Relativistic Calculation of Specific Mass Shifts for Ar +, Ni, Kr +, and Ce + using a multi-configuration Dirac-Fock Approach

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1 Relativistic Calculation of Specific Mass Shifts for Ar +, Ni, Kr +, and Ce + using a multi-configuration Dirac-Fock Approach Warren F. Perger and Muhammad Idrees Departments of Electrical Engineering and Physics, Michigan Technological University, Houghton, MI , USA Department of Chemistry and Physics, Southeastern Louisiana University, Hammond, LA Short Title: Relativistic Calculation of Specific Mass Shifts Classification Numbers: Gs, Jv, Tz Received: Abstract. An extension of the multi-configuration Dirac-Fock (MCDF) program for calculation of the specific mass shift (SMS) is described. The SMS values obtained from this relativistic program are shown to exhibit a trend: the higher the Z-value of the atom, the better the agreement with experiment and the greater the discrepancy with the non-relativistic results. The various modes (average level, optimal level, etc.) for achieving an approximate wavefunction, and their impact on the relativistic SMS values, are explored. Comparisons are made with other theoretical SMS values as well as with experiment for Ar +, Ni, Kr +, and Ce + with new results reported for each atom. 1. Introduction Systematics of nuclear properties are currently being revealed by increasingly accurate laser spectroscopic studies of isotope shifts (Bromley, 1987; Vadla et al., 1987; Lorenzen and Niemax, 1982). These isotope shifts arise from the finite nuclear mass (mass shifts) and finite nuclear volume (field shift). The finite nuclear volume gives rise to a field polarization correction and is not the subject of this article. The finite nuclear mass gives rise to two shifts: a reduced mass (or often called normal mass ) shift and a specific mass shift (SMS) (also referred to as a mass polarization shift) (Fischer and Mielczarek, 1983). From the viewpoint of experiment, these effects are not measured separately. The reduced mass is straightforward to calculate; the SMS can be obtained from either a high-quality King plot or from atomic theory. From the viewpoint of atomic theory, there are two primary, inter-related phenomena acting simultaneously, which must be included in order to interpret the experimental SMS results correctly. These phenomena are correlation and, for heavy atoms, relativistic effects. Regarding correlation, because the SMS effect involves the difference of two individual level shifts, it places a relatively stringent demand on the quality of the correlation included in the wavefunction expansion. 1

