Isotope shift of the 3 2S1/2−2 2S1/2 transition in lithium and the nuclear polarizability

Isotope Shift of the 32S1=2-22S1=2Transition in Lithium and the Nuclear Polarizability M. Puchalski, 1 A. M. Moro, 2 and K. Pachucki 1 1 Institute of Theoretical Physics, Warsaw University, Hoz ˙a 69, 00-681 Warsaw, Poland 2 Departamento de FAMN, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain (Received 29 July 2006; published 25 September 2006) High precision calculation of the isotope shift of the 32S1=2-22S1=2transition in lithium is presented. The wave function and matrix elements of relativistic operators are obtained by using recursion relations. Apart from the relativistic contribution, we obtain the nuclear polarizability correction for 11Li. The resulting difference of the squared charge radii 11Li-7Li based on the measurements of Sa ´nchez et al. [Phys. Rev. Lett. 96, 033002 (2006)] is r2 ch 0:15781fm2, which significantly differs from the previous evaluation. DOI: 10.1103/PhysRevLett.97.133001 PACS numbers: 31.30.Jv, 21.10.Ft The recent advances in precise spectroscopy of atomic systems make possible the determination of nuclear charge radii from isotope shift measurements [1–4]. In spite of the fact that the nuclear size is 5 orders of magnitude smaller than the atomic size, the achieved experimental precision for transition frequencies allows one to determine nuclear charge radii much more accurately than from electron scattering measurements. This, however, requires the theoretical calculations to be performed with high relative precision, for example, at least 106for lithium isotopes. The accuracy of theoretical predictions achieved for hydrogen [5] leads to the determination of the proton charge radius [5,6], which is far more accurate than from the electron scattering measurements. A similar experimental accuracy achieved for deuterium allowed one to determine very accurately the deuteron radius [1]. Surprisingly, the atomic measurements lead to a slightly different value from the electron scattering determination, which stimulated a reanalysis of electron scattering data. What makes the deuteron different from other typical nuclei is its low binding energy of about 2.226 MeV. Such a soft nucleus is distorted by a surrounding electron, which leads to a shift in atomic transition frequencies. Even smaller is the twoneutron separation energy in 11Li [7], which indicates the possible significance of nuclear structure effects on the isotope shift. In this Letter, we present significantly improved calculations of finite nuclear mass contributions to the isotope shift in the lithium 32S1=2-22S1=2transition. Such calculations have already been performed by Yan and Drake in Refs. [8–10] and were used to determine nuclear charge radii of various lithium isotopes on the basis of recent isotope shift measurements [3,4]. Our result for the relativistic recoil correction is about 10 times smaller than that reported in Refs. [8,9]. Apart from the known nonrelativistic, leading relativistic, and QED corrections, we include higher order recoil corrections and the nuclear polarizability effect Epol, the last being significant for the 11Li nucleus. Finally, we use our combined results to calculate new nuclear charge radii for the lithium isotopes, taking the experimental isotope shift results from Refs. [3,4]. Let us denote =M. The expansion of an energy level in the fine structure constant is E; m2E2m4E4m5E5 m6E6O7Epol Efs;(1) where Efs is the nuclear finite size correction, and m,M, are the electron, nucleus, and the reduced mass, respectively. In the following, we obtain these expansion coefficients. The nonrelativistic Hamiltonian of the lithium atom in atomic units is HH1HM;(2) H1X ap2 a 2Z raX a>b 1 rab ;(3) HMX a p2 a 2MX a>b ~ pa~ pb M:(4) We solve the Schro ¨dinger equation for H1and treat HM perturbatively. Following Ref. [11], the Schro ¨dinger wave function is represented in the Hylleraas basis set rn1 23rn2 31rn3 12rn4 1rn5 2rn6 3ew1r1w2r2w3r3;(5) with all combinations of ni, such that P6 i1ni and 3;...;12. Details can be found in Ref. [12]. The matrix elements of the nonrelativistic Hamiltonian can all be expressed by a Hylleraas integral f fn1;n 2;n 3;n 4;n 5;n 6Zd3r1 4Zd3r2 4Zd3r3 4ew1r1w2r2w3r3rn11 23 rn21 31 rn31 12 rn41 1rn51 2rn61 3;(6) with integer values of ni. We calculate fanalytically for values of ni0;1in sextuple precision and use recursion PRL 97, 133001 (2006) PHYSICAL REVIEW LETTERS week ending 29 SEPTEMBER 2006 0031-9007=06=97(13)=133001(4) 133001-1 ©2006 The American Physical Society relations for