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• W3204146107 abstract "The leading terms in the 1/Z expansion of the two-electron Bethe logarithm are calculated for the states 1s' ' S o , ls2s 'S1, ls2s %,, ls2p 'P, and ls2p 3P, by the use of a novel finite basis set method. The resulting QED terms are combined with other relativistic and mass polarisation corrections to obtain total transition frequencies. The results are compared with recent measurements in helium-like ions from Lif to Fez4+. Recent high-precision measurements of the ls2s 3S1-ls2p 3PJ ( J = 0,1,2) transition frequencies in high-Z two-electron ions (Davis and Marrus 1977, Holt et al 1980, DeSerio et a1 1981, Buchet et al 1981, Stamp et a1 1981, Livingston and Hinterlong 1982) have stimulated considerable interest in the theoretical calculation of relativistic and quantum electrodynamic (QED) effects in these ions (Goldman and Drake 1983, Hata and Grant 1983a, b, c, 1984). (For a review, see Drake 1982.) Since the non-relativistic energy difference increases only as Z, compared with a2Z4 and a3Z4 In (aZ) for the relativistic and QED corrections, the corrections become rapidly more important with increasing Z. For example, at Z = 20, they are about 20% and 1 ' / o of the total, respectively. The experimental transition frequency for Cli5+ determines the two-electron Lamb shift to an accuracy of *0.65% (DeSerio et a1 1981) (assuming that other contributions are accurately known), which is more accurate than corresponding measurements in high-Z one-electron ions. The purpose of this letter is to present new calculations for the Bethe logarithms of the states ls2s 'S, ls2s 3S, ls2p'P and ls2p3P, and to compare the resulting transition frequencies with experiment. Following Kabir and Salpeter (19571, the lowest order (in a ) two-electron QED correction is (in atomic units, with 1 au = a2mc2) (1) where Z is the nuclear charge and a = 1.137.03596 is the fine-structure constant. The above includes all terms of O(Z4aa'), but neglects terms of O ( Z 3 a 3 ) which are proportional to (S(rI2)) (Kabir and Salpeter 1957). The latter terms do not contribute to the energy shifts of triplet states in LS coupling because ( S ( r I 2 ) ) vanishes. The principal uncertainty in the evaluation of (1) is the value of the two-electron Bethe EL,2 =$Za3{ln(Zcr)-2+ln[Z2 Ryd/&(nLS)]+&(S(r,) + S(r2)) 5 Permanent address: Department of Physics, University of Windsor, Windsor, Ontario N9B 3P4, Canada. 0022-3700/84/070197+06$02.25 @ 1984 The Institute of Physics L197 L198 Letter to the Editor logarithm defined by in the dipole acceleration form where t+bo is the wavefunction for the nLS two-electron configuration, t = ZZirt/r; and the sums are over all intermediate states. The use of standard methods involving discrete variational basis sets to evaluate (2) leads to non-convergent results because of the large contribution from highly excited states. Accurate calculations have been attempted for the ground state with Z up to 10 (Schwartz 1961, Aashamar and Austvik 1976), and estimates have been made for the low-lying excited states of He and Li' (Suh and Zaidi 1965, Ermolaev 1975). For other cases, it has become customary to use the lowest-order hydrogenic approximation (DeSerio et a1 1981, Hata and Grant 1983b) In(Eo(nLs)) = (1 + 61,0/n3)-1(1n ~ ~ ( 1 s ) + n P 3 In ~ ~ ( n l ) ) (3) where eo(nl) is the hydrogenic Bethe logarithm for nuclear charge Z= 1. Recently, Hata and Grant (1984) have devised a semi-empirical fitting procedure to obtain improved values for In( E ( nLS)). In the present work, we write the two-electron Bethe logarithm in the form In &(nLS) = A / B where A and B are the numerator and denominator of (2) respectively, and insert the 1/Z expansions A = Z4[Ao +AIZ-' + 2(ln Z)(Bo+ BIZ-') +. . .] B = z~(B,+ B~z' +. . .). (4) ( 5 ) The coefficients in the expansion of B can be obtained from the identity (in atomic units) B = 2 ~ Z ( 8 ~ ( r , ) + 8 ' ( r ~ ) ) , The exact values of Bo and B1 for the S states are Bo( 1 lS) = 4 Bo(2 'S) =$ Bo(2 3s) =$ Bl( 1 'S) = -$+ 3 In 2 = -2.670558 B1(2 ' S ) = (-4130+6879 In 3-6720 In 2)/37=-0.562686 B1(2 'S) = (-4402+7647 In 3-7104 In 2)/37= -0.422967. The above B1 values were obtained with the aid of matrix elements tabulated by Cohen and Dalgarno (1961). Our numerical values for the P states are Bo(2 'P) = 2 Bo(2 3P) = 2 B1(2 'P) = 0.0436903 1) and Bl(2 'P) =-0.17190190(3). The value of A. is now determined by the condition Ao/Bo=ln E ~ ( ~ L S ) . Only A l requires significant additional calculation. Using l / r12 as a first-order perturbation, it is given by (Goldman and Drake 1983, Ermolaev and Swainson 1983) A l =c [2t$:?, tclo In AEO,/AEO,+Itb:?,1'AEt,(l-ln AEO,)/(AEL)'] (6) m Letter to the Editor L199" @default.
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• W3204146107 date "1984-01-01" @default.
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• W3204146107 title "Two-electron lamb shifts and 1s2s 3S1-1s2p 3PJ transition frequencies in helium-like ions [1]" @default.
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