Q E D Effects in Atoms
48 CHAPTER 3 QED EFFECTS IN ATOMS is given by
<P;1(r) =
J
eii5r( f ( -fi) -
1 ) <Pc(p) z e 1joo f (
-v )
- 1 ipr d = - m -.::2 e p p 1fr - P= Ze
loa
l m f(O - 1 eVEr d�1fr o �
(3.25)
After angular i ntegration a nd sufficient changes of variables, the i magi nary part of the electric form factor of the electron [1 0 1]
ca n be i nserted. The fi nal i ntegration gives <P;I(r)
= -
a <Pc (r) 1floa� [(
1 -1 t2 - 1 t
X
[
log ( t2 - 1 ) + log�
+ t12]
e-2rmt(3.26)
(3.27) Here, A. is a low-frequency cut-off parameter. To reproduce the results of ref. [102, 103] . a suitable replacement for the second logarith m is 4 1og( 1/Za + 0.5) where A. was selected [24] of the order of the electron binding energy A. rv (Zafm. The electrica l potentia l is fu rthermore not applicable for very small distances r « Zarc. This is taken i nto account by a small distance cut-off coefficient
mr /(mr + 0.07 Z2a2 ) . The final expression for the electric form-factor reads
with
a
J
e-2trm[ (
1)
<Pe1(r) = - A(Z, r) -<Pc( r) dt Jt2=1 1 - -2 2
1f 1 t2 - 1 t
x
[
log ( t2 - 1 ) + 4 1og(
Z1a + 0. 5)]
-�
+ .mr
A (Z, r
)
= An(Z) mr+ 0.07 Z2a2 .
(3.28)
(3.29)
A consistent calculation of the low-frequency contribution to the nonlocal self energy operator usi ng Coulom b or para metric G reen 's fun ctions is a complicated task [24] . lt is much easier and a lso sufficient to fit this contribution usi ng a
3. 3. COMPUTATIONAL DETAILS 49
para metric potential <t>1ow(
r
). To reprod uce the p-level radiative energy shifts[102, 103] . the following expression ca n be used
(3.30) Here a8 is the Bohr radius a nd is given by B(Z)
=
0.074 + 0.35Za .3 . 3 Com putational deta ils
Dirac-Hartree- Fock ( D H F ) calculations were carried out using the D i rac-Cou lomb H a m i ltonian ( i n atomic un1ts) , as descri bed before
(3.31)
with a a nd {3 denoting the Dirac matrices in the sta ndard D i rac representa tion . The equations were solved numerica l ly using a modified code of the pro g ram system G RASP [104] . For heavy elements, the 1s-shel l rad i us is very sma ll
(
(rhs
:::::: 500 fm ) and as a consequence, the i nfl uence of the fin ite nuclear sizeis an im porta nt contribution to the total energy. The electrostatic potential of the nucleus Vnuc was therefore m odeled by a two-para meter Ferm i-type charge d istri bution [105, 106]
Q( r)
=Qo
,
1
+
exp [(r- a)/b] where a a nd b were extracted from ref. [105].(3.32) Due to the sma l l size of the quantum electrodynamic (QED) corrections, they are treated as a perturbation [107] . A fu lly self-consistent im plementation of these effects within the GRAS P code is currently i n progress. The m ajor correction to the non-relativistic Cou lomb term stems from the Breit operator, which was treated within the Coulomb-gauge [96, 108, 109]
fi.1
·fi2
. _ - _ - exp(iw12
r12
) - 19w.c( 1 , 2) = -
exp(tw12 r12 ) -
(a1
· V'I)(a2
·\72) ---,2,---
rl2
W12 r12
5
0
CHAPTER 3. QED EFFECTS IN A TOMS where w12 is the energy of the virtual (transversal) excha nge photon . The first term is the retarded G aunt term (G I) and the second term arises from the choice of the Coulom b gauge i nstead of the Feynman ga uge. This is known as the re tarded ga uge term . This correction to the electron-electron i nteraction a ccou nts for magnetic i nteraction and retardation to the order a2 , a nd only i ncludes theexcha nge of a si ngle (left-rig ht), virtual photon as depicted i n Fig. 3. 1 .
The (other) rad iative corrections are ca lculated by a non loca l radiative potenti a l , which i s split i nto a n self-energy a n d a vacuu m polarization part
!J. Evp = (W I <Pvp ( r ) I W)
� (W I <Pu ( r )
+
<PwK ( r ) + <PKs( r ) I W) . (3.34)This i m pl ies t hat the energy shift is calculated as an expectation va lue of a ra diative one-electron potential using the eigenfunctions of the D H F operator. I n genera l the rad iative perturbation i s a series expansion i n the two para meters a a nd Za, where the powers of a descri be the order of the Q E D corrections a nd Z a describes the order of relativistic corrections to the energy levels
[107].
lt is known that the latter expansion works quite wel l for lighter elements, but it is less than clear how well it works for elements with high nuclear charge such as the su perheavy elements whereZ
a :S1.
For the vacuum polarization the potential is well known . By util izing perturbation theory for the polarization operator
P(
-p2 ) , the energy contribution of lowest order ( a ( Za ) ) is g iven by the Ueh l i ng potentia l (3.22) , where the virtual electron positron pair is a llowed to propagate freely (Fig . 3.2). The Uehling term g ives typica l ly more than90%
of the VP i n hydrogen- like atoms. I n presence of the nuclear Coulomb field the electron and positron wave functions become d istorted . Wich mann a nd K roll[110]
have considered the vacuu m polarization of order a in a strong Coulo m b field (Fig. 3.2) and have shown that the polarization charge density is an a n a lytic function of Za for IZa:l :S1,
a ( Za:?
[(
3 1r27) 1
<PwK ( r)=
- -((3)+ - - - -+
27r((3) 1r 26 9
r - - + -6((3) + - - - r+ O ( r2 ) .
1f3(
1f4 1f2)
]
416
6 (3.35)3. 3. COMPUTA TIONAL DETAILS 51 form . Therefore, j ust the crucial Feyn man diagrams are presented i n Fig . 3.3.
Figure 3 . 2 : Feyn m a n d i agrams for vacuum polarization o f order a ( Za ) Uehling ( l eft) and of order a ( Za )3 Wich mann and K roll ( right) .
Figure 3 . 3 : Vac u u m polarization of order a2 (Za) ( Ka l len-Sabry ) .
The calculation of the self-energy operator L:( r, r',
E)
shown i n Fig. 3.4 is more com plicated and rather tedious. The problem ca n, however, be divided i nto two parts. I n the first part the electron interacts with a high frequency virtual photon where the nuclear Coulomb field needs to be i ncluded only in fi rst order. The second part represents the interaction with a low freq uency photon .52 CHA PTER 3. QED EFFEC TS IN ATOMS