ni>1[13]. Results for nonrelativistic energies obtained with the largest number of basis functions, namely, 9576, are so far the lowest ones in the literature E22S1=27:478 060 323 890;(7) E32S1=27:354 098 421 380;(8) and finite mass corrections are presented in Table I. The high quality of the nonrelativistic wave function allows one to obtain accurate results for matrix elements, for example, X a 3ra3S X a 3ra2S 0:106 108 0:(9) Relativistic corrections are obtained from the Breit- Pauli Hamiltonian, which includes here finite mass corrections [8] E4X a~ p4 a 8Z 23ram M Z 2pi aij ra ri arj a r3 aX b pj bM X a>b3rab1 2pi aij rab ri abrj ab r3 ab pj bM ;(10) where h...iMdenotes a matrix element, calculated with the wave function that includes the finite nuclear mass. Matrix elements for E4involve extended Hylleraas integrals, where one nibecomes negative. High precision calculations of Hylleraas integrals with inverse powers of raand rab are quite difficult and are one of the principal achievements of this work. We use the integration by parts identities to express f1;0;0; 0;0;0in terms of onedimensional integrals which are performed with 48 digit accuracy, and f1;n 2;n 3;n4;n 5;n 6are obtained by recursion relations [14]. Similarly, f0;0;0;1;n 5;n 6are expressed as a one-dimensional integral which is performed numerically, and fn1;n 2;n 3;1;n 5;n 6are obtained by recursion relations [12]. Results for relativistic recoil corrections are presented in Table I. We note a strong cancellation between reduced mass, mass polarization, and the direct electron-nucleus Breit interaction. Our result of 0.038 MHz is more than 10 times smaller than the previous one in Refs. [8,9], but it is in agreement with the recent recalculation by Yan and Drake [15], which gives 0.024(19) MHz. Leading QED recoil corrections E5have been studied in Ref. [16] and calculated for lithium in Refs. [9,10]. They can be represented (in atomic units) as: E519 30 2lnlnk0M4Z 3X a 3raM 164 15 14 3lnX a>b 3rabM 7 6X a>b 1 r3 abM m M2 3ln62 98 3lnk0X a 3ra 7Z2 6X a 1 r3 aOm M2;(11) where lnk0is the Bethe logarithm and 1=r3has implicit subtraction of ln", with "being the cutoff and the Euler constant. 1=r3terms and lnk0including mass polarization corrections have been calculated by Yan and Drake in Refs. [9,10], and we use their results here. The complete m5contribution is presented in Table I. It is slightly different from that in Refs. [9,10], due to including the additional term from reduced mass scaling of 1=r3 ab in Eq. (11). Considering higher order recoil corrections m2=M6, they are known exactly only for hydrogenic systems [17– 19]. Some of them are known to be proportional to the Dirac delta function, such as the radiative or radiative recoil, so the extension to lithium is simple. Pure recoil corrections are more complicated. There are state dependent terms which are known for hydrogen [17,19], but the extension to more than one-electron atoms has not yet been achieved. We neglect them here and associate roughly 25% uncertainty. This correction is already quite small, so the approximate treatment is sufficient. The formula we use for E6is E6 m3Z2427 96 2 ln2 m MZ235 36 448 2722 ln2 63 2 m MZ34 ln2 7 2X a 3ra;(12) and numerical results using Eq. (9) are presented in Table I, in agreement with the estimation from Ref. [9]. Apart from relativistic and QED effects, there is a finite nuclear size and nuclear polarizability which contribute to the isotope shift. Since one uses the isotope shift measurement to determine nuclear charge radius, one shall estimate the effect of nuclear polarizability, which we find here to be significant for 11Li. For this, we assume that the interaction of the nucleus with the electromagnetic field can be described as follows: Hint qA0~ d~ E~ ~ Bq 6hr2ir2A0:(13) TABLE I. Contributions to the Li isotope shift =M. Contribution 32S1=2-22S1=2 11Li-7Li (MHz) m20.133 764 842 25 104.489(21) m220.123 659 8 2:968 m40.003 78(3) 0.038 m51:43 0:104 m638:100:0205 Epol 0.039(4) PRL 97, 133001 (2006) PHYSICAL REVIEW LETTERS week ending 29 SEPTEMBER 2006 133001-2 The dominating nuclear excitations are E1transitions by the electric dipole coupling ~ d~ E[20]. The energy shift due to the two-photon exchange in the temporal gauge is Epol ie2 20Zd! 2Zd3k 23!2ik kikk !2 !2k2 jl kjkl !2 !2k2 Trj1 p6 k6 mii1 p6 k6 mj0I 4 hNjdk1 ENHN!dljNi;(14) where 20m3hPa3rai,pm; ~ 0, and we used plane wave approximation for the electrons, since the characteristic photon momentum kis much larger than the inverse Bohr radius. After performing kintegration and replacing !iw, one