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Progress in helium fine-structure calculations and the fine-structure constant G.W.F. Drake
Abstract: The long-term goal of this work is to determine the fine-structure constant α from a comparison between theory and experiment for the fine-structure splittings of the helium 1s2p 3 PJ states. All known terms of order α 5 a.u. (α 7 mc2 ) arising from the electron– electron interaction, and recoil corrections of order α 4 µ/M a.u. are evaluated and added to previous tabulation. The predicted energy splittings are ν0,1 = 29 616.946 42(18) MHz and ν1,2 = 2291.154 62(31) MHz. Although the computational uncertainty is much less than ±1 kHz, there is an unexplained discrepancy between theory and experiment of 19.4(1.4) kHz for ν1,2 . PACS Nos.: 31.30Jv, 32.10Fn Résumé : L’objectif à long terme de ce travail est la détermination de la constante de structure fine α à partir d’une comparaison entre la théorie et les mesures pour l’écartement de structure fine des états 1s2p3 PJ dans l’hélium. Nous évaluons tous les termes connus à l’ordre α 5 a.u. (α 7 mc2 ) provenant des interactions électron–électron et les corrections de recul d’ordre α 4 µ/M a.u. et nous les ajoutons aux valeurs antérieurement tabulées. Les écartements prédits en énergie sont ν0,1 = 29 616,946 42(18) MHz et ν1,2 = 2291,154 62(31) MHz. Même si l’incertitude de calcul est beaucoup moindre que ±1 kHz, il reste une différence inexpliquée de 19,4(1,4) kHz pour ν1,2 . [Traduit par la Rédaction]
1. Introduction As first pointed out by Schwartz [1], a comparison between theory and experiment for the finestructure splittings in the 1s2p 3 PJ manifold of states in helium (see Fig. 1) has the potential to provide an accurate determination of the fine-structure constant α = e2 /c, provided that both theory and experiment can be done to sufficiently high accuracy. Since the fine-structure splitting is (in lowest order) proportional to α 2 , one simply adjusts α to bring theory and experiment into agreement. With this in mind, Schwartz’s original suggestion stimulated a sequence of early calculations [2–4] and measurements [5], resulting in the value α −1 = 137.036 08(13), accurate to 900 parts per billion (ppb) (1 billion = 109 ). Since that time, the accuracy of other methods of determining α has improved dramatically, as shown
Received 11 September 2002. Accepted 16 September 2002. Published on the NRC Research Press Web site at http://cjp.nrc.ca/ on 29 November 2002. G.W.F. Drake. Department of Physics, University of Windsor, Windsor, ON N9B 3P4, Canada (e-mail:
[email protected]). Can. J. Phys. 80: 1195–1212 (2002)
DOI: 10.1139/P02-111
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Fig. 1. Fine-structure splittings in the helium 1s2p 3 PJ state.
Fig. 2. Determinations of the fine-structure constant. The data are from (a) Mohr and Taylor [6], (b) the Cs recoil measurement from Wicht et al. [7], and (c) present work.
in Fig. 2 [6]. The most accurate determination (±4 ppb) is from the anomalous magnetic moment (g −2) of the electron, but the interpretation of the measurement requires that quantum electrodynamics (QED) be correct. The solid state measurements based on the quantum Hall and ac Josephson effects yield α directly, but they are not in good agreement with each other. An interesting photon recoil measurement of the ratio h/MCs for cesium atoms determines α in terms of the Rydberg constant R∞ and various mass ratios from the equation [7] mp MCs h 2R∞ −2 α = (1) c me mp MCs Similarly, the ratio h/mn for neutrons determines α from simultaneous measurements of their momentum and wavelength. The value based on muonium hyperfine structure is most closely related to the present work, but the accuracy is lower than the other measurements. The significance of all this work is that measurements of α by a variety of different methods provide a sensitive test of the consistency of physics across a range of energy scales and physical phenomena. The resurgence of interest in fine-structure splittings in helium since 1990 stems from the development of new variational techniques to solve the two-electron Schrödinger equation to extremely high precision [8, 9], coupled with progress in calculating the higher order corrections. These developments have stimulated several high-precision measurements [10–13] at the ±1 kHz level of accuracy or better. The purpose of the present paper is to discuss progress toward a measurement of α from helium fine structure at the ±16 ppb level of accuracy. As shown in Fig. 2, this requires an accuracy of ±1 kHz out ©2002 NRC Canada
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of ν0,1 29 617 MHz for the large J = 0 → 1 interval. The small interval ν1,2 2291 MHz is also important in providing a test of theory that is relatively insensitive to the value of α. The basic strategy is to start from accurate variational wave functions for the nonrelativistic helium atom, and to include relativistic and QED corrections by perturbation theory. The result is an expansion of the form so EJ = α 2 B2,0 + α 3 B3,0 + α 4 B2,0 G B2,0 + α 4 B4,0 + α 5 ln(Zα)−2 B5,1 soo ss + B5,1 + 2α 5 ln(Zα)−2 B2,0 G A3,1 + α 5 ln α B5,1 + α 5 B5,0 + 2α 5 B2,0 G A3,0 + O(α 6 ) (2) where G denotes the reduced Green’s function for second-order contributions, and Bm,n denotes an operator associated with a factor of α m [ln(Zα)−2 ]n for operators of the spin–orbit (so) type, and α m [ln(α)]n for operators of the spin–other–orbit (soo) or spin–spin (ss) type. The leading term B2,0 is then the Breit–Pauli interaction, and expectation values of this and the other higher order operators are calculated with respect to the nonrelativistic wave function, as discussed in the following sections. Similarly, matrix elements of the operator α m [ln(Zα)−2 ]n Am,n give the two-electron QED energy shift of that order. In addition, each term has an expansion of the form µ µ 2 (0) (1) (2) + + Bm,n + ··· (3) Bn,m Bm,n = Bm,n M M due to the effects of finite nuclear mass M, and µ = mM/(m + M) is the reduced electron mass. In reduced mass atomic units, the unit of length is aµ = (m/µ)a0 , where a0 is the Bohr radius, and the unit of energy is e2 /aµ = 2(µ/m)R∞ . In these units, the effects of finite nuclear mass come from both the mass scaling of each matrix element, and the perturbing effect of the mass polarization operator −(µ/M)∇1 · ∇2 in the complete three-body Hamiltonian H =−
∇12 ∇2 Z Z 1 µ − 2 − − + − ∇1 · ∇ 2 2 2 r1 r2 r M
(4)
where r = |r| and r = r1 − r2 . For simplicity of notation, the additional expansion in powers of µ/M will not be displayed explicitly, but the contributions will be listed separately in the tables. The paper is organized as follows. Section 2 presents a brief description of the variational wave functions used in the calculations. Section 3 contains a discussion of the various operators in (2) and a tabulation of their matrix elements. To make the origin of the various contributions explicit, this section contains a complete recapitulation of past work, together with new results for the terms of order α 5 a.u. and α 4 µ/M a.u. Finally, Sect. 4 presents a comparison of the calculated fine-structure splittings with experiment, and a discussion of the unexpectedly large differences between theory and experiment.
