HEAVY PARTICLE COLLISION PROCESSES. Alain Dubois

Transcription

1 HEAVY PARTICLE COLLISION PROCESSES Alain Dubois Laboratoire de Chimie Physique - Matière et Rayonnement Université Pierre et Marie Curie - CNRS Paris FRANCE

2 Heavy particle collision processes I - Introduction II - Three models for electron transfer a) The Thomas mechanism b) The Bohr-Lindhardt model c) Resonant capture at low energy in homonuclear one electron collision systems III - The quasi-molecular formalism IV - The travelling asymptotic state basis set expansion : the intermediate energy domain V - Capture in ion-molecule collisions

3 I - Introduction Electronic processes : A q+ + B* excitation A q+ + B A (q-1)+ + B + capture (transfert) A q+ + B + + e - ionization ex: H + + H(2s) H + + H(2p) H + + H(1s) H(1s) + H + H(2p) + H + H + + H + + e - (ε k )

4 He 2+ + He(1s 2 ) He + + He + I - Introduction Li 3+ + He Li 2+ + He + Li + + He 2+ H + + H - (1s 2 ) H(nl) + H(n l ) H - (1s 2 ) + H + H + + H + e - (ε k ) Kr Ar Kr Ar n+ + n e -

5 I - Introduction atomic units m e e h 4π ε = 1 Velocity of ligth in vacuum c = 1/α 137 u.a.

6 Typical structure and collision parameters for H + H (H 2 ) system I - Introduction energy threshold velocity * caracteristic time collision time ** de Broglie wavelength E V thres = 2eE ev µ E t > h T coll = a / v λ db = h / µv electronic transitions 10 ev ( 1 a.u.) v > m.s -1 ( a.u.) τ s ( 1 a.u.) T coll < s ( 10 a.u.) λ db l < m ( 0.2 a.u.) vibrationnal transitions 10-1 ev ( 10-2 a.u.) v > m.s -1 ( a.u.) τ s ( 10 2 a.u.) T coll < s ( 100 a.u.) λ db l < m ( 2 a.u.) rotationnal transitions 10-3 ev ( 10-4 a.u.) v > m.s -1 ( a.u.) τ s ( 10 4 a. u.) T coll < s ( 10 3 u.a.) λ db l < 10-9 m ( 20 a.u.) * µ=m H /2 = kg 10 3 a.u. ** collision zone a 5 Å ( 10 a.u.)

7 I - Introduction Cross sections for the inelastic scattering of protons by atomic hydrogen in the ground state. Cross sections for the inelastic scattering of protons by He + ions in the ground state. C= capture, E=excitation, I=ionization E kev/amu = 25 v 2 au from Charge Exchange and the Theory of Ion-Atom Collisions, Bransden and McDowelll

8 I - Introduction Cross sections for the inelastic scattering of protons by helium in the ground state. Cross sections for the inelastic scattering of protons by He + ions in the ground state. C= capture, E=excitation, I=ionization E kev/amu = 25 v 2 au from Charge Exchange and the Theory of Ion-Atom Collisions, Bransden and McDowell

9 Heavy particle collision processes I - Introduction II - Three models for electron transfer a) The Thomas mechanism b) The Bohr-Lindhardt model c) Resonant capture at low energy in homonuclear one electron collision systems III - The quasi-molecular formalism IV - The travelling asymptotic state basis set expansion : the intermediate energy domain V - Capture in ion-molecule collisions

10 IIa - The Thomas mechanism A v 0 z v 0 M a, +Z a e r e - m e, -e B B A+e - M b, +Z b e single scattering mechanism => σ C 1 / v 0 12 (matching of a part of the electron momentum distribution with v 0 ) double scattering mechanism classical model proposed by Thomas in 1927 Proc. Roy. Soc. London A 114, 561 (1927) * target nucleus and electron assumed at rest (high velocity approx => v 0 >> 1 a.u.) * 2 binary collisions between charged particles (Rutherford scattering) - 1st stage: between A and e - to give v 0 to the electron - 2nd stage: between e - and B to bring e - in the direction of the moving projectile A

