Calculations of charge exchange cross sections for some ion-atom and ion-molecule systems

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1 Calculations of charge exchange cross sections for some ion-atom and ion-molecule systems Ismanuel Rabadán Departamento de Química, Universidad Autónoma de Madrid May / 42

2 Coworkers Luis Méndez TCAM group Luis Errea Clara Illescas Henok Getahun Pablo Martínez Francisco Guzmán (ADAS-EU) Bernard Pons (Bordeaux) Emese Rozsályi (Debrecen) 2 / 42

3 Charge Exchange and ionization N 2 : possible scavenger gas for ITER Tabarés et al Nucl. Fusion 45 (2005) L27L31 H + 2 +H : ab initio calculations? Li (and Li + ): coating material Phys. Rev. A (2008); Apicella et al J. Nucl. Mater (2007) H(1s), H(n=2) and B 5+ : diagnostic tools J. Phys. B (2010) 3 / 42

4 Charge Exchange and ionization N 2 : possible scavenger gas for ITER Tabarés et al Nucl. Fusion 45 (2005) L27L31 H + 2 +H : ab initio calculations? Li (and Li + ): coating material Phys. Rev. A (2008); Apicella et al J. Nucl. Mater (2007) H(1s), H(n=2) and B 5+ : diagnostic tools J. Phys. B (2010) 3 / 42

5 Charge Exchange and ionization N 2 : possible scavenger gas for ITER Tabarés et al Nucl. Fusion 45 (2005) L27L31 H + 2 +H : ab initio calculations? Li (and Li + ): coating material Phys. Rev. A (2008); Apicella et al J. Nucl. Mater (2007) H(1s), H(n=2) and B 5+ : diagnostic tools J. Phys. B (2010) 3 / 42

6 Charge Exchange and ionization N 2 : possible scavenger gas for ITER Tabarés et al Nucl. Fusion 45 (2005) L27L31 H + 2 +H : ab initio calculations? Li (and Li + ): coating material Phys. Rev. A (2008); Apicella et al J. Nucl. Mater (2007) H(1s), H(n=2) and B 5+ : diagnostic tools J. Phys. B (2010) 3 / 42

7 In this talk... Ion-atom collisions A q+ + B A q+ + B Electronic excitation A (q 1)+ + B +1 Single electron capture (SEC) A q+ + B + + e Ionization 4 / 42

8 In this talk... Ion-atom collisions A q+ + B A q+ + B Electronic excitation A (q 1)+ + B +1 Single electron capture (SEC) A q+ + B + + e Ionization 4 / 42

9 In this talk... Ion-molecule collisons A q+ + BC Electronic and vibronic excitation A q+ + BC A (q 1)+ + BC +1 A q+ + BC +1 + e Single electron capture (SEC) Ionization A q+ + B + C Dissociation B q+ + AC Reactive charge exchange 5 / 42

10 In this talk... Ion-molecule collisons A q+ + BC Electronic and vibronic excitation A q+ + BC A (q 1)+ + BC +1 A q+ + BC +1 + e Single electron capture (SEC) Ionization A q+ + B + C Dissociation B q+ + AC Reactive charge exchange 5 / 42

11 1 Introduction 2 Theoretical treatment Quantum Mechanical treatment Semiclassical Eikonal treatment Classical Eikonal formalism 3 Progress results H + +Li B 5+ +H(n=1,2) H + +N 2 H + 2 +H 6 / 42

12 Group activity Study of ion-atom and ion-molecule collisions Energy range Low Intermediate High kev/u > 10 Process Elastic & CE CE & ionization Ionization Treatment Full quantal Semiclassical/CTMC CTMC 7 / 42

13 Quantum mechanical treatment Simplified system: 2 nuclei + 1 electron e A CMN r R B H = 1 2µ 2 R 1 2µ e 2 e + V (r, R) 8 / 42

14 Quantum mechanical treatment Boundary conditions: Elastic (or excitation) channel e A R a r a B Ψ Φ A i (r a )e ik i R a + Φ A f (r a)f if (ˆR a ) eik f R a f R a 9 / 42

15 Quantum mechanical treatment Boundary conditions: Elastic (or excitation) channel e A R a r a B Ψ Φ A i (r a )e ik i R a + Φ A f (r a)f if (ˆR a ) eik f R a f R a Boundary conditions: charge exchange channel e A R b r b B Ψ f Φ B f (r b)f if (ˆR b ) eik f R b R b 9 / 42

