- Email: [email protected]
Relativistic and QED effects in highly charged ions
Nuclear Instruments and Methods in Physics Research B 98 (1995) l-9
Beam Interactions with Materials & Atoms
EISEVIER
Relativistic and QED effects in highly charged ions E. Lindroth Department of Physics, Chalmers Uniuersity of Technology, University of Gothenburg, S-412 96 GGteborg, Sweden
Abstract We review the recent progress in calculations of high-2 many-electron ions. The relativistic many-body perturbation theory approach is discussed together with the recent development of calculations of two-particle effects beyond it. Theoretical results for several ionic systems are compared with experiments. We discuss also the treatment of effects which vanish in the non-relativistic limit, especially concerning the bound state corrections to the electron g-factor.
1. Introduction We see today a promising experimental development where more and more accurate experiments are made on a wide range of ionic systems. Accurate measurements on highly charged systems can be used to test our understanding of quantum electrodynamics (QED) in the presence of strong fields. Although the accuracy obtainable on hydrogen-like systems has increased significantly recently, most accurate experiments are still done on systems with more than one electron. A careful theoretical treatment of the electron-electron interaction is thus very important. Accordingly, the most recent development on the theoretical side [1,2] concerns two-particle effects which are beyond the standard relativistic many-body perturbation theory (RMBPT) approach [3,4]. In Fig. 1 I have tried to classify the different approaches to very accurate calculations. Quantum electrodynamics is the underlying theory, thus it incorporates also the non-relativistic and relativistic effects which we know are quite good approximations. However, somewhat unprecisely, we often let QED effects refer only to those effects which cannot be described without a full QED treatment. This is also done in this article except for Fig. 1 where QED stands for the whole treatment. In the low 2 region it is possible to make calculations in form of (&-expansions. The g-factor for the free electron is, e.g., calculated extremely accurately by Kinoshita with an cr-expansion. Also the Lamb shift calculations were earlier done in this way until Mohr [5-7] essentially solved the
problem of the self-energy of hydrogenlike systems with a point nucleus. Recently, this work was extended to finitenucleus hydrogenic systems [S]. Calculations of the vacuum polarization in strong fields has been done by Soff and Mohr [9] and an extensive tabulation of the Lamb shift over the whole periodic table has been performed by Johnson and Soff [lo]. For helium and other systems with more than one electron, the problem is to determine the non-relativistic correlation effects accurately enough so that the uncertainty in those does not ruin the effort to calculate the QED effects. In the late fifties Sucher [11,12] used Hylleraas wave functions to describe helium non-relativistically and then he evaluated all corrections of relativistic and QED nature as first order perturbations. An (oZ)-expansion was thus used only for these effects. The same idea has been carried out to perfection by Drake [13] who today has extremely accurate non-relativistic wave functions. With this method it is possible to get very accurate results in the low Z-region. The Hylleraas method, to determine the wave functions, is however limited to two- or perhaps three-electron systems. QED in Atomic Systems one electron
more than one electron aZ expansion
high Z Blundell Gmt
and Ouiney
Fig. 1. Classification of different approaches to accurate calcula* E-mail [email protected], present address: Department of Atomic Physics, Stockholm University, S-104 05 Stockholm, Sweden.
tions. For low Z very accurate results can be obtained with crZ-expansions. In the high Z region, however, the approach must be non-perturbative in aZ.
0168-583X/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0168-583X(95)00021-6
1. FUNDAMENTAL ASPECTS
E. Lindroth /Nucl. Instr. and Meth. in Phys. Res. B 98 (1995) 1-9
2
Very recently methods have been developed which try to combine the ability to treat many-electron systems with a rigorous QED treatment. These methods are based on relativistic many-body perturbation theory (RMBPT). Although perturbation theory carried on to all orders can produce accurate results the methods used today cannot compete with the method of Drake. However, as soon as we leave the neutral systems even a few orders in the perturbation expansion is enough as we will soon see. This paper will briefly, Sections 12-14, discuss RMBPT, its origin and limitations. The recent development concerning effects beyond RMBPT is also discussed and new calculations are compared with experiments. Effects which would not exist with a non-relativistic treatment offer a possibility to study the effect of relativistic and radiative effects unmasked. This possibility will be discussed in Section 5. Forbidden transitions are one such example and bound state corrections to the electron g-factor for an S,,, system is another.
2. Relativistic (RMBPT)
many-body
perturbation
theory
Since the mid-eighties schemes to treat many-body systems completely relativistically have been developed by Johnson et al. [4,14,15], who use an order by order perturbation expansion, and by Lindgren et al. [3,16-191, who use a coupled-cluster approach where certain effects are treated to all orders. Recently relativistic coupled-cluster calculations have also been performed by Ilyabaev and Kaldor [20]. A commonly used alternative approach is the multi-configurational Dirac-Fock method, developed by Desclaux and Indelicate [21,22] and by Grant [23], but this method will not be discussed in this article. All these calculation schemes use a relativistic Hamiltonian with a two-particle interaction projected onto positive energy states. This Hamiltonian can be understood as follows. In non-relativistic theory the starting point is the many-electron Schrodinger equation
In a relativistic treatment the one-particle Schriidinger equation, the first two terms on the right-hand side of Eq. (l), is naturally replaced with the one-particle Dirac equation e2
which then have positive energies. If a positron were present it could be annihilated by one of the atomic electrons and the problem is that the Dirac equation does not know that no positrons exist in vacuum. A form of the electron-electron interaction as in Eq. (1) would result in an unstable atom since two electrons which interact can undergo an Auger-like process where one electron is sent into the positive continuum and one into the negative continuum. This problem can only be solved if we explicitly specify that there are no positrons in vacuum. We thus use a second-quantized representation of the electron-electron interaction [24] and surround it with so called projection operators, A+, which give zero if they work on negative energy states and unity if they work on positive energy states H= Chi+ i
c h+A; (&-isj
+Bi,)h:h:.
(3)
In addition to the ordinary Coulomb interaction between the electrons an additional term, B,,, is added which will be discussed below, see Eq. (5). Note that the notation “positive” and “negative” energy state refer to a certain one-particle description, and its meaning will change if we change the one-particle Hamiltonian from the hydrogen-like to e.g. the Hartree-Fock Hamiltonian. Thus the eigenvalues of H, Eq. (3), depend on the choice of one particle Hamiltonian and the projected interaction can never be the final answer. We know, however, that it is correct to order (Y’ Ry and if we want to go beyond that we have to consider that the field between the electrons can produce virtual electron-positron pairs and then, but only then, are there positrons to annihilate. 2.1. Breit interaction Above the relativistic Hamiltonian was discussed in analogy with the non-relativistic Hamiltonian. Another way is to start directly from quantum electrodynamics (QED) in the so called Furry representation, i.e. from a bound state description (a more detailed discussion can be found in e.g. Ref. [15]). In Coulomb gauge the exchange of one virtual photon give rise to an eJfectiuepotentiaZ[25,26] I
Z
h=ccu~p+pmc2-47iET,
(2) 0
but the expression for the electron-electron interaction remains to be specified. The first problem which we encounter is that the Dirac equation, Eq. (21, provides both positive and negative energy solutions. Unoccupied negative energy states correspond to occupied positron states,
An alternative expression is obtained in Feynman gauge, but although that expression looks simpler it will give problems in higher orders [27], as will be discussed below.
E. Lindroth/Nucl.
Instr. and Meth. in Phys. Res. B 98 (1995) 1-9
If the wavelength of the virtual photon, (- c/o), is large compared to the distance between the electrons, rij, the expression in Eq. (4) can be simplified. A series expansion of cos((w 1c)rij) is then possible and it gives the ordinary Coulomb interaction plus the Breit interaction;
-
e2
1
(Yi. aj
( cYi. rij)( aj. fij)
rij
2rij
2r:
----
4Tg,
i
(5) I
The Coulomb interaction accounts for the electrostatic interaction between the electrons, but in a relativistic treatment the electrons also have spin and thus there is magnetic interaction between them as well. Also the retardation of the electromagnetic field should be accounted for. In a first approximation this is done through the Breit interaction, but if retardation is to be included completely, the series expansion of cos((o ( c)ri,) above has to be avoided. The expression in Eq. (5) for the electron-electron interaction is, however, correct to order (Y* Ry and is what is used in the projected Hamiltonian, i.e.
(“i.‘ij)Ccrj”rj)
Afh+
-
2r:
I I
(6)
Ii
which is the basis for RMBFT. Two particle effects beyond this have only recently been considered [ 1,2] and it is these effects which one would like new accurate experiments to test.
