Do atomic electrons stay bound in changing plasma environment?

Do atomic electrons stay bound in changing plasma environment? T. N. Chang 1 and T. K. Fang 2 1 Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, U.S.A. 2 Department of Physics, Fu Jen Catholic University, Taipei, Taiwan, 242, ROC We present a detailed theoretical study on the disappearance of bound states for two-electron atoms subject to external plasma environment. In contrast to a number of recent theoretical works, our analysis leads to the conclusion that the electron-electron Coulomb interaction between atomic electrons should not be modified by a long-range screening similar to the one-electron screened Coulomb potential for the atomic electrons. Our quantitative estimates have resolved the qualitative disparity on the disappearance of the bound states between the one- and two-electron atoms suggested implicitly by other recent theoretical studies. We also point out a few interesting problems for further study. PACS number(s); 32.70.Cs, 31.15.vj, 31.15.ag, 52.20.Dq 1

I. INTRODUCTION Theoretical study of atomic processes in plasma environment has attracted considerable renewal interests recently [1-11] following a number of earlier works [12-14] based on the application of the Debye-Hückel model [15] for a classical electron-ion collisionless plasma under thermodynamical equilibrium. Quantitative atomic data have been compiled for the bound-bound transitions, the photoionization, and the electron-atom collision for one- and two-electron atoms [1-14]. Almost all the recent theoretical estimations [1-11] employed a screened Coulomb potential (or, the Yukawa potential) for an atomic electron subject to a one-electron Hamiltonian, i.e., ( ; ) = 2 2 2 (1) where is the electron momentum and is the nuclear charge. The screening parameter is known as the Debye-Hückel length (or Debye length). It characterizes the screening of the nuclear charge experienced by the atomic electron due to the plasma in the outer region of an atom and is given in Bohr radius by (see also, e.g., Eq. (2.5) of [14]) = 1 304 10 8 ( ) 1 2 (2) where and are the plasma temperature (on Kelvin scale) and density (in 3 ), respectively. The value of ranges over many orders of magnitude, i.e., from 10 2 for the plasma in solar core to over 10 4 for the ones in intergalactic medium. Since the plasma temperature usually varies only 3 to 4 orders in magnitude (e.g., from 10 2 to 10 6 ), the value of (or, the degree of penetration of the ion-electron gas into the atom) depends more on the variation of the plasma density. For the one-electron atom, the most interesting feature resulting from the screened Coulomb potential is the disappearance of the bound state as becomes smaller than the corresponding critical screening length of that particular state (see, e.g., Tables III and IV of [12]). In other words, the potential can only attract a decreasingly finite number of bound states below the ionization threshold as decreases [12-14]. For the low-density plasmas such as the ones in interstellar medium or even for those generated in low power gas discharge, the Debye length may be a few orders of magnitude larger than the size of the atoms and should have little effect on the atomic processes. Since the plasma frequency for electron (i.e., 1 2 ) is also relatively small and the period of plasma oscillation is substantially longer than the time scale of a typical atomic process, the plasma environment effect on atomic transition is also expected to be minimal. On the other hand, for the high density plasma system such as the ones in the Sun and Sun-like stars, the Debye length is of the order of 10 11. 2

Consequently, the effect of the Debye screening could change significantly the atomic transitions for the astrophysical relevant elements. That, in turn, could have serious implication to the opacity data, which is of critical importance to the development of high quality equation of state for such systems [16]. For the laser-generated dense plasma at an electronic temperature of a few hundred and a density of 10 22 10 24 3 [17], the Debye length would be comparable to the size of the atoms. Although the laser plasma is far from a thermodynamically equilibrium system, a detailed theoretical estimate based on the Debye-Hückel model may help to better understand qualitatively the physics behind the observed spectra. Alternatively, a number of approaches investigating the quantum plasmas have also been proposed recently [18-20]. In particular, Shukla and Eliasson [18, 19] have derived a screening potential in terms of a characteristic length subject to a stationary charge in the plasma located at =0intheformof ( ; )= 2 ( ) (3) where the Shukla-Eliasson length isexpressedintermsoftheelectronplasmafrequency, i.e., =( ) 1 2 2 or, more conveniently, in terms of the plasma density (in 3 )by =1 802 10 4 1 (5) 1 4 It should be noted that unlike the Debye length, does not depend on the plasma temperature and =7 24 10 3 ( 1 2 )1 2 (6) from Eqs (2) and (5). In fact, and could easily differ by several orders of magnitude at the same plasma density. For example, could be comparable to the size of an atom at a density of 10 12 10 16 3 whereas the Debye length is still several orders of magnitude larger. As a result, the Shukla-Eliasson screening may start affecting the atomic processes at a much lower plasma density than the Debye screening. For the multi-electron atoms (e.g.,, and ),anumberofrecentworks[3,6-8] appear to suggest that many of the singly excited states remain bound even when is smaller than the average size of an isolated atom in the same singly bound excited states. (4) 3