11 IIa - The Thomas mechanism A v 0 A v 0 z e - v 0 θ = 60 B v0 e - Rem: sinθ tgθ L = cosθ + τ with = m e /M a or M a / m e * 1st stage the electron gets v 0 and is scattered at 60 (insures energy and momentum conservation) The probability for A to pass an electron at a distance b when travelling the distance l is dp 1 = 2π b db l Z a (N density of atom B, b impact parameter) with b = cotg θ/2 (Rutheford) so that dp 1 µ v 2 v 5 v=v 0 dv

12 IIa - The Thomas mechanism A v 0 A v 0 z e - v 0 B θ = 60 v0 e - θ = 60 Rem: sinθ tgθ L = cosθ + τ with = m e /M a or M a / m e * 2nd stage the electron scattered by the target nucleus B and escape with the same velocity in the same direction than the projectile dp 2 b db Z b with b = cotg θ /2 (Rutheford) so that dp 2 µ v 2 v 4 dθ

13 IIa - The Thomas mechanism So that finally dv dθ dp= dp 1 dp 2 v 9 with v v 0 so that u the relative velocity of e - with respect to A and µ u 2 < 2 Z a /r so that dv is such v 2 dv dω v = 4/3 π (2 Z a / µ r) 3/2 1 dp v 11 valid in the relativistic regime (!) but the Thomas peak (angle) an be observed sinθ m tgθ al = = e sinθ 3 1/2 m e 0.5 mrad for A= H + cosθ + M a / m e M a 2 M a

14 IIa - The Thomas mechanism

15 Heavy particle collision processes I - Introduction II - Three models for electron transfer a) The Thomas mechanism b) The Bohr-Lindhardt model c) Resonant capture at low energy in homonuclear one electron collision systems III - The quasi-molecular formalism IV - The travelling asymptotic state basis set expansion : the intermediate energy domain V - Capture in ion-molecule collisions

16 Another classical model working at intermediate to high energy proposed by Bohr and Lindhardt in 1954 K. Dan. Vid. Sel. Mat. Phys. Medd. 28, 7 (1954) IIb - The Bohr-Lindhardt model * for fully stripped ion A colliding on hydrogenic ion B * Capture can take place if two conditions are satisfied : - 1st condition : release of e - by B - 2nd condition : capture of the e - by A -10 y m = " Z b Z a + Z b R V(y m ) = " 1 R ( Z a + Z b )2 Energy of the electron on B E n = " 1 2 Z b 2 n 2-15 Z b = 1 Z a = 6 + extra attraction term from A, in average the total potential energy acting on the electron (at y from B on the internuclear line R) " Z a R R V(y) V(y) = " Z b y " Z a R " y R 0.5 R 0.75 R 1R B y A

17 * 1st condition : release the initial energy of the electron equal to the height of the barrier IIb - The Bohr-Lindhardt model R 1 = 2n2 (Z b + 2 Z a Z b ) " 4n2 Z a 2 3/2 Z b Z b if Z a >> Z b * 2nd condition : capture the electron has a kinetic energy v 2 /2 with respect to the moving A: capture can occur if the attraction -Z a /R balances this term that is when R < R 2 with Then R 2 = 2Z a v 2 - at high energy where R 1 > R 2 the geometrical cross section is σ C 1 = π R 2 2 but ionization may occur => σ C weighted by the probability of ionizing the electron before capture takes place, approximate by the ratio between collision time R 2 /v to the periodic time of the electron in the initial Bohr orbit a n /v n $ " C 2 R 1 = # R 2 1 & %& v v n a n ' ) () = 8# Z $ a & % 3 v n a n ' ) v *7 ( a n = n 2 /Z b v n = Z b /n

18 Z a 2 v n Z b - at low energy where R 2 > R 1 ( ) " C 2 = # R 2 1 $ 16# n4 Z a a v 2 < 1 or 16# Z n 3 a 2 2 Z b v n Comparison with experimental data from Knudsen, Haugen, and Hvelplund, Phys. Rev. A 23, 597 (1981))

19 Heavy particle collision processes I - Introduction II - Three models for electron transfer a) The Thomas mechanism b) The Bohr-Lindhardt model c) Resonant capture at low energy in homonuclear one electron collision systems III - The quasi-molecular formalism IV - The travelling asymptotic state basis set expansion : the intermediate energy domain V - Capture in ion-molecule collisions

