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Physics Letters B 552 (2003) 35-40

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Nuclear-polarization effect to the hyperfine structure in heavy multicharged ions

A.V. Nefiodova b, G. Pluniena, G. Soffa

a Institut für Theoretische Physik, Technische Universität Dresden, Mommsenstraße 13, D-01062 Dresden, Germany b Petersburg Nuclear Physics Institute, 188300 Gatchina, St. Petersburg, Russia

Received 14 November 2002; accepted 24 November 2002

Editor: J.-P. Blaizot

Abstract

We have investigated the correction to the hyperfine structure of heavy multicharged ions, which is connected with the nuclear-polarization effect caused by the unpaired bound electron. Numerical calculations are performed for hydrogenlike ions taking into account the dominant collective nuclear excitations. The correction defines the ultimate limit of precision in accurate theoretical predictions of the hyperfine-structure splittings. © 2002 Elsevier Science B.V. All rights reserved.

PACS: 12.20.Ds; 31.30.Gs; 32.10.Fn; 32.10.Dk

During the last decade a significant progress has been achieved in investigations of the hyperfine structure (hfs) in heavy multicharged ions. To date, accurate measurements of hfs splittings were performed for a number of elements by using optical spectroscopy at the experimental storage ring at GSI in Darmstadt and at the Super-EBIT at the Lawrence Livermore National Laboratory [1-6]. One of the major purposes of these investigations consists in testing non-trivial effects of bound-state QED in intense nuclear fields with an accuracy on the level of a few percent. Experiments on the hfs splitting allow one also to probe internal nuclear structure, in particular, the magnetic moment distribution within the nucleus (the so-called Bohr-Weisskopf effect). Though a rough estimate of the latter effect can be obtained in the framework of the single-nucleon model, its accurate calculation with the use of a microscopic nuclear theory still has to be performed.

Two attempts to calculate the extended magnetization distribution taking into account nuclear many-body corrections should be mentioned here. The first approach is based on the dynamic correlation model [7], which allows to include the excited-core configurations in the nuclear wave function. Numerical calculations performed in the framework of this model provide accurate values for the nuclear magnetic moments. However, no satisfactory agreement with experimental results for the hfs splittings has been achieved [8]. The second approach, which also takes into account the nuclear core polarization by the unpaired nucleon, is based on the theory of finite Fermi

E-mail address: nefiodov@theory.phy.tu-dresden.de (A.V. Nefiodov).

0370-2693/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0370-2693(02)03093-9

systems [9]. In this case, the correction due to the nuclear magnetization distribution calculated for hydrogenlike bismuth turned out to be too small compared with the corresponding value deduced from the experiment.

At present it appears to be unlikely to obtain an accurate theoretical value for the Bohr-Weisskopf correction. However, one can eliminate it to a large extend in combined measurements of the hfs splittings in H- and Li-like ions [10]. For the ground state of 209Bi80+, two independent calculations have lead to similar predictions, 797.1(2) meV [11] and 797.15(13) meV [12], respectively. While both theoretical results agree with the experimental value of 820(26) meV [4], further efforts to measure the predicted splitting with higher precision have not been successfull.

In the present Letter we evaluate a correction, which has not been previously considered in calculations of the hyperfine structure. The correction is connected with the nuclear-polarization effect caused by the unpaired bound electron. While the corresponding contribution is relatively small in the case of hydrogenlike ions, it turns out to be non-negligible for accurate theoretical predictions of the hfs splittings in Li-like heavy ions. Since the uncertainty of our calculation is comparable with the magnitude of the nuclear-polarization effect itself, the latter cannot be completely eliminated by extracting the Bohr-Weisskopf contribution in accordance with Shabaev's idea [10]. The effect under consideration sets a natural limit up to which one can test bound-state QED, even if a specific difference of the hfs splittings of H- and Li-like ions is introduced [13].

