This study aims to present the resonances in positro–atom scattering with kappa-distribution plasmas. During the past few years, the applicability of Lorentzian (or kappa) distribution functions in astrophysical and space environments, such as solar-wind, solar corona, planetary magnetospheres, planetary nebula, supernova magnetospheres has been studied extensively^[1–8] (references therein). In astrophysical and space plasma environments, the velocity distributions of inner particles can be well-described by non-Maxwellian or Lorentzian distribution. The kappa distribution function which was first shown by Vasyliunas^[9] can be described in the following form:^[10–20] 1 where κ is the spectral index of the plasma, m and v are the mass and velocity of the plasma electron, E_κ = [(1−3/2κ)E_T] is the characteristic energy in generalized Lorentzian plasma, E_T = k_B T_e, k_B is the Boltzmann constant, T_e is the electron temperature, and n is the number density. The effective potential for two-body system in kappa-distribution plasma having charges q_a and q_b can be written as 2 where μ_κ is the effective screening parameter in kappa-distribution plasma, and μ_κ can be written as , μ_D = 1/λ_D is the Debye screening parameter, λ_D is the standard Debye length in Maxwellian plasma. In the limit κ → ∞, the kappa distribution functions become a simple Maxwellian function.

In this work, we are mainly interested in describing the behavior of resonance parameters (positions and widths) of positronic helium and positronic lithium under the influence of kappa-distribution plasmas. Resonances in positro–helium system using the model potential approach were first studied by Kar and Ho^[21], who used the stabilization method (SM). Ren et al.^[22] have performed a three-body calculation for the S-wave resonances in the positro–helium scattering in the hyperspherical coordinates by using the SM. Yu et al.^[23] reported their investigation on resonances in e⁺–He scattering by using the momentum-space coupled-channel optical method. The first theoretical study of resonance phenomena for positro–lithium scattering in a vacuum was carried out by Ward et al.^[24], who used the close-coupling approach. Roy and Ho^[25,26] calculated the resonance parameters in positro–lithium scattering using the SM based on the model potential approach and Hylleraas-type basis functions. Han et al.^[27] calculated S-wave resonances in positronic lithium using the SM in the framework of hyperspherical coordinate. Liu et al.^[28] have calculated the resonances for positronic Li using the momentum-space coupled-channel optical method. In plasma environments, the recent calculations of S-wave resonances in positro–helium system and positro–lithium system under the Debye screening were performed by Ghoshal and Ho^[29,30] using SM.

In the present investigation, we use the stabilization method^[31–33] to extract the S-wave resonance parameters (both positions and widths) for the proposed systems below the Ps(2s) threshold. Hylleraas-type wave functions are used to represent the correlation effects between the outer electron, the positron and the core. We obtain the resonance parameters for different values of spectral index κ and Debye screening parameter μ_D. All of the calculations are performed on IBM machines by using quadruple precision arithmetic in the CENT operating system.

The non-relativistic Hamiltonian for positro–helium and positro–lithium systems in kappa-distribution plasma is given by^{[14,15,21,25,26]} 3 with 4 5 where r₁ and r₂ denote, respectively, the radial coordinate of electron and positron with respect to core. The V_e⁻ (r₁ ) and V_e⁺ (r₂ ) are used to represent the model potentials for interactions between the active electron and the respective ionic cores, and the positron and the respective cores, respectively. The well-tested explicit expressions for V_e⁻ (r₁ ) and V_e⁺ (r₂) can be considered as^[34–41] 6 7 where A = Z − 1. In this study, we set B = 1.6559, C = 1.7212 for e⁺–Li^[25] system and B = C = 2.601 for e⁺–He^[21] system. Using these model potentials, the energies for the (1s2s ¹S, 1s2p ¹P) states of helium and (1s²2s ²S, 1s²2p ²P) states of lithium obtained from this calculations for free-atomic system are (−0.14595, −0.12548) a.u. (the unit a.u. is short for atomic unit) and (−0.19814, −0.12993) a.u., respectively, which are in good agreement with the experimental values (−0.14595, 0.12383) a.u.^[21] and (−0.19814, −0.13024) a.u..^[25] It is important to mention here that we set the same values of the parameters B and C in the model potentials for the free-atomic systems and the screening environments as recommended in the earlier studies.^{[38,39,42,43]}

