Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.gao.spb.ru/russian/psas/ffk2013/tfiles/shabaev_ffk_13.pdf Äàòà èçìåíåíèÿ: Mon Nov 25 12:35:09 2013 Äàòà èíäåêñèðîâàíèÿ: Fri Feb 28 03:18:11 2014 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï

Model operator approach to calculations of the Lamb shifts in relativistic many-electron atoms
Vladimir Shabaeva in collaboration with Ilya Tupitsyna and Vladimir Yerokhinb
a b

St. Petersburg State University

St. Petersburg State Polytechnical University

FFK, Oct 7-11, 2013 ­ p.1/26


Outline of the talk
· Introduction · Per turbation theory for the QED calculations in the Furry picture · SchrÆdinger-like equation for a relativistic atom in the framework of QED · Lowest-order approximation: Dirac-Coulomb-Breit Hamiltonian

· Model operator approach to the Lamb shift in many-electron atoms · Calculations with the model self-energy operator in many-electron systems · Conclusion

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Introduction
Quantum electrodynamics in the external field approximation (Furry picture of QED) High-Z few-electron ions N Z, where Z is the nuclear charge number and N is the number of electrons. To zeroth-order approximation: (-i + m + VC (r ) ) (r) = (r) Interelectronic-interaction and QED effects: Interelectronic interaction 1 , Binding energy Z In uranium: Z = 92, Z 0.7. QE D ( Z )2 . Binding energy

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Introduction
Relativistic many-electron atoms and ions The interelectronic interaction is not small and must be taken into account at the zero-order level: VC Veff = VC + Vscr , where Vscr describes approximately the electron-electron interaction effects. Therefore, to zeroth order: (-i + m + Veff (r ) ) (r) = (r)

In higher orders, besides the interelectronic-interaction and QED effects, one must add the interaction with -Vscr .
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Green function
Standard (2N-time) QED Green function for an N-electron atom:
G(x1 , . . . x ; x1 , . . . xN ) = 0|T (x ) · · · (x ) (xN ) · · · (x1 )|0 , N 1 N

where x = (t, x), (x) is the electron-positron field operator in the Heisenberg representation, and (x) = 0 . In the interaction representation: G(x , . . . x ; x1 , . . . xN ) 1 N = where m e µ [ in (x), in (x)] HI (x) = [ in (x)µ , in (x)]Ain (x) - 2 2 is the interaction Hamiltonian.
FFK, Oct 7-11, 2013 ­ p.5/26

0|T in (x ) · · · in (x ) in (xN ) · · · in (x1 ) exp {-i 1 N 0|T exp {-i d4 z HI (z )}|0

d4 z HI (z )}|0

,


Two-time Green function
We introduce the two-time Green function: G(t , t) G(t = t = · · · t t ; t1 = t2 = · · · tN t) 1 2 N

t = t = · · · = t t 1 2 N The Fourier transform: 11 G (E ) (E - E ) = 2 i N !
-



t1 = t2 = · · · = tN t

dt dt exp (iE t - iE t)G(t , t) .

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Per turbation theor y for quasidegenerate levels
We consider s degenerate or quasidegenerate states. The projector on the unper turbed states:
s s

P

(0 )

=
k =1

P

(0 ) k

=
k =1

uk uk .

We project G (E ) on the space s formed by the s unper turbed states: g (E ) = P
(0 )

(0 )

G (E )P

(0 )

.

We consider a contour in the complex E plane: r r r rr

r

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Per turbation theor y for quasidegenerate levels
We introduce operators K and P by K 1 2 i d E E g (E ) ,


P

1 2 i

d E g (E ) .


The energies and the wavefunctions are determined from the equations: K vk = Ek P vk , The solvability condition yields: d e t (K - E P ) = 0 .
vk P vk = k k

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¨ Schrodinger-like equation for a relativistic atom
These equations can be transformed to the SchrÆdinger-like equation: H k = Ek k ,
1 1

k k =
1

kk



,

where H P - 2 K P - 2 and k P 2 vk . The energy levels are determined from the equation: det(H - E ) = 0 . The space of the quasidegenerate states can be extended to the space + that includes all positive-energy states whose energies are smaller than the pair-creation threashold: r r r rrr E
(0 ) (0 )

E

(0 )

+ 2mc

2

In this picture the photon spectra are omitted.
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¨ Schrodinger-like equation for a relativistic atom
The operators K and P are constructed by per turbation theory: K P The operator H is H=H where H H H
(0 ) (1 ) (0 )

= =

K P

(0 )

+K +P

(1 )

+K

(2 )

(0 )

(1 )

+P

(2 )

+ ··· .