2. Nonrelativistic wave functions Since the fine-structure splittings are dominated by the α 2 B2,0 term in (2), this term must be known to a few ppb or better. The basic technique used to achieve this accuracy is to expand the nonrelativistic wave function in a doubled Hylleraas variational basis set of the form φ0 (r1 , r2 ) = aSH φSH (r1 , r2 ) +
A B χi,j,k (αA , βA ) + ai,j,k χi,j,k (αB , βB ) − exchange (5) ai,j,k
i+j +k≤" i,j,k
A,B and aSH are linear variational where φSH (r1 , r2 ) is the screened hydrogenic wave function, the ai,j,k parameters found by diagonalization, and the pairs αA , βA and αB , βB are separately optimized nonlinear parameters appearing in the basis functions defined by (for the case of 3 Po states) j
χi,j,k (α, β) = r1i r2 r k exp(−αr1 − βr2 )r2 cos(θ2 )
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Can. J. Phys. Vol. 80, 2002 Table 1. Convergence of the nonrelativistic eigenvalue for the 1s2p 3 P state of helium. "
N
E(")
Ratioa
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Extrap. P.&S.b
104 145 197 265 346 446 559 692 836 1000 1173 1366 1566 1786 2011
−2.133 164 179 530 632 839 −2.133 164 189 081 366 112 −2.133 164 190 499 111 339 −2.133 164 190 742 324 637 −2.133 164 190 771 336 101 −2.133 164 190 777 945 275 −2.133 164 190 778 975 927 −2.133 164 190 779 220 305 −2.133 164 190 779 266 289 −2.133 164 190 779 279 286 −2.133 164 190 779 281 816 −2.133 164 190 779 282 811 −2.133 164 190 779 283 045 −2.133 164 190 779 283 150 −2.133 164 190 779 283 181 −2.133 164 190 779 283 202(5) −2.133 164 190 779 283 16(3)
6.74 5.83 8.38 4.39 6.41 4.22 5.31 3.54 5.14 2.54 4.24 2.25 3.34
a b
Ratio = [E("−2)−E("−1)] . [E("−1)−E(")] Pachucki and Sapirstein [14].
The two sets of nonlinear parameters allow two physically distinct sets of distance scales to be represented in the problem, one for the asymptotic part of the wave function, and one for short-range correlation effects (see ref. 8 for further details). This additional flexibility also plays an important role in maintaining the numerical stability of the basis set against linear dependence problems. Table 1 shows the convergence of the nonrelativistic eigenvalue as a function of the parameter " in (5) controlling the size of the basis set. The final result is slightly more accurate than the value obtained by Pachucki and Sapirstein [14] using a pseudorandom basis set and multiple precision arithmetic. The present results were obtained with standard quadruple precision (about 32 decimal digit) arithmetic.
3. Contributions to the fine-structure splittings 3.1. Terms of order α 2 The operator B2,0 in (2) stands for the Breit interaction terms [15] µ 4 µ 3 m B2,0 = H1 + H2 + H4 + Hso + Hsoo + Hss + (2 + so + 2Hso ) m m M
(7)
where 1 4 (p + p24 ) 8 1 1 1 1 p1 · p2 + 3 r · (r · p1 )p2 H2 = − 2 r r Z Z H4 = π δ(r1 ) + δ(r2 ) − δ(r) 2 2
Z 1 1 Hso = (r1 × p1 ) · σ 1 + 3 (r2 × p2 ) · σ 2 4 r13 r2 H1 =
(8) (9) (10) (11) ©2002 NRC Canada
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Table 2. Convergence of reduced matrix elements for the spin-dependent Breit interaction (10−3 α 2 a.u.). N
Hso
Hsoo
Hss
so
104 145 197 265 346 446 559 692 836 1000 1173 1366 1566 Extrap.
−34.659 788 7697 −34.659 291 4412 −34.659 222 9477 −34.659 214 2013 −34.659 209 6215 −34.659 208 5331 −34.659 207 9268 −34.659 207 4496 −34.659 207 4185 −34.659 207 4443 −34.659 207 4311 −34.659 207 4224 −34.659 207 4216 −34.659 207 4215(1) (0.003 ppb) −34.659 207 420(5)
51.477 881 8136 51.477 935 5933 51.478 041 5795 51.478 117 1609 51.478 087 0316 51.478 079 0068 51.478 076 6660 51.478 075 9633 51.478 075 7536 51.478 075 6499 51.478 075 6213 51.478 075 6052 51.478 075 5979 51.478 075 5945(33) (0.064 ppb) 51.478 075 590(5)
−22.520 333 0764 −22.520 188 3873 −22.520 168 7513 −22.520 180 7015 −22.520 169 8121 −22.520 167 1745 −22.520 166 2292 −22.520 166 0104 −22.520 165 9315 −22.520 165 8876 −22.520 165 8771 −22.520 165 8685 −22.520 165 8656 −22.520 165 8646(10) (0.044 ppb) −22.520 165 860(5)
99.557 675 1445 99.553 845 5741 99.554 459 5353 99.554 542 1928 99.554 519 0091 99.554 495 5507 99.554 498 1056 99.554 500 3599 99.554 499 7132 99.554 499 6588 99.554 499 6883 99.554 499 7185 99.554 499 6876 99.554 499 69(3)
P.&S.a a
Pachucki and Sapirstein [14].
