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Current status of Lamb shift predictions for heavy hydrogen-like ions
15 December 1997
PHYSICS LETTERS A
Physics Letters A 236 ( 1997) 329-338
Current status of Lamb shift predictions for heavy hydrogen-like ions T. Beier a,l, P.J. Mohrb, H. PerssonC, G. Pluniena, M. Greiner a, G. Soffa a Institut j?ir Theoretische Physik, Technische Universitiit Dresden, Mommsenstrasse 13, D-01062 Dresden, Germany b National Institute of Standards and Technology, Atomic Physics Division, Gaithersburg, MD 20899, USA ’ Department of Physics, Chalmers University of Technology and the University of Giiteborg, SE-412 96 Gdteborg, Sweden
Received 17 July 1997; revised manuscript received 6 October 1997; accepted for publication 17 October 1997 Communicated by B. Fricke
Abstract to the Lamb shift of lSl+ 2S1/2, and 2Pyz states in hydmgen-me i$$Au, due to un~alc~at~ two-photon self-energy diagrams and insufficiently known nuclear parameters are estimated. @ 1997 Elsevier Science B.V. A sums
of the various connations
ZPb , ‘~Th, and ZU is given. Unce~nties PACS: 31.10.+2; 31,30.-i; 31.3O.J~ Keywords: Lamb shift; Heavy hydrogen-lie
ions
1. Introduction Heavy few-electron ions provide a unique testing ground for quantum electrodynamics (QED) of strong external fields. The average magnitude of the nuclear electric field in the 1S state in @‘+ is (1st [Ellis) = 2.3 x lOI V/m. The Lamb shift affects the ground state of heavy hydrogen-like ions much more than low-2 systems (Fig. 1). QED effects that are only of minor importance in low-2 systems become a notable contribution to the total level shift in high-2 systems (Fig. 2). In recent years the investigation of these highly charged ions has become a major field of research for both experimental and theoretical physics. On the ex~~rnent~ side the new Super-EBIT at Livermore and the SWESR at GSI in Darrnstadt provide unique facilities. At the latter facility the ground state Lamb shifts of hy~ogen-like gold and uranium were measured to be 202 f 8 eV [ 1J and 470 rir 16 eV [2] , respectively, and the measurements are expected to become at least one order of magnitude more precise in the near future. Such measurements would also probe theoretical predictions of quantum electrodynamic (QED) effects that include more than one photon line in their Feynman diagrams. Theoretically, considerable progress toward the final evaluation of all the diagrams has been made since the last comprehensive tabulation of Lamb shift ’ E-mail: [email protected]. 0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PIISO375-9601(97)00810-4
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L.&em
A 236 (1997) 329-338
IO
nuclear charge number 2
nuclear charge number 2
Fig. I. Ratio of the Lamb shift to the total pointnucleus binding energy for the lSl/2 state of hydrogen-like ions. For large 2, the slope of the curve increases mainly due to the nuclear size correction. Fig. 2. Contribution of the two-loop vacuum polarization ladder diagram to the total Lamb shift of the lSl/z state in hydrogen-like ions. The negative of tbe ratio is given. This and the other contributions of order a2 become more important for high-i: systems.
predictions in one-electron systems was published by Johnson and Soff [ 35. The one-photon diagrams have been reconsidered. For the self-energy an improved c~culation scheme was derived [4] which accurately takes into account the finite extent of the nucleus [ 51. The Wichmann-Kroll contribution to the vacuum polarization was calculated with high precision both using an analytical approach [ 61 and a numerical B-spline method [ 71 for the evaluation of the Green function. The development of the partial wave renormalization technique 18,9] and the use of B-spline methods for the c~cula~on of wave functions [ 101 made it possible to compute the contribution of nearly all of the two-photon diagrams. In this Letter, we provide figures for all known contributions to the one-electron Iamb shift for a few systems of particular experimental relevance. Our intention is not only to show the theoretical input needed to make predictions with the precision shown in the following, but also to provide a set of figures that might serve to stimulate more precise measurements. We do not describe the calculation schemes we used, but refer to the original articles. However, we will briefly mention the unce~nty of each of the figures presented and its source, thus leading to a better underst~ding of problems in theoretical predictions of QED effects.
2. Discussion of numerical values In Tables l-4 we present all known con~butions to the level shifts of the lSt/2, 2&/z, and 2P1/a states of hydrogen-like lg7Au 2esPb, 232Th, and 238U. For uranium, this table has already been published by Persson et al, [ 113. Similar’results for the 1S state of hydrogen-like uranium were given by Mohr [ 123. The table presented here is slightly different both due to new c~culations for the two-loop vacuum election diagram and due to improved numerical results for the nuclear polarization. In the following we point out the methods used in obtaining the figures presented in the tables. The binding energy of an electron in the field of a point nucleus is simply obtained by solving the Dirac equation. As a result the binding energy EB is found to be
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331
Table 1 One-electron Lamb shift con~butions for Is7AURA+.Values are given in eV w/z
Binding energy EB (point nucleus) Correction: Finite nuclear size Self energy (order cu) VP: Uehling con~bution VP: Wichmann-Kroll contribution Total vacuum polarization (order a) SESE (a) (b) (c) VPVP (a) (ladder diagrams) VPVP (b) (Kilbn-Sabry contibution -I- h.o.) VPVP (c) ( Klillen-Sabry contibution) SEVP (a) (b) (c) S(VP)E Radiative recoil (estimate) Reduced mass Relativistic recoil Total recoil Nuclear polarization Sum of corrections Resulting total binding energy Lamb shift (theory)
-93459.89 49.13 196.68 -41.99 1.79 -40.20 -0.07 -0.05 -0.29 0.42 0.05 0.00 0.26 0.08 0.34 -0.02 205.99 -93253.90 205.73
2pi/2
-23925.06
-23925.06
8.32 33.58 -6.71 0.27 -6.44 uncalculated -0.01 -0.01 -0.04 0.07 0.01 0.00 0.07 0.02 0.09 0.00 35.57 -23889.49 35.50
0.07 0.00 0.07 0.00 2.90 -23922.16 2.83
-26059.84
-26059.84
0.65 2.94 -0.80 0.05 =G?i!? 0.00 0.00 -0.01 0.00 0.00 0.00
Table 2 One-electron lamb shift contributions for *“*Pbst+. Values are given in eV
Binding energy EB (point nucleus) Correction: Finite nuclear size Self-energy (order a) VP: Uehling contribution VP: Wichmann-Kroll contribution Total vacuum polarization (order a) SESE (a> (b) (c) VPVP (a) (ladder diagrams) VPVP (b) ( Kallkn-Sabry contribution 4 h.o.) VPVP (c) (Kahn-Sabry contribution) SEVP (a) (b) (c) S(VP)E Radiative recoil (estimate) Reduced mass Relativistic recoil Total recoil Nuclezu polarization Sum of corrections Resulting total binding energy Lamb shift (theory)
-101581.37 67.25 226.33 -50.70 2.29 -48.41 -0.09
-0.07 -0.34 0.53 0.07 0.00 0.27 0.10 0.37 0.00 245.64 -101335.73 245.37
11.68 39.29 -8.27 0.35 -7.92 uncalculated -0.01 -0.01 -0.05 0.09 0.02 0.00 0.07 0.02 0.09 0.00 43.17 -26016.67 43.10
1.oo 3.91 -1.08 0.07 -1.01 0.00 0.00 -0.01 0.01 0.00 0.00
0.07 0.00 0.07 0.00 3.97 -26055.87 3.90
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Table 3 One-electron
Lamb shift contributions
Letters A 236 (1997) 329-338
for 23*Th*9+. Values are given in eV lSI/Z
Binding energy En (point nucleus) Correction: Finite nuclear size Self-energy (order (u) VP: Uehling contribution VP: Wichmann-Kroll contribution Total vacuum polarization (order a) SESE VPVP VPVP VPVP
(a) (a) (b) (c)
(b) (c) (ladder diagrams) (Kllltn-Sabry contribution (Ktilltn-Sabry contribution)
-125655.61 160.52 325.02 -82.88 4.28 -78.60
+ h.o.)
