A Quantum-Classical Approach for the Study of Cascade Processes in Exotic Hydrogen Atoms

Transcription

1 PSAS 28 International Conference on Precision Physics of Simple Atomic Systems Windsor, July 21-26, 28 A Quantum-Classical Approach for the Study of Cascade Processes in Exotic Hydrogen Atoms M.P. Faifman and L.I. Men'shikov Russian Research Center Kurchatov Institute, Moscow, Russia

2 The atom formation (initial stage): {pμ, pk } (n, l, E ) x + H ( px ) + e, x = K nle, μ n m 2l + 1 ~, n 14, n 3, W, E ~1 ev. x (μ) (K) 2 me n Cascade processes (de-excitation stages): elastic scattering (n' = n, l' = l), ( px ) nl + H ( px ) n'' l + H Stark transitions (n' = n, l' l), Coulomb transitions (n' n), ( px ) + H ( px ) + p+ e Auger transitions ( n ' < n), nl n' l ' ) nl ( n' l ' γ ( px px ) + radiative transitions ( n' < n). Accompanying processes: weak decay (μ ), nuclear absorption (K ).

3 n ini ~ (m x /m e ) 1/2 Stark mixing n l nl' n' l' Collisional de-excitation: Coulomb and Auger transitions n=5 5 n=4 5 n=3 5 n=2 5 2S 2P Radiation transitions n=1 5 1S Cascade in exotic hydrogen atom

4 The general problem: (px) nl +H all final states The existing approaches to solve this problem: Quantum Mechanics (QM) methods: three-body problem; multi-channel Coulomb problem (n 2 ~1 1 muonic/kaonic states); total and differential cross sections (and the lack of the complete set of them). Classical Mechanics (CM) description: three Coulomb charged planet problem (classical collisions); natural description of multi-quantum Coulomb transitions (Δ= n n'>1); possibility to take into account protons chemical binding in H 2 molecule. Good argument for solution of the QM problem by the CM methods is successful description of the electron charge exchange in collisions of multi-charged ions with hydrogen atoms (R. Olson and A. Salop, 1976): differences between calculated and experimental cross-sections are about ~2%. Another argument is the Bohr Correspondence Principle: the CM results coincide with the same in QM at large n.

5 Mechanisms of (px ) nl exotic atoms acceleration ( px ) ( E) + p ( px ) ( E ') + p nl nl ' ' Coulomb transitions (n' < n, E' > E), ( px ) ( E) + H ( px ) ( E ') + p + e nl n ' l ' Auger transitions ( n ' < n, Δ = n n ), 1 2 ( px ) nl + H C ' + e, C ' ( px) n ' l ' p Auger capture( n ' = n 1, Δ = n1 n2 < ), C ' C'' + γ, ( px) n " l " C" ( px) p radiative deexcitation n < n n" l" ( ' ), + p predissociation ( n" < n' = n 1).

6 Quantum-Classical Monte Carlo method Proposed scheme of cascade calculations: Radiative transitions are considered by QM methods; Collisions are considered by methods of CM; Auger processes are treated semiclassically. The processes of Auger capture are negligible for heavy exotic atoms (e.g., pk ), which become more and more energetic during the cascade due to multi-quantum Coulomb transitions. How Auger processes is important for light exotic atoms (pμ)?

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8 Block-scheme of exotic atom cascade in hydrogen Output: cross-sections of Coulomb, Stark and Auger transitions; kinetic energy distributions; decay characteristics of the exotic molecular complex; cascade time in the exotic atom; Doppler broadening of the atomic {nl}-state; X-ray yields.

