Model operator approach to calculations of the Lamb shifts in relativistic many-electron atoms

Transcription

1 Model operator approach to calculations of the Lamb shifts in relativistic many-electron atoms Vladimir Shabaev a in collaboration with Ilya Tupitsyn a and Vladimir Yerokhin b a St. Petersburg State University b St. Petersburg State Polytechnical University FFK, Oct 7-11, 2013 p.1/26

2 Outline of the talk Introduction Perturbation theory for the QED calculations in the Furry picture Schrödinger-like equation for a relativistic atom in the framework of QED Lowest-order approximation: Dirac-Coulomb-Breit Hamiltonian Model operator approach to the Lamb shift in many-electron atoms Calculations with the model self-energy operator in many-electron systems Conclusion FFK, Oct 7-11, 2013 p.2/26

3 Introduction Quantum electrodynamics in the external field approximation (Furry picture of QED) High-Z few-electron ions N Z, where Z is the nuclear charge number and N is the number of electrons. To zeroth-order approximation: ( i α +mβ +V C (r))ψ( r) = εψ( r) Interelectronic-interaction and QED effects: Interelectronic interaction Binding energy 1 Z, QED Binding energy α(αz)2. In uranium: Z = 92, αz 0.7. FFK, Oct 7-11, 2013 p.3/26

4 Introduction Relativistic many-electron atoms and ions The interelectronic interaction is not small and must be taken into account at the zero-order level: V C V eff = V C +V scr, where V scr describes approximately the electron-electron interaction effects. Therefore, to zeroth order: ( i α +mβ +V eff (r))ψ( r) = εψ( r) In higher orders, besides the interelectronic-interaction and QED effects, one must add the interaction with V scr. FFK, Oct 7-11, 2013 p.4/26

5 Green function Standard (2N-time) QED Green function for an N-electron atom: G(x 1,...x N;x 1,...x N ) = 0 Tψ(x 1) ψ(x N)ψ(x N ) ψ(x 1 ) 0, where x = (t,x), ψ(x) is the electron-positron field operator in the Heisenberg representation, and ψ(x) = ψ γ 0. In the interaction representation: G(x 1,...x N;x 1,...x N ) = 0 Tψ in(x 1) ψ in (x N )ψ in(x N ) ψ in (x 1 )exp{ i d 4 z H I (z)} 0 0 T exp{ i d 4 z H I (z)} 0, where H I (x) = e 2 [ψ in(x)γ µ,ψ in (x)]a µ in (x) δm 2 [ψ in(x),ψ in (x)] is the interaction Hamiltonian. FFK, Oct 7-11, 2013 p.5/26

6 Two-time Green function We introduce the two-time Green function: G(t,t) G(t 1 = t 2 = t N t ;t 1 = t 2 = t N t) t 1 = t 2 = = t N t t 1 = t 2 = = t N t The Fourier transform: G(E)δ(E E ) = 1 2πi 1 N! dtdt exp(ie t iet) G(t,t). FFK, Oct 7-11, 2013 p.6/26

7 Perturbation theory for quasidegenerate levels We consider s degenerate or quasidegenerate states. The projector on the unperturbed states: P (0) = s k=1 P (0) k = s u k u k. k=1 We project G(E) on the space Ω (0) s formed by the s unperturbed states: g(e) = P (0) G(E)P (0). We consider a contour Γ in the complex E plane: Γ FFK, Oct 7-11, 2013 p.7/26

8 Perturbation theory for quasidegenerate levels We introduce operators K and P by K 1 2πi Γ de Eg(E), P 1 2πi Γ de g(e). The energies and the wavefunctions are determined from the equations: Kv k = E k Pv k, v k Pv k = δ k k The solvability condition yields: det(k EP) = 0. FFK, Oct 7-11, 2013 p.8/26