1 1 1 Hsoo = r × p2 · (2σ 1 + σ 2 ) − 3 r × p1 · (2σ 2 + σ 1 ) 4 r3 r
1 3 8 1 − π δ(r) + 3 σ 1 · σ 2 − 5 (σ 1 · r)(σ 2 · r) Hss = 4 3 r r Z 2 = − 2
(12) (13)
1 1 (p1 + p2 ) · p1 + 3 r1 · [r1 · (p1 + p2 )]p1 r1 r1 1 1 + (p1 + p2 ) · p2 + 3 r2 · [r2 · (p1 + p2 )]p2 r2 r2
so
Z = 2
1 1 r1 × p 2 · σ 1 + 3 r2 × p 1 · σ 2 3 r1 r2
(14)
(15)
Only the spin-dependent terms Hso , Hsoo , Hss , and so contribute in lowest order to the fine-structure splittings, but the remaining terms H1 , H2 , H4 , and 2 all contribute to the second-order Breit terms. The terms so and 2 come from the transformation of the pairwise Breit interactions to centre-of-mass plus relative coordinates. The results in Table 2 show that the matrix elements have converged to much better than the required 1 ppb accuracy. The tabulated reduced matrix elements are related to the ν0,1 and ν1,2 transition frequencies by ν0,1 = −(1/6)(Hso + Hsoo ) + (1/2)Hss
(16)
ν1,2 = −(1/3)(Hso + Hsoo ) − (1/5)Hss
(17)
For future reference, all the operators of spin–orbit type (tensors of rank 1) contribute to the two intervals in the ratio 1:2, while the operators of spin–spin type (tensors of rank 2) contribute in the ratio −5:2. ©2002 NRC Canada
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Can. J. Phys. Vol. 80, 2002 Table 3. Summary of contributions to the fine-structure intervals of helium. Units are MHz. Term
Source
ν0,1
ν1,2
α B2,0 α 2 (µ/M)δM B2,0 (2) B2,0 α 2 (µ/M)2 δM 3 α B3,0 α 3 (µ/M)δM B3,0 α 4 B4,0 α 4 B2,0 G B2,0 so α 5 ln(Zα)−2 B5,1 soo α 5 ln αB5,1 ss α 5 ln αB5,1 5 α ln(Zα)−2 2B2,0 G A3,1 Subtotal (1996)
Eq. (7) Eq. (18) Eq. (18) Eq. (19) Eq. (19) Table 4 Table 6 Eq. (24) Eq. (25) Eq. (26) Eq. (27)
29 564.595 37 −0.830 97 0.000 80 54.707 86 −0.003 82 −3.335 19(3) 1.727 58(4) 0.031 82 −0.011 09 0.019 36 0.042 50 29 616.944 22(4)
2 317.231 86 3.009 64 −0.000 08 −22.548 22 0.003 21 1.533 93(5) −8.040 38(5) 0.063 64 −0.022 19 −0.007 74 −0.043 80 2 291.179 87(7)
α 4 (µ/M)δM B4,0 α 4 (µ/M)δM B2,0 G B2,0 α 4 (µ/M)4,0 α 4 (α/π)δanom B2,0 G B2,0 (1) + 2α 5 B2,0 G A3,0 E5,0 (2) E5,0 2-photon QED (3) 3-photon QED E5,0 anom E5,0 X E5,0 2-e cross diags. Total
Table 4 Table 6 Table 5 Table 6 Refs. 14 and 20 Table 7 Table 8 Table 9 Table 10
0.001 79 −0.010 81(5) −0.001 35 0.008 95 −0.012 30(16) −0.013 82 −0.000 68 0.000 60 0.029 82 29 616.946 42(18)
0.001 46 0.010 19(11) −0.001 85 −0.026 02 −0.001 58(27) −0.001 55 −0.000 62 −0.000 25 −0.008 13 2 291.154 62(31)
Experiment
Refs. 10 and 27
29 616.950 9(9)
2 291.174 0(14)
0.004 5(9)
0.019 4(14)
2
Difference
The total contributions are listed in Table 3 for the case of infinite nuclear mass. The value used for α is updated to the 1998 CODATA value α −1 = 137.035 999 76(50), and the conversion factor to MHz is 2α 2 cR∞ = 350 377.0776(25) MHz. The reduced mass ratio for 4 He is µ/M = 1.370 745 62 × 10−4 . The additional finite nuclear mass corrections of order α 2 µ/M and α 2 (µ/M)2 come from two sources: (i) the mass-scaling factors and the so term in (7), and (ii) mass polarization corrections to the wave function. The former terms are µ 2 µ mass = (18) B2,0 (−Hso + 3Hsoo + 3Hss − 2so ) (−Hso − 3Hsoo − 3Hss + so ) + M M and the latter contribution is obtained by repeating the entire calculation for the finite nuclear mass case and taking differences. The total corrections of order α 2 µ/M and α 2 (µ/M)2 , denoted in the tables by (2) α 2 µ/MδM B2,0 and α 2 (µ/M)2 δM B2,0 respectively, are listed separately in Table 3. 3.2. Terms of order α 3 Terms of order α 3 come entirely from the leading term in the anomalous magnetic moment correction g − 2 = α/π + 2a2 (α/π )2 + 2a3 (α/π )3 + · · · . The terms correspond to the matrix elements of the effective operator
1 2 µ B3,0 = Hso + Hsoo δS,S + Hss + −2Hso − 2Hsoo δS,S + so /2 (19) π 3 M The factor δS,S means that Hsoo does not contribute to singlet–triplet mixing terms. Numerical values for the anomalous magnetic moment term and the finite mass correction are listed in Table 3. ©2002 NRC Canada
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(i) Table 4. Douglas–Kroll terms α 4 B4,0 , i = 1, . . . , 15, and finite mass corrections, in units of kHz.
i 1 2 3 4 5 6 7 8 9 10 11 12a 12b 13 14
mass mass Matrix element ν0,1 ν0,1 ν1,2 ν1,2 4 3Zα 1 ∇12 φ0 | 3 σ 1 · (r1 × p1 )|φ0 2091.66 0.75 4183.32 1.51 8 r1 1 153.16 −0.12 306.32 −0.24 –Zα 4 φ0 | 3 3 σ 1 · (r1 × r)(r · p2 )|φ0 r r1 4 1 Zα φ0 | 3 3 (σ 1 · r)(σ 2 · r1 )|φ0 95.19 −0.42 −38.07 0.17 2 r r1 4 1 α 322.39 −0.01 644.77 −0.03 φ0 | 4 σ 1 · (r × p2 )|φ0 24 r α 1 – φ0 | 6 (σ 1 · r)(σ 2 · r)|φ0 757.11 −0.08 −302.85 0.03 24 r iα 1 ∇ 2 φ0 | σ 1 · (p1 × p2 )|φ0 2.06 0.07 4.12 0.14 4 4 1 r 3iα 1 ∇12 φ0 | 3 (r · p2 )σ 1 · (r × p1 )|φ0 60.24 −0.21 120.47 −0.41 4 r 1 −2465.76 0.42 −4931.52 0.84 α 4 ∇12 φ0 | 3 σ 1 · [r × (3p2 /4 − 5p1 /8)]|φ0 r 3iα 4 1 φ0 | 5 σ 1 · [r × (r · p2 )p1 )]|φ0 −80.90 0.01 −161.79 0.02 8 r 3α 4 2 1 – −7334.97 2.20 2933.99 −0.88 ∇1 φ0 | 5 (σ 1 · r)(σ 2 · r)|φ0 2 r iα 4 2 1 ∇1 φ0 | 3 (σ 1 · r)(σ 2 · p1 )|φ0 1266.05 −0.30 −506.42 0.12 4 r iα 4 2 1 – ∇1 φ0 | 3 [(σ 1 · r)(σ 2 · p2 ) + (σ 2 · r)(σ 1 · p2 )]|φ0 1975.28 −0.55 −790.11 0.22 8 r 3iα 4 2 1 ∇1 φ0 | 5 (σ 1 · r)(σ 2 · r)(r · p2 )|φ0 12.76 −0.02 −5.10 0.01 8 r α4 1 – φ0 | 3 (σ 1 · p2 )(σ 2 · p1 )|φ0 −96.33 0.00 38.53 0.00 16 r 3α 4 1 – φ0 | 5 (σ 2 · r × (σ 1 · r × p1 )p2 |φ0 −41.58 0.03 16.63 −0.01 16 r Subtotal −3283.63 1.78 1512.29 1.48 anom α 4 B4,0
15 Total
−51.55
0.02
−3335.19
1.79
21.64 −0.01 1533.93
1.46
