Polarizabilities and Hyperpolarizabilities for Lithium Atoms in Debye Plasmas

Transcription

1 Commun. Theor. Phys. 62 (2014) Vol. 62, No. 6, December 1, 2014 Polarizabilities and Hyperpolarizabilities for Lithium Atoms in Debye Plasmas KANG Shuai ( ), 1, HE Juan ( ), 1 XU Ning ( ), 1 and CHEN Chang-Yong ( ) 2 1 School of Physics and Mechanical & Electrical Engineering, Zunyi Normal College, Zunyi , China 2 Department of Physics, Shaoguan University, Shaoguan , China (Received May 14, 2014; revised manuscript received July 31, 2014) Abstract The effects of plasma environments on energies, oscillator strengths, polarizabilities and hyperpolarizabilities for lithium atom have been calculated by combining the l-dependent model potential of free lithium atom and linear variation method based on B-spline basis functions. The influence of plasma on lithium atom is represented by the Debye screened potential, which describes effectively the averaged effect of the plasma environment on atomic spectra. The results are in agreement with other reported ones. PACS numbers: Vy, Dk, Pf Key words: plasmas, polarizabilities, B-spline, Debye screened potential, Lithium atom 1 Introduction Atomic processes in plasma environments have attracted numerous attentions because of people s lasting interest in the solar system, stellar space, as well as the search for the laser-produced hot and dense plasmas. [1 41] In such environments, the interaction between valence electrons and the atomic nucleus is screened. The influence of plasma on the embedded atoms can be represented by a certain models. For example, to describe the influence about one-electron atoms in plasma environments, the Yukawa-type electron-nuclear attraction potential, the self-consistent-field confined atomic model, and the super-configuration method have been proposed by Rogers et al., [1] Nguyen et al., [2] and Blenski et al., [3] respectively. To account for the electron correlation for multi-electron atoms in a plasma environments, the full configuration interaction method has been introduced by Lopez et al. [4] and Saha et al. [5] For atoms in stronglycoupled plasma environments, the ion-sphere model is a widely used one. [6 11] While for atoms in weakly-coupled plasma environments, the effect of plasma can be represented by the Debye screening potential, which describes effectively the averaged effect of the plasma environment on atomic spectra. Under the classical theory of Debye screening potential model, the potential between two charges changes from Coulomb potential to Yukawa-type model one. The screening parameter in Debye potential can be determined by the electron density and electron temperature of the plasma. [42] A number of theoretical investigations on atoms under Debye screening potential have been performed. Okutsu et al. has reported the spectral properties of the low-lying singlet states of a helium atom and those of the lowlying doublet states of a lithium atom in laser plasmas. [12] The importance of Debye screening incalculating partition functions in thermodynamics, astrophysical plasma diagnostics, and, astrophysical observations have discussed by Saha et al. [13] and Sil et al. [14] Kar have reported the dynamic dipole polarizability of the helium atom. [15] Bhatia et al., [16] and Jiao et al. [17] have reported the effect of plasma screening for H in Debye plasma environments. Qi et al. [18] has reported the ground-state energies, transition energies, and oscillator strengths for hydrogen atom in Debye plasma environments. Recently, Li et al. [40] and Ning et al. [41] have reported some results for lithium and sodium atoms in plasma environments. In their work, a simple model potential was adopted to describe the free lithium and sodium atoms, and then was modelled using Debye screening approach to describe the effect of plasma. In this paper, we are mainly interested to report the behavior of the dipole polarizability, quadrupole polarizability, octupole polarizability, and scalar hyperpolarizability of lithium atom embedded in Debye plasmas. The polarizability and hyperpolarizability for an atomic system play an