16 Quantum mechanical treatment Common reaction coordinate: Thorson y Delos (1978) To order O(µ 1 ): if r a r b : if r a r b : ( µ k i ξ µ a ) 1/2 k i R a ( ) µ 1/2 k j ξ k j R b µ a ξ = R + 1 µ s(r, R) with s(r, R) = f (r, R)r 1 2 f 2 (r, R)R 10 / 42

17 Quantum mechanical treatment Molecular expansion HΨ J = EΨ J Ψ J (r, ξ) = k χ J k (ξ)φ k(r, ξ) Clamped-nuclei Born-Oppenheimer Hamiltonian: H elec (r, R)Φ k (r, R) = ɛ k (R)Φ k (r, R) 11 / 42

18 Quantum mechanical treatment Second order differential equations [ (2µ) 1 2 ξ + (E ɛ j) ] χ J j + (2µ) 1 k [ 2Mjk ξ + < Φ j 2 ξ Φ k > ] χ J k = 0 with the non-adiabatic couplings: M jk = φ j ξ φ k + φ j (s q ) (s q ) φ k 12 / 42

19 Cross sections Numerical solution of the differential equations: χ J k (ξ) are obtained. Computation of the S matrix. Cross sections: σ ij = π k 2 i (2J + 1) Sij J 2 J 13 / 42

20 Cross sections Numerical solution of the differential equations: χ J k (ξ) are obtained. Computation of the S matrix. Cross sections: σ ij = π k 2 i (2J + 1) Sij J 2 J 13 / 42

21 Cross sections Numerical solution of the differential equations: χ J k (ξ) are obtained. Computation of the S matrix. Cross sections: σ ij = π k 2 i (2J + 1) Sij J 2 J 13 / 42

22 Semiclassical formalism Straight-line nuclear trajectories R = b + vt v X R b Z 14 / 42

23 Semiclassical formalism Straight-line nuclear trajectories: R = b + vt Eikonal equation: [ ] H elec i Ψ(r, t) = 0 t r Molecular expansion: Ψ(r, t) = exp [iu(r, t)] j ( t ) a j (t)φ j (r, R) exp i ɛ j dt 0 15 / 42

24 Semiclassical formalism Straight-line nuclear trajectories: R = b + vt Eikonal equation: [ ] H elec i Ψ(r, t) = 0 t r Molecular expansion: Ψ(r, t) = exp [iu(r, t)] j ( t ) a j (t)φ j (r, R) exp i ɛ j dt 0 15 / 42

25 Semiclassical formalism Straight-line nuclear trajectories: R = b + vt Eikonal equation: [ ] H elec i Ψ(r, t) = 0 t r Molecular expansion: Ψ(r, t) = exp [iu(r, t)] j ( t ) a j (t)φ j (r, R) exp i ɛ j dt 0 15 / 42

26 Cross sections σ ij (v) = 2π 0 b P ij (b, v)db P ij (b, v) = lim t φ j (r)d j (r, t) Ψ 2 = lim t a j (t; b, v) 2 Sudden approximation: σ sud = 0 dρ σ(ρ) χ(ρ) 2 16 / 42

27 Eikonal CTMC The projectile follows straight-line trajectories: R = b + vt Electronic motion: ensemble of N ( 10 5 ) trajectories {r j } with a Classical distribution function ρ(r, p, t) = 1 N N δ(r r j )δ(p p j ) j=1 that satisfies the Liouville s equation: ρ t = [ρ, H] 17 / 42

28 Hamilton s equations ṙ j = H p j ; ṗ j = H r j Electron trajectories are obtained by solving the Hamilton s equations with the classical Hamiltonian: H(r, p) = p2 2 V t Z p r p 18 / 42

29 Multi-centre pseudo-potential for N 2 V 2c (r N1, r N2 ) = V N (r N1 ) + V N (r N2 ) V N = 7 N N r N N N r N (1 + αr N ) exp ( 2αr N ) OM3 α = OM4 α = OM5,6 α = OM7 α = N N = 6.5 ɛ k (model potential) ɛ k (SCF) < a.u. 19 / 42