To represent the correlation effect between the electron, positron and the core, we consider the Hylleraas-type basis functions in the form of^[44–46] 8 where l_i, m_i, and n_i are positive integers or zero, and α is the nonlinear variational parameter, N is the number of basis terms. Here, N has been conveniently chosen to be the well-known relation N = (ω+1)(ω+2)(ω+3)/6.^[47–50] Note that we use the wave-functions(8) to calculate the S-state only and the total angular momentum is zero for the S-wave.

To calculate the Ps(2s) and He⁺(1s) threshold energy, we employ the Slater-type wavefunctions: 9 Exploiting the model potential approach, we estimate the 1s²2s ²S state energy of Li by using wavefunctions (9).

In the present calculation, we have employed the SM^{[21,22,25–27,29–33,50–53]} to extract the resonance states of positro–helium and positro–lithium systems in kappa-distribution plasma. To extract resonance parameters with using the SM, in the first step, one needs to determine the energy levels precisely. In this work, we calculate the energy levels by diagonalizing the Hamiltonian with basis set (8). After calculating the energy levels E(α, μ, κ) for varying α for different values of μ and κ, we construct the stabilization diagrams by plotting E(α, μ, κ) versus α for fixed μ and κ. The resonance positions can be identified by observing the slowly decreasing energy levels at avoided crossings (e.g., as marked in red, see Fig. 1) in the stabilization diagrams. To extract the resonance energy E_r and width Γ for a particular resonance state, we calculate the density of the resonance states by using the well-known formula^[32] 10 where the index i is the i-th varied value for α_i, and the index n is for the n-th resonance. The α_i−1 and α_{i + 1} are the (i − 1)-th and (i + 1)-th varied α values next to α_i. Once the density of resonance states has been calculated by using formula (10), we plot the density of states versus E(α, μ, κ) and use the standard Lorentzian fitting formula having the baseline offset y₀ and total area A 11 where E_r is the center of the peak and Γ denotes the full width at the half height of the peak. It is interesting to point out that we obtain the resonance energy E_r and total width Γ by observing the best fitting in all the fittings at the avoided crossings for a particular resonance. In this way, we obtain the resonance parameters (E_r, Γ) for each κ and μ. We perform the calculations with ω = 13 and N = 560 for different α values within 0.2–0.4. Figures 1 and 2 show the stabilization diagram and the best fitting of the density resonance state below the Ps(2s) threshold in positro–helium system in kappa-distribution plasma for κ = 6, μ_D = 0.01.

Fig. 1.

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	Fig. 1. (color online) Stabilization plot for the lowest S-wave resonance in positro–helium system in kappa-distribution plasma for κ = 6, μD = 0.01 (in unit a.u.). Red circles indicate avoided crossings.

Fig. 2.

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	Fig. 2. (color online) The best fitting (solid line) of calculated density of resonance states (circles) to the Lorentzian form for S-wave resonance of positro–helium system below Ps(2s) threshold in kappa distribution plasma for κ = 6, μD = 0.01 (in unit a.u.).

Similarly, Figures 3 and 4 show the stabilization diagram and the best fitting of the density resonance state below the Ps(2s) threshold in positro–lithium system in kappa-distribution plasma for κ = 6, μ_D = 0.01. All of the values of resonance parameters obtained from this work for different κ and μ_D are shown in Tables 1 and 2 and Figs. 5–8. To show the convergence with increasing the number of terms in the basis functions, we also include the results for ω = 12 and N = 455 for selected screening parameters in Tables 1 and 2.

Fig. 3.

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	Fig. 3. (color online) Stabilization plot for the lowest S-wave resonance of positro–lithium system in kappa-distribution plasma for κ = 6, μD = 0.01 (in unit a.u.).

Fig. 4.