SEVP (a) (b) (c) S( VP)E Radiative recoil (estimate) Reduced mass Relativistic recoil Total recoil Nuclear polarization Sum of corrections Resulting total binding energy Lamb shift (theory)
Table 4 One-electron
Lamb shift contributions
SESE VPVP VPVP VPVP
-32443.85
-32443.85
29.92 59.14 - 14.35 0.70 - 13.65 uncalculated -0.03 -0.03 -0.09 0.18 0.02 0.00 0.08 0.04 0.12 -0.02 75.56 -32368.29 75.48
(order (I )
(b) (c) (a) (ladder diagrams) (b) ( K&hen-Sabry contribution + h.o.) (c) (K;illtn-Sabry contribution)
IS,/2
2Slj2
- 132279.96
-34215.49
198.82 355.05 -93.58 4.99 -88.60
(a)
SEVP (a) (b) (c) S(VP)E Radiative recoil (estimate) Reduced mass Relativistic recoil Total recoil Nuclear polarization Sum of corrections Resulting total binding energy Lamb shift (theory)
2h/2
3.29 8.04 -2.39 0.16 -2.23 0.00 0.00 -0.02 0.02 0.00 0.00 0.08 0.00 0.08 0.00 9.18 -32434.67 9.10
for 238U9’+. Values are given in eV
Binding energy Ea (point nucleus) Correction: Finite nuclear size Self-energy (order a) VP: Uehiing contribution VP: Wichmann-Kroll contribution Total vacuum polarization
-0.18 -0.13 -0.54 0.98 0.11 0.00 0.30 0.17 0.47 -0.13 407.52 -125248.09 407.22
w/2
-0.22 -0.15 -0.60 1.14 0.13 0.00 0.30 0.2 1 0.51 -0.19 465.89 -131814.07 465.59
31.77 65.42 - 16.46 0.82 -15.64 uncalculated -0.04 -0.03 -0.10 0.21 0.02 0.00 0.08 0.05 0.13 -0.03 87.71 -34127.78 87.63
2PI 12
-34215.49 4.42 9.55 -2.90 0.21 -2.70 0.00 0.00
-0.02 0.02 0.00 0.00 0.08 0.01 0.09 0.00 11.36 -34204.13
11.28
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333
)2]-“2l}.
(1)
We use a = l/137.036 throughout our copulations. The value of the Rydberg energy in eV at present is known to a relative precision of only 3 x 10p7. Therefore, the absolute binding energy values of the tables are uncertain by the same amount. This, however, is a total uncertainty in scaling which does not affect any comparison between the values presented here. The sum of all nuclear and QED corrections to the Dirac value for a point nucleus is termed the Lamb shift of a single state in analogy to the classical Lamb shift which describes the splitting between the 2P’/2 state and the 2s’~~ state. However, by convention [ 131 there is one exception to this rule: the reduced mass correction, which reads
(2)
&E,,,, = -AEn, m-!-M
with m and M the masses of electron and nucleus, respectively, is not included in the Lamb shift. It results from the sep~ation of the classical two-body problem into center-of-mass and relative coordinates. In particular, it is propo~ional to En and therefore does not contribute to the “classical” 2P’,2-2S’/2 Lamb shift. For this reason, it is also not included in the one-level Lamb shift although it is by no means sufficient to treat the finite nuclear mass classically in high-Z systems, as we will point out below. However, we adopt the convention in use. Therefore, the rows “Sum of corrections” and “Lamb shift (theory)” are listed separately, although they differ only by this reduced mass correction. The finite nuclear size correction is not a QED effect, but simply accounts for the finite extent of the nuclear charge distribution which yields a potential that differs from the point charge Coulomb potential inside the nucleus. Therefore, wave functions and binding energies are slightly altered, as discussed by Franosch and Soff [ 141. A suitable model for the nuclear charge distribution is the two-parameter Fermi distribution, which reads N p(r)
= 1
(3)
+exp[(r-c)/a]’
where c denotes the half-density radius and a indicates the radial distance over which the charge density decreases appreciably. The skin thickness 1 = 4 ln3a is defined as the radial distance over which the charge density declines from 90% to 10% of its value at the origin. N is chosen to normalize p according to M 4n-
dr = eZ
p(r)r2
,
(4)
I
0
In Table 5 we give the parameters we employed to obtain the entries for the finite nuclear size correction. Instead of c we give the rms radius
(r2j1j2= (
rr4p(r) dr/
0
fur2p(r)
dr)li2,
(5)
0
which is always fixed when comparing different models for the charge distribution. Note that all current calculations consider the charge distributions as being spherically symmetric and do not consider deformations of the nucleus. To account for uncertainty from this effect as well as for the model dependence of the two-parameter Fermi distribution, we also calculated the binding energy employing a homogeneously charged sphere of the same rms radius for the nuclear charge distribution. The difference
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334 Table 5 Nuclear charge distribution
parameters
and resulting
Letters A 236 (1997) 329-338
binding energies for the levels under consideration tg7Au
(r*)‘/*
2S112 states
a .!$j (Fermi distribution) En (Horn. sphere) Difference Size uncertainty Es (Fermi distribution)
2P,/z states
Es (Horn. sphere) Difference Size uncertainty En (Fermi distribution)
lS1/2 states
En (Horn. sphere) Difference Size uncertainty
*08Pb
5.437 f 0.011 0.535 -93410.76
fm fm fm eV
-93410.67 0.09 0.16 -23916.74 -23916.72 0.02 0.03 -23924.41 -23924.41 0.00 0.00
eV eV eV eV eV eV eV eV eV eV eV
5.505 + 0.001 0.5334 - 101514.12 -101513.98 0.14 0.02 -26048.16 -26048.14 0.02 0.00 -26058.84 -26058.84 0.00 0.00
238 ”
232Th fm fm fm eV eV eV eV eV eV eV eV eV eV eV eV
5.802 fm f 0.004 fm 0.5110 fm - 125495.09 eV - 125494.79 eV 0.30 eV 0.16 eV -32413.93 eV -32413.87 0.06 0.03 -32440.56 -32440.55 0.01 0.00
eV eV eV eV eV eV eV
5.860 0.002 0.5046 -132081.14eV -132080.78 0.36 0.10 -34177.72
eV eV eV eV
-34177.64 0.08 0.02 -34211.07 -34211.07 0.00 0.00
eV eV eV eV eV eV eV
f
fm fm fm
between the binding energies obtained with both models is also indicated in Table 5. In the case of Au, Th, and U this value might well serve as an estimate of the “nuclear shape” uncertainty. In the case of Pb, however, it should be considered as an overestimate, as 208Pb is a double-magic nucleus and can very well be considered as being spherical in shape and described by a Fermi distribution. However, none of the models accounts for any internal structure of the nucleus and the charge is always assumed to be distributed homogeneously. Uncertainties in knowledge of the rms radius also cause an uncertainty in the prediction of the binding energy. In Table 5 we present the current error estimates of the rms radii together with their influence on the binding energy. The subsequent rows in Tables l-4 contain the values of the energy shifts caused by self-energy and vacuum polarization of order LY.The calculations for the self-energy were all carried out using the methods of Refs. [4,5]. For the nuclear charge distribution, a uniform sphere of the given rms radius was assumed. Earlier investigations [ 51 have shown that at this level of precision no deviations occur when employing other reasonable