2 Regarding relativistic effects, because current experiments are entering the heavy atom regime, where relativistic effects are important, e.g. krypton (Schuessler et al., 1992), and since SMS effects can be appreciable in heavy atoms (Bauche, 1974; Bauche and Crubellier, 1970), ab initio calculations that include relativistic effects will be necessary. For krypton (Schuessler et al., 1992), the size of the relativistic effects is believed to be smaller than the experimental accuracy; however, the experimental accuracy is improving, and more accurate atomic theory will eventually be required to continue to extract new physics. Pseudorelativistic Hartree-Fock SMS calculations originated with the work of Wilson in series of papers on various atoms (Wilson, 1978b; Wilson, 1978a; Wilson, 1987). Therefore, owing to the comparatively few SMS studies that employ relativistic effects in a Dirac-Fock framework and the existence of Dirac-Fock programs (Grant et al., 1980; Dyall et al., 1989; Desclaux, 1975), development of the necessary atomic physics computer programs and experience with their use appears in order. The theoretical investigation of the specific mass shift dates back to the work done by Vinti on magnesium (Vinti, 1939) and boron (Vinti, 1940). Since that time, there have been a variety of methods employed for different atoms: Pekeris and coworkers (Pekeris, 1962; Schiff et al., 1965; Accad et al., 1971) performed an accurate calculation of the SMS in helium, but their method was not readily extended to heavier atoms. Flannery and Stewart (Flannery and Stewart, 1963) using open-shell wave functions, and Prasad and Stewart (Prasad and Stewart, 1966) using the 45-parameter wave functions of Weiss, calculated the SMS for lithium and compared their results with the experimental value obtained by Hughes (Hughes, 1955). Chambaud, Levy, and Stacey (Chambaud et al., 1984) used a Gaussian basis set in a multi-configurational approach for lithium, while Mårtensson and Salomonson (Mårtensson and Salomonson, 1982) used many-body perturbation theory and compared to the experimental results of Lorenzen and Niemax (Lorenzen and Niemax, 1982). King (King, 1989) used an extensive Hylleraas-type basis set for high precision calculations of the Li 2 S ground state SMS. Bauche (Bauche, 1969; Bauche, 1974) performed Hartree-Fock calculations of the SMS in the transition between the two lowest-lying states for a large number of atoms. Keller (Keller, 1973) and Labarthe (Labarthe, 1973) used a multi-configurational Hartree-Fock (MCHF) approach to include second-order correlation effects. An MCHF calculation of the SMS by Froese Fischer and Mielczarek (Fischer and Mielczarek, 1983), using gradient and Slater forms for the SMS operator, revealed that agreement between the two operator forms was possible if the total wave function was nearly exact. While the results of that work suggested that the Slater form was the most accurate, subsequent studies (Fischer, 1990; Fischer et al., 1991) have shown the gradient form to be superior and it is the one that has been used in the current study. Parpia, et al (Parpia et al., 1992b) recently presented representative results of their SMS program which uses the relativistic GRASP2 program (Parpia et al., 1992a). The use of the various modes, average-level (AL) and optimal-level (OL), with their extended counterparts, EAL and EOL, have been developed for addressing correlation effects in the multi-configuration Dirac-Fock (MCDF) program 2

3 of Grant, et al (Grant et al., 1980) and its successors, GRASP (Dyall et al., 1989) and GRASP2 (Parpia et al., 1992a). Prior work with these modes has suggested that the MCDF(AL) mode provided accurate estimates of energy levels at a computation cost much less than the MCDF(OL), but is generally inferior to MCDF(OL) for transition properties, such as oscillator strengths, in neutral atoms or atoms with only a few ionized electrons (Grant et al., 1976; Rose et al., 1978). This has more recently been confirmed for Zn I, Rb VIII, Hg I, and Rn VII with small basis sets (Migdalek and Stanek, 1990). As will be shown in the current work, this is also true for the SMS. 2. Calculation of the Specific Mass Shift The specific mass shift for an atomic state may be expressed as an expectation value of the shift operator (in gradient form, atomic units h = m = 1) (Hughes and Eckart, 1930) ô = 2mR M M ( p i p j ). (1) i<j By taking the matrix element of this operator for an atomic state function ψ obtained by some method (Hartree-Fock, CI, etc.) for a given isotope, A, then repeating for another isotope, B, one obtains the SMS by taking the difference between these two matrix elements. That is to say, equation (1) gives the shift of an atomic level of a given isotope; repeating the calculation for another isotope and taking the difference between the two shifts produces the desired SMS result that can be compared with experiment. In equation (1), m and M denote the mass of an electron (equal to 1 in atomic units) and the nucleus, respectively, R M is the Rydberg constant appropriate for the atomic mass in question (defined in equation (3) below), and p i is the momentum of the i th electron. Note that the SMS operator in equation (1) is non-relativistic, which should be sufficient for our purposes; however, relativistic expressions for nuclear effects in atomic spectra have been derived elsewhere (Stone, 1961; Stone, 1963; Parpia et al., 1992b) and will be used in future versions of our SMS package. As is apparent from equation (1), the SMS effect involves a momentum correlation between two electrons; as such, there is a strong resemblance with the electron-electron correlation effect. Stone (Stone, 1959) first pointed out that the angular analogy between the p i p j scalar product in the SMS operator and the [C (k) i C (k) j ] scalar product in the inter-electronic electrostatic operator when k=1 allows one to calculate the SMS merely by changing the values of the G 1 Slater (exchange) integrals for the total atomic energy (Bauche and Champeau, 1976). Thus, the problem of accurately calculating the SMS requires a treatment of correlation effects typically beyond that of simple Hartree- (or Dirac-) Fock. The expectation value is described in terms of the Vinti integrals (Vinti, 1939) as δν SMS = ( RA M n M A R B M n M B ) 2m a,b X ab J 2 (a, b) c,d X cd J 2 (c, d). (2) 3