3.3. Terms of order α 4 As indicated in (2), terms of order α 4 a.u. come from both a second-order Breit term, and a collection (i) of 14 spin-dependent operators B4,0 , i = 1, . . . , 14 first derived by Douglas and Kroll [2]. There is also a fifteenth term corresponding to anomalous magnetic moment corrections of order (α/π )2 to the lowest order Breit terms given by the effective operator anom B4,0
2 1 a2 (Hso + 23 Hsoo )δS,S + ( 18 + a2 )Hss =2 π
(20)
with a2 = −0.328 478 965. The other 14 operators are as listed in Table 4. The original derivation of Douglas and Kroll has been repeated by Zhang [16], and again by Pachucki [17] using the techniques of nonrelativistic quantum electrodynamics (NRQED). The matrix elements were first calculated by Daley et al. [18], and to higher accuracy by Yan and Drake [19]. The numerical results listed in Table 4 can, therefore, be taken as well-established. Table 4 also includes the finite nuclear mass corrections coming from both mass polarization and the mass scaling of the various operators. All these terms have been checked independently by Pachucki and Sapirstein [20]. ©2002 NRC Canada
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Can. J. Phys. Vol. 80, 2002 Table 5. Finite mass corrections 4,0 of order α 4 µ/M a.u. to Douglas and Kroll terms, in units of kHz. Term 1 2 3 4 5 6 7 8
Matrix element Zα 4 µ 2 i ∇ φ0 | σ 1 · (p1 × p2 )|φ0 – 4 M 1 r1 Zα 4 µ 2 i ∇ φ0 | 3 σ 1 · [r1 × (r1 · (p1 + p2 ))p1 ]|φ0 4 M 1 r1 r1 Zα 4 µ 2 ∇ φ0 |σ 1 · 3 × (p1 + p2 )|φ0 3 4 M 1 r1 r µ Zα 4 φ0 |σ 1 · × (p1 + p2 )|φ0 M r 1 r3 r1 r µ –Zα 4 φ0 |σ 1 · 3 × 3 r1 · (p1 + p2 )|φ0 M r r1 r1 r2 2 4 µ φ0 |σ 1 · 3 × 3 r1 · p1 |φ0 Z α M r1 r2 r1 Z2 α4 µ φ0 |σ 1 · 4 × (p1 + p2 )|φ0 – 2 M r1 r2 r1 Z2 α4 µ φ0 |σ 1 · 3 σ 2 · 3 |φ0 – 4 M r2 r1
Total
ν0,1
ν1,2
0.480
0.959
0.165
0.330
−1.103 −2.206 0.129
0.268
0.042
0.085
–0.236 –0.472 −0.471 −0.941 −0.354
0.141
−1.347 −1.846
In addition, there are several new operators of order α 4 µ/M derived by Zhang [16], including a three-body term [21]. They are analogous to the Stone terms 2 and so that come from transforming the Douglas and Kroll operators to centre-of-mass plus relative coordinates [22]. However, a rederivation of the same terms by Pachucki and Sapirstein [23] yielded a difference of sign for term 6 of Table 5. The operators and their expectation values are listed in Table 5, with the sign for term 6 reversed to agree with Pachucki and Sapirstein. The terms correspond to the sum of contributions from eqs. (343) and (344) of ref. 16, and eq. (18) of ref. 21. The second-order Breit term in (2) is calculated from a discrete variational representation of the resolvent operator in the form G =
|nn| 1 = (E0 − H0 ) E0 − E n n
(21)
where the states |n and eigenvalues En represent sets of pseudostates obtained by diagonalizing the nonrelativistic Hamiltonian H0 in discrete variational basis sets, and the prime on the summation indicates that the initial state corresponding to E0 is to be omitted. In LS coupling, the contributing symmetries for the intermediate states are 3 Po , 1 Po , 3 Do , 3 Do , and 3 F1o . In practice, matrix diagonalization can be replaced by the equivalent but much more efficient process of solving the sets of inhomogeneous algebraic equations (k,k) (k) (k,0) H0 − E0 O (k,k) φ1 = − B2,0 − B2,0 O (k,0) φ0 (22) (k)
for the components of the column vector φ1 , where k = 1, . . . , 5 denotes basis sets having one of the (k,k) is the square matrix of the nonrelativistic Hamiltonian (with or five symmetry types listed above, H0 without mass polarization included) in the kth basis set and O (k,k) is the corresponding matrix of overlap (k,0) integrals. On the right-hand side, B2,0 and O (k,0) are the corresponding rectangular matrices of the Breit interaction and overlap integrals connecting the kth intermediate state basis set with the initial (k,k) state φ0 . The prime indicates that for the case k = 1, where the inverse of the matrix [H0 −E0 O (k,k) ] becomes singular since E0 is an eigenvalue, one member of the basis set is to be omitted to obtain a ©2002 NRC Canada
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State
ν0,1
3
−4894.40(4) −0.47(4) 6595.64(1) −10.31(1) 26.33(1) −0.02 0.00 0.00 0.00 0.00 1727.58(4) −10.81(4)
P P 3 D 1 D 3 F Total 1
mass ν0,1
anom ν0,1
ν1,2
mass ν1,2
anom ν1,2
−14.40 23.28 0.07 0.00 0.00 8.95
−1569.66(4) −6595.64(1) 50.51(1) 22.20 52.22(4) −8040.38(5)
−0.07(3) 10.31(1) −0.02 0.05 −0.07(10) 10.19(11)
−3.20 −23.28 0.16 0.05 0.24 −26.02
nonredundant set of equations. The spin-dependent (SD) part of the second-order energy is then given by 1 (k) B2,0 B = φ1 |B2,0 |φ0 SD (23) 2,0 (E0 − H0 ) SD where the subscript SD indicates that only the spin-dependent combinations of terms from B2,0 in (22) and (23) are to be retained. The omission of spin-independent parts avoids technical complications arising from the appearance of divergent combinations of δ-function terms that do not contribute to the fine-structure splittings. The second-order terms, as originally calculated by Hambro [3] and Lewis and Serafino [4], turned out to be slowly convergent. This problem was solved by Yan and Drake [19] when they realized that the basis set for the sum over intermediate states must include terms more singular at the origin than those required for the nonrelativistic wave function φ0 . They obtained dramatically improved convergence by including terms of the form χi,j,k /r2 and χi,j,k /r so that, for example, the basis set contains terms that behave as r 0 cos(θ ) at the origin for the p electron, instead of the usual r 1 cos(θ ) for hydrogenic wave functions (see (6)). The necessity for