Cite this Article
Zhang Yong-Hui, Tang Li-Yan, Zhang Xian-Zhou, Shi Ting-Yun. Calculations of the dynamic dipole polarizabilities for the Li+ ion. Chinese Physics B, 2016, 25(10): 103101  
Permissions
Calculations of the dynamic dipole polarizabilities for the Li+ ion
Zhang Yong-Hui1, 2, Tang Li-Yan2, †, , Zhang Xian-Zhou1, Shi Ting-Yun2
Department of Physics, Henan Normal University, Xinxiang 453007, China
State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China

 

† Corresponding author. E-mail: lytang@wipm.ac.cn

Project supported by the National Basic Research Program of China (Grant No. 2012CB821305) and the National Natural Science Foundation of China (Grant Nos. 11474319, 11274348, and 91536102).

Abstract
Abstract

The B-spline configuration-interaction method is applied to the investigations of dynamic dipole polarizabilities for the four lowest triplet states (2 3S, 33S, 23P, and 33P) of the Li+ ion. The accurate energies for the triplet states of n3S, n3P, and n3D, the dipole oscillator strengths for 23S(33S) → n3P, 23P(33P) → n3S, and 23P(33P) → n3D transitions, with the main quantum number n up to 10 are tabulated for references. The dynamic dipole polarizabilities for the four triplet states under a wide range of photon energy are also listed, which provide input data for analyzing the Stark shift of the Li+ ion. Furthermore, the tune-out wavelengths in the range from 100 nm to 1.2 μm for the four triplet states, and the magic wavelengths in the range from 100 nm to 600 nm for the 23S → 33S, 23S → 23P, and 23S → 33P transitions are determined accurately for the experimental design of the Li+ ion.

PACS: 31.15.ac;31.15.ap;32.10.Dk
Keyword:triplet states of Li+ ion;dynamic dipole polarizabilities;tune-out wavelength;magic wavelength
1. Introduction

Since the relativistic and QED corrections in the Li+ ion should be larger than in helium due to the Z4 scaling of the energy, the studies of the Li+ ion are of great interest because of its wide applications. For example, the combination between the experimental measurements and the theoretical calculations of the 23S hyperfine splitting for the Li+ ion provides a sensitive test of the hyperfine atomic structure theory,[1–3] and the investigation of the 23PJ fine structure of the Li+ ion can be used to determine the fine structure constant.[4–8] In addition, the high-precision measurement of the 23S → 23P transition for the Li+ ion isotopes could be used to test the relativistic and QED corrections,[2,3,9] to extract the relative nuclear charge radius,[10] to test the time dilation and verify the special relativity.[11]

The dynamic dipole polarizabilities of ions, which describe the distortion of the electronic charge distribution in the presence of an oscillating electric field with angular frequency ω, play an important role in the calculations of the van der Waals C6 coefficient,[12,13] in the determinations of refractive index and dielectric constants of ionic crystals,[14] in the construction of a core-polarization model for atoms and molecules,[15,16] and in the analysis of the Stark shifts.[17–19] Especially, the tune-out and magic wavelengths,[18,20,21] which are extracted from the dynamic dipole polarizabilities, become important parameters for experimental design of ion trapping. For example, recently, Liu et al. suggested that the magic wavelengths of ions pave a way to build an all-optical trapped ion[17,22–24] to eliminate the second-order of Ac Stark shift.

At present, there many works reporting the calculation of the static dipole polarizability of the ground-state Li+ ion.[25–31] For example, Zhu et al. computed the nonrelativisitc polarizabilities by using Hylleraas basis,[29] Johnson and Cheng got the polarizability by exploiting the B-spline relativistic configuration interaction (CI) method to solve the Dirac–Coulomb–Breit equation,[30] and Lim et al. obtained the polarizability by using the fully relativistic coupled-cluster method.[32] For the dynamic polarizabilities of the Li+ ion, Chung applied the variational perturbation method into the calculation of the ground state dynamic dipole polarizabilities,[33] and Glover and Weinhold obtained the rigorous upper and lower bounds for dynamic dipole polarizabilities with the laser frequency up to the first excitation threshold of the ground state.[25] However, compared with the ground-state polarizabilities, there are several calculations of polarizabilities for the triplet states of the Li+ ion. For example, the static dipole polarizability for the 23S and 23P states can be found in Refs. [34]–[36], only two works[37,38] about the dynamic polarizabilities for the 23S state were published, and there has been no work reported for the dynamic dipole polarizabilities of other triplet states of the Li+ ion in the literature.

