Nuclear polarization effects in atoms and ions
NNuclear polarization effects in atoms and ions
V. V. Flambaum , , , I. B. Samsonov , H. B. Tran Tan , and A. V. Viatkina , School of Physics, University of New South Wales, Sydney 2052, Australia Helmholtz Institute Mainz, GSI Helmholtzzentrum f¨ur Schwerionenforschung, 55099 Mainz, Germany and Johannes Gutenberg University Mainz, 55099 Mainz, Germany
In heavy atoms and ions, nuclear structure effects are significantly enhanced due to the overlapof the electron wave functions with the nucleus. This overlap rapidly increases with the nuclearcharge Z . We study the energy level shifts induced by the electric dipole and electric quadrupolenuclear polarization effects in atoms and ions with Z ≥
20. The electric dipole polarization effectis enhanced by the nuclear giant dipole resonance. The electric quadrupole polarization effect isenhanced because the electrons in a heavy atom or ion move faster than the rotation of the deformednucleus, thus experiencing significant corrections to the conventional approximation in which they‘see’ an averaged nuclear charge density. The electric nuclear polarization effects are computednumerically for 1 s , 2 s , 2 p / and high ns electrons. The results are fitted with elementary functionsof nuclear parameters (nuclear charge, mass number, nuclear radius and deformation). We constructan effective potential which models the energy level shifts due to nuclear polarization. This effectivepotential, when added to the nuclear Coulomb interaction, may be used to find energy level shifts inmulti-electron ions, atoms and molecules. The fitting functions and effective potentials of the nuclearpolarization effects are important for the studies of isotope shifts and nonlinearity in the King plotwhich are now used to search for new interactions and particles. I. INTRODUCTION
Hydrogen-like ions represent a powerful tool for study-ing various aspects of quantum electrodynamics (QED)and physics beyond the Standard Model (SM). Sincethese systems are, at least at the electronic level, freefrom many-body interactions, the spectra of hydrogen-like ions may be calculated with high accuracy, see, e.g.,Refs. [1, 2] for a review. Corrections to the electronicenergy levels in these ions including nuclear recoil, nuclearfinite size corrections, one- and two-loop QED corrections(Lamb shift) and nuclear polarization effects have beenidentified by the ever increasing accuracy of modern ex-perimental techniques, see, e.g., Refs. [3–5]. In this paper,we study the effects of nuclear polarization on the energyspectra of hydrogen-like ions with Z ≥
20. We also seekto extend the formalism for hydrogen-like ions to the caseof multielectron atoms.The effects of nuclear polarization are significantly en-hanced in a heavy ion because its s and p / electronwave functions have sizable overlaps with the nucleus.The technique for computing corrections to atomic spec-tra due to the electric nuclear polarization was developedin a series of papers [6–10]. Therein, it was demonstratedthat the leading corrections to an energy level arise frommixing with the nuclear electric giant dipole resonancestate (E1) and mixing with nearby nuclear rotationalstates (E2). The latter mechanism may play a significantrole in deformed nuclei where the effect is enhanced byclose nuclear rotational levels: in very heavy atoms, theseintervals become smaller than typical energy intervals forvirtual electron excitations. The goal of this paper isto find the corrections to the atomic energy levels dueto these mechanisms for all medium and heavy atoms.The results are presented in terms of simple interpolationformulas which depend on the nuclear charge Z , nucleon number A , nuclear radius R and quadrupole deformationparameter β .An important feature of a nuclear giant dipole reso-nance is that its energy and transition strength are, to agood approximation, monotonic functions of the atomicnumber Z and mass number A . As a result, the en-ergy levels shifts caused by virtual nuclear giant dipoleresonance transitions should also be well-described byfunctions which are monotonic in these parameters. Inthis paper, we numerically calculate these shifts for avariety of ions with ( Z ≥
20) and fit the results withelementary functions of Z , A and R . These interpolatingfunctions describe the corresponding energy shifts in allheavy nuclei with a good accuracy: the error is under2%, as compared with the results of direct numericalcalculations.In a similar way, we fit the results of numerical cal-culations for the energy shifts due to nuclear rotationalE2 polarization. The error of the interpolating functionsin this case is also under 2%. Note that because thetransition strengths of the nuclear rotational E2 transi-tions have strong dependence on the nuclear deformationparameter β , which changes significantly even betweenneighboring nuclei, the energy shifts are non-monotonicfunctions of Z and A . This behaviour of the energy shiftsdue to the nuclear polarization is expected to give signif-icant contributions to the nonlinearity of the King plotfor isotope shifts [11].In multielectron atoms and molecules, it is convenientto describe the corrections to the spectra in terms of ef-fective interactions V L ( r ) which should be added to thenuclear Coulomb potential. By definition, the expectationvalues of the potentials (cid:104) V L ( r ) (cid:105) are equal to the energyshifts due to the dipole and quadrupole nuclear polariza-tions. These potentials may be useful in the study of thenonlinearity of the King’s plot [12–16], which provides a r X i v : . [ phy s i c s . a t o m - ph ] J a n information about physics beyond the SM [11, 17–21].For example, the nonlinearity may be interpreted as amanifestation of a new boson mediating electron-nucleusinteraction. The study of such non-SM nonlinear devia-tions would strongly benefit from the careful accountingof SM contributions to the isotope shift nonlinearity. Thecurrent paper provides important estimate for the con-tributions due to the nuclear structure effects in atomicspectra which would be subtracted from experimentaldata to identify the strength of new, non-SM interactions.The rest of the paper is organized as follows. In the nextsection, we review the scalar and tensor nuclear polariz-abilities and estimate the effect of the latter in the spectraof heavy atoms. In Sec. III, we calculate specific energyshifts in the spectra of medium and heavy hydrogen-likeions and multielectron atoms due to the scalar nuclearpolarization. The results for ions are represented in termsof interpolating formulas which reproduce these energyshifts with good accuracy. The effective potentials thatproduce the energy shifts in multieletron atoms are thesubject of Sec. IV. In Sec. V, we summarize and discussour findings.In this paper, we use natural units wherein (cid:126) = c = 1.Nuclear energies are denoted by the Latin letter E whereasfor atomic energy levels we use the Greek letter ε . II. TENSOR NUCLEAR POLARIZABILITYCONTRIBUTION TO THE HYPERFINESPLITTING OF ATOMIC ENERGY LEVELS
In this section, we compare the effects of the hyperfinesplitting of atomic energy levels due to nuclear tensor po-larizability with those due to nuclear electric quadrupolemoment. We start with a short review of the nuclearquadrupole moment and its contribution to the atomichyperfine structure. Then we consider similar contribu-tions from the nuclear tensor polarizability and estimatethe ratio between parameters of these two effects.
