Radiative corrections in highly ionized atoms

Nuclear Instruments and Methods North-Holland, Amsterdam

Section II. Fundamental

in Physics

problems

RADIATIVE CORRECTIONS

Research

459

B9 (1985) 459-464

in atomic structure IN HIGHLY IONIZED ATOMS

Peter J. MOHR J. W. Gibbs Laboratory,

Physics Department,

Yale Uniuersity, New Haven, CT 06511, USA

A review of the theory of radiative corrections in highly ionized atoms is given. Comparison of theory and experiment in these atoms provides an important test of the theory of strongly bound electrons. For one-electron atoms, energy level separations predicted by quantum electrodynamics among the n = 1 and n = 2 states of hydrogen-like atoms with nuclear charge Z in the range lo-40 have recently been tabulated on the basis of a complete nonperturbative calculation of the self energy. In highly ionized atoms with more than one electron, the hydrogenic radiative level shifts provide a first approximation to the few-electron radiative level shifts. Uncalculated corrections due to multielectron effects in the radiative corrections give the largest known uncertainty in the theory of two-electronatoms. Various approaches to this problem and the prospects for further improvements are described.

1. Introduction This paper is a review of radiative corrections in highly ionized atoms. It is organized in three parts. The first part is a review of the basic formalism. The second part concerns tests of QED in highly ionized one-electron atoms. And finally, something will be said about two-electron tests in highly ionized atoms.

2. Basic formalism The next few formulas summarize external field quantum electrodynamics in the Furry picture [l]. One starts with a field operator that is expanded in destruction operators for electrons b,, and creation operators for positrons d,* multiplied by wavefunctions that are solutions of the Dirac equation, with positive energy and negative energy, respectively: G(x)

= C&&,(x) lI+

+ Cd,*@“,(X), n-

where +n(x)=+,(x)exp(-iE,,t/tz), and &,(x) solution of the coordinate space Dirac equation [-itica.V

+ V(x)+j3pmc2-&]&(x)=0.

(1) is a (2)

One of the calculational difficulties is the fact that these expressions involve the bound state electron propagator, which is more complicated than the free electron propagator. An approach employed by Wichmann and Kroll [4] and Brown et al. [S] in calculations is to write the propagator as a contour integral over the Dirac Coulomb Green’s function which in turn is written as a sum over eigenfunctions of angular momentum. The calculations then require evaluation of the radial Green’s functions, which has been done by various methods. For example, the self energy represented by the Feynman diagram in fig. la has been evaluated with this general approach. Cheng and Johnson [6] and Soff et al. [7] have done calculations for both the potential of a finite size nucleus and the Coulomb potential by numerically integrating the differential equation satisfied by the radial Green’s functions. I have done a Coulomb evaluation based on power series and asymptotic expansions of the radial Green’s functions [8]. In another context, Drake has approximated the radial Green’s functions with a discrete basis method [9]. Erickson has done an approximate analytic evaluation of the self energy based on a power series in Za [lo]. I have done a numerical Coulomb calculation for the n = 2 states [ll].

The interaction Hamiltonian H,(x)

= -&[?(x)Y, - ;smc*[S(x),

#(x)&(x) $(x)1

(3)

contains the coupling between the radiation field A,, and the current of the electron-positron field, and a mass renormalization term. That interaction determines the radiative level shifts, and explicit expressions are obtained by applying the prescription of Gell-Mann and Low [2] and Sucher [3] that relates energy level shifts AE,, to S-matrix elements or Feynman diagrams. 0168-583X/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(0)

(b)

Fig. 1. Feynman diagrams for the lowest order (a) self energy and (b) vacuum polarization. II. ATOMIC