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	Fig. 4. (color online) The best fitting (solid line) of calculated density of resonance states (circles) to the Lorentzian form for S-wave resonance of positro–lithium system below Ps(2s) threshold in kappa distribution plasma for κ = 6, μD = 0.01 (in unit a.u.).

Table 1.

Table 1.

Table 1. Values of S-wave resonance energy Er and width Γ for positro–helium system below the Ps(2s) threshold in kappa-distribution plasma for selected spectral index κ and Debye screening parameter μD. . μD κ He+ (1s 2S) He (1s2 1S) Er Γ Ps(2s) threshold a.u. a.u. a.u. eV (10−4) a.u. meV a.u. eV 0.01 2 −1.96558 −2.85211 −0.059885 22.494 2.48 6.75 −0.046852 22.849 4 −1.97644 −2.86839 −0.064983 22.503 2.50 6.80 −0.051466 22.871 6 −1.97798 −2.87070 −0.065718 22.504 2.50 6.80 −0.052143 22.873 8 −1.97860 −2.87163 −0.066016 22.504 2.50 6.80 −0.052419 22.874 −0.066002a 22.505a 2.50a 6.80a 10 −1.97894 −2.87214 −0.066177 22.504 2.50 6.80 −0.052568 22.875 0.02 2 −1.93161 −2.80116 −0.044997 22.437 2.30 6.26 −0.034144 22.733 4 −1.95309 −2.83337 −0.054212 22.478 2.43 6.61 −0.041884 22.814 6 −1.95614 −2.83795 −0.055579 22.483 2.45 6.67 −0.043066 22.823 8 −1.95738 −2.83980 −0.056135 22.484 2.45 6.67 −0.043549 22.827 10 −1.95805 −2.84080 −0.056437 22.485 2.45 6.67 −0.043813 22.829 0.03 2 −1.89807 −2.75089 −0.032109 22.333 2.01 5.47 −0.023920 22.556 4 −1.92994 −2.79866 −0.044312 22.433 2.29 6.23 −0.033584 22.725 6 −1.93448 −2.80547 −0.046189 22.444 2.32 6.31 −0.035124 22.745 8 −1.93632 −2.80823 −0.046957 22.448 2.34 6.37 −0.035759 22.753 −0.046941a 22.449a 2.31a 6.29a 10 −1.93732 −2.80972 −0.047376 22.450 2.34 6.37 −0.036105 22.757 0.04 2 −1.86496 −2.70127 −0.021356 22.176 1.64 4.46 −0.015849 22.326 4 −1.90700 −2.76427 −0.035351 22.366 2.10 5.71 −0.026431 22.608 6 −1.91300 −2.77327 −0.037611 22.386 2.15 5.85 −0.028205 22.642 8 −1.91543 −2.77691 −0.038542 22.393 2.18 5.93 −0.028942 22.655 10 −1.91675 −2.77889 −0.039051 22.397 2.19 5.96 −0.029346 22.661 0.05 2 −1.83227 −2.65230 −0.012807 21.966 1.26 3.43 −0.009671 22.051 4 −1.88425 −2.73019 −0.027376 22.274 1.87 5.09 −0.020320 22.466 6 −1.89170 −2.74134 −0.029884 22.307 1.95 5.31 −0.022217 22.515 8 −1.89471 −2.74585 −0.030927 22.319 1.98 5.39 −0.023013 22.534 −0.030909a 22.320a 1.91a 5.20a 10 −1.89634 −2.74830 −0.031498 22.326 1.99 5.42 −0.023451 22.545 0.06 2 −1.79999 −2.60396 −0.006515 21.700 0.98 2.67 −0.0051722 21.736 4 −1.86171 −2.69640 −0.020413 22.158 1.61 4.38 −0.015158 22.301 6 −1.87056 −2.70967 −0.023034 22.207 1.71 4.65 −0.017083 22.369 8 −1.87415 −2.71505 −0.024136 22.225 1.75 4.76 −0.017900 22.395 10 −1.87609 −2.71796 −0.024745 22.235 1.77 4.82 −0.018351 22.409 a Present results with ω = 12 and N = 455.

