Ab initio calculations of oscillator strengths and lifetimes of low-lying states in Mo VI
Narendra Nath Dutta, Gopal Dixit, B. K. Sahoo, Sonjoy Majumder
aa r X i v : . [ phy s i c s . a t o m - ph ] J u l Ab initio calculations of oscillator strengths and lifetimes oflow-lying states in Mo VI
Narendra Nath Dutta , Gopal Dixit , B. K. Sahoo and Sonjoy Majumder Department of Physics and Meteorology, Indian Instituteof Technology-Kharagpur, Kharagpur-721302, India Theoretical Physics Divison, Physical Research Laboratory, Ahmedabad-380009, India (Dated: November 2, 2018)Relativistic coupled-cluster (RCC) calculations have been performed to estimate theelectromagnetic forbidden transition probabilities, oscillator strengths and lifetimesof many low-lying states of five times ionized molybdenum (Mo VI). Contributionsfrom the Breit interaction up to the first order of perturbation have been examined.Our results are in good agreement with the available other reported theoretical andexperimental results. A long lifetime about 4 . s of the first excited state, 4d D / , has been predicted which can be a very useful criteria in the doping process ofthin films. Correlations trends from various RCC terms to the transition amplitudecalculations are discussed. I. INTRODUCTION
Electromagnetic forbidden, both magnetic dipole (M1) and electric quadrupole (E2),transitions of Mo VI are important for temperature and density estimations of tokamakplasmas [1, 2], especially in the collision-radiative model [3, 4]. Long lifetimes of metastablestates are dominated by these forbidden transitions and these states are generally difficult toobserve in the laboratory plasmas due to strong collisions. However, these forbidden tran-sitions of Mo VI have been observed in laboratory in electron spin resonance experiment[5] and therefore, they must be one of the sources of density estimations in astrophysicalplasmas where collisions are very low due to high dilute interstellar medium [6]. Accurate es-timation of abundances of molybdenum in the atmosphere of the evolved stars are importantto understand the stellar nucleosynthesis [7].Hexavelent molybdenum, isoelectronic to rubidium with 4p
4d as ground state configu-ration, is generated by electron impact in the atomic collision process. The electron-impactionization of multiply charged Mo ions, relevant to astrophysics and laboratory plasma re-search, have also been investigated [8]. Recently, Fisker et al. have given the possibilityof the origin of the lightest isotope of molybdenum in proton rich type II supernova [9].The necessity of accurate estimation of allowed dipole transition strengths to find out theirmixing these effects in the dipole forbidden transitions in Mo VI is explicitly discussed byT. Yamamoto [10]. Again, the transition strength between the fine structure states of 4dlavel can reflect the electronic structure of Mo VI in crystal [11].A few calculations have been carried out to study the on electric dipole (E1) transitions inMo VI over the last few decades using the mean-field theory [12, 13]. More recently, J. Reader[15] has estimated the E1 transition probabilities among low-lying states by estimatingtransition strengths in the semiempirical approach with the experimental excitation energies.For this single reference system, Mo VI, we have performed relativistic coupled-cluster(RCC) calculation with single (S), double (D) and partial triple (T) excitations in theframework of Fock space multi-reference (FSMR). Both the excitation energies and transitionprobabilities are determined using this RCC method using which lifetimes of many low-lyingstates are estimated.