A. Hyperfine splitting due to nuclear quadrupoledeformation
In this subsection, we review the well-known resultsconcerning the contributions of the nuclear quadrupolemoment to the hyperfine level splitting [22, 23]. Althoughthe information in this subsection is not new, it is usefulfor the next subsection, where we will estimate analogouscontributions due to nuclear tensor polarizability.Let E ηI be a nuclear energy corresponding to a state | ηIM (cid:105) . Here, I is the nuclear spin, M is the magneticquantum number and η denotes all other relevant quan-tum numbers. By definition, the quadrupole moment ofthe nucleus is a second-rank tensor of the form Q ij = 3 Q I (2 I − (cid:18) I i I j + I j I i − I ( I + 1) δ ij (cid:19) , (1) where Q is the expectation value of the electric quadrupoleoperator ˆ Q ij = e (3 r i r j − r δ ij ) in the stretched state | ηI, M = I (cid:105) . Here, e is the charge of the proton. Thisquantity Q may be related to the intrinsic nuclear electricquadrupole moment Q (the quadrupole moment in therotating body frame) via [23, 24] Q = I (2 I − I + 1)(2 I + 3) Q . (2)The intrinsic quadrupole moment Q may, in turn, be re-lated to the nuclear radius R and the nuclear quadrupoledeformation parameter β [25] via Q = 3 √ π eZR β . (3)The values of quadrupole moments for different nuclei aretabulated in Ref. [26]. In the nuclear droplet model, thenuclear radius is described by the formula R = 1 . A / fm . (4)Within a model where the nucleus behaves as a de-formed three-dimensional harmonic oscillator with fre-quencies ω x = ω y (cid:54) = ω z , one may derive an alternativerepresentation for the quadrupole moment (3) [27]: Q = 25 eZR ¯ ω (cid:18) ω z − ω x (cid:19) , (5)where ¯ ω = ( ω x + ω y + ω z ) is a mean frequency whichmay be estimated using the phenomenological formula[24] ¯ ω = 41 A − / MeV . (6)In an atom, the nuclear quadrupole moment is known tocontribute to the hyperfine energy level splitting [22, 23]as ∆ ε hfs ,Q = B Q K ( K + 1) − I ( I + 1) J ( J + 1)2 I (2 I − J (2 J − , (7)where J is the electronic total angular momentum quan-tum number, F is the atomic total angular momentumquantum number, which is the vector sum of I and J , and K ≡ F ( F + 1) − I ( I + 1) − J ( J + 1). The coefficient B Q is proportional to the nuclear quadrupole moment Q andthe expectation value of 1 /r calculated with electronicradial wave functions, B Q = eQ (cid:10) r − (cid:11) C IJ . (8)Here C IJ is a coefficient which takes into account theintegration over angular variables. We do not specify theexplicit value of this coefficient here, as it will drop outfrom the final result. B. Scalar and tensor nuclear polarizabilities
When an external electric field E is applied to a nucleus,the nuclear energy levels E ηI are shifted due the quadraticStark effect. These shifts may be written as ∆ E ηI = − α ij E i E j where α ij is, by definition, the electric nuclearpolarizability tensor, α ij ≡ − (cid:88) η (cid:48) I (cid:48) M (cid:48) (cid:104) ηIM | d i | η (cid:48) I (cid:48) M (cid:48) (cid:105)(cid:104) η (cid:48) I (cid:48) M (cid:48) | d j | ηIM (cid:105) E ηI − E η (cid:48) I (cid:48) . (9)Here d = (cid:80) Ak =1 q k r k is the nuclear electric dipole oper-ator, q k is the nucleon charge which appears due to therecoil effect, q k = eN/A for proton and q k = − eZ/A forneutron.The symmetric tensor α ij may be decomposed into atrace, α = (cid:80) k α kk /
3, and a traceless part, α (T) ij = α ij − δ ij (cid:80) k α kk /
3. Conventionally, the components α and α (T) ij are referred to as the scalar and tensor polarizabilities ,respectively, see, e.g., Ref. [22].The scalar polarizability will be considered in Sec. III.In this section, we will focus on the tensor part. Simi-larly to the nuclear quadrupole moment (1), α (T) ij may beexpressed in terms of the nuclear spin operator ˆ I as α (T) ij = 3 α I (2 I − (cid:18) I i I j + I j I i − I ( I + 1) δ ij (cid:19) , (10)where the coefficient α is the value of α (T) zz calculatedwith the stretched state | ηII (cid:105) . Note that Eq. (10) definesthe tensor polarizability α in the laboratory frame. Thecorresponding value in the rotating body frame α (0)2 isrelated to α via an equation similar to (2) α = I (2 I − α (0)2 ( I + 1)(2 I + 3) . (11)The quantity α (0)2 may be estimated in the deformedthree-dimensional harmonic oscillator model with frequen-cies ω x = ω y (cid:54) = ω z which was used to derive Eq. (5). Incartesian coordinates, the nuclear states have the form | n x n y n z (cid:105) ≡ | n x (cid:105)| n y (cid:105)| n z (cid:105) where n x,y,z are the quantumnumbers in respective directions. With these functions,the diagonal components of the electric nuclear polariz-ability (9) read α (0) ii = 2 A (cid:88) k =1 q k (cid:88) n (cid:48) = n ± (cid:104) n | r k,i | n (cid:48) (cid:105)(cid:104) n (cid:48) | r k,i | n (cid:105) E ( i ) n (cid:48) − E ( i ) n , (12)where E ( i ) n = ω i ( n + ) are the harmonic oscillator energies.Note that for any particular i , ω i is assumed to be thesame for all nucleons. The computation of the matrixelements in Eq. (12) is elementary [28]. Using the identity (cid:80) Ak =1 q k = e N Z/A , one finds α (0)2 = 23 α (0) zz −
13 ( α (0) xx + α (0) yy )= 2 N Z A e m p (cid:18) ω z − ω x (cid:19) , (13) where m p is nucleon mass.Since the tensors (1) and (10) have the same structure,at the atomic level, the operator (10) produces hyper-fine energy level splitting analogous to (7), but with theconstant B Q replaced by B α = e α (cid:10) r − (cid:11) C IJ . (14)An estimate of the shifts due to tensor nuclear polarizabil-ity may thus be obtained by computing the ratio B α /B Q .Note that the power of r in Eq. (14) is different fromthat in Eq. (8) because the operators (1) and (10) havedifferent dimensions. Making use of Eqs. (5) and (13), wefind B α B Q = 5 N e Am p ¯ ω R (cid:10) r − (cid:11) (cid:104) r − (cid:105) . (15)The expectation values of the operators r − and r − inEq. (15) receive main contributions from the near-nucleusregion, where r (cid:28) a B /Z / . In this region, the screeningof the Coulomb field of the nucleus is negligible and theelectron radial wave functions may be well approximatedby the Bessel functions [29], f njl ( r ) = c njl r (cid:104) ( γ + κ ) J γ ( x ) − x J γ − ( x ) (cid:105) ,g njl ( r ) = c njl r ZαJ γ ( x ) , (16)where x ≡ (cid:112) Zr/a B , γ = √ κ − Z α and κ =( − j − l +1 / ( j + 1 / c njl may be found in Ref. [29]. For our purpose,the explicit value of this constant is not needed since itcancels out in the ratio (15).With the wave functions (16), the expectation value ofthe r − p operator may be found analytically for p < γ , (cid:104) r − p (cid:105) c njl = 132 (cid:18) Za B (cid:19) p − Γ( p − / γ − p ) √ π Γ( p )Γ( p + 2 γ ) × (cid:104) γ + p (5 + p ( p −
4) + 4 Z α ) − Z α + κ (4 p − − κ ( p − p − − (cid:105) . (17)Here we have used the wave functions (16) with j > / s / and p / states. The equation (17)allows us to estimate the ratio of the expectation valuesoperators r − and r − for heavy atoms ( Z >
70) in the p / state (cid:104) r − (cid:105)(cid:104) r − (cid:105) = ξZa B , ξ ≈ . . (18)Substituting this relation into Eq. (15), we find (cid:12)(cid:12)(cid:12)(cid:12) B α B Q (cid:12)(cid:12)(cid:12)(cid:12) = 5 ξN Ze Aa B m p ¯ ω R ≈ . × − Z , (19)where we have used Eqs. (4) and (6) and assumed theapproximations N ≈ . Z and A ≈ . Z for heavy nuclei.Numerically, for heavy atoms with Z ∼ − and is smaller for lighterelements. The effect of the tensor nuclear polarizabilityon electronic spectra is nearly four orders in magnitudesmaller than that of the quadrupole nuclear moment. Asa result, in many cases, the nuclear tensor polarizabilityeffect may be neglected. In the next section, we will focuson the effects of scalar nuclear polarizability. III. ENERGY SHIFT DUE TO SCALARNUCLEAR POLARIZABILITY IN MEDIUM ANDHEAVY HYDROGEN-LIKE IONS
In this section, we study the energy level shifts inmedium and heavy hydrogen-like ions due the nuclear po-larization induced by the electron-nucleon interaction. InSec. III A, we review the necessary theoretical backgrounddeveloped in the papers [6–10]. In the subsequent sub-sections, we present and discuss the results of numericalcalculations of these effects.
