Dynamic polarizabilities for the low lying states of Ca+
aa r X i v : . [ phy s i c s . a t m - c l u s ] A p r Dynamic polarizabilities for the low lying states of Ca + Yong-Bo Tang , , Hao-Xue Qiao , Ting-Yun Shi and J. Mitroy Department of Physics, Wuhan University, Wuhan 430072, P. R. China State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, P. R. China ∗ and School of Engineering, Charles Darwin University, Darwin NT 0909, Australia (Dated: April 16, 2013)The dynamic polarizabilities of the 4 s , 3 d and 4 p states of Ca + , are calculated using a relativisticstructure model. The wavelengths at which the Stark shifts between different pairs of transitionsare zero are computed. Experimental determination of the magic wavelengths can be used toestimate the ratio of the f d J → p J ′ and f s / → p J ′ oscillator strengths. This could prove valuablein developing better atomic structure models and in particular lead to improved values of thepolarizabilities needed in the evaluation of the blackbody radiation shift of the Ca + ion. PACS numbers: 31.15.ac, 31.15.ap, 34.20.Cf
I. INTRODUCTION
The dynamic polarizability of an atom or ion gives ameasure of the energy shift of the atom or ion when im-mersed in an electromagnetic field [1–3]. For any givenstate, one can write∆ E = − α d ( ω ) F , (1)where α d ( ω ) is the polarizability of the quantum stateat frequency ω , and F is a measure of the strength ofthe AC electromagnetic field. The value of the dynamicpolarizability in the ω → ∗ Electronic address: [email protected] rely on a precise determination of the strength of a staticelectric field or the intensity of a laser field. This makesit possible to determine the magic wavelengths to a highdegree of precision [9, 18–22].There have been a number of theoretical studies ofthe properties of the low-lying Ca + ion [23–29] by 3 dif-ferent research groups. One of these groups [24] used anon-relativistic approach while the other two groups usedexplicitly relativistic formulations [26–29]. One of thesingular features about the relativistic calculations aresignificant differences between predictions of the prop-erties of spin-orbit doublets. The relativistic all-ordermany body perturbation theory method predicts rela-tively small non-geometric differences between the linestrengths of the 3 d J and 4 p J spin-orbit doublets [23, 29].The relativistic coupled cluster approach typically givesmuch larger differences [26]. One of the secondary aimsof the present work is to shed light on these differences.The present manuscript reports calculations of the dy-namic polarizabilities of the five lowest states of Ca + .The Hamiltonian used is a fully relativistic version ofa semi-empirical fixed core potential that has been suc-cessfully applied to the description of many one andtwo electron atoms [30–33]. While there are many dif-ferences in the technical detail, the underlying philoso-phy and the effective Hamiltonian for the valence elec-tron are essentially the same once the relativistic mod-ifications are taken into account. Magic wavelengthsfor the 4 s → d / , / and 4 s → p / , / transitionsare given. The dynamic polarizability of the groundCa + (4 s ) state is dominated by the 4 s → p J transitionsand its accuracy is largely dependent on the accuracyof the transition matrix elements connecting the 4 s and4 p J states. The description of transitions involving theCa + (3 d ) state is complicated by the effect that the 3 d electrons have on the core electrons. The 3 d orbitals havethe smallest h r i expectation values of any of the valenceelectrons and this does distort the wavefunctions for theoutermost core electrons [34, 35]. One consequence ofthis are greater uncertainties in the calculation of transi-tion matrix elements involving the 3 d J states [27, 29, 35]All results reported in this paper are given in atomicunits with the exception of the lifetimes which are givenin seconds. The value adopted for the speed of light is c = 137 . II. FORMULATION AND ENERGIESA. Solution of the Dirac-Fock equation for closedshell atomic system
