Fine structure of helium and light helium-like ions

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1 Fine structure of helium and light helium-like ions Krzysztof Pachucki and Vladimir A. Yerokhin Abstract: Calculational results are presented for the fine-structure splitting of the 3 P state of helium and helium-like ions with the nuclear charge Z up to 0. Theoretical predictions are in agreement with the latest experimental results for the helium fine-structure intervals as well as with the most of the experimental data available for light helium-like ions. Comparing the theoretical value of the 3 P 0 3 P interval in helium with the experimental result [T. Zelevinsky et al. Phys. Rev. Lett. 95, 0300 (005, we determine the value of the fine-structure constant α with an accuracy of 3 parts per billion.. Introduction The fine structure splitting of the 3 P state in helium plays a special role in atomic spectroscopy because it can be used for an accurate determination of the fine structure constant α. This fact was first pointed out by Schwartz in 964 [. The attractive features of the fine structure splitting in helium as compared to other atomic transitions are, first, the long lifetime of the metastable 3 P J levels (roughly two orders of magnitude larger than that of the p state in hydrogen and, second, the relative simplicity of the theory of the fine structure. Schwartz s suggestion stimulated a sequence of calculations [, 3, 4, 5, which resulted in a theoretical description of the helium fine structure complete up to order mα 6 (or α 4 Ry and a value of α accurate to 0.9 ppm [6. The present experimental precision for the fine-structure intervals in helium is sufficient for a determination of α with an accuracy of 4 ppb [7, 8. In order to match this level of accuracy in theoretical description of the fine structure, the complete calculation of the next-order, mα 7 contribution and an estimation of the higher-order effects is needed. The work towards this end started in 990s and extended over two decades [9, 0,,, 3, 4, 5, 6, 7, 8. In 006, the first complete evaluation of the mα 7 correction to the helium fine structure was reported [9. However, the numerical results presented there disagreed with the experimental values by more than 0 standard deviations. In our recent investigations [0,, we recalculated all effects up to order mα 7 to the fine structure of helium and performed calculations for helium-like ions with nuclear charges Z up to 0. The calculations were extensively checked by studying the hydrogenic (Z limit of individual corrections and by comparing them with the results known from the hydrogen theory. We found several problems in previous studies. As a result, the present theoretical predictions are in agreement with the latest experimental data for both fine-structure intervals in helium, as well as with the most of experimental Krzysztof Pachucki. Institute of Theoretical Physics, University of Warsaw, Hoża 69, Warsaw, Poland Vladimir A. Yerokhin. Institute of Physics, University of Heidelberg, Philosophenweg, D-690 Heidelberg, Germany and Gesellschaft für Schwerionenforschung, Planckstraße, D-649 Darmstadt, Germany and Center for Advanced Studies, St. Petersburg State Polytechnical University, Polytekhnicheskaya 9, St. Petersburg 955, Russia data available for light helium-like ions. Comparison of our theoretical prediction for the 3 P 0 3 P interval in helium (accurate to 57 ppb with the experimental value [7 (accurate to 4 ppb determines the value of the fine structure constant α with an accuracy of 3 ppb. The calculation of the mα 7 correction for the fine-structure splitting of light helium-like atoms was reported in our recent Letter [. In this paper, we present an extended description of the mα 7 correction and a detailed term-by-term comparison of our results with independent calculations by Drake [7 for helium and by Zhang et al. [ for helium-like ions.. The spin-dependent