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Oct 5, 1995 - The difference between our calculations and JBS's mainly comes from that we ..... [ 6 ] W. R. Johnson, S. A. Blundell, and J. Sapirstein, Phys. Rev ...

CHINESE JOURNAL OF PHYSICS

VOL. 33, NO. 5

OCTOBER 1995

Numerical Test of Lamb Shift Formulas for Li-Isoelectronic Sequence Sy-Sang Liaw Department of Physics, National Chung Hsing University, Taichung, Tazwan 402, R.O.C.

(Received August 24, 1995)

We use forty B-spline basis functions to solve the Brueckner equation. The energy levels and transition amplitudes thus obtained have included the electron-electron interaction up to second order. The third order corrections are added by perturbation. The results plus the contributions from the Breit interaction and mass polarization give accurate theoretical values for energies. We then extract the contribution of radiative corrections by taking the difference of these theoretical values and available experimental data. We show that the Lamb-shift formulas derived by Feldman and Fulton do yield correct order of magnitude for Li-isoelectronic sequence. PACS. 31.15.Ar - Ab initio calculations. PACS. 31.30. Jv - Relativistic and quantum electrodynamic effects in atoms and molecules. PACS. 31.25.-v - Electron correlation calculations for atoms and molecules.

I. INTRODUCTION The development of computer techniques has enabled atomic theorists to do accurate structure calculations for systems with few electrons. The particular interest is to test QED for these systems. The system which is most intensively studied is the helium isoelectronic sequence [l-4]. For systems with three electrons, the lithium-isoelectronic sequence, there are also several calculations [5-81 which have treated the electron-electron correlation very accurately. With these accurate results we might begin to test the QED effects for these ions. In our previous work [9,10] we have developed a systematic approach to calculate atomic structures of N + 1 electron systems, where N is the number of electrons of a nondegenerate core. The approach is based on the Green’s function formalism. The starting point of our calculation is the Brueckner approximation, which is a straightforward extension of the Dirac-Fock approximation. Namely, the Brueckner approximation is a self-consistent approximation up to second order to the exact Dyson equation, while the 505

@ 1995 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA

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Dirac-Fock approximation is self-consistent in first order only. If only the energy levels are concerned, the formalism is same with the many-body perturbation theory (MBPT) (see the comparison of the Brueckner approximation with MBPT in Ref. [ll]). We have used the Brueckner approximation to produce very good results for some atoms [9,10]. The accuracy of these results are comparable with the other accurate calculations. The third order corrections to the Brueckner approximation have also been given explicitly by us [ll]. With these corrections we have produced the best theoretical values for the life-times of 4p levels of Ca+ [12]. In this article we will calculate the energies of the lowest three states of Li-like ions in the Brueckner approximation. We then add the third order corrections and the effects of the Breit interaction and mass polarization [6]. Assuming that the non-radiative effects having been taken care in this way, the difference between these values and experimental values are then used to give a numerical test for the Lamb-shift formulas proposed by Feldman and Fulton [13]. II. FORMULAS

The Brueckner equation and other formulas computed here have been given explicitly in different sources. We briefly describe these formulas and cite relevant references in this section. II-l. The Brueckner Equation

The Brueckner equation is given in Ref. [9], Eq. (3): (EV - HI+) = C(+J) + c(2)]V) )

(1)

where Hu is a hydrogenic Hamiltonian; E, is the term energy of orbital 1~). C(l) and Ct2) are the first and second order self-energy. They are explicitly given in Ref. [9]. We expand wavefunctions in terms of forty B-spline basis functions and solve the Brueckner equation self-consistently. We first solve the DF equation, which is obtained simply by dropping the term with second order self-energy in the Bureckner equation, to generate the complete set of DF wavefunctions. Once the complete set of the DF wavefunctions are obtained, we add the second order self-energy and start iteration. It takes only two or three iterations before the eigenvalues converge to six significant digits. II-2. The Third Order Correction

The explicit form of the leachg third order coorections to the Brueckner energies is given in Ref. [ll], Eq. (4.3). This contribution to energies, denoted by Ec3), is mostly computationally demanding because that each term in E c3) is a product of three matrix