A. Theoretical background
It is well known that nuclear polarization due to theelectron-nucleon interaction contributes to the electronicLamb shift, see, e.g., Ref. [1] for a review. In light atomsand ions this effect is very small [30, 31], but it becomessignificant in heavy atoms and ions, where the electronwave functions have sizable overlap with the nucleus and may thus be considerably affected by the nuclear structure.In this section we study these energy shifts in hydrogen-like medium and heavy ions. The effects in multieletronatoms will be discussed in Sec. IV. Motivated by futurestudy of the nonlinearity of King’s plot [13] induced bythe nuclear polarization, we will mainly focus on theeven-even nuclei with vanishing nuclear spin.The atomic energy shifts due to nuclear polarizationare well-understood within the framework of QED andmay be represented by a two-photon exchange betweenthe electron and an unpaired nucleon in the nucleus. Thisprocess may be illustrated by the Feynman diagrams inFig. 1.
FIG. 1. Contributions to electronic self energy due to nuclearpolarization. Solid and double solid lines correspond to theelectron and nucleon propagators, respectively, while the wavylines represent the photon propagators.
As mentioned above, we are considering nuclei withvanishing angular momentum in the ground state, L = 0.The excited nuclear states may, on the other hand, havearbitrary angular momentum L and energy E L . The ini-tial electronic state may be characterised by its principalquantum number n , its orbital angular momentum l andits total angular momentum j . The electronic energieswill be denoted by ε nlj . The atomic energy level shiftdue to the processes presented in Fig. 1 was calculated inRef. [10] and reads∆ ε nlj = − α π (2 L + 1) B ( EL ; L → (cid:88) j (cid:48) (2 j (cid:48) + 1) (cid:18) j (cid:48) j L − (cid:19) × (cid:32)(cid:88) n (cid:48) l (cid:48) |(cid:104) nlj | F L | n (cid:48) l (cid:48) j (cid:48) (cid:105)| ε n (cid:48) l (cid:48) j (cid:48) − ε nlj + E L + (cid:90) − m e −∞ |(cid:104) nlj | F L | εj (cid:48) (cid:105)| ε − ε nlj − E L dε + (cid:90) ∞ m e |(cid:104) nlj | F L | εj (cid:48) (cid:105)| ε − ε nlj + E L dε (cid:33) , (20)where B ( EL ; L →
0) is the reduced transition probabilityfor nuclear electric transitions from an excited state withangular momentum L to the ground state and F L ( r ) is aradial function of the form F L ( r ) = 4 π (2 L + 1) R L (cid:18) r L R L +10 Θ( R − r )+ R L r L +1 Θ( r − R ) (cid:19) , ( L ≥ . (21)which behaves like 1 /r L +1 outside the nucleus and like r L inside. The function (21) represents a regularization ofthe 1 /r L +1 potential to the case of an extended nucleus. The three terms in the second line in Eq. (20) corre-spond to the contributions from intermediate electronicstates in discrete, negative energy continuum and posi-tive energy continuum spectra, respectively. The matrixelements in these terms are defined as (cid:104) A | F L | B (cid:105) ≡ (cid:90) ∞ dr r F L ( f A f B + g A g B ) , (22)where f A,B and g A,B are, respectively, upper and lowerDirac radial wave functions of the electron.It was pointed out in Refs. [7, 8] that the intermediateelectronic states in the discrete spectrum give negligiblecontribution to the energy shift as compared to the contri-butions from the lower and upper continua. Indeed, theradial integral in Eq. (22) receives the main contributionfrom the vicinity to the surface of the nucleus, where theradial function (21) peaks. Thus, the dominant contribu-tions come from small distances, i.e., from states with highenergies in the continuous spectrum. Moreover, the dis-crete spectrum terms in Eq. (20) are suppressed by largedenominators because ε n (cid:48) l (cid:48) j (cid:48) − ε nlj (cid:28) E L . Therefore, inour calculations below, we will ignore the contributionsfrom the discrete spectrum.In Eq. (20), the leading contributions come from thelow- L transitions while the higher- L terms are suppressedbecause the corresponding electron wave functions havea small overlap with the nucleus. Therefore, to a gooddegree of accuracy, it is sufficient to consider only theterms with L = 1 and L = 2, the former corresponds toa nuclear giant electric dipole resonance while the lattermay be interpreted as a contribution from nuclear rotationassociated with the collective nuclear quadrupole momentin a deformed nucleus. In the following subsections, wewill consider these two contributions separately. B. Contribution from giant electric dipoleresonance transition
The giant electric dipole resonance nuclear transitionscorrespond to L = 1 in Eq. (20),∆ ε nlj = − α π B ( E (cid:88) j (cid:48) (2 j (cid:48) + 1) (cid:18) j (cid:48) j − (cid:19) × (cid:18)(cid:90) − m e −∞ |(cid:104) nlj | F | εj (cid:48) (cid:105)| dεε − ε nlj − E GR + (cid:90) ∞ m e |(cid:104) nlj | F | εj (cid:48) (cid:105)| dεε − ε nlj + E GR (cid:19) , (23)where the energy of giant dipole resonance E GR in a heavynucleus is given by [25, 32]: E GR = 95(1 − A − / ) A − / MeV . (24)The transition probability B ( E ≡ B ( E
1; 1 →
0) inEq. (23) for giant electric dipole resonance transitionsmay be estimated using the Thomas-Reiche-Kuhn sumrule [25], giving B ( E ≡ B ( E
1; 1 →
0) = 38 π Z ( A − Z ) e AE GR m p . (25)Using the formula (23) with the nuclear transitionstrength (25) we numerically calculate the energy shiftsfor 1 s , 2 s and 2 p / states in hydrogen-like ions. The ra-dial integrals (22) are calculated numerically using knowncontinuum and discrete state Dirac radial wave functions,taking into account the finite size of the nucleus. Theintegrals over dε in Eq. (23) is also evaluated numerically.We consider hydrogen-like ions with nuclear chargesranging from Z = 20 (Calcium) to Z = 98 (Californium),and extend the results to superheavy elements up to Z = 136. For each Z , ions with different A are alsoconsidered. It should be noted that although we restrictourselves to even-even nuclei, Eq. (23) also applies tonuclei with odd A , only in this case the structure of atomicenergy levels is more complicated due to the hyperfineinteractions. On the other hand, extending the currentcomputation to odd- A nuclei proves to be convenient forfitting the results, see Eqs. (26) and (27) below.The results of our numerical calculation are presentedin Table VII. In Table I, we compare our results withthose published earlier [7, 8] for certain heavy elements.This table shows that our numerical methods provide theaccuracy within 5% of earlier publications [7, 8]. Z A ∆ ε s ∆ ε ref1 s ∆ ε s ∆ ε ref2 s ∆ ε p ∆ ε ref2 p /
82 208 18 . . . . . −
90 232 39 . . . . .
80 0 . . . . . . .
192 236 47 . . . . . .
192 238 48 . . . . . .
098 250 84 . . . . . .
498 252 85 . . . . . . ε are given inunits of meV. One of goals in this paper is to establish an analyticaldependence ∆ ε = ∆ ε ( Z, A ), which should give the en-ergy shifts for medium and heavy ions including isotopicdependence. We find that to a high degree of accuracy,the isotopic dependence of the energy shifts is linear. Foreach Z we consider the isotope with the atomic number A ( Z ) = [ − .