mα 7 correction The mα 7 correction to the fine-structure splitting of a twoelectron atom can be conveniently separated into four parts, [ E (7 mα 7 E (7 = mα 7 E (7 log + E(7 first + E(7 sec + E(7 L. [ The first term in the brackets above combines all terms with lnz and lnα [0,,, 4, 9, [ E (7 Z log = ln[(z α 3 i p δ 3 (r p σ 4 ( σ ( σ δ 3 3 (r i p δ 3 (r p σ + 8Z H (4 [ δ 3 3 (E 0 H 0 (r + δ 3 (r, [ where r = r r, H 0 and E 0 are the Schrödinger Hamiltonian and its eigenvalue, and H (4 is the spin-dependent part of the Breit-Pauli Hamiltonian, H (4 = [ ( σ 4 r 3 + σ r p ( σ + σ r p + Z ( r 4 r 3 p σ + r r 3 p σ + ( σ σ 4 r 3 3 σ r σ r r 5. [3 The second part of E (7 is induced by effective Hamiltonians to order mα 7. They were derived by one of us (K.P. in Re. unknown 99: 8 (00 DOI: 0.39/Zxx-xxx c 00 NRC Canada

2 unknown Vol. 99, 00 [9, 0. (The previous derivation of this correction by Zhang [0, turned out to be not entirely consistent. The result is E (7 first = H Q + H H + H (7,amm. [4 The Hamiltonian H Q is induced by the two-photon exchange between the electrons, the electron self-energy and the vacuum polarization. It is given by [9 H Q = Z 9 80 i p δ 3 (r p σ [ ( σ ( σ δ 83 3 (r 30 + lnz [ i p δ 3 (r p σ 0 lnz 5 8 π r 7 ( σ r( σ r 3 4 π i p r 3 p σ. [5 Here, the terms with ln Z compensate the logarithmic dependence implicitly present in expectation values of singular operators /r 3 and /r 5, so that matrix elements of H Q do not have any logarithms in their /Z expansion. The singular operators are defined though their integrals with the arbitrary smooth function f, d 3 r f( r lim r3 ǫ 0 and d 3 r r 7 ( r i r j δij lim ǫ 0 d 3 r 3 r [ r 7 [ d 3 r θ(r ǫ r3 + 4 π δ 3 (r(γ + lnǫ f( r, [6 f( r (r i r j δij + 4 π 5 δ3 (r(γ + lnǫ 3 r θ(r ǫ ( i j δij 3 f( r, where γ is the Euler constant. The effective Hamiltonian H H represents the anomalous magnetic moment (amm correction to the Douglas-Kroll mα 6 operators and is given by [9 [7 H H = Z 4 p r r 3 p σ 3 Z 4 r r 3 r r 3 σ ( r p + 3 Z 4 r r 3 σ r r p σ i p r 3 σ + r 4 r p σ 3 4 r 6 r σ r σ r 4 p r 3 p σ r 4 p r 3 p σ Z 4 r r 3 r 3 r p r p σ + 3 i 4 r 5 r ( r p p σ 3 8 r 5 r ( r p σ p σ 8 r 3 p σ p σ + 6 p r 5 r σ r σ 3 i r 8 p r 3 σ p σ + i 8 p [ r r 3 σ p σ + ( r σ ( p σ 3r r σ r σ r p 4 p σ p r r 3 p + 8 p σ [ p σ r p r r r 5 σ. [8 The Hamiltonian H (7 is the mα 7 amm correction to the Breit-Pauli Hamiltonian, H (7 = ( ( σ π 3 r 3 + σ r p p c3 ( r + Z π 3 + π 3 r 3 ( σ σ p σ + r r 3 3 σ r σ r r 5 r 3 p σ c 3 (c c + c 3, [9 where c = /, c = and c 3 = are the expansion coefficients of the freeelectron amm in powers of (α/π. The third part of E (7 is given by the second-order matrix elements of the form [9 E sec (7 = + H (4 (E 0 H 0 H(5 nlog [ H (4 + H (4 n (E 0 H 0 H(5, [0 where H (5 nlog is the effective Hamiltonian responsible for the nonlogarithmic mα 5 correction to the energy, H (5 nlog = 7 6 π r Z 45 H (4 n [ δ 3 (r + δ 3 (r, [ is the spin-independent part of the Breit-Pauli Hamiltonian (with the term δ 3 (r omitted since it does not contribute c 00 NRC Canada

3 3 in our case, n = 8 (p4 + p4 + Z π [ δ 3 (r + δ 3 (r ( δ ij pi r + ri r j r 3 p j, [ H (4 and H (5 is the mα 5 amm correction to H (4, H (5 = ( ( σ 4π r 3 + σ r p p + Z ( r 4π r 3 + ( σ σ 4π r 3 p σ + r r 3 3 σ r σ r r 5 p σ. [3 The fourth part of E (7 is the contribution induced by the emission and reabsorption of virtual photons of low energy. It is denoted as E (7 L and interpreted as the relativistic correction to the Bethe logarithm. The expression for E (7 L reads [5 E (7 L = 3 π δ ( p + p (H 0 E 0 ln [ (H0 E 0 Z ( p + p + i ( r Z 3 π r 3 + r r 3 σ [ ( + σ (H0 E 0 r ln Z r 3 + r r 3, [4 where δ... denotes the first-order perturbation of the matrix element... by H (4, implying perturbations of the referencestate wave function, the reference-state energy, and the electron Hamiltonian. 