Ionization Potentials and Quantum Defects of 1s 2 np 2 P Rydberg States of Lithium Atom

Commun. Theor. Phys. (Beijing, China) 50 (2008) pp. 733 737 c Chinese Physical Society Vol. 50, No. 3, September 15, 2008 Ionization Potentials and Quantum Defects of 1s 2 np 2 P Rydberg States of Lithium Atom CHEN Chao Department of Physics, Beijing Institute of Technology, Beijing 100081, China (Received October 26, 2007; Revised December 6, 2007) Abstract In this work, ionization potentials and quantum effects of 1s 2 np 2 P Rydberg states of lithium are calculated based on the calibrated quantum defect function. Energy levels and quantum defects for 1s 2 np 2 P bound states and their adjacent continuum states are calculated with the R-matrix theory, and then the quantum defect function of the 1s 2 np (n 7) channel is obtained, which varies smoothly with the energy based on the quantum defect theory. The accurate quantum defect of the 1s 2 7p 2 P state derived from the experimental data is used to calibrate the original quantum defect function. The new function is used to calculate ionization potentials and quantum effects of 1s 2 np 2 P (n 7) Rydberg states. Present calculations are in agreement with recent experimental data in whole. PACS numbers: 31.25.Jf, 31.15.Pf, 32.30.Bv Key words: ionization potential, quantum effect, R-matrix theory 1 Introduction Recently, some advances have been made in theoretical and experimental research into the high l Rydberg states of the lithium atom. [1 5] As the simplest alkali-mental element, Rydberg states of lithium consist of a two-electron core and a distant Rydberg electron, and are nearly hydrogenic. However, the more subtle features of their spectra deviate from hydrogenic features especially for lower-l states. Such difference between lithium and the hydrogenic spectra arises from the interaction between the excited electron and the 1s 2 core instead of a point Coulomb charge. The most obvious effect is the depression of energies for lower-l states, which arises from the polarization and penetration of the core by the Rydberg electron. For Rydberg states of lithium with lower l (l= s, p) quantum numbers, there are some experimental measurements reported in the literature, [6 9] but it then occurs to us that the information in the literature concerning theoretical studies for Rydberg states of lithium with lower l quantum numbers is rather scarce, so the research from the theoretical aspect is interesting in itself. The variational method and configuration-interaction (CI) theory have been very successful for low-lying excited states of atomic systems. However, in the calculation of Rydberg states, one encounters great difficulties. Countless Rydberg states and continuum states near the threshold must be taken into account in CI and variational calculations, and the problem of convergence induced by rounding errors is a formidable task. Therefore it is difficult to obtain accurate energy levels of Rydberg states with the traditional variational method. The quantum defect theory (QDT) [10 14] has been developed and applied successfully in the field of spectroscopy and collision phenomena in atomic and molecular systems. In the framework of the QDT, infinite Rydberg states and adjacent continuum states are classified as channels. For the 1s 2 nl state of lithium, only the single channel QDT is needed to deal with energy levels of the Rydberg states. Physical parameters of the channel (e.g., the quantum defect µ) are smooth functions of the energy, so Rydberg energy levels can be calculated through the quantum defect function (QDF), which depends weakly on the energy. In this work, the QDF is used to calculate