+ ··· ,

+H

(1 )

+H

(2 )

+ ··· ,

= = =

K K K

(0 ) (1 )

, 1 -P 2 1 -P 2
(1 )

(2 )

(2 )

3 +P 8

1 - K (0 ) P (1 ) , 2 1 (0 ) (2 ) 1 (1 ) (1 ) 1 (1 ) (2 ) (0 ) K -KP -PK -KP 2 2 2 3 (0 ) (1 ) (1 ) 1 (1 ) (0 ) (1 ) (1 ) (0 ) PK +KPP +PKP . 8 4
(1 )

K

(0 )

(1 )

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Interelectronic-interaction operator
One-photon exchange contribution to the Hamiltonian H :

In the space + we get
( i , j , k , l > 0 )

(0 )

hi

nt

=
i = j,k = l

1 |i j i j | [I (i - k ) + I (j - l )]|k l k l | , 2

where I ( ) = e2 D ( ) , 0 = (1, ), D ( ) is the photon propagator, and i is the one-electron Dirac energy.
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Dirac-Coulomb-Breit Hamiltonian
Taking hint in the Coulomb gauge at zero energy transfer ( = i - k = 0) and summing over atomic electrons leads to the Dirac-Coulomb-Breit Hamiltonian: H=
(+) (+) i

hD + i
i< j

(ViC + ViB ) j j

(+)

,

where

is the projector on the positive-energy states, Z VC (r ) = - r ,

hD = i · pi + mi + VC (ri ) , i

· 1 ViC = rj , ViB = - irij j + 2 (i · i )(j · j )rij . j j i To account for the nonzero energy transfer, one should simply replace ViC + ViB by the interelectronic-interaction operator hint derived above. j j In the Feynman gauge, to get the Hamiltonian to the same accuracy, one has to account for the higher-order photon exchange diagrams.
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Lamb shift operator for a relativistic atom
The QED contributions to the Hamiltonian H :

hQ

ED

= =

hSE + hV
( i , k > 0 )

P

i ,k

|i i |

1 SE ( (i ) + SE (k )) + V 2

VP

|k k | ,

where SE (i ) and V VP are the renormalized self-energy (SE) and vacuum-polarization (VP) operators, respectively. Details of the two-time Green function method and the derivation of these formulas can be found in [V.M. Shabaev, Phys. Rep., 2002; JPB, 1993].
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Lamb shift operator for a relativistic atom
The dominant par t of the VP contribution is represented by the Uehling potential: 2 - Z 3
2

VU

ehl

(r ) =

dr 4 r (r )
0 1

t2 - 1 1 d t (1 + 2 ) 2t t2

[exp (-2m(c/ )|r - r |t) - exp (-2m(c/ )(r + r )t)] , â 4mr t where is the fine structure constant and |e|Z (r ) is the density of the nuclear charge distribution ( (r )dr = 1). Evaluation of the remaining Wichmann-Kroll potential is a much more difficult problem [G. Soff and P.J. Mohr, PRA, 1988; N.L. Manakov et al., JETP, 1989]. To a good accuracy, it can be calculated with the help of the approximate formulas derived in [A.G. Fainshten et al., JPB, 1991].

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Model self-energy operator for a relativistic atom
Let us now consider the SE operator:
( i , k > 0 ) SE

h

=
i ,k

|i

1 SE i | [ (i ) + SE (k )]|k k | . 2

We represent hSE as a sum of local and nonlocal par ts. The local par t is given by
E VlSc = o

A exp (-r /C )P ,

where P is the projector on the states with the given value of = (-1)j +l+1/2 (j + 1/2), the constant A is chosen to reproduce the SE shift for the lowest energy level at the given in the corresponding H-like ion, and C = /(mc).

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Model self-energy operator for a relativistic atom
SE We restrict the active space of the remaining SE operator, hSE - Vloc , to to the basis functions {i (r)}n 1 which, having the same angular i= par ts as the H-like functions {i (r)}n 1 , are localized at smaller i= distances. With these functions, we approximate the one-electron SE operator as follows n

hSE = Vl

SE oc

+
i ,k = 1

| i B

ik

k | ,

where the matrix Bik has to be determined to reproduce the diagonal and non-diagonal SE corrections with the H-like wave functions. This leads to the equations
n