II. THEORY AND METHOD OF CALCULATIONSA. Theory
The oscillator strength for E1 transition from | Ψ f i to | Ψ i i is given as f fi = 23 g f ∆ E fi × | D fi | , (2.1)where ∆ E fi is the excitation energy between the upper and lower states and g f = 2 J f + 1is the degeneracy factor of the upper state with total angular momentum J f .The single particle reduced matrix elements for the E1, E2 and M1 transition operatorsare given in [21]. The emission transition probabilities (in sec − ) for the E1, E2 and M1channels from states f to i can be expressed as A E fi = 2 . × λ (2 j f + 1) S E , (2.2) A E fi = 1 . × λ (2 j f + 1) S E , (2.3) A M fi = 2 . × λ (2 j f + 1) S M , (2.4)where S O = |h Ψ f | O | Ψ i i| is the transition strength for the coressponding operator O (ina.u.) and λ (in ˚A) is the corresponding transition wavelength.The lifetime of a particular excited state i can be computed by the reciprocal of the totaltransition probability, P j A ij (in sec − ), arising from all possible states j due to spontaneouselectromagnetic transitions, i.e. τ i = 1 P j A ij . (2.5) B. Fock Space Multi-reference RCC theory
The FSMRCC method is one of the most powerful highly correlated many-body ap-proaches due to its all order structure to account the correlation effects [16]. The FSMRCC,which is mainly meant for multi-reference systems, is used here for the one valence electronand has been described in details elsewhere [16–19]. Here we present the method briefly.We first consider the Dirac-Coulomb Hamiltonian for a closed-shell N electron systemwhich is given by H = N X i =1 h c ~α i · ~p i + βmc + V Nuc ( r i ) i + N X i In the present calculation, the radial wavefunctions of DF orbitals of closed shell Mo VIIare obtained using Gaussian type orbitals (GTO) basis with finite nuclear size as discussedin our earlier paper [22]. We have used universal basis set, where the exponent α i is relatedwith two parameters α and β , same for all the symmetries, expressed as α i = α β i − . (3.1)We have considered α and β as 0.00625 and 2.72, respectively, after obtaining best fit of thebound orbital energies and evaluating the expectation values of different radial functions ( r , r , 1 /r ) generated with GTOs and GRASP2 [23]. In the DF calculations, we have taken22, 20, 17, 15, and 12 number of GTOs for s, p, d, f, and g type symmetries, respectively,to generate the atomic orbitals. In Fig. 1, we have given the relative errors obtained fordifferent orbitals in the calculations of these quantities using the above chosen parameters.Since these errors are very small, it shows that there is a good agreement between results. Weassume that both the bound and continuum orbitals generated using the above parameterswill describe well both inside and outside of the nucleus. Therefore, we have considered allthe orbitals obtained using GTOs for the rest of the calculations.The number of the DF orbitals for different symmetries used in the present calculationis based on convergent criteria of core correlation energy of Mo VII for which it satisfiesnumerical completeness. The number of DF orbitals considered for s, p, d, f, and g type -0.010.000.010.020.030.040.050.060.070.080.090.100.110.120.130.14 FIG. 1: The relative energies and expectation values of r , r and 1 /r of the DF GTO orbitals tothe DF GRASP orbitals. symmetries in the RCC calculations are 12, 11, 10, 9 and 8, respectively; and among them9, 8, 7, 5, and 5 are bound orbitals, respectively, including all the core orbitals. The T amplitudes are first determined by solving the closed shell RCC equations for the closed-shell system (Mo VII), then S amplitudes are solved from the open-shell equations for thesingle-valence states of Mo VI.Table I summarizes the calculated excitation energies (EE) and fine structure splitting(FS) of low-lying excited states and their comparison with the recent experimental results[15]. The average deviation is around 0.5% for EE. We have also presented the contributionfrom the partial triple excitations to the EE (Etriple), which is around 0.3% to the total EE.We have examined the first order excitation energy corrections due to Breit interaction usinglarge scale relativistic CI calculations. Maximum contribution is coming for 5s state, whichis around +2%, whereas, contributions to 5p and 4f states are around +0 . 4% and − . cm − consistent with the result obtained by TABLE I: Excitation energies (EE) and fine structure splitting (FS) (in cm − ) of Mo VI and theircomparison with experimental results. Contribution from absolute value of partial triple excitationto the EE (Etriple) are also presented.EE FS EEtripleState CC Exp. a CC Exp. a D / D / S / P / P / F / F / D / D / S / P / P / a Ref. [15]. Pan and Beck [14]. Since, the contributions due to Breit interaction are relatively small, wedo not consider them here selfconsistently to evaluate wavefunctions.Since the transition rate is proportional to the square of the transition amplitude, there-fore precise description of the wavefunction is necessary due to one order higher dependanceon wavefunctions than energy. In Table II, we compare the E1 transition amplitude in bothlength and velocity gauges for few transitions. We find a good agreement between them,which is one of the characteristics to judge the accuracy of the wavefunctions.In Table III, we compare our ab initio oscillator strength values correspond to E1 tran-sitions with the recent semi-empirical calculations by Pan and Beck [14] and by Reader[15]. Reader has obtained wavefunctions using fitting parameters by comparing calculatedand experimental energies, whereas, Pan and Beck have used relativistic CI method for theavailable transitions. Our calculated values of