3. Beyond RMBPT When the Hamiltonian in Eq. (6) is used as the basis for many-body perturbation theory the expression in Eq. (5) is often used as the perturbation (alternatively part of the electron-electron interaction can be accounted for in the single particle Hamiltonian and the rest treated as a perturbation). An iterative scheme then gives ladder diagrams, indicated in Fig. 2a, but diagrams where the photons are allowed to cross each other, indicated in Fig. 2b, will not be accounted for. Again to order (Y* Ry only the ladder diagrams contribute. The crossed photons enter in order a3Z3 Ry. However, this is only true when Coulomb gauge is used. In the alternative Feynman gauge, where the two-particle interaction actually has a simpler expression, crossed photons contribute already in order (Y’ Ry [27]. Since crossed photons are more complicated to evaluate, Feynman gauge is less suitable for calculations of manyelectron systems in most regions of the periodic table. The crossed photons is one example of an effect outside conventional RMBPT. Other examples already discussed are the effects of virtual electron-positron pairs and retardation beyond the Breit interaction. The former effect contributes in order cr3Z3 Ry and beyond. For equivalent
(a)
(b)
Cc)
(4
Hi’: 0
0
(e)
(0
Fig. 2. Two-photon Feynman diagrams which involve more than one electron. Relativistic many-body perturbation theory accounts approximately for the ladder diagram, diagram (a), what is neglected is the retardation of the electromagnetic field beyond the Breit interaction and virtual electron-positron pair effects. Contributions from diagram (b), crossed photons, are also neglected in RMBPT. The terms neglected in RMBPT from diagram (a) and (b) are here referred to as Araki-Sucher terms. The diagrams (c)-(f) represent the two-electron radiative effects, also called the screening of the Lamb shift. electrons, as for the helium-like ground state, this is also true for the latter effect. For non-equivalent electrons, however, the retardation beyond the Breit interaction affects also, through the exchange term, the expectation value of the electron-electron interaction. This contribution enters in order a4Z5 Ry and can be quite important [28,29]. Since the cr3Z3-effects discussed here first were evaluated, in the non-relativistic limit, by Araki 1301 and Sucher [11,12] they are sometimes referred to as the Araki-Sucher terms. It should be noted, however, that a subset of the a3Z3-terms calculated by Araki and by Sucher are included already in RMBPT when the expression in Eq. (5) is evaluated in second order. Other terms hitherto neglected are the radiative effects. The most important is the self-energy and the next in importance is the vacuum polarization, which both contribute in order a3Z4 Ry and beyond, for hydrogen-like systems. In many-electron systems we have also two-particle effects of radiative nature, sometimes called the screening of the Lamb shift, which contribute in order cu3Z3 Ry and beyond, see Figs. 2c-2f. 3.1. A Z-expansion Although Z-expansions are not the basis for any modern calculations it is illustrative to summarize the contributions to the binding energy of many-electron systems as follows; E = ~nrel + AEtel + AEQED (7) The contributions to the first term on the right-hand side of Eq. (7) have the following Z-dependence 1 1 Enre’ = a,Z’ + a,Z + a32 + a42 + . . .
(8)
1. FUNDAMENTAL ASPECTS
E. Lindroth / Nucl. Instr. and Meth. in Phys. Res. B 98 (1995) l-9
4
3.2. Comparison with experiments
These are the non-relativistic parts which more or less always dominate the binding energies. The first two terms, the hydrogen-like energy and the expectation value of electron-electron interaction, are trivial. The rest, which we can call correlation, are always approximated. For few electron systems correlation can however be accurately calculated. For every order in the perturbation expansion another factor of l/Z enters, thus it converges faster for highly ionized systems. The contributions to the second term on the right-hand side of Eq. (7) have different Z-dependences AE”’ = a2Z2
b,Z2 + b,Z + b2Zo + b3;
+ . .
A few years ago Schweppe et al. performed an experiment on lithium-like uranium [31], where the splitting of the 2~-2pt,~ levels was measured very accurately, with an error of only 0.1 eV. This stimulated several groups to do accurate calculations and results in good agreement have been presented e.g. by Johnson et al. and by Lindgren et al. The result from these calculations are presented in Table 1. The new achievements concerned especially the so called screening of the Lamb shift [32-341. The self-energy is a complicated effect for which there is no simple potential. Since the work of Mohr the hydrogen-like selfenergy, for any Z, can be considered as well known. However, that procedure assumes a Coulomb potential from the nucleus and it cannot easily be extended to any potential. Only recently was it possible to evaluate the self-energy in a screened nuclear potential. The self-energy results given in Table 1 include the screening, which is of order - 1.6 eV since the hydrogen-like result is 55.9 eV [lo]. Although the methods in Refs. [35] and [36] are rather different the results agree well. Since the experimental uncertainty is only 0.1 eV an account of the screening is necessary in order to obtain agreement. The comparison in Table 1 includes also a few effects of the same order as the experimental uncertainty. The nuclear polarization, calculated by Plunien et al. [37,38], is one of them and the cross term between the vacuum polarization and the self-energy is another. The latter is the only calculated second order Lamb shift effect. There are several such effects which are assumed to be of the same order of magnitude. The so called Breit-Coulomb reducible correction is one of the Araki-Sucher terms, see above, and the other terms of this class are expected to be of the same order. The agreement with experiment is very good although several effects of magnitude comparable to the experimental accuracy are missing and might be fortuitous. The lacking terms have not yet been calculated for lithium-like systems, but very recently calculations of the
. (9
Here only the leading relativistic corrections are listed. Together with certain terms of order a3 Ry (from the two-electron part only), a4 Ry and beyond they are accounted for by RMBPT. Again we have the leading correction to the hydrogen-like result and then the leading correction to the expectation value of the electron-electron interaction, which are well known and rather trivial. The third term is the first relativistic correction to the correlation. Finally we have the last term on the right-hand side of Eq. (7), ~EQno=c
1
a3z4+c
2
a3z3
. .
(10)
the radiative parts and the non-radiative parts (Araki-Sucher terms) which are outside the RMBPT treatment. The leading contribution is due to the hydrogen-like self-energy and vacuum polarization, which are quite well understood. The contributions of order a3Z3 Ry are effects which involve more than one electron. Among these are the self-energy screening, the effects of crossed photons and of negative energy states. The part of the binding energy which is not that well understood, or tested, is the a3Z3 Ry contributions. Since any binding energy is dominated by non-relativistic effects it is clear that accurate experiments are needed to really test the calculations.
Table 1 Transition energy for the 2p,,,-2s,,,
transition in Li-like uranium (eV)
RMBRISelf energy (including screening) Vacuum polarization (including screening) Nuclear recoil and polarization Mixed self energy-vacuum polarization Breit-Coulomb reducible correction Total Experiment
a
’ Schweppe
et al. [31].
correction
Lindgren et al. [35]
Blundell et al. [36]
322.33(15) - 54.32(3) 12.59(l) 0.10 -0.19 0.04
322.41 - 54.24(2) 12.56(l) 0.10 _
280.55(20) 280.59(9)
280.83(10)
E. Lindroth/Nucl. Table 2 Two-electron 2
to the ground state energy of helium like ions (eV)
RMBPT a
32 54 74 83 a b ’ d e
contribution
Instr. and Meth. in Phys. Res. B 98 (1995) 1-9
Total ’
QED
1st order
2nd order
Araki-Sucher
567.6 1036.5 1586.9 1897.5
-5.2 - 6.9 -9.4 - 11.0
0.03 0.16 0.55 0.85
’
Two-electron
Experiment Super-Ebit
Lamb shift d 561.9 1028.1 1574.6 1882.7
-0.5 - 1.6 -3.4 -4.5
f f f i
0.5 1.6 3.4 4.5
562.5 1027.2 1579. 1878.
b
+ 1.6 + 3.5 * 15 + 14
I. Lindgren private communication. Obtained with an extended nucleus with uniform charge distribution. R.E. Marrs, S.R. Elliott and T. StBhlker, unpublished. Lindgren et al. [Z]. The two-electron (screening) Lambshift is only estimated, more detailed calculations are under way. 1. Lindgren private communication. The unceratinty is estimated to be of the order of the two-ele. .L:on Lamb shift which is yet not calculated from first principles.
Araki-Sucher terms have appeared for the ground state of helium [1,2]. The ground state of helium-like systems is attractive for calculations, but usually a direct comparison with experiment is not possible. However, in a new experiment from the Super-Ebit [39] the two-particle contributions to helium-like systems have been measured directly. The experiment measures the wavelength of the emitted photon when electrons recombine with bare ions and with hydrogen-like ions, respectively. The difference in photon energy corresponds directly to the two-electron part of the binding energy of the helium-like ground state. A comparison between the experimental results and preliminary calculations is shown in Table 2. The agreement is good, but the experimental uncertainty is still too large to test the more recent theoretical achievements, i.e. the Araki-Sucher terms in the third column. The two-electron Lamb shift, i.e. the screening of the Lamb shift, is here on the edge of observation.
