit nuclear interaction to get a rectilinear orbits from the equation of motion for the nuclei

Defining initial unperturbed eigen function of the electron on A To make an expansion of y(r,t)in the state f(s) of A

After reaction: To make an expansion of  (r,t)in the state  f (s) of B

As it is the cross section describing the capture of an electron from state i of A to state jof B is given by The initial conditions being Finally:

So the radial part of the exchange amplitude for the n 0 l 0 m 0 -n 1 l 1 m 1 transition has the form 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 The H-like wave functions of an optical electron in the initial and final states: P nl H is the radial wave function of a hydrogen atom and Coulomb interaction potential has the form :

 Scattering amplitude  Effective charge of target(He):Ejection electron energy:  Projectile velocity: n  Minimum momentum transfer: q min = （ I p + e ） / n I p is binding energy of the projectile electron, e is energy of the ejected electron,  The final state: to solve radial equation in coulomb field  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. A q+ A (q-1)+