3. Results: helium The summary of individual contributions to the finestructure intervals of helium is given in Table. Numerical results are presented for the large ν 0 and the small ν intervals, defined by ν 0 = [ E( 3 P 0 E( 3 P /h, [5 ν = [ E( 3 P E( 3 P /h. [6 We note that the style of breaking the total result into separate entries used in Table differs from that used in the summary tables of the previous papers by K.P. et al. [9, 0. Particulary, the lower-order terms listed in Table III of Ref. [9 and in Table II of Ref. [0 contained contributions of higher orders, whereas in the present work, the entries in Table contain only the contributions of the order specified. A term-by-term comparison with the independent calculation by Drake [7 is made whenever possible. We observe good agreement between the two calculations for the lowerorder terms, namely, for the mα 4, mα 5, and mα 6 corrections. However, for the recoil correction to order mα 6, our results differ from Drake s ones by about 0.5 khz for both intervals. The reason for this disagreement seems to be different for the large and the small intervals. For the large interval, the deviation is due to the recoil operator part, whereas for the small interval, it is mainly due to the mass polarization part (see discussion in Ref. [0. Our estimates of the uncalculated higher-order effects for helium are much larger than those in the previous studies [6, 7. The previous estimates amounted to significantly less than khz for both intervals and were based on some logarithmic contributions to order mα 8 that were identified by analogy with the hydrogen fine structure. We now believe that the dominant mα 8 contribution might be of the relativistic origin. Our estimates of ±.7 khz for both intervals were obtained by multiplying the mα 6 contribution for the sum of ν 0 + ν by the factor of (Zα. Our result for the ν 0 interval of helium agrees well with the experimental values [7, 3, 4. For the ν interval, our theory is by about σ away from the values obtained in Re. [7, 5 but in agreement with the latest measurement by Hessels and coworkers [8. Assuming the validity of the theory, we combine the theoretical prediction for the ν 0 interval in helium with the experimental result [7 and obtain the following value of the fine structure constant, α (He = (39 theo (6 exp, [7 which is accurate to 3 ppb and agrees with the more precise results of Re. [6, 7, Results: helium-like ions Table gives the summary of individual contributions to the fine-structure intervals of helium-like atoms with the nuclear charge number Z up to 0. We choose to present results for the intervals ν 0 and ν 0 ν 0 + ν, and not for ν 0 and ν, as is customary. The reason to consider ν 0 is that this interval is c 00 NRC Canada

4 4 unknown Vol. 99, 00 Table. Summary of individual contributions to the fine-structure intervals in helium, in khz. The parameters [ are α = (94, cr = ( khz, and m/m = The values by Drake are taken from Table 3 of Ref. [7. The label (+m/m indicates that the corresponding entry comprises both the non-recoil and recoil contributions of the specified order in α. Term ν 0 ν Ref. mα 4 (+m/m a a [7 mα 5 (+m/m [7 mα ( ( (7 [7 mα 6 m/m (5 9.80( [7 mα 7 log(zα b 5.87 b [7 mα 7, nlog mα 8 ±.7 ±.7 Total theory ± ±.7 Experiment (70 c (35 f (0 d (5 c (9 e (0 g a the original result was scaled to the present value of α. b the original result was altered by the substitution ln(α ln(zα in the terms proportional to ln(α, in order to comply with the present result for the logarithmic mα 7 contribution. c Ref. [7. d Ref. [3. e Ref. [4. f Ref. [8. g Ref. [5. c 00 NRC Canada

5 5 Table. Individual contributions to the fine-structure intervals of helium-like atoms, in MHz/Z 4. The label (+m/m indicates that the corresponding entry comprises both the non-recoil and recoil contributions of the specified order in α. For Z = 3, 7, and 0, a term-by-term comparison is made with the previous calculation by Zhang et. al. [. The results of Ref. [ for the leading mα 4 correction were scaled to the present value of α. The deviation for the mα 7 (log contribution is due to the difference in the expressions for this term. Z mα 4 (+m/m mα 5 (+m/m mα 6 mα 6 m/m mα 7 (log mα 7 (nlog Total Ref. ν ( ( ( [ ( ( ( ( (3 [ ( ( ( [ ν ( ( ( [ ( ( ( ( (3 [ ( ( ( [ c 00 NRC Canada