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507

elements of the electron-electron interaction potential and has quintet summation over the intermediate orbitals. We have included the contributions from all the orbitals with angular momentum 1 5 4. In evaluating Et31 we first carry out the angular reduction for each term. This was done by using a graphical method [14] and checked by using the symbolic software: Maple. 11-3. Breit Interaction and Mass Polarization In the Brueckner approximation and its third order correction we take the electronelectron interaction potential to be A, the leading term in Coulomb gauge, to calculate the energies. Since the radiative effects are of order of 1% or less of the term energies, we have to include the effects of the exchange of one transverse photon, namely, the Breit interaction. In addition, the effects due to the finite mass of the nucleus and mass polarization are also significant in this level of accuracy and cannot be neglected. The formulas for Breit interaction and mass polarization effect are given in Ref. [6]. Because they are small, calculations of these effects using the Brueckner wavefunctions are not necessary for the accuracy we need. We simply quote results of Ref. [6] (which are calculated using the DF wavefunctions). 11-4. Lamb Shifts

Feldman and Fulton [13] have derived the radiative corrections to the energies of a system with a core plus or minus one electron. They gave explicit formulas for the lowest order (o”) corrections, and claimed that the formulas should be accurate to better than 3% for 3 5 2 < 10. We adopt these formulas (Eqs. (2.3)(2.22) of Ref. [15]) for calculations of theoretical Lamb shifts. III. TABULATION OF RESULTS

The results of our calculations of term energies for some Li-isoelectronic sequence are given in Table I. Columns 2, 3, and 4 are the Brueckner energies, the third order corrections, and the effects of the Breit interaction plus mass polarization respectively. The sum of these three terms is given in column 5. Our theoretical results are compared with the MBPT calculations of Johnson, Blundell, and Sapirstein (JBS [S]) which are listed in column 6. Our results are consistent with, and in general a little bit larger than, the results of JBS. The difference between our calculations and JBS’s mainly comes from that we have included the second order contributions in a self-consistent way while JBS consider the pure second order contributions in the DF bases. (See Ref. [ll] for more details of the comparison between our formalism and MBPT.)

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TABLE I. Energy Levels of n = 2 states for Li-like ions. EB’

2s 2Pl/2 2P3/2

2s 2Pl/2 2P3/2

2s 2P1/2 2P3/2

2s 2P1/2 2P3/2

2s 2Pl/2 2P3/2

2s 2Pl/2 2P3/2

2s 2P, /2 2P3/2

2s 2Pl/2 2P312

2s 2Plj2 2P312

E(3)

-0.197984 -0.130040 -0.130036

-0.000100 -0.000138 -0.000138

-0.669096 -0.523423 -0.523378

-0.000127 -0.000269 -0.000269

-1.393839 -1.173242 -1.173039

-0.000128 -0.000306 -0.000306

-2.370160 -2.076066 -2.075467

-0.000124 -0.000306 -0.000306

-3.597637 -3.230737 -3.229344

-0.000117 -0.000295 -0.000295

-5.076282 -4.636917 -4.634136

-0.000109 -0.000278 -0.000278

-6.806294 -6.294559 -6.289553

-0.000102 -0.000261 -0.000261

-8.787980 -8.203887 -8.195538

-0.000096 -0.000246 -0.000246

-42.561662 -41.240520 -41.045587

-0.000058 -0.000149 -0.000147

B+RM 2=3 0.000018 0.000010 0.000008 2=4 0.000059 0.000041 0.000028 2=5 0.000120 0.000109 0.000062 Z=6 0.000221 0.000233 0.000122 2=7 0.000350 0.000421 0.000208 .2=8 0.000523 0.000695 0.000332 z=9 0.000737 0.001065 0.000495 2 = 10 0.001034 0.001562 0.000719 z = 20 0.008970 0.016653 0.007372

Total

JBS [6] -0.198076

-0.198066 -0.130168 -0.130166

-0.130148 -0.130147

-0.669164 -0.523651 -0.523619

-0.669192 -0.523657 -0.523628

-1.393847 -1.173439 -1.173283

-1.393889 -1.173482 -1.173328

-2.370063 -2.076139 -2.075651

-2.370114 -2.076209 -2.075721

-3.597404 -3.230611 -3.229431

-3.597458 -3.230695 -3.229515

-5.075868 -4.636503 -4.634082

-5.075923 -4.636578 -4.634159

-6.805659 -6.293755 -6.289319

-6.805712 -6.293854 -6.289416

-8.787042 -8.202571 -8.195065

-8.787093 -8.202671 -8.195162

-42.552750 -41.224016 -41.038362

-42.552751 -41.224132 -41.038469

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TABLE II. The 2s - 2p,/z Lamb shifts of Li-like ions.