4. Many electron ions Helium- and lithium-like systems are few-electron ions where the many-body effects are still limited, but it is also interesting to see how well more complicated systems can be treated. Alkali-like ions and copper-like ions have been studied by Blundell et al. [40-42,361, where the later references include also the screening of the Iamb shift. From atomic calculations it is known that beryllium can be considered as the first real many-body system. Due to the strongly interacting outer pair of electrons it is considerably more complicated than lithium. Examples of all order coupled-cluster calculations for beryllium-like systems are shown in Table 3. Even here the screening of the Lamb shift is necessary in order to get agreement with experiment [43,44]. The screening is here estimated with an approximative method [22]. In addition the RMBPT calculation includes a first estimate of the correlation correction to the Lamb shift. Since the (2s2)‘S, ground state of beryllium like ions has a large admixture of the (2~~)’
configurations the Lamb shift contribution is corrected accordingly. This correction to the ground state binding energy affects the splitting between the 2%, and 21,3P, levels with 0.00196 a.u. (430 cm-‘) and 0.00605 a.u. (1328 cm-‘) for iron and molybdenum respectively. As can be seen in Table 3 effects of this size are for most transitions one order of magnitude larger than the experimental accuracy. This is an indication that more severe tests of our ability to combine radiative and many-electron
Table 3 Comparison of theory and experiment for the (2~2p~‘~~P,-(2s~)‘S, splittings in beryllium-like iron and beryllium-like molybdenum
(2s2p)3P,-(2s%, RMBPT ’ (a.u.) d Screening of the Lamb shift ’ (a.u.) Total (a.u.) Total (cm-‘) Experiment
Fez2 +
Mo3’+
1.72513 k 0.00007 0.00227 f 0.00040
3.29924 + 0.00020 0.00755 + 0.00130
1.72740 + 0.00041
3.30679 + 0.00135
379118+90 37914OrtZO”
725751+ 300 725758+158b
3.42645 * 0.00020 0.00204 + 0.00035
9.12633 + 0.00030 0.00673 f 0.00115
3.42849 + 0.00040
c 9.13306 + 0.00120
752459 f 90 752372 f 57 a
2004464 f 300 2003847+1200b
(cm-’ ) (2s2pY P, -(Zs%S, RMBPT ’ (a.u.1 d Screening of the Lamb shift ’ (a.u.1 Total (a.u.) Total (cm Experiment (cm-’ )
’)
a Hinnov [44]. b Denne et al. [43]. ’ Lindroth and Hvarfner [19]. d 1 a.u. (56Fe) = 2.194725X lo5 cm-’ using the value R, = 109736.2 cm-’ (M = 55.9349) for the Rydberg constant. 1 a.u. c9sMo) = 2.194734X lo5 cm-’ using the value R, = 109736.7 cm - ’ (M = 97.9055) for the Rydberg constant. ’ Indelicate and Desclaux [22].
1. FUNDAMENTAL
ASPECTS
E. Lindroth /Nucl. Instr. and Meth. in Phys. Res. B 98 (199.5) l-9
6
effects could probably be carried out in e.g. beryllium-like systems.
5. Effects which vanish in the non-relativistic
limit
We can also test our understanding of QED and relativity on effects which are forbidden non-relativistically. Calculations of such effects do not suffer from the problem with large non-relativistic contributions which have to be carefully calculated before a meaningful comparison can be done. An example of a pure relativistic effect is the possibility of so called forbidden transitions. Electric dipole transitions between singlet and triplet states constitute one kind of such transitions and it has been studied e.g. in beryllium-like systems by beam-foil spectroscopy. The transition rate scale then as a!‘Z”. In spite of long term theoretical efforts, including very recent work [45,46] the agreement between theory and experiment is, however, still not really satisfying. The effect on the forbidden transitions rates from the inclusion of Breit interaction is significant [45,46], but the possible effects from radiative contributions are not studied yet. A review of experimental results and agreement with theory can be found in Ref. 1471. Since many years there has been an interest in the forbidden magnetic dipole transition, 23S, + l’s,, in helium-like systems. The transition rate scale here as CI’Z’~. A recent experiment from the TSR storage ring in Heidelberg [48] measured the decay rate of the 23S,-state in helium-like carbon with a relative error of only 0.2 percent. The agreement with the calculation by Drake [49] is here very good. This is in spite of the fact that the calculation in Ref. [49] is done with an aZ-expansion of the relativistic effects and that radiative effects are not considered. The excellent agreement indicates the importance of electron correlation over relativistic effects in this region of Z. A fully correlated relativistic calculation has been performed for helium-like argon [17]. For this more highly charged ion relativistic effects are more important resulting in a deviation from the result by Drake [49] by 1.4%. However, no experiment of such an accuracy has been performed on argon. Another quantity which does not exist in the non-relativistic limit is the bound state corrections to the electron g-factor, which is discussed in some detail below. 5.1. The electron
g-factor
we get in addition the Schwinger correction which is of order - a/n. The experimental result from Dehmelt et al. is -2 X (1 + 0.001 159 652 188c4.3)) [50], whereas the theoretical result by Kinoshita is - 2 X (1 + 0.001 159,652 140(5.3x4.1)(27)) [51]. The first two uncertainties in the theoretical value come from the numerical errors in the o-expansion and the third and largest term comes from the uncertainty in the fine-structure constant. For bound electrons the g-factor changes slightly. These changes are due to relativistic and radiative effects only. For s-states in hydrogenlike systems (neglecting nuclear spin and nuclear size effects) we can write; (gy),?=
+cu3~2..
e
P = &G’.
(11)
Classically g, would be - 1, whereas Dirac’s relativistic theory gives a value of -2. From the full QED treatment
+;$
. (12) Ia
where E is the relativistic binding energy. The third term on the right-hand side is due to bound state relativistic corrections, while the last term on the right-hand side indicates the presence of bound state radiative corrections [52]. The latter have been studied theoretically in the non-relativistic limit by Hegstrom and Grotch [53,54], but no calculations exist for highly charged hydrogenlike systems. Experiments on highly charged hydrogenlike systems are planned, but results are still not available. However, accurate experimental results from ion traps have been presented for singly charged ions e.g. for Be+, Mg’, and Ba+ [55-581. For these systems the bound state relativistic corrections, the third term on the right-hand side of Eq. (12), cannot be given in analytical form since it is affected also by the electron-electron interaction. For these singly ionized systems the bound state radiative corrections are at the limit of being visible. Slightly more accurate experiments or a modest increase in charge would probably be enough to reveal these effects. This requires though a careful theoretical treatment of the bound state relativistic corrections. Although calculations in the nonrelativistic limit have been successfully performed for light elements [59,60] even medium heavy elements, as neutral rubidium, are strongly influenced by relativistic effects beyond that [61,59]. However, an accurate, fully correlated, relativistic treatment is possible as shown in Ref. 1621, and discussed below. It should also be noted that for higher charge states the electron-electron effects are relatively less important, making the isolation of the bound state radiative corrections easier. X1.1.
One of the more important tests of QED is the successful prediction of the magnetic moment of the free electron
-2(1+0.0011596-..
Theory
The interaction
is considered H”
= chm
I
between the atom and a magnetic field through the addition of a term
= xecai.Ai
(13)
to the relativistic no-pair Hamiltonian Eq. (3). In Eq. (3) the two-particle terms, the Coulomb as well as the Breit
E. Lindroth /Nucl. Instr. and Meth. in Phys. Res. B 98 (1995) 1-9
interaction, are surrounded with projection operators onto positive energy states. In addition there are contributions from virtual pairs which have to be considered already in order (Ye, see Ref. [62]. Radiative effects are, however, omitted and the effect of this omission has also been discussed in Ref. [62]. If the magnetic field is assumed to be homogeneous over the extension of the atom the vector potential A can be written A = -;(rxB)
(14)
and thus hm=
-~(nXr).B=~iJZr(nC’}‘.B.
(15)
Classically the energy for a magnetic dipole in a magnetic field is given as the scalar product between the dipole and the field. The magnetic dipole created by the motion of a charged particle is proportional to its angular momentum and we write E=
-p.B=
-p&j.B=
-gi!fj.B,
(16)
where the gj factor, with a classical value of - 1, has been introduced. Eqs. (15) and (16) can now be combined to give an expression for the gj factor. gj( j) = -(mcJZir{cuC1}l).
