Dynamical adiabatic theory of atomic...

Dynamical adiabatic theory of atomic collisions

T. P. Grozdanov*E. A. Solov’ev&

*Institute of Physics, University of Belgrade&Bogoliubov Laboratory of Theoretical Physics , JINR, Dubna

SPIG2020-25.08.2020.

T. P. Grozdanov* E. A. Solov’ev& SPIG2020-25.08.2020. SPIG2020-25.08.2020. 1 / 15

Ratko Janev (1939-2019)(1965 - 1972) Instutute of Nuclear Sciences ,Vinca - Belgrade

(1972 - 1987) Institute of Physics, Belgrade

(1987-1999) International Atomic EnergyAgency, Vienna

(1999 - 2019) Macedonian Academy ofSciences and Arts, Skopje

(1999 - 2000) National Institute of FusionScience, Tokyo

(2000 - 2002) Juelich Reserch Center,Germany

(2005 -2019) Institute of Applied Physicsand Computational Mathematics, Beijing


(1972 - 1987) Institute of Physics, Belgrade

Founder of the Group for Theory of AtomicCollisions:

High energy atomic collisions - DževadBelkić (Karolinska Institute, Stockholm)

Ion - surface interactions - NatašaNedeljković (Faculty of Physics, Belgrade)

Low energy atomic collisions - TaskoGrozdanov (Institute of Physics, Belgrade),Predrag Krstić (Institute for AdvancedComputational Science , Stony Brook)

My first and the last paper with R. Janev:Physical Review A 17 (1978), 880Eur. Phys. J. D (2018) 72: 14


Dynamical adiabatic theory of atomic collisions

Overview

Introduction

Dynamical adiabatic representation

Hidden crosings in HeH2+ system

Application to electron capture process: H++He(1s)+ →H(1s)+He2+

Concluding remarks


Introduction

MotivationIncompatibility of standard (i.e. Born-Oppenheimer) adiabatic basis with physicalboundary conditions in slow atomic collisions

Solutions for impact-parameter formulation:

”Electronic translational factors” attached to basis functions [D.R. Bates and R.McCarroll 1958]

Non-stationary scaling of length - Dynamical adiabatic basis. [E.A. Solov’ev 1976, 1982](1) Boundary conditions.-All non-adiabatic couplings Wij (R) = 〈i|∂/∂R|j〉 → 0 as R →∞.(In the standard adiabatic basis some Wij (R)→ const as R →∞.)

.(2) Rotational transitions.-Transformed into radial transitions in rotating frame.(In standard adiabatic - need numerical close-coupling calculations.)

.(3) Ionization process.-Described using a basis of the complete discrete orthogonal wave-packets.

Solution for quantum formulation:

Hyperspherical coordinates (E.A. Solov’ev and S.I. Vinitsky, 1985)


Dynamical adiabatic representation

r

Z B

Z A

v B

v A

ρ

e

R

O

(i) Non-stationary scaling of electronic coordinates:

q = Ôz [ϕ(t)]r/R(t),

Ôz [ϕ(t)] - rotation matrix to molecular (rotating withinternuclear axis) frame.(ii) Introducing new wavefunction f (q, τ) (in a.u.):

Ψ(r, t) = R−3/2 exp

[ir2

2R

dR

dt

]f (q, τ),

exp

[ir2

2R

dR

dt

]|Rr−1j →∞

= exp[i(vj · rj +1

2v2j t)], j = A,B.

(iii) A new time-like variable: dτ = dt/R(t)2.

Modified time dependent Schrödinger equation

H(τ)f (q, τ) = i ∂f (q,τ)∂τ

,

H(τ) = − 12

∆q − R(τ)(

ZA|q+αq̂1|

+ ZB|q−βq̂1|

)+ ωL3 +

12ω2q2

ω = ρv , v = |vA − vB |,

R(τ) = ρ/ cosωτ, L3 = −i(q1

∂∂q2− q2 ∂∂q1

), Π3(q3 → −q3)


Adiabatic expansion

f (q, τ) =∑j

gj(τ)Φj(q,R(τ), ω) exp

(−i∫ τ

0Ej(R(τ

′), ω)dτ ′)

Dynamical adiabatic basis depends on two parameters: ω = ρv and real

(or complex) values of R

H(R, ω)Φj(q,R, ω) = Ej(R, ω)Φj(q,R, ω)