4 The index a refers to the pair of quantum numbers nκ for orbital a (and similarly for b, c, and d), the prime is used to indicate the upper state, and X ab is the angular coefficient between orbitals a and b. The adjusted Rydberg constant, R A, is determined by: R R A = [1 + m/m A ] cm 1, (3) where M n = m, R = cm 1 and M A is the atomic weight of isotope A (40 M n, for example, in Ar + ) and similarly for R B. The relativistic version of the Vinti integral is given by (Parpia et al., 1992b) J(a, b) = 0 P na κ a (r) ( d dr κ ) a(κ a + 1) κ b (κ b + 1) P nb κ 2r b (r) dr ( d + Q na κ a (r) 0 dr κ ) a(κ a 1) κ b (κ b 1) Q nb κ 2r b (r) dr. (4) ( ) P The single-particle orbitals,, used in equation (4) are Dirac-Fock orbitals. Q The angular part of equation (2), X, is given by the same angular coefficient of the corresponding G 1 Slater integral, and can therefore be obtained conveniently from the GRASP2 code MCP package (Dyall et al., 1989). To approximate the correlation, one may use a multi-configuration approach, i.e., expand the total atomic state wave function as a superposition of determinantal configuration state functions with configuration mixing coefficients, c i : ψ = nconf i=1 c i Φ(γ i JMπ), (5) where Φ(γJMπ) is a configuration state function with a specific coupling scheme, J is the total angular momentum, π is the parity, nconf is the number of configurations, and M = J,... + J. Note that J(i, j) refers to the Vinti integral between orbital i with quantum numbers n and κ and orbital j with quantum numbers n and κ while J refers to the total angular momentum; the use of J for both these purposes is common in the literature. The total SMS contribution to level r is then given by: SMS = nclosed i=1 i 1 j=1 q r i q r j [C(κ i, k, κ j )] 2 ( [J(i, j)] 2 ) + nmct ja=1 c sr c tr q t jb nclosed ia=1 ( q r ia) [C(κ ja, k, κ ia )] 2 J(ja, ia)j(b, c)( 1) n + nmcp j=1 nconf i=1 X st j c sr c tr [ J(ia, ic)j(ib, id)], (6) where the 3 terms are the contributions from core-core, core-valence, and valencevalence, respectively. The angular quantum number is given by κ i for orbital i, C(κ i, k, κ j ) is the Wigner 3-J coefficient for orbitals i and j with multipole moment 4

