CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES.
ADIABATIC APPROACH
Inga Yu. Tolstikhina
P.N.Lebedev Physical Institute, Russian Academy of SciencesMoscow, Russia
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
Theoretical approaches and methods of calculations
Charge exchange in collisions of hydrogen and helium with low-z impurities (Data for the Pellet Charge eXchange (PCX) active diagnostics)
Influence of the isotope effect on the charge exchange in slow collisions of Li, Be, C and W ions with H, D and T
Effect of the electron-nuclei interaction on the internuclear motion in slow ion-atom collisions
General formulation of the problem
Adiabatic approximation: an asymptotic expansion of the solution in the small parameter v
The time-dependent Schrödinger equation
Boundary conditions:
Calculation of the principal terms of the asymptotic of the expansion coefficients
ψ(r , t )=∑p
gp( t )φ p(r , R )exp(− i∫t
E p(R ( vt ' ))dt ' )
H (R)φ p(r , R )=E p(R)φ p(r , R )
H (R )ψ(r , t )=i ∂ψ(r , t )∂ t
r set of electronic coordinates
H(R) electronic Hamiltonian of diatomic quasi-moleculeR=R(vt) the inter-nuclear separationv initial relative nuclear velocity
Epmolecular potential curves adiabatic terms
P pq=limt→ ∞
|g p(t )|2 , lim
t→ −∞g p( t )=δpq
R→ ∞ , E p(R)→ E p(a) , φ p(r , R)→ φ p
(a) σ pq=2π∫0
∞
Ppq(ρ)ρdρ
g p(t), v→ 0
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
Adiabatic approximation: radial and rotational transitions
2 4 6 8 10
-1
Rad
4d
Rot
Rot
Rot
Rad
4f
3d
3d
3d
3p
3p
3s
2p
2p
2s
Li3+
+ H(1s)
Li2+(n=2)
E, a
.u.
R, a.u.
n=3
n=4
United atom Separated atoms
H(1s)
Li2+(n=3)
Rad
Adiabatic theory of transitions in slow collisions, E. Solov’ev Sov. Phys. Uspekhi, 32, 1989
Li3++ H(1s) → Li2+(nl)+ H+CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
Code ARSENY is based on the method of hidden crossingsInput: Z1, Z2; nl; E; basis size; reduced mass
1. Calculates adiabatic potential curves of two Coulomb center problemin complex R-plane
Li3+ + H(1s)
0 5 10 15 20 25
1
2
3
4
5
Nef
f(R
)
R
Separatedted atoms
Li (n=4)
Li (3s,3p,3d)
Li (2s,2p)
H (1s)
Li (1s)
United atom
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
Code ARSENY
2. Searches all branch points and calculates the corresponding Stueckelberg parameter
R the inter-nuclear separation
p , q the set of quantum numbers of the final and initial atomic states
Rc a complex branch point
Ep , Eq energies of the final and initial atomic states
V(R,b) the radial internuclear velocity
b the impact parameter
Δpq=|Im ∫Re Rc
Rc
[ Ep (R )−Eq( R )]dR
V (R,b )|
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
Code ARSENY
3. Calculates the probability as a function of L (nuclear angular momentum) for the entire set of nonadiabatic transitions is calculated as
4. The S-matrix is calculated as a product of elementary S-matrices for the individual transitions induced by the separated branch points
Ppq = e−2 Δpq
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
Code ARSENY
5. The cross sections are calculated as a sum over L:
elastic scattering
inelastic scattering
M the reduced mass of nuclei
Eq(∞) the energy levels of separated atoms
ε the energy of the system in the center of masses
σqq =π
Kq2 ∑
L=0
∞(2 L+1 ) |1 −Sqq
(L)|2
σ pq =π
Kq2 ∑
L=0
∞(2 L+1 ) |Spq
(L)|2
Kq = √ 2 M (ε −Eq (∞ ))
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
Code ARSENY
Large Helical Device (LHD)CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
Charge exchange in collisions of Hydrogen and Helium with low-z impurities
Pellet Charge eXchange (PCX) diagnostics
In the local active diagnostic method, referred to as Pellet Charge eXchange (PCX), an ablating solid impurity pellet is used as a dense target for electron capture by fast ions of a fusion plasma
ncl (x) – cloud density function
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
Pellet Charge eXchange (PCX) diagnostics
The Polystyrene (C8H8 )n is used as an ablating solid impurity pelletin the PCX experiment on LHD.