(17)
From Eq. (17) the result for a free electron is gDirac = - 2. Corrections to this value have two main sources. First rudiuliue effects, omitted in Eq. (17), give the Schwinger correction of approximately -0.002319, for a free electron. Secondly the expression (17) gives relativistic corrections to the gj factor for a bound electron, which are not present in the non-relativistic limit. The relativistic corrections scale as Z2a2 and are of the order 10m4 or 1O-5 in light systems. Finally there are also bound stute
7
radiative corrections, these are smaller then the bound state relativistic corrections by one order of the fine-structure constant. If the expression in Eq. (17) is used directly, together with relativistic wave functions, it will thus give the non-relativistic factor of -2 and, in addition to that, the bound state corrections. Due to the relative smallness of these corrections the requirement on the numerical accuracy will be very high. In addition; the absence of non-relativistic bound state corrections is not explicit in Eq. (17) but has to be achieved by numerical means. It would certainly be desirable to have an operator which instead gave the corrections directly. Such an operator was also derived in Ref. [62]. 5.1.2. Results A fully relativistic treatment has been applied to a few systems and the contributions from different classes of effects are summarized and compared with experiments [55-581 in Table 4. For Be+ around 95% of the final result is given by the sum of the Dirac-Fock value and the subset of many-body effects which can be included via the random phase approximation (RPA). Correlation beyond the RPA-like diagrams contribute 1.5% and the total effect of the Breit interaction is 2.3% for Be+. For Ba+ the situation is very different. The lowest order result even shows the wrong sign. After inclusion of the RPA chains, which are completely dominated by the polarization of the 5p orbitals, the result is 20% above the experimental result. The remaining contributions come from the Breit interaction and the correlation due to the Coulomb interaction which contribute with 10% each of the final result. The theoretical uncertainties are estimated contributions from radiative corrections, which are probably less than 1% as discussed in Ref. [62], the pure numerical uncertainty and the uncertainty caused by the truncation of the
Table 4 Contributions
to the bound state corrections
to the g, factor from different
classes of diagrams.
The corrections
electron value, g, = - 2.002 319 304 377 (91, and obtained as 6g = gi - g,. The results are displayed
Lowest order, Dirac-Fock A Dirac-Fock-Breit RPA-like diagram . A Dirac-Fock-Breit Breit RPA-like diagrams Coulomb correlation A Dirac-Fock-Breit Breit correlation Total Experiment
Be+
Mg+
Ba’
4.932 -0.124 0.477 - 0.023 0.35 1 0.090 - 0.004 - 0.046
5.217 0.125 0.110 - 0.003 0.613 0.474 - 0.001 0.004
5.539 0.708 - 26.455 0.462 0.824 1.985 0.089 - 0.330
5.653 f 0.050 5.694 f 0.033 a
6.54 f 0.06 6.52 f 0.03 b
are relative to the free
as 6g X 10’
- 17.18 + 0.3 - 17.13 + 0.12 c
a Wineland et al. [55,56]. b Bollinger et al. [57]. ’ Knab et al. [58].
1. FUNDAMENTAL
ASPECTS
8
E. Lindroth/Nucl.
Instr. and Meth. in Phys. Res. B 98 (1995) l-9
partial wave expansion of the two-particle interactions. For Be+ the latter is small since many terms could be kept in the expansion. The lowest order result is in addition the dominant contribution and higher order effects show a stable behavior. An estimated uncertainty of a few percent in the many-body part of the calculation gives an overall uncertainty of around 0.1%. Thus an estimated error of 1% should not be too optimistic. The behavior of Ba+ is considerably less stable. The lowest order result has the wrong sign and the situation is restored only after inclusion of RPA effects. We assume that the overall uncertainty due to numerical uncertainties and truncations could lead to an error of around 1%. To account also for possible radiative contributions we estimate the total error to two percent.
6. Conclusions The interesting experimental development which have resulted in many studies of ions of varying charge states has been accompanied by a promising theoretical development. For simple systems such as helium- or lithium-like systems accurate calculations can today be done including two-particle radiative corrections and other effects which are beyond a standard relativistic treatment of the electron-electron interaction. For the near future we can expect calculations of this kind to be carried out also for more complicated systems. For transition rates and atomic properties in general the theoretical treatment has not yet reached the same accuracy. However, with increasing experimental and theoretical accuracy, especially concerning completely relativistic effects, there is a lot to learn also in this field.
Acknowledgements I am grateful to R.E. Marrs, S.R. Elliott and T. Stijhlker for allowing me to show their results prior to publication. I have benefitted from discussions with Prof. I. Lindgren, Dr. A.-M. M%rtensson-Pendrill, Dr. S. Salomonson and Dr. H. Persson. Financial support was received from the Swedish Natural Science Research Council (NFR).
References t11 S.A. Blundell, P.J. Mohr, W.R. Johnson
and J. Sapirstein, Phys. Rev. A 48 (19931 2615. El I. Lindgren, H. Persson and S. Salomonson, Phys. Rev. A accepted (1994). t31 S. Salomonson and P. Qster, Phys. Rev. A 40 (1989) 5548. [41 W.R. Johnson, S. Blundell and J. Sapirstein, Phys. Rev. A 37 (1988) 307.
[51 P.J. Mohr, Ann. Phys. (NY) 88 (1974) 26.
[d P.J. Mohr, Ann. Phys. (NY) 88 (1974152. [71 P.J. Mohr, Phys. Rev. A 46 (1992) 4421.
[81P.J. Mohr and G. Soff, Phys. Rev. Lett. 70 (1993) 158. [91 G. Soff and P.J. Mohr, Phys. Rev. A 38 (1988) 5066. DOI W.R. Johnson and G. Soff, Atom. Data Nucl. Data Tables 33 (1985) 405.
Hll J. Sucher, Phys. Rev. 109 (1958) 1010. WI J. Sucher, thesis, Coulombia University 1957. I131 G.W.F. Drake, Can. J. Phys. 66 (19881 586. [I41 S.A. Blundell, W.R. Johnson, Z.W. Liu and J. Sapirstein, Phys. Rev. A 40 (198912233.
D51 J. Sapirstein, Phys. Scripta T 46 (1993) 52.
b51 E. Lindroth, Phys. Rev. A 37 (1988) 316. [171 E. Lindroth
and S. Salomonson, Phys. Rev. A 41 (1990) 4659. [181E. Lindroth, H. Persson, S. Salomonson and A.-M. Mittensson-Pendrill, Phys. Rev. A 45 (1992) 1493. I191 E. Lindroth and J. Hvarfner, Phys. Rev. A 45 (1991) 2771. La E. Ilyabaev and U. Kaldor, Phys. Rev. A 47 (1993) 137. Dll J.P. Desclaux, Computer Physics Communications 9 (1975) 31. La P. Indelicate and J.P. Desclaux, Phys. Rev. A 42 (1990) 5139. [231 K.G. Dyall et al., Comput. Phys. Commun. 55 (19891425. t241 I. Lindgren and J. Morrison, Atomic Many-Body Theory, Series on Atoms and Plasmas, 2nd ed. (Springer, Berlin, 19861. [25] I. Lindgren, in: Program on Relativistic, Quantum Electrodynamics, and Weak Interaction Effects in Atoms (AIP Conf. Ser. No. 189, ITP Santa Barbara, 1989). [26] I. Lindgren, Nucl. Instr. and Meth. B 31 (19881 102. [27] I. Lindgren, J. Phys. B 23 (1990) 1085. [28] S.A. Blundell, W.R. Johnson and J. Sapirstein, Phys. Rev. A 41 (1990) 1698. [29] P. Indelicate and E. Lindroth, Phys. Rev. A 46 (1992) 2426. [30] H. Araki, Prog. Theor. Phys. 17 (1957) 619. [31] J. Schweppe et al., Phys. Rev. Lett. 66 (1991) 1434. [32] P. Indelicate and P.J. Mohr, Theoretica Chemica Acta 80 (1991) 207. [33] S.A. Blundell and N.J. Snyderman, Phys. Rev. A 44 (1991) R1427. [34] H. Persson, I. Lindgren and S. Salomonson, Phys. Rev. A 47 (1993) R4555. r351 I. Lindgren et al., J. Phys. B 26 (1993) L503. [361 S.A. Blundell, Phys. Rev. A 47 (1993) 1790. [371 G. Plunien, B. Miiller, W. Greiner and G. Soff, Phys. Rev. A 39 (1989) 5428. [381 G. Plunien, B. Miller, W. Greiner and G. Soff, Phys. ‘Rev. A 43 (1991) 5853. [391 R.E. Marrs, S.R. Elliott and T. Stohlker, unpublished. [401 S.A. Blundell, D.S. Guo, W. Johnson and J. Sapirstein, Atom. Data Nucl. Data Tables 37 (1987) 103. [411 W.R. Johnson, S.A. Blundell and J. Sapirstein, Phys. Rev. A 42 (1990) 1087. [421 S.A. Blundell, Phys. Rev. A 46 (1992) 3762. I431 B. Denne, G. Magyar and J. Jacquinot, Phys. Rev. A 40 (1989) 3702. Ml E. Hinnov, referenced as private communication in Ref. [43]. [451 A. Ynnerman and C. Froese-Fischer, to be published.
