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

Proton scattering from excited states of atomic hydrogen + some

• ther processes

A S Kadyrov and I B Abdurakhmanov, Sh U Alladustov, J J Bailey, I Bray Curtin University, Australia 2nd 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) § C6+ + H(1s) § p + He(1s2) § H(1s) + H(1s)

1-centre semiclassical CC approach

A lab frame: the origin at the target, z-axis ! " v and x-axis ! " b Projectile position " R(t) = " b + " Z = " b + " vt The w.f. is a solution to SC TDSE i ∂Ψ(! r,t) ∂t = (HT +VP)Ψ(! r,t) Expand Ψ in terms of pseudostates of HT Ψ(! r,t) = aα

α

∑

(t)exp(−iεαt)φα (! r) ! R ! vt ! b y z x ! r

1

1-centre semiclassical approach

Then we get i! aα (t) = exp[i(εα − εβ )t]aβ

β

∑

(t)Dαβ Dαβ = φα − 1 R(t) + 1 | " R(t)− " r

i | i

∑

φβ In matrix form i! a = Da Pseudostates φβ HT φα = δ βαεα

Conventional 2-centre CC approach

In 1-centre case we used Ψ(! rA,t) = aα

α

∑

(t)φα

A(!

rA)e−iεαt It is a solution to TDSE i ∂Ψ(! rA,t) ∂t = (H A +VB)Ψ(! rA,t)

• Now we take into account electron capture
• We need a 2-centre expansion

Conventional 2-centre CC approach

2-centre expansion Ψ(! r,t) = aα

α

∑

(t)φα

A(!

rA)e−iεαt + bβ

β

∑

(t)φβ

B(!

rB)e

−iεβt

There are 2 problems We write TDSE in c.m. frame i ∂Ψ(! r,t) ∂t = (T!

r +V)Ψ(!

r,t) However, this does not solve the problem. The wave function does not satisfy boundary conditions.

Electronic translational factors

2-centre expansion safisfying the boundary conditions Ψ(! r,t) = aα

α

∑

(t)φα

A(!

rA)e−iεαt+iπα

A(!

r,t)−iv2t/8 +

bβ

β

∑

(t)φβ

B(!

rB)e

−iεβt+iπβ

B(!

r,t)−iv2t/8

where πα

A(!

r,t) and π β

B(!

r,t) are arbitrary functions. The only condition is that when | t |→ ∞ πα

A(!

r,t) → − 1

2

! v! r and π β

B(!

r,t) → 1

2

! v! r Bates and McCarroll (1958): electronic translational factors (ETF)

Science of ETFs

• There is a non-uniqueness problem
• Choice of ETFs and their optimisation (using variational techniques)

become elaborate science

• Types of ETFs:

n common n state-dependent n plane-wave n non-PW etc

• Many papers and reviews have been published
• Bates and McCarroll (1958) solution was incomplete
• We believe there is a better solution

2 problems with the standard approach

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(! r,t) = aα

α

∑

(t)φα

A(!

rA)e−iεαt+iπα

A(!

r,t)−iv2t/8 +

bβ

β

∑

(t)φβ

B(!

rB)e

−iεβt+iπβ

B(!

r,t)−iv2t/8

2nd mistake is inTDSE § 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

problem

What is the solution?

• The correct 1-centre expansion should look like

Ψ(! r,t) = aα

α

∑

(t)φα

A(!

rA)e−iεαt ⇒ " Ψ(! r,t) = aα

α

∑

(t)φα

A(!

rA)ei

! kα ! σ

• Both satisfy the semi-classical TDSE

i ∂Ψ(! r, ! b,t) ∂t = (T!

r +V)Ψ(!

r, ! b,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?

• This w.f. does not satisfy TDSE

i ∂ ! Ψ(" r,t) ∂t ≠ (T"

r +V) !

Ψ(" r,t)

• But satisfies the full TISE (E − H) !

Ψ = 0

• The correct 2-centre expansion is

! Ψ(" r,t) = aα

α

∑

(t)φα

A("

rA)ei

" kα " σ +

bβ

β

∑

(t)φβ

B("

rB)e

i " kβ " ρ

How does ETF appear?

