CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH Inga Yu. Tolstikhina
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1 CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH Inga Yu. Tolstikhina P.N.Lebedev Physical Institute, Russian Academy of Sciences Moscow, Russia
2 Theoretical approaches and methods of calculations Charge exchange in collisions of hydrogen and helium with low-z impurities (Data for the Pellet Charge exchange (PCX) active diagnostics) Influence of the isotope effect on the charge exchange in slow collisions of Li, Be, C and W ions with H, D and T Effect of the electron-nuclei interaction on the internuclear motion in slow ion-atom collisions
3 General formulation of the problem The time-dependent Schrödinger equation H (R ) ψ( r, t )=i ψ( r, t ) t r set of electronic coordinates H(R) electronic Hamiltonian of diatomic quasi-molecule R=R(vt) the inter-nuclear separation v initial relative nuclear velocity Adiabatic approximation: an asymptotic expansion of the solution in the small parameter v t ψ( r,t )= g p ( t ) φ p ( r, R )exp( i E p ( R ( vt ' )) dt ' ) p H (R) φ p ( r, R )=E p (R) φ p ( r, R ) Calculation of the principal terms of the asymptotic of the expansion coefficients g p (t), v 0 Boundary conditions: (a) p R, E p ( R) E, φ p (r, R) φ molecular potential curves adiabatic terms Ep P pq =lim g p (t ), lim g p ( t )=δ pq t t (a) p σ pq = π P pq (ρ)ρ d ρ 0
4 Adiabatic approximation: radial and rotational transitions Separated atoms United atom n=4 Li (n=3) 4d Rad 3d 3s 3p Rot n=3-1 3p 4f 3d H(1s) Rad Rot E, a.u. 3d Li (n=) Rad s Rot p 3 Li H(1s) p 4 6 R, a.u. 8 Li3 H(1s) Li(nl) H Adiabatic theory of transitions in slow collisions, E. Solov ev Sov. Phys. Uspekhi,
5 Code ARSENY Code ARSENY is based on the method of hidden crossings Input: Z Z; nl; E; basis size; reduced mass 1. Calculates adiabatic potential curves of two Coulomb center problem in complex R-plane Separatedted atoms United atom 5 Neff (R) Li (n=4) Li (3s,3p,3d) 4 Li3 H(1s) H (1s) 3 Li (s,p) Li (1s) R
6 Code ARSENY. Searches all branch points and calculates the corresponding Stueckelberg parameter R Δ pq= Im c [ E p ( R ) E q ( R )] V (dr R,b ) Re R c R the inter-nuclear separation p,q the set of quantum numbers of the final and initial atomic states Rc a complex branch point Ep, Eq energies of the final and initial atomic states V(R,b) the radial internuclear velocity b the impact parameter
7 Code ARSENY 3. Calculates the probability as a function of L (nuclear angular momentum) for the entire set of nonadiabatic transitions is calculated as P pq = e Δ pq 4. The S-matrix is calculated as a product of elementary S-matrices for the individual transitions induced by the separated branch points
8 Code ARSENY 5. The cross sections are calculated as a sum over L: elastic scattering inelastic scattering π σ qq = Kq π σ pq = Kq ( L1 ) 1 S(qqL) L=0 ( L1 ) S(pqL) L=0 K q = M ( ε Eq ( )) M the reduced mass of nuclei Eq( ) the energy levels of separated atoms ε the energy of the system in the center of masses
9 Charge exchange in collisions of Hydrogen and Helium with low-z impurities Pellet Charge exchange (PCX) diagnostics Large Helical Device (LHD)