9 The basic parameters of the problem: The mean distances between atoms: R where = N 6ϕ» 1, 1/3 1/3 ϕ = N N N = cm au /, (6 1..). The radii of the Kepler muon orbits where μ = mm /( m + m μ The "initial data sphere'' radius: The free path length: λ p μ p ). r «R. The typical collision length: λ ~ f n = ( πr N) ; 2 1 c 2 ( n~5): rn = n / μ.15, R = R + 2 r ; R = 2 5; R ; n n n λ f 1 5 =» 1. λ πr N R ϕ 3 3 c

10 λ f» λ c, Free flight and radiative transitions (pμ) nl (pm) n'l' + γ i.e., it is possible to neglect the radiative transitions during the collision The transitions (nl) (n'l') are described by the quantum mechanical system of equations dns () t = Γ sns() t + Γ s' sns' (), t Ns() t = 1, dt n' > n, l' nl where s ( n, l), Γ = Γ, l' = l± 1. The initial conditions: The longevity of free s n' < n, l' N flight: s' s () N () = δ δ. s nl nn ll 1 1 t = ln, ξ (,1). 2 πrn v ξ i i

11 Initial data sphere ρ 1 (t), ρ 2 (t), ρ K (t) vector-coordinates of two protons and muon; ρ impact parameter; R 1 = R c.m. ρ 2. μ r 1 c.m. P 1 Z t μ r 1 P 1 Z t > v R 1 ρ P 1 R Y K ρ Κ ρ 1 ρ 2 Y P 2 P 2 X X

12 (pμ) nl +H (pμ) n'l' + H - 3-body problem in Classical Mechanics 1 r12 m ˆ ˆ μv& μ = Fμ1+ Fμ2, F12 =+ f( r 2 12) r12, r12 =, r12 r12 1 r m ˆ ˆ 1v& 1 = Fμ 1+ F12, F = = 1 r & = = ˆ ˆ = μ1 μ1 f ( ρ 2 μ1) rμ1, rμ1, rμ1 rμ1 μ 2 m2v2 Fμ 2 F1 2, Fμ2 f ( ρ 2 μ2) rμ2, rμ2, rμ2 rμ2 5 5 rμ1 rμ2 4 4 μ1 =, μ2 =, = rμ1 + rμ 2, rij = ri rj, ( i, j = ρ ρ σ ) (1,2, μ). σ σ 2 2R f( R) = (1 + 2R+ 2 R ) e is the electron screening factor. The initial conditions (at t = ): m m r = R + r, r = R r, r =, 1 μ μ μ1 1 μ1 2 mμ m1 m m r& = v+ v, r& = v v, r& =. 1 μ μ μ1 1 μ1 2 mμ m1 The end of collision stage: fulfilment of the condition r > R. 12

13 As a result the transition n l f E ( n, l, E ) ( n, l, E 2 μ μr& μ1 1 =, l = l, ε =, 2ε 2 r f f f f r = r i i i f f f The rate of the Coulomb transition ni nf : 2l + 1 N λ v σ ; σ( ) π. 2 i ( nl) i ( nl) f nn' = N nn' ni nf = R 2 ll ni Nto t i f μ1 μr&, if the final state is the p μ atom, μ1 μ1 1 μr&, if the final state is the p μ atom, μ2 μ2 2 m μ1 mμr& μ + m1r& 1 2 m μ1 = m μ2 mμr& μ + m 2r& 2 2 m μ 2 2 2, for the p μ atom, 1, for the p μ atom, 2 ) takes place:

14 Coulomb de-excitation λ n, n-1, 1 11 s (pμ) n + H (pμ) n-1 + H 25 Korenman, Pomerantsev, Popov 26 Kravtsov and Mikhailov Present work ,1 1 1 E cm, ev

15 Charge exchange reaction 2 (dμ) n + H (pμ) n-1 + D λ n, n-1, 1 11 s Kravtsov and Mikhailov Present work ,1 1 1 E cm, ev