terms of this type can be understood from the analytic properties of the exact solutions to the one-electron perturbation equation for the Breit interaction. The convergence for terms involving pi4 or δ(r i ) from the Breit interaction terms H1 and H4 was further improved by transforming H1 and H4 to less singular forms, as described in Appendix A. Except for the addition of finite mass and anomalous magnetic moment corrections, and slight improvements in accuracy, the results in Table 6 are the same as those obtained by Yan and Drake [19]. The second-order terms and the anomalous magnetic moment corrections agree with those obtained by Pachucki and Sapirstein [20]. The finite-mass corrections in Table 6 come from three sources. The first is a mass-scaling factor of (µ/M)6 for cross terms involving H1 , and (µ/M)5 for all the others. The second is the masspolarization correction obtained by repeating the calculation with the (µ/M)∇1 · ∇2 included explicitly in the Hamiltonian (4) and taking differences. The third is from the appearance of new cross terms coming from the finite-mass corrections (µ/M)(2 + so + 2Hso ) in (7). For example, the three contributions to ν0,1 from 3 P intermediate states are 3.25 − 3.66 − 0.06 = −0.47 kHz, and the contributions from 1 P intermediate states are −4.52 − 7.66 + 1.87 = −10.31 kHz, respectively. The corresponding contributions to ν1,2 are 1.17 − 2.15 + 0.91 = −0.07 kHz and 4.52 + 7.66 − 1.87 = 10.31 kHz, respectively. 3.4. Terms of order α 5 ln α The calculation of terms beyond those of order α 4 requires a derivation from QED of nonrelativistic operators analogous to the Douglas and Kroll terms. This task was achieved for terms of order α 5 ln α and α 5 in a series of papers by Zhang [16,24], and Zhang and Drake [25], starting from a fully covariant form of the Bethe–Salpeter equation. The equivalent nonrelativistic operators in coordinate space of order α 5 ln(Zα) and α 5 ln α were subsequently confirmed by Pachucki [17]. Numerical values were ©2002 NRC Canada
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calculated by Zhang et al. [26]. As shown in (2), the contributions are so α 5 ln(Zα)−2 B5,1 = −2Zα 5 ln(Zα)−2 φ0 |δ(r1 )
1 σ 1 · r1 × p1 |φ0 r12
1 σ 1 · r × p1 |φ0 r2 15 5 1 ss = α ln αφ0 |δ(r) 4 σ 1 · rσ 2 · r|φ0 α 5 ln αB5,1 2 r together with the second-order term 2α 5 ln(Zα)−2 B2,0 G A3,1 soo = −9α 5 ln αφ0 |δ(r) α 5 ln αB5,1
(24)
(25) (26)
(27)
where A3,1 is the operator A3,1 =
4 Z [δ(r1 ) + δ(r2 )] 3
(28)
whose expectation value α 3 ln(Zα)−2 φ0 |A3,1 |φ0 gives the logarithmic part of the leading term in the two-electron Lamb shift. The second-order term can, therefore, be understood as the Breit interaction correction to the electron density at the nucleus in the standard Bethe–Salpeter expression for the twoelectron Lamb shift (only the spin-dependent part of B need be retained). Although this term nominally scales as Z 6 ln(Zα)−2 , the leading term vanishes in a 1/Z expansion, and so the leading Z dependence is Z 5 ln(Zα)−2 [26]. It is interesting and instructive to compare the above result with the corresponding hydrogenic Lamb shift. The leading j -dependent term from electron self-energy in the hydrogenic Lamb shift is E(nlj ) =
1 α 5 Z 6 ln(Zα)−2 (1 − n−2 )δj,1/2 δl,1 3π n3
(29)
(5,1)
This coincides with the term α 5 ln(Zα)−2 φ0 |Bso |φ0 in the limit where φ0 is represented by a jj coupled product of hydrogenic wave functions. This is the only term of order α 5 ln α that scales as Z 6 with nuclear charge. The other terms scale as Z 5 or lower, and so represent two-electron corrections to (5,1) the hydrogenic Lamb shift. Note that the hydrogenic matrix elements of Bso vanish for s electrons because the operator is proportional to l·s, and for electrons with l ≥ 2 because of the δ function. The (5,1) matrix elements are nonvanishing only for p electrons. Bso thus represents the operator whose matrix elements give the leading j dependence of the hydrogenic electron self-energy. The numerical values for all four contributions are as listed in Table 3. The line labeled Subtotal (1996) corresponds to the total theoretical splittings as of 1996 [26], except for an adjustment of α to the 1998 CODATA value. The operators up to this point have been calculated at least twice by different authors, and can be considered to be well established. 3.5. Terms of order α 5 Nonrelativistic operators for the terms of order α 5 have been derived by Zhang [16, 24], and Zhang and Drake [25]. They come from a set of 28 time-ordered diagrams representing both exchange and radiative processes [26]. Some of the results have been extended, and in some cases corrected, by Pachucki and Sapirstein [14, 20], as discussed in the following paragraphs. A combination of their results with ours gives a complete account of the known terms, but most of the terms have not been checked by independent calculations. Because of the considerable complexity of the calculations, the results should still be taken as somewhat tentative until they have received independent verification. ©2002 NRC Canada
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1205 (2) Table 7. Two-photon QED electron–electron self-energy terms E5,0 of order 5 α in units of kHz.
Term E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15 E16 E17 Total
Matrix element
ν0,1