20 Resonant capture (C) vs. elastic scattering (E) IIc - Resonant transfer at low E E : H A + + H B (1s) > H A + + H B (1s) C: H A + + H B (1s) > H A (1s) + H B + at low velocities in a semiclassical treatment (R R(t) = b + v t) so that with " i "#(r,t) "t H e = " 1 2 # r " 1 r a " u # " 2p$ u = H e #(r,t) " 1 r b (+ 1 R ) E(R) (a.u.) s! g 2p! u 2p" u H A + + H B (1s) H A (1s) + H B + " g # " 1s$ g H e " g / u (r,r) = E g / u (R) " g / u (r,r) s! g H 2 + " g / u -2 are orthonormalized Internuclar distance (a.u.)

21 "(r,t) = a g (t)# g (r,r) e $i% 1s t + a u (t)# u (r,r) e $i% 1s t IIc - Resonant transfer at low E " g (r,r) # 1 with 2 ($ 1s(r b ) + $ 1s (r a )) when R > " u (r,r) # 1 2 ($ (r ) % $ (r )) 1s b 1s a so that "(r,t) = 1 2 (a g(t) + a u (t)) # 1s (r b ) e $i% 1s t (a g(t) $ a u (t)) # 1s (r a ) e $i% 1s t a E 1s (t) a C 1s (t) when R > ESdt => i d dt a g / u(t) = a g / u (t) (E g / u (R) " # 1s ) with initial conditions a g ("#) = a u ("#) = 1 2

22 IIc - Resonant transfer at low E a g / u (t) = 1 2 & t exp "i E ) ' % [ g / u (R') " # 1s ] dt' * ( + "$, a g / u (t,b) since R'= b 2 + v 2 t' 2 => P C (b) = a 1sC (+ ) 2 = % 1 sin 2 & ' 2 +# $ "# ( [ E g (R) " E u (R)]dt) * oscillates with unit amplitude versus * b at fixed v or * v (E) at fixed b(θ) $ % ( but " C (E) = 2# b P C (b) db does not) 0 from Lockwood and Everhart, Phys. Rev. 125, 567 (1962)

23 Heavy particle collision processes I - Introduction II - Three models for electron transfer a) The Thomas mechanism b) The Bohr-Lindhardt model c) Resonant capture at low energy in homonuclear one electron collision systems III - The quasi-molecular formalism IV - The travelling asymptotic state basis set expansion : the intermediate energy domain V - Capture in ion-molecule collisions

24 III - Quasi molecular formalism To solve generally the tdse and i "#(r,t) "t ' = H e #(r,t) with H e = " 1 2 # " 1 r r a "(r,t) = a i (t) # i (r )exp($i dt' E i (R(t'))) & t $% " " 1 r b (+ 1 R ) one gets to coupled differential equations for the probability amplitudes a j (t) involving only dynamical couplings : " j d dt " i which can be written as v R " j # #R " i + i $ " j L y " i radial couplings inducing transitions between states with same symmetry (Σ <-> Σ, Π<-> Π, ) rotational couplings inducing transitions between states with different symmetry (Σ <-> Π, )

25 the basis set molecular functions can be chosen to III - Quasi molecular formalism * such that H e φ j (r,r) = E j (R) φ j (r,r) adiabatic representation * or to minimize some of the dynamical couplings diabatic representations Best choice? adiabatic representation vs. diabatic representation E E < φ i φ j > R x internuclear distance R < φ i φ j > R x internuclear distance R

26 1 kev θ L = ev elastic capt. (1s) exc. (n=2) ev 2s! g p" u E(R) (a.u.) -1 2p! u H + + H(1s) s! g H 2 + from Houver et al, J. Phys. B 7, 1358 (1974) Internuclar distance (a.u.)