In the present study, the CI method based on B-spline functions developed in our previous work[39] is extended to calculate the dynamic dipole polarizabilities for the four lowest triplet states of the Li+ ion. The tune-out wavelengths for the 23S, 23P, 33S, and 33P states are determined, and the magic wavelengths for the 23S → 33S, 23S → 23P, and 23S → 33P transitions are tabulated for reference. Atomic units are used throughout this paper unless specifically mentioned.

2. Structure calculations
2.1. B-spline CI method

Taking the Li+ ion as a Coulomb three-body system, choosing the atomic nucleus as particle 0 and two electrons as particle 1 and particle 2, respectively, the Hamiltonian can be written as

where r12 = |r1 − r2| is the distance between two electrons.

Recently, significant progress has been made in the variational calculations for two-electron systems by using the B-spline CI basis set,[39–42] which has the following functional form,

where ηij is a normalization constant, the Clebsch–Gordan coefficients 〈ℓimi;ℓjmj|LM〉 and 〈1/2msi;1/2msj|SMS〉 represent ℓ–ℓ and s–s coupling, and is the i-th eigen-wavefunction of the single-electron Schrödinger equation with eigen-energy of ɛi. The radial wavefunction Ri(r) is expanded as the B-spline functions,[39]

Using the variational method, the expectation value of the Hamiltonian can be simplified to the following configuration equation,

where λ and ci j are respectively the eigen-energies and the expansion coefficients of eigen-wavefunction for the Li+ ion, the expression of the potential-energy matrix element Vijkℓ can be found in our previous paper about helium.[39]

The present CI calculation is performed by fixing the box size R0 = 150 a.u. (the unit a.u. is short for atomic unit), the nonlinear parameter γ = R0 × 0.042, and the order of the B-spline k = 7. The convergence test of the energies and oscillator strengths are performed as the number of B-spline N and partial waves ℓmax increased. The maximum N and ℓmax used in our calculation are respectively 40 and 10.

2.2. Energies and oscillator strengths

Figure 1 is a diagram about the nonrelativistic triplet energy levels of the Li+ ion, where some important transitions are labeled with arrows. Compared with our previous work of Ref. [39], the energies and oscillator strengths have similar convergent styles to helium.[39]

Fig. 1. Low-lying triplet energy levels of the Li+ ion.

Table 1 lists final extrapolated energies for n3S, n3P, and n3D states of the Li+ ion with the main quantum number of n up to 10. It is clearly seen that all the energies have at least six significant digits and can compare well with the experimental values from NIST.[43] Especially for the 23S and 23P states, the energies are respectively −5.1107273(1) a.u. and −5.0277156(1) a.u., which are in excellent agreement with the Hylleraas values of Cann et al.[44] and much more accurate than the results of Refs. [45] and [46]. For the 33S and 33P states, present values of −4.7520764(2) a.u. and −4.7304597(2) a.u. have seven significant digits compared with the values of Ref. [44].

Table 1.

Comparison of the energies (in unit a.u.) for the Li+ ion. The experimental data were taken from the National Institute of Science and Technology (NIST) tabulation.[43] The numbers in the parentheses are the computational uncertainties.

.

The extrapolated oscillator strengths for some selective dipole transitions are presented in Table 2, and comparisons with other published values are also listed in this table. All of our results are in good agreement with the explicitly correlated wavefunction calculations of Ref. [44]. For the 23S → 23P transition, our result of 0.3079405(3) is in perfect agreement with the value of 0.3079402 in Ref. [47] and more accurate than the value of 0.307944 by two orders of magnitude.[44] For the 23P → 33D transition, the present result agrees well with the values of Refs. [47] and [48]. Even for the dipole transitions involved in the unnatural-parity states, both the oscillator strengths of 23P → 23Pe and 33P → 33Pe transitions have four significant digits.

Table 2.

Comparison of the oscillator strengths for the Li+ ion. The numbers in the parentheses represent computational uncertainties.