α5 2 r p φ0 |σ 1 · 3 × p1 |φ0 −1.614 2π5 1 r α r p2 φ0 |σ 1 · 3 × p2 |φ0 – −0.618 8π 1 r r 29 − 12 ln 2 φ0 |δ(r)σ 1 · 2 × p1 |φ0 α5 0.631 5 r α5 r – φ0 |σ 1 · 3 × p1 (p1 · p2 )|φ0 −0.267 8π5 r r α φ0 | 3 · p1 σ 1 · (p1 × p2 )|φ0 0.220 4π5 r α r 3 φ0 |σ 1 · 5 × [r · (r · p1 )p2 ]p1 |φ0 −0.011 4π5 r α 1 φ0 | 3 σ 1 · p1 σ 2 · p2 |φ0 – −0.448 8π5 r α 1 φ0 | 3 σ 1 · p1 σ 2 · p1 |φ0 – 0.112 16π5 r α 1 φ0 | 5 σ 1 · {r × [σ 2 · (r × p2 )]p1 }|φ0 –3 0.193 8π5 r α 1 φ0 | 5 σ 1 · {r × [σ 2 · (r × p1 )]p1 }|φ0 3 0.598 16π5 r α 1 p12 φ0 | 5 σ 1 · rσ 2 · r|φ0 15 −10.649 16π r 5 α 1 3 φ0 | 5 σ 1 · r(r · p1 )σ 2 · p 1 |φ0 −1.866 8π5 r α 1 p2 φ0 | σ 1 · rσ 2 · p2 |φ0 4.588 4π5 1 r 3 α i –3 p12 φ0 | 5 σ 1 · rσ 2 · r(r · p2 )|φ0 0.030 8π r 1 9α 5 φ0 |δ(r) 4 σ 1 · rσ 2 · r|φ0 −4.722 r −13.822
ν1,2 −3.227 −1.237 1.262 −0.534 0.440 −0.021 0.179 −0.045 −0.077 −0.239 4.259 0.746 −1.835 −0.012 1.889 1.548
The terms of order α 5 correspond to the terms α 5 B5,0 and 2α 5 B2,0 G A3,0 in (2). The latter term is a second-order QED contribution, with A3,0 defined by 4 11 3 1 7 A3,0 = Z [δ(r1 ) + δ(r2 )] (30) + − − 3 24 8 5 6π r 3 As with A3,1 , the expectation value of the first three terms in A3,0 gives the Lamb shift arising from electron self-energy, anomalous magnetic moment, and vacuum polarization respectively, and the last term is the Araki–Sucher QED correction to the electron–electron interaction. The terms in the operator B5,0 can be grouped according to QED
X B5,0 = B5,0 + B5,0
(31)
QED
X comes from photon exchange diagrams. where B5,0 comes from radiative QED diagrams, and B5,0
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Can. J. Phys. Vol. 80, 2002 Table 8. Three-photon QED electron–electron self-energy (3) of order α 5 in units of kHz. terms E5,0 Term E18 E19 E20 E21 E22 Total
Matrix element ν0,1 ν1,2 5 Zα r2 φ0 |σ 1 · 3 × p1 |φ0 – −0.460 −0.920 4π rr2 5 r2 r Zα φ0 |σ 1 · 3 × 3 r · p1 |φ0 0.089 0.178 4π r r2 Zα 5 r2 r φ0 |iσ 1 · 3 × 3 |φ0 0.000 0.000 4π r r2 5 r r2 Zα φ0 |σ 1 · 3 σ 2 · 3 |φ0 −0.088 0.035 4π r r2 5 r Zα r φ0 |σ 1 · 3 σ 2 · 3 |φ0 −0.220 0.088 16π r r −0.679 −0.619
Beginning with the radiative QED terms, the contributions from electron self-energy, vacuum polarization, and vertex corrections can be grouped according to (1)
QED
(2)
(3)
anom E5,0 = E5,0 + E5,0 + E5,0 + E5,0 QED
(32)
QED
where E5,0 = α 5 B5,0 , and similarly for the other terms. These terms are related to a list of 25 operators with expectation values E1 to E25 derived by (1) Zhang [24] as follows: E5,0 = 2i=1 Ei are the electron–nucleus terms E1 = −
Zα 5 2 1 p1 φ0 | 3 σ 1 · (r1 × p1 )|φ0 4π r1
(33)
91 δ(r1 ) (34) φ0 | 2 σ 1 · (r1 × p1 )|φ0 E2 = −2Zα ln(Zα) − β (nSL, Z) + 120 r1 17 (2) including the spin-dependent Bethe logarithm term β (nSL, Z); E5,0 = i=3 Ei are electron– 22 (3) electron terms arising from two-photon diagrams; E5,0 = i=18 Ei are electron–electron terms 25 anom arising from three-photon diagrams; and E5,0 = i=23 Ei is the anomalous magnetic moment contribution for a free electron. The discussion of these terms is complicated by the fact that Zhang’s derivation has only partially been checked by Pachucki and Sapirstein [14, 20]. In particular, Pachucki and Sapirstein have obtained (1) the electron–nucleus terms contained in E5,0 , but with 91/120 in place of 31/120, and with an apparently + E ). In the different combination of expressions for the spin-dependent Bethe logarithm (their EL1 L2 (1) present work, we take the calculation of Pachucki and Sapirstein for E5,0 to be correct, and evaluate 5
−2
(2)
(3)
anom . The results are listed in the remaining electron–electron terms contained in E5,0 , E5,0 , and E5,0 Tables 7 to 9, and the totals included in Table 3. Many of the terms are already known from the Douglas and Kroll operators, but E6 , E7 , E8 , E12 , E18 , and E20 are new. Each of these terms involves algebraically complicated combinations of highly singular integrals. General methods have been developed to cancel the singularities and obtain the finite residual parts, as will be described in a future publication. A useful check is to verify that, for example, the operator for E11 − 2E12 is Hermitian, since the E11 part is already known from the Douglas and Kroll terms.
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1207 anom anomalous magnetic moment terms of order α 5 Table 9. Contributions to E5,0 in units of kHz. The coefficients are a1 = 1/2, a2 = −0.328 478 965, a3 = 1.181 241 456.
Term E23 E24 E25 Total
Matrix element ν0,1 ν1,2 α 3 r1 Zα 2 a3 φ0 |σ 1 · 3 × p1 |φ0 −0.360 −0.719 π r 1 α 3 r –2α 2 a3 φ0 |σ 1 · 3 × p1 |φ0 0.356 0.712 π r 2 3 α α σ1 · σ2 3σ 1 · rσ 2 · r (a1 a2 + a3 )φ0 | − |φ0 0.603 −0.241 2 π r3 r5 0.600 −0.248
X discussed by Zhang comes from the nonrelativistic and relativistic The final group of terms E5,0 parts of the exchange diagrams. The total contribution can be written in the form
15 X E5,0 = α 5 9 (Qso + Nso /6) − (35) (Qss + Nss /6) + α 5 φ0 |Oso + Oss |φ0 2
where the nonlogarithmic parts of the O operators are Oso = 12 ln 2 −
265 12
− 43 π
Oss = − 25 π + 9 ln 2 −
δ(r) r2
σ 1 ·(r × p1 ) +
δ(r)
8i ˆ 1 )p2 ] δ(r)σ 1 ·[rˆ × (r·p 9
2i δ(r) ˆ 2 ·rˆ + ˆ 2 − 3 σ 2 ·rˆ r)·p ˆ 1 σ 1 ·rσ σ 1 ·r(σ r2 9 r2 and the Q and N terms are defined by 1 1 Qso = lim Nso [ln(.) + γ ] + σ 1 · r × p1 4π r(.)5 .→0 1 1 ˆ ˆ Qss = lim Nss [ln(.) + γ ] + σ · rσ · r 1 2 4π r(.)5 .→0 δ(r) Nso = σ · r × p 1 1 r2 δ(r) ˆ 2 · rˆ Nss = σ 1 · rσ r2 4115 96
(36) (37)
(38)
(39) (40) (41)
where γ is Euler’s constant, and . is the radius of a sphere about the point r = 0 that is omitted from the range of integration. As discussed in Appendix B, the final matrix elements are independent of both . and γ . As can be seen from the results listed in Table 10, the contributions from these terms are quite large.