Z

Ep - E, (theory)

3 4 5 6 7 8 9 10 20

0.067898

0.067906

0.145513 0.220408 0.293924 0.366793 0.439365 0.511904 0.584471 1.328734

0.14548 0.22034 0.29381 0.36662 0.43912 0.51150 0.58390 1.32156

Ep - E, (experiment)

LS”““P”

(Liaw) LS”““P”

-0.000008 0.00003 0.00007 0.00011 0.00017 0.00024 0.00040 0.00057 0.00717

(JBS)

0.000022 0.00006 0.00007 0.00010 0.00014 0.00022 0.00036 0.00052 0.00706

LS (FF) 0.000001 0.00001 0.00003 0.00006 0.00012 0.00022 0.00035 0.00053 0.00780

Our purpose is to test the explicit formulas of the radiation corrections for Liisoelectronic sequence, given by Feldman and Fulton (FF [13]). Column 2 of Table II gives our results of the difference between 2s and 2p1/z term energies taken from Table I. Column 3 is experimental value for the difference of these two states. The difference of column 2 and 3 must then be due to the radiative corrections, assuming the electron-electron correlation has been accounted very accurately by the Bureckner approximation plus its corrections listed in Table I. We therefore have these “experimental” Lamb shifts listed in column 4 of Table II. Column 5 we list the corresponding results when the JBS instead of our term energies are used. Our “experimental” Lamb shifts agree with those ot JBS except for the cases: 2 = 3 and 2 = 4. We next calculate the Lamb shifts for the Li-like ions based on the formulas of FF. The calculations are straight forward. (See Ref. [15] for details.) The results are listed in column 6 of Table II. They are the same as those calculated by Ref. [15] up to the significant digits listed in Table II. Compared the Lamb-shifts calculated based on the formulas ot FF with “experimental” Lamb shifts as described above, we see that the Lamb shift formulas of FF give the right order of magnitude. For 2 = 3 and 2 = 4, theoretical values ot F F are closer to our “experimental” values than JBS’s. Blundell, Johnson, Lie, and Sapirstein (BJLS [7]) has also done an “all-order” calculation for Li and Be+. If their results were used, we obtain “experimental” Lamb shifts 0.000011 and 0.00002 for 2 = 3 and 2 = 4 respectively, which are also in better agreement with theoretical values of FF than JBS’s. The Lamb shifts between 2s and 2p3/z are very close to those of 2s - 2pr/2. Their numerical values similar to Table II are listed in Table III. It is possible to take the difference between the results of Table II and ‘Iable III (JBS did so) to see the size of radiative corrections to the fine-structure intervals. However, we feel that the present calculation is

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TABLE III. The 2s - 2p,,, Lamb shifts of Li-like ions.

z 3 4 5 6 7

8 9 10 20

EP - E, (theory) Ep - E, (experiment) LS”““P” (Liaw) LS”““P” 0.067900 0.145545 0.220564 0.294412 0.367973 0.441786 0.516340 0.591977 1.516075

0.067908 0.14551 0.22050 0.29430 0.36780 0.44154 0.51595 0.59141 1.50769

--0.000008 0.00003 0.00006 0.00011 0.00017 0.00025 0.00039 0.00057 0.00839

(JBS)

0.000022 0.00006 0.00008 0.00010 0.00014 0.00022 0.00037 0.00052 0.00753

LS (FF) 0.000001 0.00001 0.00003 0.00006 0.00012 0.00021 0.00033 0.00051 0.00732

not accurate enough to give significant values for this purpose. We notice that the 2s - 2p112 Lamb shifts are smaller than the 2s - 2psi2 Lamb shifts based on FF’s formulas (column 6. Tables II and III.) This is originated from Eq. (2.11) of Ref. [15]. Our ab initio calculation of correlations shows, however, the reversal. Namely, the Lamb shofts of 2s - 2p,i2 are larger than those of 2s - 2p312. This signifies that the FF’s formulas might need to be modified, in particular, for large 2 2 8. We can also calculate the transition amplitudes between s and p states. The results are listed in Table IV. The transition amplitudes are gauge-invariant in the approximation. The third order corrections have only tiny contributions and decrease as 2 increases. We also list results of two available experiments and the “all-order” calculation of BJLS. Our results are in very good agreement with those of BJLS. Since the relative contribution of the third order corrections gets smaller as 2 increases, we expect our results are more accurate in high Z cases. Unfortunately, there is no accurate experimental values for high 2 along Li-isoelectronic sequence. IV. CONCLUSION Our general conclusion is that the Lamb shift formulas derived by Feldman and Fulton [13] give the right order of magnitude for the Lamb shifts of the bound states of Liisoelectronic sequence. The numerical test for any Lamb-shift formulation relies on accurate calculation of electron-electron correlation without radiative corrections. In particular, for 2 = 3 and Z = 4 in Li-isoelectronic sequence, the Lamb shifts are less than 0.01% of the correlation energies, our calculations based on the Brueckner approximation plus the third order corrections are barely accurate enough. A calculation of perturbative type that