6 6 unknown Vol. 99, 00 free from effects of the 3 P P mixing, which strongly affect the ν 0 and ν intervals. As a result of the absence of the mixing effects, all corrections to ν 0 starting with the order of mα 6 demonstrate a weaker Z dependence as compared to those to ν 0 and ν. The most drastic difference occurs for the α 6 m /M correction: for Z = 0, this correction for ν 0 is by 3 orders of magnitude smaller than that for ν 0. The uncertainty of the theoretical values specified in Table is solely due to uncalculated higher-order effects. Its estimation for helium was already discussed. For helium-like ions, we obtain the uncertainty by multiplying the mα 6 contribution for the corresponding interval by the factor of (Zα. So, our error estimates are typically by a factor of /Z smaller for the ν 0 interval than for the ν 0 (or, equivalently, ν interval. For Z = 3, 7, and 0, Table presents a term-by-term comparison with the previous calculation by Zhang et. al. [. We observe excellent agreement for the mα 4 and mα 5 corrections. For the mα 6 correction, the agreement is excellent in all cases except for the ν 0 interval and Z = 7 and 0, where a small deviation is present. The results of the two calculations for the mα 7 (log correction are different, but this is explained by the difference in the expressions for this term. If we use the same expression as in Ref. [, excellent agreement is found again. In Fig. we plot our numerical results for the mα 7 correction as a function of the nuclear charge number Z, together with the fit of the /Z expansion and with the asymptotical high-z limit of this correction. The form of the /Z expansion and the values of the first coefficient(s are known. For the ν 0 interval, the leading term scales as Z 6 and is calculated for hydrogen in Ref. [9. For the ν interval, there are additional Z 7 and Z 6 contributions due to the triplet-singlet mixing, which are obtained by expanding the following expression in /Z, P H (4 3 P δe mix = E 0 ( 3 P E 0 ( P. The resulting asymptotic form of the nonlogarithmic mα 7 correction is E (7,nlog (ν 0 /Z 7 = /Z + O(/Z, [8 E (7,nlog (ν 0 /Z 6 = O(/Z. [9 By fitting the /Z expansion of our numerical data for E (7,nlog, we were able to reproduce well the values of the coefficients given above, which served as an important check of our calculations. The comparison of the present theoretical predictions with experiment data is summarized in Table 3. The agreement is very good in most cases. The only significant discrepancy is for Be +, where the difference of.7 standard deviations (σ is observed for ν and that of 3.5 σ, for ν 0. It is important that for both the ν 0 and ν intervals there are experimental results available for helium-like ions, whose accuracy significantly exceeds the theoretical errors. These are the measurement of ν 0 in helium-like nitrogen by Thompson at al. [30 and that of ν in helium-like fluorine by Myers et al. [3. Good agreement of the present theory with these experimental results suggests that the theoretical errors (i.e., the uncalculated higher-order effects were reasonably estimated. It is unfortunate that there are no experimental results with comparable accuracy available for the ν 0 interval. Since ν 0 is not affected by the triplet-singlet mixing effects, accurate experimental results for this interval in light helium-like ions would yield an improved estimate for the uncalculated higher-order effects in helium, thus increasing accuracy of determination of α from the helium fine structure. Comparing theoretical and experimental results for the fine structure of helium-like ions, one should keep in mind that the present calculation is