In the following, we shall consider a hydrogenlike ion with a non-zero-spin nucleus, so that the total angular momentum of an atom F is defined by the coupling of the nuclear spin I with the total angular momentum of an electron j. The hfs energy shift due to the magnetic-dipole interaction is given by

AE^F) = ^[F(F + 1) - /(/ + 1) - j(j + 1 )]A„K. (1)

Here AnK is the hfs constant, which depends on the electron state characterized by the standard set of quantum numbers. For electron states with j = 1 /2, the hfs splitting between the levels with F = I + 1/2 and F = I — 1/2 is just AEnK = (I + 1 /2)AnK. Since the nuclear size is rather small with respect to the radius of the electron orbit, the hyperfine structure can be fairly understood in the framework of the external-field approximation. This allows to treat the magnetic field of a nucleus as a perturbing potential in calculations of the hfs splitting.

Employing the Dirac equation for the electron in the external field of an infinitely heavy pointlike nucleus, one obtains (h = c = 1) [14-16]

D 3 m2 giK [2ic(nr + y) — N]

A®=a(aZf-----—, V-(2)

mpj(j + 1) jV4y(4y2 ~ 1)

where a = e2 is the fine-structure constant (e > 0), gj is the nuclear g factor, nr = n — \k\ is the radial quantum number, n is the principal quantum number, k = (j + l/2)(-ly+l+V2, y = Jk2 - (aZ)2, N = V (nr + y)2 + («Z)2, and in and mp are the electron and proton masses, respectively. Because of various nuclear and QED effects, the experimental value for the hfs constant deviates from its Dirac prediction (2). To describe the extended nucleus, one usually employs the following parametrization

AenK = ADk(1 — 8nK)(1 — €nK), (3)

where the corrections SnK and enK account for the nuclear charge and magnetic moment distributions within the nucleus, respectively [17-20]. In addition to the approximation (3), the radiative corrections should be taken into account. All leading QED effects for the hyperfine structure have been independently calculated by different groups and the numerical results are consistent [21]. For Li-like ions, one can develop a perturbation theory with respect to the parameter 1/Z, which accounts for corrections arising from the electron-electron interaction [11,22]. The recoil correction, which is due to the finite nuclear mass, is negligibly small for heavy ions.

Here we consider a correction AAnK to the hfs constant (3) due to the nuclear-polarization effect, which is caused by the bound electron. More precisely, a core-polarization part of the effect is considered only, which is due to collective nuclear excitations. The corresponding single-nucleon contributions should be generally considered

beyond the external-field approximation. However, they are assumed to be completely negligible. To describe nuclear polarization, we adopt a relativistic field theoretical approach, which incorporates the many-body theory for virtual nuclear excitations within bound-state QED for atomic electrons [23]. This approach has been successfully applied in calculations of nuclear-polarization effects to the Lamb shift [23-25] and to the bound-electron g factor [26]. To some extent, the present formulae are quite similar to those derived in Ref. [26]. The correction under consideration may be represented by the sum of contributions, which are referred to as the irreducible, the reducible, and the vertex parts.

The irreducible contribution to the hfs constant can be written in terms of a multipole decomposition as follows:

nK lit ;'(/' + 1) v

J J L>0 « ,Ki 2

2 {nK\FL\niKi){niKi\FL\tiK) — £«iki — sgn(e«i,q )&l

where = e/(2mp) is the nuclear magneton, enK is the one-electron energy, mL = EL — E0 are the nuclear excitation energies with respect to the ground-state energy E0 of the nucleus and B(EL) = B(EL; 0 ^ L) are the corresponding reduced electric transition probabilities. The sum over n1 runs over the entire Dirac spectrum, while the sum over k1 is restricted to those intermediate states, where l +11 + L is even. A two-component radial vector (r \«k) is determined by

(r\«K ) = ( PnK(r)\ (5)

where P«K(r) = rg«K(r) and Q«K(r) = rf«K(r), with g«K(r) and f«K(r) being the upper and lower components of the Dirac wave function, respectively. The radial shape parametrizing the nuclear transitions is carried by the functions [23-25]