The calculation methodology is as follows. The firststep involves a Dirac-Fock (DF) calculation of the Ca ground state. The DF calculation begins with the equa-tion N X i H Di + N X i 1) + V ( r i ) . (3)where c is the speed of light, p is the momentum operator,and α and β are the Dirac matrices [36].The orbitals of the DF wave function, ψ ( r ), can bewritten as ψ ( r ) = 1 r (cid:18) g nκ ( r )Ω κm (ˆ r ) if nκ ( r )Ω − κm (ˆ r ) (cid:19) , (4)where g nκ ( r ) and f nκ ( r ) are the large and small compo-nents, Ω κm (ˆ r ) and Ω − κm (ˆ r ) correspond to the angularcomponents. The radial Dirac equation for an orbitalcan be expressed schematically as (cid:18) V ( r ) + V DF ( r ) − c ( ddr − κr ) c ( ddr + κr ) − c + V ( r ) + V DF ( r ) (cid:19) (cid:18) g nκ ( r ) f nκ ( r ) (cid:19) = ε (cid:18) g nκ ( r ) f nκ ( r ) (cid:19) , (5)where V DF is called the Dirac-Fock potential, and V ( r )is the interaction potential between the electron and thenucleus. A Fermi nuclear distribution approximation isusually adopted for many-electron atomic system.The single particle orbitals are written as linear combi-nations of analytic basis functions and so the method ofRoothaan [37, 38] is used to recast the DF equations intoa set of matrix equations. The functions chosen are B-splines with Notre-Dame boundary conditions [39]. Thelarge and small components are expanded in terms ofa B-spline basis of k order defined on the finite cavity[0 , R max ], g nκ ( r ) = N X i =1 C g,ni B i,k ( r ) (6) f nκ ( r ) = N X i =1 C f,ni B i,k ( r ) . (7) TABLE I: Theoretical and experimental energy levels (inHartree) for some of the low-lying states of Ca + . The en-ergies are given relative to the energy of the Ca core. Theexperimental data were taken from the NIST tabulation [43].Level DF DFCP Experiment [43]4 s / -0.4166315 -0.4362777 -0.43627763 d / -0.3308695 -0.3740834 -0.37408273 d / -0.3307597 -0.3738074 -0.37380624 p / -0.3099986 -0.3214966 -0.32149664 p / -0.3090889 -0.3204818 -0.32048105 s / -0.1933158 -0.1983486 -0.19858764 d / -0.1687383 -0.1751536 -0.17729894 d / -0.1686641 -0.1750622 -0.17721145 p / -0.1567656 -0.1603178 -0.16046885 p / -0.1564329 -0.1600612 -0.1601123 The finite cavity is set as a knots sequence, { t i } , satis-fying an exponential distribution [40, 41]. The specificsof the grid were that R max = 60 a and 50 B-splines oforder k = 7 were used to represent the single particlestates. Using the Galerkin method and MIT-bag-modelboundary conditions [39], the DF equations were solve byiteration until self-consistency was achieved. The single-electron orbital (Koopmans) energies of the closed shellCa ion agreed with those computed with the GRASP92program [42] to better than 10 − a.u. B. Polarization potential The effective potential of the valence electron with thecore is then written V core = V dir ( r ) + V exc ( r ) + V pol ( r ) . (8)The direct and exchange interactions of the valenceelectron with the DF core were calculated exactly.The ℓ -dependent polarization potential, V pol , was semi-empirical in nature with the functional form V pol ( r ) = − X ℓj α core g ℓj ( r )2 r | ℓj ih ℓj | . (9)The coefficient, α core is the static dipole polarizabilityof the core and g ℓj ( r ) = 1 − exp (cid:0) − r /ρ ℓ,j (cid:1) is a cutofffunction designed to make the polarization potential fi-nite at the origin. The static dipole polarizability corewas set to α core = 3 . 26 a.u. [29]. The cutoff parame-ters, ρ ℓ,j were tuned to reproduce the binding energiesof the ns ground state and the np J , nd J excited states.Values of the cutoff parameters are ρ , / = 1 . a , ρ , / = 1 . a , ρ , / = 1 . a , ρ , / = 1 . a ,and ρ , / = 1 . a . The cutoff parameters for ℓ ≥ a . Table I givesthe calculated B-spline and experimental energies com-ing from [43]. The calculations with the core-polarizationpotential are identified as the Dirac-Fock plus core polar-ization (DFCP) model. Differences between DFCP and TABLE II: Comparison of the electric dipole (E1), electricquadrupole (E2) reduced matrix elements of several interestedstates of the Ca + ion.Transition DFCP MBPT-SD RCCDipole4 s / − p / s / − p / s / − p / s / − p / d / − p / d / − p / d / − p / d / − p / d / − p / d / − p / d / − f / d / − f / d / − f / p / − s / p / − d / p / − d / p / − d / s / − d / s / − d / experimental energies mostly occur in the fourth digitafter the decimal point.One of the interesting aspects of Table I concerns thespin-orbit splitting of the 4 p J and 5 p J states. The polar-ization potential parameters ρ , / and ρ , / were tunedto give the correct spin-orbit splitting of the 4 p J states.Making this choice resulted in the spin-orbit splittingsfor the 5 p J states also being very close to experiment. III. TRANSITION MATRIX ELEMENTS ANDASSOCIATED QUANTITIESA. Reduced Matrix Elements The dipole matrix elements were computed with amodified transition operator [30, 46, 47], e.g. r C = r C − (cid:0) − exp( − r /ρ ) (cid:1) / α core r C r (10)The cutoff parameter, ρ used in Eq. (10) was set to ρ =( ρ ℓ a ,j a + ρ ℓ b ,j b ) / a, b refer to the initial and finalstates of the transition.The static quadrupole polarizability of the Ca coreis needed for the calculation of the lifetimes of the 3 d J states. It was set α q, core = 6 . 936 a.u. [48].There have been a number of previous calculations ofreduced matrix elements and polarizabilities for the low-lying states of Ca + . The semi-empirical configuration in-teraction plus core polarization (CICP) can be regarded TABLE III: Comparison of the line strengths ratios for tran-sitions involving various spin-orbit doublets. The notation4 s / − p / / means the line strength ratio defined by di-viding 4 p / line strength by the 4 p / line strength.Transition DFCP MBPT-SD RCC4 s / − p / / s / − p / / d / − p / / p / − d / / s / − d / / p / − d / / p / − d / as a non-relativistic predecessor of the present calcula-tion [24, 30]. Another method used is the relativisticall-order single-double method where all single and dou-ble excitations of the Dirac-Fock (DF) wave function areincluded to all orders of many-body perturbation theory(MBPT-SD) [27, 29, 49]. There have also been calcula-tions using the relativistic coupled cluster (RCC) method[26]. The RCC and MBPT-SD approaches have manycommon features [50–52]. Atomic parameters computedusing the RCC approach have on a number of occasionshad significant differences with independent calculations[29, 53–55].The reduced matrix elements between the various lowlying states are the dominant contributor to the polar-izabilities of the 4 s , 3 d and 4 p levels. These are givenin Table II and compared with the results from otherrecent calculations. The ratio of line strengths for spin-orbit doublets is also interesting to tabulate since theycan reveal the extent to which dynamical effects (as op-posed to geometric effects caused by the different angularmomenta) are affecting the matrix elements. Some linestrength ratios are given in Table III.The variation between the DFCP, MBPT-SD and RCCmatrix elements listed in Table III does not exceed 5%.The DFCP matrix elements are usually closer to theMBPT-SD calculations than the RCC matrix elements.A better indication of the differences between the DFCP,MBPT-SD and RCC calculations is gained by examina-tion of the line strength ratios listed in Table III. TheDFCP line strength ratios are within 1% of the valuesthat would be expected simply due to the angular mo-mentum factors alone. The ratios are in very good agree-ment with the MBPT-SD ratios. It should be noted,that the line strength ratios for the resonant transitionof potassium have been measured to be very close to 2.0[17] and DFCP and MBPT-SD calculations also predictline strength ratios very close to 2.0 [17, 56].By way of contrast, RCC matrix element ratios ex-hibit about 4% differences from the geometric ratios. Onewould expect the RCC matrix element ratios to be muchcloser to the MBPT-SD ratio given the close formal sim-ilarities between the RCC and MBPT-SD approaches.The RCC matrix element ratios listed in Table III alsoshow significant differences from the geometric ratio forthe 4 s → d J transitions. The DFCP and MBPT-SDratios lie within 1% of the geometric ratios. It should benoted that a similar situation exists for the 5 s − d / / line strength ratios of Sr + with RCC calculations