Local emission of atoms :
( ) ( ) ( ) ( )0, v ,i i iE F E n f Eg =r r r
F0 (E ) neutral fraction
fi ( E, r ) proton distribution function
The relevant elementary charge exchange processes are
H+ + Ck+ H+ + H0 H0 + Ck+
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
Data for the Pellet Charge eXchange (PCX) diagnostics
Collision energies range:1 keV/a.m.u. – 1 MeV/a.m.u.
VF = the velocity of the active electron / the collisional velocity
Adiabatic region
(VF is larger than unity)
Advanced adiabatic approach (E.A. Solov’ev)
Code ARSENY
Born region
(VF is less than unity)
Normalized Brinkman-Kramers (BK) approximation in the impact parameter representation
Code CAPTURE (I.Yu.Tolstikhina, V.P.Shevel’ko)
Coulomb Distorted Wave (CDW) approximation (I.M.Cheshire)
Codes CDW & CDW2 (Dz. Belkic)
Theoretical approaches and methods of calculations
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
reaction k
H+ + Lik+ → H0 + Li(k+1)+H0 + Lik+ → H+ + Li(k-1)+He+ + Lik+ → He0 + Li(k+1)+He2+ + Lik+ → He+ + Li(k+1)+He0 + Lik+ → He+ + Li(k-1)+He+ + Lik+ → He2+ + Li(k-1)+
0, 1, 2 1, 2, 3 0, 1, 2 0, 1, 2 1, 2, 3 1, 2, 3
H+ + Bek+ → H0 + Be(k+1)+H0 + Bek+ → H+ + Be(k-1)+He+ + Bek+ → He0 + Be(k+1)+He2+ + Bek+ → He+ + Be(k+1)+He0 + Bek+ → He+ + Be(k-1)+He+ + Bek+ → He2+ + Be(k-1)+
0, 1, 2, 3 1, 2, 3, 4 0, 1, 2, 3 0, 1, 2, 3 1, 2, 3, 4 1, 2, 3, 4
H+ + Bk+ → H0 + B(k+1)+H0 + Bk+ → H+ + B(k-1)+He+ + Bk+ → He0 + B(k+1)+He2+ + Bk+ → He+ + B(k+1)+He0 + Bk+ → He+ + B(k-1)+He+ + Bk+ → He2+ + B(k-1)+
0, 1, 2, 3, 4 1, 2, 3, 4, 5 0, 1, 2, 3, 4 0, 1, 2, 3, 4 1, 2, 3, 4, 5 1, 2, 3, 4, 5
H+ + Ck+ → H0 + C(k+1)+H0 + Ck+ → H+ + C(k-1)+He+ + Ck+ → He0 + C(k+1)+He2+ + Ck+ → He+ + C(k+1)+He0 + Ck+ → He+ + C(k-1)+He+ + Ck+ → He2+ + C(k-1)+
0, 1, 2, 3, 4, 5 1, 2, 3, 4, 5, 6 0, 1, 2, 3, 4, 5 0, 1, 2, 3, 4, 5 1, 2, 3, 4, 5, 6 1, 2, 3, 4, 5, 6
H+ + Nek+ → H0 + Ne(k+1)+H0 + Nek+ → H+ + Ne(k-1)+He+ + Nek+ → He0 + Ne(k+1)+He2+ + Nek+ → He+ + Ne(k+1)+He0 + Nek+ → He+ + Ne(k-1)+He+ + Nek+ → He2+ + Ne(k-1)+
0, 1, 2, 3, 4, 5, 6, 7 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 0, 1, 2, 3, 4, 5, 6, 7, 8, 91, 2, 3, 4, 5, 6, 7, 8, 9, 101, 2, 3, 4, 5, 6, 7, 8, 9, 10
H0, H+, He0, He+, He++ Li, Be, B, C, Ne
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
10-1 100 101 102 10310-19
10-18
10-17
10-16
10-15
10-14
Experiment M.B.Shah et al., J.Phys.B, 11, L233 (1978) W.Seim et al., J.Phys.B, 14, 3475 (1981)
Theory N.Toshima, Phys.Rev.A, 50, 3940 (1994) ARSENY CAPTURE CDW Approx. Formula
, c
m2
E/m, keV/a.m.u.