important role in the studies of atomic collision processes, interactions between matter and electromagnetic fields, the long-range interactions among atoms, Supported by the National Natural Science Foundation of China under Grant No , the Science and Technology Foundation of Guizhou Province under Grant Nos. J[2012]2345, LKZS[2012]02, and LKZS[2014]05, the Special Fundation of Governor of Guizhou Province for Science and Technology and Education Talents under Grant No. [2012]87, the Key Project of Education Department of Guizhou Province under Grant No. KY(2013)171, the Doctor Foundation of Zunyi Normal College under Grant No. 2012BSJJ17, and the Key Disciplines of Guizhou Province under Grant No. QXWB[2013]18 kngshuai@tom.com c 2014 Chinese Physical Society and IOP Publishing Ltd

2 882 Communications in Theoretical Physics Vol. 62 and other physicsal and chemistrical areas. To calculate the polarizability and hyperpolarizability, we introduce a method for computing the energy and wavefunctions for lithium atom with linear variation based on B- splines basis functions, in which the valence electron for the free lithium atom is described by l-dependent model potential. [43] The introduced method is similar to the reported one by Li et al. [40] and Ning et al., [41] but the form of model potential for free lithium atom. The linear variation based on B-splines basis functions can be found in the Refs. [44 46]. A number of theoretical investigations on atomic systems based on B-spline basis functions have been reported. [44 50] Specially, the polarizability calculations for a certain atomic systems using B-spline basis sets have been reported. [51 56] Very recently, the highorder dispersion coefficients for alkali-metal atoms have been reported combined with the linear variation based on B-splines basis functions and the l-dependent model potential for free alkali-metal atoms. [57] 2 Theory and Method In this paper, we consider l-dependent model potential method to represent the free Li atom. In the model potential method, in which Li atomic system is composed of the valence electron and the atomic ion core, the motion of the valence electron can be expressed in the form of V l (r) = Z l(r) α c 6 [1 e (r/rc) ], (1) r 2r4 where α c is the polarizability of the positive ion core. Atomic units are used throughout the paper. The radial charge Z l (r) is expressed as Z l (r) = 1 + (z 1) e a1r r(a 3 + a 4 r) e a2r, (2) where z = 3 is the nuclear charge of the neutral atom and r c is the cut-off radius introduced to truncate the unphysical short-range contribution of the polarization potential near the origin. For a given atom, a 1, a 2, a 3, a 4, α c and r c are parameters. The l-dependent model potential parameters of lithium atoms are listed in Table 1, which is given in Ref. [43]. For the system of lithium atom in plasma environment, we adopt the Debye screening approach to incorporate plasma screening effect on the electron-ion interaction as V (r, λ D ) = V l (r) e r/λ D, (3) where λ D is the Debye screening length, which is a function of electron density and electron temperature and can be obtained by [κt /4π(1 + Z)n] 1/2. Here n is the number density, T, the temperature, and Z, the nuclear charge of the plasma. The value of λ D is associated with the screened strength of the plasma environment. Detail applications of the adopted Debye screening approach for multi-electron systems in plasma environment can be found from the article [37] and book. [58] Combining model potential approach of free atoms and the Debye screening approach have also been adopted by other authors. [36,40 41] Since the Hamiltonian of the valence electron for lithium atom in Debye plasma environment is spherical symmetric, the angular wavefuctions are the spherical harmonics functions, Y lm (θ, φ). Therefore, for a given angular momentum l, the radial Hamiltonian can be represented by Ĥ r = 1 [ d ( 2r 2 r 2 d ) ] l(l + 1) + V (r, λ D ). (4) dr dr In our calculation, the radial wavefunctions are expanded using the B-spline basis as Ψ(r) = i C i r l B i,k (r), (5) where C i are the expansion coefficients, and B i,k (r), the