30 Multi-centre pseudo-potential for N 2 V 2c (r N1, r N2 ) = V N (r N1 ) + V N (r N2 ) V N = 7 N N r N N N r N (1 + αr N ) exp ( 2αr N ) OM3 α = OM4 α = OM5,6 α = OM7 α = N N = 6.5 ɛ k (model potential) ɛ k (SCF) < a.u. 19 / 42

31 Microcanonical initial distribution Random values of φ and cos θ r = rmax 2 ( β ), β=random and rmax fulfills V (θ, φ, r max ) = E j Electron in the perihelion: [ ] θ p = tan 1 1 tan θ cos (φ φ p ) ; φ p random The Hamilton s equations are integrated until t = t 0 + 2π random 20 / 42

32 Microcanonical initial distribution Random values of φ and cos θ r = rmax 2 ( β ), β=random and rmax fulfills V (θ, φ, r max ) = E j Electron in the perihelion: [ ] θ p = tan 1 1 tan θ cos (φ φ p ) ; φ p random The Hamilton s equations are integrated until t = t 0 + 2π random 20 / 42

33 Microcanonical initial distribution Random values of φ and cos θ r = rmax 2 ( β ), β=random and rmax fulfills V (θ, φ, r max ) = E j Electron in the perihelion: [ ] θ p = tan 1 1 tan θ cos (φ φ p ) ; φ p random The Hamilton s equations are integrated until t = t 0 + 2π random 20 / 42

34 Microcanonical initial distribution Random values of φ and cos θ r = rmax 2 ( β ), β=random and rmax fulfills V (θ, φ, r max ) = E j Electron in the perihelion: [ ] θ p = tan 1 1 tan θ cos (φ φ p ) ; φ p random The Hamilton s equations are integrated until t = t 0 + 2π random 20 / 42

35 Many-electron interpretation One-electron transition probabilities Capture: p cap k = N cap /N Ionization: p ion k = N ion /N Elastic: pk el = 1 pcap k pk ion Independent event model (IEVM) Crothers and McCarrol (JPB 1987); Janev et al (JPB 1995) P SEC = 2 5 k=1 p cap k ; P SI = 2 5 k=1 p ion k 21 / 42

36 Anisotropy treatment σ X (v) = 1 4π db dωp X (b, v, Ω) σx 1 + σx 2 + σx / 42

37 H + +Li H+Li + (Errea et al 2008) Orbital energies (hartree) R (a 0 ) R (a 0 ) 23 / 42

38 H + +Li H+Li σ (10-16 cm 2 ) σ (10-16 cm 2 ) E (ev/amu) E (ev/amu) 24 / 42

39 B 5+ +H(n=2) B 4+ +H + (Guzmán et al 2010) 13 n eff B 4+ (n=10) B 4+ (n=9) B 5+ +H(n=2) B 4+ (n=8) B 4+ (n=7) B 4+ (n=6) B 4+ (n=5) Be 5+ +H(1s) B 4+ (n=4) B 4+ (n=3) B 4+ (n=2) R (a 0 ) 25 / 42

40 B 5+ +H(n=2) B 4+ +H σ (10-16 cm 2 ) 10 1 B 5+ + H(n=2) CTMC B 5+ + H(n=2) OEDM ionization (n=2) CTMC E (kev/amu) 26 / 42

41 B 5+ +H(n=2) B 4+ +H + 1e-13 1e-14 n=6 n=8 n=7 ionization 1e-15 n=5 total CX σ (cm 2 ) 1e-16 1e-17 n=4 n=3 1e-18 n=2 1e E (kev/amu) 27 / 42

42 1 B 5+ +H(1s) potential 0 Energy (a.u./10-3 ) B 5+ +H(1s) C 4+ (1s 2 )+H(1s) R (a.u.) 28 / 42

43 Elastic and EC collisions B 5+ +H(1s) (Barragán et al 2010) E 1/2 σ (Å 2 ev 1/2 ) 36 (25) (1,27) 34 (26) (1,28) (29) (31) (30) (32) (0,34) (0,35) (0,36) (0,37) (38) (39) Cross section (10-13 cm 2 ) (25) (1,27) (1,28) (29) (30) (31) (32) 1-state calculation 12-state calculation σ SLL + σ g σ SLL (0,34) (0,35) Energy (10-3 ev) LZL model Energy (10-3 ev) (0,36) (37) (38)(39) 29 / 42