H0 + Li3+ → H+ + Li2+
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
20 40 60 80 100 120 140 160 180
101
102 = 0.88 = 0.85 = 0.82
H0 C
ount
Rat
e, a
.u.
E, keV
20 40 60 80 100 120 140 160 180
102
103 ECH Target plasma: Hydrogen ne(0) = 0.4x10
13 cm-3, Te(0) = 4.8 keV = 0.88 = 0.85 = 0.82
f i(E)
, a.u
.
E, keV
20 40 60 80 100 120 140 160 180
101
102 = 0.89 = 0.86 = 0.83
H0 C
ount
Rat
e, a
.u.
E, keV
20 40 60 80 100 120 140 160 180
102
103 = 0.89 = 0.86 = 0.83
f i(E)
, a.u
.
E, keV
ECH, NBI 1 Target plasma: Hydrogen ne(0) = 0.5x10
13 cm-3, Te(0) = 3.2 keV
Einj NBI 1
Local PCX Neutral Spectra and Calculated Proton Energy Distributions
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
Electric Charge State Changing Collisions of Hydrogen and Helium with Low-Z Impurity Particles
Part I. Charge Exchange Processes
I.Yu. Tolstikhina1, P.R. Goncharov2, T. Ozaki2, S. Sudo2, N. Tamura2 and V.Yu. Sergeev3
1 P.N. Lebedev Physical Institute, Moscow, Russia2 National Institute for Fusion Science, Toki, Gifu, Japan3 St.Petersburg Polytechnical University, St.Petersburg, Russia
NIFS-DATA-102April 2008
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
N. Stolterfoht, R. Cabrera-Trujillo, Y. Ohrn, E. Deumens, R. Hoekstra, and J. R. SabinPHYSICAL REVIEW LETTERS 99, 103201 (2007)
He2++ H, D, T → He+ + H +, D +, T + 100 eV/amu
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
Influence of the isotope effect on the charge exchange in slow collisions of Li, Be, C and W ions with H, D and T
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
Influence of the isotope effect on the low-temperature plasma characteristics
Isotope effect: 5 – 500 eV/amu
near-wall plasmas diverter plasmas
plasma facing components: Li, Be, C, W
neutralization population of excited levels
ions charge distributionradiative cooling particle transport
Li q+ + H, D, T Li (q-1)+ + H+, D+, T+
Be q+ + H, D, T Be (q-1)+ + H+, D+, T+
C q+ + H, D, T C (q-1)+ + H+, D+, T+
W q+ + H, D, T W (q-1)+ + H+, D+, T+
Inga Yu. Tolstikhina, Daiji Kato, and V. P. Shevelko, Phys. Rev. A 84 (2011) 012706Inga Yu. Tolstikhina et al., J. Phys. B: At. Mol. Opt. Phys., 45, (2012) 145201
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
Theoretical method: rotational transitionsin close collisions (Re[R] = 0)
Rmax=( l+ 1
2)
2
Z1 +Z 2
ϕ=arccos (sin χ2 + ρRmax cos χ2 )
R=ρcos χ
2
cos ϕ−sin χ2
scattering angle
internuclear axis rotation angle
for < Rmax and Rmax> Rclmb :
i ȧm−Em am +i ∑m'=−l
l⟨φnlm|
∂∂ t
|φnlm' ⟩ am'=0
χ=2 arctan ( Z1Z2μ ρ v2)R
clmb
Z2
Z1
Rmax
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
0.1 1
10-23
10-21
10-19
10-17
10-15 Li
3+ + H, D, T Li2+ + H+, D+, T+
, c
m2
E, keV/amu
T D H H** H*
* w/o PR
**Seim W. et al., J. Phys. B, 14, 3475 (1981)
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
0.1 110
-24
10-22
10-20
10-18
10-16
10-14
Be3+
+ H, D, T Be2+ + H+, D+, T+
, c
m2
E, keV/amu
T D H H*
*w/o PR
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
