Relativistic calculations of the lifetimes and hyperfine structure constants in 67 Zn +
Gopal Dixit, H. S. Nataraj, B. K. Sahoo, R. K. Chaudhuri, Sonjoy Majumder
aa r X i v : . [ phy s i c s . a t o m - ph ] O c t Relativistic calculations of the lifetimes and hyperfine structure constants in Zn + Gopal Dixit , H. S. Nataraj , B. K. Sahoo , R. K. Chaudhuri , and Sonjoy Majumder Department of Physics, Indian Institute of Technology-Madras, Chennai-600 036, India Indian Institute of Astrophysics, Bangalore-34, India Max Planck Institute for the Physics of Complex Systems, D-01187 Dresden, Germany (Dated: November 23, 2018)This work presents accurate ab initio determination of the magnetic dipole (M1) and electricquadrupole (E2) hyperfine structure constants for the ground and a few low-lying excitedstates in Zn + , which is one of the interesting systems in fundamental physics. The coupled-cluster (CC) theory within the relativistic framework has been used here in this calculations.Long standing demands for a relativistic and highly correlated calculations like CC can beable to resolve the disagreements among the lifetime estimations reported previously for a fewlow-lying states of Zn + . The role of different electron correlation effects in the determinationof these quantities are discussed and their contributions are presented. I. INTRODUCTION
The quantum information processing (QIP) is one of the interesting areas in physics which isgaining momentum both in theoretical and experimental fronts in the recent years. Mostly, thesingle valence ions particularly the ones with S / ground states are being chosen for QIP studies[1] to encode qubits into the hyperfine levels. These levels are chosen due to their relatively longlifetimes against spontaneous decay rates and long phase coherence because of their small energyseparations. The hyperfine structure studies help us understand the nuclear structure of an atomand its influence on the short range wavefunctions correctly [2].The rapid progress in the development of technology involving laser cooling and ion trappinghas made possible to bring these theoretical ideas to fruition and singly ionized zinc (Zn + ) is oneof the recent important inclusion in that family [3]. Though there are a few studies of radiativelifetimes of Zn + in the literature [4], its hyperfine structures are not studied so far limited to thebest of our knowledge. Here, we have carried out the magnetic dipole ( A ) and electric quadrupole( B ) hyperfine structure studies of Zn + for principle quantum number n=4 states which is wantedfor the QIP studies as mentioned above.Zn is also one of the important elements in astrophysics, especially for the understanding of thepost-main sequence evolution of the chemically peculiar stars, in which Zn is either scarce (if notnon-existing) or over abundant [5]. The high resolution spectra obtained from GHRS onboard Hub-ble Space Telescope has provided vital informations about its abundances [6]. Applications of theradiative transitions of this ion in cosmology, stellar dynamics, interstellar medium, nucleosynthesisetc. have been discussed extensively in the literature [4, 5, 7, 8].It seems from the reported results that, there exits disagreements in the lifetime estimationsamong the experimental measurements and various theoretical calculations. One distinct featureof the lifetime table is the order of the 4 D fine structure states, i.e., the lifetime of the 4 D / stateshould be less than the lifetime of 4 D / state according to both experiments available [9, 10]. Thiswas not found in any of the ab initio studied so far [9, 11, 12, 13]. Our calculated lifetimes for the4 D / and 4 D / states which are reported here, have the same order as well as in good agreementwith the experimental results. II. THEORY