exhibit-ing much larger differences due to non-geometric effectsthan other calculations [57]. The feature common to theDFCP and MBPT-SD methods is that they use large B-spline basis sets and calculated quantities are expectedto be independent of basis set effects. One possible causefor the different RCC matrix element ratios lies in thegaussian basis set used to represent virtual excitations inthe RCC calculation. This point will be addressed laterwhere polarizabilities are discussed. TABLE IV: Lifetime of the 3 d / and 3 d / levels of Ca + (insec). The 3 d / : 3 d / lifetime ratio is also given.Source τ d / τ d / RatioDFCP 1.143(1)(s) 1.114(1) 1.0260MBPT-SD [29] 1.196(1)(s) 1.165(11) 1.0266RCC [26] 1.185(7) 1.110(9) 1.0675MCHF [34] 1.160 1.140 1.0175Experiment [23] 1.176(11) 1.168(7) 1.007(15)Experiment [58] 1.17(5) 1.09(5) 1.073(90)Experiment [59] 1.064(17)Experiment [60] 1.111(46) 0.994(38) 1.118(80) B. Lifetimes The two most important lifetimes for the Ca + clock[61–65] are the lifetimes of the 3 d J and 4 p J levels.The 3 d J states decay to the ground state in an electricquadrupole transition with lifetime of about 1.1 sec [23].The 4 p J states experience electric dipole transitions toboth the 3 d J and 4 s states. Table IV gives the lifetimesof the 3 d J states while Table V gives the lifetimes of the4 p J states. All DFCP lifetimes were computed using ex-perimental energy differences.The most recent experiment for the 3 d J lifetimes givea ratio of 1 . ± . 015 sec for the 3 d / and 3 d / states.This suggests that the 4 s → d J matrix element ra-tios should be close to the values expected from angularmomentum coupling considerations. Older experiments[58, 60] give ratios further from unity, but in these casesthe uncertainties are much larger.The lifetimes of the 4 p J states depend on two transi-tions, these are the 4 s -4 p J and 3 d J ′ -4 p J transitions, withthe 4 s -4 p J transition being the most important. The life-times and branching ratios for the 4 p J states are given inTable V. It not possible to reconcile the theoretical andexperimental lifetimes at the 1% level. The two most re-cent experiments [66, 67] gave lifetimes that are 2% largerthan the DFCP lifetimes and 3% larger than the MBPT-SD lifetimes. Older Hanle effect experiments [68, 69] gave TABLE V: Lifetimes (in nsec.) of the 4 p and 4 p states.The 4 p : 4 p lifetime ratio is also given. The quantity R gives fraction of the total decay rate arising from the indicatedtransition.Level DFCP MBPT-SD RCC Expt.[29] [26]4 p ( ns ) 6.94(1) 6.88(6) 6.931 7.098(20) [66]7.07(7) [67]4 p ( ns ) 6.75(1) 6.69(6) 6.881 6.926(19) [66]6.87(6) [67]6.72(2) [68]6.61(30) [69] R (4 p − s ) 0.9324 0.9374(74) R (4 p − d ) 0.0676 0.0626(5) R (4 p − s ) 0.9313 0.9340 0.9350(62) 0.9347(3) [70] R (4 p − d ) 0.0069 0.00667 0.00666(4) 0.00661(4) [70] R (4 p − d ) 0.0617 0.0593 0.0583(4) 0.0587(2) [70]Ratio 1.0281 1.0284 1.0073 1.025(3) [66]1.029(14) [67] lifetimes closer to the MBPT-SD and DFCP lifetimes.Measurements of the branching ratios of the 4 p / stateyield a picture where the MBPT-SD calculations largelyagree with experiment while the DFCP tends to overes-timate the contributions of the decays to the 3 d J levels.Another area of partial agreement between theory andexperimental occurs for the 4 p / : 4 p / lifetime ratio.The DFCP, MBPT-SD and experimental ratios rangefrom 1.025 to 1.030, with the RCC calculation again pro-viding an outlier at 1.0073. IV. POLARIZABILITIESA. Static Polarizabilities The static dipole and quadrupole polarizabilities arecalculated by the usual sum-rule α ( ℓ ) = X i f ( ℓ ) gi ε gi (11)where the f ( ℓ ) gi are the absorption oscillator strengths and ε gi is the excitation energy of the transition. Static dipolepolarizabilities for the 4 s , 4 p J and 3 d J states are listedin Table VI. All polarizabilities were computed using ex-perimental energy differences.The most important polarizability is that of the 4 s ground state and there is only a 1% variation betweenthe DFCP, MBPT-SD and CICP static dipole polariz-abilities. The DFCP polarizability is smaller than theMBPT-SD polarizability because the DFCP 4 s − p J matrix elements are smaller. The RCC calculation ofthe dipole polarizability is the clear outlier at 73.0 a.u.