0.01 0.1 1
10-29
10-27
10-25
10-23
10-21
10-19
10-17
10-15
, c
m2
E, keV/amu
T D H H*
* w/o PR
C + H+, D
+, T
+ C+ + H, D, T
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
0.110
-21
10-20
10-19
10-18
10-17
10-16
10-15
T D H H* H**
* w/o PR
** experimentM. Imai, Kyoto University
W+ + H, D, T W + H+, D+, T+
, c
m2
E, keV/amu
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
W3++ He(1s2) → W2++ He(1s)
0 20 40 60 80 100 120
-5
-4
-3
-2
-1
n=6
n=11n=10n=9
n=8
n=7
final state W7+
E, a.u.
R, a.u.
initial state He(1s)
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
W3++ He(1s2) → W2++ He(1s)
0 50 100 150
2x10-15
4x10-15
6x10-15
8x10-15
K.Soejima, experiment total cross section, theory
(cm2)
E (eV/amu)
n=8
R. Cabrera-Trujillo, J. R. Sabin, Y.Ohrn, E. Deumens, and N. Stolterfoht PHYSICAL REVIEW A 83, 012715 (2011)
Coulomb potential and screened Coulomb potential are purely repulsive and can not reproduce negative scattering angles found in the END calculations
He2++ H, D, T → He+ + H +, D +, T +
Effect of the electron-nuclei interaction on the internuclear motion in slow ion-atom collisions
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
The internuclear motion should be described by the Born-Oppenheimer (BO)potential corresponding to the entrance collision channel
The interaction between two bare nuclei is described by the Coulomb potential
where R is the internuclear distance.
In the BO approximation the internuclear interaction is described by
where E(R) is the electronic energy in the field of the nuclei fixedin space at a distance R and E (R) = E0 .
Inga Yu. Tolstikhina and O. I. Tolstikhin, Phys. Rev. A 92 (2015) 042707
V C(R)=Z1 Z2
R,
V BO(R)=V C(R)+E (R)−E0 ,
Internuclear potential and trajectory
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
He2++ H, D, T → He+ + H +, D +, T +
Internuclear potential and trajectory
V BO(R)=V C(R)+E (R)−E0 ,
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
The classical scattering angle for a given internuclear potential V (R) as a function of the impact parameter ρ and collision energy E is given by
where energy of the internuclear motion in the center-of-mass frame, reduced mass of the nuclei, v is their initial relative velocity, distance of closest approach defined by the equation F(Rmin) = 0.
For the Coulomb potential we have
θ(R)=π−∫Rmin
∞ 2ρd RR2 F (R )
,
F (R)=√1− ρ2R2−V (R)E ,μ=M 1 M 2/(M1+M 2)Rmin
θC(ρ , E)=2arctan ( Z1 Z22ρ E )
E=μ v2/2
Internuclear potential and trajectory
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
Internuclear potential and trajectory
He2++ H, D, T → He+ + H +, D +, T +
Collision energy 50 eV/amu
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
He2++ H, D, T → He+ + H +, D +, T +
CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH HYDROGEN ISOTOPES. ADIABATIC APPROACH
Internuclear potential and trajectory
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