Proton scattering from excited states of atomic hydrogen + some - PowerPoint PPT Presentation

Proton scattering from excited states of atomic hydrogen + some other processes A S Kadyrov and I B Abdurakhmanov, Sh U Alladustov, J J Bailey, I Bray Curtin University, Australia 2 nd RCM, IAEA, Vienna 2019

Outline ① Single-centre semiclassical close-coupling (CC) approach ② Difficulties associated with two-centre semiclassical CC approach ③ Two-centre semiclassical convergent close-coupling (CCC) approach ④ Total and various differential cross sections for ionisation and electron capture in following collisions p + H(n=2) § § C 6+ + H(1s) § p + He(1s 2 ) § H(1s) + H(1s)

1-centre semiclassical CC approach " A lab frame: the origin at the target , z -axis ! " v and x -axis ! b " " " " b + " R ( t ) = b + Z = Projectile position vt The w.f. is a solution to SC TDSE ! ! vt b i ∂Ψ ( ! ! = ( H T + V P ) Ψ ( ! r , t ) R r , t ) ∂ t x z Expand Ψ in terms of pseudostates of H T ! y r 1 Ψ ( ! ( t )exp( − i ε α t ) φ α ( ! ∑ r , t ) = a α r ) α

1-centre semiclassical approach Then we get ∑ a α ( t ) = exp[ i ( ε α − ε β ) t ] a β i ! ( t ) D αβ β D αβ = φ α − 1 1 ∑ R ( t ) + φ β " R ( t ) − " | r i | i a = Da In matrix form i ! Pseudostates φ β H T φ α = δ βα ε α

Conventional 2-centre CC approach In 1-centre case we used Ψ ( ! A ( ! ∑ r A ) e − i ε α t r A , t ) = ( t ) φ α a α α It is a solution to TDSE i ∂Ψ ( ! = ( H A + V B ) Ψ ( ! r A , t ) r A , t ) ∂ t • Now we take into account electron capture • We need a 2-centre expansion

Conventional 2-centre CC approach 2-centre expansion Ψ ( ! A ( ! B ( ! r A ) e − i ε α t + ∑ ∑ − i ε β t r , t ) = ( t ) φ α ( t ) φ β a α b β r B ) e α β There are 2 problems We write TDSE in c.m. frame i ∂Ψ ( ! r + V ) Ψ ( ! r , t ) = ( T ! r , t ) ∂ t However, this does not solve the problem. The wave function does not satisfy boundary conditions.

Electronic translational factors Bates and McCarroll (1958): electronic translational factors (ETF) 2-centre expansion safisfying the boundary conditions Ψ ( ! A ( ! B ( ! r , t ) − iv 2 t /8 + B ( ! ∑ A ( ! ∑ − i ε β t + i π β r , t ) − iv 2 t /8 r A ) e − i ε α t + i π α r , t ) = ( t ) φ α ( t ) φ β a α b β r B ) e α β A ( ! B ( ! where π α r , t ) and π β r , t ) are arbitrary functions. The only condition is that when | t | → ∞ A ( ! v ! ! B ( ! v ! ! π α r , t ) → − 1 r and π β r , t ) → 1 r 2 2

Science of ETFs o There is a non-uniqueness problem o Choice of ETFs and their optimisation (using variational techniques) become elaborate science o Types of ETFs: n common n state-dependent n plane-wave n non-PW etc o Many papers and reviews have been published o Bates and McCarroll (1958) solution was incomplete o We believe there is a better solution

2 problems with the standard approach § Bates and McCarroll (1958) solution was incomplete § There is no need for an ad-hoc solution using as ETF § The reason for the problem was 2-fold 1st problem 1s mistake appears in the attempt to represent the 2nd centre w.f. in the same form as the w.f. of the 1st centre Ψ ( ! A ( ! B ( ! r , t ) − iv 2 t /8 + B ( ! ∑ A ( ! ∑ − i ε β t + i π β r , t ) − iv 2 t /8 r A ) e − i ε α t + i π α r , t ) = ( t ) φ α ( t ) φ β a α b β r B ) e α β 2nd mistake is inTDSE problem