The one-electron reduced matrix elements corresponding to E1, M1 and E2 transitions are givenin these papers [14, 15]. The emission transition probabilities (in sec − ) for the E1, E2 and M1channels from states f to i are given by A E i,f = 2 . × λ (2 j f + 1) S E , (2.1) A E i,f = 1 . × λ (2 j f + 1) S E , (2.2) A M i,f = 2 . × λ (2 j f + 1) S M , (2.3)where S = |h Ψ f | O | Ψ i i| is the transition strength for the operator O (in a.u.) and λ (in(˚A)) isthe corresponding transition wavelength. The lifetime of a particular state is the reciprocal of thetotal transition probability arising from all possible spontaneous electromagnetic transitions fromthe state to all the lower energy levels. τ i = 1 A i . (2.4)The interaction between the electromagnetic multipole moments of the electrons and the elec-tromagnetic field created at the site of the nucleus is termed as hyperfine interaction and thecorresponding Hamiltonian is given by [16] H hfs = X k M ( k ) · T ( k ) , (2.5)where M ( k ) and T ( k ) are the spherical tensor operators of rank k in the nuclear and electronicspaces, respectively. The k =1 and 2 terms of the expansion represent the magnetic dipole andelectric quadrupole interactions, respectively.The diagonal hyperfine interaction constants can be written as [16] A = µ I IJ h γJ J | T ( ) | γJ J i = µ I IJ J J - J J h J || T ( ) || J i , (2.6)and B = 2 Q h γJ J | T ( ) | γJ J i = 2 Q J J -J 0 J h J || T ( ) || J i , (2.7)where I , J are the total angular momentums of nucleus and electrons; µ I and Q are magneticdipole and electric quadrupole moments of the nucleus, respectively. The T ( ) and T ( ) operatorsare defined as T ( ) = X i − ie √ r − i α i · C ( ˆ r i ) (2.8)and T ( ) = X i − er − i C q ( ˆ r i ) , (2.9)where, C kq = p π/ (2 k + 1) Y kq with Y kq being the spherical harmonic functions.In the first-order perturbation theory, the hyperfine energy E hfs ( J ) of the fine-structure state | J M J i is the expectation value of the corresponding hyperfine interaction Hamiltonians in thatstate. The energies corresponding to the magnetic dipole and electric quadrupole hyperfine tran-sition are defined as E M = AK/ , (2.10)and E Q = B K ( K + 1) − I ( I + 1) J ( J + 1)2 I (2 I − J (2 J − , (2.11)where K = 2 h I · J i = F ( F + 1) − I ( I + 1) − J ( J + 1) with F = I + J . Here we have neglectedhigher order hyperfine interactions.The basic formalism of the valence universal coupled-cluster (CC) method was developed morethan two decades before [17, 18, 19, 20] however a suitably relativistic version of this approach hasbeen successfully employed to obtain the various properties accurately in different single valenceatomic systems only recently [21, 22, 23, 24, 25]. Here we just outline the method applied in thiscalculation of the wavefunctions of Zn + accurately.The single valence CC theory extended for the relativistic framework and is based on the no-virtual-pair approximation with Dirac-Fock orbitals [18]. The concept of the common vacuumfor both the closed-shell N and open-shell N + 1 electron systems allows to formulate a directmethod of excitation energies. The dynamical electron correlation effects are introduced throughthe valence-universal wave-operator Ω v [17, 18] for the state with v as the valence orbital is writtenin the normal ordered form as, Ω v = e T { e S v } , (2.12)where cluster operator T represents excitations from the occupied core orbitals of the closed shellsystem Zn ++ and S represent the core-valence and valence-valence excitations. Dominant amongthese correlations are pair correlations and core polarizations. The Dirac-Coulomb Hamiltoniandressed with the excitation cluster operators T and S v are then diagonalized within the model spaceconstructed from the core and valence orbitals to obtain the desired eigenvalues and eigenvectors[20]. In this work, a leading order triple excitations are included in the open shell CC amplitudeevaluation by an approximation that is similar in spirit to CCSD(T) method [26].The expectation value of any operator O can be expressed in the CC method as O = h Ψ v | O | Ψ v ih Ψ v | Ψ v i = h Φ v |{ S v † } e T † Oe T { S v }| Φ v ih Φ v |{ S v † } e T † e T { S v }| Φ v i . (2.13) III. RESULTS AND DISCUSSIONS
We have used Gaussian-type orbitals (GTO) to calculate the DF wavefunctions | Φ DF i as givenin [27] using the basis functions of the form [28, 29, 30] G i,k ( r ) = r k i e − α i r (3.1)where k = 0 , , , , .... for s, p, d, f ..... type orbital symmetries respectively. The large and smallcomponents of the relativistic GTOs satisfy the kinetic balance condition [31]. The exponents aredetermined by the even tempering condition; i.e., for each symmetry exponents are assigned as α i = α β i − i = 1 , , .....N (3.2)where N is the number of basis functions for the specific symmetry. In this calculation, we haveused α = 0 . β = 2 .