the ionization potentials and quantum effects of 1s 2 np 2 P Rydberg states of lithium. Ionization potentials and quantum defects for 1s 2 np 2 P bound states and their adjacent continuum states are calculated using the R-matrix theory. [15 17] By analyzing quantum defects of 1s 2 np 2 P low-lying Rydberg states derived from various theoretical and experimental data, the 1s 2 7p 2 P state is chosen as the initial state to obtain the QDF of the 1s 2 np channel using the results of the R-matrix calculation, which vary smoothly with the energy based on the QDT. The R- matrix method provides lower limits of quantum defects, therefore the quantum defect of the 1s 2 7p 2 P state derived from the experimental data [6] is used to calibrate the original QDF. The final QDF is used to calculate ionization potentials and quantum effects of the 1s 2 np 2 P (n 7) Rydberg states. The present results are in agreement with recent experimental data [8] in whole. It is indicated that the combination of the R-matrix method and the QDT is a very efficient method to study high Rydberg states of lithium with lower l quantum numbers. Calculations in this work should provide reference for more experimental investigations in future. 2 Theory The R-matrix theory has been described in detail in Refs. [15] [17], so a brief description is given here. In the R-matrix theory, a value a of the radial variable r is chosen such that exchange interactions between the scattered electron and target electrons are negligible for r a, where a is the R-matrix box radius. Within the reaction zone (r a), the interactions between the scattered electron and target electrons involve static electron-electron The project supported by National Natural Science Foundation of China under Grant No. 10404017, and the Basic Research Foundation of Beijing Institute of Technology

734 CHEN Chao Vol. 50 screening, dynamic polarizations, etc. It is a many-body problem, which is solved variationally for the whole system to obtain the logarithmic derivative boundary matrix R(E). Outside the reaction zone (r a), the scattered electron feels mainly the Coulomb potential. The present R-matrix code allows us to take into account the long-range static polarization potentials. The excited electron moves in the long-range multipole potential of the (N 1)-electron core, whose wave function is obtained by matching the boundary conditions. Therefore, the wave functions ψ k of the energy eigenstates for the system can be expanded as ψ k = A a ijk Φ i u ij (r) + b jk ϕ j, (1) ij j where A is the antisymmetrization operator which accounts for electron exchange between target electrons and the scattered electron, Φ i are the channel functions obtained by coupling the target-state wave functions with angular and spin functions of the scattered electron, the continuum orbitals u ij represent the motion of the scattered electron, and ϕ j are formed from the bound orbitals to ensure completeness and include the short-range correlation effects. Started with the logarithmic derivative boundary matrix R(E), physical eigenchannel parameters (eigenquantum defects µ and orthogonal transformation matrix U ia ), and corresponding eigenchannel wave functions are calculated directly by performing an eigenchannel treatment of the R-matrix theory. [18,19] Based on the compact set of eigenchannel parameters, atomic perturbed discrete Rydberg series, auto-ionizing states, and their adjacent continuum can be treated in an analytical unified manner without any numerical integration outside the R-matrix box. The detailed description of the eigenchannel treatment of R-matrix theory was given in Ref. [18]. According to the QDT, the