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

ETF

2-centre semi-classical equations

NB: Compare with 1-centre case: i! a = D Aa

• 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
• 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

−1.0 − 〈ϕκlϕT

nl〉ϕT nl(r)

κ −1.0 −0.5 0.0 0.5 〈ϕκlϕT

nl〉ϕT nl(r)

κ=0.68587 a.u., l=1 Wave packet Laguerre Coulomb − − 〈ϕκlϕT

nl〉ϕT nl(r)

κ

Wave-packet continuum discretisation

10-1 100 101 102 103 energy (eV) energy wave Laguerre Laguerre Laguerre bins packets s-states p-states d-states

φil

WP(r) =

1 wi dkϕkl(r)

ki−1 ki

∫

φ jl

WP HT φil WP = δ jiεi

Coulomb function

− − 〈ϕκ ϕ 〉ϕ κ − − 〈ϕκ ϕ 〉ϕ κ −1.0 − 20 40 60 80 100 120 140 160 r (a.u.) 〈ϕκ ϕ 〉ϕ κ

• Advantages of WP: there are 3

ψ !

k φ f = 2 π (−i)leiσ lbnl(k)Ylm( ˆ

k) bnl(k) = drϕkl(r)

∞

∫

ϕn

WP(r) =

1 wn

Ionisation amplitude

• Surface-integral formulation of scattering theory

Kadyrov et al., Ann Phys 324 (2009) 1516:

T post = Φ0

− !

H − E Ψ i

+

≈ Φ0

− IN

! H − E

( )IN Ψ i

+

= ! qf ,ψ !

k IN

" H − E

( )Ψ i

N+ ≡

ψ !

k φn n=1 N

∑

φn, ! qf " H − E Ψ i

N+

= ψ !

k φf

! Tfi for k 2 / 2 = εf T post ≠ ! qf , ! k V Ψ i

+

Kadyrov et al, PRL 101 (2008) 230405

Breakup amplitude including ECC

• Surface-integral formulation of scattering theory:

T post = Φ0

− !

H − E Ψ i

+ ≈ 〈Φ0 −(IN T +IM P ) |

! H − E |(IN

T +IM P )Ψ i +〉

≡ 〈Φ0

−IN T |

! H − E |Ψ i

NM+〉 + 〈Φ0 −IM P |

! H − E |Ψ i

NM+〉

Thus the breakup amplitude splits into two: direct ionisation (DI) and electron capture to continuum (ECC) TT = ! qf ,ψ !

k T IN

! H − E

( )Ψ i

NM+ = ψ ! k T φf T

! Tfi

T for k 2 / 2 = εf

T P = ! qf ,ψ !

p P IP

! H − E

( )Ψ i

NM+ = ψ ! p P φf P

! Tfi

T for p2 / 2 = εf

where ψ !

k T and ψ ! p P are the continuum states of target and projectile.

p + H(n=2)

Density matrix*

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

C6+ + H(1s) ionisation: test

10−6 10−5 10−4 10−3 10−2 10−1 100 50 100 150 200 250 300 350 dσion/dε (10−16 cm2/eV) ε (eV) Ein = 400 keV/amu DS ECC Tot

10 100 1000 10000 Projectile Energy (KeV) 20 40 60 80 Cross section ( 10

• 16 cm

2 )

FBA: Analytic FBA: WP-CCC

e-capture and ionisation: convergence

10 20 30 40 50 60 100 101 102 103 cross section, σtec (10−16cm2) projectile energy (keV/amu) l-convergence lmax = 0 lmax = 1 lmax = 2 lmax = 3 lmax = 4 lmax = 5 lmax = 6 5 10 15 20 25 30 101 102 103 104 cross section, σion (10−16cm2) projectile energy (keV/amu) l-convergence lmax = 0 lmax = 1 lmax = 2 lmax = 3 lmax = 4 lmax = 5 lmax = 6

electron capture ionisation

Electron capture and ionisation

C6+-H DDCS at 1 MeV/amu

Exp: Tribedi et al., Phys Rev A 63, 062723 (2001)