62 + 2 . Z ] , (26)and write the mass number of the other isotopes as A = A ( Z ) + ∆ A . (27)In the series decomposition of energy shift ∆ ε , it is suf-ficient to keep only linear terms in ∆ A , ∆ ε ( Z, A ) =∆ ε ( Z )+∆ ε ( Z )∆ A . The functional behaviour of ∆ ε ( Z )is presented by dots in Fig. 2 for 1 s , 2 s and 2 p / states.These graphs show that − ∆ ε ( Z ) grows approximatelyexponentially with Z . It may be verified that ∆ ε ( Z )shows similar behaviour. Therefore, we use the followingfitting functions to approximate the energy shift, In Eq. (26), the square brackets denote the rounding to thenearest integer. This formula covers most of the stable isotopes(when exist) for each Z in the region 20 ≤ Z ≤
98. We note alsothat the choice of A here is entirely for convenience. Differentchoices require different fitting functions and parameters but areotherwise equivalently legitimate.
20 40 60 80 100 - - - -
200 Z Δ ϵ ( m e V ) ( a )
20 40 60 80 100 - - -
50 Z Δ ϵ ( m e V ) ( b )
20 40 60 80 100 - - - - Δ ϵ ( m e V ) ( c ) p / FIG. 2. Shifts of 1 s , 2 s and 2 p / energy levels in medium and heavy hydrogen-like ions (dots) due to nuclear polarizationthrough E1 nuclear transitions. Solid lines represent the best fit of these shifts with the function (28). ∆ ε ( Z, ∆ A ) = − (cid:2) exp( a + a Z + a Z ) Z a + exp( b + b Z + b Z ) Z b ∆ A (cid:3) meV , (28)with fitting parameters a , , , and b , , , presented inTable II. With these parameters, equation (28) reproducesthe numerical results in Table VII with an accuracy under2% for 20 ≤ Z ≤
98. As a demonstration, the functions(28) are plotted in Fig. 2 for 1 s , 2 s and 2 p / states.In computing the energy level shifts in Table VII, weemployed the empirical formula (4) for the nuclear radius.This formula is, however, only approximate, and exper-imental values of R may have deviations from Eq. (4), R = R + δR . (29)To take into account such deviations, we modify Eq. (28)as ∆ ε ( Z, A, R ) = ∆ ε ( Z, A, R ) + δ R ε ( Z ) δR , (30)where ∆ ε ( Z, A, R ) is given by Eq. (28), and the correc-tion term δ R ε ( Z ) may be approximated by the function δ R ε ( Z ) = exp( c + c Z + c Z ) Z c meV / fm . (31)The coefficients c , , , in Eq. (31) are computed nu-merically and collected in Table II. We stress that thecorrections due to variations of the nuclear radius are im-portant within the study of possible nonlinearity of King’splot [12–16] and physics beyond the SM [11, 17–21].We note that here we consider only the giant dipoleresonance nuclear transitions with the nuclear energy (24)and reduced transition probability (25). In certain nuclei,such as Th, there can be additional E1 transitionsfrom the ground state to low lying levels with typicalenergy in the keV range. As a result, the contributionsof such transitions to the overall energy shift are, to acertain degree, enhanced when compared to other typicaltransitions with energy in the MeV range. However, s s p / a − . − . − . a − . × − − . × − − . × − a . × − . × − . × − a b − . − . − . b . × − . × − . × − b . × − . × − . × − b c − . − . − . c − . × − − . × − − . × − c . × − . × − . × − c s , 2 s and 2 p / energy levelshifts in hydrogen-like ions due to nuclear giant electric dipoleresonance. we point out that the probability B ( E
1) of these lowlying transitions is three to four orders of magnitudesmaller than that of the giant dipole resonance. Therefore,contributions from isolated low-lying E1 nuclear energylevels are still negligible.Finally, we point out that the accuracy of calculationof the energy shift with Eq. (23) strongly depends onthe value of the nuclear reduced transition probability(25). The latter formula provides an approximate, av-erage description of giant dipole resonance transitions,and particular isotopes may have considerable deviationsfrom this formula. For such isotopes one can improvethe accuracy of calculations of the energy shift by apply-ing the correcting coefficient B exact ( E /B ( E B exact ( E
1) is the exact value of the nuclear reduced tran-sition probability found from experiments and B ( E
1) isthe approximate value calculated with the use of Eq. (25).In particular, the values of B exact ( E
1) may be derivedfrom the photonuclear cross-section data collected, e.g.,in Ref. [33].
C. Contribution from nuclear rotational transition
A spinless nucleus with quadrupole deformation mayhave collective E2 excitations from the ground state intothe rotational band. The energies of these transitions aretypically on the order of a few dozens keV, that is, muchlower than the energy of giant electric dipole resonancetransition which is about a dozen of MeV. As a result, theshifts due to nuclear rotational transitions receive sizableenhancements.It is worth noting that nuclear transitions to higherrotational states give minor contributions to the atomicenergy level shifts [8], because the reduced transitionprobability B ( E
2; 2 →
0) decreases rapidly for such states,and there is additional suppression from higher nuclearenergy in the denominator. Therefore, in this paper weconsider only the nuclear transition from the ground stateto the lowest rotational state with L = 2. In this case,Eq. (20) may be written as∆ ε nlj = − α π B ( E (cid:88) j (cid:48) (2 j (cid:48) + 1) (cid:32) j (cid:48) j − (cid:33) × (cid:18)(cid:90) − m e −∞ |(cid:104) nlj | F | εj (cid:48) (cid:105)| dεε − ε nlj − E rot + (cid:90) ∞ m e |(cid:104) nlj | F | εj (cid:48) (cid:105)| dεε − ε nlj + E rot (cid:19) , (32)where E rot is the energy of the lowest nuclear rotationallevel.The transition probability B ( E ≡ B ( E
2; 2 →
0) maybe expressed via the intrinsic nuclear quadrupole moment(5) as B ( E
2; 2 →
0) = Q / (16 π ) (see, e.g., Ref. [25]),which gives B ( E ≡ B ( E
2; 2 →
0) = 15 (cid:18) π (cid:19) Z e R β . (33)Numerical values of the nuclear deformation parameter β as well as the reduced transition probability B ( E
2) maybe found in, e.g., Ref. [34]. In this reference, the valuesof the reduced transition probability are calculated fromknown values of the lifetime of the excited nuclear state.Lifetimes of the excited nuclear states may be found, e.g.,in [35].According to an empirical rule [25], the energy of thefirst excited 2 + nuclear rotational state is connected toits transition probability from the ground state via E rot B ( E ≈ Z A − e MeV × fm , (34)which, along with Eq. (33), allow us to express the energy E rot in terms of macroscopic nuclear parameters, E rot ≈ π AR β GeV × fm . (35)It is worth noting that this formula is applicable to de-formed nuclei with 0 . (cid:46) β (cid:46) . β which is, ingeneral, a nonmonotonic function of Z and A . Moreover,experimental data of B ( E
2) collected, e.g., in Ref. [34]have some deviation from formula (33). As a result, weshall keep B ( E
2) as a free parameter and calculate thequantity ∆ ε nlj /B ( E β is the rotationalenergy E rot in the denominator in Eq. (32). However,the energy integrals in Eq. (32) vary slowly for differentvalues of E rot given by Eq. (35) because these integralsreceive dominant contributions from ε ≈ m e whereas E rot (cid:46) β = 0 .
27 to estimate the rotationalenergy E rot in the denominator in Eq. (32).We numerically calculate the quantity ∆ ε nlj /B ( E
2) for1 s , 2 s and 2 p / energy levels in hydrogen-like ions withnuclear deformations 0 . (cid:46) β (cid:46) .
35. The results of thesecalculations are collected in Table VIII. Comparisons withknown results are presented in Table III.
Z A ∆ ε s B ( E
2) ∆ ε ref1 s B ( E
2) ∆ ε s B ( E
2) ∆ ε ref2 s B ( E
2) ∆ ε p B ( E
2) ∆ ε ref2 p B ( E
90 230 0 .
602 0 .
605 0 .
114 0 .