99. The number of basis functions used in the present calculationare 32, 32, 30, 25, 20 for l = 0, 1, 2, 3, 4 symmetries, respectively. TABLE I: Radiative lifetimes(ns) for different low-lying states of Zn + .State Experiment Other theories This work4 P / P / S / D / D / We report our calculated lifetime results along with the available calculated and measuredresults in Table I. It is apparent from the table that, there are large disagreements among theearlier results and also the measurements are not very precise. In our previous work (here afterreferred to as paper I [32]), we have presented ionization energies, allowed and forbidden transitionamplitudes of the same system considered in the present work and their astrophysical applicationsare emphasized [32]. Our results in paper I are in excellent agreement with the experimentalmeasurements. With this spirit we have computed the lifetime calculations in the present workwhich show moderately good agreement with the available experimental results. There are tworecent experiments for the lifetime estimations of the fine structure states of 4 D [9, 10] and bothshow that, the lifetime of the 4 D / state is shorter than 4 D / state. However, many of the earliertheoretical calculations in the literature show the opposite trend, whereas, our CCSD(T) results FIG. 1: Decay channels for the first few low-lying excited states of Zn + . The lines of different typescorrespond to different electromagnetic (multipole) transitions. not only show the same trend as that of the experimental results, but also the ratio of their lifetimesis in an excellent agreement with these two measurements. The large transition rate of the 4 d / to 4 p / state (1167893370 sec − ) compared to the allowed transitions from the 4 d / state to thelower energy states i.e. 4 p / (126252430 sec − ) and to 4 p / (642890811 sec − ) made this orderof lifetime, as seen in Fig. 1.The computed values of the magnetic dipole hyperfine structure constants ( A ) for the groundstate and a few low-lying excited states of Zn + are given in Table II. Neither the calculationsnor the measurements of the hyperfine constants A for all the states, except for the groundstate, considered here are available in the literature known to our knowledge. The importantmany-body correlation contributions to the total A h values are included in our work throughrelativistic CC theory. We have used µ I = 0.87547 and I = 5 / A are quite large and vary from (2-430)% among the different low-lyingstates. The core correlation effects are significant inthe 4 p / , 4 d / and 4 d / states; especiallyfor the last two states these effects are larger than the DF contributions. The lowest order paircorrelation and core polarization effects tabulated here highlight their important contributionswhich are comparable to the DF contributions, especially to note is the cancellation effect of corepolarization in the case of 4 p / state. The large effects of S † v ¯ OS v (4.93 MHz for 4 d / and 40.0MHz for 4 s state) are observed in these cases. We have used the expression (2.10) to calculatethe ground state hyperfine energy separation which turns out to be 7018.743 MHz. Panigrahyet al. also have calculated the same using relativistic linked-cluster many body perturbationtheory and get 7.2 GHz [34] which is in good agreement with our result. The hyperfine energy TABLE II: Magnetic dipole hyperfine constant ( A ) of different low-lying states of Zn + in MHz.State DF Core Correlation Pair Correlation Core Polarization Norm Total4 S / P / P / D / D / B ) of different low-lying states of Zn + in MHz.State DF Core Correlation Pair Correlation Core Polarization Norm Total4 P / D / D / separation lies in the microwave region of the electromagnetic spectrum, which suggest that Zn + can be proposed as the new frequency standard in microwave region, however it needs furtherinvestigation about its stability and accuracy of estimation etc.The computed values of the electric quadrupole hyperfine structure constants ( B ) for a fewlow-lying excited states are given in Table III. In this calculation the electric quadrupole momentof the nucleus, Q = 0.150 is used [33]. It may be noted that the effects of pair correlation andcore polarization effects are stronger than the core correlation effects; in particular, the corepolarization effects for the D states are stronger than the DF contributions. IV. CONCLUSION
In this work, we have determined the hyperfine structure constants A and B of the ground stateand a few low-lying excited states in Zn + using the relativistic coupled-cluster theory. We havealso calculated the hyperfine energy separation for the ground state, which is 7018.743 MHz. Thereis no experimental result available for the hyperfine energy separation for the ground state, whichseems to be an important candidate for QIP studies. Also, Zn + can be considered as one of thepromising candidates for the frequency standard in the microwave region. We have also determinedthe lifetimes of the low-lying state sin Zn + , which are in good agreement with experimental results.Especially our calculated lifetimes of 4 D fine structure states explain the same trend as observedin the experiments [9, 10] i.e., the lifetime of 4 D / state is shorter than the 4 D / state, unlikemany other theoretical results which show opposite trend and also the ratio of their lifetimes in ourcalculation is in excellent agreement with the experimental result.This suggests that the relativisticCC method applied in the present work and our numerical approach in obtaining the wavefunctionsof the system considered are more accurate and reliable. V. ACKNOWLEDGMENT
We greatly acknowledge Prof. B. P. Das, Indian institute of Astrophysics, Bangalore andProf. Debashis Mukherjee, Indian Association of Cultivation for Science, Kolkata for the helpfuldiscussions. [1] Ozeri R. et al. 2007
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