quantum defect µ should be a smooth function of the energy, and which can be expressed in the following form µ(ε) = µ 0 + µ 0ε + µ 0ε 2, (2) where ε is the energy relative to the first ionization threshold. In this work, energy levels and quantum defects for 1s 2 np 2 P bound states and their adjacent continuum states of lithium are calculated by the R-matrix theory, and then the QDF of the whole channel is obtained, which varies smoothly with the energy based on Eq. (2). For lithium, the quantum defect µ can be determined from the formula µ = n 1, (3) 2ε where n is the principal quantum number. The second term on the right-hand side of Eq. (3) denotes the effective principal quantum number. Combining Eq. (2) with Eq. (3), an equation can be constructed to calculate the energy levels of 1s 2 np 2 P Rydberg states of lithium. This equation is in the following form 1 ε + 2[n µ(ε)] 2 = 0. (4) The advantage of the R-matrix theory is that the whole channel can be treated in an analytical unified manner and gives the tendency of the quantum defect varying smoothly with the energy. The appropriate initial state will be selected to obtain the QDF of the channel, and then the more accurate quantum defects of this initial state derived from the experimental data will be used to calibrate the original QDF; namely, using the difference of quantum defects of the initial state between the R-matrix method and the experimental data to calibrate the coefficient µ 0 in Eq. (2). The final QDF is used to calculate energy levels and quantum defects of 1s 2 np 2 P Rydberg states. Present theoretical predictions will be compared with the data derived from experimental measurements in the literature. 3 Results and Discussions Energy levels and quantum defects for the 1s 2 2 P states and their adjacent continuum states of lithium are firstly calculated by using the R-matrix method introduced in the preceding section. For the R-matrix calculation of lithium, the following set of basis orbitals is used which is calculated from the CIV3 code. [20] The target set is 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s, 5p, 5d, 5f, where nl (n 4, l 3) are spectroscopy orbitals, 5l( l 3) are polarized pseudo-orbitals including static polarization effects. [19,20] In order to obtain accurate results, 27 target functions are chosen carefully, and long-range multipole potentials are taken into account sufficiently. Then energy levels and quantum defects for 1s 2 np 2 P bound states and their adjacent continuum states are calculated. In this work, negative values of energy levels ε from the R-matrix calculation give ionization potentials of various states. Ionization potentials and quantum defects of 1s 2 np 2 P states of lithium calculated from R-matrix method are listed in Tables 1 and 2. It is easy to find that the quantum defect increases smoothly with n. Note that the reduced Rydberg constant of Li of 109 728.73 cm 1 is adopted in present calculations. The traditional variational method has the advantage in the calculation of low-lying excited states, so the results derived from the variational calculations are used to compare with present R-matrix calculations. Ionization potentials and quantum defects of 1s 2 np 2 P (n = 2 9) states of lithium from different theories and the experiments are also listed in Table 1 for comparisons. Theoretical quantum defects are calculated through Eq. (3), and experimental quantum defects are derived from differences between energy levels of 1s 2 np 2 P states and the first ionization threshold. According to the QDT, the quantum defect should be a smooth function of the energy, so the accuracy of the ionization potential can be checked by reviewing the corresponding quantum defect. The regularities of the quantum defect varying with the energy based on the data in Table 1 are illustrated in Fig. 1. Quantum defects calculated by the R-matrix method are the least group data in Fig. 1, which increase smoothly with the energy. Quantum defects derived from the full core plus correlation (FCPC) calculations [21 24]