27 N 3+ + H(1s) H(1s) N 2+ (3s)

28 N 3+ + H(1s)

29 I - Introduction Heavy particle collision processes II - Three models for electron transfer a) The Thomas mechanism b) The Bohr-Lindhardt model c) Resonant capture at low energy in homonuclear one electron collision systems III - The quasi-molecular formalism IV- The travelling asymptotic state basis set expansion : the intermediate energy domain V - Capture in ion-molecule collisions

30 IV - Travelling AO expansion approach To solve the tdse one can choose purely diabatic (asymptotic) states and = H e #(r,t) with H e = " 1 2 # r + Vb (r + pr(t))+ V a (r " qr(t)) one gets rid of asymptotic spurious couplings and non galilean invariance e.g. N i "#(r,t) "t "(r,t) = $ b n (t) # b n (r,t) + $ a m (t) # a m (r,t) " b n (r,t) = # b n (r b )e $ipr v. r e $i% a nt$i 1 2 p2 v 2 t M " a m (r,t) = # a m (r a )e +iqr v. r e $i% b mt$i 1 2 q2 v 2 t h x " i x (r x ) = (# 1 2 $ r x + V b (r x))" i x (r x ) = % i x " i x (r x ) b m ("#) = $ m,i a n ("#) = 0 A r b = r + pb r + pv r t q v -qr(t) R(t) With such travelling states expansions and making use of the relations " = " + pv r. r " # "t r "t rb = " $ qv r.# r r b "t r "t ra r a " r b b n # rb " n' r pr(t) -p v B R(t) r b = r " qb r " qv r t

31 IV - Travelling AO expansion approach => Coupled differential equations for the probability amplitudes with, e.g. i S c = V c c = d dt c S ba kl = " b k e ir v. r a " l c = " $ $ $ $ $ $ $ # b 1 M b N a 1 M a M % ' ' ' ' ' ' ' & e #i($ a l #$ b m +(q#p)v 2 )t = " b k e ir v. r m " a l e #i($ a l #$ b m )t V ba kl = " b k V b e ir v. r a " l r m = 1 2 (r a + r b ) e #i($ a l #$ b m +(q#p)v 2 )t = " b k V b e ir v. r m " a l e #i($ a l #$ b m )t and galilean invariance is fulfilled but a lot of computations to be done

32 Heavy particle collision processes I - Introduction II - Three models for electron transfer a) The Thomas mechanism b) The Bohr-Lindhardt model c) Resonant capture at low energy in homonuclear one electron collision systems III - The quasi-molecular formalism IV - The travelling asymptotic state basis set expansion : the intermediate energy domain V - Capture in ion-molecule collisions

33 Good textbooks about collisions Quantum Collision theory, C.J. Joachain (North-Holland) Introduction to the theory of ion-atom collisions, M.R.C. McDowell and J.P. Coleman (North-Holland) Charge exchange and the Theory of Ion-Atom Collisions, B.H. Bransden and M.R.C. McDowell (Oxford Science Pub.) Molecular Collision Theory, M. Child (Academic Press, NY) 1974.

34 Ion-molecule collisions Electronic capture (1 electron system) (resonant system) Recent experiments Bräuning et al, J. Phys. B (2001) Total capture Reiser et al, HCI-2002, Caen Orientation effets (COLTRIMS)

35 Capture cross sections Franck-Condon approximation

36 Effect of vibrational dof large values of R AB

37 Differential cross sections vs. R AB

38 Capture cross sections

39 Unknown exp. vibr. distribution! Example : Franck-Condon distribution H 2 + hv H e-

40 Orientation effects P v B P v B Θ m Θ m = 90 A P v A Θ m = 0 A B

41 Differential cross sections vs. Θ m General tendancy : importance of 90 Inversion for He 2+ at v = 0.3 a.u. Steric factor Dynamical effect

42 He 2+ v = 1.0 a.u. Θ m = 90

43 He 2+ v = 1.0 a.u. Θ m = 0

44 He 2+ v = 1.0 a.u. Θ m = 90 Θ m = 0

45 Oscillations T 1/2 = T osc / a.u.

46 Oscillations Quantal interferences Resonance : v = R AB /T 1/2 = 0.28 a.u. T 1/2 = π/δe 7.25 a.u.

47 He 2+ v = 0.3 a.u. Θ m = 0

48 He 2+ Θ m = 0 v = 0.3 a.u. v = 1.0 a.u.