112 0 . . .
710 0 .
708 0 .
136 0 .
133 0 . . .
705 0 .
702 0 .
135 0 .
132 0 . . .
15 1 .
12 0 .
235 0 .
213 0 . . .
14 1 .
07 0 .
234 0 .
213 0 . . ε/B ( E
2) aregiven here in units of meV / fm . For each ion, we consider several isotopes to determinethe isotopic dependence of the energy shift with respectto the quantity ∆ A as defined in Eq. (27). We find thatthese results may be approximated with the exponentialfunction,∆ ε ( Z, ∆ A ) B ( E
2) = (cid:2) − exp(˜ a + ˜ a Z + ˜ a Z ) Z ˜ a + exp(˜ b + ˜ b Z + ˜ b Z ) Z ˜ b ∆ A (cid:3) meV10 fm , (36)where the values of the coefficients ˜ a , , , and ˜ b , , , arecollected in Table IV. The function (36) with ∆ A = 0 isplotted in Fig. 3 for 1 s , 2 s and 2 p / energy level shifts.
30 40 50 60 70 80 90 100 - - - - - Δ ϵ B ( E )( m e V f m ) ( a )
30 40 50 60 70 80 90 100 - - - -
500 Z Δ ϵ B ( E )( m e V f m ) ( b )
30 40 50 60 70 80 90 100 - - - - - -
50 Z Δ ϵ B ( E )( m e V f m ) ( c ) p / FIG. 3. Shifts of 1 s , 2 s and 2 p / energy levels in heavy hydrogen-like ions (dots) due to nuclear polarization through rotationalE2 transitions. Solid lines represent the best fit of these shifts with the function (36).1 s s p / ˜ a − . − . − . a − . × − − . × − − . × − ˜ a . × − . × − . × − ˜ a b − . − . − . b − . × − − . × − − . × − ˜ b . × − . × − . × − ˜ b c − . − . − . c − . × − − . × − − . × − ˜ c . × − . × − . × − ˜ c s , 2 s and 2 p / energy levelshifts in hydrogen-like ions due to nuclear E2 rotational tran-sitions in deformed nuclei. Analogously to Eqs. (30) and (31), we find the correc-tions due to the deviation of the actual nuclear radius(29) from the approximate formula (4) in the form δ R ε ( Z ) B ( E
2) = exp(˜ c + ˜ c Z + ˜ c Z ) Z ˜ c meV10 fm , (37)where the coefficients ˜ c , , , are given in Table IV for 1 s ,2 s and 2 p / states.We point out that the results of this section readilygeneralize to superheavy elements. In tables VII and VIIIwe present estimates of energy shifts for such elements upto Z ≤ IV. EFFECTIVE POTENTIAL FOR SCALARNUCLEAR POLARIZATION CORRECTIONS INMEDIUM AND HEAVY ATOMS
In Sec. III, we calculated shifts of lowest energy levelsin heavy hydrogen-like ions. This calculation may be per-formed with high accuracy because it makes use of exactelectron wave functions in the discrete and continuousspectra. A generalization of these results to multielectron ions and neutral heavy atoms is hindered by many-bodyeffects which are usually taken into account within themany-body theory based on the relativistic Hartree-Fockbasis states. Precision calculation of energy level shifts inmultielectron atoms and ions due to electric polarizationof the nucleus goes beyond the scope of this paper, as itrequires special numerical methods and computer codeswhich take into account many-body effects.In this section, however, we will demonstrate that theeffect of nuclear polarization may be taken into accountby an effective potential which, when added to the un-perturbed Hamiltonian, gives the same atomic energylevel shifts as have been found in the previous section. Given that this potential is local and has a simple form,it may be added to the nuclear Coulomb potential andincorporated into numerical calculations of the spectraof multielectron ions and atoms including calculations ofthe isotope shifts. Such numerical computation will begiven elsewhere.
A. General properties of the effective potential
Recall that the effect of nuclear polarization due tothe electron-nucleon interaction is well described by QEDquantum corrections corresponding to Feynman graphsin Fig. 1. In this process, the electron-nucleus interactionis essentially non-local as it is based on one-loop quantumeffects with virtual electronic states having arbitrary highenergy. At large distance, however, this interaction shouldreduce to a local four-point vertex. This dictates the large-distance asymptotic behaviour of the effective potentialfor this interaction, V L ( r ) | r →∞ ∼ r − L − , where L = 1for E1 and L = 2 for E2 nuclear transitions, respectively.The coefficient of proportionality in this relation may be Rigorously the energy shift may be presented as expectation valueof a non-local (integration) self-energy operator Σ( r, r (cid:48) , E ) whichat large distances becomes an ordinary local polarization poten-tial V ( r ) = − ¯ α E e r . Such approach with the operator Σ( r, r (cid:48) , E )added to the Hartree-Fock Hamiltonian has been developed to cal-culate correlation corrections due to interaction between valenceand core electrons [36]. deduced from Eq. (20). Indeed, at large distance from thenucleus, where the electron energy is small as comparedwith the energy of nuclear transitions, the energy shiftshould be proportional to the nuclear polarizability, ∆ ε ∝ α EL , where α EL = 8 π L + 1 B ( EL ; L → E L (38)is the scalar nuclear polarizability due to the EL nucleartransition with energy E L . Therefore, we fix the asymp-totic behaviour of the effective potential in the form V L ( r ) | r →∞ → − e α EL r L +2 . (39)Let b be a characteristic distance at which the effectiveelectron-nucleus interaction becomes non-local such thatit cannot be described by the asymptotic formula (39).Although there may be different ways to extend the effec-tive potential to the region r < b which would have thesame asymptotic behaviour (39), we find it suitable todefine the effective potential as V L ( r ) = − e α EL r L +2 + b L +2 . (40)In this case, the parameter b may be thought of as ancut-off parameter below which the effective potential isnearly constant.We stress that b is the only free parameter in theeffective potential (40). This parameter is, however, notuniversal in the sense that it should take into accountspecific nuclear properties such as nuclear charge Z , massnumber A , nuclear radius R and nuclear deformation β , b = b ( Z, A, R , β ) . (41)Moreover, this parameter may be different for s and p / states as well as for E1 and E2 nuclear transitions. Below,we describe the procedure that will allow us to find thedependence (41) and will apply it to E1 and E2 nucleartransitions.Let ∆ ε ( L )Ψ be an energy level shift in an atom or ionin a state Ψ due to nuclear polarization induced by EL nuclear transition. In particular, for hydrogen-like ions,the values of such shifts are calculated in the previoussection and presented in Tables VII and VIII. Given thevalue of this energy level shift, we require that the effectivepotential (40) should yield the same value,∆ ε ( L )Ψ = (cid:104) Ψ | V L | Ψ (cid:105) . (42)This equation allows us to find the value of the freeparameter b in the effective potential (40) for a givenatom or ion in the state Ψ. The variety of values ofthis parameter for different Z , A , R and β sets up thefunction (41). It is natural to expect that this functionshould vary in the range R < b (cid:28) a B /Z . It is important to note that the effective potential (40)should be applied to multielectron atoms and ions. Inthis case, the parameter b must be found from Eq. (42) inwhich the energy shift in the left-hand side is calculatedwith a valence-electron wave function Ψ. In heavy atoms,this wave function (and the corresponding energy shift)may be