No. 3 Ionization Potentials and Quantum Defects of 1s 2 np 2 P Rydberg States of Lithium Atom 735 are greater than the R-matrix method results and less than experimental results. But for the 1s 2 9p 2 P state, the FCPC result is less than the R-matrix result, and the trend inverses when n 7, which proves that the traditional variational method is difficult to calculate Rydberg states accurately. The trend of the experimental quantum defect of Johansson [6] is good but no 1s 2 np 2 P (n > 7) Rydberg states is measured. Based on the above discussion, the 1s 2 7p 2 P state is chosen as the initial state to obtain the QDF of the 1s 2 np channel by using quantum defects of finite 1s 2 np 2 P (n > 7) bound states and phase shifts of 1s 2 εp 2 P continuum states from the R- matrix calculation, which varies smoothly with the energy as µ(ε) = 0.046 883 + 0.054 545 4ε + 0.290 189ε 2. The quantum defect derived from the FCPC calculations is a little less than the true value for low-lying excited states, and the deviation of the FCPC result to the experimental result [6] is less than 0.1% for the 1s 2 7p 2 P state, therefore the experimental quantum defect for the 1s 2 7p 2 P state is credible and can be used to calibrate the original QDF. The final QDF is µ(ε) = 0.047 283 + 0.054 545 4ε + 0.290 189ε 2, (5) which can be used to calculate ionization potentials and quantum effects of 1s 2 np 2 P (n 7) Rydberg states of lithium. Quantum defects of 1s 2 np 2 P series of lithium as a function of the energy obtained from the R-matrix method and the QDF, i.e., equation (5) is illustrated in Fig. 2. It is shown that quantum defects obtained from the QDF vary smoothly with the energy to the threshold. Ionization potentials and quantum defects of 1s 2 np 2 P (n = 7 60) Rydberg states of lithium from different theories and the experiment are listed in Table 2. The present QDF utilizes the advantage of the R-matrix method in calculating high Rydberg states and continuum states, and then obtains more accurate ionization potentials and quantum defects by the calibration from the accurate experimental data of the lower Rydberg state. It is clearly seen from Table 2 that the QDF calculations agree with results of Haq et al. [8] in whole for ionization potentials, and discrepancies are very tiny and oscillate near zero; the greatest discrepancy is only 0.131% for n = 51. The quantum defect is the very important physical parameter used to calculate energies of Rydberg states. The present method can give precise quantum defects of high Rydberg states of lithium, which increase smoothly with n and excel results of Haq et al. [8] in whole trend. It is indicated that the present QDF is an effective method to calculate ionization potentials and quantum effects of 1s 2 np 2 P Rydberg states of lithium. Fig. 1 Quantum defects of 1s 2 np 2 P low-lying excited states of lithium as a function of the energy by different theories and the experiment. Fig. 2 Quantum defects of 1s 2 np 2 P series of lithium as a function of the energy by the R-matrix method and the QDF. Table 1 Ionization potentials (I.P.) and quantum defects (µ) of 1s 2 np 2 P (n = 2 9) states of lithium from different theories and the experiment (in a.u.). I.P. µ n R-matrix FCPC Expt. a R-matrix FCPC Expt. a 2 0.130 238 0 0.130 242 5 0.130 245 27 0.040 63 0.040 67 0.040 686 1 3 0.057 235 7 0.057 237 8 0.057 239 01 0.044 36 0.044 41 0.044 445 7 4 0.031 972 6 0.031 975 6 0.031 975 97 0.045 46 0.045 65 0.045 668 5 0.020 372 7 0.020 374 3 0.020 375 0 0.045 94 0.046 14 0.046 22 6 0.014 105 2 0.014 106 1 0.014 106 8 0.046 18 0.046 38 0.046 52 7 0.010 340 6 0.010 341 6 0.010 341 8 0.046 35 0.046 71 0.046 75 8 0.007 904 0 0.007 904 6 0.046 47 0.046 76 9 0.006 237 2 0.006 237 1 0.046 55 0.046 49 a Ref. [6].