k order B-spline basis functions. The B-spline functions are piece-polynomials determined uniquely in the interval [0, R max ] by the order k and the knot sequence, which is adopted as 0 = r 1 = r 2 = = r k r k+1 r nr+1 = r nr+2 = = r nr+k = R max. (6) Namely, we let r = 0 be k multiple knots, r = R max be k 1 multiple knots, other B-spline knots are distributed exponentially in the intervals (0, R max ). For details, readers can refer to Refs. [44] and [46]. In our early work, this method has been proven effective. [47 50] In this calculation, the 9th order B-splines (k = 9) is applied. The Schrödinger equation can be written in matrix form HC = ESC, (7) where E is the eigenvalue, C the corresponding eigenvector, H is the Hamiltonian matrix, and S is the overlap matrix with respect to the B-spline basis set used. The energy eigenvalues and the corresponding eigenstates can be obtained by diagonalizing the Hamiltonian matrix. To calculate polarizability for lithium atom, we use the 2 l -pole static polarizability relation, which is a sum over all intermediate states including the continuum state, can be expressed in form of f (l) nk α l = (E n E 0 ) 2, (8) n where f (l) nk is the 2l -pole oscillator strength between the initial Ψ 0 and the intermediate Ψ n states, E 0, the groundstate energy, and E n, the corresponding n-th intermediate energy. To calculate scalar hyperpolarizability for lithium atom, we use the method reported by Tang et al., [51] which can be expressed in form of

3 No. 6 Communications in Theoretical Physics 883 where T (L a, L b, L c ) = kmn 2L 128π2 1 γ 0 = ( 1) G 0 (L, L a, L b, L c )T (L a, L b, L c ), (9) 3 2L + 1 L al b L c n 0 L T 1 ml a ml a T 1 nl b nl b T 1 kl c kl c T 1 n 0 L [E k (L c ) E n0 (L)][E n (L b ) E n0 (L)][E m (L a ) E n0 (L)] n δ(l, L b )( 1) 2L La Lc 0 L T 1 ml a 2 [E m m (L a ) E n0 (L)] k G 0 (L, L a, L b, L c ) = ( ) ( 1 1 K1 1 1 K2 (0, K 1, K 2 ) K 1K 2 { } { } { 1 1 K1 1 1 K2 K2 K 1 0 L L b L a L L b L c L L L b n 0 L T 1 kl c 2 [E k (L c ) E n0 (L)] 2, (10) ) ( K1 K ) }, (11) where T 1 is the dipole transition operator, (a, b, c) = 2a + 1 2b + 1 2c + 1. The 3j-symbol and 6j-symbol are also adopted. In particular, for the case 2S state of lithium, the scalar hyperpolarizability can be expressed as [ γ 0 = 128π2 1 2 ] T (1, 0, 1) T (1, 2, 1). (12) Table 1 The l-dependent model potential parameters of lithium atom. α c a 1 a 2 a 3 a 4 r c l = l = l Results and Discussions We calculate the ground energies for lithium atom under different Debye screening parameters using the proposed method. The calculated results are compared in Table 2 with those available in the literature. From Table 2, it is clear that our numerical results for the case of λ D, which correspond to the case of free lithium atom, are in good agreement with the reported results. [59 60] It is noticed that in the works of Sahoo et al., [36] Li et al., [40] Ning et al., [41] and Qi et al., [18] the adopted model potentials for free Li atom are identical, which is different from our l-dependent model potential, so that all their results are exactly the same values. For the case of λ D being finite, our results are a little larger than those of Sahoo et al. and Li et al. With the decrease of λ D, the differences between our results and theirs increase. For example, when λ D = 100 a.u., the differences between our values and those of Sahoo and Li are not more than 0.5%. In contrast, when λ D = 4 a.u., the differences reach 2.8%. The results obtained by our calculations about the ground-state energies E 2s, transition energies E np,2s, and oscillator strengths f 2s 2p, f 2s 3p, and f 2s 4p as functions of Debye screening length for lithium atom under Debye plasmas are presented in Table 3. We compare our results with the data of Ning et al. From Table 3, for ground-state energies E 2s, our calculated results are a little larger than theirs, while for transition energies E np,2s are about 1% difference. It is also evident that for all cases the oscillator strengths of f 2s 2p are greater than theirs about 1%, while those of f 2s 3p and f 2s 4p are less than theirs about 10%. Table 2 The ground energies for lithium atom under different Debye screening parameters. λ D Present work Sahoo et al. [36] Li et al. [40] Qi et al. [18] Ning et al. [41] Bachau et al. [60] Bashkin et al. [59]