quite different from the exact 1 s , 2 s and 2 p / Dirac wave functions employed in Sect. III for hydrogen-like ions. We will apply these functions only in the region r (cid:28) a B /Z / where the effects of the nuclear polarizationare significant. In this region, calculation with the ap-proximate radial wave functions (16) would have a goodaccuracy.One has to keep in mind that the approximate wavefunctions (16) correspond to the model of point-like nu-cleus. In heavy atoms, however, finite nuclear size correc-tions are significant. To take such effects into account inthe leading order we will use the wave functions (16) onlyoutside the nucleus, i.e., for R < r < a B /Z / , whileinside the nucleus, 0 ≤ r ≤ R , these functions may beextended as f s ( r ) = c ns ( − γ ) J γ ( x ) − x J γ − ( x ) R ,g s ( r ) = c ns rR ZαJ γ ( x ) , (43)where c ns is the normalization constant and x ≡ (cid:112) ZR /a B .Note also that in atoms it is sufficient to consider only s -electron wave functions since higher waves give minorcorrections due to nuclear polarization. Indeed, as isseen from Tables VII and VIII, the contribution fromthe 2 p / wave is from one to two orders in magnitudesmaller than that from 2 s wave. Therefore, we will restrictourselves to specifying the effective potential for s wavesonly. From the comparison of the potentials for 2 p / and 2 s / electrons we see that the difference between the2 p / and 2 s / is not significant, so for the approximatecalculation of a relatively small contribution of the p / potential one may use the s -wave potential. B. On the effective potential for light atoms
Although in this paper we study nuclear polarizationcorrections to the spectra of medium and heavy atoms,in this section we briefly consider the effective potentialin light atoms. We present this result only for a demon-stration of the procedure of derivation of the effectivepotential which will be applied to medium and heavyatoms in subsequent subsections. In the case of lightatoms, the corrections due to nuclear polarization werefound analytically in Ref. [30]. Therefore, the procedure ofconstructing the effective potential is considerably simplerand more transparent in this case.We restrict ourselves only to E1 nuclear transitionswhich are taken into account by the potential (40) with0 L = 1, V = − e α E r + b . (44)In this potential, we have to determine the cut-off param-eter b as a function of nuclear parameters.Light atoms may be well-described by non-relativisticwave functions. Let φ ( r ) be a wave function of valence s electron in a light atom. In this state, the expectationvalue of the operator (44) may be found analytically, (cid:104) s | V | s (cid:105) ≈ − πe ¯ α E | φ (0) | (cid:90) ∞ r drr + b = − π e ¯ α E | φ (0) | √ b . (45)Here we have taken into account that the s -wave functionvaries slowly inside the nucleus, so that | φ ( r ) | may be ap-proximated by the electron density at the nucleus | φ (0) | .The expectation value (45) should be matched with theatomic energy s -level shift due to nuclear polarizabilitycalculated in Ref. [30]:∆ ε = − m e e | φ (0) | ¯ α E (cid:20)
196 + 5 ln (cid:18) ωm e (cid:19)(cid:21) . (46)Here ¯ ω is an average nuclear excitation energy in E ω given by Eq. (6) in He and heavier nuclei. Equation∆ ε = (cid:104) s | V | s (cid:105) yields the value of the cut-off parameter b in light elements: b ≈ π √ / ω/m e )] m e . (47)Eq. (47) allows us to estimate the value of the cut-offparameter for light elements. In particular, for He andLi, b ≈
105 fm. As we will show below, in heavy atomsthe value of the cut-off parameter is smaller but of thesame order.
C. Effective potential due to giant electric dipoleresonance
In Sect. III B, we calculated shifts of 1 s , 2 s and 2 p / energy levels in medium and heavy hydrogen-like ions.These data, however, do not apply to neutral heavy atomsin which energy level shifts come from ns electronic or-bitals with n >
2. The energy level shifts in neutral atomswith ns valence electrons will be denoted as ∆ ε ns in thissection. In Table IX we present the results of calculationsof ∆ ε ns in medium and heavy atoms with 20 ≤ Z ≤ ns electron wave functions (16) extendedto the inside of the nucleus as in Eq. (43). Note that wedo not specify the normalization coefficients c ns in thesefunctions since our final result for the effective potential will be independent from these values. Therefore, in Ta-ble IX we present the values of the dimensionless quantity∆ ε ns /c ns .In this section we consider the effects of nuclear po-larization due to giant electric dipole resonance nucleartransition in medium and heavy atoms. In this case, theeffective potential (40) reads V = − e α E r + b , α E = 8 πB ( E E GR , (48)where B ( E
1) is the reduced transition probability (25)and E GR is the energy of giant electric dipole resonancetransition (24). The expectation value of the operator(48) in the ns state reads (cid:104) ns | V | ns (cid:105) = − e α E (cid:90) f ns ( r ) + g ns ( r ) r + b ns r dr , (49)where f ns and g ns are radial wave functions (16) extendedto the inside of the nucleus as in Eqs. (43).Recall that the value of the parameter b in the effectivepotential (48) (denoted by b ns in what follows) should befound from Eq. (42). In the case of giant dipole resonancenuclear transitions, the energy in the left-hand side inEq. (42) is given by ε ns presented in Table IX while theright-hand side is given by Eq. (49). As a result, we have1 c ns (cid:90) f ns ( r ) + g ns ( r ) r + b ns r dr = − E GR πe B ( E
1) ∆ ε ns c ns . (50)Eq. (50) defines the parameter b ns for each value of theenergy shift ∆ ε ns /c ns . We solve this equation numer-ically and present the values of the parameter b ns inTable IX. As is seen from this table, the parameter b ns is a monotonic function of Z , b ns = b ( Z ). It is con-venient to approximate this function by the exponent, b ( Z ) = exp( λ + λ Z + λ Z ) Z λ , where the best fit forthe parameters λ , , , is presented in Table V.The dependence of the parameter b ns on the massnumber A and nuclear radius R may by taken into accountin the same way as in Sect. III B: For each Z we fix A ( Z ) as in Eq. (26) and consider ∆ A deviations from A ( Z ) (27). Then, for each isotope we fix the nuclearradius by Eq. (4) and consider small deviation from thisvalue, Eq. (29). The parameter b is now considered up tolinear terms in ∆ A and δR , b ns ≡ b ( Z, A, R ) = b ( Z ) + b ( Z )∆ A + b ( Z ) δR . It is convenient to approximate thefunctions b , , ( Z ) by exponents as follows: b ( Z, A, R ) = (cid:2) exp( λ + λ Z + λ Z ) Z λ + exp( ν + ν Z + ν Z ) Z ν ∆ A (cid:3) fm+ exp( τ + τ Z + τ Z ) Z τ δR . (51)Here λ , , , , ν , , , and τ , , , are fitting parameterswith numerical values given in Table V. In this table, Explicit values of these coefficients are presented, e.g., in Ref. [29]. b s , b s and b p / givenin Table IX. These values are given for comparison. Inparticular, it is seen that the functions b ns and b s arevery close, while the 1 s and 2 p / state are described byslightly different functions. s p / λ .
88 9 . λ − . × − . × − λ . × − . × − λ − . − . ν .
05 4 . ν − . × − − . × − ν . × − . × − ν − . − . τ .
