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Department of Physics St. Petersburg State University

Relativistic calculation of inner-shell atomic processes in low-energy ion-atom collisions
Yury Kozhedub, Ilya Tupitsyn and Vladimir Shabaev

Outline Intoduction and Motivation Theoretical Description and Numerical Results One-electron case Many-electron case Summary and Outlook


Introduction
Heavy few-electron ions provides possibility to test of QED at extremely strong electromagnetic fields

maximum E field in petawatt laser pulses


Introduction
Point nucleus: E 1s= mc 2 1 -( Z )2 . Existence of point charge with value more than Z > 1 is forbidden [GДtner et al., 1981].

The 1s level dives into the negative-energy continuum at Zcrit ~173 [S.S. Gershtein, Ya.B. Zeldovich, 1969; W. Pieper, W. Greiner, 1969] .


Introduction: super-heavy quasi-molecules
Super-critical field could be achieved in collision of two heavy ions with Z1+Z2 >173.

Diving time period is about 10-21 sec. Spontaneous e+e- pair creation time is about 10

-19

sec [MЭller et al., 1972].


Time-dependent equation
Features of the investigated process: Low-energy ions: ~ 6 MeV/u for U Relativistic electron: ve ~ (Z )c me « M
nucl

Nuclei (RA, RB) move according to the Rutherford trajectory

The time-dependent many-electron two-center Dirac equation (in a.u.):

i

d ( x1 , x 2 , ... , x N , t ) dt

=H ( x 1 , x 2 , ... , x N , t ) ( x 1 , x 2 , ... , x N , t ) ,
( x i , x j) ,

H=


i

hD ( x i ) +

1 2


i j

V

e- e

D 2 h = c ( p ) + mc + V

AB

( ) , r

V

AB

( )= V r

( A) nucl

( r A )+ V

( B) nuc l

(r B) ,

where

,

r are the Dirac matrices, and rA= - R A ,

r r B= - R B

.


One-electron case
The time-dependent one-electron two-center Dirac equations (in a.u.):
i d D =h ( , t ) , r dt h = c ( ) + mc + V p
D 2 ( A) nucl

( r A )+ V

(B ) nucl

(r B)

The coupled-channel approach:

( , t )= r



i

C i (t ) i ( ) r

{

i



j

S

dC j ( t )
ij

dt
t -

=



j

( H ij - T ij ) C j ( t )
0

lim C (t )=C

D H ij = ih j , T ij =i i j , S i j = i j . t


Finite Basis Expansion
( , t )= r



= A , B



C

,

(t )

,

( - R ( t )) r



,

- the Dirac and Dirac-Sturm orbitals, localized on each ion.



nkm

( , )= r

(

P nk ( r ) r i Q nk ( r ) r



km

( , ) ( , )



- km

)

;

( j + 1 / 2) 1k j =k-1 / 2, l = j + 2 k

k =(-1 )l

+ j +1 / 2

The large Pnk and small Qnk radial components are obtained by solving numerically the Dirac equation in the center field potential V(r)

{

c- c

( (

dk 2 + Q nk ( r ) +( V ( r ) + c ) P nk ( r )= nk P nk ( r ) dr r

dk + P nk ( r )+( V ( r )- c 2 ) Q nk ( r )= nk Q nk ( r ) dr r

) )


Monopole approximation
Monopole approximation enables partly accounting for the potential of the second ion in constructing the basis functions. For example, the potential of the center A is given by
V
A

r = V

A nucl

r V

B mon

r ,

where (for the point nucleus case)
1 4 ZB - r R AB ZB r d = AB ZB - R r - r R AB R AB

V

B mon

r =-



{


Energies of the 1+ ground state
The 1+ state energy of the U the internuclear distance R. Rcrit(1s) =25.5 fm Rcrit(1+)=34.7 fm Rnucl = 5.9 fm
2 183 +

of quasimolecules

quasimolecule as a function of

I.I. Tupitsyn, Y.S. Kozhedub et al., PRA 2010


Critical Distances
Critical Distances Rc (fm)
Point nucleus Z 88 90 92 This work 24.27 30.96 38.43 Others 24.24a 30.96a 38.4b 38.42a 36.8 94 96 98 46.58 55.38 64.79
c a a a

Extended nucleus This work 19.91 27.06 34.74 Others 19.4 26.5 34.7 34.3 42.6
d d b d f

