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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.
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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
T. P. Grozdanov* E. A. Solov’ev& SPIG2020-25.08.2020.
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(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
T. P. Grozdanov* E. A. Solov’ev& SPIG2020-25.08.2020.
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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
T. P. Grozdanov* E. A. Solov’ev& SPIG2020-25.08.2020.
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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)
T. P. Grozdanov* E. A. Solov’ev& SPIG2020-25.08.2020.
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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)
T. P. Grozdanov* E. A. Solov’ev& SPIG2020-25.08.2020.
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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.
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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 )
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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)
T. P. Grozdanov* E. A. Solov’ev& SPIG2020-25.08.2020.
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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 )
T. P. Grozdanov* E. A. Solov’ev& SPIG2020-25.08.2020.
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TRANSITION PROBABILITIES: H+ + He+
(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 )
T. P. Grozdanov* E. A. Solov’ev& SPIG2020-25.08.2020.
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TRANSITION PROBABILITIES: H+ + He+
(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
T. P. Grozdanov* E. A. Solov’ev& SPIG2020-25.08.2020.
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TRANSITION PROBABILITIES: H+ + He+
(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
T. P. Grozdanov* E. A. Solov’ev& SPIG2020-25.08.2020.
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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]
T. P. Grozdanov* E. A. Solov’ev& SPIG2020-25.08.2020.
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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.
T. P. Grozdanov* E. A. Solov’ev& SPIG2020-25.08.2020.
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