E. Lindroth /Nucl. Instr. and Meth. in Phys. Rex B 98 (1995) 1-9 [46] [47] [48] [49] [50]
S. Fritzsche and I.P. Grant, to be published. E. Trgbert, Phys. Scripta 48 (1993) 699. H.T. Schmidt et al., Phys. Rev. Lett. 72 (1994) 1616. G.W.F. Drake, Phys. Rev. A 3 (1971) 908. R.S.V. Dyck, P.B. Schwinberg and H.G. Dehmelt, Phys. Rev. Lett 59 (1987) 26. [51] T. Kinoshita and D.R. Yennie, in: Quantum Electrodynamics, ed. T. Kinoshita (World Scientific, Singapore, 1990). [52] The change in the bound state relativistic corrections due to the anomalous part of the magnetic moment of a free electron is here included among the bound state radiative corrections. This particular term amounts to -0.001 159 6.. . fE/mc’ and is thus not just equal to the product of the free particle radiative correction and the relativistic corrections. [53] R.A. Hegstrom, Phys. Rev. 184 (1969) 17.
9
[54] H. Grotch and R.A. Hegstrom, Phys. Rev. 4 (1971) 59. [55] D.J. Wineland, J.J. Bollinger and W.M. Itano, Phys. Rev. Lett. 50, 628 (1983). [56] The result gj = -2.002 262 36 (31) is obtained from Ref. [55] when the most recent values for the mass ratios; mp /m, = 1836.152 701(37) and m(gBe’>/m, = 8.946 534 45 (41) are used. [57] J.J. Bollinger, J.M. Gilligan, W.M. Itano, F.L. Moore and D.J. Wineland, private communication. [58] H. Knab, K. Knoll, F. Scheerer and G. Werth, Z. Phys. D 26 (1993) 205. [59] L. Veseth, Phys. Rev. 22 (1980) 803. [60] L. Veseth, J. Phys. B 16 (1983) 2891. [61] V.V. FIambaum, LB. Khriplovich and O.P. Sushkov, Phys. Len. A 67 (1978) 177. [62] E. Lindroth and A. Ynnerman, Phys. Rev. A 47 (1993) 961.
1. FUNDAMENTAL
ASPECTS
Beam Interactions with Materials & Atoms
EISEVIER
Relativistic and QED effects in highly charged ions E. Lindroth Department of Physics, Chalmers Uniuersity of Technology, University of Gothenburg, S-412 96 GGteborg, Sweden
Abstract We review the recent progress in calculations of high-2 many-electron ions. The relativistic many-body perturbation theory approach is discussed together with the recent development of calculations of two-particle effects beyond it. Theoretical results for several ionic systems are compared with experiments. We discuss also the treatment of effects which vanish in the non-relativistic limit, especially concerning the bound state corrections to the electron g-factor.
1. Introduction We see today a promising experimental development where more and more accurate experiments are made on a wide range of ionic systems. Accurate measurements on highly charged systems can be used to test our understanding of quantum electrodynamics (QED) in the presence of strong fields. Although the accuracy obtainable on hydrogen-like systems has increased significantly recently, most accurate experiments are still done on systems with more than one electron. A careful theoretical treatment of the electron-electron interaction is thus very important. Accordingly, the most recent development on the theoretical side [1,2] concerns two-particle effects which are beyond the standard relativistic many-body perturbation theory (RMBPT) approach [3,4]. In Fig. 1 I have tried to classify the different approaches to very accurate calculations. Quantum electrodynamics is the underlying theory, thus it incorporates also the non-relativistic and relativistic effects which we know are quite good approximations. However, somewhat unprecisely, we often let QED effects refer only to those effects which cannot be described without a full QED treatment. This is also done in this article except for Fig. 1 where QED stands for the whole treatment. In the low 2 region it is possible to make calculations in form of (&-expansions. The g-factor for the free electron is, e.g., calculated extremely accurately by Kinoshita with an cr-expansion. Also the Lamb shift calculations were earlier done in this way until Mohr [5-7] essentially solved the
problem of the self-energy of hydrogenlike systems with a point nucleus. Recently, this work was extended to finitenucleus hydrogenic systems [S]. Calculations of the vacuum polarization in strong fields has been done by Soff and Mohr [9] and an extensive tabulation of the Lamb shift over the whole periodic table has been performed by Johnson and Soff [lo]. For helium and other systems with more than one electron, the problem is to determine the non-relativistic correlation effects accurately enough so that the uncertainty in those does not ruin the effort to calculate the QED effects. In the late fifties Sucher [11,12] used Hylleraas wave functions to describe helium non-relativistically and then he evaluated all corrections of relativistic and QED nature as first order perturbations. An (oZ)-expansion was thus used only for these effects. The same idea has been carried out to perfection by Drake [13] who today has extremely accurate non-relativistic wave functions. With this method it is possible to get very accurate results in the low Z-region. The Hylleraas method, to determine the wave functions, is however limited to two- or perhaps three-electron systems. QED in Atomic Systems one electron
more than one electron aZ expansion
high Z Blundell Gmt
and Ouiney
Fig. 1. Classification of different approaches to accurate calcula* E-mail [email protected], present address: Department of Atomic Physics, Stockholm University, S-104 05 Stockholm, Sweden.
tions. For low Z very accurate results can be obtained with crZ-expansions. In the high Z region, however, the approach must be non-perturbative in aZ.
0168-583X/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0168-583X(95)00021-6
1. FUNDAMENTAL ASPECTS
E. Lindroth /Nucl. Instr. and Meth. in Phys. Res. B 98 (1995) 1-9
2
Very recently methods have been developed which try to combine the ability to treat many-electron systems with a rigorous QED treatment. These methods are based on relativistic many-body perturbation theory (RMBPT). Although perturbation theory carried on to all orders can produce accurate results the methods used today cannot compete with the method of Drake. However, as soon as we leave the neutral systems even a few orders in the perturbation expansion is enough as we will soon see. This paper will briefly, Sections 12-14, discuss RMBPT, its origin and limitations. The recent development concerning effects beyond RMBPT is also discussed and new calculations are compared with experiments. Effects which would not exist with a non-relativistic treatment offer a possibility to study the effect of relativistic and radiative effects unmasked. This possibility will be discussed in Section 5. Forbidden transitions are one such example and bound state corrections to the electron g-factor for an S,,, system is another.
2. Relativistic (RMBPT)
many-body
perturbation
theory
Since the mid-eighties schemes to treat many-body systems completely relativistically have been developed by Johnson et al. [4,14,15], who use an order by order perturbation expansion, and by Lindgren et al. [3,16-191, who use a coupled-cluster approach where certain effects are treated to all orders. Recently relativistic coupled-cluster calculations have also been performed by Ilyabaev and Kaldor [20]. A commonly used alternative approach is the multi-configurational Dirac-Fock method, developed by Desclaux and Indelicate [21,22] and by Grant [23], but this method will not be discussed in this article. All these calculation schemes use a relativistic Hamiltonian with a two-particle interaction projected onto positive energy states. This Hamiltonian can be understood as follows. In non-relativistic theory the starting point is the many-electron Schrodinger equation
In a relativistic treatment the one-particle Schriidinger equation, the first two terms on the right-hand side of Eq. (l), is naturally replaced with the one-particle Dirac equation e2
which then have positive energies. If a positron were present it could be annihilated by one of the atomic electrons and the problem is that the Dirac equation does not know that no positrons exist in vacuum. A form of the electron-electron interaction as in Eq. (1) would result in an unstable atom since two electrons which interact can undergo an Auger-like process where one electron is sent into the positive continuum and one into the negative continuum. This problem can only be solved if we explicitly specify that there are no positrons in vacuum. We thus use a second-quantized representation of the electron-electron interaction [24] and surround it with so called projection operators, A+, which give zero if they work on negative energy states and unity if they work on positive energy states H= Chi+ i
c h+A; (&-isj
+Bi,)h:h:.