STRUCTURE

P.J. Mohr / Radtatiue corrections in high!v ionized atom

460

3. One-electron atoms To test the basic theory, we compare theory and experiment in this section. The separations that have been measured accurately at high Z are the Lamb shift, which is the difference in energy between the 2S,,, and from the 2P 2P, ,z states, and the X-ray transitions states to the ground state. The latter have been measured with enough accuracy to be sensitive to the radiative corrections. The theoretical contributions to the Lamb shift are the self energy which has been discussed above, the vacuum polarization, the nuclear size effect, and smaller corrections such as higher order radiative corrections, reduced mass, and relativistic recoil effects; as far as we know, any other corrections are still smaller [12]. The self energy is given by

diction for the one-electron high-Z Lamb shift. Fig. 2 shows a comparison of theory and experiment. The four measurements at high Z, 15-18, are recent experiments and they all seem to be in reasonable agreement with theory. For these results the differences between theory and experiment have the same sign, but it is not clear if that is a real effect. The other test of QED in high-Z one-electron atoms considered here is the comparison of theory and experiment for the n = 2 to n = 1 level separations. For these separations, there are additional contributions to the theory. First of all, as is well known, the 2S,,, state and the 2P,,, state in the Dirac theory are completely degenerate, but for n = 2 to n = 1 transitions the Dirac energy separation is the main effect. The Dirac energy level is a well known simple function of the principle quantum number n and Dirac’s angular momentum quantum number K: -l/2

(Za)2

where F is a relatively slowly varying function of Z. The vacuum polarization (fig. 1b) has a similar form, varying approximately as ( Za)4 for S states E’2’-~~H(Za)mc2. VP -

The function

H can be broken up as

H(Za)=H,(Za)+H,(Za)+....

(6)

where the subscript indicates the order of the external Coulomb field in that term. Only first, third and higher odd order terms survive. The first term H, is just the Uehling potential contribution. which was derived in 1935 (131. The higher order terms were examined by Wichmann and Kroll in 1956 [4]. They wrote analytic expressions for the third and higher order terms, and one can derive from those expressions a power series in Za for H, for the Lamb shift [12] AH,( Za) = (Za)*[0.0567

n-

]a]+(+

(Za)*)“*]

2

I

mc2.

(9) Another necessary effect is the reduced mass correction. This is negligible for the Lamb shift at high Z, but for n = 2 to n = 1 transitions with very high accuracy it must be included. The lowest order effect is obtained by replacing the electron mass by the reduced mass of the electron-nucleus system. The correction is the difference l)(E,-mc2)

+--

E,,,=

=-

l+;,m(Eo-mc2).

- 0.1374( Za)

+0.0580(Za)21n(Za)-2+

. ..I.

(7)

For Z 5 40, we can approximate the vacuum polarization by the sum of the Uehling potential correction plus the power series for H,. The nuclear size correction accounts for the fact that the potential for an actual finite sized nucleus is different from the potential for a point nucleus. The correction to the Lamb shift is approximately [12] AE,,=

[I

+1,19(~a)~]&(za)~(~)*~m~~.

(8)

and R is the rms charge where s = [l - ( Za)2]‘/2 radius of the nucleus. Combining these corrections with the smaller corrections mentioned earlier, one obtains a theoretical pre-

Fig. 2. Comparison tron high-2 from refs.

of theory and experiment

for the one-elec-

Lamb shift. Theory from ref. [12] and experiments

[ 14-221.

P.J. Mohr / Radiative corrections in highly ionized atoms

Table 1 Comparison of theory and experiment for the IS,,,Z=

2Fi,2 transitions in atoms with Z=16,

461

17, 18, and 26, in units of lo3 cm-‘.