4. Results and discussion The results of the previous section complete the evaluation and tabulation of all known contributions to the fine-structure splittings up to order α 5 and α 4 µ/M a.u. They represent an update of the results published previously by the Windsor group [26] and extended by Pachucki and Sapirstein [14, 20]. There is no disagreement on the numerical values of the terms evaluated by both groups. As discussed by Pachucki and Sapirstein, the next higher order α 6 a.u. contributions can be expected to be considerably less than ±1 kHz, and the computational uncertainties for the known terms are around ±0.3 kHz. In principle, one should, therefore, be able to adjust α so as to bring theory and experiment into agreement ©2002 NRC Canada
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Can. J. Phys. Vol. 80, 2002 Table 10. Contributions to X5,0 from photon exchange diagrams of order α 5 in units of kHz. Matrix element
ν0,1
φ0 |Oso |φ0 4.039 φ0 |Oss |φ0 23.395 9[Qso + Nso /6] −2.457 −15[Qss + Nss /6]/2 4.841 Total 29.818
ν1,2 8.078 −9.357 −4.913 −1.936 −8.128
Table 11. Summary of measurements and theory for the 2 3 P intervals of helium. Units are MHz. Authors Experiment George et al. [10] Storry et al. [27] Minardi et al. [12] Castillega et al. [11] Wen [13] Frieze et al. [5] Theory Present work
ν0,1
ν1,2
29 616.950 9(9)
2 291.1740(14)
29 616.949 7(20) 29 616.959(3) 29 616.936(8) 29 616.864(36)
2 291.174(15) 2 291.175 9(10) 2 291.198(8) 2 291.196(5)
29 616.946 42(18)
2 291.154 62(31)
for the large ν0,1 interval, and thus determine a new value for the fine-structure constant. The value obtained in this way is α −1 = 137.035 989 3(23) As shown in Fig. 1, this value is substantially below the high-precision g − 2 result, but it agrees well with the ac Josephson result. However, the comparison between theory and experiment in Tables 3 and 11 shows that for the small ν1,2 interval, the difference of 19.4 kHz from the measurement of George et al. [10], or even more for the other measurements, is much larger than that which can be accommodated by any reasonable adjustment in α. Assuming that both the measurements and the calculations up to this point are correct, it seems clear that there must be an additional spin-dependent contribution that has not yet been taken into account. As an example indicating what would be required, the large coefficient −265/12 in Oso (see (36)) would have to increase in magnitude to about −60 to remove the discrepancy for ν1,2 . An adjustment by a factor of two or so to the other coefficients of order α 5 , and in particular to the E5 term in Table 7 (see ref. 14 for further discussion) would not be sufficient to resolve the discrepancy. One could also look for other physical effects that might be playing a role. For example, spindependent corrections to the energy shift due to finite nuclear size could, in principle, be important, but detailed calculations show that the effect is less than 0.1 kHz. In summary, theory has developed to the point that it is now very close to providing the necessary accuracy to determine an atomic physics value for α from fine-structure splittings in helium. The comparison between theory and experiment is already providing tests of QED at unprecedented levels of accuracy for a three-body system. However, the remaining discrepancy of about 19 kHz for ν1,2 must be resolved before a meaningful value for α can be derived. ©2002 NRC Canada
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Acknowledgements The author is grateful to Drs. Krzysztof Pachucki and Jonathan Sapirstein for pointing out their different sign for term 6 in Table 5, and other numerical differences in Table 5. The numerical differences have all been satisfactorily resolved. Research support by the Natural Sciences and Engineering Research Council of Canada, and by SHARCnet (Shared Hierarchical Academic Research Computing) through grants from CFI, OIT, and ORDCF is gratefully acknowledged.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.
C. Schwartz. Phys. Rev. 134, A1181 (1964). M. Douglas and N.M. Kroll. Ann. Phys. (N.Y.), 82, 89 (1974). L. Hambro. Phys. Rev. A: Gen. Phys. 5, 2027 (1972); 6, 865 (1972); 7, 479 (1973). M.L. Lewis and P.H. Serafino. Phys. Rev. A: Gen. Phys. 18, 867 (1978). W. Frieze, E.A. Hinds, V.W. Hughes, and F.M.J. Pichanick. Phys. Rev. A: Gen. Phys. 24, 279 (1981) and earlier references therein. P.J. Mohr and B.N. Taylor. J. Chem. Phys. Ref. Data, 28, 1713 (1999); Rev. Mod. Phys. 72, 351 (2000). A. Wicht, J.M. Hensley, E. Sarajlic, and S. Chu. Proc. 6th Int. Symposium on frequency standards. Edited by P. Gill. World Scientific, Singapore. 2002. p. 193. G.W.F. Drake and A.J. Makowski. J. Opt. Soc. Am. B: At. Mol. Opt. Phys. 5, 2207 (1988). Z.-C. Yan and G.W.F. Drake. Phys. Rev. A, 46, 2378 (1992). M.C. George, L.D. Lombardi, and E.A. Hessels. Phys. Rev. Lett. 87, 173002 (2001). J. Castillega, D. Livingston, A. Sanders, and D. Shiner. Phys. Rev. Lett. 84, 4321 (2000). F. Minardi, G. Bianchini, P.C. Pastor, G. Giusfredi, F.S. Pavone, and M. Inguscio. Phys. Rev. Lett. 82, 1112 (1999). J. Wen. Ph.D. thesis. Harvard University. K. Pachucki and J. Sapirstein. J. Phys. B: At. Mol. Opt. Phys. 33, 5297 (2000). H.A. Bethe and E.E. Salpeter. Quantum mechanics of one- and two-electron atoms. Springer-Verlag, Berlin. 1957. T. Zhang. Phys. Rev. A: At. Mol. Opt. Phys. 54, 1252 (1996). K. Pachucki. J. Phys. B: At. Mol. Opt. Phys. 32, 137 (1999). J. Daley, M. Douglas, L. Hambro, and N.M. Kroll. Phys. Rev. Lett. 29, 12 (1972). Z.-C. Yan and G.W.F. Drake. Phys. Rev. Lett. 74, 4791 (1995). K. Pachucki and J. Sapirstein. J. Phys. B: At. Mol. Opt. Phys. 35, 1783 (2002). T. Zhang. Phys. Rev. A: At. Mol. Opt. Phys. 56, 270 (1997). A.P. Stone. Proc. Phys. Soc. (London), 77, 786 (1961); 81, 868 (1963). K. Pachucki and J. Sapirstein. J. Phys. B: At. Mol. Opt. Phys. (2003). In press. T. Zhang. Phys. Rev. A: At. Mol. Opt. Phys. 53, 3896 (1996). T. Zhang and G.W.F. Drake. Phys. Rev. A: At. Mol. Opt. Phys. 54, 4882 (1996). T. Zhang, Z.-C. Yan, and G.W.F. Drake. Phys. Rev. Lett. 77, 1715 (1996). C.H. Storry, M.C. George, and E.A. Hessels. Phys. Rev. Lett. 84, 3274 (2000). H.A. Bethe and E.E. Salpeter. Quantum mechanics of one- and two-electron atoms. Springer-Verlag, Berlin. 1957. p. 182. G.W.F. Drake. In Long range Casimir forces: Theory and recent experiments on atomic systems. Edited by F.S. Levitt and D. Micha. Plenum Press, New York. 1993. pp. 146–147. J. Hiller, J. Sucher, and G. Feinberg. Phys. Rev. A: Gen. Phys. 18, 2399 (1978). G.W.F. Drake. Nucl. Instrum. Methods Phys. Res. B, 31, 7 (1988).