4 884 Communications in Theoretical Physics Vol. 62 Table 3 The ground-state energies E 2s, transition energies E, and oscillator strengths f 2s 2p, f 2s 3p, and f 2s 4p for lithium atom under different Debye screening parameters. λ D E 1s Transition Transition Energies Oscillator Strengths Ning et al. [41] Present work Ning et al. [41] Present work Ning et al. [41] Present work s 2p s 3p s 4p s 2p s 3p s 4p s 2p s 3p s 4p s 2p s 3p s 4p s 2p s 3p s 2p s 3p s 2p s 3p s 2p s 2p s 2p Table 4 The static dipole polarizabilities α 1, quadrupole polarizabilities α 2, octupole polarizabilities α 3, and scalar hyperpolarizabilities γ 0 for lithium atom under different Debye screening parameters. The numbers in the parentheses denote powers of ten. λ D α 1 α 2 α 3 γ 0 Li et al. [40] Ning et al. [41] Present work Li et al. [40] Present work Li et al. [40] Present work (2) (2) (2) (3) (3) (4) (4) 3.231(3) 3.39(3) a 3.0(3) b 2.9(3) c 3.93(3) d 3.45(3) e 3.060(3) f (2) (2) (2) (3) (3) (4) (4) 3.305(3) (2) (2) (2) (3) (3) (4) (4) 3.430(3) (2) (2) (3) (3) (4) (4) 3.837(3) (2) (2) (2) (3) (3) (4) (4) 7.052(3) (2) (2) (2) (3) (3) (4) (4) (3) (2) (2) (2) (3) (3) (4) (4) (3) (2) (2) (2) (3) (3) (4) (4) (3) (2) (2) (3) (3) (4) (4) (3) (2) (2) (2) (3) (3) (4) (4) (3) (2) (2) (2) (3) (3) (4) (4) (3) (2) (2) (2) (3) (3) (4) (4) (3) a: Ref. [61], b: Ref. [62], c: Refs. [63 64], d: Ref. [65], e: Ref. [6], f: Ref. [51]. The oscillator strengths of oscillator strengths f 2s 2p, f 2s 3p, and f 2s 4p as functions of Debye screening parameters are also presented in Fig. 1. Figure 1 shows that the oscillator strength of f 2s 2p transition increases when 1/λ D < and then decreases rapidly. Figure 1 shows that the oscillator strengths of f 2s 3p transition decreases when

5 No. 6 Communications in Theoretical Physics 885 1/λ D < and then increases, and that of f 2s 4p transition decreases with the increasing of 1/λ D. Fig. 1 The oscillator strengths f 2s 2p, f 2s 3p, and f 2s 4p as functions of the screening parameter 1/λ D for lithium atom under Debye screening plasmas. Fig. 2 The static dipole polarizability, quadrupole polarizability, and octupole polarizability as functions of the screening parameter 1/λ D for lithium atom under Debye screening plasmas. (a) dipole polarizability α 1, (b) quadrupole polarizability α 2, and (c) octupole polarizability α 3. Fig. 3 The static hyperpolarizability as functions of the screening parameter 1/λ D for lithium atom under Debye screening plasmas. Our calculations of the static dipole polarizabilities α 1, quadrupole polarizabilities α 2, and octupole polarizabilities α 3, and scalar hyperpolarizabilities γ 0 for lithium atom under different Debye screening parameters are shown in Table 4. The static dipole polarizability, quadrupole polarizability, and octupole polarizability as functions of the screening parameter 1/λ D for lithium atom under Debye screening plasmas are also shown in Fig. 2, while scalar hyperpolarizability γ 0 as a function of the screening parameter 1/λ D is also shown in Fig. 3. For dipole polarizability α 1, a little difference exists between our values and those of Ning et al. and Li et al. [40] For most chosen cases, α 1 has no more than 3% difference between our values and their results. As far as we

6 886 Communications in Theoretical Physics Vol. 62 know, the best value for dipole polarizability of free Li atom is a.u., which was reported by Yan. [67] From Table 4, for the case of λ D, only 0.07% difference exists between our numerical result and result of Yan et al. [67] However, for this case, the differences between the result of Ning et al. and that of Yan et al. and between the result of Li et al. and that of Yan et al. are 0.23% and 0.30%, respectively. These comparisons illustrate that our results should be more reliable than theirs. For the scalar hyperpolarizability γ 0 of free lithium atom, our result is also compared with the reported result