Relaxation effect and radiative corrections in many-electron atoms Andrei Derevianko∗ and Boris Ravaine Department of Physics, University of Nevada, Reno, Nevada 89557 W. R. Johnson† Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556 (Dated: February 2, 2008) 4 0 We illuminate the importance of a self-consistent many-body treatment in calculations of vac- 0 uum polarization corrections to the energies of atomic orbitals in many-electron atoms. Including 2 vacuum polarization in the atomic Hamiltonian causes a substantial re-adjustment (relaxation) of n theelectrostatic self-consistent field. Theinducedchangein theelectrostatic energiesissubstantial a for states with the orbital angular momentum l > 0. For such orbitals, the relaxation mechanism J determines the sign and even the order of magnitude of the total vacuum polarization correction. 0 This relaxation mechanism is illustrated with numerical results for the Cs atom. 1 PACSnumbers: 31.30.Jv,31.15.Ne,31.15.Md,31.25.Eb ] h p Compared to hydrogenic one-electron systems, calcu- when the presence of other electrons is accounted for. - lation of radiative corrections for many-electron atoms In particular, we consider vacuum polarization (VP) m brings in an additional layer of complexity: a strong corrections to energies of atomic states. To the leading o Coulomb repulsion between the electrons. The problem orderinαZ theVPmaybeaccountedforbyintroducing at isespeciallychallengingforneutralmany-electronatoms, the Uehling potential UVP(r) into the atomic Hamilto- . wheretheinteractionofanouter-shellelectronwithother nian. This potential is attractive, and for a hydrogen- s c electrons is comparable to its interaction with the nu- like ion the resulting VP corrections to the energies are i cleus. At the same time, a reliable calculation of radia- always negative. For a complex atom, we find by con- s y tive corrections for a heavy neutral system is required trast that, for orbitals with l > 0, the total correction h in evaluation of the parity non-conserving(PNC) ampli- is positive. Briefly, the reason for such a counterintu- p tude in the 55-electron 133Cs atom. Here it has been itive effect is due to a readjustment of atomic orbitals [ only recently realized that the sizes of radiative correc- whentheUVP(r)potentialisaddedtotheself-consistent 1 tions [1, 2, 3, 4, 5] are comparable to the experimental Dirac-Hartree-Fock (DHF) equations. The innermost 1s v error bar [6] of 0.35% and, together with the Breit cor- orbitals are “pulled in” by the short-ranged VP poten- 3 rection [7], dramatically affect agreement (or disagree- tial, leading to a decrease of the effective nuclear charge 4 ment [8]) with the Standard Model of elementary parti- seen by the outer orbitals and thus to an increase of the 0 cles. electrostatic energy of these orbitals. Since for orbitals 1 0 Asystematicapproachtotheproblemofradiativecor- with l > 0, overlap with UVP(r) and thus the lowest 4 rections in strongly correlatedsystems is to start from a order correction are small, the resulting indirect “relax- 0 Furryrepresentationbasedonaself-consistentelectronic ation”contributiondominatesthetotalVPcorrectionto / potential [9]. This potential takes into account the fact the energies. In the following we will present numerical s c that an electron moves in an average field created by results supporting this relaxation mechanism. Atomic si boththenucleusandotherelectrons. Basedonthisidea, units (~=|e|=me ≡1) are used throughout. y a program of calculating radiative corrections to PNC Because of our interest in PNC in Cs, below we illus- h amplitudes have been put forth by Sapirstein et al. [5]. trate the relaxationeffect with numerical results for this p Kuchiev and Flambaum [3] and Milstein et al. [4] pur- atom; however, the relaxation mechanism is also appli- : v sue a more qualitative approach using an independent- cableinthecasesofothermany-electronatoms. Wealso i electron approximation. We believe that the question notice that the relaxation mechanism described here is X of an interplay between correlations and radiative cor- similar to that observed in calculations of the Breit cor- ar rections is yet to be addressed. While here we do not rections [10, 11]. compute the PNC corrections, we illuminate a situation The conventional many-electron Hamiltonian may be where disregardingcorrelationswould leadto a substan- represented as tialerrorindeterminingradiativecorrection: aradiative 1 1 correctionchanges sign and eventhe orderof magnitude H = h0(i)+ , (1) 2 r Xi Xi6=j ij where the single-particle Dirac Hamiltonian is ∗Electronic address: [email protected]; h0(i)=c(αi·pi)+βic2+Vnuc(ri). (2) URL:http://unr.edu/homepage/andrei †Electronic address: [email protected]; The nuclear potential Vnuc(r) is obtained from the nu- URL:http://www.nd.edu/~johnson clear charge distribution ρnuc(r); which is we approxi- 2 mate by the Fermi distribution nucleus of charge Z reads ρnuc(r)= 1+exp[ρ(0r−c)/a], (3) UVp.Pc.