94 10 . τ − . × − . × − τ . × − − . × − τ − . − . b for s and p / effective potentials in multielectron atoms due to nuclear E1electric dipole giant resonance. To summarize, the function (51) with the parameters λ , , , , ν , , , and τ , , , given in Table V specifies theeffective potential (48). This effective potential allowsone to calculate the energy level shifts with error under1% as compared with the results presented in Table IX. D. Effective potential due to electric quadrupolenuclear polarization
In the derivation of the effective potential which takesinto account electric quadrupole nuclear polarization cor-rections in the atomic spectra we will follow the sameprocedure as in Sect. IV C. Namely, we start with thecalculation of the energy shifts ∆˜ ε ns using the ns valenceelectron wave functions (16) as the initial and final elec-tronic states in Eq. (32). The results of these calculationsare presented in Table X.In the case of nuclear polarization due to the rotationalnuclear transitions the effective potential (40) reads V = − e ¯ α E r + ˜ b , (52)where ˜ b is the effective cut-off parameter and¯ α E = 8 π B ( E E rot , ¯ E rot = 50 keV , (53) is the modified E2 nuclear polarizability. In contrast withthe conventional nuclear polarizability (38), it has fixedenergy ¯ E rot = 50 keV which corresponds to typical energyof the lowest rotational state in deformed heavy nuclei.Using this fixed energy in Eq. (53) appears more conve-nient in the effective potential (52) because the physicalenergy (35) depends on the deformation parameter β which changes non-monotonically with Z . Indeed, thepotential (52) defined via the modified nuclear polariz-ability (53) is a monotonic function of Z and, thus, it issuitable for modelling atomic energy level shifts due tothe nuclear rotational transitions given in Table X.The expectation value of the effective potential in the ns state is (cid:104) ns | V | ns (cid:105) = − πB ( E e E rot (cid:90) f ns + g ns r + ˜ b r dr , (54)where the integral is calculated with the use of the wavefunctions (16) extended to the inside of the nucleus asin Eq. (43). The cut-off parameter ˜ b should now befound upon matching the expectation value of the effectiveoperator (54) with the energy shifts ∆˜ ε ns given in Table X,1 c ns (cid:90) f ns + g ns r + ˜ b r dr = − E rot πe ∆˜ ε ns c ns B ( E . (55)The value of the parameter ˜ b may be found by solvingthis equation numerically for each given isotope. Wepresent these values in Table X. In the same table wegive also the values of this parameter corresponding tothe 1 s , 2 s and 2 p / energy level shifts in hydrogen-likeions from Table VIII. We point out that the values of theparameter b ns are very close to the corresponding valuesof b s and especially b s that confirms the consistency ofthe definition of the effective potential (52).The numerical values of the parameter ˜ b ns in Table Xdefine the function ˜ b ns = ˜ b ( Z, A, R ). This function maybe approximated by˜ b ( Z, A, R ) = (cid:2) ˜ λ + ˜ λ Z + exp(˜ ν + ˜ ν Z + ˜ ν Z )∆ A (cid:3) fm+ exp(˜ τ + ˜ τ Z + ˜ τ Z ) Z τ δR , (56)where the best fit for the parameters ˜ λ , , , ˜ ν , , and˜ τ , , is given in Table VI.To summarize, in this section we found the effectivepotential (52) with the parameter ˜ b given by Eq. (56).For medium and heavy elements, this effective potentialreproduces the atomic energy level shifts presented inTable X with error under 1%. For superheavy elements,the error is within 5%.2 s p / ˜ λ
135 148˜ λ − . − . ν − .
172 7 . × − ˜ ν − . × − − . × − ˜ ν . × − . × − ˜ τ .
93 2 . τ − . × − − . × − ˜ τ . × − . × − ˜ τ .
386 0 . λ , , ˜ ν , , and˜ τ , , , which specify the cut-off parameter (56) as a functionof Z , A and R . V. CONCLUSIONS
In this paper, we studied the effects of electric nuclearpolarization in the spectra of medium and heavy atomsand ions. These effects manifest themselves in electronenergy levels shifts and isotope shifts or in contributions tothe hyperfine structure. Although in neutral atoms sucheffects are small, they are strongly enhanced in heavyhydrogen-like ions. Indeed, the s / and p / electronwave functions are known to be significantly enhancednear a heavy nucleus [29], so that corrections due tonuclear polarization are observable.Recall that the tensor nuclear polarizability is respon-sible for contributions to the hyperfine structure. Weobserve that the effective operator describing the contri-butions from the tensor polarizability has the same tensorstructure as the quadrupole nuclear moment. Therefore,it is natural to compare the contributions from these oper-ators to the atomic hyperfine structure. We show that theeffect from the tensor nuclear polarizability is nearly threeorders of magnitude weaker than that from the electricquadrupole nuclear moment. Although in neutral atomsthis effect is rather unobservable, it may be noticeablein heavy hydrogen-like ions where the hyperfine energysplitting is on order of 1 eV. In this paper, we estimatedthe order of magnitude of this effect while accurate calcu-lations of contributions from the tensor polarizability inparticular atoms are left for further studies.The scalar nuclear polarizability is responsible foratomic energy levels shifts. The method of calculatingthese shifts was developed in a series of papers [6–10]where these shifts were found for a limited number ofhydrogen-like ions. We employ this method and extendthe results to include hydrogen-like ions with 20 ≤ Z ≤ Z , A and nuclear radius. Errors inthe atomic calculations are smaller than 5%.Energy level shifts in some superheavy hydrogen-likeions ( Z = 106 , , , , Z approximatelyexponentially. Therefore, this functions may be taken inthe form ε ( Z ) = ε e aZ , with parameters ε and a chosento reproduce exactly values of ε ( Z ) for two nearby ionswhere we have performed the calculations.We consider separately contributions from nuclear giantelectric dipole resonance transitions (E1) and rotationaltransitions (E2). In the latter case, the energy shiftsdepend also on the nuclear quadrupole deformation pa-rameter β . By comparing our results with the earliercalculations in some heavy hydrogen-like ions [7, 8] wefind that the obtained formulae provide energy shifts inheavy elements with error of atomic calculations under7%. The accuracy in the nuclear parameters, which weuse, is defined in the referenced papers.We also study the dependence of the energy shiftson the variation of nuclear radius ∆ R . Indeed, certainisotopes may have deviations of the nuclear radius fromthe general rule (4), and the energy shifts are sensitiveto such deviations. This may lead to nonlinearity inKing’s plot for isotope shifts and imitate effects of newinteractions. Note that such nonlinearity was observedrecently in Yb isotopes [37] and interpreted in termsof new interaction beyond the Standard Model. Ourcalculation opens the way for systematic study of thiseffect in a wide range of atoms and ions.We point out that the calculation of corrections dueto nuclear polarizability in the spectra of hydrogen-likeions may be performed with a good accuracy because inthis calculation one can use exact Dirac wave functionin discrete and continuous spectra. A generalization ofthese results to multielectron ions and neutral atoms ishindered by many-body effects which are usually takeninto account using many-body theory based on the rela-tivistic Hartree-Fock basis states. In order to facilitatesuch calculations in future works, in this paper we developan effective potential which models the corrections due tothe nuclear polarizability. This potential has simple localform (40) with one parameter b which is approximatedby the functions (51) and (56) such that the expectationvalue of this potential gives correct energy level shifts forhydrogen-like ions and for s electrons in a many-electronatoms. As a result, to calculate the nuclear polarizationeffect in many-electron atoms and ions, one simply has toadd this potential to the nuclear Coulomb interaction andthen solve for the self-consistent Hartree-Fock equations.Incorporation into the calculation of the correlation cor-rections is straightforward with the use of Hartree-Fockbasis states.The effect is dominated by the nuclear polarizationpotential for s electrons, with a much smaller contribu-tion from the slightly different potential for p / electrons.The difference between 2 s and ns potentials is very small,therefore, one can use 2 p / potential for all p / electrons.Moreover, the difference between the p / and s poten-tials is not significant. Potentials in all waves have thesame long distance asymptotic but have different cut-offparameters b . We have checked that in heavy atoms the3difference between parameters b in 2 s and 2 p / potentialsis small. 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Parity nonconservation in atomic phe-nomena (Gordon and Breach, 1991).[30] K. Pachucki, D. Leibfried, and T. W. H¨ansch, Phys. Rev.A , R1 (1993).[31] K. Pachucki, M. Weitz, and T. W. H¨ansch, Phys. Rev. A , 2255 (1994).[32] B. Hoffmann, G. Baur, and J. Speth, Z. Phys. , 57(1984).[33] T. Kawano et al. , Nucl. Data Sheets , 109 (2020).[34] S. Raman, C. W. G. Nestor, Jr, and P. Tikkanen, Atom.Data Nucl. Data Tabl. , 1 (2001).[35] National Nuclear Data Center, Nuclear structure & decaydata, .[36] V. A. Dzuba, V. V. Flambaum, P. G. Silvestrov, and O. P.Sushkov, J. Phys. B , 1399 (1987).[37] I. Counts, J. Hur, D. P. L. Aude Craik, H. Jeon, C. Leung,J. C. Berengut, A. Geddes, A. Kawasaki, W. Jhe, andV. Vuleti´c, Phys. Rev. Lett. , 123002 (2020). Z A − ∆ ε s − ∆ ε s − ∆ ε p / δ R ε s δ R ε s δ R ε p / (meV) (meV) (meV) (meV/fm) (meV/fm) (meV/fm)20 40 3 . × − . × − . × − . × − . × − . × −
20 44 4 . × − . × − . × −
26 54 1 . × − . × − . × − . × − . × − . × −
26 60 1 . × − . × − . × −
34 72 6 . × − . × − . × − . × − . × − . × −
34 80 7 . × − . × − . × −
42 92 0 .