736 CHEN Chao Vol. 50 Table 2 A comparison of ionization potentials and quantum defects of 1s 2 np 2 P states of lithium from different theories and the experiment, and the data in parentheses are the relative differences between experimental values and the QDF calculations for ionization potentials (in cm 1 ). I.P. µ n R-matrix QDF Expt. a R-matrix QDF Expt. a 7 2 269.31 2 269.58 0.046 35 0.046 750 0 8 1 734.60 1 734.78 0.046 47 0.046 870 1 9 1 368.80 1 368.92 0.046 55 0.046 954 1 10 1 107.59 1 107.68 0.046 62 0.047 015 1 11 914.592 914.659 0.046 66 0.047 060 7 12 767.971 768.022 0.046 70 0.047 095 7 13 653.975 654.015 0.046 72 0.047 123 0 14 563.598 563.630 0.046 75 0.047 144 8 15 490.739 490.764 490.95(0.038%) 0.046 76 0.047 162 5 0.05 16 431.145 431.167 431.34(0.040%) 0.046 78 0.047 177 0 0.05 17 381.783 381.801 381.90(0.026%) 0.046 79 0.047 189 0 0.049 18 340.437 340.452 340.67(0.064%) 0.046 80 0.047 199 1 0.053 19 305.461 305.474 305.61(0.045%) 0.046 81 0.047 207 6 0.051 20 275.611 275.622 275.73(0.039%) 0.046 82 0.047 215 0 0.05 21 249.941 249.99(0.020%) 0.047 221 3 0.049 22 227.689 227.72(0.014%) 0.047 226 7 0.049 23 208.281 208.40(0.057%) 0.047 231 5 0.055 24 191.253 191.27(0.009%) 0.047 235 7 0.048 25 176.231 176.34(0.062%) 0.047 239 4 0.053 26 162.912 162.98(0.042%) 0.047 242 7 0.052 27 151.048 151.17(0.081%) 0.047 245 6 0.058 28 140.434 140.48(0.033%) 0.047 248 2 0.052 29 130.900 130.90(0%) 0.047 250 6 0.047 30 122.306 122.39(0.069%) 0.047 252 7 0.057 31 114.531 114.60(0.060%) 0.047 254 6 0.062 32 107.474 107.57(0.089%) 0.047 256 4 0.061 33 101.050 101.06(0.010%) 0.047 257 9 0.048 34 95.1855 95.27(0.089%) 0.047 259 4 0.062 35 89.8169 89.87(0.059%) 0.047 260 7 0.057 36 84.8900 84.96(0.082%) 0.047 261 9 0.07 37 80.3576 80.33( 0.034%) 0.047 263 1 0.04 38 76.1788 76.18(0.002%) 0.047 264 1 0.047 39 72.3177 72.26( 0.080%) 0.047 265 1 0.031 40 68.7428 68.70( 0.062%) 0.047 265 9 0.034 41 65.4266 65.41( 0.025%) 0.047 266 8 0.041 42 62.3447 62.30( 0.072%) 0.047 267 5 0.031 43 59.4756 59.51(0.058%) 0.047 268 2 0.059 44 56.8000 56.87(0.123%) 0.047 268 9 0.073 45 54.3010 54.27( 0.057%) 0.047 269 5 0.033 46 51.9634 51.94( 0.045%) 0.047 270 1 0.036 47 49.7736 49.73( 0.088%) 0.047 270 6 0.025 48 47.7193 47.70( 0.040%) 0.047 271 2 0.036 49 45.7896 45.75( 0.086%) 0.047 271 6 0.025 50 43.9746 43.98(0.012%) 0.047 272 1 0.049 51 42.2654 42.21( 0.131%) 0.047 272 5 0.012 52 40.6540 40.67(0.039%) 0.047 272 9 0.056 53 39.1331 39.11( 0.059%) 0.047 273 3 0.03 54 37.6959 37.71(0.037%) 0.047 273 6 0.055 55 36.3364 36.36(0.065%) 0.047 274 0 0.063 56 35.0492 35.04( 0.026%) 0.047 274 3 0.038 57 33.8292 33.83(0.002%) 0.047 274 6 0.046 58 32.6718 32.66( 0.036%) 0.047 274 9 0.034 59 31.5728 31.56( 0.041%) 0.047 275 2 0.033 60 30.5283 30.51( 0.060%) 0.047 275 4 0.027 a The data are derived from Table 1 of Haq et al. [8] Table 3 The limit quantum defect of 1s 2 np 2 P series of lithium from different experiments and the theory. Author Haq et al. [8] Lorenzen and Niemax [25] Goy et al. [7] Bushaw et al. [9] This work µ 0.051(4) 0.047 263 0.047 170(2) 0.047 20(2) 0.047 283