16 Auger processes (pμ) nl +H [(pμ) n'l' + p] +e A on Γ R ( theory by Bukhvostov and Popov,1982) The rate of Auger transiti 11/2 A 1,1n 2 Γ n ( R) = ψ ( R), at n< n 5/2, μ A A Γ ( R) =Γ ( R), at n> n, n A n n ( ) : 2R 2 e ψ ( R) =, n = ( μ / IH), IH is the ionization energy of H. π The probability of the Auger process: W = 1 exp( p ), A dpa A =Γ ( r12), r12 = r1 r2. dt It is necessary to put at the moment t = t of the Auger transition: the screening factor f( R) = 1, n' = n 1, l' = l 1. The condition r 12 >R corresponds to the fact of the [p 1 μp 2 ] molecule decay. A

17 (pμ) nl +H [(pμ) n'l' p]+e 14 τ (pμ) n'l' +p 12 1 at small τ 1 no complex formation φ =.1 f n (τ) at large τ 3 complex formation ~1% n=3 n=4 n=5 τ, (a.u.) Decay events time distributions of the muonic pμp complexes formed in Auger capture processes.

18 (pμ) nl +H [(pμ) n'l' p]+e 1 (pμ) n"l" +p E d f n (E d ) 1,1,1 φ =.1 n=5 n=4 n=3 1E E d, ev Distributions of pμ atoms over kinetic energy differences E d =E pμ(nl) -E pμ(n'l') gained after Auger process.

19 (pμ) nl +H [(pμ) n'l' p]+e,25,2 E d,τ (pμ) n"l" +p F n (E d,τ),15,1,5 n=5 φ =.1, 1 2 E d, ev τ, a.u. Distribution of muonic complexes pμp decay events over times τ and differences of kinetic energy E d of pμ atoms : small τ, large E d Coulomb transitions; large τ and E d predissociation (~1% of events)

20 X-ray yields in the muonic hydrogen 1, Lauss et al.,1998; Bregant et al., 1998; Anderhub et al., Theory Jensen, Markushin, 22 Present work,8 pμ K α K x, yield,6,4,2 K β, K γ 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3,1,1 1 φ, density

21 pμ atomic cascade time 1 Jensen, Markushin, 22 present work PSI (exp), 21 PSI (exp), 26 τ, ns 1 1,1 τ = [ W ( ] C t) 1 dt 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3,1,1 1 φ, density (LHD)

22 f n (E) 1 1 1,1 (pμ) n n=1 n=2 n=3 n=4 n=5 n=6,1 φ =.1 1E-3 1E-4,1, E, ev Kinetic energy distributions of (pμ) atoms in n-state at density φ=.1 LHD.

23 Fractions of pk atoms in the different energy intervals pk - n=4 φ =.3 f n (E) E, ev, % 2% 65% 11%

24 K α yield of X-rays in pk atoms at Γ 2p =.4 mev and Γ 2p =2 mev Iwasaki et al., 1998 Γ 2p =.4 mev (Leon and Bethe,1962): Jensen, Markushin, 22 Faifman, Men'shikov,22 present work Y, (%) Kα Γ 2p =2 mev (Ivanov et al., 24): present work E-5 1E-4 1E-3,1,1 1 φ, density (LHD)

25 X-ray yields of kaonic hydrogen atoms 4 K α 3 Y, (%) 2 Iwasaki et al., K α 1 K β K γ 1E-5 1E-4 1E-3,1,1 φ, density (LHD)

26 Summary A quantum-classical code for ab initio calculations of cascade in exotic hydrogen atoms is developed. This code does not use any fit parameters, and seems to be more accurate than the calculation scheme requiring a sewing procedure. The analysis of the kinetics of cascade processes in muonic and kaonic hydrogen atoms leads to conclusion, which is important for simplifying the cascade calculations: Auger acceleration is negligible for all exotic hydrogen atoms. The obtained results have demonstrated good agreement between theory and experiment. The developed code enables to carry out calculations (with sufficient accuracy ~ 2% and less) of main characteristics of cascade processes: cross-sections of Coulomb, Stark and Auger transitions; kinetic energy distributions; cascade time in the exotic atom; Doppler broadening of the atomic states; X-ray yields.