(3)
In addition to the ordinary Coulomb interaction between the electrons an additional term, B,,, is added which will be discussed below, see Eq. (5). Note that the notation “positive” and “negative” energy state refer to a certain one-particle description, and its meaning will change if we change the one-particle Hamiltonian from the hydrogen-like to e.g. the Hartree-Fock Hamiltonian. Thus the eigenvalues of H, Eq. (3), depend on the choice of one particle Hamiltonian and the projected interaction can never be the final answer. We know, however, that it is correct to order (Y’ Ry and if we want to go beyond that we have to consider that the field between the electrons can produce virtual electron-positron pairs and then, but only then, are there positrons to annihilate. 2.1. Breit interaction Above the relativistic Hamiltonian was discussed in analogy with the non-relativistic Hamiltonian. Another way is to start directly from quantum electrodynamics (QED) in the so called Furry representation, i.e. from a bound state description (a more detailed discussion can be found in e.g. Ref. [15]). In Coulomb gauge the exchange of one virtual photon give rise to an eJfectiuepotentiaZ[25,26] I
Z
h=ccu~p+pmc2-47iET,
(2) 0
but the expression for the electron-electron interaction remains to be specified. The first problem which we encounter is that the Dirac equation, Eq. (21, provides both positive and negative energy solutions. Unoccupied negative energy states correspond to occupied positron states,
An alternative expression is obtained in Feynman gauge, but although that expression looks simpler it will give problems in higher orders [27], as will be discussed below.
E. Lindroth/Nucl.
Instr. and Meth. in Phys. Res. B 98 (1995) 1-9
If the wavelength of the virtual photon, (- c/o), is large compared to the distance between the electrons, rij, the expression in Eq. (4) can be simplified. A series expansion of cos((w 1c)rij) is then possible and it gives the ordinary Coulomb interaction plus the Breit interaction;
-
e2
1
(Yi. aj
( cYi. rij)( aj. fij)
rij
2rij
2r:
----
4Tg,
i
(5) I
The Coulomb interaction accounts for the electrostatic interaction between the electrons, but in a relativistic treatment the electrons also have spin and thus there is magnetic interaction between them as well. Also the retardation of the electromagnetic field should be accounted for. In a first approximation this is done through the Breit interaction, but if retardation is to be included completely, the series expansion of cos((o ( c)ri,) above has to be avoided. The expression in Eq. (5) for the electron-electron interaction is, however, correct to order (Y* Ry and is what is used in the projected Hamiltonian, i.e.
(“i.‘ij)Ccrj”rj)
Afh+
-
2r:
I I
(6)
Ii
which is the basis for RMBFT. Two particle effects beyond this have only recently been considered [ 1,2] and it is these effects which one would like new accurate experiments to test.
3. Beyond RMBPT When the Hamiltonian in Eq. (6) is used as the basis for many-body perturbation theory the expression in Eq. (5) is often used as the perturbation (alternatively part of the electron-electron interaction can be accounted for in the single particle Hamiltonian and the rest treated as a perturbation). An iterative scheme then gives ladder diagrams, indicated in Fig. 2a, but diagrams where the photons are allowed to cross each other, indicated in Fig. 2b, will not be accounted for. Again to order (Y* Ry only the ladder diagrams contribute. The crossed photons enter in order a3Z3 Ry. However, this is only true when Coulomb gauge is used. In the alternative Feynman gauge, where the two-particle interaction actually has a simpler expression, crossed photons contribute already in order (Y’ Ry [27]. Since crossed photons are more complicated to evaluate, Feynman gauge is less suitable for calculations of manyelectron systems in most regions of the periodic table. The crossed photons is one example of an effect outside conventional RMBPT. Other examples already discussed are the effects of virtual electron-positron pairs and retardation beyond the Breit interaction. The former effect contributes in order cr3Z3 Ry and beyond. For equivalent
(a)
(b)
Cc)
(4
Hi’: 0
0
(e)
(0
Fig. 2. Two-photon Feynman diagrams which involve more than one electron. Relativistic many-body perturbation theory accounts approximately for the ladder diagram, diagram (a), what is neglected is the retardation of the electromagnetic field beyond the Breit interaction and virtual electron-positron pair effects. Contributions from diagram (b), crossed photons, are also neglected in RMBPT. The terms neglected in RMBPT from diagram (a) and (b) are here referred to as Araki-Sucher terms. The diagrams (c)-(f) represent the two-electron radiative effects, also called the screening of the Lamb shift. electrons, as for the helium-like ground state, this is also true for the latter effect. For non-equivalent electrons, however, the retardation beyond the Breit interaction affects also, through the exchange term, the expectation value of the electron-electron interaction. This contribution enters in order a4Z5 Ry and can be quite important [28,29]. Since the cr3Z3-effects discussed here first were evaluated, in the non-relativistic limit, by Araki 1301 and Sucher [11,12] they are sometimes referred to as the Araki-Sucher terms. It should be noted, however, that a subset of the a3Z3-terms calculated by Araki and by Sucher are included already in RMBPT when the expression in Eq. (5) is evaluated in second order. Other terms hitherto neglected are the radiative effects. The most important is the self-energy and the next in importance is the vacuum polarization, which both contribute in order a3Z4 Ry and beyond, for hydrogen-like systems. In many-electron systems we have also two-particle effects of radiative nature, sometimes called the screening of the Lamb shift, which contribute in order cu3Z3 Ry and beyond, see Figs. 2c-2f. 3.1. A Z-expansion Although Z-expansions are not the basis for any modern calculations it is illustrative to summarize the contributions to the binding energy of many-electron systems as follows; E = ~nrel + AEtel + AEQED (7) The contributions to the first term on the right-hand side of Eq. (7) have the following Z-dependence 1 1 Enre’ = a,Z’ + a,Z + a32 + a42 + . . .
(8)
1. FUNDAMENTAL ASPECTS
E. Lindroth / Nucl. Instr. and Meth. in Phys. Res. B 98 (1995) l-9
4
3.2. Comparison with experiments
These are the non-relativistic parts which more or less always dominate the binding energies. The first two terms, the hydrogen-like energy and the expectation value of electron-electron interaction, are trivial. The rest, which we can call correlation, are always approximated. For few electron systems correlation can however be accurately calculated. For every order in the perturbation expansion another factor of l/Z enters, thus it converges faster for highly ionized systems. The contributions to the second term on the right-hand side of Eq. (7) have different Z-dependences AE”’ = a2Z2
b,Z2 + b,Z + b2Zo + b3;
+ . .
A few years ago Schweppe et al. performed an experiment on lithium-like uranium [31], where the splitting of the 2~-2pt,~ levels was measured very accurately, with an error of only 0.1 eV. This stimulated several groups to do accurate calculations and results in good agreement have been presented e.g. by Johnson et al. and by Lindgren et al. The result from these calculations are presented in Table 1. The new achievements concerned especially the so called screening of the Lamb shift [32-341. The self-energy is a complicated effect for which there is no simple potential. Since the work of Mohr the hydrogen-like selfenergy, for any Z, can be considered as well known. However, that procedure assumes a Coulomb potential from the nucleus and it cannot easily be extended to any potential. Only recently was it possible to evaluate the self-energy in a screened nuclear potential. The self-energy results given in Table 1 include the screening, which is of order - 1.6 eV since the hydrogen-like result is 55.9 eV [lo]. Although the methods in Refs. [35] and [36] are rather different the results agree well. Since the experimental uncertainty is only 0.1 eV an account of the screening is necessary in order to obtain agreement. The comparison in Table 1 includes also a few effects of the same order as the experimental uncertainty. The nuclear polarization, calculated by Plunien et al. [37,38], is one of them and the cross term between the vacuum polarization and the self-energy is another. The latter is the only calculated second order Lamb shift effect. There are several such effects which are assumed to be of the same order of magnitude. The so called Breit-Coulomb reducible correction is one of the Araki-Sucher terms, see above, and the other terms of this class are expected to be of the same order. The agreement with experiment is very good although several effects of magnitude comparable to the experimental accuracy are missing and might be fortuitous. The lacking terms have not yet been calculated for lithium-like systems, but very recently calculations of the
. (9
Here only the leading relativistic corrections are listed. Together with certain terms of order a3 Ry (from the two-electron part only), a4 Ry and beyond they are accounted for by RMBPT. Again we have the leading correction to the hydrogen-like result and then the leading correction to the expectation value of the electron-electron interaction, which are well known and rather trivial. The third term is the first relativistic correction to the correlation. Finally we have the last term on the right-hand side of Eq. (7), ~EQno=c
1
a3z4+c
2
a3z3
. .
(10)
the radiative parts and the non-radiative parts (Araki-Sucher terms) which are outside the RMBPT treatment. The leading contribution is due to the hydrogen-like self-energy and vacuum polarization, which are quite well understood. The contributions of order a3Z3 Ry are effects which involve more than one electron. Among these are the self-energy screening, the effects of crossed photons and of negative energy states. The part of the binding energy which is not that well understood, or tested, is the a3Z3 Ry contributions. Since any binding energy is dominated by non-relativistic effects it is clear that accurate experiments are needed to really test the calculations.