before. From Table 4, our value is greater than that of Tang et al., [51] that of Pipin et al., [62] that of Kassimi et al., [63] and that of Thakkar et al., [64] while less than that of Laughlin et al., [65] that of Jaszunski et al., [66] and that of Cohen et al. [61] From Table 4, it is also clear that with the decreasing of λ D, the dipole polarizability α 1, quadrupole polarizability α 2, octupole polarizability α 3, and scalar hyperpolarizability γ 0 increase with different speed. The dipole polarizability increases only in magnitude of value, while quadrupole polarizability, octupole polarizability, and scalar hyperpolarizability increase in order of magnitude. It is also shown in Figs. 2 and 3 that the increasing speeds of dipole polarizability, quadrupole polarizability, octupole polarizability, and scalar hyperpolarizability increase with the increasing of 1/λ D. 4 Summary In this paper we have calculated the ground energies, oscillator strengths, transition energies, polarizabilities, and hyperpolarizabilities for lithium atom under different Debye screening parameters by combining the l- dependent model potential of free lithium atoms and linear variation method based on B-spline basis functions. The calculation and comparison for free lithium atoms show that our numerical results agree very well with the reported theoretical prediction and the experimental measurement values. [59 60,67] The behaviors of oscillator strength, polarizability, and hyperpolarizability for lithium atom under Debye plasma environments with the change of screening parameter are nearly the same as the reported ones. [40 41] There are sound reasons that our reported results here are useful in the interpretation of the spectral properties for lithium atoms in laboratorial and astrophysical plasma environments. In addition, the proposed method should be applied to other alkali metal atoms in plasma environments. It is therefore believed that our results are useful references for the communities in plasma, atomic, and chemical physics. References [1] F.J. Rogers, H.C. Grabsoke, Jr., and D.J. Harwood, Phys. Rev. A 1 (1970) [2] H. Nguyen, M. Koenig, D. Benredjem, M. Caby, and G. Coulaud, Phys. Rev. A 33 (1986) [3] T. Blenski, A. Grimaldi, and F. Perrot, Phys. Rev. E 55 (1997) R4889. [4] X. Lopez, C. Sarasola, and J.M. Ugalde, J. Phys. Chem. A 101 (1997) [5] B. Saha, T.K. Mukherjee, P.K. Mukherjee, and G.H.F. Diercksen, Theor. Chem. Acc. 108 (2002) 305. [6] J.C. Stewart and K.D. Pyatt, Jr., Astrophys. J. 144 (1966) [7] B.F. Rozsnyai, Phys. Rev. A 43 (1991) [8] J. Basu and D. Ray, Phys. Rev. E 83 (2011) [9] X. Li and F. B. Rosmej, Phys. Rev. A 82 (2010) [10] Y.D. Jung, Phys. Plasmas 5 (1998) [11] Y.D. Jung and H.D. Jeong, Phys. Rev. E 54 (1996) [12] H. Okutsu, T. Sako, K. Yamanouchi, and G.H.F. Diercksen, J. Phys. B 38 (2005) 917. [13] B. Saha, P.K. Mukherjee, and G.H.F. Diercksen, A&A 396 (2002) 337. [14] A.N. Sil and P.K. Mukherjee, Int. J. Quantum Chem. 102 (2002) [15] S. Kar, Phys. Rev. A 86 (2012) [16] A.K. Bhatia and C. Sinha, Phys. Rev. A 86 (2012) [17] L.G. Jiao and Y.K. Ho, Phys. Rev. A 87 (2013) [18] Y.Y. Qi, Y. Wu, and J.G. Wang, Phys. Plasmas 16 (2009) [19] S. Kar and Y.K. Ho, Phys. Rev. A 71 (2005) [20] Z. Wang and P. Winkler, Phys. Rev. A 52 (1995) 216. [21] C.S. Lam and Y.P. Varshni, Phys. Rev. A 27 (1983) 418. [22] P. Winkler, Phys. Rev. E 53 (1996) [23] S. Nakai and K. Mima, Rep. Prog. Phys. 67 (2004) 321. [24] S. Kar and Y.K. Ho, Int. J. Quantum Chem. 106 (2006) 814. [25] S. Kar and Y.K. Ho, New J. Phys. 7 (2005) 141. [26] S. Kar and Y.K. Ho, Phys. Rev. E 70 (2004) [27] D. Ray and P.K. Mukherjee, Eur. Phys. J. D 2 (1998) 89. [28] D. Ray and P.K. Mukherjee, J. Phys. B: At. Mol. Opt. Phys. 31 (1998) [29] B. Saha, T.K. Mukherjee, and P.K. Mukherjee, Chem. Phys. Lett. 373 (2003) 218. [30] L.B. Zhao and Y.K. Ho, Phys. Plasmas 4 (2004) [31] C. Stubbins, Phys. Rev. A 48 (1993) 220. [32] B.L. Whitten, N.F. Lane, and J.C. Weisheit, Phys. Rev. A 29 (1984) 945. [33] J.S. Yoon and Y.D. Jung, Phys. Plasmas 3 (1996) 329. [34] S. Kang, Y.C. Yang, J. He, F.Q. Xiong, and N. Xu, Centr. Eur. J. Phys. 311 (2013) 584.