j,l = 1

i |j B

jl

l |k 1 E ((i ) + (k )) - VlSc |k . o 2
FFK, Oct 7-11, 2013 ­ p.16/26

= i |


Model self-energy operator for a relativistic atom
Now let us consider the choice of the functions {i (r)}n 1 . We i= construct them using the H-like wave functions multiplied with the factor l (r ) = exp (-2Z (r /C )/(1 + l)), where l = | + 1/2| - 1/2 is the orbital angular momentum of the state under consideration. In what follows, we restrict the basis functions by ns, np1/2 , np3/2 , nd3/2 , and nd5/2 states with the principal quantum number n 3 for the s states and n 4 for the p and d states, and put 1 i (r) = (I - (-1)si )li (r )i (r) , 2 where I is the identity matrix, is the standard Dirac matrix, the index si = ni - li enumerates the positive energy states at the given , and ni is the principal quantum number.
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Model self-energy operator for a relativistic atom
Thus, the model SE operator is given by hS
E

=

Vl

SE oc

+

1 4

i ,k

j,l

(I - (-1)si )li (r )|i
SE oc

1 â((D ) )ij j | ((j ) + (l )) - Vl 2 â(D -1 )lk k |lk (r )(I - (-1)sk ),
t -1

|l

where the summations run over ns states with the principal quantum number n 3 and over np1/2 , np3/2 , nd3/2 , and nd5/2 states with n 4, li (r ) D and si = ni - li .
FFK, Oct 7-11, 2013 ­ p.18/26

= =

ik

exp (-2Z (r /C )/(1 + li )) , 1 i |(I - (-1)si )li (r )|k , 2


Self-energy matrix elements with H-like wave functions
To complete the construction of the model SE operator, one needs to 1 evaluate the matrix elements ik i | 2 ((i ) + (k ))|k with the H-like wave functions. To perform such calculations we used the method described in [V.A. Yerokhin and V.M. Shabaev, PRA, 1999; V.A. Yerokhin, K. Pachucki, and V.M. Shabaev, PRA, 2005]. The results of the calculations are conveniently expressed in terms of the function Fik (Z ) defined by i
k

( Z )4 1 i | [(i ) + (k )]|k = Fik (Z )mc2 , 2 (ni nk )3/2

where ni and nk are the principal quantum numbers of the i and k states, respectively. For the diagonal matrix elements, these results are in a good agreement with the calculations performed in [P.J. Mohr,
PRA, 1992; P.J. Mohr and Y.-K. Kim, PRA, 1992; T. Beier, P.J. Mohr, H. Persson, and G. Soff, PRA, 1998].

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Self-energy matrix elements with H-like wave functions
Self-energy matrix elements Fik (Z ), defined by wave functions for extended nuclei. Z 10 30 60 90 120 F
1s 1s
4 ( Z ) (ni nk )3/2

ik

1 i | 2

[(i ) + (k )]|k = F
2s 2s

Fik (Z )mc2 , with H-like F F

F

3s 3s

F

1s 2s

1s 3s

2s 3s

4 .6 5 4 2 2 .5 5 1 8 1 .6 8 2 0 1 .4 7 2 1 1 .7 3 3 5

4 .8 9 4 4 2 .8 3 8 6 2 .0 9 2 3 2 .1 4 3 1 3 .1 2 5 6

4 .9 5 2 4 2 .8 9 3 7 2 .1 4 1 0 2 .1 7 0 2 3 .0 2 9 5

4 .7 9 6 1 2 .7 0 8 4 1 .8 7 9 5 1 .7 6 1 5 2 .2 7 5 3

4 .8 1 4 5 2 .7 2 3 5 1 .8 8 8 6 1 .7 6 0 7 2 .2 2 9 4

4 .9 3 2 5 2 .8 7 4 8 2 .1 2 4 2 2 .1 6 2 5 3 .1 1 2 5
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Calculations with the model SE operator
To demonstrate the efficiency of the method, we applied it to calculations of the Lamb shifts in neutral alkali metals, Cu-like ions, superheavy atoms, and Li-like ions. Ab initio calculations of the Lamb shift in alkali metals were performed [J. Sapirstein and K.T. Cheng, PRA, 2002] in the potential U (r ): Zeff (r ) V (r ) = - , r where


Zeff (r ) = Z

nuc

(r ) - r

0

81 1 t (r ) + x r t dr 2 r> 32


1/3

and t = v + c is total (valence plus core) electron charge density. The choice x = 0 corresponds to the Dirac-Har tree potential, x = 2/3 gives the Kohn-Sham potential, and x = 1 is the Dirac-Slater potential.
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Self energy in neutral alkali metals
Self-energy function F (Z ), defined by E neutral alkali metals in different potentials. Ato m Na 3s
1/2 SE