(r)= 32παrZ Z ∞dt t2−1(cid:18)t12 + 21t4(cid:19)exp(cid:20)−2αrt(cid:21) . 1 p (5) where ρ is the normalization constant, c and a are the 0 This potential must be folded with the nuclear charge nuclear parameters. In our the numerical example for 133Cs, we use c=5.6748 fm and a=0.52 fm. distribution, A common starting point for describing a multi- electron atom is the self-consistent field method. Here UVP(r)=Z dr′ρnuc(|r−r′|)UVp.Pc.(r′) . the many-body wave-function is approximated by a Slater determinant constructed from single-particle or- We approximated ρ (r) with the Fermi distribution, nuc bitals (bi-spinors) uk(r). The orbitals are obtained by Eq. (3). In the numerical evaluation of the extended- solving self-consistently the eigenvalue equations nucleusUehling potential,we employedthe routine from Ref. [12]. The Uehling potential U (r) generated by (h +U )u (r)=ε u (r), (4) VP 0 DHF k k k the Cs nucleus is shownin Fig.1. Notice that the actual range of this potential is a few nuclear radii (instead of where U is the traditional DHF potential which de- DHF Comptonwavelengthλ ≈384fm),becausethepotential pendsontheorbitalsoccupiedintheSlaterdeterminant. e forapoint-likecharge,Eq.(5),divergeslogarithmicallyas The DHF energies for the core and several valence or- r → 0; therefore the folded potential U is dominated bitals of Cs are listed in Table I. VP by the contributions accumulated inside the nucleus. TABLE I: Vacuum polarization corrections to binding ener- 0.00 gies in neutral Cs (Z = 55). Here ε are the DHF en- nlj ergies, δε(n1l)j are the expectation values of the Uehling po- -0.02 tential (Eq.(6)), and δεDHF are the VP corrections with the correlations included (Enqlj.(8)). All quantities are given in eV -0.04 M afotromxi×c u10nyit.s, 1a.u. = 27.21138eV, and notation x[y] stands U(r), VP-0.06 Orbital ε δε(1) δεDHF -0.08 nlj nlj nlj core orbitals -0.10 1s −1330.396958 −2.853[−1] −2.782[−1] 1/2 2s1/2 −212.597116 −3.392[−2] −3.267[−2] 0 10 20 30 40 50 2p1/2 −199.428898 −1.510[−3] 5.406[−4] r, fm 2p3/2 −186.434858 −1.650[−4] 1.690[−3] nuclear radius 3s −45.976320 −6.868[−3] −6.581[−3] 1/2 3p −40.448097 −3.339[−4] 1.987[−4] 1/2 3p3/2 −37.893840 −3.719[−5] 4.609[−4] FIG. 1: Uehling potential for 133Cs. Notice that the radius 3d3/2 −28.309043 −1.839[−7] 4.531[−4] oftheinnermost1sorbitalisabout103 fm,muchlargerthan 3d5/2 −27.774710 −4.370[−8] 4.425[−4] theeffective range of the VPpotential. 4s −9.514218 −1.457[−3] −1.397[−3] 1/2 4p −7.446203 −6.726[−5] 8.097[−5] 4p1/2 −6.920865 −7.506[−6] 1.355[−4] How does one compute the VP corrections δεk to the 4d3/2 −3.485503 −3.440[−8] 1.153[−4] energies of the atomic orbitals? Below we consider two 3/2 4d −3.396788 −8.100[−9] 1.129[−4] possibilities: (i) lowest-order perturbative treatment, 5/2 5s −1.490011 −2.057[−4] −2.050[−4] 5p11//22 −0.907878 −7.773[−6] 2.035[−5] δε(k1) =huk|UVP|uki, (6) 5p −0.840312 −8.395[−7] 2.757[−5] 3/2 valence states and (ii) the self-consistent approach. Indeed, as in 6s −0.127380 −1.054[−5] −1.159[−5] Ref. [2], the VP potential may be introduced into the 1/2 6p −0.085616 −1.942[−7] 2.284[−7] DHF equations, 1/2 6p −0.083785 −2.180[−8] 4.513[−7] 3/2 7s −0.055190 −2.896[−6] −3.143[−6] (h +U +U′ )u′(r)=ε′u′(r), (7) 1/2 0 VP DHF k k k 7p −0.042021 −6.957[−8] 8.150[−8] 1/2 7p −0.041368 −7.873[−9] 1.606[−7] and a set of new energies ε′ and orbitals u′(r) is ob- 3/2 k k tained. Notice that the DHF potential is modified as well, since it depends on the new set of the occupied or- The polarization of the vacuum by the nucleus mod- bitals u′(r). The correlated VP correctionto the energy k ifies the nuclear electric field seen by the electrons. To of the orbital k is simply theleadingorderinαZ,theVPmaybeconvenientlyde- scribedwiththeUehlingpotential,whichforapoint-like δεDHF =ε′ −ε . (8) k k k 3 Additionally, we carried out an independent corre- The figure 2 may be interpreted in the following way: latedcalculationin the frameworkofthe linearizedCou- the s orbitals are “pulled in” by the attractive Uehling pled DHF approximation[13], which is equivalent to the potential