Table 1. Values of S-wave resonance energy Er and width Γ for positro–helium system below the Ps(2s) threshold in kappa-distribution plasma for selected spectral index κ and Debye screening parameter μD. .

Table 2.

Table 2.

Table 2. Values of S-wave resonance energy Er and width Γ for positro–lithium system below the Ps(2s) threshold in kappa-distribution plasma for selected spectral index κ and Debye screening parameter μD. . μD κ Li (1s22s 2S) Er Γ Ps(2s) threshold a.u. a.u. eV (10−4) a.u. meV a.u. eV 0.01 2 −0.18102 −0.063902 3.187 1.92 5.22 −0.046852 3.651 4 −0.18633 −0.068972 3.193 1.94 5.28 −0.051466 3.670 6 −0.18709 −0.069703 3.194 1.94 5.28 −0.052143 3.672 8 −0.18739 −0.070000 3.194 1.94 5.28 −0.052419 3.673 −0.069885a 3.197a 1.94a 5.28a 10 −0.18756 −0.070161 3.195 1.94 5.28 −0.052568 3.673 0.02 2 −0.16498 −0.049131 3.152 1.76 4.79 −0.034144 3.560 4 −0.17502 −0.058268 3.177 1.89 5.14 −0.041884 3.623 6 −0.17648 −0.059625 3.180 1.91 5.20 −0.043066 3.630 8 −0.17707 −0.060177 3.181 1.91 5.20 −0.043549 3.633 10 −0.17739 −0.060477 3.181 1.91 5.20 −0.043813 3.635 0.03 2 −0.14994 −0.036368 3.090 1.31 3.56 −0.023920 3.429 4 −0.16421 −0.048452 3.150 1.73 4.71 −0.033584 3.555 6 −0.16630 −0.050313 3.156 1.78 4.84 −0.035124 3.569 8 −0.16715 −0.051075 3.159 1.79 4.87 −0.035759 3.575 −0.050957a 3.162a 1.70a 4.63a 10 −0.16761 −0.051488 3.160 1.80 4.90 −0.036105 3.578 0.04 2 −0.13586 −0.025686 2.998 0.632 1.72 −0.015849 3.266 4 −0.15387 −0.039578 3.110 1.46 3.97 −0.026431 3.468 6 −0.15654 −0.041816 3.122 1.54 4.19 −0.028205 3.492 8 −0.15763 −0.042739 3.126 1.56 4.24 −0.028942 3.502 10 −0.15822 −0.043242 3.129 1.58 4.30 −0.029346 3.507 0.05 2 −0.12269 −0.017105 2.873 0.079 0.21 −0.009671 3.075 4 −0.14398 −0.031674 3.056 1.06 2.88 −0.020320 3.365 6 −0.14718 −0.034160 3.075 1.20 3.27 −0.022217 3.400 8 −0.14848 −0.035195 3.083 1.25 3.40 −0.023013 3.414 −0.035076a 3.086a 1.25a 3.40a 10 −0.14919 −0.035762 3.087 1.28 3.48 −0.023451 3.422 0.06 2 −0.11037 −0.010418 2.720 0.0079 0.021 −0.0051722 2.863 4 −0.13452 −0.024745 2.987 0.561 1.53 −0.015158 3.248 6 −0.13820 −0.027357 3.016 0.763 2.08 −0.017083 3.296 8 −0.13970 −0.028455 3.027 0.835 2.27 −0.017900 3.314 10 −0.14052 −0.029060 3.033 0.886 2.41 −0.018351 3.324 a Present results with ω = 12 and N = 455.

Table 2. Values of S-wave resonance energy Er and width Γ for positro–lithium system below the Ps(2s) threshold in kappa-distribution plasma for selected spectral index κ and Debye screening parameter μD. .