In particular, for the ground state of, it concludes that even when is substantially smaller than the average size of the ground state of the isolated, its ground state remains bound [7]. Such implication is intuitively and qualitatively different from the disappearance of the bound states for the hydrogen atoms in plasma environment. The main objective of this paper is to point out that such qualitative disparity between the disappearance of the bound states in one- and two-electronatoms results from ana priori application of the Debye screening to the electron-electron interaction that is similar to the one for the one-electron orbit. It has perhaps unintentionally, but, effectively led to a potential for the atomic electrons that is significantly more attractive than it should be by reducing greatly the Coulomb repulsion between atomic electrons. Sec. II outlines the theory and calculation procedure for this study. Results and discussions are presented in Sec. III. II. THEORY AND CALCULATION PROCEDURE Following the Debye-Hückel original approximation, the potential ( ) due to an electron-ion plasma at a distance far from a force center (e.g., an atomic nucleus) outsideaninnerdebyesphereofradius can be derived from the Gauss law 5 2 ( ) = ( ) (7) where is the dielectric constant of the electron-ion gas and is its total charge density at. Assuming a charge density of at = withazeropotential ( = ) =0 and following the Boltzmann distribution, the charge densities of the positive charge and the negative charge at could be expressed as + ( ) = ( ) and ( ) = + ( ), respectively, where is the Boltzmann constant and the absolute temperature. The net total charge density at is then given by ( ) = ( ( ) ( ) )= 2 sinh ( ( ) ) (8) Assuming, in addition, that the potential energy is relatively small comparing to the kinetic energy, Eq. (7) for the potential ( ) in the outer region of the Debye sphere could then be approximated by the linear Poisson-Boltzmann equation 5 2 ( ) =( 1 2 ) ( ) (9) where is the Debye length defined by the charge density ( )andthe temperature outside the Debye sphere. The potential inside the Debye sphere can also be derived from the Gauss law and takes the form of ( ) = +Constant (10) 4

with a nucleus charge located at = 0. By matching the two potentials and and their first-order derivatives at =, one gets [13, 14] ( ) = 2 ( 1 1 ) + ( ) = (11) ( ) = 2 ( + ) In the limit when 0, Eq. (11) reduces approximately to the screened Coulomb potential employed in most of the recent theoretical atomic calculations. A more detailed atomic structure calculation subject to external plasma environment should replace the Yukawa potential in Eq. (1) by a potential ( ) given either by the Debye screening from Eq. (11) for an electron-ion collisionless plasma or proposed by Shukla and Eliasson in Eq. (3) for a nominally quantum plasma, i.e., ( ; )= 2 2 + ( ) (12) when or is comparable to the size of the atom. An alternative screened Coulomb potential for Debye screening with four parameters was introduced by Wang and Winkler [9] in their theoretical calculation. With the modified one-electron screening potential, the N-electron Hamiltonian for an atom in plasma environment can now be expressed in terms of ( ) given in Eq. (12) as ( ; )= X = 1 ( ; )+ X 2 (13) where = represents the separation between the atomic electrons and. The less attractive nature of the static screened Coulomb potential for an atomic electron in the field of the nuclear charge could be understood easily since close to the atomic nucleus, there is a non-negligible presence of the free-moving electrons from external plasma with their relatively high mobility. However, to justify a similar Coulomb screening between two atomic electrons would require an unlikely qualitative picture, i.e., a substantial presence of the positive ions between atomic electrons, in spite of the relatively low mobility for the much heavier ions. In addition, since the screened potentials discussed above are derived from Gauss law by assuming a stationary nuclear charge located at = 0, a similar approach could not be applied directly to the freemoving atomic electrons. In other words, there is no compelling reason to modify the electron-electron interaction by the same Coulomb screening. In fact, by including the Debye-like screening in the electron-electron interaction, the repulsive force between 5