9 Schrödinger-like equation for a relativistic atom These equations can be transformed to the Schrödinger-like equation: Hψ k = E k ψ k, ψ k ψ k = δ kk, where H P 1 2 KP 1 2 and ψ k P 1 2v k. The energy levels are determined from the equation: det(h E) = 0. The space of the quasidegenerate states can be extended to the space Ω (0) + that includes all positive-energy states whose energies are smaller than the pair-creation threashold: Γ E (0) E (0) +2mc 2 In this picture the photon spectra are omitted. FFK, Oct 7-11, 2013 p.9/26

10 Schrödinger-like equation for a relativistic atom The operators K and P are constructed by perturbation theory: K = K (0) +K (1) +K (2) +, P = P (0) +P (1) +P (2) +. The operator H is H = H (0) +H (1) +H (2) +, where H (0) = K (0), H (1) = K (1) 1 2 P(1) K (0) 1 2 K(0) P (1), H (2) = K (2) 1 2 P(2) K (0) 1 2 K(0) P (2) 1 2 P(1) K (1) 1 2 K(1) P (1) P(1) P (1) K (0) K(0) P (1) P (1) P(1) K (0) P (1). FFK, Oct 7-11, 2013 p.10/26

11 Interelectronic-interaction operator One-photon exchange contribution to the Hamiltonian H: In the space Ω (0) + we get h int = (ε i,ε j,ε k,ε l >0) i j,k l ψ i ψ j ψ i ψ j 1 2 [I(ε i ε k )+I(ε j ε l )] ψ k ψ l ψ k ψ l, where I(ω) = e 2 α ρ α σ D ρσ (ω), α ρ γ 0 γ ρ = (1,α), D ρσ (ω) is the photon propagator, and ε i is the one-electron Dirac energy. FFK, Oct 7-11, 2013 p.11/26

12 Dirac-Coulomb-Breit Hamiltonian Taking h int in the Coulomb gauge at zero energy transfer (ω = ε i ε k = 0) and summing over atomic electrons leads to the Dirac-Coulomb-Breit Hamiltonian: H = Λ (+)[ ] (Vij C +Vij) B Λ (+), i h D i + i<j where Λ (+) is the projector on the positive-energy states, h D i = α i p i +mβ i +V C (r i ), V C (r) = αz r, Vij C = r α [ ij, Vij B = α αi α j r ij ] ( i α i )( j α j )r ij. To account for the nonzero energy transfer, one should simply replace Vij C +VB ij by the interelectronic-interaction operator hint derived above. In the Feynman gauge, to get the Hamiltonian to the same accuracy, one has to account for the higher-order photon exchange diagrams. FFK, Oct 7-11, 2013 p.12/26

13 Lamb shift operator for a relativistic atom The QED contributions to the Hamiltonian H: h QED = h SE +h VP = (ε i,ε k >0) i,k ψ i ψ i [ 1 2 (ΣSE (ε i )+Σ SE (ε k ))+V VP] ψ k ψ k, where Σ SE (ε i ) and V VP are the renormalized self-energy (SE) and vacuum-polarization (VP) operators, respectively. Details of the two-time Green function method and the derivation of these formulas can be found in [V.M. Shabaev, Phys. Rep., 2002; JPB, 1993]. FFK, Oct 7-11, 2013 p.13/26

14 Lamb shift operator for a relativistic atom The dominant part of the VP contribution is represented by the Uehling potential: V Uehl (r) = αz 2α 2 3π 0 dr 4πr ρ(r ) 1 dt (1+ 1 t2 1 2t 2) t 2 [exp( 2m(c/ ) r r t) exp( 2m(c/ )(r+r )t)] 4mrt, where α is the fine structure constant and e Zρ(r) is the density of the nuclear charge distribution ( ρ(r)dr = 1). Evaluation of the remaining Wichmann-Kroll potential is a much more difficult problem [G. Soff and P.J. Mohr, PRA, 1988; N.L. Manakov et al., JETP, 1989]. To a good accuracy, it can be calculated with the help of the approximate formulas derived in [A.G. Fainshten et al., JPB, 1991]. FFK, Oct 7-11, 2013 p.14/26