carried out for a spinless nucleus, whereas the experimental results listed in Table 3 were performed for non-zero nuclear spin isotopes. For a nucleus with spin, the hyperfine splitting can usually be evaluated separately and employed for an experimental determination of the fine structure. This procedure, however, ignores the mixing between the hyperfine and the fine splittings. So, more accurate calculations should account for both effects simultaneously. In summary, the theory of the fine structure of helium and light helium-like ions is now complete up to orders mα 7 and α 6 m /M. Theoretical predictions agree with the latest experimental results for helium, as well as with most of the experimental data for light helium-like ions. A combination of the theoretical and experimental results for the 3 P 0 3 P interval in helium yields an independent determination of the fine structure constant α accurate to 3 ppb. Support by NIST through Precision Measurement Grant PMG 60NANB7D653 and by the Helmholtz Gemeinschaft (Nachwuchsgruppe VH-NG-4 is gratefully acknowledged. References. C. Schwartz, Phys. Rev. 34, A8 (964.. M. Douglas and N. Kroll, Ann. Phys. (NY 8, 89 ( L. Hambro, Phys. Rev. A 5, 07 ( L. Hambro, Phys. Rev. A 6, 865 ( L. Hambro, Phys. Rev. A 7, 479 ( M. L. Lewis and P. H. Serafino, Phys. Rev. A 8, 867 ( T. Zelevinsky, D. Farkas, and G. Gabrielse, Phys. Rev. Lett. 95, 0300 ( J. S. Borbely, M. C. George, L. D. Lombardi, M. Weel, D. W. Fitzakerley, and E. A. Hessels, Phys. Rev. A 79, (R ( Z.-C. Yan and G. W. F. Drake, Phys. Rev. Lett. 74, 479 ( T. Zhang, Phys. Rev. A 54, 5 (996.. T. Zhang, Phys. Rev. A 53, 3896 (996.. T. Zhang, Z.-C. Yan, and G. W. F. Drake, Phys. Rev. Lett. 77, 75 ( T. Zhang, Phys. Rev. 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7 7 0.8 (nlog [ khz/z E ν 0 (nlog [ khz/z E ν Z 0 00 Z Fig.. Nonlogarithmic mα 7 correction to the fine-structure intervals of helium-like atoms, for the ν 0 interval (left and for the ν 0 interval (right. Black dots denote the numerical results, solid line stands for the fit of the /Z expansion, and the dashed (red line indicates the asymptotic high-z results. Note that the results in different graphs are scaled by different factors. It is Z 7 for the ν 0 interval and Z 6, for ν 0. The different Z scaling is the consequence of the triplet-singlet mixing effects. 3. G. Giusfredi, P. C. Pastor, P. D. Natale, D. Mazzotti, C. de Mauro, L. Fallani, G. Hagel, V. Krachmalnicoff, and M. Inguscio, Can. J. Phys. 83, 30 ( M. C. George, L. D. Lombardi, and E. A. Hessels, Phys. Rev. Lett. 87, 7300 ( J. Castillega, D. Livingston, A. Sanders, and D. Shiner, Phys. Rev. Lett. 84, 43 ( D. Hanneke, S. Fogwell, and G. Gabrielse, Phys. Rev. Lett. 00, 080 ( M. Cadoret, E. de Mirandes, P. Cladé, S. Guellati-Khélifa, C. Schwob, F. Nez, L. Julien, and F. Biraben, Phys. Rev. Lett. 0, 3080 ( A.-M. Jeffrey, R. E. Elmquist, L. H. Lee, and R. F. Dziuba, IEEE Trans. Inst. Meas. 46, 64 ( U. Jentschura and K. Pachucki, Phys. Rev. A 54, 853 ( J. K. Thompson, D. J. H. Howie, and E. G. Myers, Phys. Rev. A 57, 80 ( E. G. Myers, H. S. Margolis, J. K. Thompson, M. A. Farmer, J. D. Silver, and M. R. Tarbutt, Phys. Rev. Lett. 8, 400 ( E. Riis, A. G. Sinclair, O. Poulsen, G. W. F. Drake, W. R. C. Rowley, and A. P. Levick, Phys. Rev. A 49, 07 ( T. J. Scholl, R. Cameron, S. D. Rosner, L. Zhang, R. A. Holt, C. J. Sansonetti, and J. D. Gillaspy, Phys. Rev. Lett. 7, 88 ( T. P. Dinneen, N. Berrah-Mansour, H. G. Berry, L. Young, and R. C. Pardo, Phys. Rev. Lett. 66, 859 ( N. J. Peacock, M. F. Stamp, and J. D. Silver, Phys. Scr. T8, 0 (984. c 00 NRC Canada

8 8 unknown Vol. 99, 00 Table 3. Comparison of theoretical and experimental results for the fine-structure intervals of helium-like ions. Units are MHz for Li + and cm for other atoms. Z Present theory Experiment Ref. ν ( (66 [ ( (5 [ (9 6.03(8 [ ( (7 [ (3 59. (. [ ( (. [35 ν ( ( [3 ν ( (6 [ ( (5 [ ( (6 [ (4 6.3(7 [ ( (.5 [35 c 00 NRC Canada