10 Pellet Charge exchange (PCX) diagnostics In the local active diagnostic method, referred to as Pellet Charge exchange (PCX), an ablating solid impurity pellet is used as a dense target for electron capture by fast ions of a fusion plasma ncl (x) cloud density function
11 Data for the Pellet Charge exchange (PCX) diagnostics The Polystyrene (C8H8 )n is used as an ablating solid impurity pellet in the PCX experiment on LHD. Local emission of atoms : g ( E, r ) = F0 ( E ) ni ( r ) vi f i ( E, r ) F0 (E ) neutral fraction fi ( E, r ) proton distribution function The relevant elementary charge exchange processes are H Ck H H0 H0 Ck
12 Theoretical approaches and methods of calculations Collision energies range: 1 kev/a.m.u. 1 MeV/a.m.u. VF = the velocity of the active electron / the collisional velocity Adiabatic region Advanced adiabatic approach (E.A. Solov ev) (VF is larger than unity) Code ARSENY Born region Normalized Brinkman-Kramers (BK) approximation in the impact parameter representation (VF is less than unity) Code CAPTURE (I.Yu.Tolstikhina, V.P.Shevel ko) Coulomb Distorted Wave (CDW) approximation (I.M.Cheshire) Codes CDW & CDW (Dz. Belkic)
13 H0, H, He0, He, He reaction Li, Be, B, C, Ne k H H0 He He He0 He Lik Lik Lik Lik Lik Lik H0 H He0 He He He Li(k1) Li(k-1) Li(k1) Li(k1) Li(k-1) Li(k-1) 0, 0, 0, H H0 He He He0 He Bek Bek Bek Bek Bek Bek H0 H He0 He He He Be(k1) Be(k-1) Be(k1) Be(k1) Be(k-1) Be(k-1) 0, 0, 0, H H0 He He He0 He Bk Bk Bk Bk Bk Bk H0 H He0 He He He B(k1) B(k-1) B(k1) B(k1) B(k-1) B(k-1) 0, 0, 0, 4, 4, 4, H H0 He He He0 He Ck Ck Ck Ck Ck Ck H0 H He0 He He He C(k1) C(k-1) C(k1) C(k1) C(k-1) C(k-1) 0, 0, 0, 4, 4, 4, 4, 5, 4, 4, 5, 5, H H0 He He He0 He Nek Nek Nek Nek Nek Nek H0 H He0 He He He Ne(k1) Ne(k-1) Ne(k1) Ne(k1) Ne(k-1) Ne(k-1) 0, 0, 0, 4, 4, 4, 4, 5, 4, 4, 5, 5, 5, 6, 5, 5, 6, 6, 6, 7, 6, 6, 7, 7, 7 8, 7, 7, 8, 8, 9, 8, 8, 9, 9, 9 9
14 H0 Li3 H Li , cm Experiment M.B.Shah et al., J.Phys.B, 1 L33 (1978) W.Seim et al., J.Phys.B, 14, 3475 (1981) Theory N.Toshima, Phys.Rev.A, 50, 3940 (1994) ARSENY CAPTURE CDW Approx. Formula E/m, kev/a.m.u. 3
15 Local PCX Neutral Spectra and Calculated Proton Energy Distributions H Count Rate, a.u. 1 = 0.89 = 0.86 = 0.83 H C ount R ate, a.u. = 0.88 = 0.85 = E, kev ECH 3 ne(0) = 0.4x cm, Te(0) = 4.8 kev = 0.89 = 0.86 = 0.83 ECH, NBI 1 Target plasma: Hydrogen = 0.88 = 0.85 = ne(0) = 0.5x 13-3 cm, Te(0) = 3. kev fi(e), a.u. fi(e), a.u. 10 E, kev Target plasma: Hydrogen 13 0 Einj NBI E, kev E, kev
16 Electric Charge State Changing Collisions of Hydrogen and Helium with Low-Z Impurity Particles Part I. Charge Exchange Processes I.Yu. Tolstikhina P.R. Goncharov T. Ozaki S. Sudo N. Tamura and V.Yu. Sergeev3 P.N. Lebedev Physical Institute, Moscow, Russia National Institute for Fusion Science, Toki, Gifu, Japan 3 St.Petersburg Polytechnical University, St.Petersburg, Russia 1 NIFS-DATA- April 008
17 Influence of the isotope effect on the charge exchange in slow collisions of Li, Be, C and W ions with H, D and T N. Stolterfoht, R. Cabrera-Trujillo, Y. Ohrn, E. Deumens, R. Hoekstra, and J. R. Sabin PHYSICAL REVIEW LETTERS 99, 301 (007) 0 ev/amu He H, D, T He H, D, T