charge distributions of the same rms value. Numerically, the stability of the values has also been established to all digits cited here [ 51. The vacuum polarization of order (Yis split into two parts. The lowest order Uehling contribution is obtained from the Uehling potential, which can be specified without any problem for any spherically symmetric charge distribution [ 15- 181. The Wichmann-Kroll contribution contains all higher orders of an expansion in Zcu. Its calculation requires elaborate numerical techniques. Recently, both Soff and Mohr [6] as well as Persson et al. [ 71 carried out calculations based on a partial wave decomposition of the electron propagator. Their results agree within 0.01 eV for the lSi/z states of all systems under consideration. We present values calculated by the technique of Persson et al. [ 71, which includes a higher number of partial waves. Again a homogeneously charged sphere is adopted as the model for the nuclear charge distribution. Up to the current experimental precision, the finite size correction together with the self-energy and vacuum polarization of order LYare sufficient to describe the Lamb shift in heavy one-electron systems. All subsequent values in Tables 1-4 have not yet been tested by experiment. The next five rows in the tables contain the values of the QED contributions of second order in LY,which are displayed in Fig. 3. The graphs in this figure correspond to the entries in the table with the same label. The contribution of the diagrams in the first row is the most difficult one to calculate and has not yet completely been evaluated. Only a few results exist [ 191, and these are obtained by a computation scheme that is not fully
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335
b) VPVP
a)
(;-@
W
~
4
b
Fig. 3. Feynman diagrams of all QED con~butions diagram (the wavy lines). Thedouble lines indicate which indicates whether a diagram consists of two mixture of both. The letters (a) to (c) are used to
d”
b)
SEVP
4
S(VP)E
which contribute to the Lamb shift of order 02, which means two photon lines in the electrons or positrons propagating in the field of the nucfeus. We use a naming scheme diagrams which am self-energy-like (SFSE) or vacuum ~l~~tion”iike (VPVP) or a distinguish between diagmms only.
gauge invariant. The major difficulty in computing these contributions is the divergences which occur when two of the vertices meet when integrating over space-time coordinates of the vertices. This problem is difficult to handle when working with bound state propagators. This missing contribution is by far the largest uncertainty in the table. We estimate its maximum influence to about 0.5% of the total Lamb shift value given at the end of the table, which amounts to about 2 eV for the lS’/2 state of Ug’+. In contrast to this, the numbers of the next four rows - VPVP (a), VPVP (b), VPVP (c), and SEVP (a)-(c) - are much easier to obtain. If only one photon line is attached to the bound fermion state, this interaction can be fo~ulated as an additional potential. As this potential is considered to be small compared to that of the nucleus, the influence of the co~es~nding diagram can be obtained in first order ~~urbation theory. This is the case for both the vacuum polarization of order EY,where this potential is split into Uehling and Wichmann-Kroll parts, and also for the VPVP (b) , (c) graphs, named after Kiillen and Sabry who considered them first [ 201. For these graphs a potential was formulated by Blomqvist [21] and Fullerton and Rinker [ 161, which can be used to obtain the contribution for point-like [22] as well as for extended nuclei [23]. However, in this potential the electron-positron pairs of the loops are considered to be free fermions, which implies the use of a single instead of a double line in the corresponding graphs. This is called the Uehling approximation. Only the VPVP (b) diagram has been calculated to all orders in ZCXand this was done only recently [24). For the VPVP (c) graph currently only the Uehling approximation is available. The values given in the corresponding row are obtained by subtracting the Uehling approximation of the VPVP (b) contribution from the values given
336
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in Ref. [ 231, because only the total is given in this reference. However, the influence of higher order terms is assumed to result in almost no change of these values compared to other uncertainties discussed in this article. For the VPVP (b) contribution the current all order calculation influenced only the last digit as compared to the original K&hen-Sabry value. The vacuum pol~i~tion potential of order LYcan also be used to obtain rnodi~~ wave functions, if it is included in the Dirac equation together with the Coulomb potential of the nucleus. The difference between this energy value and that without this additional potential yields the influence of vacuum polarization ladder diagrams, from which the do~nant contribution of order ty has to be subtracted to obtain the value co~~pon~ng to the VPVP (a) graph [ 71, also termed a two-loop ladder diagram (see Fig. 1). A somewhat similar approach can be used for the self-energy vacuum polarization contributions SEVP (a)-(c) [25]. VPVP (a) as well as SEVP (a)-(c) have been calculated completely, and remaining numerical uncertainties are expected to be less than 0.01 eV in each case. The remaining QED diagram of order ty2 is the self-energy graph with a vacuum polarization loop in the photon line. It was calculated recently independently by Mallampalli et al. [ 261 and by Persson et al. [ 111. Both approaches were carried out in the Uehling approximation. To generate our values, we used the technique of Persson et al. which is based on the partial wave renormalization technique similar to the one developed for the ordinary self-energy [ 81. For details we refer to Ref. [ 111. As the effect itself is rather small, any calculation beyond the Uehling approximation will not affect the values presented here to more than 0.01 eV. Therefore, we expect that all QED contributions to the one-electron Lamb shift in the ground states of heavy systems are known to at least this precision and only the missing two-photon self-energy remains as a major source of uncertainty. In light systems, a contribution is important which is termed the “radiative recoil effect”. It describes the modification of self-energy and vacuum polarization of order (Ydue to the finite nuclear mass and has to be considered to obtain accurate theoretical values for the Lamb shift in hydrogen [ 131. An evaluation to all orders in ZCYhas not yet been done, as the effect is thought to be completely negligible in heavy systems. We simply estimate its order by multiplying {self-energy + vacuum polarization) by m/(m + M). For low 2, an exact formula of the leading order in ZCYis given by Eq. (28.62) of Ref. [27]. The analytical coefficients in this formula are of order one, and therefore our estimate seems reliable. It results in contributions of less than 0.005 eV for all nuclei and states under consideration, and therefore we expect the effect to be negligible for high-2 systems. The reduced mass contribution was already mentioned. In heavy systems the mutual motion of nucleus and electron has to be considered relativistic&y and therefore another approach to the recoil effects has to be utilized. Recently Artemyev et al. [28] solved that problem by describing the total recoil effect of order m/M as the interaction of two spin-$ Dirac particles. For the nucleus this is only an approximation, but it yields the correct result of order m/M to all orders in Za. What is termed “relativistic recoil” is obtained by subtracting the reduced mass effect from this value, contrary to earlier approaches to the same correction [3]. For the ions under consideration here, the relativistic recoil correction is of the same magnitude as its reduced mass counterpart. The rem~ning entry of “nuclear polarization” is the most uncert~n of a11those in the tables. Whereas the interaction of a bound fermion with virtual photons or virtual electron-positron pairs can be well described by quantum electrodynamics, this is not true for any interaction of the electron with virtual states of the nucleus itself. Al~ough this i~ter~tion can be considered by formulae and diagrams similar to those of pure QED, both experimental parameters and ~sumptions about nuclear models enter the c~culation [29]. In particular, it is not easy to find any satisfactory way to handle one-particle excitations in odd-A nuclei. Therefore, we considered only the most dominant giant resonances for rg7Au. For Th and U, we refer to Refs. 130,293, where the field th~retical approach to nuclear pol~ization is given in detail. Note that a factor of (217) -’ was omitted in some of the earlier calculations which causes a difference between those values and the ones presented here. As the number of nuclear states included in any calculation may also vary, the outcome of nuclear polarization
T. Beier et al./Physics Letters A 236 (1997) 329-338
337
calculations should be taken as an estimate only. In addition, some of the parameters which enter the calculations have to be deduced from experiments, e.g., transition strengths. They are not very precisely known, and therefore the uncertainty of any nuclear polarization contribution should be considered as being about 25% of the value itself. One might consider with skepticism the zero values presented for the nuclear polarization in *08Pb. They are due to another aspect of nuclear polarization, where the nucleus also interacts with a virtual electron-positron pair. This effect was considered recently [ 3 11, and it was found that for lead, due to collective monopole vibrations of this particular nucleus, this effect almost totally cancels that of the ordinary nuclear polarization. Therefore, the nuclear polarization effects which otherwise limit very precise Lamb shift predictions are almost completely negligible for *08Pb, making this ion especially suitable for the most precise theoretical predictions.
3. Final discussion We have presented the current status of theoretical predictions for the Lamb shift in a few heavy hydrogenlike ions of particular experimental interest. Apart from the missing contribution of the two-photon self-energy, currently all quantum electrodynamic corrections to the Dirac binding energy value are expected to be known to a precision of about 0.01 eV or better. It is clear from our numerical values that any further improvement in computing these QED contributions is meaningless as long as no better estimates for the nuclear effects that also enter the Lamb shift predictions exist. From this point of view, the *08Pb ion is the one for which the most precise predictions can be made. However, for all of the systems under consideration here, predictions which are uncertain by less than 1 eV will be possible as soon as the missing two-photon self-energy values become available. Currently, work is under way to calculate these values as well.
Acknowledgement The authors are grateful to Martin G.H. Gustavsson, Christian R. Hofmann, I. Lindgren, Leonti N. Labzowsky, Sten Salomonson, and Per Sunnergren for valuable discussions. For financial support we thank the Bundesministerium fur Bildung und Forschung (BMBF), the Deutsche Forschungsgemeinschaft (DFG), and the GSI (Darmstadt) . In particular, we are grateful to the Svenska Institutet (SI) and the Deutsche Akademisthe Austauschdienst (DAAD) for the possibility of carrying out this investigation aided by a number of mutual research visits. Two of us (PM. and HI?) gratefully acknowledge support by the Alexander-von-Humboldt foundation.
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1lo] W.R. Johnson, S.A. Blundell, J. Sapirstein, Phys. Rev. A 37 ( 1988) 307. [ 111 H. Persson, I. Lindgren, L. Labzowsky, T. Beier, G. Plunien, G. Soff, Phys. Rev. A 54 ( 1996) 2805. ] 121 PJ. Mohr, Nucl. Ins& Meth. El 87 (1994) 232. [ 131 J.R. Sapirstein, D.R. Yennie, in: Quantum Electrodyn~ics, ed. T. Kinoshita (World Scientific, Singapore, 1990) p. 560. [ 141 T. Franosch, G. Soff, Z. Phys. D IS (1991) 219. [ 151 K.-N. Huang, Phys. Rev. A 14 (1976) 1311. [ 161L.W. Fullerton, G.A. Rinker, Jr., Phys. Rev. A 13 (1976) 1283. [ 171 S. Klarsfeld, Phys. Len. B 66 (1977) 86. [ 181 D.J. Hylton, Phys. Rev. A 32 (1985) 1303. [ 191 A. Mitrushenkov, L. Labzowsky, I. Lindgmn, H. Persson, S. Salomonson, Phys. L&t. A 200 (1995) 51; L.N. Labzowsky, A.O. Mitrushenkov, Phys. Rev. A 53 ( 1996) 53. [20] G. KBlICn,A, Sabry, Mater. Fys. Medd. Dan. Vid. Selsk. 29 (1955) 17. [ 211 J. Blomqvist, Nucl. Phys. B 48 ( 1972) 95. [22] T. Beier, G. Soff, Z. Phys. D 8 (1988) 129. [ 231 SM. Schneider, W. Greiner, G. Soff, J. Phys. B 26 ( 1993) L.529. [ 241 G. Phmien, T. Beier, G. Soff, H. Persson, 2. Phys. D, submitted. [ 2.51I. Lindgren, H. Fersson, S. S~omonson, V. Karasiev, L. Labzowsky, A. Mi~shenkov, M. Tokman, J. Phys. B 26 f 1993) L503 [ 261 S. MalI~p~li, J. Sapirstein, Phys. Rev. A 54 (1996) 2714. 1271 PJ. Mohr, in: Atomic Molecular, and Optical Handbook, ed. G.W.F. Drake (AIP, Woodbury, New York, 1996) p, 341. 1281 A.N. Anemyev, V.M. Shabev, V.A. Yerokhin, Phys. Rev. A 52 ( 1995) 1884; J. Phys. B 28 ( 1995) 5201. (291 G. Phmien, G. Soff, Phys. Rev. A 51 ( 1995) 1119; A 53 ( 1996) 4614. [ 301 L.N. Labzowsky, A.V. Nefiodov, Phys. Lett. A 188 ( 1994) 371; A.V. Nefiodov, L.N. Lahzowsky, G. Plunien, G. Soff, Phys. Lea. A 222 ( 1996) 227. [ 311 L.N. Lahzowsky, A.V. Nefiodov, G. Plunien, T. Beier, G. Soff. J. Phys. B 29 (1996) 3841.