Table 1 Transition energy for the 2p,,,-2s,,,
transition in Li-like uranium (eV)
RMBRISelf energy (including screening) Vacuum polarization (including screening) Nuclear recoil and polarization Mixed self energy-vacuum polarization Breit-Coulomb reducible correction Total Experiment
a
’ Schweppe
et al. [31].
correction
Lindgren et al. [35]
Blundell et al. [36]
322.33(15) - 54.32(3) 12.59(l) 0.10 -0.19 0.04
322.41 - 54.24(2) 12.56(l) 0.10 _
280.55(20) 280.59(9)
280.83(10)
E. Lindroth/Nucl. Table 2 Two-electron 2
to the ground state energy of helium like ions (eV)
RMBPT a
32 54 74 83 a b ’ d e
contribution
Instr. and Meth. in Phys. Res. B 98 (1995) 1-9
Total ’
QED
1st order
2nd order
Araki-Sucher
567.6 1036.5 1586.9 1897.5
-5.2 - 6.9 -9.4 - 11.0
0.03 0.16 0.55 0.85
’
Two-electron
Experiment Super-Ebit
Lamb shift d 561.9 1028.1 1574.6 1882.7
-0.5 - 1.6 -3.4 -4.5
f f f i
0.5 1.6 3.4 4.5
562.5 1027.2 1579. 1878.
b
+ 1.6 + 3.5 * 15 + 14
I. Lindgren private communication. Obtained with an extended nucleus with uniform charge distribution. R.E. Marrs, S.R. Elliott and T. StBhlker, unpublished. Lindgren et al. [Z]. The two-electron (screening) Lambshift is only estimated, more detailed calculations are under way. 1. Lindgren private communication. The unceratinty is estimated to be of the order of the two-ele. .L:on Lamb shift which is yet not calculated from first principles.
Araki-Sucher terms have appeared for the ground state of helium [1,2]. The ground state of helium-like systems is attractive for calculations, but usually a direct comparison with experiment is not possible. However, in a new experiment from the Super-Ebit [39] the two-particle contributions to helium-like systems have been measured directly. The experiment measures the wavelength of the emitted photon when electrons recombine with bare ions and with hydrogen-like ions, respectively. The difference in photon energy corresponds directly to the two-electron part of the binding energy of the helium-like ground state. A comparison between the experimental results and preliminary calculations is shown in Table 2. The agreement is good, but the experimental uncertainty is still too large to test the more recent theoretical achievements, i.e. the Araki-Sucher terms in the third column. The two-electron Lamb shift, i.e. the screening of the Lamb shift, is here on the edge of observation.
4. Many electron ions Helium- and lithium-like systems are few-electron ions where the many-body effects are still limited, but it is also interesting to see how well more complicated systems can be treated. Alkali-like ions and copper-like ions have been studied by Blundell et al. [40-42,361, where the later references include also the screening of the Iamb shift. From atomic calculations it is known that beryllium can be considered as the first real many-body system. Due to the strongly interacting outer pair of electrons it is considerably more complicated than lithium. Examples of all order coupled-cluster calculations for beryllium-like systems are shown in Table 3. Even here the screening of the Lamb shift is necessary in order to get agreement with experiment [43,44]. The screening is here estimated with an approximative method [22]. In addition the RMBPT calculation includes a first estimate of the correlation correction to the Lamb shift. Since the (2s2)‘S, ground state of beryllium like ions has a large admixture of the (2~~)’
configurations the Lamb shift contribution is corrected accordingly. This correction to the ground state binding energy affects the splitting between the 2%, and 21,3P, levels with 0.00196 a.u. (430 cm-‘) and 0.00605 a.u. (1328 cm-‘) for iron and molybdenum respectively. As can be seen in Table 3 effects of this size are for most transitions one order of magnitude larger than the experimental accuracy. This is an indication that more severe tests of our ability to combine radiative and many-electron
Table 3 Comparison of theory and experiment for the (2~2p~‘~~P,-(2s~)‘S, splittings in beryllium-like iron and beryllium-like molybdenum
(2s2p)3P,-(2s%, RMBPT ’ (a.u.) d Screening of the Lamb shift ’ (a.u.) Total (a.u.) Total (cm-‘) Experiment
Fez2 +
Mo3’+
1.72513 k 0.00007 0.00227 f 0.00040
3.29924 + 0.00020 0.00755 + 0.00130
1.72740 + 0.00041
3.30679 + 0.00135
379118+90 37914OrtZO”
725751+ 300 725758+158b
3.42645 * 0.00020 0.00204 + 0.00035
9.12633 + 0.00030 0.00673 f 0.00115
3.42849 + 0.00040
c 9.13306 + 0.00120
752459 f 90 752372 f 57 a
2004464 f 300 2003847+1200b
(cm-’ ) (2s2pY P, -(Zs%S, RMBPT ’ (a.u.1 d Screening of the Lamb shift ’ (a.u.1 Total (a.u.) Total (cm Experiment (cm-’ )
’)
a Hinnov [44]. b Denne et al. [43]. ’ Lindroth and Hvarfner [19]. d 1 a.u. (56Fe) = 2.194725X lo5 cm-’ using the value R, = 109736.2 cm-’ (M = 55.9349) for the Rydberg constant. 1 a.u. c9sMo) = 2.194734X lo5 cm-’ using the value R, = 109736.7 cm - ’ (M = 97.9055) for the Rydberg constant. ’ Indelicate and Desclaux [22].
1. FUNDAMENTAL
ASPECTS
E. Lindroth /Nucl. Instr. and Meth. in Phys. Res. B 98 (199.5) l-9
6
effects could probably be carried out in e.g. beryllium-like systems.
5. Effects which vanish in the non-relativistic
limit
We can also test our understanding of QED and relativity on effects which are forbidden non-relativistically. Calculations of such effects do not suffer from the problem with large non-relativistic contributions which have to be carefully calculated before a meaningful comparison can be done. An example of a pure relativistic effect is the possibility of so called forbidden transitions. Electric dipole transitions between singlet and triplet states constitute one kind of such transitions and it has been studied e.g. in beryllium-like systems by beam-foil spectroscopy. The transition rate scale then as a!‘Z”. In spite of long term theoretical efforts, including very recent work [45,46] the agreement between theory and experiment is, however, still not really satisfying. The effect on the forbidden transitions rates from the inclusion of Breit interaction is significant [45,46], but the possible effects from radiative contributions are not studied yet. A review of experimental results and agreement with theory can be found in Ref. 1471. Since many years there has been an interest in the forbidden magnetic dipole transition, 23S, + l’s,, in helium-like systems. The transition rate scale here as CI’Z’~. A recent experiment from the TSR storage ring in Heidelberg [48] measured the decay rate of the 23S,-state in helium-like carbon with a relative error of only 0.2 percent. The agreement with the calculation by Drake [49] is here very good. This is in spite of the fact that the calculation in Ref. [49] is done with an aZ-expansion of the relativistic effects and that radiative effects are not considered. The excellent agreement indicates the importance of electron correlation over relativistic effects in this region of Z. A fully correlated relativistic calculation has been performed for helium-like argon [17]. For this more highly charged ion relativistic effects are more important resulting in a deviation from the result by Drake [49] by 1.4%. However, no experiment of such an accuracy has been performed on argon. Another quantity which does not exist in the non-relativistic limit is the bound state corrections to the electron g-factor, which is discussed in some detail below. 5.1. The electron
g-factor
we get in addition the Schwinger correction which is of order - a/n. The experimental result from Dehmelt et al. is -2 X (1 + 0.001 159 652 188c4.3)) [50], whereas the theoretical result by Kinoshita is - 2 X (1 + 0.001 159,652 140(5.3x4.1)(27)) [51]. The first two uncertainties in the theoretical value come from the numerical errors in the o-expansion and the third and largest term comes from the uncertainty in the fine-structure constant. For bound electrons the g-factor changes slightly. These changes are due to relativistic and radiative effects only. For s-states in hydrogenlike systems (neglecting nuclear spin and nuclear size effects) we can write; (gy),?=
+cu3~2..
e
P = &G’.
(11)
Classically g, would be - 1, whereas Dirac’s relativistic theory gives a value of -2. From the full QED treatment
+;$
. (12) Ia
where E is the relativistic binding energy. The third term on the right-hand side is due to bound state relativistic corrections, while the last term on the right-hand side indicates the presence of bound state radiative corrections [52]. The latter have been studied theoretically in the non-relativistic limit by Hegstrom and Grotch [53,54], but no calculations exist for highly charged hydrogenlike systems. Experiments on highly charged hydrogenlike systems are planned, but results are still not available. However, accurate experimental results from ion traps have been presented for singly charged ions e.g. for Be+, Mg’, and Ba+ [55-581. For these systems the bound state relativistic corrections, the third term on the right-hand side of Eq. (12), cannot be given in analytical form since it is affected also by the electron-electron interaction. For these singly ionized systems the bound state radiative corrections are at the limit of being visible. Slightly more accurate experiments or a modest increase in charge would probably be enough to reveal these effects. This requires though a careful theoretical treatment of the bound state relativistic corrections. Although calculations in the nonrelativistic limit have been successfully performed for light elements [59,60] even medium heavy elements, as neutral rubidium, are strongly influenced by relativistic effects beyond that [61,59]. However, an accurate, fully correlated, relativistic treatment is possible as shown in Ref. 1621, and discussed below. It should also be noted that for higher charge states the electron-electron effects are relatively less important, making the isolation of the bound state radiative corrections easier. X1.1.