[26]. The good agreement between the DFCP, CICP and TABLE VI: Dipole and quadrupole polarizabilities (in a.u.) for low-lying states of the Ca + ion. Non-relativistic quadrupolepolarizabilities are not given for states with ℓ > 0. The RCC-STO results are those from Ref. [26] that used a Slater typeorbital basis to represent virtual excitations. α (0)1 α ( t )1 α State DFCP Others DFCP Others DFCP Others4 s / f -sums [72] 906(5) RCC [45]74.3 RCC-STO [26]4 p / − − p / − − d / − − − − − d / − − − − − − − MBPT-SD polarizabilities does not necessarily imply a1% reliability in these polarizabilities since the calcu-lations give lifetimes for the 4 p J states that are 2-3%smaller than experiment.The variation between the DFCP, MBPT-SD andCICP estimates of the 3 d J state polarizabilities do notexceed 1.0 a.u. The difference in the polarizabilities forthe two members of the spin-orbit doublet is only 0.2 a.u.The polarizabilities of the 4 p J states are close to zerowith the polarizability of the 4 p / state being about 1.8a.u. larger than the polarizability of the 4 p / state.The polarizability is small because the downward transi-tions to the 4 s / and 3 d J states have negative oscillatorstrengths which result in cancellations in the oscillatorstrength sum. This is evident in Tables VII and VIIIwhich show the breakdown of the different contributionsto the polarizabilities from the oscillator strength sumrule.The comparisons of the polarizabilities suggest thatthe basis set used in the RCC calculations [26] could beimproved. The recommended results for the RCC calcu-lation are those computed with the gaussian basis. How-ever, RCC calculations performed using a Slater typeorbital basis [26] give polarizabilities that are in muchbetter agreement with the MBPT-SD and DFCP polar-izabilities. TABLE VII: The contributions of individual transitions tothe polarizabilities of the 4 s / and 4 p / states at the magicwavelengths. The numbers in brackets are uncertainties inthe last digits of the energy or wavelength calculated by in-troducing 2% uncertainties into the most important matrixelements. ω (a.u.) 0 0.0659561(11247) 0.1152981(4) 0.1238091(303) λ (nm) ∞ s / p / − − p / p / − p / − p / s / − − s / − d / − d / − − B. Dynamic polarizabilities and magic wavelengths The dynamic dipole polarizability of a state at photonenergy ω is defined α ( ω ) = X i f (1) gi ε gi − ω (12) TABLE VIII: The contributions of individual transitions to the polarizabilities of the 4 s / and 4 p / states at the magicwavelengths. These results assume non-polarized light. The numbers in brackets are uncertainties in the last digits calculatedby assuming certain matrix elements have ± 2% uncertainties. ω (a.u.) 0 0.0663204(11651) 0.1149923(4) 0.1232650(511) 0.0677517(11210) 0.1151251(3) λ (nm) ∞ s / p / − − − p / p / − p / − − p / Average m j = 1 / m j = 1 / m j = 1 / m j = 3 / m j = 3 / s / − − − s / − d / − d / d / − d / − − − ( a . u . ) (a.u.) FIG. 1: (color online) Dynamic polarizabilities of the 4 s / and 4 p / states of the Ca + ions. Magic wavelengths areidentified by arrows. The dipole polarizability has a tensor component forstates with states with J > / 2. This can be written α T1 ( ω ) = 6 (cid:18) J g (2 J g − J g + 1)6( J g + 1)(2 J g + 3) (cid:19) / × X J i ( − J g + J i (cid:26) J g J i J g (cid:27) f (1) gi ε gi − ω (13) TABLE IX: Pseudo-spectral oscillator strength distributionused in the computation of the dynamic polarizability of theCa core. Energies are given in a.u.. i ε i f i The polarizability for a state with non-zero angular mo-mentum J depends on the magnetic projection M g : α ,M g = α + α T M g − J g ( J g + 1) J g (2 J g − . (14)The dynamic polarizabilities includes contributionsfrom the core which is represented by a pseudo-oscillatorstrength distribution [31, 73, 74] which is tabulated in Ta-ble IX. The distribution is derived from the single particleenergies of a Hartree-Fock core. Each separate ( n, ℓ ) levelis identified with one transition with a pseudo-oscillatorstrength