No. 3 Ionization Potentials and Quantum Defects of 1s 2 np 2 P Rydberg States of Lithium Atom 737 The R-matrix method gives the lower limit of the limit quantum defect at the threshold of the 1s 2 np 2 P series of lithium as 0.046 883, and the calibrated one through the QDF is 0.047 283. The limit quantum defect of 1s 2 np 2 P series of lithium from different experiments and the theory are listed in Table 3 for comparisons. The experimental limit quantum defects are all obtained by extrapolating the finite np series data. The experimental limit quantum defect is reported in Ref. [8] as 0.051(4), which is overestimated. Lorenzen and Niemax [25] extrapolated the data of Johansson [6] and obtained the value as 0.047 263 based on a modified Ritz relation, which agrees with the present calculation through the QDF well. The limit quantum defect of the 1s 2 np series is calculated as 0.047 170(2) by results of 1s 2 np 1/2 series and 1s 2 np 3/2 series from Goy et al. [7] The latest experimental data reported by Bushaw et al. [9] is 0.047 20(2), which also agrees with the present calculation well. In summary, ionization potentials and quantum effects of 1s 2 np 2 P Rydberg states of lithium are calculated by using the QDF which is based on the R-matrix theory. Although quantum defects calculated by the R-matrix method have not been converged absolutely yet, the advantage of the R-matrix method in calculating high Rydberg states and continuum states is utilized to obtain the QDF of the channel, and then the calibration from the accurate experimental value of lower Rydberg states can provide the final QDF and more accurate quantum defects. Present calculations are in agreement with recent experimental data in whole. The research in this work indicates that the combination of the R-matrix method and the QDT is a very efficient method to study high Rydberg states of lithium with lower l quantum numbers. Present calculations should provide the reference for more experimental investigations in future. Acknowledgments The author expresses his gratitude to Prof. J.M. Li for his discussions. References [1] N.E. Rothery, C.H. Storry, and E.A. Hessels, Phys. Rev. A 51 (1995) 2919. [2] R.J. Drachman and A.K. Bhatia, Phys. Rev. A 51 (1995) 2926. [3] C.H. Storry, N.E. Rothery, and E.A. Hessels, Phys. Rev. A 55 (1997) 128. [4] A.K. Bhatia and R.J. Drachman, Phys. Rev. A 55 (1997) 1842. [5] A.K. Bhatia and R.J. Drachman, Phys. Rev. A 60 (1999) 2848. [6] I. Johansson, Ark. Fys. 15 (1959) 169. [7] P. Goy, J. Liang, M. Gross, and S. Haroche, Phys. Rev. A 34 (1986) 2889. [8] M.A. Haq, S. Mahmood, M. Riaz, R. Ali, and M.A. Baig, J. Phys. B 38 (2005) S77. [9] B.A. Bushaw, W. Nörtershäuser, G. W. F. Drake, and H.-J. Kluge, Phys. Rev. A 75 (2007) 052503. [10] M.J. Seaton, Mon. Not. R. Astron. Soc. 118 (1958) 504. [11] M.J. Seaton, Rep. Prog. Phys. 46 (1983) 167. [12] U. Fano and C.M. Lee, Phys. Rev. Lett. 31 (1973) 1573. [13] C.M. Lee and K.T. Lu, Phys. Rev. A 8 (1973) 1241. [14] C.M. Lee, Phys. Rev. A 10 (1974) 584. [15] P.G. Burke, A. Hibbert, and W. D. Robb, J. Phys. B 4 (1971) 153. [16] K.A. Berrington, P.G. Burke, J.J. Chang, et al., Comput. Phys. Commun. 8 (1974) 149. [17] K. A. Berrington, P. G. Burke, M. L. Dourneuf, W. D. Robb, K. T. Taylor, and L. VoKy, Comput. Phys. Commun. 14 (1978) 346. [18] J.M. Li, L. VoKy, Y.Z. Qu, et al., Phys. Rev. A 55 (1997) 3239. [19] J. Yan, Y.Z. Qu, L. VoKy, and J.M. Li, Phys. Rev. A 57 (1998) 997. [20] A. Hibbert, Comput. Phys. Commun. 9 (1975) 141. [21] K.T. Chung, Phys. Rev. A 44 (1991) 5421. [22] Z.W. Wang, X.W. Zhu, and K.T. Chung, Phys. Rev. A 46 (1992) 6914. [23] Z.W. Wang, X.W. Zhu, and K.T. Chung, J. Phys. B 25 (1992) 3915. [24] Z.W. Wang, X.W. Zhu, and K.T. Chung, Phys. Scr. 47 (1993) 65. [25] C.J. Lorenzen and K. Niemax, Phys. Scr. 27 (1983) 300.