=

( Z ) n3

4

F (Z )mc2 , for

Method
SE v |Vloc |v v | H SE | v Exacta SE v |Vloc |v v | H SE | v Exacta SE v |Vloc |v v | H SE | v Exacta

x = 0 0 .1 6 6 0 .1 7 0 0 .1 6 9 0 .0 1 8 7 0 .0 2 2 9 0 .0 2 2 8 0 .0 0 4 7 0 .0 0 6 9 0 .0 0 6 8

x = 1/3 0 .1 6 3 0 .1 6 8 0 .1 6 7 0 .0 1 9 3 0 .0 2 3 7 0 .0 2 3 6 0 .0 0 5 2 0 .0 0 7 6 0 .0 0 7 5

x = 2/3 0 .1 7 6 0 .1 8 3 0 .1 8 1 0 .0 2 3 0 0 .0 2 8 4 0 .0 2 8 3 0 .0 0 6 7 0 .0 0 9 9 0 .0 0 9 8

x = 1 0 .2 1 4 0 .2 2 4 0 .2 2 3 0 .0 3 2 0 0 .0 3 9 7 0 .0 3 9 6 0 .0 1 0 2 0 .0 1 5 1 0 .0 1 5 0
FFK, Oct 7-11, 2013 ­ p.22/26

Rb 5s

1/2

Fr 7s

1/2

a

J. Sapirstein and K.T. Cheng, PRA, 2002.


Self energy in Cu-like ions
Self-energy contribution to the 4s - 4p energies in Cu-like ions, in eV. Io n Yb 4 1 + Transition 4s - 4p1/2 4s - 4p3/2 4p1/2 - 4d3/ 4p3/2 - 4d3/ 4p3/2 - 4d5/ 4s - 4p1/2 4s - 4p3/2 4p1/2 - 4d3/ 4p3/2 - 4d3/ 4p3/2 - 4d5/
1/2

and 4s - 4p

3/2

transition

2 2 2

U6

3+

2 2 2

Model SE operator -1.29 -1.21 -0.10 -0.18 -0.14 -4.24 -4.32 -0.87 -0.79 -0.63

Exacta -1.28 -1.21 -0.11 -0.18 -0.14 -4.24 -4.33 -0.88 -0.79 -0.65

a

J. Sapirstein and K.T. Cheng, PRA, 2002.

FFK, Oct 7-11, 2013 ­ p.23/26


Self energy in superheavy atoms
Self-energy contribution to the binding energy of the valence electrons in Rg and Cn, in eV. In this work, the per turbation theory (PT) value is obtained by averaging the model SE potential with the Dirac-Fock wave function of the valence electron, while the DF value is obtained by including this potential into the DF equations. Ato m Rg Valence electron 7s Method PT DF Welton meth. Local SE pot. PT DF This work -0.088 -0.105
I. Goidenko, EPJD, 2009

-0.089 -0.102

Ot h e r works -0.087a -0.084b -0.089c

Cn
a

7s

-0.101 -0.105

-0.103 -0.110

L. Labzowsky et al., PRA, 1999; b P. Indelicato et al., EPJD, 2007; c C. Thierfelder and P. Schwerdtfeger, PRA, 2010.
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Screened self energy in Li-like ions
Screened self energy for the 2s, 2p1/2 , and 2p3/2 states of Li-like ions, in eV. The Kohn-Sham (KS) and Dirac-Fock (DF) results are obtained using the model SE potential approach. Z 20 Sta te 2s 2p1/2 2p3/2 2s 2p1/2 2p3/2 2s 2p1/2 2p3/2 KS -0.047 -0.009 -0.012 -0.50 -0.13 -0.14 -2.35 -0.97 -0.65
b

50

83

DF -0.045 -0.008 -0.011 -0.49 -0.12 -0.14 -2.25 -0.98 -0.61

PTa -0.044 -0.008 -0.013 -0.48 -0.12 -0.16 -2.32 -1.07 -0.75

PTb -0.046 -0.008 -0.013

-2.26 -1.07 -0.76

a

Y.S. Kozhedub et al., PRA, 2010;

J. Sapirstein and K.T. Cheng, PRA, 2011.

FFK, Oct 7-11, 2013 ­ p.25/26


Conclusion

· SchrÆdinger-like equation for a relativistic many-electron atom can be derived from the first principles of QED by the two-time Green function method. · The QED contribution can be approximated by a model operator, which provides a very simple and efficient tool for evaluation of the Lamb shifts in many-electron atoms and ions [V. M. Shabaev, I. I. Tupitsyn, and V. A. Yerokhin, Phys. Rev. A 88, 012513 (2013)].

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