Appendix A: Calculation of second-order p4 and δ(r) cross terms This Appendix discusses some of the technical details involved in the calculation of second-order cross terms containing the p14 operator in H1 and δ(r 1 ) in H4 . A direct attempt to calculate the second©2002 NRC Canada
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order quantities E2 (H1 ,Hx ) = α 4 φ0 |H1 G Hx + Hx G H1 |φ0
(A.1)
where Hx stands for Hso , Hsoo , Hss , or so leads to results that are slowly convergent with basis-set size. For example, our uncertainty for a direct calculation of E2 (H1 , Hss ) is about 40 kHz. For diagonal expectation values, a standard method to improve convergence is to replace the operator p14 +p24 by a less singular form involving the operator p12 p22 through the indentity (in reduced mass units of e2 /aµ ) [28,29] (p12 + p22 )|φn = fn |φn
(A.2)
valid for exact wave functions, where fn is the state-dependent quantity fn = En − V −
µ p ·p M 1 2
(A.3)
and V =−
Z 1 Z − + r1 r2 r
(A.4)
is the potential. A double application of (A.2), with appropriate allowance for commutators, then leads to the result [29] H1 |φn = −
1 2 2fn − p12 p22 |φn 4
(A.5)
This expression can be used as it stands for diagonal matrix elements, but for second-order energies, the state dependence of fn must be taken into account in the sum over intermediate states. This can be (0) done by writing the second-order energies in the form E2 (H1 , Hx ) = E2 (H1 , Hx ) + E2 (H1 , Hx ) where
α 4 φ0 |2f02 − p12 p22 |φn φn |Hx |φ0 φ0 |Hx |φn φn |2f02 − p12 p22 |φ0 (0) + E2 (H1 ,Hx ) = − 4 E0 − E n E0 − E n n=0
(A.6) is the f0 part and E2 (H1 ,Hx ) = −
α 4 φ0 |2fn2 − 2f02 |φn φn |Hx |φ0 4 E0 − E n
(A.7)
n=0
is the correction. In the expansion µ fn2 − f02 = (En2 − E02 ) − 2(En − E0 ) V + p 1 · p 2 M
(A.8)
the (En2 − E02 ) term does not contribute due to orthogonality, and the sum over the second term can be completed by closure with the result E2 (H1 , Hx ) = −α 4 [φ0 |V Hx |φ0 − φ0 |V |φ0 φ0 |Hx |φ0 µ + φ0 |p 1 · p 2 Hx |φ0 − φ0 |p 1 · p 2 |φ0 φ0 |Hx |φ0 M
(A.9)
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As a numerical example and check, a direct calculation of E2 (H1 , Hsoo ) for the finite nuclear mass case (excluding the mass scaling factor of (µ/m)6 ) yields the particularly well-converged value (0) −1038.174(19) kHz for the ν0,1 interval. In comparison, (A.6) gives E2 (H1 , Hsoo ) = −1629.262(1) µ kHz (including 1.034 kHz from the − M p1 · p2 term in f0 ), and the two groups of terms of (A.9) for E2 (H1 , Hsoo ) contribute 590.916 MHz and 0.180 MHz, respectively, for a total of −1038.166(1) kHz. The agreement with the directly calculated value is sufficient to verify that all terms are correctly accounted for. The corresponding total for the case of infinite nuclear mass is −1038.4913(3) kHz. The difference of 0.325(1) kHz is the correction due to mass polarization. Similarly, the accuracy of matrix elements of δ(r 1 ) + δ(r 2 ) can be improved by about a factor of 20 by use of the Hiller–Sucher–Feinberg (HSF) [30] global operator, generalized to include the effects of nuclear recoil. The generalized operator, in Hermitian form, is [29, 31] 1 Z 1 ∂r 1 2 µ 1 1 ∇1 · ∇2 − 2 r 1 · (r 1 · ∇1 )∇2 − 2 r 1 · ∇2 − 2 − 3 l1 + D1 = 2π r12 r ∂r1 Mr1 r1 r1 r1 − rˆ 1 · ∇1 , H (A.10) 1 (A.11) rˆ 1 · ∇1 , H ≡ D¯ 1 − 2π such that, for exact wave functions, φ0 |δ(r 1 )|φ0 = φ0 |D1 |φ0 in units of aµ−3 . The operator rˆ 1 · ∇1 coincides with the operator ∂/∂r1 defined by HSF. The final commutator term does not contribute to diagonal expectation values, but it does contribute to the off-diagonal matrix elements in the secondorder perturbation sum π Zα 4 φ0 |δ(r 1 ) + δ(r 2 )|φn φn |Hx |φ0 E2 (H4 , Hx ) = 2 E0 − E n n=0
φ0 |Hx |φn φn |δ(r 1 ) + δ(r 2 )|φ0 (A.12) + E0 − E n After the operator replacement δ(r 1 ) + δ(r 2 ) → D¯ 1 + D¯ 2 − [ˆr 1 · ∇1 + rˆ 2 · ∇2 , H ]/(2π ), the sum over states for the commutator term can be completed by closure with the result
π Zα 4 φ0 |D¯ 1 + D¯ 2 |φn φn |Hx |φ0 φ0 |Hx |φn φn |D¯ 1 + D¯ 2 |φ0 + E2 (H4 , Hx ) = 2 E0 − E n E0 − E n n=0 1 φ0 |[ˆr 1 · ∇1 + rˆ 2 · ∇2 , Hx ]|φ0 + (A.13) 2π As a numerical example and check, a direct calculation of E2 (H4 , Hsoo ) from (A.12) for the infinite mass case yields 2329.555(5) kHz for the 0 → 1 transition, while the two groups of terms in (A.13) contribute 3474.8265(1) and −1145.2786 kHz, respectively. The total of 2329.5479(1) kHz agrees with the direct calculation, but is considerably more accurate. The mass polarization correction is 1.1007(1) kHz.
Appendix B: Calculation of singular integrals The operators in Tables 7 to 10, and especially the most singular operators Qso and Qss defined by (38) and (39), require the evaluation of radial integrals of the form r+r2 ∞ ∞ rarb In (a, b) = r dr r2 dr2 r1 dr1 1 n2 exp(−αr1 − βr2 ) (B.1) r |r−r2 | . 0 ©2002 NRC Canada
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in the limit . → 0, and for n ≤ 5. These can all be calculated by repeated differentiation with respect to α and β of the starting integral 2 In (−1, −1) = 2 α − β2
∞ .
e−βr − e−αr dr r n−1
(B.2)
∂ a+b In (−1, −1) ∂α a ∂β b
(B.3)
such that In (a − 1, b − 1) = (−1)a+b
For n ≥ 3, the integrals contain terms that diverge as ln ., as well as progressively higher powers of 1/. for n = 4 and 5. A detailed analysis shows that terms involving powers of 1/. cancel, leaving the residual part I˜n (−1, −1) =
2(−1)n α n−2 (ln α + Cn ) − β n−2 (ln β + Cn ) 2 2 (n − 2)!(α − β )
(B.4)
−1 where Cn = γ + ln . − n−2 i=1 i , and γ is Euler’s constant. All of the final results are independent of γ . This, in fact, provides a valuable check that the integrals are calculated correctly. Also, all the final matrix elements are independent of ln ., provided that the cancellation between the two terms in the definition of Qso and Qss is carried out. Even with these cancellations, the matrix elements of Qso and Qss are sensitive to the term −11/6 in C5 = γ + ln . − 11/6. Equation (B.4) and its derivatives become numerically unstable when α ≈ β. These problems can be avoided by use of the very useful equations a+b ln(α/β) α−β a!b! a+b ∂ (−1) = 2 F1 1, a + 1; a + b + 2; ∂α a ∂β b α − β (a + b + 1)β b α a+1 α β −α a!b! (B.5) = 2 F1 1, b + 1; a + b + 2; (a + b + 1)β b+1 α a β where 2 F1 (a, b; c; z) is a hypergeometric function. With simple algebraic rearrangements and use of the chain rule for differentiation, all the numerically unstable integrals can be expressed in terms of these functions. The effect is to replace finite but numerically unstable expressions by infinite expansions that are stable and rapidly convergent.
©2002 NRC Canada