18 Influence of the isotope effect on the low-temperature plasma characteristics Isotope effect: ev/amu near-wall plasmas diverter plasmas plasma facing components: Li, Be, C, W neutralization population of excited levels ions charge distribution radiative cooling particle transport Li q H, D, T Li (q-1) H, D, T Be q H, D, T Be (q-1) H, D, T C q H, D, T C (q-1) H, D, T W q H, D, T W (q-1) H, D, T Inga Yu. Tolstikhina, Daiji Kato, and V. P. Shevelko, Phys. Rev. A 84 (011) Inga Yu. Tolstikhina et al., J. Phys. B: At. Mol. Opt. Phys., 45, (01) 14501
19 Theoretical method: rotational transitions in close collisions (Re[R] = 0) for < Rmax and Rmax> Rclmb : l i a m Em am i φnlm t φnlm' am' =0 m'= l Z scattering angle Z1 Z χ= arctan μ ρ v ρ cos ( ) R= cos ϕ sin ( χ ρ χ cos Rmax ) χ Rclmb internuclear axis rotation angle ϕ=arccos sin χ Z1 Rmax 1 ( l ) R max= Z 1 Z
20 Li H, D, T Li H, D, T T D H H** H*, cm -15 R * w/o P **Seim W. et al., J. Phys. B, 14, 3475 (1981) E, kev/amu
21 -14 3, cm Be H, D, T Be H, D, T T D H H* *w/o P E, kev/amu R
22 , cm C H,D,T C H, D, T T D H H* * w/o P E, kev/amu 1 R
23 , cm -15 W H, D, T W H, D, T T D H H* H** R -0-1 * w/o P ** experiment M. Imai, Kyoto University 0.1 E, kev/amu
24
25
26 W3 He(1s) W He(1s) E, a.u. initial state He(1s) -1 n=11 n= n=9 - n=8 n=7-3 n=6-4 final state W R, a.u.
27 W3 He(1s) W He(1s) (cm ) 8x -15 6x -15 4x -15 x -15 K.Soejima, experiment total cross section, theory n= E (ev/amu)
28 Effect of the electron-nuclei interaction on the internuclear motion in slow ion-atom collisions R. Cabrera-Trujillo, J. R. Sabin, Y.Ohrn, E. Deumens, and N. Stolterfoht PHYSICAL REVIEW A (011) Coulomb potential and screened Coulomb potential are purely repulsive and can not reproduce negative scattering angles found in the END calculations He H, D, T He H, D, T
29 Internuclear potential and trajectory The internuclear motion should be described by the Born-Oppenheimer (BO) potential corresponding to the entrance collision channel The interaction between two bare nuclei is described by the Coulomb potential Z1Z V C ( R)=, R where R is the internuclear distance. In the BO approximation the internuclear interaction is described by V BO ( R)=V C ( R) E ( R) E 0, where E(R) is the electronic energy in the field of the nuclei fixed in space at a distance R and E (R ) = E0. Inga Yu. Tolstikhina and O. I. Tolstikhin, Phys. Rev. A 9 (015) 04707
30 Internuclear potential and trajectory He H, D, T He H, D, T V BO ( R)=V C ( R) E ( R) E 0,
31 Internuclear potential and trajectory The classical scattering angle for a given internuclear potential V (R) as a function of the impact parameter ρ and collision energy E is given by θ(r)=π R min ρ d R, R F (R ) ρ V ( R) F ( R)= 1, E R where E=μ v / μ=m 1 M /( M 1 M ) R min energy of the internuclear motion in the center-of-mass frame, reduced mass of the nuclei, v is their initial relative velocity, distance of closest approach defined by the equation F(Rmin) = 0. For the Coulomb potential we have ( ) Z1 Z θc (ρ, E)=arctan ρ E
32 Internuclear potential and trajectory He H, D, T He H, D, T Collision energy 50 ev/amu
33 Internuclear potential and trajectory He H, D, T He H, D, T
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