equal to the number of electrons in the shell.The excitation energy is set by adding a constant to theKoopmans energies and adjusting the constant until thecore polarizability from the oscillator strength sum ruleis equal to the known core polarizability of 3.26 a.u. Thecore polarizabilities of any two states effectively canceleach other when the polarizability differences are com-puted.The dynamic polarizabilities for the 4 s / and 4 p / states of Ca + are shown in Figure 1. The first magic ( a . u . ) (a.u.) s p , |m j | = 3/2 p , |m j | = 1/2 , |m j | = 3/2 , |m j | = 1/2 = 0.1232650 = 0.1151250 ( a . u . ) (a.u.) FIG. 2: (color online) Dynamic polarizabilities of the 4 s / and 4 p / states of Ca + . Magic wavelengths are identified byarrows. ( a . u . ) (a.u.) , |m j | = 5/2 , |m j | = 3/2 , |m j | = 1/2 , Average FIG. 3: (color online) Dynamic polarizabilities of the 4 s / and 3 d / states of Ca + . Magic wavelengths are identified byarrows. wavelength occurs at ω = 0 . p / -3 d / tran-sition. Magic wavelengths are identified at λ = 690 . s -4 p J states. The 368.015 nm magic , |m j | = 5/2 , |m j | = 3/2 , |m j | = 1 /2 , Average ( a . u . ) (a.u.) FIG. 4: (color online) Dynamic polarizabilities of the 4 s / and 3 d / states of Ca + . Magic wavelengths is identified bycircles and arrows. wavelength occurs near the energy for the 4 p / − s / transition. The dominant contributions to polarizabili-ties at the magic wavelengths are listed in Table VII. The4 s polarizability is dominated by the 4 s / -4 p J transi-tions with the next largest contribution coming from thecore. However, the 4 p / polarizability has significantcontributions from the transitions to the 4 s , 5 s and 3 d / states. A magic wavelength experiment would give infor-mation about the 4 p / state, but would not give detailedinformation about any individual matrix element. Anexperiment that measured all three magic wavelengthscould conceivably be able to extract information aboutindividual line strengths, however it should be noted thattwo of the transitions are in the ultraviolet.The dynamic polarizabilities of the 4 s / and 4 p / states of Ca + are shown in Figure 2. These figures as-sume non-polarized light. Figure 2 only has two magicwavelengths below ω = 0 . 125 a.u. Transitions to the ns / states make no contribution to the 4 p / state po-larizability. This is evident from Table VIII which de-tails the breakdown of different transitions to the polar-izability. The magic wavelength at 395.775 nm for the4 p / ,m =3 / magnetic sub-level can give an estimate ofthe contribution to the np / polarizability arising fromexcitations to the nd J levels.The 4 s / and 3 d / polarizabilities are shown in Fig-ures 3 and 4. The 3 d / ,m polarizabilities are shown forall magnetic sub-levels and also for the average polariz-ability. Magic wavelengths occur when the photon energygets close to the excitation energies for the 3 d / → p J transitions and the 4 s / → p J transitions. Figure 3shows the 4 s / and 3 d / polarizabilities at photon en-ergies between 0.02 and 0.07 a.u. Precise values of themagic wavelengths and the breakdown of the polarizabil-ity into different components can be found in Table X.Two of the magnetic sub-levels have magic wave-lengths at infrared frequencies, namely λ = 1338 . d / po-larizability are dominated by the 3 d / → p / transi-tion which constitutes about 88% of the polarizability.The measurement of these magic wavelengths provides amethod to determine the f s / → p J to f d / → p / oscil-lator strength ratios. Suppose all the remaining compo-nents of the 3 d / polarizability can only be estimated toan accuracy of 10%. The overall net uncertainty in theremaining terms would be less than 1.5%.There are also an additional magic wavelengths thatcan potentially be measured. The 4 s dynamic polar-izability goes through zero as the wavelength passesthrough energies needed to excite the 4 s → p / and4 s → p / transitions. Figure 4 shows the polarizabili-ties for the 4 s and 3 d / at energies near the 4 s → p J excitation energies. The 3 d / polarizabilities are typi-cally small in magnitude in this wavelength