5 k = 1, q j i is the occupation number of orbital i in configuration j, and c sr is the configuration mixing coefficient between eigenstates s and r. In the core-valence term, the integers ja and jb are the orbital labels read from the GRASP2 MCT package and if ja jb then b = ia, c = jb, n = 2; else b = ja, c = ia, and n = 1 (s and t are also read from the MCT subroutine). In the valence-valence term, Xj st is the MCP coefficient and s and t are read from the GRASP2 MCP package. The number of closed orbitals is given by nclosed, nconf is the number of configurations, and nmct is the number of MCT coefficients, as listed from a call to the GRASP2 MCT subroutine. Equation 6 for the SMS has been programmed as a subroutine for the GRASP2 code and several Dirac-Fock (single configuration) calculations have been performed. To clearly show the steps for calculating the specific mass shifts reported here, we will use as an example a single-configuration Dirac-Fock calculation for each of the two levels in the Ar + 3p 4 3d 2 G 9/2 3p 4 4p 2 F 7/2 transition, for isotopes M A = 36 M n and M B = 40 M n. The steps used are then: 1. Calculate orbitals and configuration mixing coefficients for the first atomic level, the 3p 4 3d 2 G 9/2. The SMS, as evaluated from equation (6), is SMS = a.u. (see Table 2). 2. Repeat for the 3p 4 4p 2 F 7/2 level. The SMS value for this level is SMS = a.u. (see Table 1). 3. Evaluate the Rydberg constants, R A and R B from equation (3). 4. The δν SMS is then: δν SMS = 2m ( RA R ) B M A M B (SMS SMS ) cm 1. (7) For our example, this translates to cm 1, or GHz, using MHz per cm 1. This compares with the non-relativistic Hartree-Fock value of GHz and the experimental value of ± GHz from Eichhorn, et al. (Eichhorn et al., 1982). The various terms are given in Table 1 for the 2 F 7/2 SMS and Table 2 for the 2 G 9/2 SMS in order to permit a detailed comparison with the corresponding terms given by Eichhorn, et al. Note that because our approach uses relativistic orbitals, there is not a oneto-one correspondence of terms, i.e. our J(2 p,1s) + J(2p,1s) terms correspond to their J(2p,1s) term (relativistic orbitals are distinguished with an unbarred letter denoting j = l + 1; a barred one j = l 1 ). A few other points are noteworthy 2 2 concerning the numerical accuracy of our SMS program. We calculated J(i, j) and compared it with J(j, i) and found them to agree to typically 11 significant figures. The orthogonalities of the single-particle orbitals were always 10 7 or better. Also, we have used a Fermi nucleus for all of our calculations. 3. Results We have made SMS runs for several other atoms, primarily to see if any relativistic effects can be observed. We chose atoms (and levels) based upon 5

6 Table 1: Dirac-Fock calculation of the 2 F 7/2 specific mass shift in Ar + (an unbarred letter denotes j = l ; a barred one j = l 1 2 ). Angular coefficient Orbitals J 2 (a.u.) Contribution (a.u.) p, 1s p, 2s Core p, 1s Core p, 2s s, 2 p s, 2p p, 1s p, 2s p, 3s Core p, 1s Valence p, 2s p, 3s p, 1s p, 2s p, 3s Total Table 2: Dirac-Fock calculation of the 2 G 9/2 specific mass shift in Ar + (an unbarred letter denotes j = l ; a barred one j = l 1 2 ). Angular coefficient Orbitals J 2 (a.u.) Contribution (a.u.) p, 1s p, 2s Core p, 1s Core p, 2s s, 2 p s, 2p p, 1s p, 2s p, 3s Core p, 1s Valence p, 2s p, 3s d, 3p Val-Val p, 3d Total