.
2.3. Dynaimc dipole polarizabilities

When an atom or ion is exposed at the external electric field with the photon energy of ω, the dynamic dipole polarizabilities for the magnetic sub-level |NgLgMg〉 with the main quantum number Ng, angular momentum quantum number Lg, and magnetic quantum number Mg, can be expressed as

where and are the dynamic scalar and tensor dipole polarizabilities, respectively, they can be written as the summation of all the intermediate states including the continuum[39]

Using the energies and oscillator strengths of the Li+ ion, the dynamic dipole polarizabilities can be calculated. Table 3 presents a convergence test of the static dipole polarizability for the 23S state of the Li+ ion. It is seen clearly that under the same number of partial wave ℓmax, the static dipole polarizability is decreased as the number of B-spline N increased, while the result is increased as ℓmax increased under the same N. This kind of convergence style for the Li+ ion is similar to our previous work of the static dipole polarizabilities for helium.[39] Combining the different convergent trends of the static dipole polarizability as N and ℓmax, an extrapolated value of 46.87980(1) is given in the last line of Table 3, which is in good agreement with the Hylleraas value of 46.8798023(7) .[36] Comparing our extrapolated result with the variation-perturbation result of 46.8798 of Chen,[34] the first five digits are the same. For other triplet 23P, 33S, and 33P states, the final extrapolated values of static dipole polarizabilities are listed in the first line of the Table 4. The present result of the scalar static dipole polarizability for the 23P state agrees with the Hylleraas value of −5.790965718(2) (Refs. [35] and [36]) at the level of about 3-ppm accuracy.

Table 3.

Convergence of the static dipole polarizabilities (in unit a.u.) for the 23S state of the Li+ ion as the number of B-spline N and partial waves ℓmax increased. The number in parenthesis for the extrapolated value gives the computational uncertainty.

.
Table 4.

Dynamic dipole polarizabilities (in unit a.u.) of the 23S, 23P, 33S, and 33P states for the Li+ ion. The numbers in parentheses represent computational uncertainties.

.

The dynamic dipole polarizabilities for the 23S state of the Li+ ion for some selective photon energy of 0 ≤ ω ≤ 0.38 a.u. are listed in Table 4. The figures in parentheses represent computational uncertainties. It can be seen that all of the present values have at least four significant figures except the result of 918.6(2) for ω = 0.38 a.u., where the photon energy lies near the second resonance excitation threshold of 0.38031 a.u. Comparing our results with the values of Glover and Weinhold,[37] all of our values lie within the upper and lower limits of their values. In addition, Table 4 also lists the dynamic dipole polarizabilities for the 23P, 33S, and 33P states. For the 33S state, present values have 5–7 significant digits with ω ≤ 0.13 a.u. For the 23P and 33P states, all of the results for the scalar and tensor dipole polarizabilites have at least four significant digits except values for the photon energy ω near the resonance transition position or ionization threshold.

2.4. Tune-out and magic wavelengths

The tune-out wavelength is defined as the wavelength which makes the dynamic dipole polarizability of a single state disappear.[49,50] The magic wavelength is defined as the wavelength which makes the dynamic dipole polarizabilities of the upper and lower energy levels of a transition equal.[51–53] The dynamic dipole polarizabilities for the lowest four triplet states of the Li+ ion are also plotted in Figs. 2–7 as the photon energy ω increased. The crossing points between a curve and the horizontal zero line are labeled as tune-out wavelengths, which denoted as a solid magenta circle. The crossing points between two curves are the magic wavelengths, which are denoted as a blank red circle. The vertical dashed lines are the resonance transition positions.