What is the solution? • The correct 1-centre expansion should look like Ψ ( ! A ( ! Ψ ( ! A ( ! r A ) e − i ε α t ⇒ " ! ! ∑ ∑ σ r , t ) = ( t ) φ α r , t ) = ( t ) φ α r A ) e i k α a α a α α α • Both satisfy the semi-classical TDSE ! i ∂Ψ ( ! ! r + V ) Ψ ( ! r , b , t ) = ( T ! r , b , t ) ∂ t • But " Ψ also satisfies the full (exact) TISE ( E − H ) " Ψ = 0

How does temporal factor emerge? Since 𝑨 = 𝑤𝑢

What is the solution? • The correct 2-centre expansion is Ψ ( " A ( " B ( " " " σ + " " ∑ ∑ ! ρ r , t ) = ( t ) φ α ( t ) φ β i k β r A ) e i k α a α b β r B ) e α β • This w.f. does not satisfy TDSE Ψ ( " i ∂ ! Ψ ( " r , t ) r + V ) ! ≠ ( T " r , t ) ∂ t • But satisfies the full TISE ( E − H ) ! Ψ = 0

How does ETF appear? ETF • These 2 terms were introduced ad-hoc to fix the problem • In our approach they appear naturally • Details: Abdurakhmanov etal, PRA 97, 032707 (2018)

2-centre semi-classical equations • Inserting ! Ψ into TISE ( E − H ) ! Ψ = 0 and using semi-classical approximation we get the same result as we would get using PW ETFs a = D A a NB: Compare with 1-centre case: i ! • Thus there is no SC TDSE when rearrangment inlcuded • Riley and Green (1971): PW ETFs are optimal for atomic orbitals • Because there is no choice

− κ − 〈ϕ κ �ϕ 〉ϕ 〈ϕ κ �ϕ 〉ϕ 〈ϕ κ �ϕ 〉ϕ Wave-packet continuum discretisation − κ − κ − 1.0 − 10 3 10 2 0.5 energy (eV) nl (r) nl (r) nl (r) 10 1 nl 〉ϕ T nl 〉ϕ T nl 〉ϕ T 0.0 〈ϕ κ l �ϕ T 〈ϕ κ l �ϕ T 〈ϕ κ l �ϕ T 10 0 κ =0.68587 a.u., l =1 − 0.5 − κ Wave packet 10 -1 Laguerre Coulomb − 1.0 − 1.0 energy wave Laguerre Laguerre Laguerre 0 20 40 60 80 100 120 140 160 bins packets s-states p-states d-states r (a.u.) 1 k i ∫ φ il WP ( r ) = dk ϕ kl ( r ) o Advantages of WP: there are 3 w i k i − 1 Coulomb function π ( − i ) l e i σ l b nl ( k ) Y lm ( ˆ 2 k ) ψ ! k φ f = WP = δ ji ε i WP H T φ il − κ φ jl 1 ∞ ϕ n ∫ WP ( r ) = b nl ( k ) = dr ϕ kl ( r ) 0 w n −

Ionisation amplitude ! ! T post ≠ + q f , k V Ψ i o Surface-integral formulation of scattering theory Kadyrov et al. , Ann Phys 324 (2009) 1516: Kadyrov et al, PRL 101 (2008) 230405 − ! T post = Φ 0 + H − E Ψ i ! − I N ( ) I N Ψ i + ≈ Φ 0 H − E " " ! N φ n , ! N + ≡ ( ) Ψ i N + = q f , ψ ! k I N H − E ∑ q f H − E Ψ i ψ ! k φ n n = 1 T fi for k 2 / 2 = ε f ! = ψ ! k φ f