Table 1 shows the He⁺(1s ²S) state energies, He(1s^{2 1}S) energies, the lowest resonance parameters for positro–helium system and the Ps(2s) threshold energies in kappa-distribution plasma in terms of κ and μ_D. Similarly, Table 2 displays the Li(1s²2s ²S) state energies, the lowest resonance parameters for positro–lithium system and the Ps(2s) threshold energies in kappa-distribution plasmas in terms of κ and μ_D. In the tables, the energies are listed both in atomic units (a.u.) and electron volts (eV), and the resonance widths are presented in units a.u. and meV. To convert energies into eV, we estimate the energies of ground states of the respective atomic systems. As our model potential is related to the He⁺ core, the energy of the He⁺ core (as shown in Table 1 for each screening parameter) should be added to our results (as shown in Table 1) to obtain the total energy of a resonance. The total resonance energies E_r and Ps(2s) threshold are converted into eV as measured from the He(1s^{2 1}S) energies. We calculate the He (1s^{2 1}S) energies for each screening parameters by using the Hylleraas-type basis functions in Eq. (8) within the framework of Ritz variational method. The He (1s^{2 1}S) eigen energies are listed in Table 1. In Table 2, we convert the resonance energies E_r and Ps(2s) threshold into eV relative to Li(1s²2s ²S) state energies. We calculate the Li(1s²2s ²S) energies by using the Slater-type orbitals (9) in the framework of model potential approach and the Li(1s²2s ²S) energies for different screening parameters are listed in Table 2. Figures 5 and 6 display the surface plots of resonance energy E_r and width Γ for positro–helium system each as a function of spectral index κ and Debye screening parameter μ_D. Figures 7 and 8 exhibit the same plots for positronic Li. It is seen evident from the figures and tables that the resonance parameters (in unit of eV) increase with κ increasing and decreases with μ_D increasing.

Fig. 5.

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	Fig. 5. (color online) Resonance energy for positro–helium system as a function of spectral index κ and Debye screening parameter μD (in unit eV).

Fig. 6.

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	Fig. 6. (color online) Resonance width for positro–helium system as a function of spectral index κ and Debye screening parameter μD (in unit meV).

Fig. 7.

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	Fig. 7. (color online) Plots of resonance energy for positro–lithium system as function of spectral index κ for different values of Debye screening parameter μD (in unit eV).

Fig. 8.

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	Fig. 8. (color online) Plots of resonance width for positro–lithium system as function of spectral index κ for different values of Debye screening parameter μD (in unit meV).

For the free-atomic system, the resonance parameters (E_r, Γ) (in atomic units) of positro–helium system estimated from this calculation are (−0.076505, 0.00025) which is in agreement with the resonance parameters (−0.07653, 0.00025), (−0.07654, 0.00023), (0.07667, 0.00143), and (−0.076495, 0.00023) reported respectively by Kar and Ho,^[21] Ren et al.,^[22] Yu et al.,^[23] Ghoshal and Ho.^[29] The resonance parameters (E_r, Γ) (in atomic units) of positron-lithium system obtained from this work are (−0.080476, 0.000194) which are comparable to the results (−0.080495, 0.000205), (−0.080658, 0.000202), and (−0.080454, 0.000193) reported, respectively, by Roy and Ho,^[25] Han et al.,^[27] Ghoshal and Ho.^[30] Apart from this theoretical work, positronic He and Li are also of experimental interest.^[54,55] For kappa-distribution plasma environments, there are no other theoretical researches of e⁺–He and e⁺–Li systems in the literature for comparison. At present, the theoretical results might be used for astrophysical plasma diagnostic purposes. In the future, positronic He or Li may be of experimental interest due to importance of kappa-distribution plasma in astrophysics and stellar atmosphere.

In this work, we have investigated the S-wave resonances of positro–helium and positro–lithium systems in kappa-distribution plasma based on a simple and powerful variational stabilization method. The Hylleraas-type wave functions are used to represent the correlation effect between the charged particles. A model potential approach is used to represent the interactions between the outer electron, the positron and the core. We employ the Lorentzian fitting technique to extract resonance parameters. With applicability of Lorentzian (kappa-distribution) astrophysical plasmas^[1–8] and with recent progress of atomic process in kappa-distribution plasmas,^[11–20] it is expected that our findings will contribute to further studies in positro–atom scattering phenomenon.