CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH … · 2016. 12. 16. · Adiabatic theory of transitions in slow collisions, E. Solov’ev Sov. Phys. Uspekhi, 32, 1989 Li3++ H(1s)

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


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


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 )|


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


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 (∞ ))


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


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+


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


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


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+


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


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


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


Influence of the isotope effect on the charge exchange in slow collisions of Li, Be, C and W ions with H, D and T


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


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


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)


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


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


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


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)


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


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


He2++ H, D, T → He+ + H +, D +, T +


V BO(R)=V C(R)+E (R)−E0 ,


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




He2++ H, D, T → He+ + H +, D +, T +

Collision energy 50 eV/amu


He2++ H, D, T → He+ + H +, D +, T +



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CHARGE EXCHANGE IN SLOW COLLISIONS OF IONS WITH … · 2016. 12. 16. · Adiabatic theory of transitions in slow collisions, E. Solov’ev Sov. Phys. Uspekhi, 32, 1989 Li3++ H(1s)

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