15 Model self-energy operator for a relativistic atom Let us now consider the SE operator: h SE = (ε i,ε k >0) i,k ψ i ψ i 1 2 [ΣSE (ε i )+Σ SE (ε k )] ψ k ψ k. We represent h SE as a sum of local and nonlocal parts. The local part is given by V SE loc = κ A κ exp( r/λ C )P κ, where P κ is the projector on the states with the given value of κ = ( 1) j+l+1/2 (j +1/2), the constant A κ is chosen to reproduce the SE shift for the lowest energy level at the given κ in the corresponding H-like ion, and λ C = /(mc). FFK, Oct 7-11, 2013 p.15/26

16 Model self-energy operator for a relativistic atom We restrict the active space of the remaining SE operator, h SE V SE to to the basis functions {φ i (r)} n i=1 which, having the same angular parts as the H-like functions {ψ i (r)} n i=1, are localized at smaller distances. With these functions, we approximate the one-electron SE operator as follows h SE = V SE loc + n i,k=1 φ i B ik φ k, where the matrix B ik has to be determined to reproduce the diagonal and non-diagonal SE corrections with the H-like wave functions. This leads to the equations loc, n ψ i φ j B jl φ l ψ k j,l=1 = ψ i [ 1 2 (Σ(ε i)+σ(ε k )) Vloc SE ] ψk. FFK, Oct 7-11, 2013 p.16/26

17 Model self-energy operator for a relativistic atom Now let us consider the choice of the functions {φ i (r)} n i=1. We construct them using the H-like wave functions multiplied with the factor ρ l (r) = exp( 2αZ(r/λ C )/(1+l)), where l = κ+1/2 1/2 is the orbital angular momentum of the state under consideration. In what follows, we restrict the basis functions by ns, np 1/2, np 3/2, nd 3/2, and nd 5/2 states with the principal quantum number n 3 for the s states and n 4 for the p and d states, and put φ i (r) = 1 2 (I ( 1)s i β)ρ li (r)ψ i (r), where I is the identity matrix, β is the standard Dirac matrix, the index s i = n i l i enumerates the positive energy states at the given κ, and n i is the principal quantum number. FFK, Oct 7-11, 2013 p.17/26

18 Model self-energy operator for a relativistic atom Thus, the model SE operator is given by h SE = V SE loc (I ( 1) s i β)ρ li (r) ψ i i,k j,l ((D t ) 1 ) ij ψ j [ 1 2 (Σ(ε ] j)+σ(ε l )) Vloc SE ψl (D 1 ) lk ψ k ρ lk (r)(i ( 1) s k β), where the summations run over ns states with the principal quantum number n 3 and over np 1/2, np 3/2, nd 3/2, and nd 5/2 states with n 4, ρ li (r) = exp( 2αZ(r/λ C )/(1+l i )), D ik = 1 2 ψ i (I ( 1) s i β)ρ li (r) ψ k, and s i = n i l i. FFK, Oct 7-11, 2013 p.18/26

19 Self-energy matrix elements with H-like wave functions To complete the construction of the model SE operator, one needs to evaluate the matrix elements Σ ik ψ i 1 2 (Σ(ε i)+σ(ε k )) ψ k with the H-like wave functions. To perform such calculations we used the method described in [V.A. Yerokhin and V.M. Shabaev, PRA, 1999; V.A. Yerokhin, K. Pachucki, and V.M. Shabaev, PRA, 2005]. The results of the calculations are conveniently expressed in terms of the function F ik (αz) defined by Σ ik ψ i 1 2 [Σ(ε i)+σ(ε k )] ψ k = α π (αz) 4 (n i n k ) 3/2F ik(αz)mc 2, where n i and n k are the principal quantum numbers of the i and k states, respectively. For the diagonal matrix elements, these results are in a good agreement with the calculations performed in [P.J. Mohr, PRA, 1992; P.J. Mohr and Y.-K. Kim, PRA, 1992; T. Beier, P.J. Mohr, H. Persson, and G. Soff, PRA, 1998]. FFK, Oct 7-11, 2013 p.19/26