FL(r) =

rL . RL

&(Ro-r) + -£-&(.r-Ro)

r>L+1 v 0 y rL+1

(2L + 1)RL

in the case of multipole excitations with L > 1 and

F0(r) =

&(R0 — r) (7)

for monopole excitations, respectively. Here R0 is an average radius assigned to the nucleus in its ground state. In Eq. (4), the matrix element is given by

(a\FL\b) = J drFL(r)[Pa(r)Pb(r) + Qa(r)Qb(r)\

The perturbed vector (r\nic), which follows as

/ I—\ \ K\aXr \HK) / I / \ /Q\

(r\nic) = > -(r\n K), (9)

can be evaluated analytically by means of the generalized virial relations for the Dirac equation [27] (see also Refs. [22,28]). InEq. (9), ax is the Pauli matrix. The reducible contribution reads

pj r -ln9 / M 1/- I 77 T I *-J , l/-, \ 2

AAlf = -—A5? y B(EL) Y Tc'1,1 4tt "k ^ L i\i

L^0 «i ,Ki 2

)kL °J

nK\FL\niKi}2__^

[e«K — e«lKl — sgn(e«lKl )rnL\2

where A^K is the hfs constant given by Eq. (3). The sum I +1\ + L again should be even.

The nuclear core-polarization correction to the hfs constant due to the vertex part is conveniently represented as the sum of a pole term

^^ruc —

egIIXNK y B(EL) y (2jl + 1)3/2 4tt VTCT+WTTT) ^ ¿ViiOi + 1)

2 Í j1 j1 1 j j L

(niKi\axr 2\niKi){niKi\FL\nK)

[bhk - BniKi - sgn(£n1K1 )ml]2

and of a residual term

AAJS? —

a \plegi /xnk

ñ^jürWfTT)

yBiEQj^yjiñTid;,

2 2 QJ

j ¿L0 j jLO h-\ 11

, 1 -Jl 2

n\,n2 ,K2

j1 j2 n (nK\FL\n1K1)(n2K2\FL\nK) (n1K1\oxr 2\n2K2)

j j L] enK &n2K2

- Sgn(Sn2K2 )&>L e.

respectively. Here AAnK accounts for the terms with n1 = n2 and k1 = k2 in the sums over intermediate states. The prime in the sum in Eq. (12) indicates that en1K1 = en2K2 when k1 = k2, i.e., the pole contribution is supposed to be omitted. In Eqs. (11) and (12), the value l +11 + L has to be even. A second condition in Eq. (12) is that the sum l1 +12 should be even as well. The total nuclear core-polarization contribution to the hfs constant is determined by the sum of all contributions given by Eqs. (4), (10), (11), and (12).

In Table 1, we present numerical results for some hydrogenlike ions, which are of particular experimental interest. The calculations were performed taking into account a finite set of dominant collective nuclear excitations. To estimate the nuclear parameters, ml and B(EL), in the case of nearly spherical nucleus of 289Bi, we employed experimental data corresponding to the low-lying vibrational levels in neighboring even-even isotope of 208Pb, which were deduced from nuclear Coulomb excitation. In the case of giant resonances, we utilized phenomenological energy-weighted sum rules [30], which are assumed to be concentrated in single resonant states. In the present calculations, contributions due to monopole, dipole, quadrupole, and octupole giant resonances have been taken into account. The infinite summations over the entire Dirac spectrum were performed by the finite basis set method. Basis functions are generated via B splines including nuclear-size effects [31].