range. Themagic wavelength arises more from the the cancellationof the 4 p / and 4 p / contributions to the 4 s dynamicpolarizability than from the cancellation between the 4 s and 3 d / dynamic polarizabilities. Measurement of themagic wavelength here is in some respects in analogousto a measurement of the longest tune-out wavelengthfor neutral potassium [56]. Zero field shift wavelengthsmeasured in the spin-orbit energy gap of the resonanttransition are strongly dominated by the large and op-posite polarizability contributions of the two members ofthe spin-orbit doublet [56, 75]. This makes it possibleto accurately determine the oscillator strength ratio, i.e. f s → p / : f s → p / , of the two transitions comprising thespin-orbit doublet. ( a . u . ) (a.u.) , |m j | = 3/2 , |m j | = 1/2 Average FIG. 5: (color online) Dynamic polarizabilities of the 4 s / and 3 d / states of Ca + . Magic wavelengths are identified byarrows. Table XI identifies the magic wavelengths associatedwith the 4 s → d / energy interval. The situation hereis similar to the situation for the 4 s → d / magic wave- lengths. However, there are three magic wavelengths inthe infrared region of the spectrum. This transition hasan additional magic wavelength since the 3 d / ,m =1 / state, unlike the 3 d / ,m =1 / state, also undergoes un-dergoes a transition to the 4 p / state. The polarizabil-ity difference in the 0.02 to 0.07 a.u. energy range isplotted in Figure 5. The 3 d / polarizability is domi-nated by the 3 d / → p J transition and a magic wave-length measurement can be used to make an estimateof the 3 d / → p J line strength relative to the 4 s dy-namic polarizability. The 3 d / ,m =1 / polarizability at850.335 nm has large contributions from the 4 p / and4 p / states since it lies between the excitation ener-gies of these of states. Measurement of the 850.335nm and 1308.590 nm wavelengths together would giveestimates of the 3 d / → p / line strengths and the f d / → p / : f d / → p / ratio. A measurement of themagic wavelengths in the vicinity 395 nm provides wouldpermit a determination of the f s → p / : f s → p / ratio. C. Uncertainties An uncertainty analysis has been done for all the magicwavelengths presented in the preceding sections. Thisanalysis was aimed at making an initial estimate of howuncertainties in the matrix elements of the most impor-tant transitions would translate to a shift in the magicwavelengths. The primary purpose of the uncertaintyanalysis is to define reasonable limits to help guide anexperimental search for the magic wavelengths identifiedin this paper.In the case of the 4 s → p J polarizability differences,the 4 s → p J , 4 p J → s , 4 p J → d J and 4 p J → d J matrix elements were all changed by 2% and the magicwavelengths recomputed. The matrix elements involvingthe different spin-orbit states of the same multiplet wereall given the same scaling. A variation of ± 2% was cho-sen by reference to the difference of the DFCP matrixelements with the experimental or the MBPT-SD ma-trix elements. The estimate of a 2% uncertainty in the4 s → p J matrix element can be regarded as a conserva-tive estimate.The 4 s → d J polarizability difference is predomi-nantly determined by the 4 s → p J and 3 d J → p J matrix elements. So variations of ± 2% in these two tran-sitions were used in determining the uncertainties in themagic wavelengths.There are a number of magic wavelengths which arerelatively insensitive to changes in the matrix elementsof a multiplet. One of these wavelengths is the 850 nmwavelength for the 4 s − d / interval and the others arethe magic wavelengths near 395 nm. These wavelengthsarise due to cancellations in the polarizabilities due totwo transitions of a spin-orbit doublet. In the case of the850 nm magic wavelength, the relevant transitions arethe 3 d / → p J transitions.The sensitivity of the magic wavelengths near 395 nmto changes in the transition matrix elements depends onthe overall size of the polarizabilities of the 4 p J and 3 d J levels. When these are large due to transitions otherthan the 4 s → p J transition, then the 395 nm magicwavelength shows higher