atomic electrons would be reduced greatly; thus, leaving the atom in a bound state when or is smaller than the average size of an isolated atom in that particular bound state. The numerical results presented in Sec. III are calculated with the B-spline based configuration interaction (BSCI) method which has been applied successfully to a large number of atomic structure properties. Details of the theoretical approach, the computation procedures, and a large number of its applications have already been presented elsewhere [21]. The energies of the bound states and the oscillator strengths between various states are calculated typically in this study with a basis set of atomic orbitals representing over 10,000 two-electron configurations. III. RESULTS AND DISCUSSIONS Our calculated results are in excellent agreement with the most accurate non-relativistic theoretical results [22] in the absence of plasma environment, i.e., at =10 12. For the excitation energies, the agreement is about 10 4 or better and the oscillator strengths to three to four digits. In addition, the agreement between length and velocity results in oscillator strength is typically at 0 1% or better. As we pointed out earlier, the main objective of this work is to understand the qualitative disparity in the disappearance of bound states between the one- and twoelectron atoms implied from the recent theoretical estimates. We will show that such difference indeed results from the electron-electron Debye screening included in those calculations for the two-electron atoms. To that end, the top plot of Fig. 1 presents the comparison between our results with Hamiltonian given by Eq. (13) with =0 and the ones by Eq. (1) of [6] using an electron-electron interaction term in the form of. The -dependent threshold energies for + (1 ) are identical from both calculations as they should be. The energy for the 1 2 1 ground state from the present calculation intersects the threshold energy at a substantially larger (greater than 1 5 ) than a value of less than 0 5 from [6]. We should note that our results are in qualitative agreement with a somewhat limited calculation by Mukherjee et al which employed the one-electron Yukawa potential without the Debye screening in the electron-electron interaction (see, the results of model B shown in Fig. 3 of [2].) With the Shukla-Eliasson potential, the result given in [8] shows that the ground state remains bound at =0 75 whereas the results from the present calculation with Eq. (13) suggest that the ground state disappears at a much larger value of that is close to 2 (as shown in the bottom plot of Fig. 1). 6

A more rigorous application of the Debye-Hückel model using Eq. (11) should lead to a more negative + (1 ) ground state energy since the attractive potential in the outer region with is enhanced by the factor. Fig. 2 shows a substantial + change in the + (1 ) threshold energies with =1 0 and our calculated 1 2 1 ground state energy intersects the + threshold at around 1 0. This suggests a critical length approximately at 1 0 with a qualitative feature similar to the oneelectron atoms. In contrast, the estimated ground state energies and the + threshold energies from [6], with a Debye screened electron-electron potential, remain far apart at =1 0. In other words, the estimates from [6] and [8] imply that the ground state will remain bound at a distance smaller than the average size of an isolated atom until = 0 45 or 0 75, respectively. The top plot of Fig. 3 shows the results of the present calculation for atom using screened Coulomb potential. Our estimated critical length is slightly greater than 17 (close to the value of ) where the energy of the 1 ground state intersects the (1 ) threshold energy. This suggests the disappearance of the ground state at a value of much greater than a value of that is smaller than 1 4 suggested by [7]. Again, this difference is due to the use of Debye screening for the electron-electron interaction which has the effect of reducing substantially the repulsive interaction between atomic electrons. Our results are in qualitative agreement with the pair-function calculation by Wang and Winkler which employed the Yukawa potential without the Debye screening in the electron-electron interaction (see, Table I of [9]). The bottom plot of Fig. 3 shows that the ground state disappears at a greater (close to 35 ) than the one calculated with the screened Coulomb potential. Thisisconsistentwiththefact that the one-electron bound state energies subject to Shukla-Eliasson potential are less negative than the ones due to the screened Coulomb potential. Fig. 4 presents our results on the energy variation of the =2and3singly excited states as changes. We have nominally chosen = 2.Forthe =2singly excited states, their disappearances occur at values ranging from 7 for the 1 2 3 state to slightly less than 12 for the 1 2 1 state. Again, the results from [6] suggest that the = 2 excited states remain bound even when is as small as 1,i.e.,when is substantially smaller than the radius of the isolated atom. The disappearance of the singly excited = 3 states starts at as large as 26 for the 1 3 1 state to around 19 for the 1 3 3 state. Our investigation concludes clearly that there is no qualitative disparity between the disappearances of the bound states of the one- and two-electron atoms. Similar to the ground state of, our results are in qualitative agreement with the pair-function calculation by Wang and Winkler (see, Table IV and Fig. 1 of [9]). We also note that the nominal critical lengths from [9] are greater than 7