PHYSICS LETTERS A
Physics Letters A 236 ( 1997) 329-338
Current status of Lamb shift predictions for heavy hydrogen-like ions T. Beier a,l, P.J. Mohrb, H. PerssonC, G. Pluniena, M. Greiner a, G. Soffa a Institut j?ir Theoretische Physik, Technische Universitiit Dresden, Mommsenstrasse 13, D-01062 Dresden, Germany b National Institute of Standards and Technology, Atomic Physics Division, Gaithersburg, MD 20899, USA ’ Department of Physics, Chalmers University of Technology and the University of Giiteborg, SE-412 96 Gdteborg, Sweden
Received 17 July 1997; revised manuscript received 6 October 1997; accepted for publication 17 October 1997 Communicated by B. Fricke
Abstract to the Lamb shift of lSl+ 2S1/2, and 2Pyz states in hydmgen-me i$$Au, due to un~alc~at~ two-photon self-energy diagrams and insufficiently known nuclear parameters are estimated. @ 1997 Elsevier Science B.V. A sums
of the various connations
ZPb , ‘~Th, and ZU is given. Unce~nties PACS: 31.10.+2; 31,30.-i; 31.3O.J~ Keywords: Lamb shift; Heavy hydrogen-lie
ions
1. Introduction Heavy few-electron ions provide a unique testing ground for quantum electrodynamics (QED) of strong external fields. The average magnitude of the nuclear electric field in the 1S state in @‘+ is (1st [Ellis) = 2.3 x lOI V/m. The Lamb shift affects the ground state of heavy hydrogen-like ions much more than low-2 systems (Fig. 1). QED effects that are only of minor importance in low-2 systems become a notable contribution to the total level shift in high-2 systems (Fig. 2). In recent years the investigation of these highly charged ions has become a major field of research for both experimental and theoretical physics. On the ex~~rnent~ side the new Super-EBIT at Livermore and the SWESR at GSI in Darrnstadt provide unique facilities. At the latter facility the ground state Lamb shifts of hy~ogen-like gold and uranium were measured to be 202 f 8 eV [ 1J and 470 rir 16 eV [2] , respectively, and the measurements are expected to become at least one order of magnitude more precise in the near future. Such measurements would also probe theoretical predictions of quantum electrodynamic (QED) effects that include more than one photon line in their Feynman diagrams. Theoretically, considerable progress toward the final evaluation of all the diagrams has been made since the last comprehensive tabulation of Lamb shift ’ E-mail: [email protected]. 0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PIISO375-9601(97)00810-4
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L.&em
A 236 (1997) 329-338
IO
nuclear charge number 2
nuclear charge number 2
Fig. I. Ratio of the Lamb shift to the total pointnucleus binding energy for the lSl/2 state of hydrogen-like ions. For large 2, the slope of the curve increases mainly due to the nuclear size correction. Fig. 2. Contribution of the two-loop vacuum polarization ladder diagram to the total Lamb shift of the lSl/z state in hydrogen-like ions. The negative of tbe ratio is given. This and the other contributions of order a2 become more important for high-i: systems.
predictions in one-electron systems was published by Johnson and Soff [ 35. The one-photon diagrams have been reconsidered. For the self-energy an improved c~culation scheme was derived [4] which accurately takes into account the finite extent of the nucleus [ 51. The Wichmann-Kroll contribution to the vacuum polarization was calculated with high precision both using an analytical approach [ 61 and a numerical B-spline method [ 71 for the evaluation of the Green function. The development of the partial wave renormalization technique 18,9] and the use of B-spline methods for the c~cula~on of wave functions [ 101 made it possible to compute the contribution of nearly all of the two-photon diagrams. In this Letter, we provide figures for all known contributions to the one-electron Iamb shift for a few systems of particular experimental relevance. Our intention is not only to show the theoretical input needed to make predictions with the precision shown in the following, but also to provide a set of figures that might serve to stimulate more precise measurements. We do not describe the calculation schemes we used, but refer to the original articles. However, we will briefly mention the unce~nty of each of the figures presented and its source, thus leading to a better underst~ding of problems in theoretical predictions of QED effects.