34.7 43.13 52.10 61.61

46.57 55.37 64.79

d

61.0d 61.1f

V. Lisin et al., PRL 1977 c J. Rafelski and B. MЭller, PL 1976 f B. MЭller and W. Greiner, ZN 1975
a

b

A. Artemyev et al., JPB 2010 d V. Lisin et al., PL 1980

I.I. Tupitsyn, Y.S. Kozhedub et al., PRA 2010


U (1s)-U

91+

92+

E=6.0 MeV/u

Charge-transfer probability as a function of the impact parameter b I.I. Tupitsyn, Y.S. Kozhedub et al., PRA 2010


Many-electron case
i d ( x1 ,... , x N , t ) dt =H eff ( x 1 , ... , x N , t ) ( x 1 , ... , x N , t )
H eff = h
i eff i

Independent particle model:

{

t -

dt 0 lim ( i ( t )- i ( t ))=0
1 N!

i

d i (t )

= heff i ( t ) i

( x 1 , ... , x N )=





1 ( x 1 ) N ( x 1 )

1 ( x N ) N ( x N )




Dirac-Kohn-Sham equation
Dirac-Kohn-Sham equation

d i =h dt ( ) r

DKS

( , t ) r

hDKS=c ( ) + mc 2 + V p V

AB

AB

( )=V H [ ]+ V xc [ ] r V H [ ]=V
A nucl

( r A) + V
3

B nucl

( r B )+V C [ ]

V

nucl

3 nucl ( r ' ) ( )= d r ' r - r ' r

(r ' ) V C [ ]= d r ' - r ' r

V xc [ ] is the exchange-correlation potential
in the Perdew-Zunger parametrization Perdew and Zunger, PRB 23, 5048 (1981)


Inclusive probability
P P
f 1 ,..., f

N

= i ( x 1 , ... , x N , t = ) f ( x 1 , ... , x N ) =det (
nn '

2

f 1 ,..., f

N

) n , n ' = 1, ... , N = f nf n '
( x , x ' )=

nn '

i
i

N

( x , t = ) i ( x ' , t = )

P P

f 1 ,..., f

q

=
f


q+ 1

P
N

< ...< f

f 1 ,... , f

N

q
f 1 ,..., f

q

=det (

nn '

) n , n ' = 1, ... , q


Numerical calculation
Test system: Ne(1s22s22p6) - F8+(1s) Ne(1s22s22p6) - F6+(1s22s) S. Hagmann et al., PRA 1982; 1986; 1987 W. Fritsch and C.D. Lin, PRA 1985 A. Toepfer et al., PLA 1987 B. Thies et al., PLA 1989

Experiment:

Theoretical calculations:

I.I. Tupitsyn, Y.S. Kozhedub et al., PRA 2012


Ne(1s 2s 2p )-F8+(1s)
230 keV/u

2

2

6

The probability of Ne K-shell-vacancy production as a function of the impact parameter b I.I. Tupitsyn, Y.S. Kozhedub et al., PRA 2012


Ne(1s 2s 2p )-F8+(1s)
130 keV/u

2

2

6

The probability of Ne K-shell-vacancy production as a function of the impact parameter b I.I. Tupitsyn, Y.S. Kozhedub et al., PRA 2012


Ne(1s 2s 2p )-F (1s 2s)
6+

2

2

6

2

The probability of Ne K-shell-vacancy production as a function of the impact parameter b


Xe-Xe53+(1s)

The probability of Xe K-shell-vacancy production as a function of the impact parameter b I.I. Tupitsyn, Y.S. Kozhedub et al., PRA 2012


Xe-Xe53+(1s)

The probability of Xe K-shell-vacancy production as a function of the impact parameter b I.I. Tupitsyn, Y.S. Kozhedub et al., PRA 2012


Summary

A new method employing the Dirac-Sturm (Dirac-Fock-Sturm) basis functions for evaluation of various electron-excitation processes in low-energy heavy-ion collisions has been developed Systematic calculations of inner-shell atomic processes in low-energy ion-atom collisions have been carried out Relativistic effects are investigated


Collaboration

Siegbert Hagmann Alexandre Gumberidze Christophor Kozhuharov Thomas StЖhlker

GЭnter Plunien

Andrey Surzhykov Anton Artemyev


Many thanks to

Thank you very much for your attention!