7 their Z-number and the availability of other published results. The atoms and their levels are: 1. Ne 8+, ground state only (Fischer and Mielczarek, 1983). 2. Ar +, 3p 4 3d 2 G 9/2 3p 4 4p 2 F 7/2 (Eichhorn et al., 1982). 3. Ni, 3d 8 4s 2 3 F 4 3d 9 4s 3 D 3, (Bauche and Champeau, 1976). 4. Kr +, 4p 4 4d 4 D 7/2 4p 4 5p 4 P 5/2, (Schuessler et al., 1992). 5. Ce +, 4f 2 6s 4 H 4f5d 2 4 H, (Wilson, 1978b). We have summarized our Dirac-Fock results, previously published theoretical values, and experimental values in Table 3. To interpret these results, several key points must be mentioned. In the helium-like Ne 8+ run, our value of GHz = x10 5 a.u. compares very well with the result of Parpia, et al. (Parpia et al., 1992b) of x10 5 a.u.. This is as expected because in both cases, the approximate wave function was expressed as: Ψ >= n max n=1 n κ= 1,1, 2,2... c nκ Φ((nκ) 2 J = 0) >, (8) where n max = 2 and the c nκ are the configuration mixing coefficients of equation (5). The non-relativistic value of GHz was obtained from Fischer and Mielczarek by taking their value of cm 1 (Table 3 of that reference) and converting it to GHz. It is interesting to note that this was a four configuration calculation, and our value was obtained using the MCDF-OL mode of the GRASP2 program. Table 3: Comparison of Dirac-Fock SMS results with other non-relativistic Hartree-Fock calculations for various atoms (in GHz). Atom nconf Z This work Other work % diff. Experiment Ne a 0.59 Ar b ±.008 b Ni c c Kr d ±.008 d Ce e f nconf = number of relativistic configurations used a Fischer and Mielczarek (1983), M A = M n b Eichhorn et al. (1982), M A = 36 M n, M B = 40 M n c Bauche and Champeau (1976), M A = 60 M n, M B = 62 M n d Schuessler et al. (1992), M A = 84 M n, M B = 86 M n e Wilson (1978b), M A = 140 M n, M B = 142 M n f Champeau and Verges (1976), For N i, we compare our single-configuration Dirac-Fock with the Hartree-Fock result of Bauche and Champeau where their value of GHz was obtained by 7

8 taking the value of a.u. from Table VIII (Bauche and Champeau, 1976) as the quantity (SMS SMS ) in equation (7). The experimental value was read from Fig. 4 of that reference. The Ce + case compares the non-relativistic result of Wilson (Wilson, 1978b) but he also reported a pseudo-relativistic value of 26.7 a.u. (0.800 GHz), which compares well with our value. The Kr + case is interesting for several reasons. The result of Schuessler, et al, quoted in Table 3 was a single-configuration, Hartree-Fock value (and our value was correspondingly from a single-configuration Dirac-Fock calculation). However, they also performed a fairly extensive multi-configuration Hartree-Fock calculation (with roughly 40 configurations for each level) and obtained a value of GHz, pulling it in closer agreement to the experimental value of GHz. This would suggest that, as has been noted in many of the references given earlier, correlation is likely to be an important contribution to the SMS matrix element. Consequently, our value of GHz, falling in nearly perfect agreement with the experimental value, would appear to be at least partly fortuitous. Pursuing this line of reasoning, we examined the effect of the various modes (AL, OL, etc.) on the SMS Kr + matrix element. We made three pairs of runs (one run requiring a separate calculation for each of the two levels) using the OL, EAL, and AL modes and the results are shown in Table 4. Table 4: Comparison of Energies and SMS results for Kr + for different MCDF modes. Mode E( 4 P ) (cm 1 ) E( 4 D) (cm 1 ) δν SMS (GHz) OL 124, , EAL 132, , AL 130, , Experiment 133, a 120, a ±.008 b a Moore (1971) b Schuessler et al. (1992) Thus, as expected, the EAL and AL modes gave energies in good agreement with experiment. Furthermore, the behavior of the SMS resembles that of dipole transition matrix elements in that the OL method produced a far superior result. The conclusions that are suggested by the results presented here would be that, as can be seen from Table 3, there is a clear trend towards a growing discrepancy between the relativistic and non-relativistic results as Z increases. Furthermore, in all cases the relativistic values are closer to experiment than their non-relativistic counterparts. However, because these results are all from single- or few-configuration calculations, and because the SMS effect is strongly correlation dependent, it would be imprudent to conclude that it is presently more than a trend; more calculations with more configurations are clearly in order. 8

9 Acknowledgments The authors gratefully acknowledge the importance of discussions with and thorough critique of the manuscript by Mike Wilson and Liesl Neale to the results obtained in this article. 9