Concluding, we have evaluated a correction to the hyperfine structure in heavy multicharged ions, which is connected with the nuclear core-polarization effect caused by the unpaired bound electron. The correction is

Table 1

For various hydrogenlike ions, the nuclear spin/parities In, the nuclear magnetic dipole moments i (in units of the nuclear magneton) [29], the nuclear-polarization contributions to the hfs constant AAjj for K-shell electron, and the nuclear-polarization corrections to the ground-state hfs splitting AEjj are tabulated. Column (a): contributions from low-lying collective nuclear modes using experimental values for nuclear excitation energies ml and electric transition strengths B(EL); (b) contributions from giant resonances employing empirical sum rules [30]; (c) total effect. The negative value of the nuclear magnetic moment for uranium indicates that the level with F = I — 1/2 lies above the one with F = I + 1/2. The numbers in parentheses are powers of ten

In ¡ (nm) (a) AA]S (meV) (b) (c) AE1s (meV)

159Tb64+ 3/2+ 2.014 0.25(-2) 0.26(-2) 0.51(-2) 0.10(-1)

165Ho66+ 7/2 - 4.132 0.38(-3) 0.27(-2) 0.31(-2) 0.12(— 1)

175Lu70+ 7/2+ 2.2327 0.13(-2) 0.22(-2) 0.35(-2) 0.14(—1)

187Re74+ 5/2+ 3.2197 0.25(-2) 0.66(-2) 0.91(-2) 0.27(—1)

203-^80+ 1/2+ 1.62226 0.16(-2) 0.29(-1) 0.31(-1) 0.31(-1)

209Bi82+ 9/2- 4.1106 0.81(-3) 0.10(-1) 0.11 (—1) 0.55 (— 1)

235y91 + 7/2 ~ -0.38 -0.361-2) —0.29(—2) —0.65(—2) 0.261-1)

exhausted over distances of the order of the nuclear size and it is enhanced due to a singular behavior of the hyperfine-interaction operator. The uncertainty of our calculation can be as large as the nuclear-polarization effect itself. It yet determines the natural limitation for testing higher-order QED corrections in future experiments aiming for accurate hfs measurements. In the case that the experimental value for the ground-state hfs splitting is used to eliminate the Bohr-Weisskopf effect [10], the latter cannot be separated from the effect we have considered. Accordingly, the utmost precision for theoretical hfs predictions in lithiumlike heavy ions is determined by the nuclear-polarization correction to the ground state in hydrogenlike ions. In particular, for the ground state in 209Bi82+, the nuclear-polarization effect contributes on the level of about 0.05 meV. This implies that although numerical calculations of bound-state QED corrections provide sufficiently stable results, the conservative estimate of uncertainties for the hfs splitting in 209Bi80+ quoted in Ref. [11] appears to be more realistic rather then the one predicted in Ref. [12]. It is also worth noting that because of a similar scaling dependence of nuclear and radiative corrections upon the principal quantum number, significant cancellations of almost all corrections, except for the screened QED contribution, occur in a specific difference of the hfs splittings in H- and Li-like heavy ions [13]. In this case, the nuclear-polarization effect might be most relevant for determining the uncertainties of accurate theoretical predictions.

Acknowledgements

A.N. is grateful for financial support from RFBR (Grant No. 01-02-17246) and from the Alexander von Humboldt Foundation. G.S. and G.P. acknowledge financial support from BMBF, DFG and GSI.

References

[1] I. Klaft, S. Borneis, T. Engel, B. Fricke, R. Grieser, G. Huber, T. Kühl, D. Marx, R. Neumann, S. Schröder, P. Seelig, L. Völker, Phys. Rev. Lett. 73 (1994) 2425.

[2] J.R. Crespo Lopez-Urrutia, P. Beiersdorfer, D.W. Savin, K. Widmann, Phys. Rev. Lett. 77 (1996) 826.

[3] J.R. Crespo Lopez-Urrutia, P. Beiersdorfer, K. Widmann, B.B. Birkett, A.-M. Martensson-Pendrill, M.G.H. Gustavsson, Phys. Rev. A 57 (1998) 879.

[4] P. Beiersdorfer, A.L. Osterheld, J.H. Scofield, J.R. Crespo Lopez-Urrutia, K. Widmann, Phys. Rev. Lett. 80 (1998) 3022.