2. Discussion of numerical values In Tables l-4 we present all known con~butions to the level shifts of the lSt/2, 2&/z, and 2P1/a states of hydrogen-like lg7Au 2esPb, 232Th, and 238U. For uranium, this table has already been published by Persson et al, [ 113. Similar’results for the 1S state of hydrogen-like uranium were given by Mohr [ 123. The table presented here is slightly different both due to new c~culations for the two-loop vacuum election diagram and due to improved numerical results for the nuclear polarization. In the following we point out the methods used in obtaining the figures presented in the tables. The binding energy of an electron in the field of a point nucleus is simply obtained by solving the Dirac equation. As a result the binding energy EB is found to be
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Letters A 236 (1997) 329-338
331
Table 1 One-electron Lamb shift con~butions for Is7AURA+.Values are given in eV w/z
Binding energy EB (point nucleus) Correction: Finite nuclear size Self energy (order cu) VP: Uehling con~bution VP: Wichmann-Kroll contribution Total vacuum polarization (order a) SESE (a) (b) (c) VPVP (a) (ladder diagrams) VPVP (b) (Kilbn-Sabry contibution -I- h.o.) VPVP (c) ( Klillen-Sabry contibution) SEVP (a) (b) (c) S(VP)E Radiative recoil (estimate) Reduced mass Relativistic recoil Total recoil Nuclear polarization Sum of corrections Resulting total binding energy Lamb shift (theory)
-93459.89 49.13 196.68 -41.99 1.79 -40.20 -0.07 -0.05 -0.29 0.42 0.05 0.00 0.26 0.08 0.34 -0.02 205.99 -93253.90 205.73
2pi/2
-23925.06
-23925.06
8.32 33.58 -6.71 0.27 -6.44 uncalculated -0.01 -0.01 -0.04 0.07 0.01 0.00 0.07 0.02 0.09 0.00 35.57 -23889.49 35.50
0.07 0.00 0.07 0.00 2.90 -23922.16 2.83
-26059.84
-26059.84
0.65 2.94 -0.80 0.05 =G?i!? 0.00 0.00 -0.01 0.00 0.00 0.00
Table 2 One-electron lamb shift contributions for *“*Pbst+. Values are given in eV
Binding energy EB (point nucleus) Correction: Finite nuclear size Self-energy (order a) VP: Uehling contribution VP: Wichmann-Kroll contribution Total vacuum polarization (order a) SESE (a> (b) (c) VPVP (a) (ladder diagrams) VPVP (b) ( Kallkn-Sabry contribution 4 h.o.) VPVP (c) (Kahn-Sabry contribution) SEVP (a) (b) (c) S(VP)E Radiative recoil (estimate) Reduced mass Relativistic recoil Total recoil Nuclezu polarization Sum of corrections Resulting total binding energy Lamb shift (theory)
-101581.37 67.25 226.33 -50.70 2.29 -48.41 -0.09
-0.07 -0.34 0.53 0.07 0.00 0.27 0.10 0.37 0.00 245.64 -101335.73 245.37
11.68 39.29 -8.27 0.35 -7.92 uncalculated -0.01 -0.01 -0.05 0.09 0.02 0.00 0.07 0.02 0.09 0.00 43.17 -26016.67 43.10
1.oo 3.91 -1.08 0.07 -1.01 0.00 0.00 -0.01 0.01 0.00 0.00
0.07 0.00 0.07 0.00 3.97 -26055.87 3.90
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Table 3 One-electron
Lamb shift contributions
Letters A 236 (1997) 329-338
for 23*Th*9+. Values are given in eV lSI/Z
Binding energy En (point nucleus) Correction: Finite nuclear size Self-energy (order (u) VP: Uehling contribution VP: Wichmann-Kroll contribution Total vacuum polarization (order a) SESE VPVP VPVP VPVP
(a) (a) (b) (c)
(b) (c) (ladder diagrams) (Kllltn-Sabry contribution (Ktilltn-Sabry contribution)
-125655.61 160.52 325.02 -82.88 4.28 -78.60
+ h.o.)
SEVP (a) (b) (c) S( VP)E Radiative recoil (estimate) Reduced mass Relativistic recoil Total recoil Nuclear polarization Sum of corrections Resulting total binding energy Lamb shift (theory)
Table 4 One-electron
Lamb shift contributions
SESE VPVP VPVP VPVP
-32443.85
-32443.85
29.92 59.14 - 14.35 0.70 - 13.65 uncalculated -0.03 -0.03 -0.09 0.18 0.02 0.00 0.08 0.04 0.12 -0.02 75.56 -32368.29 75.48
(order (I )
(b) (c) (a) (ladder diagrams) (b) ( K&hen-Sabry contribution + h.o.) (c) (K;illtn-Sabry contribution)
IS,/2
2Slj2
- 132279.96
-34215.49
198.82 355.05 -93.58 4.99 -88.60
(a)
SEVP (a) (b) (c) S(VP)E Radiative recoil (estimate) Reduced mass Relativistic recoil Total recoil Nuclear polarization Sum of corrections Resulting total binding energy Lamb shift (theory)
2h/2
3.29 8.04 -2.39 0.16 -2.23 0.00 0.00 -0.02 0.02 0.00 0.00 0.08 0.00 0.08 0.00 9.18 -32434.67 9.10
for 238U9’+. Values are given in eV
Binding energy Ea (point nucleus) Correction: Finite nuclear size Self-energy (order a) VP: Uehiing contribution VP: Wichmann-Kroll contribution Total vacuum polarization
-0.18 -0.13 -0.54 0.98 0.11 0.00 0.30 0.17 0.47 -0.13 407.52 -125248.09 407.22
w/2
-0.22 -0.15 -0.60 1.14 0.13 0.00 0.30 0.2 1 0.51 -0.19 465.89 -131814.07 465.59
31.77 65.42 - 16.46 0.82 -15.64 uncalculated -0.04 -0.03 -0.10 0.21 0.02 0.00 0.08 0.05 0.13 -0.03 87.71 -34127.78 87.63
2PI 12
-34215.49 4.42 9.55 -2.90 0.21 -2.70 0.00 0.00
-0.02 0.02 0.00 0.00 0.08 0.01 0.09 0.00 11.36 -34204.13
11.28
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Letters A 236 (1997) 329-338
333
)2]-“2l}.