closer to the nucleus. As a result, screening of random-phase approximation (RPA). This approxima- the nuclear charge by the inner orbitals becomes more tion describes a linear response of the atomic orbitals efficient. For example, the modification of the effective to the perturbing interaction, i.e. the VP potential. Nu- charge felt by the n=2 electrons is simply the area un- mericalvaluesobtainedfromthelinearizedcoupledDHF der the δρ (r) curve, accumulated between r = 0 and el calculations were in close agreement with the full DHF theradiusoftheshell(r ≈0.08a ); fromFig.2itisclear 0 results. that the induced modification of the effective charge for Thenumericalresultsofourcalculationsarepresented the n = 2 shell has a negative sign. Such an enhanced in Table I. While analyzing this Table, we observe that screeningleadstoareducedattractionoftheelectronsby the lowest order corrections, δε(1), are always negative, thenucleusandtotheincreaseintheenergyoftheouter k reflectingthefactthattheUehlingpotentialisattractive electrons. From Table I, we see that this indirect relax- (see Fig. 1). Owing to the short-ranged nature of VP, ationcontributiontothe energymaybewellcomparable and the fact that only the s-orbitals have a significant tothedirectVPcorrection,δε(1). Whileforl=0orbitals k overlap with the nucleus, the corrections to the energies the direct correction gives a reasonable estimate, for all of l = 0 orbitals are much larger than those for l > orbitals with l > 0, the neglect of the relaxation would 0 orbitals. As to the correlated corrections, they differ leadto evenqualitativelyincorrectresult. Moreover,the quite substantially from the lowest order-corrections. A higher the orbital angularmomentum, the smaller is the comparison of Eq. (7) and Eq. (4) reveals the origin of direct correction, and the more important is the relax- this discrepancy: the perturbation, in addition to the ation mechanism. For example, for 4d orbitals the VP Uehling potential, contains a difference between the two correctioninthelowestorderisfourordersofmagnitude DHF potentials smaller than the correlated result. δU =U +(U′ −U ) . (9) VP DHF DHF For orbitals with l > 0, where the first term above is small,the modificationofthe DHF potentialcontributes 0.00 significantly to the VP energy corrections. The modificationofthe DHFpotentialinducedbythe -0.02 vacuum polarization is clearly a many-body effect, not ) r presentin hydrogen-likesystem. Such aneffect has been (el -0.04 explored before, for example in calculations of the Breit -3 corrections [10, 11], and it is commonly referred to as a -0.06 el(r) x 10 relaxation mechanism. Let us illustrate this relaxation mechanism. Denoting the correction to the occupied or- -0.08 n=1 bital wave functions as χ (r)=u′(r)−u (r), we write n=2 a a a -0.10 (UD′HF−UDHF)(r)≈ 0.00 0.02 0.04 0.06 0.08 0.10 χ†(r′) 1 u (r′)dr′+ r Z a |r−r′| a Xa 1 u†(r′) χ (r′)dr′−exchange, Xa Z a |r−r′| a FIG.2: Perturbation of theelectronic radial charge distribu- tion δρ (r) (solid line) for Cs atom due to vacuum polariza- el where we discarded contributions non-linear in χ (r), tion by the nucleus. We also show the unperturbed density a and “exchange” denotes non-local part of the perturba- ρel(r) multiplied by a factor of 10−3 (dashed line). The min- tion. Thefirsttwo(direct)termscanbeinterpretedasan ima of ρel(r) correspond to positions of the electronic shells, marked on the plot by their values of the principal quantum electrostaticpotentialproducedbyaperturbationδρ (r) el numbern. in the radial electronic density ρ (r)=− 1 u†(r)u (r). To summarize, here we illuminated the importance el 4πr2 a a of the self-consistent many-body treatment in calcu- Xa lations of vacuum polarization corrections. Including We plot both the electronic density ρ (r) and the VP- the VP Uehling potential into the atomic Hamiltonian el induced perturbation δρ (r) in Fig. 2. The minima of causesre-adjustment(relaxation)oftheelectrostaticself- el ρ (r) correspond to positions of the electronic shells, consistent field. The induced change in the electrostatic el marked on the plot by their values of principal quantum energiesis substantialfor states with the orbitalangular number n. momentuml >0. As illustratedin ournumericalresults 4 forCs,therelaxationmechanismdeterminesthesignand work of W.R.J. was supported in part National Science even the order of magnitude of the total VP correction Foundation Grant No. PHY-01-39928. for orbitals with l>0. 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