16

17

18

26

Dirac energy Reduced mass Lamb shift

21135.830 - 0.361 -6.168

23870.087 - 0.373 - 7.597

26772.502 - 0.366 - 9.240

56104.05 - 0.54 - 32.16

Total energy *) Experiment

21129.300(4) 21126.3(2.9) bt

23862.117(S) 23862.4(8) =) 23862.7(B) d,

26762.896(5) 26763 ‘) 26762(4) n

56071.35(2) 560X$6) g,

a) Ref. [12]. b, Ref. [23]. ‘) Ref. [24]. d, Ref. (251. e, Ref. [26]. 0 Ref. [27]. g, Ref. [28].

where M is the mass of the nucleus. There is an additional smaller relativistic reduced mass effect derived by Breit and Brown EL;,

_@hmc2 8n4

M

(11)

’

In addition, it is necessary to include Lamb shift effects described above. The sum of these contributions for the 2P,,z to IS,/, transition is compared to experiment in table 1. Recent experiments are accurate enough to be sensitive to the QED effects at about the 10% level. For the 2P3,, to lS,,, transition, the tests are at the same level of accuracy.

4. Two-electron

atoms

In two-electron atoms, measurements have been made with sufficient accuracy to see radiative corrections in the 23S,-23P0 and 23S,-23P, level splittings. There have been meas~ements of the energy of transitions from n = 2 to n = 1, but we shall concentrate on the fine structure splittings. Is it possible to check QED in high-Z two-electron systems? In helium the relative size of the radiative corrections to the total energy of the 23S,-23P, separation is very small: 0.002%. But at high Z, the radiative corrections become a larger fraction of the energy splitting. For example, at Z = 20, the ratio is about 1%. There are various approaches to the theory of twoelectron atoms. One way of writing down the theory is in terms of a series expansion in (Y.This approach starts with the nonrelativistic expression for the energy levels derived from the two-electron Schrodinger equation, in the form of some function of Z times (Y’. This function of Z has been evaluated numerically and as an expansion in powers of l/Z [29-321. The relativistic corrections have a similar form. There is a factor 1y4 times another function of Z which can also be evaluated [29]. The energy level thus has the form E= [a*f(Z)+a”g(Z>+

. ..]w&

(12)

In the limit of large Z, the leading term in each of these

expansions gives just the hydrogenic energy levels. This approach has a long tradition; Layzer and Bahcall [33], Dalgarno and Stewart 1341, and Doyle [35], studied it; Ermolaev and Jones [36] made a systematic evaluation of energy levels in highly ionized two-electron atoms based on this approach, I have used it [37], and Drake has done work on this approach [38]. For the radiative corrections one can write a similar expansion that starts with the lowest order terms in a, which in this case happen to be a5 and c&m, where each term is multiplied by some function of Z. AE=

[a5h(Z)+aSlnruh’(Z)+

. ..]mc’.

03)

Kabir and Salpeter [39], Sucher 1401, and Araki [41] have derived operators that give these functions when evaluated with Schrijdinger wave functions. Radiative corrections of higher order have not been completely derived from the Bethe-Salpeter equation; Sucher gives some of the higher order terms [40]. Thus the h’s are the expressions that have to be evaluated. Recent work has been done by Goldman and Drake who calculated the first correction in an expansion in l/Z for the Bethe sum, which is contained in eq. (13). In this approach, the neglected corrections are of relative order Za, so the uncertainties grow at high Z. To the extent that the two-electron and one-electron radiative corrections are similar, one can expect the higher order terms to be numerically important. Other methods, more oriented toward high-Z reiativistic atoms, include the relativistic random phase approximation which has been employed by Johnson and Lin [42] to calculate energy levels. In an accompanying paper in these proceedings, Grant discusses multiconfiguration Dirac-Fock methods, which take into account relativity for ionized atoms [43]. An alternative method of dealing with relativistic effects and radiative corrections is to apply perturbation theory directly in the formulation presented earlier [3&l-47]. One can apply this formalism to the two-electron atom, and simply draw Feynman diagrams with more and more photon loops to get higher order corrections to the energy levels. In this approach, the lowest order expression for the energy is some function of Za, which is known exactly to all orders in Za. It is just the II. ATOMIC STRUCTURE

P.J. Mohr / Radiative corrections in highly ionized atoms

462

lated to all orders in Za; that is one of the things that needs to be done in order to have a complete calculation in this scheme. But one can get the first two powers in ( Za)2 from existing calculations:

-Q

23s,-23Pa:

(b)

(a)

(cl

AEc4’ = (y2mc2 [ -0.0256 Pe

Fig. 3. Feynman diagrams of order a for two-electron atoms.