One of the more important tests of QED is the successful prediction of the magnetic moment of the free electron
-2(1+0.0011596-..
Theory
The interaction
is considered H”
= chm
I
between the atom and a magnetic field through the addition of a term
= xecai.Ai
(13)
to the relativistic no-pair Hamiltonian Eq. (3). In Eq. (3) the two-particle terms, the Coulomb as well as the Breit
E. Lindroth /Nucl. Instr. and Meth. in Phys. Res. B 98 (1995) 1-9
interaction, are surrounded with projection operators onto positive energy states. In addition there are contributions from virtual pairs which have to be considered already in order (Ye, see Ref. [62]. Radiative effects are, however, omitted and the effect of this omission has also been discussed in Ref. [62]. If the magnetic field is assumed to be homogeneous over the extension of the atom the vector potential A can be written A = -;(rxB)
(14)
and thus hm=
-~(nXr).B=~iJZr(nC’}‘.B.
(15)
Classically the energy for a magnetic dipole in a magnetic field is given as the scalar product between the dipole and the field. The magnetic dipole created by the motion of a charged particle is proportional to its angular momentum and we write E=
-p.B=
-p&j.B=
-gi!fj.B,
(16)
where the gj factor, with a classical value of - 1, has been introduced. Eqs. (15) and (16) can now be combined to give an expression for the gj factor. gj( j) = -(mcJZir{cuC1}l).
(17)
From Eq. (17) the result for a free electron is gDirac = - 2. Corrections to this value have two main sources. First rudiuliue effects, omitted in Eq. (17), give the Schwinger correction of approximately -0.002319, for a free electron. Secondly the expression (17) gives relativistic corrections to the gj factor for a bound electron, which are not present in the non-relativistic limit. The relativistic corrections scale as Z2a2 and are of the order 10m4 or 1O-5 in light systems. Finally there are also bound stute
7
radiative corrections, these are smaller then the bound state relativistic corrections by one order of the fine-structure constant. If the expression in Eq. (17) is used directly, together with relativistic wave functions, it will thus give the non-relativistic factor of -2 and, in addition to that, the bound state corrections. Due to the relative smallness of these corrections the requirement on the numerical accuracy will be very high. In addition; the absence of non-relativistic bound state corrections is not explicit in Eq. (17) but has to be achieved by numerical means. It would certainly be desirable to have an operator which instead gave the corrections directly. Such an operator was also derived in Ref. [62]. 5.1.2. Results A fully relativistic treatment has been applied to a few systems and the contributions from different classes of effects are summarized and compared with experiments [55-581 in Table 4. For Be+ around 95% of the final result is given by the sum of the Dirac-Fock value and the subset of many-body effects which can be included via the random phase approximation (RPA). Correlation beyond the RPA-like diagrams contribute 1.5% and the total effect of the Breit interaction is 2.3% for Be+. For Ba+ the situation is very different. The lowest order result even shows the wrong sign. After inclusion of the RPA chains, which are completely dominated by the polarization of the 5p orbitals, the result is 20% above the experimental result. The remaining contributions come from the Breit interaction and the correlation due to the Coulomb interaction which contribute with 10% each of the final result. The theoretical uncertainties are estimated contributions from radiative corrections, which are probably less than 1% as discussed in Ref. [62], the pure numerical uncertainty and the uncertainty caused by the truncation of the
Table 4 Contributions
to the bound state corrections
to the g, factor from different
classes of diagrams.
The corrections
electron value, g, = - 2.002 319 304 377 (91, and obtained as 6g = gi - g,. The results are displayed
Lowest order, Dirac-Fock A Dirac-Fock-Breit RPA-like diagram . A Dirac-Fock-Breit Breit RPA-like diagrams Coulomb correlation A Dirac-Fock-Breit Breit correlation Total Experiment
Be+
Mg+
Ba’
4.932 -0.124 0.477 - 0.023 0.35 1 0.090 - 0.004 - 0.046
5.217 0.125 0.110 - 0.003 0.613 0.474 - 0.001 0.004
5.539 0.708 - 26.455 0.462 0.824 1.985 0.089 - 0.330
5.653 f 0.050 5.694 f 0.033 a
6.54 f 0.06 6.52 f 0.03 b
are relative to the free
as 6g X 10’
- 17.18 + 0.3 - 17.13 + 0.12 c
a Wineland et al. [55,56]. b Bollinger et al. [57]. ’ Knab et al. [58].
1. FUNDAMENTAL
ASPECTS
8
E. Lindroth/Nucl.
Instr. and Meth. in Phys. Res. B 98 (1995) l-9
partial wave expansion of the two-particle interactions. For Be+ the latter is small since many terms could be kept in the expansion. The lowest order result is in addition the dominant contribution and higher order effects show a stable behavior. An estimated uncertainty of a few percent in the many-body part of the calculation gives an overall uncertainty of around 0.1%. Thus an estimated error of 1% should not be too optimistic. The behavior of Ba+ is considerably less stable. The lowest order result has the wrong sign and the situation is restored only after inclusion of RPA effects. We assume that the overall uncertainty due to numerical uncertainties and truncations could lead to an error of around 1%. To account also for possible radiative contributions we estimate the total error to two percent.
6. Conclusions The interesting experimental development which have resulted in many studies of ions of varying charge states has been accompanied by a promising theoretical development. For simple systems such as helium- or lithium-like systems accurate calculations can today be done including two-particle radiative corrections and other effects which are beyond a standard relativistic treatment of the electron-electron interaction. For the near future we can expect calculations of this kind to be carried out also for more complicated systems. For transition rates and atomic properties in general the theoretical treatment has not yet reached the same accuracy. However, with increasing experimental and theoretical accuracy, especially concerning completely relativistic effects, there is a lot to learn also in this field.
Acknowledgements I am grateful to R.E. Marrs, S.R. Elliott and T. Stijhlker for allowing me to show their results prior to publication. I have benefitted from discussions with Prof. I. Lindgren, Dr. A.-M. M%rtensson-Pendrill, Dr. S. Salomonson and Dr. H. Persson. Financial support was received from the Swedish Natural Science Research Council (NFR).
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S. Fritzsche and I.P. Grant, to be published. E. Trgbert, Phys. Scripta 48 (1993) 699. H.T. Schmidt et al., Phys. Rev. Lett. 72 (1994) 1616. G.W.F. Drake, Phys. Rev. A 3 (1971) 908. R.S.V. Dyck, P.B. Schwinberg and H.G. Dehmelt, Phys. Rev. Lett 59 (1987) 26. [51] T. Kinoshita and D.R. Yennie, in: Quantum Electrodynamics, ed. T. Kinoshita (World Scientific, Singapore, 1990). [52] The change in the bound state relativistic corrections due to the anomalous part of the magnetic moment of a free electron is here included among the bound state radiative corrections. This particular term amounts to -0.001 159 6.. . fE/mc’ and is thus not just equal to the product of the free particle radiative correction and the relativistic corrections. [53] R.A. Hegstrom, Phys. Rev. 184 (1969) 17.
9
[54] H. Grotch and R.A. Hegstrom, Phys. Rev. 4 (1971) 59. [55] D.J. Wineland, J.J. Bollinger and W.M. Itano, Phys. Rev. Lett. 50, 628 (1983). [56] The result gj = -2.002 262 36 (31) is obtained from Ref. [55] when the most recent values for the mass ratios; mp /m, = 1836.152 701(37) and m(gBe’>/m, = 8.946 534 45 (41) are used. [57] J.J. Bollinger, J.M. Gilligan, W.M. Itano, F.L. Moore and D.J. Wineland, private communication. [58] H. Knab, K. Knoll, F. Scheerer and G. Werth, Z. Phys. D 26 (1993) 205. [59] L. Veseth, Phys. Rev. 22 (1980) 803. [60] L. Veseth, J. Phys. B 16 (1983) 2891. [61] V.V. FIambaum, LB. Khriplovich and O.P. Sushkov, Phys. Len. A 67 (1978) 177. [62] E. Lindroth and A. Ynnerman, Phys. Rev. A 47 (1993) 961.
1. FUNDAMENTAL
ASPECTS