theonessuggestedbythepresentcalculation. This is consistent with the shorter ground state nominal critical length for the one with =1 than the one with =0 shown respectively in Fig. 2 and the top plot of Fig. 1. Fig. 5 compares the energy variations of and for a number of bound excited states from calculations with screened Coulomb potential and Shukla-Eliasson potential. Our results suggest that the bound excited states are pushed into the continuum at a larger screened length due to the Shukla-Eliasson potential than the screened Coulomb potential. Again, this is qualitatively similar to the and ground states discussed above. As for the atomic transition, our calculation shows that there is indeed a decrease in the energy differences between the bound states and a small variation in the corresponding oscillator strength. For instance, the excitation energy for the transition from the 1 2 1 ground state to the 1 2 1 excited state decreases by about 6% together with a small reduction (about 10%) in the oscillator strength as decreases to the limit when the 1 2 1 state disappears. This is qualitatively consistent with the observed red shift in the emission spectra for the He-like atoms [3, 17]. Our results are in qualitative agreement with a somewhat limited calculation by Lopez et al which employed the Yukawa potential without the Debye screening in the electron-electron interaction (see, e.g., the results with = 0 from [11].) Although there is minor quantitative difference between the present calculation and other calculations with the screened electron-electron interaction [5], there is little qualitative difference in the variation of the transition energy and oscillator strength as or decreases. More detailed quantitative results and comparison with the results from other calculations will be presented elsewhere. The disappearance of the singly excited bound states due to the plasma environment may also affect substantially the presence of the doubly excited resonances in the ground state photoionization spectra. Since the spectra of such resonances, for isolated atoms, are dictated primarily by the one-electron transitions from the 1 or 2 configurations mixed in the initial state to the doubly excited 2, 2, or 2 configurations in the final states, the transition amplitude leading to the doubly excited resonances would be greatly impacted by the disappearance of the corresponding 1 1 bound states as or decreases. The lack of low-lying singly bound excited states could also reduce substantially the dielectronic recombination rate. The other interesting question for further exploration is what happens to the 1 2 1 1 1 transition when the individual bound excited state is pushed into the continuum, if it indeed occurs, as or decreases. In conclusion, we present in this paper the Hamiltonian for a multi-electron atom subject to external plasma environment that differs from the ones employed in other 8

recent calculations. In particular, we have concluded that there is no theoretical ground to include a long-range Coulomb screening in the electron-electron interaction between the atomic electrons. Our quantitative analysis has led to a more intuitively expected qualitative picture on the disappearance of the bound states. Additional detailed quantitative calculations are required for a more definitive account of the critical length of each excited state for multielectron atoms. We have also pointed out a number of other interesting problems for further investigations and the potential impact on the development of high quality equation of motion for high density solar plasma. ACKNOWLEDGMENTS This work was supported by the National Science Council in Taiwan under the grant nos. NSC 99-2112-M-030-003 and NSC 100-2119-M-007-001. One of us (TNC) wishes to acknowledge a few in-depth discussions with Werner Däppen. References [1] A. N. Sil and P. K. Mukherjee, Int. J. Quantum. Chem. 102, 1061 (2005). [2]P.K.Mukherjee,J.Karwowski,andG.H.F.Diercksen,Chem.Phys.Lett.363, 323 (2002). [3]H.Okutsu,T.Sako,K.Yamanouchi,andG.H.F.Diercksen,J.Phys.B38, 917 (2005). [4] S.B.Zhang,J.G.Wang,andR.K.Janev,Phys.Rev.A81, 032702 (2010); Phys. Rev. Lett. 104, 023203 (2010); Y. Y. Qi, Y. Wu, J. G. Wang, and Y. Z. Qu, Phys. Plasmas 16, 023502 (2009). [5] S. Sahoo and Y. K. Ho, J. Quant. Spectrosc. Radiat. Trans. 111, 52 (2010); S. Kar and Y. K. Ho, J. Quant. Spectrosc. Radiat. Trans. 109, 445 (2008); S. Kar and Y. K. Ho, Phys. Plasmas 15, 013301 (2008). [6] S. Kar and Y. K. Ho, Int. J. Quantum Chem. 106, 814 (2006). [7] A.GhoshalandY.K.Ho,J.Phys.B42, 175006 (2009). [8] A.GhoshalandY.K.Ho,J.Phys.B42, 075002 (2009). [9] Z. Wang and P. Winkler, Phys. Rev. A52, 216 (1995). 9