Relation to standard adiabatic eigenvalues: Ej(R, ω = 0) = εj(R)R2

Numerical method: Lagrange-mesh method [T.P. Grozdanov and E.A. Solov’ev,2013,2014]

Prolate spheroidal coordinates {ξ, η, φ}: Nξ × Nη mash points (ξi = hxi , ηj )(i = 1, ...,Nξ, j = 1, ...,Nη), related to zeros of the Laguerre and Legendre polynomials:LNξ (xi ) = 0 and PNη (ηj ) = 0,

hπ3m (φ) =

{[(1 + δm,0)π]−1/2 cosmφ for π3 = 1(π)−1/2 sinmφ for π3 = −1.

m = |m1| = 0, 1, ...,MMatrix elements by using Gaussian quadratures - all analytic.


HeH2+ SYSTEM (ZA = 1,ZB = 2). ELECTRON CAPTURE PROCESSES

He2+ + H(1s)→ He+(n = 2) + H+ [T.P. Grozdanov and E.A. Solov’ev, 2015]

H+ + He+

(1s)→ H(1s) + He2+ [T.P. Grozdanov and E.A. Solov’ev, 2018]

Effective united-atom principal quantum

number:

NUA(R, ω) = (ZA + ZB)R[−2E(R, ω)]−1/2

0 2 4 6 8 1 01 . 0

1 . 5

2 . 0

2 . 5

3 . 0

3 . 5

3 p σ3 d σ

2 s σ

2 p π

2 p σ

_ _ _ ω=0_____ ω=1 π3=1

NUA j(R

,ω)

R e R

1 s σ

j = 1

2

34

5

6

H + + H e + ( 1 s )

H ( 1 s ) + H e 2 +

H + + H e + ( n = 2 )

I m R = 0

Hidden crosssings: Q- and L3- series of branch

points

0 2 4 60

2

4

6

L 3 ( 2 - 3 )

Q ( 1 - 2 )

L ( 1 )3 ( 3 - 4 )

ω=6

ω=0

ω=6

ω=0

ω=4.5

ω=0 ω=0

ω=0

ω=6

Im(R

)

R e ( R )

ω=6 Q ( 2 - 3 )

L ( 2 )3 ( 3 - 4 )


TRANSITION PROBABILITIES

Example: one branch point, two paths

[R.K. Janev, J. Pop-Jordanov, E.A. Solov’ev, 1997]

P =∣∣∣∑k=1,2 A(k) exp{−iφ(k) − in(k)π/2}∣∣∣2,

A(k) =√

1− p√p, φ(k) = Re∫Lk

E(R(t))dt, n(k) = ±1p = e−2s , s =

∣∣∣Im ∫Lk E(R(t))dt∣∣∣− Stueckelberg parameterP = 4e−2s(1− e−2s)sin2(φ(1)/2− φ(2)/2)


TRANSITION PROBABILITIES: H+ + He+

(1s)→ H(1s) + He2+

“X = vt - complex plane”, R2 = X 2 + ρ2

- 6 - 4 - 2 00

1

2

3

4

L 3 ( 2 - 3 )

L ( 1 )3 ( 3 - 4 )

E c m = 1 4 k e V ( v = 0 . 8 3 7 )

ω3=2.51 ( b 3 = 3 )ω2=1.05 ( b 2 = 1 . 2 5 )ω1=0.34 ( b 1 = 0 . 4 )

2 - 4

ω=ω2

ω=ω1

Q ( 1 - 2 )

1 - 4

ω=ω3

ω=0ω=0

ω=ω3

ω=ω21 - 2

ω=0

2 - 4

2 - 3ω=ω1

ω=ω3

ω=0

L ( 2 )3 ( 3 - 4 )

ω=ω3

ω=0

ω=ω3Q ( 2 - 3 )

Im(X)

R e ( X )

ji = 1→ jf = 2

P1,2 =∣∣∣∑9k=1 A(k)1,2 exp{−iφ(k)1,2 − in(k)1,2π/2}∣∣∣2,

φ(k)1,2 = Re

∫C

(k)1,2

E(R,ω)