10 References Accad, Y., Pekeris, C. L., and Schiff, B. (1971). Phys. Rev Bauche, J. (1969). PhD thesis, Paris. Bauche, J. (1974). J. Physique Bauche, J. and Champeau, R. J. (1976). Advances in Atomic and Molecular Physics Bauche, J. and Crubellier, A. (1970). J. Physique Bromley, D. A., editor (1987). Nuclear Radii and Moments of Unstable Nuclei Treatise on Heavy Ion Physics, volume 8, New York. Plenum. Chambaud, G., Levy, B., and Stacey, D. N. (1984). J. Phys. B.: At. Mol. Phys Champeau, R. J. and Verges, J. (1976). Physica 83C Desclaux, J. P. (1975). Comput. Phys. Commun Dyall, K. G., Grant, I. P., Johnson, C. T., Parpia, F. A., and Plummer, E. P. (1989). Comput. Phys. Commun Eichhorn, A., Elbel, M., Kamke, W., Quad, R., and Bauche, J. (1982). Z. Phys. Phys. A: Atoms and Nuclei Fischer, C. F. (1990). Phys. Rev. A Fischer, C. F., Godefroid, M., and Vaeck, N. (1991). Phys. Rev. A (submitted). Fischer, C. F. and Mielczarek, L. S. (1983). J. Phys. B.: At. Mol. Phys Flannery, M. R. and Stewart, A. L. (1963). Proc. Phys. Soc Grant, I., McKenzie, B. J., and Norrington, P. H. (1980). Comput. Phys. Commun Grant, I. P., Mayers, D. F., and Pyper, N. C. (1976). J. Phys. B: At. Mol. Opt. Phys Hughes, D. S. and Eckart, C. (1930). Phys. Rev Hughes, R. H. (1955). Phys. Rev Keller, J. C. (1973). J. Phys. B.: At. Mol. Phys King, F. W. (1989). Phys. Rev. A Labarthe, J. J. (1973). J. Phys. B.: At. Mol. Phys

11 Lorenzen, C. J. and Niemax, K. (1982). J. Phys. B.: At. Mol. Phys. 15 L139. Mårtensson, A. M. and Salomonson, S. (1982). J. Phys. B.: At. Mol. Phys Migdalek, J. and Stanek, M. (1990). Phys. Rev. A Moore, C. E. (1971). Atomic energy levels, Nat. Bur. Stand. (U.S.) Circ No. 35, Vol. I ( 1 H 21 V ). U.S. GPO, Washington, D.C. Parpia, F. A., Grant, I. P., and Fischer, C. F. (1992a). (unpublished). Parpia, F. A., Tong, M., and Fischer, C. F. (1992b). Physical Review A Pekeris, C. L. (1962). Phys. Rev Prasad, S. S. and Stewart, A. L. (1966). Proc. Phys. Soc Rose, S. J., Grant, I. P., and Pyper, N. C. (1978). J. Phys. B: At. Mol. Opt. Phys Schiff, B., Lifson, H., Pekeris, C. L., and Rabinowitz, P. (1965). Phys. Rev. 140 A1104. Schuessler, H. A., Alousi, A., Idrees, M., Li, Y. F., Buchinger, F., Evans, R. M., and Fischer, C. F. (1992). Phys. Rev. A Stone, A. P. (1959). Proc. Phys. Soc Stone, A. P. (1961). Proc. Phys. Soc Stone, A. P. (1963). Proc. Phys. Soc Vadla, C., Obrebski, A., and Niemax, K. (1987). Opt. Commun Vinti, J. P. (1939). Phys. Rev Vinti, J. P. (1940). Phys. Rev Wilson, M. (1978a). Phys. Lett. 65A Wilson, M. (1978b). Physica 95B Wilson, M. (1987). Phys. Lett. A

Calculation of the Isotope Shifts on 5S 1/2 4D 3/2,5/2 Transitions of 87,88 Sr +

Commun. Theor. Phys. (Beijing, China) 37 (22) pp 76 7 c International Academic Publishers Vol. 37, No. 6, June 5, 22 Calculation of the Isotope Shifts on 5S /2 4D 3/2,5/2 Transitions of 87,88 Sr + LI Yong,,2