(1)
We use a = l/137.036 throughout our copulations. The value of the Rydberg energy in eV at present is known to a relative precision of only 3 x 10p7. Therefore, the absolute binding energy values of the tables are uncertain by the same amount. This, however, is a total uncertainty in scaling which does not affect any comparison between the values presented here. The sum of all nuclear and QED corrections to the Dirac value for a point nucleus is termed the Lamb shift of a single state in analogy to the classical Lamb shift which describes the splitting between the 2P’/2 state and the 2s’~~ state. However, by convention [ 131 there is one exception to this rule: the reduced mass correction, which reads
(2)
&E,,,, = -AEn, m-!-M
with m and M the masses of electron and nucleus, respectively, is not included in the Lamb shift. It results from the sep~ation of the classical two-body problem into center-of-mass and relative coordinates. In particular, it is propo~ional to En and therefore does not contribute to the “classical” 2P’,2-2S’/2 Lamb shift. For this reason, it is also not included in the one-level Lamb shift although it is by no means sufficient to treat the finite nuclear mass classically in high-Z systems, as we will point out below. However, we adopt the convention in use. Therefore, the rows “Sum of corrections” and “Lamb shift (theory)” are listed separately, although they differ only by this reduced mass correction. The finite nuclear size correction is not a QED effect, but simply accounts for the finite extent of the nuclear charge distribution which yields a potential that differs from the point charge Coulomb potential inside the nucleus. Therefore, wave functions and binding energies are slightly altered, as discussed by Franosch and Soff [ 141. A suitable model for the nuclear charge distribution is the two-parameter Fermi distribution, which reads N p(r)
= 1
(3)
+exp[(r-c)/a]’
where c denotes the half-density radius and a indicates the radial distance over which the charge density decreases appreciably. The skin thickness 1 = 4 ln3a is defined as the radial distance over which the charge density declines from 90% to 10% of its value at the origin. N is chosen to normalize p according to M 4n-
dr = eZ
p(r)r2
,
(4)
I
0
In Table 5 we give the parameters we employed to obtain the entries for the finite nuclear size correction. Instead of c we give the rms radius
(r2j1j2= (
rr4p(r) dr/
0
fur2p(r)
dr)li2,
(5)
0
which is always fixed when comparing different models for the charge distribution. Note that all current calculations consider the charge distributions as being spherically symmetric and do not consider deformations of the nucleus. To account for uncertainty from this effect as well as for the model dependence of the two-parameter Fermi distribution, we also calculated the binding energy employing a homogeneously charged sphere of the same rms radius for the nuclear charge distribution. The difference
I: Beier et al/Physics
334 Table 5 Nuclear charge distribution
parameters
and resulting
Letters A 236 (1997) 329-338
binding energies for the levels under consideration tg7Au
(r*)‘/*
2S112 states
a .!$j (Fermi distribution) En (Horn. sphere) Difference Size uncertainty Es (Fermi distribution)
2P,/z states
Es (Horn. sphere) Difference Size uncertainty En (Fermi distribution)
lS1/2 states
En (Horn. sphere) Difference Size uncertainty
*08Pb
5.437 f 0.011 0.535 -93410.76
fm fm fm eV
-93410.67 0.09 0.16 -23916.74 -23916.72 0.02 0.03 -23924.41 -23924.41 0.00 0.00
eV eV eV eV eV eV eV eV eV eV eV
5.505 + 0.001 0.5334 - 101514.12 -101513.98 0.14 0.02 -26048.16 -26048.14 0.02 0.00 -26058.84 -26058.84 0.00 0.00
238 ”
232Th fm fm fm eV eV eV eV eV eV eV eV eV eV eV eV
5.802 fm f 0.004 fm 0.5110 fm - 125495.09 eV - 125494.79 eV 0.30 eV 0.16 eV -32413.93 eV -32413.87 0.06 0.03 -32440.56 -32440.55 0.01 0.00
eV eV eV eV eV eV eV
5.860 0.002 0.5046 -132081.14eV -132080.78 0.36 0.10 -34177.72
eV eV eV eV
-34177.64 0.08 0.02 -34211.07 -34211.07 0.00 0.00
eV eV eV eV eV eV eV
f
fm fm fm
between the binding energies obtained with both models is also indicated in Table 5. In the case of Au, Th, and U this value might well serve as an estimate of the “nuclear shape” uncertainty. In the case of Pb, however, it should be considered as an overestimate, as 208Pb is a double-magic nucleus and can very well be considered as being spherical in shape and described by a Fermi distribution. However, none of the models accounts for any internal structure of the nucleus and the charge is always assumed to be distributed homogeneously. Uncertainties in knowledge of the rms radius also cause an uncertainty in the prediction of the binding energy. In Table 5 we present the current error estimates of the rms radii together with their influence on the binding energy. The subsequent rows in Tables l-4 contain the values of the energy shifts caused by self-energy and vacuum polarization of order LY.The calculations for the self-energy were all carried out using the methods of Refs. [4,5]. For the nuclear charge distribution, a uniform sphere of the given rms radius was assumed. Earlier investigations [ 51 have shown that at this level of precision no deviations occur when employing other reasonable charge distributions of the same rms value. Numerically, the stability of the values has also been established to all digits cited here [ 51. The vacuum polarization of order (Yis split into two parts. The lowest order Uehling contribution is obtained from the Uehling potential, which can be specified without any problem for any spherically symmetric charge distribution [ 15- 181. The Wichmann-Kroll contribution contains all higher orders of an expansion in Zcu. Its calculation requires elaborate numerical techniques. Recently, both Soff and Mohr [6] as well as Persson et al. [ 71 carried out calculations based on a partial wave decomposition of the electron propagator. Their results agree within 0.01 eV for the lSi/z states of all systems under consideration. We present values calculated by the technique of Persson et al. [ 71, which includes a higher number of partial waves. Again a homogeneously charged sphere is adopted as the model for the nuclear charge distribution. Up to the current experimental precision, the finite size correction together with the self-energy and vacuum polarization of order LYare sufficient to describe the Lamb shift in heavy one-electron systems. All subsequent values in Tables 1-4 have not yet been tested by experiment. The next five rows in the tables contain the values of the QED contributions of second order in LY,which are displayed in Fig. 3. The graphs in this figure correspond to the entries in the table with the same label. The contribution of the diagrams in the first row is the most difficult one to calculate and has not yet completely been evaluated. Only a few results exist [ 191, and these are obtained by a computation scheme that is not fully
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L.etters A 236 (1997) 329-338
335
b) VPVP
a)
(;-@
W
~
4
b
Fig. 3. Feynman diagrams of all QED con~butions diagram (the wavy lines). Thedouble lines indicate which indicates whether a diagram consists of two mixture of both. The letters (a) to (c) are used to
d”
b)
SEVP
4
S(VP)E
which contribute to the Lamb shift of order 02, which means two photon lines in the electrons or positrons propagating in the field of the nucfeus. We use a naming scheme diagrams which am self-energy-like (SFSE) or vacuum ~l~~tion”iike (VPVP) or a distinguish between diagmms only.