7 No. 6 Communications in Theoretical Physics 887 [35] Z. Jiang, S. Kar, and Y.K. Ho, Phys. Rev. A 84 (2011) [36] S. Sahoo and Y.K. Ho, Phys. Plasmas 13 (2006) [37] A.N. Sil, S. Canuto, and P.K. Mukherjee, Adv. Quant. Chem. 58 (2009) 115. [38] S. Sen, P. Mandal, and P.K. Mukherjee, Phys. Plasmas 19 (2012) [39] L.J. Stevanović, J. Phys. B: At. Mol. Opt. Phys. 43 (2010) [40] H.W. Li and S. Kar, Phys. Plasmas 19 (2012) [41] L.N. Ning and Y.Y. Qi, Chin. Phys. B 21 (2012) [42] P. Debye and E. Hückel, Phys. Z. 24 (1923) 185. [43] M. Marinescu, H.R. Sadeghpour, and A. Dalgarno Phys. Rev. A 49 (1994) 982. [44] F. Martin, J. Phys. B: At. Mol. Opt. Phys. 32 (1999) R197. [45] H. Bachau, E. Cormier, P. Decleva, J.E. Hansen, and F. Martin Rep. Prog. Phys. 64 (2001) [46] C. deboor, A Practical Guide to Splines, Springer, New york (1978). [47] S. Kang, J. Li, and T.Y. Shi, J. Phys. B: At. Mol. Opt. Phys. 39 (2006) [48] S. Kang, Q. Liu, H.Y. Meng, and T.Y. Shi, Phys. Lett. A 360 (2007) 608. [49] T.Y. Shi, H.X. Qiao, and B.W. Li, J. Phys. B: At. Mol. Opt. Phys. 33 (2000) L349. [50] T.Y. Shi, C.G. Bao, and B.W. Li, Commun. Theor. Phys. 35 (2001) 195. [51] L.Y. Tang, Z.C. Yan, T.Y. Shi, and J.F. Babb, Phys. Rev. A 79 (2009) [52] Y.B. Tang, C.B. Li, and H.X. Qiao, Chin. Phys. B 23 (2014) [53] Z. Ren, H. Han, T. Shi, and J. Mitroy, J. Phys. B: At. Mol. Opt. Phys. 45 (2012) [54] L.Y. Tang, Y.H. Zhang, X.Z. Zhang, J. Jiang, and J. Mitroy, Phys. Rev. A 86 (2012) [55] Y.B. Tang, H.X. Qiao, T.Y. Shi, and J. Mitroy, Phys. Rev. A 87 (2013) [56] L.Y. Tang, J.Y. Zhang, Z.C. Yan, T.Y. Shi, and J. Mitroy, J. Chem. Phys. 133 (2010) [57] S. Kang, C.K. Ding, C.Y. Chen, and X.Q. Wu, Commun. Theor. Phys. 60 (2013) 73. [58] D. Salzman, Atomic Physics in Hot Plasmas, Oxford University Press, New York (1998). [59] S. Bashkin and J.O. Stoner, Atomic Energy Levels and Grotrian Diagrams, North-Holland, Amsterdam (1975). [60] H. Bachau, P. Galan, and F. Martin, Phys. Rev. A 41 (1990) [61] S. Cohen and S.I. Themelis, J. Phys. B 38 (2005) [62] J. Pipin and D.M. Bishop, Phys. Rev. A 45 (1992) [63] N.E.B. Kassimi and A.J. Thakkar, Phys. Rev. A 50 (1994) [64] A.J. Thakkar and C. Lupinetti, in Atoms, Molecules and Clusters in Electric Fields: Theoretical Approaches to the Calculation of Electric Polarizability, Imperial College, London (2006). [65] C. Laughlin, J. Phys. B 28 (1995) L701. [66] M. Jaszunski and A. Rizzo, Int. J. Quantum Chem. 60 (1996) 487. [67] Z.C. Yan, J.F. Babb, A. Dalgarno, and G.W.F. Drake, Phys. Rev. A 54 (1996) 2824.