Fig. 2. Dynamic dipole polarizabilities (in unit a.u.) of the Li+ ion for the photon energy 0.075 ≤ ω ≤ 0.225 a.u. The solid black line denotes the dynamic polarizabilities for the 23S state, and the solid blue line represents the dynamic polarizabilities for the 33S state. The crossing points denoted as solid magenta circles are the tune-out wavelengths, and the crossing points denoted as blank red circles are the magic wavelengths. The vertical dashed lines are the resonance transition positions, and the green line is a horizonal zero line.
Fig. 3. Dynamic dipole polarizabilities (in unit a.u.) of the Li+ ion for the photon energy 0.0 ≤ ω ≤ 0.40 a.u. The solid black line denotes the dynamic polarizabilities for the 23S state, and the solid blue line represents the dynamic polarizabilities for the 23P (Mg = 0) state.
Fig. 4. Dynamic dipole polarizabilities (in unit a.u.) of the Li+ ion for the photon energy 0.0 ≤ ω ≤ 0.4 a.u. The solid black line denotes the dynamic polarizabilities for the 23S state, and the solid blue line represents the dynamic polarizabilities for the 23P (|Mg| = 1) state.
Fig. 5. Dynamic dipole polarizabilities (in unit a.u.) of the Li+ ion for the photon energy 0.050 ≤ ω ≤ 0.150 a.u. The solid black line denotes the dynamic polarizabilities for the 23S state, and the solid blue line represents the dynamic polarizabilities for the 33P (Mg = 0) state.
Fig. 6. Dynamic dipole polarizabilities (in unit a.u.) of the Li+ ion for the photon energy 0.16 ≤ ω ≤ 0.20 a.u. The solid black line denotes the dynamic polarizabilities for the 23S state, and the solid blue line represents the dynamic polarizabilities for the 33P (Mg = 0) state.
Fig. 7. Dynamic dipole polarizabilities (in unit a.u.) of the Li+ ion for the photon energy 0.025 ≤ ω ≤ 0.205 a.u. The solid black line denotes the dynamic polarizabilities for the 23S state, and the solid blue line represents the dynamic polarizabilities for the 33P (|Mg| = 1) state.

The values of the tune-out and magic wavelengths in the Figs. 2–7 are listed in Tables 5 and 6, respectively. It is noticed that most tune-out wavelengths are in the range of 110 nm–500 nm except for the tune-out wavelength of 1190.14(3) nm for the 33P (|Mg| = 1) state. All the values of magic wavelengths are in the range of 110 nm–600 nm. All the tune-out wavelengths in Table 5 have at least four significant digits, and all the present magic wavelengths in Table 6 have five significant digits. These accurate theoretical values of tune-out and magic wavelengths can be used as important parameters for experiment design to trap the Li+ ion for high-precision measurements.

Table 5.

Tune-out wavelengths λt (in unit nm) for 23S, 33S, 23P, and 33P states of the Li+ ion. The numbers in parentheses give the computational uncertainties.

.
Table 6.

Magic wavelengths λm (in unit nm) for 23S → 33S, 23S → 23P, and 23S → 33P transitions of the Li+ ion. The numbers in parentheses give the computational uncertainties.

.
3. Conclusions

The ab-initio calculations of the dynamic dipole polarizabilities for the four lowest triplet states (2 3S, 33S, 23P, and 33P) of the Li+ ion are carried out by using the B-spline CI method. The large number of accurate energies and oscillator strengths with the main quantum number n up to 10 of the Li+ ion are tabulated for theoretical references. Present static dipole polarizabilities for the 23S and 23P states are in excellent agreement with the most accurate results from variational Hylleraas basis calculations,[35,36] and present dynamic dipole polarizabilities for the 23S state that lies within the upper and lower bounds of the values from the correlated variational trial function calculation.[37] In addition, the tune-out wavelengths for the four lowest triplet states, and the magic wavelengths for the three transitions of 23S → 33S, 23S → 23P, and 23S → 33P are identified with at least four significant digits. Present polarizabilities provide theoretical references for analyzing the Stark shift of the Li+ ion. Our tune-out and magic wavelengths can be used for designing the high-precision experimental measurement on the Li+ ion in the future.