0.00 2.00 4.00 6.00 8.00 Ein = 1 MeV/amu, ε = 3 eV d2σion/dεdΩe (10−18 cm2/eV sr) 0.00 1.00 2.00 3.00 4.00 30 60 90 120 150 180 Ein = 1 MeV/amu ε = 10 eV 0.00 0.20 0.40 0.60 Ein = 1 MeV/amu ε = 40 eV 0.00 0.04 0.08 0.12 0.16 30 60 90 120 150 180 Ein = 1 MeV/amu ε = 100 eV FBA WP-CCC Tribedi ejection angle, θe (deg) FBA WP-CCC Tribedi FBA WP-CCC Tribedi ejection angle, θe (deg) FBA WP-CCC Tribedi

C6+-H DDCS at 1 MeV/amu

Exp: Tribedi et al., Phys Rev A 63, 062723 (2001)

106 105 104 103 102 101 Ein = 1 MeV/amu, θe = 15 d2σion/dεdΩe (1016 cm2/eV sr) 105 104 103 102 100 101 102 Ein = 1 MeV/amu, θe = 45 106 105 104 103 102 Ein = 1 MeV/amu, θe = 90 107 106 105 104 103 102 100 101 102 Ein = 1 MeV/amu, θe = 120 FBA WP-CCC Tribedi ejected energy, ε (eV) FBA WP-CCC Tribedi FBA WP-CCC Tribedi ejected energy, ε (eV) FBA WP-CCC Tribedi

C6+-H DDCS at 2.5 MeV/amu

0.00 1.00 2.00 3.00 4.00 5.00 Ein = 2.5 MeV/amu, ε = 3 eV d2σion/dεdΩe (10−18 cm2/eV sr) 0.00 0.10 0.20 0.30 0.40 30 60 90 120 150 180 Ein = 2.5 MeV/amu ε = 30 eV 0.00 0.04 0.08 0.12 Ein = 2.5 MeV/amu ε = 60 eV 0.00 0.02 0.04 0.06 30 60 90 120 150 180 Ein = 2.5 MeV/amu ε = 90 eV FBA WP-CCC Tribedi ejection angle, θe (deg) FBA WP-CCC Tribedi FBA WP-CCC Tribedi ejection angle, θe (deg) FBA WP-CCC Tribedi

Exp: Tribedi et al., J Phys B 31, L369 (1998)

C6+-H DDCS at 2.5 MeV/amu

Exp: Tribedi et al., J Phys B 31, L369 (1998)

107 106 105 104 103 102 101 Ein = 2.5 MeV/amu, θe = 15 d2σion/dεdΩe (1016 cm2/eV sr) 106 105 104 103 102 100 101 102 Ein = 2.5 MeV/amu, θe = 45 106 105 104 103 102 Ein = 2.5 MeV/amu, θe = 90 107 106 105 104 103 102 100 101 102 Ein = 2.5 MeV/amu, θe = 120 Tribedi FBA WP-CCC ejected energy, ε (eV) Tribedi FBA WP-CCC Tribedi FBA WP-CCC ejected energy, ε (eV) Tribedi FBA WP-CCC

Conclusions

• Developed 2-centre CCC approach to HCI-atom collisions including

ECC

• Resolved the notorious ETF problem. Details: PRA 97, 032707 (2018)
• Accurate calculations of the total and various differential cross sections

for ionisation and electron capture in p + H and C6+ + H collisions

• p + He and H + H collisions
• C6+ + H: DDCS and SDCS: good agreement at 2.5 MeV/amu
• DDCS: some disagreement when low-energy electrons are ejected

near the forward direction at 1 MeV/amu

• SDCS: some disagreement with the experiment seen in the forward

direction at 1 MeV/amu

• p + He: integrated cross sections in good agreement with experiment
• H + H: good agreement with experiment for electron-loss cross section

Acknowledgements

Co-authors: Dr Ilkhom Abdurakhmanov Dr Jackson Bailey PhD candidate Shukhrat Alladustov Prof Igor Bray This work is supported by Australian Research Council Thank you for attention!