gauge invariant. The major difficulty in computing these contributions is the divergences which occur when two of the vertices meet when integrating over space-time coordinates of the vertices. This problem is difficult to handle when working with bound state propagators. This missing contribution is by far the largest uncertainty in the table. We estimate its maximum influence to about 0.5% of the total Lamb shift value given at the end of the table, which amounts to about 2 eV for the lS’/2 state of Ug’+. In contrast to this, the numbers of the next four rows - VPVP (a), VPVP (b), VPVP (c), and SEVP (a)-(c) - are much easier to obtain. If only one photon line is attached to the bound fermion state, this interaction can be fo~ulated as an additional potential. As this potential is considered to be small compared to that of the nucleus, the influence of the co~es~nding diagram can be obtained in first order ~~urbation theory. This is the case for both the vacuum polarization of order EY,where this potential is split into Uehling and Wichmann-Kroll parts, and also for the VPVP (b) , (c) graphs, named after Kiillen and Sabry who considered them first [ 201. For these graphs a potential was formulated by Blomqvist [21] and Fullerton and Rinker [ 161, which can be used to obtain the contribution for point-like [22] as well as for extended nuclei [23]. However, in this potential the electron-positron pairs of the loops are considered to be free fermions, which implies the use of a single instead of a double line in the corresponding graphs. This is called the Uehling approximation. Only the VPVP (b) diagram has been calculated to all orders in ZCXand this was done only recently [24). For the VPVP (c) graph currently only the Uehling approximation is available. The values given in the corresponding row are obtained by subtracting the Uehling approximation of the VPVP (b) contribution from the values given
336
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in Ref. [ 231, because only the total is given in this reference. However, the influence of higher order terms is assumed to result in almost no change of these values compared to other uncertainties discussed in this article. For the VPVP (b) contribution the current all order calculation influenced only the last digit as compared to the original K&hen-Sabry value. The vacuum pol~i~tion potential of order LYcan also be used to obtain rnodi~~ wave functions, if it is included in the Dirac equation together with the Coulomb potential of the nucleus. The difference between this energy value and that without this additional potential yields the influence of vacuum polarization ladder diagrams, from which the do~nant contribution of order ty has to be subtracted to obtain the value co~~pon~ng to the VPVP (a) graph [ 71, also termed a two-loop ladder diagram (see Fig. 1). A somewhat similar approach can be used for the self-energy vacuum polarization contributions SEVP (a)-(c) [25]. VPVP (a) as well as SEVP (a)-(c) have been calculated completely, and remaining numerical uncertainties are expected to be less than 0.01 eV in each case. The remaining QED diagram of order ty2 is the self-energy graph with a vacuum polarization loop in the photon line. It was calculated recently independently by Mallampalli et al. [ 261 and by Persson et al. [ 111. Both approaches were carried out in the Uehling approximation. To generate our values, we used the technique of Persson et al. which is based on the partial wave renormalization technique similar to the one developed for the ordinary self-energy [ 81. For details we refer to Ref. [ 111. As the effect itself is rather small, any calculation beyond the Uehling approximation will not affect the values presented here to more than 0.01 eV. Therefore, we expect that all QED contributions to the one-electron Lamb shift in the ground states of heavy systems are known to at least this precision and only the missing two-photon self-energy remains as a major source of uncertainty. In light systems, a contribution is important which is termed the “radiative recoil effect”. It describes the modification of self-energy and vacuum polarization of order (Ydue to the finite nuclear mass and has to be considered to obtain accurate theoretical values for the Lamb shift in hydrogen [ 131. An evaluation to all orders in ZCYhas not yet been done, as the effect is thought to be completely negligible in heavy systems. We simply estimate its order by multiplying {self-energy + vacuum polarization) by m/(m + M). For low 2, an exact formula of the leading order in ZCYis given by Eq. (28.62) of Ref. [27]. The analytical coefficients in this formula are of order one, and therefore our estimate seems reliable. It results in contributions of less than 0.005 eV for all nuclei and states under consideration, and therefore we expect the effect to be negligible for high-2 systems. The reduced mass contribution was already mentioned. In heavy systems the mutual motion of nucleus and electron has to be considered relativistic&y and therefore another approach to the recoil effects has to be utilized. Recently Artemyev et al. [28] solved that problem by describing the total recoil effect of order m/M as the interaction of two spin-$ Dirac particles. For the nucleus this is only an approximation, but it yields the correct result of order m/M to all orders in Za. What is termed “relativistic recoil” is obtained by subtracting the reduced mass effect from this value, contrary to earlier approaches to the same correction [3]. For the ions under consideration here, the relativistic recoil correction is of the same magnitude as its reduced mass counterpart. The rem~ning entry of “nuclear polarization” is the most uncert~n of a11those in the tables. Whereas the interaction of a bound fermion with virtual photons or virtual electron-positron pairs can be well described by quantum electrodynamics, this is not true for any interaction of the electron with virtual states of the nucleus itself. Al~ough this i~ter~tion can be considered by formulae and diagrams similar to those of pure QED, both experimental parameters and ~sumptions about nuclear models enter the c~culation [29]. In particular, it is not easy to find any satisfactory way to handle one-particle excitations in odd-A nuclei. Therefore, we considered only the most dominant giant resonances for rg7Au. For Th and U, we refer to Refs. 130,293, where the field th~retical approach to nuclear pol~ization is given in detail. Note that a factor of (217) -’ was omitted in some of the earlier calculations which causes a difference between those values and the ones presented here. As the number of nuclear states included in any calculation may also vary, the outcome of nuclear polarization
T. Beier et al./Physics Letters A 236 (1997) 329-338
337
calculations should be taken as an estimate only. In addition, some of the parameters which enter the calculations have to be deduced from experiments, e.g., transition strengths. They are not very precisely known, and therefore the uncertainty of any nuclear polarization contribution should be considered as being about 25% of the value itself. One might consider with skepticism the zero values presented for the nuclear polarization in *08Pb. They are due to another aspect of nuclear polarization, where the nucleus also interacts with a virtual electron-positron pair. This effect was considered recently [ 3 11, and it was found that for lead, due to collective monopole vibrations of this particular nucleus, this effect almost totally cancels that of the ordinary nuclear polarization. Therefore, the nuclear polarization effects which otherwise limit very precise Lamb shift predictions are almost completely negligible for *08Pb, making this ion especially suitable for the most precise theoretical predictions.
3. Final discussion We have presented the current status of theoretical predictions for the Lamb shift in a few heavy hydrogenlike ions of particular experimental interest. Apart from the missing contribution of the two-photon self-energy, currently all quantum electrodynamic corrections to the Dirac binding energy value are expected to be known to a precision of about 0.01 eV or better. It is clear from our numerical values that any further improvement in computing these QED contributions is meaningless as long as no better estimates for the nuclear effects that also enter the Lamb shift predictions exist. From this point of view, the *08Pb ion is the one for which the most precise predictions can be made. However, for all of the systems under consideration here, predictions which are uncertain by less than 1 eV will be possible as soon as the missing two-photon self-energy values become available. Currently, work is under way to calculate these values as well.
Acknowledgement The authors are grateful to Martin G.H. Gustavsson, Christian R. Hofmann, I. Lindgren, Leonti N. Labzowsky, Sten Salomonson, and Per Sunnergren for valuable discussions. For financial support we thank the Bundesministerium fur Bildung und Forschung (BMBF), the Deutsche Forschungsgemeinschaft (DFG), and the GSI (Darmstadt) . In particular, we are grateful to the Svenska Institutet (SI) and the Deutsche Akademisthe Austauschdienst (DAAD) for the possibility of carrying out this investigation aided by a number of mutual research visits. Two of us (PM. and HI?) gratefully acknowledge support by the Alexander-von-Humboldt foundation.
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