R2dt,

A(k)1,2 =

∏A,B√pA√

1− pB

pA = e−2sA , sA =

∣∣∣Im ∫CA E(R,ω)R2 dt∣∣∣,A = Q(1− 2),Q(2− 3), L3(2− 3), L

(1)3 (3− 4), L

(2)3 (3− 4)

3

p L 3 ( 2 - 3 )1 - p L 3 ( 2 - 3 )

A ( 3 )1 2 = [ p Q ( 1 - 2 ) ( 1 - p L 3 ( 2 - 3 ) ) p L 3 ( 2 - 3 ) ( 1 - p L ( 2 )3 ( 3 - 4 ) ) ) p Q ( 2 - 3 ) ]1 / 2

1 - p L 3 ( 2 - 3 )1 - p L 3 ( 2 - 3 )

C ( 3 )1 , 21 - p L ( 2 )3 ( 3 - 4 )

L ( 2 )3 ( 3 - 4 )L( 2 )3 ( 3 - 4 ) L ( 1 )3 ( 3 - 4 )L

( 1 )3 ( 3 - 4 )

L 3 ( 2 - 3 ) L 3 ( 2 - 3 )

Q ( 2 - 3 ) Q ( 2 - 3 )Q ( 1 - 2 )Q ( 1 - 2 )

2

p Q ( 2 - 3 )33

322

32222

21

1p Q ( 1 - 2 ) R e ( X )

I m ( X ) ( c )

2

A ( 2 )1 2 = [ p Q ( 1 - 2 ) ( 1 - p L 3 ( 2 - 3 ) )2 ( 1 - p Q ( 1 - 2 ) ) ( 1 - p Q ( 2 - 3 ) ) ] 1 / 2

L ( 1 )3 ( 3 - 4 ) L ( 2 )3 ( 3 - 4 )L( 2 )3 ( 3 - 4 )

L 3 ( 2 - 3 )Q ( 1 - 2 )Q ( 1 - 2 )

Q ( 2 - 3 )

1 - p Q ( 2 - 3 )

32

2

Q ( 2 - 3 )

2

21L ( 1 )3 ( 3 - 4 )L 3 ( 2 - 3 )

21

1p Q ( 1 - 2 ) 1 - p Q ( 1 - 2 )R e ( X )

I m ( X )

C ( 2 )1 , 2

( b )

2 22 3 2 3

L ( 1 )3 ( 3 - 4 ) L ( 2 )3 ( 3 - 4 )L ( 2 )3 ( 3 - 4 )

L 3 ( 2 - 3 )Q ( 1 - 2 )Q ( 1 - 2 )

Q ( 2 - 3 )

1 - p Q ( 2 - 3 )

32

2

Q ( 2 - 3 )

2

21

11L ( 1 )3 ( 3 - 4 )

L 3 ( 2 - 3 )21

11 - p Q ( 1 - 2 ) p Q ( 1 - 2 )

A ( 1 )1 2 = [ ( 1 - p Q ( 1 - 2 ) ) p Q ( 1 - 2 ) ( 1 - p Q ( 2 - 3 ) ] 1 / 2

R e ( X )

I m ( X )

C ( 1 )1 , 2

( a )



(1s)→ H(1s) + He2+

Deformation of paths

- 1 . 5 - 1 . 0 - 0 . 5 0 . 00 . 0

0 . 4

0 . 8

1 . 2

L ( 1 )3 ( 3 - 4 )

E = 1 4 k e V , b = 0 . 2 5 < b 1 = 0 . 4 ,

2

Im(X)

R e ( X )

3

L 3 ( 2 - 3 )

2 3

1

Q ( 1 - 2 )

4

( a )

- 1 . 6 - 1 . 2 - 0 . 8 - 0 . 4 0 . 00 . 0

0 . 5

1 . 0

1 . 5

L 3 ( 2 - 3 )

L ( 1 )3 ( 3 - 4 )

Q ( 1 - 2 ) ( b )

Im(X)

22

R e ( X )

E = 1 4 k e Vb 1 = 0 . 4 < b = 0 . 8 < b 2 = 1 . 2 5 ,

1 2

432

- 2 - 1 00

1

2

3

L ( 1 )3 ( 3 - 4 )

L 3 ( 2 - 3 )Q ( 1 - 2 )

( c )