obtains Epol m4X a 3ram3~pol;(15) where ~pol is a kind of electric polarizability of the nucleus, which is given by the following double integral: ~pol 16 3Z1 ET dE 1 e2jhNj~ djEij2 Z1 0 dw w E E2w2 1 ?11 1?1 1 11 ?1;(16) where  12im=w p, and Eis the excitation energy of the nucleus with respect to the ground state with threshold value ET. This general formula agrees in the limit Em with that derived previously for the nuclear polarizability effect in deuterium [21]. ~pol involves a square of the transition dipole moment. This can be related to the BE1distribution which was recently accurately measured for 11Li in Ref. [20] jhj~ djEij24 3 dBE1 dE (17) in units e2fm2MeV, which explains the presence of e2in the denominator in Eq. (16). With the new data from Ref. [20] (see Fig. 1) and the two-neutron separation energy ET0:3765MeV [7], one obtains ~pol 60:96:1fm31:060:11106m3(18) and a polarizability correction to the 32S1=2-22S1=2tran- sition frequency of pol 394kHz. Polarizability correction for 9Li and lighter isotopes is expected to be at least 10 times smaller and is, thus, negligible. The last significant contribution to the isotope shift is due to the finite size of the nucleus Efs 2 3Z4m3hr2 chiX a 3ra:(19) This nuclear volume effect can now be extracted from the isotope shift measurements, to obtain nuclear charge radii. We additionally account for the leading relativistic correction 201Z2lnmrchZ to the square of the wave function at the origin. Results are summarized in Table II. All results for charge radii differences are significantly improved compared to previous determinations; for example, our result for the difference 11Li-7Li,r2 ch  0:15781fm2, can be compared to r2 ch 0:374112fm2 which is obtained using results presented in Ref. [4]. The 0123 E (MeV) 0 1 2 dB/dE (e2 fm2/MeV) Nakamura fit FIG. 1 (color online). Electric dipole line strength by Nakamura et al. [20] adapted to the new value of ETfrom Ref. [7]. TABLE II. Summary of isotope shift determination of Li charge radii, rch7Li2:39030fm [22], m7Li7:016 003 425 645u [23]; the first uncertainty of the comes from unknown higher order terms, the second uncertainty is due to the atomic mass. Mass (u)[7,24]exp (MHz) [3,4]the (MHz) r2 ch (fm2)rch (fm) 6Li 6.015 122 794(16) 11 453:9832011 452:822200.738(13) 2.540(30) 8Li 8.022 487 36(10) 8635.782(44) 8634.990(1)(1) 0:503282.282(32) 9Li 9.026 789 5(21) 15 333.272(39) 15 331.797(3)(13) 0:938262.185(33) 11Li 11.043 715 7(54) 25 101.226(125) 25 101.473(9)(21) 0.157(81) 2.423(34) PRL 97, 133001 (2006) PHYSICAL REVIEW LETTERS week ending 29 SEPTEMBER 2006 133001-3 result for the difference between 11Li and 9Li,r2 ch  1:098fm2, does not agree with that obtained on the basis of BE1data, namely, r24 3 1 Ze2Z1 ET dE dBE1 dE 0:74 fm2:(20) r2here is the square of a distance of 9Li core from the mass center. From this, one may conclude that the 9Li core is significantly perturbed by the presence of the two valence neutrons in 11Li. A similar conclusion has already been drawn in Ref. [4]. The difference in squared charge radii between 6Li and 7Li can also be obtained from the isotope shift measurement of 23S-23PJtransitions in Liby Riis et al. [25]. Taking the weighted average value and theoretical isotope shift from Ref. [26], one obtains r2 ch 0:70260fm2,in moderate agreement with the result obtained here, r2 ch  0:74213fm2. The uncertainty of numerical calculations for lithiumlike atoms is negligible in comparison to the estimation of unknown higher order terms. The pure recoil correction of order 6m2=M gives the largest theoretical uncertainty of a few kilohertz for lithium isotope shifts and will become more significant for heavier lithiumlike ions. At present, however, the dominating source of uncertainties comes from measurement of transition frequencies and from the charge radius of the reference nucleus; see Table II. Regarding the direct charge radius determination of the reference nucleus 7Li, calculations of transition frequencies are far more difficult and have been performed with sufficient precision only for hydrogenic systems. Therefore, the spectroscopic determination of the absolute charge radius can possibly be achieved by measurement of two-photon transitions in the hydrogenlike lithium ion or 2S-2Ptransitions in the muonic lithium. In summary, we have obtained high precision finite nuclear mass corrections to lithium isotope shifts and nuclear polarizability (for 11Li). The resulting nuclear charge radii are significantly improved over the former determinations. 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