44 H + +N 2 H+N V (E h ) Excitation Capture Entrance H + + N 2 (w 1 u ) H + + N 2 (a 1 Π g ) H(1s) + N 2 (B Σu ) + 2 H(1s) + N 2 (A Πu ) H(1s) + N 2 (X Σg ) H + + N 2 (X 1 + Σ g ) Expt: Gilmore R (a 0 ) 30 / 42

45 H + +N 2 H+N V (E h ) Excitation Capture Entrance H + + N 2 (w 1 u ) H + + N 2 (a 1 Π g ) H(1s) + N 2 (B Σu ) + 2 H(1s) + N 2 (A Πu ) H(1s) + N 2 (X Σg ) H + + N 2 (X 1 + Σ g ) Expt: Gilmore R (a 0 ) 0.2 Radial Coupling (a.u.) Rotational R (a 0 ) 31 / 42

46 H + +N 2 H+N Cross section (10-16 cm 2 ) Rudd (1985): Exp. capture Rudd (1985): Exp. ionization Gao (1990): Experiment Cabrera-Trujillo (2002): Theory E (kev) 32 / 42

47 H + +N 2 H+N Cross section (10-16 cm 2 ) Rudd (1985): Exp. capture Rudd (1985): Exp. ionization Gao (1990): Experiment Cabrera-Trujillo (2002): Theory CTMC ion CTMC cap E (kev) 32 / 42

48 H + +N 2 H+N Cross section (10-16 cm 2 ) Rudd (1985): Exp. capture Rudd (1985): Exp. ionization Gao (1990): Experiment Cabrera-Trujillo (2002): Theory CTMC ion CTMC cap 2, FC E (kev) 32 / 42

49 H + +N 2 H+N Cross section (10-16 cm 2 ) Rudd (1985): Exp. capture Rudd (1985): Exp. ionization Gao (1990): Experiment Cabrera-Trujillo (2002): Theory CTMC ion CTMC cap 2, FC 4, FC E (kev) 32 / 42

50 H + +N 2 H+N Cross section (10-16 cm 2 ) Rudd (1985): Exp. capture Rudd (1985): Exp. ionization Gao (1990): Experiment Cabrera-Trujillo (2002): Theory CTMC ion CTMC cap 2, FC 4, FC 6, FC E (kev) 32 / 42

51 H + +N 2 H+N Cross section (10-16 cm 2 ) Rudd (1985): Exp. capture Rudd (1985): Exp. ionization Gao (1990): Experiment Cabrera-Trujillo (2002): Theory CTMC ion CTMC cap 2, FC 4, FC 6, FC 2, Sudden E (kev) 32 / 42

52 H + 2 +H H 2 +H V (E h ) H 2 (a 3 Σ g + ) + H(1s) H 2 (X 1 Σ g + ) + H(n=2) H 2 (X 1 Σ g + ) + H(1s) R (a 0 ) 33 / 42

53 H + 2 +H H 2 +H V (E h ) H 2 (a 3 Σ g + ) + H(1s) H 2 (X 1 Σ g + ) + H(n=2) -1.4 H 3 + (gs) -1.6 H 2 (X 1 Σ g + ) + H(1s) R (a 0 ) 33 / 42

54 H + 2 +H H 2 +H V (E h ) H 2 (a 3 Σ g + ) + H(1s) H 2 (X 1 Σ g + ) + H(n=2) -1.4 H 2 + (gs)+h (gs) H 3 + (gs) -1.6 H 2 (X 1 Σ g + ) + H(1s) R (a 0 ) 33 / 42

55 H + 2 +H H 2 +H V (E h ) H 2 (a 3 Σ g + ) + H(1s) H 2 (X 1 Σ g + ) + H(n=2) -1.4 H 2 + (gs)+h (gs) H 3 + (gs) -1.6 H 2 (X 1 Σ g + ) + H(1s) R (a 0 ) 33 / 42

56 H + 2 +H H 2 +H V (E h ) -1.2 H 2 + (gs)+h (gs) H 3 + (gs) H 2 (a 3 Σ g + ) + H(1s) -1.3 H 2 (X 1 Σ g + ) + H(n=2) R (a 0 ) 33 / 42

57 H + 2 +H H 2 +H V (E h ) -1.2 H 2 + (gs)+h (gs) H 3 + (gs) H 2 (a 3 Σ g + ) + H(1s) -1.3 H 2 (X 1 Σ g + ) + H(n=2) R (a 0 ) 33 / 42