sensitivity to the changes in thematrix elements. However, the net change in the magicwavelengths for 2% changes in the matrix elements isabout 0.001 nm for the 4 s → p J interval. The sensi-tivity to 2% matrix element changes for the 4 s → d J intervals is about 0.0001 nm due to the small polarizabil-ities of the 3 d J states near 395 nm. The 850 nm magicwavelength is also relatively insensitive to changes in theoverall size of the matrix elements, with the 2% matrixelement change leading to a change of only 0.0001 nmin the magic wavelengths. The low sensitivity of magicwavelengths to the overall size of the matrix elements inthese cases means that these the magic wavelengths canbe used to give precise estimates of the matrix elementratios of the two transitions in the spin-orbit doublet.The 1338, 1309, 1074, 887 nm magic wavelengths showmuch greater sensitivity to 2% changes in the matrix ele-ments. The changes in the magic wavelengths range from3 to 80 nm. The sensitivity of the magic wavelengths tothese matrix elements is driven by the rate of changeof the 4 s and 3 d J polarizabilities with energy. A largechange in the photon energy is needed to compensate fora small change in the polarizability when dα /dω is small.The sensitivity of the magic wavelength to small changesin the matrix elements decreases as the photon energygets closer to the 3 d J → p J ′ excitation thresholds. Thehigh sensitivity of the magic wavelengths with respect tochanges in the matrix elements means it is only necessaryto measure the magic wavelength to a precision of 0.10nm to impose reasonably tight constraints on the ratiosof the 4 s → p J and 3 d J → p J ′ matrix element rations. V. CONCLUSION A relativistic semi-empirical core model is applied tothe calculation of the dynamic polarizabilities of the 4 s ,3 d J and 4 p J states of Ca + . A number of magic wave-lengths at convenient photon energies have been iden-tified for the 4 s -3 d J energy intervals. Measurement ofthese magic wavelengths can be used to determine rea- sonably accurate estimates of the 3 d J -4 p J ′ line strengthsrelative to the 4 s -4 p J line strengths. This could leadto improved estimates of the blackbody radiation shiftfor the Ca + clock transition. There is one impediment.At the moment there is a 3% spread between theoreti-cal and experimental lifetimes for the 4 p J ′ states. Thisvariation, which does not exist for the same transition inpotassium [56, 76], needs to resolved so the uncertaintyin the 4 s − p J line strengths can be reduced to 1% orbetter.There are two other relatively clean measurements ofatomic structure parameters that could be made. Mea-surement of the magic wavelength near 395 nm couldbe used to determine a value of the oscillator strength f s → p / : f s → p / ratio. This could help resolve the in-compatible predictions of this ratio by DFCP/MBPT-SD and RCC calculations. Comparisons of polarizabili-ties do suggest that the gaussian basis set used for theRCC calculations could be improved. Further, mea-surements of the two longest magic wavelengths for the3 d / ,m =1 / → s / transition could give a good esti-mate of the f d / → p / : f d / → p / ratio.The utility of measuring magic wavelengths for se-lected Ca + transitions can of course be extended to otheralkaline-earth ions, with Sr + and Ba + being obvious pos-sibilities. 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The numbers in brackets are uncertainties in the last digits calculatedby assuming certain matrix elements have ± 2% uncertainties as described in the text. ω (a.u.) 0 0.0340414(22387) 0.0424109(10654) 0.1151182(1) 0.1151184(1) 0.1151186(1) λ (nm) ∞ . . . . . s / p / − − − p / p / p / d / Average m j = 1 / m j = 3 / m j = 1 / m j = 3 / m j = 5 / p / − p / f / f / f / f / s / and 3 d / states at the magicwavelengths. These results assume non-polarized light. The numbers in brackets are uncertainties in the last digits calculatedby assuming certain matrix elements have ± 2% uncertainties as described in the text. ω λ ∞ s / p / − − p / p / p / d / Average m j = 1 / m j = 3 / m j = 1 / m j = 1 / m j = 3 / p / − − p / p / − − p / f / f /2