- 0.27( Za)2 +

.] ;

23s,-23P2: hydrogenic result, i.e., the one-electron Dirac equation result. In next order in a one obtains l/Z times another function of Za which again has no small Za assumption attached to it. This can be done for arbitrarily high ZW At low Z, as in helium, the correction term is not much smaller than the leading term. But at high Z, the series converges rapidly. For the first correction, the diagrams are shown in figs. 3a-c. The corrections from figs. 3b and 3c are exactly the same as if the second line were not present, as in fig. 1. This means that the one-electron radiative correction is a first approximation to the two-electron radiative correction. The new diagram, fig. 3a, is just the Breit interaction or the relativistic photon exchange interaction, and that can be calculated exactly numerically. It can also be calculated as a power series in (Za)‘: 23&S,-23Pa:

AE$=a2mc2[-0.0256+0.002(Za)*+...]. (15) The leading term is just the Schrodinger result, and the next term is just the relativistic correction of relative order (Za)* which is known numerically [29]. One can extract from the numerical calculations the coefficients of these terms. There are more diagrams of the same order that are discussed subsequently. In next order, from the diagram where three photons are exchanged, there is an order a3 or Z-3 correction: a3 AE(6’=-mc2[-0.0117+...]. pe za

Summarizing, this procedure yields the hydrogen levels plus corrections in ascending order of l/Z AE=

AE$)=a(Za)mc*

g [

+0.1428(

Za)* + 0.0997( Za)4 +

-0.0363(

[

fo(Za)

(Za)*mc* [

1;

23s,-23P,: AE$=a(Za)mc’

(16)

s

++/,(za)+$f4(Za)+

1,

(17)

where the functions f, approach constants as Za approaches zero. Summing the terms considered above we have AE=AE,+AE~)+AE$+AE$‘+

1 (14)

...

....

(18)

Za)2 - 0.0451( Za)4 + . . . .

This is a numerically convergent expansion, where the lowest order term is just the nonrelativistic Coulomb interaction of the two electrons. The first correction is the relativistic correction, of order a( Za)3 that was calculated by Doyle in 1969 [35]. I have calculated the next coefficient exactly to extend the power series expression. Continuing this procedure to next order in Z-‘, there are two-photon diagrams as in fig. 4 that are of order a2 or Z-*. These diagrams have not been calcu-

Fig. 4. Two-exchanged photon correction in two-electron

atoms.

Fig. 5. Comparison of theory and experiment for two-electron high-Z atoms. The filled and open symbols correspond to the 23S,-23P0 and 23S, -23Pz transitions, respectively. The experimental results are from refs. [48-531.

P.J. Mohr / Radiative

corrections tn highly Ionized atoms

where A E, is the hydrogenic energy separation. Comparison to experiment yields fig. 5. Shown in that figure are some experiments that measure the 22S,-23P,, splitting and the 23S,-23P2 splitting sufficiently accurately to check radiative corrections that are about 1% of the total. The level of agreement between theory and experiment in fig. 5 is on the order of 0.176, so these measurements are about 10% checks of the radiative corrections. What about the remaining differences? Are they a difficulty with the theory? The answer is no, because we have left out diagrams that correspond to radiative corrections and electron interactions going on simultaneously. This contribution is of the order of the difference between theory and experiment, and should be calculated for a more precise test of the theory. There is a paper in these proceedings by Gould on experiments with one- and two-electron uranium atoms [54]; these are systems where the theoretical approach described here is most appropriate. This research was supported by the National Foundation, Grant No. PHY-8403322.

Science

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II. ATOMIC

STRUCTURE

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P.J. Mohr

/

Radiarrce

corrections

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in hrghly ionized aloe

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