Im(X)41

42

E = 1 4 k e V , b = 2 > b 2 = 1 . 2 5

3

1

243

R e ( X )

2

- 3 - 2 - 1 00

1

2

3

4

Q ( 1 - 2 )

L 3 ( 2 - 3 )

L ( 1 )3 ( 3 - 4 )

( d )E = 4 0 k e V , b = 2 . 3

Im(X)

R e ( X )

1

2

2

3

2 13 5

1 6

C ( 4 , 5 )



(1s)→ H(1s) + He2+

0 1 2 31 E - 4

1 E - 3

0 . 0 1

0 . 1

1

b 2 = 1 . 2 5b 1 = 0 . 4

( b )L ( 1 )3 ( 3 - 4 )

L ( 2 )3 ( 3 - 4 )Q ( 2 - 3 )

L 3 ( 2 - 3 )

A = Q ( 1 - 2 )p A=ex

p(-2s

A)

b

0 1 2 31 E - 3

0 . 0 1

0 . 1

1

1 0

S i n g l e - p a s s p r o b a b i l i t i e s

b 2 = 1 . 2 5

L ( 1 )3 ( 3 - 4 )

L ( 2 )3 ( 3 - 4 )

A = Q ( 1 - 2 )L 3 ( 2 - 3 ) Q ( 2 - 3 )

E c m = 1 4 k e V , v = 0 . 8 3 7

s A

b

( a )b 1 = 0 . 4

S t u e c k e l b e r g p a r a m e t e r s

Phases φ(k)1,2

0 1 2 3

- 1 6

- 1 4

- 1 2

- 1 0

- 8

- 6

- 4

- 2 b 2 = 1 . 2 5

E c m = 1 4 k e V , v = 0 . 8 3 7

987 6

54 3

2

φ(k) 12

b

k = 1

b 1 = 0 . 4



(1s)→ H(1s) + He2+

0 . 0 0 . 5 1 . 0 1 . 5 2 . 00 . 0 0 0 0 0

0 . 0 0 0 0 2

0 . 0 0 0 0 4

0 . 0 0 0 0 6 2 p a t h s 9 p a t h s 5 - s t a t e s M o l . - C C E c m = 1 . 6 k e V

v = 0 . 2 8 3

bP12

(b)

b

( a )

0 1 2 30 . 0 0 0

0 . 0 0 2

0 . 0 0 4

0 . 0 0 6 ( b )E c m = 8 k e Vv = 0 . 6 3 3

bP12

(b)

b

0 1 2 30 . 0 0

0 . 0 1

0 . 0 2

0 . 0 3 ( c )E c m = 1 4 k e Vv = 0 . 8 3 7

bP12

(b)

b0 1 2 30 . 0 0

0 . 0 3

0 . 0 6( d )

E c m = 4 0 k e Vv = 1 . 4 1 5

bP12

(b)

b


CROSS SECTIONS: H+ + He+

(1s)→ H(1s) + He2+

σ1,2(Ecm) =∫∞

0 ρP1,2(ρ)dρ

2 4 6 8 1 00 . 0 0 0

0 . 0 0 5

0 . 0 1 0

0 . 0 1 5

0 . 0 2 0 E x p . 1 5 - s t a t e s M o l . C C 2 p a t h s 9 p a t h s

σ 12 (1

0-16 cm

2 )

E c m ( k e V )1 1 0 1 0 0

1 E - 5

1 E - 4

1 E - 3

0 . 0 1

0 . 1

E x p . 1 E x p . 2 9 p a t h s 2 p a t h s 5 - s t a t e s M o l . C C

σ 12 (1

0-16 cm

2 )

E c m ( k e V )

Exp.1,2 - [B. Pearth et al, 1977,1983]

5-states Mol.CC-[T.G. Winter et al,1980]


Concluding remarks

The application of hidden-crossings method for describing electronic transitions inion-atom collisions is more complicated in dynamical adiabatic theory then in thestandard adiabatic theory.

This is because one has to deal with a series of branch points in the complex R-planewhich change their positions when dynamical parameters (such as ω = ρv) are changed.

The great advantage of this method is that electronic transitions caused by relativeradial and angular motion of the nuclei can be treated on the equal footing, the propertywhich is missing in the standard adiabatic approach.

As the comparison with the close-coupling calculations show, the precision of themethod is satisfactory.


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