Breakup amplitude including ECC o Surface-integral formulation of scattering theory: − ! ! P ) | P ) Ψ i T post = Φ 0 + ≈ 〈 Φ 0 T + I M T + I M H − E Ψ i − (I N H − E |(I N + 〉 ! ! T | P | NM + 〉 + 〈 Φ 0 NM + 〉 ≡ 〈 Φ 0 − I N H − E | Ψ i − I M H − E | Ψ i Thus the breakup amplitude splits into two: direct ionisation (DI) and electron capture to continuum (ECC) ! ! T I N T for k 2 / 2 = ε f ! T T = NM + = ψ ! T φ f ( ) Ψ i T q f , ψ ! H − E T fi k k ! ! P I P T for p 2 / 2 = ε f ! T P = NM + = ψ ! P φ f ( ) Ψ i P q f , ψ ! H − E T fi p p T and ψ ! P are the continuum states of target and projectile. where ψ ! p k

p + H(n=2)

Density matrix* Abdurakhmanov et al, Plasma Phys. Control. Fusion 60 (2018) 095009

C 6+ + H(1s) ionisation: test 80 10 0 E in = 400 keV/amu FBA: Analytic DS FBA: WP-CCC ECC 10 − 1 Tot d σ ion / d ε (10 − 16 cm 2 /eV) 60 2 ) 10 − 2 -16 cm Cross section ( 10 10 − 3 40 10 − 4 20 10 − 5 10 − 6 0 10 100 1000 10000 0 50 100 150 200 250 300 350 Projectile Energy (KeV) ε (eV)

e-capture and ionisation: convergence electron capture ionisation 60 30 l -convergence l -convergence l max = 0 l max = 0 cross section, σ tec (10 − 16 cm 2 ) cross section, σ ion (10 − 16 cm 2 ) 50 25 l max = 1 l max = 1 l max = 2 l max = 2 l max = 3 l max = 3 40 20 l max = 4 l max = 4 l max = 5 l max = 5 30 l max = 6 15 l max = 6 20 10 10 5 0 0 10 1 10 2 10 3 10 4 10 0 10 1 10 2 10 3 projectile energy (keV/amu) projectile energy (keV/amu)

Electron capture and ionisation

C 6+ -H DDCS at 1 MeV/amu 8 . 00 E in = 1 MeV/amu FBA FBA 0 . 60 ε = 40 eV WP-CCC WP-CCC Tribedi Tribedi 6 . 00 0 . 40 d 2 σ ion / d ε d Ω e (10 − 18 cm 2 /eV sr) 4 . 00 0 . 20 2 . 00 E in = 1 MeV/amu, ε = 3 eV 0 . 00 0 . 00 E in = 1 MeV/amu E in = 1 MeV/amu FBA FBA ε = 10 eV ε = 100 eV WP-CCC WP-CCC 4 . 00 0 . 16 Tribedi Tribedi 3 . 00 0 . 12 2 . 00 0 . 08 1 . 00 0 . 04 0 . 00 0 . 00 0 30 60 90 120 150 180 0 30 60 90 120 150 180 ejection angle, θ e (deg) ejection angle, θ e (deg) Exp: Tribedi et al., Phys Rev A 63, 062723 (2001)

C 6+ -H DDCS at 1 MeV/amu 10 � 1 FBA FBA WP-CCC WP-CCC 10 � 2 10 � 2 Tribedi Tribedi 10 � 3 10 � 3 d 2 σ ion / d ε d Ω e (10 � 16 cm 2 /eV sr) 10 � 4 10 � 4 10 � 5 10 � 5 E in = 1 MeV/amu, θ e = 15 � E in = 1 MeV/amu, θ e = 90 � 10 � 6 10 � 6 FBA FBA WP-CCC 10 � 2 WP-CCC 10 � 2 Tribedi Tribedi 10 � 3 10 � 3 10 � 4 10 � 5 10 � 4 10 � 6 E in = 1 MeV/amu, θ e = 45 � E in = 1 MeV/amu, θ e = 120 � 10 � 5 10 � 7 10 0 10 1 10 2 10 0 10 1 10 2 ejected energy, ε (eV) ejected energy, ε (eV) Exp: Tribedi et al., Phys Rev A 63, 062723 (2001)