[5] P. Seelig, S. Borneis, A. Dax, T. Engel, S. Faber, M. Gerlach, C. Holbrow, G. Huber, T. Kühl, D. Marx, K. Meier, P. Merz, W. Quint, F. Schmitt, M. Tomaselli, L. Völker, H. Winter, M. Würtz, K. Beckert, B. Franzke, F. Nolden, H. Reich, M. Steck, T. Winkler, Phys. Rev. Lett. 81 (1998) 4824.

[6] P. Beiersdorfer, S.B. Utter, K.L. Wong, J.R. Crespo Lopez-Urrutia, J.A. Britten, H. Chen, C.L. Harris, R.S. Thoe, D.B. Thorn, E. Träbert, M.G.H. Gustavsson, C. Forssen, A.-M. Martensson-Pendrill, Phys. Rev. A 64 (2001) 032506.

[7] M. Tomaselli, Phys. Rev. C 37 (1988) 349.

[8] M. Tomaselli, T. Kühl, W. Nörtershäuser, S. Borneis, A. Dax, D. Marx, H. Wang, S. Fritzsche, Phys. Rev. A 65 (2002) 022502.

[9] R.A. Sen'kov, V.F. Dmitriev, Nucl. Phys. A 706 (2002) 351.

[10] V.M. Shabaev, in: H.F. Beyer, V.P. Shevelko (Eds.), Atomic Physics with Heavy Ions, Springer, Berlin, 1999, p. 139.

[11] V.M. Shabaev, A.N. Artemyev, O.M. Zherebtsov, V.A. Yerokhin, G. Plunien, G. Soff, Hyperfine Interact. 127 (2000) 279.

[12] J. Sapirstein, K.T. Cheng, Phys. Rev. A 63 (2001) 032506.

[13] V.M. Shabaev, A.N. Artemyev, V.A. Yerokhin, O.M. Zherebtsov, G. Soff, Phys. Rev. Lett. 86 (2001) 3959.

[14] G. Breit, Phys. Rev. 35 (1930) 1447.

[15] P. Pyykkö, E. Pajanne, M. Inokuti, Int. J. Quantum Chem. 7 (1973) 785.

[16] S.A. Zapryagaev, Opt. Spectrosk. 47 (1979) 18, Opt. Spectrosc. (USSR) 47 (1979) 9 (in English).

[17] J.E. Rosenthal, G. Breit, Phys. Rev. 41 (1932) 459.

[18] M.F. Crawford, A.L. Schawlow, Phys. Rev. 76 (1949) 1310.

[19] A. Bohr, V.F. Weisskopf, Phys. Rev. 77 (1950) 94.

[20] V.M. Shabaev, J. Phys. B 27 (1994) 5825.

[21] T. Beier, Phys. Rep. 339 (2000) 79.

[22] M.B. Shabaeva, V.M. Shabaev, Phys. Rev. A 52 (1995) 2811.

[23] G. Plunien, B. Müller, W. Greiner, G. Soff, Phys. Rev. A 43 (1991) 5853.

[24] G. Plunien, G. Soff, Phys. Rev. A 51 (1995) 1119;

G. Plunien, G. Soff, Phys. Rev. A 53 (1996) 4614, Erratum.

[25] A.V. Nefiodov, L.N. Labzowsky, G. Plunien, G. Soff, Phys. Lett. A 222 (1996) 227.

[26] A.V. Nefiodov, G. Plunien, G. Soff, Phys. Rev. Lett. 89 (2002) 081802.

[27] V.M. Shabaev, J. Phys. B 24 (1991) 4479.

[28] V.M. Shabaev, V.A. Yerokhin, Pis'maZh. Eksp. Teor. Fiz. 63 (1996) 309, JETP Lett. 63 (1996) 316 (in English).

[29] R.B. Firestone, et al. (Eds.), Table of Isotopes, Vols. 1,2, Wiley, New York, 1999.

[30] G.A. Rinker, J. Speth, Nucl. Phys. A 306 (1978) 397.

[31] W.R. Johnson, S.A. Blundell, J. Sapirstein, Phys. Rev. A 37 (1988) 307.