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TABLE IV. Transition amplitudes of n = 2 states for Li-like ions. Transition

TAB’

2P,/i ---) 2s

3.3212 4.6955

2P3/2 + 2s

TAc3) 2=3 -0.0018 -0.0001

Total

BJLS [7]

3.320 4.695

3.316 4.690

3.305” 4.674”

1.851 2.618

1.850 2.617

1.80’ 2.54b

experiment

2=4 2P,/2 ---) 2.3 2P,/2 -+ 2s

1.8514 2.6184

-0.0005 -0.0004 2=5

2P,/2 --) 2s 2P3/2 + 2s

1.2845 1.8168

-0.0002 -0.0002

1.284 1.817

Z=6 2P,/2 --) 2s 2P,/2 --) 2s

0.9847 1.3924

-0.0001 -0.0001

0.985 1.392

2=7 2P1/2 -+ 2s 2P3/2 --) 2s

0.7975 1.1282

-4 x 1o-5 -4 x 1o-5

0.798 1.128

2=8 2Pl/2 -+ 2s 2P,j2 -+ 2s

0.6706 0.9488

-2 x 1o-5 -2 x 1o-5

0.671 0.949

z=9 2P,/2 ---) 2s 2P,l2 -+ 2s

0.5781 0.8187

-2 x 1o-5 -2 x 1o-5

z= 2P1/2 + 2s 2P3/2 + 2.5

0.5086 0.7200

0.578 0.819

10

-1 x 1o-5 -1 x 1o-5

0.509 0.720

z = 20 2Pl/2 --j 2s 2P,/2 --) 2s

0.2295 0.3261

-5 x 10m7 -5 x 1o-7

0.230 0.326

a. A. Gaupp, P. Kuske, and H. J. Andra, Phys. Rev. A26, 3351 (1982). b. J. Bromander, Phys. Ser. 4, 61 (1971).

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can tell how good a Lamb-shift formalism is needs to consider at least the fourth order corrections, which is beyond the scope of present article.

ACKNOWLEDGMENT This work is supported in part by the National Science Council of the Republic of China under the Grant Contract No. NSC 83-0208-M005-019.

REFERENCES [ 11 P. Goldman and G. W. F. Drake, J. Phys. B17, L197 (1984). [ 2 ] J. Hata and I. P. Grant, J. Phys. B17, 931 (1984). [ 3 ] P. Mohr, Phys. Rev. A32, 1949 (1985). [ 4 ] D. K. McKenzie and G. W. F. Drake, Phys. Rev. A44, R6973 (1993). [ 5 ] I. Lindgren, Phys. Rev. A31, 1273 (1985). [ 6 ] W. R. Johnson, S. A. Blundell, and J. Sapirstein, Phys. Rev. A37, 2764 (1988). [ 7 ] S. A. Blundell, W. R. J oh nson, Z. W. Liu, and J. Sapirstein, Phys. Rev. A40, 2233

(1989). [ 8 ] A. W. Weiss, Can. J. Chem. 70, 456 (1992).

[ 9 ] S. S. Liaw, Phys. Rev. A48, 3555 (1993). [lo] S. S. Liaw and F. Y. Chiou, Phys. Rev. A49, 2435 (1994). [ll] S. S. Liaw, Chin. J. Phys. 32, 835 (1994). [la] S. S. Liaw, Phys. Rev. A51, R1723 (1995). [13] G. Feldman and Fulton, Ann. Phys. (N.Y.) 201, 193 (1990); G. Feldman, Fulton, and J. Ingham, Ann. Phys. (N.Y.) 219, 1 (1992). [14] See, for example, R. N. Zare, Angular Momentum (John Wiley & Sons, 1986). [15] A. Devote, G. Feldman, and T. Fulton, Phys. Rev. A47, 1503 (1993); Ann. Phys. 232, 88 (1994).

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