20 Self-energy matrix elements with H-like wave functions Self-energy matrix elements F ik (αz), defined by Σ ik ψ i 1 2 [Σ(ε i)+σ(ε k )] ψ k = α (αz) 4 π (n i n k F ) 3/2 ik (αz)mc 2, with H-like wave functions for extended nuclei. Z F 1s1s F 2s2s F 3s3s F 1s2s F 1s3s F 2s3s FFK, Oct 7-11, 2013 p.20/26

21 Calculations with the model SE operator To demonstrate the efficiency of the method, we applied it to calculations of the Lamb shifts in neutral alkali metals, Cu-like ions, superheavy atoms, and Li-like ions. Ab initio calculations of the Lamb shift in alkali metals were performed [J. Sapirstein and K.T. Cheng, PRA, 2002] in the potential U(r): V(r) = αz eff(r) r, where Z eff (r) = Z nuc (r) r 0 dr 1 [ 81 ] 1/3 ρ t (r )+x α r > 32π 2rρ t and ρ t = ρ v +ρ c is total (valence plus core) electron charge density. The choice x α = 0 corresponds to the Dirac-Hartree potential, x α = 2/3 gives the Kohn-Sham potential, and x α = 1 is the Dirac-Slater potential. FFK, Oct 7-11, 2013 p.21/26

22 Self energy in neutral alkali metals Self-energy function F(αZ), defined by E SE = α π neutral alkali metals in different potentials. (αz) 4 n 3 F(αZ)mc 2, for Atom Method x α = 0 x α = 1/3 x α = 2/3 x α = 1 Na 3s 1/2 Rb 5s 1/2 Fr 7s 1/2 v Vloc SE v v H SE v Exact a v Vloc SE v v H SE v Exact a v Vloc SE v v H SE v Exact a a J. Sapirstein and K.T. Cheng, PRA, FFK, Oct 7-11, 2013 p.22/26

23 Self energy in Cu-like ions Self-energy contribution to the 4s 4p 1/2 and 4s 4p 3/2 transition energies in Cu-like ions, in ev. Ion Transition Model SE operator Exact a Yb 41+ 4s 4p 1/ s 4p 3/ p 1/2 4d 3/ p 3/2 4d 3/ p 3/2 4d 5/ U 63+ 4s 4p 1/ s 4p 3/ p 1/2 4d 3/ p 3/2 4d 3/ p 3/2 4d 5/ a J. Sapirstein and K.T. Cheng, PRA, FFK, Oct 7-11, 2013 p.23/26

24 Self energy in superheavy atoms Self-energy contribution to the binding energy of the valence electrons in Rg and Cn, in ev. In this work, the perturbation theory (PT) value is obtained by averaging the model SE potential with the Dirac-Fock wave function of the valence electron, while the DF value is obtained by including this potential into the DF equations. Atom Valence Method This work I. Goidenko, Other electron EPJD, 2009 works Rg 7s PT a DF Welton meth. Local SE pot. Cn 7s PT DF a L. Labzowsky et al., PRA, 1999; b P. Indelicato et al., EPJD, 2007; c C. Thierfelder and P. Schwerdtfeger, PRA, b c FFK, Oct 7-11, 2013 p.24/26

25 Screened self energy in Li-like ions Screened self energy for the 2s, 2p 1/2, and 2p 3/2 states of Li-like ions, in ev. The Kohn-Sham (KS) and Dirac-Fock (DF) results are obtained using the model SE potential approach. Z State KS DF PT a PT b 20 2s p 1/ p 3/ s p 1/ p 3/ s p 1/ p 3/ a Y.S. Kozhedub et al., PRA, 2010; b J. Sapirstein and K.T. Cheng, PRA, FFK, Oct 7-11, 2013 p.25/26

26 Conclusion Schrödinger-like equation for a relativistic many-electron atom can be derived from the first principles of QED by the two-time Green function method. The QED contribution can be approximated by a model operator, which provides a very simple and efficient tool for evaluation of the Lamb shifts in many-electron atoms and ions [V. M. Shabaev, I. I. Tupitsyn, and V. A. Yerokhin, Phys. Rev. A 88, (2013)]. FFK, Oct 7-11, 2013 p.26/26