Reference
1Kowalski JNeumann RNoehte SScheffzek KSuhr HPutlitz G Z 1983 Hyp. Int. 15 159
2Clarke J Jvan Wijngaarden W A 2003 Phys. Rev. 67 012506
3Riis ESinclair A GPoulsen ODrake G W FRowley W R CLevick A P 1994 Phys. Rev. 49 207
4Pachucki KYerokhin V A 2009 Phys. Rev. 79 062516
5Zelevinsky TFarkas DGabrielse G 2005 Phys. Rev. Lett. 95 203001
6Borbely J SGeorge M CLombardi L DWeel MFitzakerley D WHessels E A 2009 Phys. Rev. 79 060503
7Pachucki KYerokhin V A 2010 Phys. Rev. Lett. 104 070403
8Zhang P PZhong Z XYan Z CShi T Y 2015 Chin. Phys. 24 033101
9Rong HGrafström SKowalski JPutlitz G zJastrzebski WNeumann R 1993 Z. Phys. 25 337
10Wijngaarden W A vanNoble G A 2008 Lect. Notes Phys. 745 111
11Botermann BBing DGeppert CGwinner GHänsch T WHuber GKarpuk SKrieger AKühl TNörtershäuser WNovotny CReinhardt SSánchez RSchwalm DStöhlker TWolf ASaathoff G 2014 Phys. Rev. Lett. 113 120405
12Arora BSahoo B K 2014 Phys. Rev. 89 022511
13Porsev S GSafronova M SDerevianko AClark C W 2014 Phys. Rev. 89 022703
14Mitroy JSafronova M SClark C W 2010 J. Phys. 43 202001
15Ward S JShertzer J 2004 Nucl. Instrum. Method Phys. Res. 221 206
16Han HZhong ZZhang XShi T 2008 Phys. Rev. 78 044701
17Liu P LHuang YBian WShao HGuan HTang Y BLi C BMitroy JGao K L 2015 Phys. Rev. Lett. 114 223001
18Tang L YBromley M W JYan Z CMitroy J 2013 Phys. Rev. 87 032507
19Farley J WWing W H 1981 Phys. Rev. 23 2397
20Safronova M SSafronova U IClark C W 2012 Phys. Rev. 86 042505
21Yu W WYu R MCheng Y JZhou Y J 2016 Chin. Phys. 25 023101
22Udem ThHolzwarth RHänsch T W 2002 Nature 416 233
23Margolis H S 2009 J. Phys. B: At. Mol. Opt. Phys. 42 154017
24Bloom B JNicholson T LWilliams J RCampbell S LBishof MZhang XZhang WBromley S LYe J 2014 Nature 506 71
25Glover R MWeinhold F 1976 J. Chem. Phys. 65 4913
26Bishop D MRérat M 1989 J. Chem. Phys. 91 5489
27Banerjee AHarbola M K 1997 Pramana. J. Phys. 49 455
28Bhatia A KDrachman R J 1997 Can. J. Phys. 75 11
29Zhu J MZhou B LYan Z C 1999 Chem. Phys. Lett. 313 184
30Johnson W RCheng K T 1996 Phys. Rev. 53 1375
31Bhatia A KDrachman R J 1997 Phys. Rev. 55 1842
32Lim I SLaerdahl J KSchwerdtfeger P 2002 J. Chem. Phys. 116 172
33Chung K T 1971 Phys. Rev. 4 7
34Chen M K 1995 J. Phys. B: At. Mol. Opt. Phys. 28 1349
35Zhu J MZhou B LYan Z C 2000 Mol. Phys. 98 529
36Yan Z CZhu J MZhou B L 2000 Phys. Rev. 62 034501
37Glover R MWeinhold F 1977 J. Chem. Phys. 66 185
38Glover R MWeinhold F 1977 J. Chem. Phys. 66 191
39Zhang Y HTang L YZhang X ZShi T Y 2015 Phys. Rev. 92 012515
40Lin Y CFang T KHo Y K 2015 Phys. Plasmas 22 032113
41Yulian V VAlejandro S 2004 J. Phys. B: At. Mol. Opt. Phys. 37 4101
42Nikolopoulos L A A 2003 Comput. Phys. Commun. 150 140
43Kramida ARalchenko YReader JNIST ASD Team2012“NIST Atomic Spectra Database”(version 5.0.0) URL http://physics.nist.gov/asd
44Cann N MThakkar A J 1992 Phys. Rev. 46 5397
45Accad YPekeris C LSchiff B 1971 Phys. Rev. 4 516
46Godefroid MVerhaegen G 1980 J. Phys. B: At. Mol. Opt. Phys. 13 3081
47Schiff BPekeris C LAccad Y 1971 Phys. Rev. 4 885
48Kono AHattori S 1984 Phys. Rev. 30 2093
49LeBlanc L JThywissen J H 2007 Phys. Rev. 75 053612
50Mitroy JTang L Y 2013 Phys. Rev. 88 052515
51Ye JVernooy D WKimble H J 1999 Phys. Rev. Lett. 83 4987
52Katori HIdo TKuwata-Gonokami M 1999 J. Phys. Soc. Jpn. 68 2479
53Safronova M SSafronova U IClark C W 2013 Phys. Rev. 87 052504