207 2 . × − . × − . × − . × − . × −
42 100 0 .
225 3 . × − . × −
50 112 0 .
583 8 . × − . × − . × − . × − . × −
50 126 0 .
649 9 . × − . × −
58 136 1 .
51 0 .
223 8 . × − .
167 2 . × − . × −
58 142 1 .
57 0 .
232 8 . × −
66 154 3 .
51 0 .
543 2 . × − .
409 6 . × − . × −
66 164 3 .
70 0 .
573 2 . × −
74 180 8 .
11 1 .
33 8 . × − .
974 0 .
159 1 . × −
74 186 8 .
33 1 .
36 8 . × −
82 204 18 . .
13 0 .
262 2 .
22 0 .
388 3 . × −
82 210 18 . .
20 0 . . .
33 0 .
792 5 .
13 0 .
963 0 . . .
38 0 . . .
07 1 .
04 6 .
34 1 .
21 0 . . .
23 1 . . . .
38 12 . .
48 0 . . . . . .
39 29 . .
44 1 . . . . . . . . . . . . . s , 2 s and 2 p / levels in hydrogen-like ions due to nuclear polarization effects. The values of the coefficient δ R ε characterizing the linear dependence of ∆ ε on thenuclear radius variation ∆ R are also presented. Z A − ∆ ε s B ( E − ∆ ε s B ( E − ∆ ε p / B ( E − δ R ε s B ( E − δ R ε s B ( E − δ R ε p / B ( E (cid:0) meV10 fm (cid:1) (cid:0) meV10 fm (cid:1) (cid:0) meV10 fm (cid:1) (cid:0) meV10 fm (cid:1) (cid:0) meV10 fm (cid:1) (cid:0) meV10 fm (cid:1)
26 54 2 .
00 0 .
259 1 . × − .
680 8 . × − . × −
26 56 1 .
96 0 .
253 1 . × −
34 74 5 .
44 0 .
721 7 . × − .
86 0 .
245 2 . × −
34 76 5 .
37 0 .
711 7 . × −
44 102 14 . .
01 3 . × − .
88 0 .
671 1 . × −
44 104 14 . .
98 3 . × −
46 108 17 . .
41 5 . × − .
48 0 .
761 1 . × −
46 110 17 . .
38 5 . × −
62 152 68 . . .
43 22 . .
34 0 . . . . . . .
71 30 . .
78 0 . . . . . .
14 41 . .
54 0 . . . . .
83 53 . .
79 0 . . . . . . .
292 234 710 136 15 . . . .
598 250 1151 235 32 . . .
198 252 1143 234 32 . . . . . . × . × . × . × . × . × . × . × . × . × . × . × . × TABLE VIII. Nuclear rotational transition contributions to the energy shifts of the 1 s , 2 s and 2 p / levels in hydrogen-like ionsdue to nuclear polarization effects. The numbers in the table need to be multiplied by the reduced nuclear transition probability B ( E ≡ B ( E
2; 2 →
0) to give the actual energy shifts. The values of B ( E
2) for different nuclei may be found, e.g., in Ref. [34].The values of the coefficient δ R ε/B ( E
2) characterizing the linear dependence of ∆ ε/B ( E
2) on the nuclear radius variation ∆ R are also presented. Z A − ∆ ε ns /c ns − b ns b s b s b p / δ R b s δ R b p / (fm) (fm) (fm) (fm)20 40 5 . × − .
91 22 .
820 44 5 . × − . × − .
52 12 .
426 60 1 . × − . × − .
02 8 . . × − . × − .
32 5 . . × − . × − .
15 4 . . × − . × − .
83 3 . . × − .
174 31.5 32.0 31.6 36.9 2 .
55 3 . .
176 31.6 32.1 31.7 37.074 184 0 .
363 28.9 29.4 29.0 33.0 2 .
37 2 . .
366 29.0 29.5 29.1 33.182 206 0 .
746 26.7 27.2 26.8 29.8 2 .
17 2 . .
752 26.7 27.3 26.9 29.990 228 1 .
53 24.7 25.3 24.9 27.2 2 .
04 2 . .
54 24.8 25.3 24.9 27.298 250 3 .
17 23.1 23.6 23.2 25.0 1 .
96 2 . .
19 23.1 23.6 23.3 25.0106 272 6 .
69 21.7 22.2 21.8 23.2 1 .
87 2 . .
72 21.7 22.2 21.9 23.3114 294 14 . .
75 1 . . . .
68 1 . . . .
61 1 . . .
54 1 . ε ns in neutral atoms with ns valence electrons and values of the cut-off parameter b ns inthe effective potential Eq. (48). For comparison, we present also values of this parameter b for 1 s , 2 s and 2 p / states ofhydrogen-like ions. Z A β − ∆˜ ε ns B ( E c ns ˜ b ns ˜ b s ˜ b s ˜ b p / δ R ˜ b s δ R ˜ b p / (10 − fm − ) (fm) (fm) (fm) (fm)26 54 0.195 0 . . .
026 56 0.239 0 . . . .
834 76 0.309 0 . . . .
344 104 0.257 0 . .
109 107 108 107 117 10 . .
046 110 0.257 0 .
108 108 108 108 11762 152 0.306 0 .
338 96.9 97.6 96.9 105 9 .
68 11 .
562 154 0.341 0 .
335 97.2 97.9 97.2 10566 162 0.341 0 .
445 94.4 95.2 94.4 102 9 .
52 11 .
266 164 0.348 0 .
441 94.7 95.4 94.7 10270 172 0.330 0 .
586 92.0 92.8 92.0 99.2 8 .
94 10 .
570 174 0.325 0 .
580 92.2 93.0 92.2 99.574 184 0.235 0 .
764 89.8 90.7 89.8 96.7 8 .
25 9 . .
757 90.0 90.9 90.0 97.090 228 0.230 2 .
36 80.3 81.5 80.3 86.0 6 .
99 8 . .
35 80.4 81.6 80.5 86.292 234 0.272 2 .
73 79.1 80.4 79.2 84.7 6 .
90 7 . .
71 79.3 80.5 79.4 84.998 250 0.299 4 .
29 75.7 77.0 75.8 80.9 6 .
54 7 . .
26 75.8 77.1 75.9 81.0106 234 0.272 2 .
73 79.1 80.4 79.2 84.7 5 .
97 6 . .
71 79.3 80.5 79.4 84.9114 234 0.272 2 .
73 79.1 80.4 79.2 84.7 5 .
58 6 . .
71 79.3 80.5 79.4 84.9122 234 0.272 2 .
73 79.1 80.4 79.2 84.7 5 .
38 5 . .
71 79.3 80.5 79.4 84.9130 234 0.272 2 .
73 79.1 80.4 79.2 84.7 5 .
08 5 . .
71 79.3 80.5 79.4 84.9136 234 0.272 2 .
73 79.1 80.4 79.2 84.7 4 .
65 5 . .
71 79.3 80.5 79.4 84.9TABLE X. Energy level shifts in neutral atoms calculated with ns valence electron wave function and values of cut-off parameter bb