oscillator strength for the 4d D / → F / TABLE II: Absolute values of E1 transition amplitude in length (D l ) and velocity (D v ) gauges forMo VI. TermUpper Lower D l D v P / → D / P / → S / P / → D / P / → S / F / → D / F / → D / F / → D / D / → P / D / → P / D / → P / S / → P / S / → P / P / → D / P / → D / P / → S / P / → D / P / → D / P / → S / transition at the DF level given in the table, agrees well with similar calculations by Zilitis[13], 1.023.Weighted oscillator strengths correspond to E1 transitions are presented in Table IV.Here we have used the length gauge values of E1 transition amplitudes and our calculatedwavelengths. All these transitions, fall in ultraviolet and visible regions, are useful forastrophysical observations and may be for laboratory researches. According to Cowan [24],if the initial states are dominated by D and final states are dominated by P , the oscillatorstrength ratio of D / → P / , D / → P / and D / → P / transitions in a given TABLE III: Oscillator strengths for E1 transitions in length form and their comparison with earlierresults. Term Present calculations Other calculationsUpper Lower DF CC [15] [14]4d D / → F / D / → F / multiplet are 6:5:1, which we find the same for 4d D → P transitions.In Table V, we present M1 and E2 transition probabilities and their corresponding wave-lengths. However most of the transitions come in ultraviolet region, there are few transitionsfall in infrared region. Though these transitions produce weak lines but they are importantparameters in astrophysical studies. As expected, transition probability for E2 transitionscome greater in value than that of M1 transitions except for the transitions fall in infraredregion.In Table VI, the contributions due to the different correlation terms like core-correlation,pair-correlation, core-polarization and important two-body contributions are estimated forfew transitions to highlight the effect of correlations. Significant correlation contributionsfrom the higher order core-polarization, like S † (0 , f ¯ OS (0 , i , are noticeable compared to thelowest order contributions. Also, contributions from two-body correlations are almost com-parable for most of the cases. For the M1 transition, 4d D / → D / , there arecancelation observed among different correlation effects. The contribution comes from core-polarization term is more compared to other term. Even in the case of 5s S / → D / M1 transition, the Dirac-Fock contribution is almost canceled by the lowest order of core-polarization, which makes the core-corelation effect more dominant to the total value oftransition matrix element.Table VII summarizes the calculated lifetime of the low-lying excited states. Recent cal-culations of the lifetimes of 5p P / and 5p P / states by Zilitis [13] are also comparedhere. Here we can see that the lifetime of the 4d D / state comes in the order of second,which suggest that Mo VI can be used for uniform doping in thin film. The lifetime of the 5s S / state is found to be of the order of microsecond due to only forbidden transition contri-butions. Lifetime of the 5 d states are larger than the 4 f states due to strong contributions0 TABLE IV: Transition wavelengths (in nm) and weighted oscillator strengths (gf) correspondingto electric dipole (E1) transitions of Mo VI.TermUpper Lower λ exp. gf5p P / → D / P / → S / P / → D / P / → D / P / → S / F / → D / F / → D / F / → D / D / → P / D / → P / D / → F / D / → P / D / → F / D / → F / S / → P / S / → P / P / → D / P / → S / P / → D / P / → S / P / → D / P / → D / P / → S / P / → D / P / → D / P / → S / TABLE V: Transition wavelengths (in nm) and transition probabilities corresponding to electricquadrupole (E2) and magnetic dipole transitions (M1) (in sec − ) of Mo VI.TermUpper Lower λ exp. A E A M D / → D / × − × − S / → D / × × − S / → D / × P / → P / × − F / → P / × F / → P / × × − F / → P / × F / → F / × − × − D / → D / × × − D / → D / × × D / → S / × × − D / → D / × D / → D / × × D / → S / × D / → D / × − × − S / → D / × × − S / → D / × S / → S / S / → D / × × − S / → D / × P / → P / × − P / → P / × × P / → F / × P / → P / × × P / → P / × P / → F / × × − P / → F / × P / → P / × − × − TABLE VI: Explicit contributions from the CCSD(T) calculations to the absolute magnitude oftransition amplitudes. Dirac-Fock Core-corr. Pair-corr. Core-polar. Core-polar. Two-body Total(lowest) (higher) contr.5p(1/2) → → → → → → TABLE VII: Radiative lifetimes (in sec.) for different low-lying states of Mo VI.Term Present calculations Other calculations a D / S / × − P / × − × − P / × − × − F / × − F / × − D / × − D / × − a Ref. [13]. from the allowed dipole transitions 4f F → D . IV. CONCLUSION Forbidden transition probabilities among the low-lying states of Mo VI relevant for astro-and plasma physics are calculated using highly correlated relativistic coupled-cluster methodfor the first time in literature to the best of our knowledge. The lifetime of the 4d D / state is found to be around 5 second, which will be useful in many physical processes.Contributions of different correlation terms are discussed and found strong effect from higherorder core-polarization. In the near future, present work will motivate experimentalists toverify our results due to its importance in many areas in physics.3 V. ACKNOWLEDGMENT We are grateful to Prof B P Das and Dr Rajat K Chaudhuri , Indian Institute of Astro-physics, Bangalore for providing the CC code. One of us (Narendra) would like to recognizethe support of Council of Scientific and Industrial Research (CSIR), India. [1] Feldman U 1981 Phys. Scr. Astrophys. J. Nucl. Fusion J. Phys. B Inst. Phys. Conf. Ser. Astrophysics J. 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