58 H + 2 +H H 2 +H Radial coupling (a.u.) 4 2 V (E h ) H 2 + (gs)+h (gs) H 3 + (gs) 0 H 2 (a 3 Σ g + ) + H(1s) H 2 (X 1 Σ g + ) + H(n=2) R (a 0 ) R (a 0 ) 34 / 42

59 H + 2 +H H 2 +H Szucs et al (1983) Peart & Bennet (1986) Peart et al (1997) Liu et al (2006) 100 σ (10-16 cm 2 ) v (cm/s) 35 / 42

60 H + 2 +H H 2 +H Szucs et al (1983) Peart & Bennet (1986) Peart et al (1997) Liu et al (2006) σ (10-16 cm 2 ) V (E h ) H 2 + (gs)+h (gs) H 3 + (gs) H 2 (a 3 Σ g + ) + H(1s) H 2 (X 1 Σ g + ) + H(n=2) v (cm/s) R (a 0 ) 36 / 42

61 H + 2 +H H 2 +H Szucs et al (1983) Peart & Bennet (1986) Peart et al (1997) Liu et al (2006) σ (10-16 cm 2 ) V (E h ) H 2 + (gs)+h (gs) H 3 + (gs) H 2 (a 3 Σ g + ) + H(1s) H 2 (X 1 Σ g + ) + H(n=2) v (cm/s) R (a 0 ) 37 / 42

62 H + 2 +H H 2 +H Szucs et al (1983) Peart & Bennet (1986) Peart et al (1997) Liu et al (2006) , σ (10-16 cm 2 ) V (E h ) H 2 + (gs)+h (gs) H 3 + (gs) H 2 (a 3 Σ g + ) + H(1s) H 2 (X 1 Σ g + ) + H(n=2) v (cm/s) R (a 0 ) 38 / 42

63 H + 2 +H H 2 +H Szucs et al (1983) Peart & Bennet (1986) Peart et al (1997) Liu et al (2006) , σ (10-16 cm 2 ) V (E h ) H 2 + (gs)+h (gs) H 3 + (gs) H 2 (a 3 Σ g + ) + H(1s) H 2 (X 1 Σ g + ) + H(n=2) v (cm/s) R (a 0 ) 39 / 42

64 Summary Large-scale quantal, semiclassical and classical calculations. Elastic, electron capture and ionization cross sections. Ion-atom collisions: B 5+ +H(n=1,2) and H + +Li Ion-molecule collisions: H + +N 2 and H +H / 42

65 References Barragán et al (2010): P. Barragán, L. F. Errea, F. Guzmán, L. Méndez, I. Rabadán, I. Ben-Itzhak. Phys. Rev. A (2010) Guzmán et al (2010): F. Guzmán, L. F. Errea, C. Illescas, L. Méndez and B. Pons. J. Phys. B (2010) Errea et al (2008): L. F. Errea, F. Guzmán, L. Méndez, B. Pons and A. Riera. Phys. Rev. A (2008) Rudd et al (1985): M. E. Rudd, T. V. Goffe and A. Itoh. Phys.Rev. A (1985) Cabrera-Trujillo et al (2002): R. Cabrera-Trujillo, Y. Öhrn, E. Deumens, J. R. Sabin, and B. G. Lindsay. Phys. Rev. A (2002) Gao et al (1990): R. S. Gao, L. K. Johnson, C. L. Hakes, K. A. Smith and R. R. Stebbings. Phys. Rev. A (1990) Gilmore (1965): F. R. Gilmore. J. Quant. Spectry. Radiative Transfer, 5, 369 (1965) 41 / 42

66 References Szücs (1983): S. Szücs, M. Karemera and M. Terao, Abstracs of XIII-ICPEAC, Berlin p-482 Liu (2006): C. L. Liu, J. G. Wang and R. K Janev. J. Phys. B (2006) Peart (1997): B. Peart, R. Padgett and D. A. Hayton. J. Phys. B (1997) Peart (1986): B. Peart and M. A. Bennett. J. Phys. B. 19 L321 (1986) 42 / 42

Theoretical treatment of Ion-Atom and Ion-Molecule Collisions

Theoretical treatment of Ion-Atom and Ion-Molecule Collisions Ismanuel Rabadán Departamento de Química, Universidad Autónoma de Madrid (Laboratorio Asociado al CIEMAT de Física Atómica y Molecular en Plasmas