[10] S. T. Dai, A. Solovyova, and P. Winkler, Phys. Rev. E64, 016408 (2001). [11] X. Lopez, C. Sarasola, and J. M. Ugalde, J. Phys. Chem. A101, 1804 (1997). [12] F. J. Rogers, H. C. Graboske, Jr., and D. J. Harwood, Phys. Rev. A 1, 1577 (1970). [13] C. A. Rouse, Phys. Rev. 163, 62 (1967). [14] H.MargenauandM.Lewis,Rev.Mod.Phys.31, 569 (1959). [15] P. Debye and E. Hückel, Phyzik Z. 24, 185 (1923). [16] W. Däppen, Astrophys. Space Sci., 328, 139 (2010). [17] A. Saemann et al, Phys. Rev. Lett. 82, 4843 (1999); Y. Leng, J. Goldhar, H. R. Griem,andR.W.Lee,Phys.Rev.E52, 4328 (1995); R. C. Elton, J. Ghosh, H. R. Griem, and E. J. Iglesias, Phys. Rev. E69, 067403 (2004). [18] P. K. Shukla and B. Eliasson, Phys. Lett. A372, 2897 (2008). [19] P. K. Shukla and B. Eliasson, Rev. Mod. Phys. 83, 885 (2011). [20] U. Gupta and A. K. Rajagopal, J. Phys. B12, L703 (1979); Yu. V. Arkhipov, F. B. Baimbetov, and A. E. Davletov, Eur. Phys. J. D8, 299 (2000); S. Ali and P. K. Shukla, Phys. Plasmas 13, 102112 (2006); G. Brodin, M. Marklund, and G. Manfredi, Phys. Rev. Lett. 100, 175001 (2008); D. Else, R. Kompaneets, and S. V. Vladimirv, Phys. Rev. E82, 026410 (2010) [21] T. N. Chang, in Many-body Theory of Atomic Structure and Photoionization, edited by T. N. Chang (World Scientific, Singapore, 1993), p. 213; T. N. Chang and T. K. Fang, Radiat. Phys. Chem. 70, 173 (2004); T. N. Chang and T. K. Fang, Phys. Rev. A52, 2638(1995);T.N.ChangandX.Tang,Phys.Rev.A44, 232 (1991); T. N. Chang, Phys. Rev. A39, 4946 (1989). [22] B. Schiff, C. L. Pekeris, andy. Accad, Phys. Rev. A4, 885 (1971); A. Kono and S. Hattori, Phys. Rev. A 29, 2981 (1984). 10

Figure Captions Figure 1: The top plot presents the energy variation of the + (1 ) threshold and the 1 2 1 ground state as functions of 1 with = 0. The results from [6] are presented for comparison. The bottom plot presents the energies of the + (1 ) threshold and the 1 2 1 ground state as functions of 1. The results from [8] are presented for comparison. Figure 2: The -variation of the + (1 ) threshold and the 1 21 ground state energy using the screened Coulomb potential with =1 0. The results from [6] are also presented for comparison. Figure 3: The energy variations of the ground state and the ionization threshold of as functions of and. Figure 4: The top plot presents the energies of the (1 2 ) 1,(1 2 ) 3,(1 2 ) 1, and (1 2 ) 3 states as functions of with =4 0. The bottom plot presents the energies of the (1 3 ) 1,(1 3 ) 3,(1 3 ) 1,and(1 3 ) 3 states as functions of with =9 0. Figure 5: The energy variations in and for a few bound excited states. 11