Introduction

Atomic data are very much needed for the modeling of plasmas [1]. They are useful for many applications in astrophysics and nuclear fusion Tokamak. Moreover, the ITER project needs accurate atomic data for a wide area of ions. The simplest multi-electron system, He-like can play an important role in providing the needed accurate atomic data. In previous work, we provided calculations of He-like neon [2].

We just focus on this study in neutral Helium. For this Nobel gas, the atom contains two protons and two electrons. The energy structure of the 1snl configuration is mainly dominated by the electron-nucleus and electron-electron Coulomb interactions. The separations of levels having belonging to the same n multiplicity and having l = s, p, or d are mainly determined by Spin-orbit interactions between the electrons. The final state is given by the ^2S+1L notation where n,l,S,L are the standard quantum numbers. The quantity 2S + 1 is the multiplicity of the term. The S and L momentums are coupled to obtain, J = S + L, for a given level. The level is denoted as ^2S+1L_J.  J being the total angular momentum.

Many experimental and theoretical studies were performed for this element. Energy levels have been published by the National Institute of Standards and Technology (NIST) and are available at their website [3]. Sow et al. [4] performed calculations for the 2s^{2 1}S, 2p^{2 1}D, 3s^{2 1}S, 3p^{2 1}D, 3d^{2 1}G, 4p^{2 1}D, 4d^{2 1}D, 4f^{2 1}I doubly excited states of 2≤ Z ≤15. The energy eigen-values of doubly excited 2pnp(¹P_e) (n=3-8) and 2pnp (³P_e) (n=2-7) bound states of a neutral helium atom were calculated under the weakly coupled plasma screening by K. Saha et al. [5]. Doubly excited states in helium are calculated by E. Lindroth [6] with a finite discrete spectrum for states with electrons in the n=2 and n=3 states. An approach to calculating the energies and widths of resonances for neutral Helium was developed on the basis of the stabilization method, the energies of 28 resonances of ⁿS symmetry with the spin multiplicities n=1, 2, 3, and 4 were calculated by I. A. Misurkin et al. [7].

Therefore, in this paper, we just interest on calculations of the singly excited energy levels for neutral Helium, namely He I. We employed the fully relativistic code Flexible Atomic Code (FAC) of Gu [8]. FAC code provides many different atomic parameters such as energy levels, transitions rates and lifetimes. In the rest of paper, we shall give, for He I, energy levels and lifetimes of the lowest 49 levels belonging to 1s² and 1snl (with n ≤ 5; 0 ≤ l ≤ n-1).

Theoretical method

We employed for our calculations the widely used FAC code of Gu [8]. We simply give here a short description of the theoretical method used by FAC.

By diagonalizing the relativistic Hamiltonian H, we get the energy levels of an N electrons atom [9]

where 𝐻_𝐷(𝑖) is the single-electron Dirac Hamiltonian. The basis states Φ_v , which are usually referred to as configuration state functions (CSF), are antisymmetric sums of products of 𝑁 one-electron Dirac spinors φ_nkm .

where 𝜒_{𝜅𝑚} is the usual spin-angular function. 𝑛 is the principal quantum number, 𝜅 is the relativistic angular quantum number and 𝑚 is the 𝑧-component of the total angular momentum j.

The approximate atomic state functions are obtained by mixing the basis states Φ_v using the same symmetries

where 𝑏_𝜈 are the mixing coefficients obtained from diagonalizing the total Hamiltonian.

Choice of Local Central Potential

To build the Hamiltonian matrix, the one-electron radial orbital must be known. According to the standard Dirac-Fock-Slater method, the large and small components, 𝑃_{𝑛𝜅} and 𝑄_{𝑛𝜅}, satisfy the coupled
Dirac equation for a local central field 𝑉 (𝑟)

where 𝛼 is the fine structure constant and 𝜀_{𝑛𝜅} are the energy eigenvalues of the radial orbitals.

Solution of Dirac Equations

The radial orbitals sought have a direct influence on the potential, so the Eq. (4) requires a self-consistent iteration. In each iteration, the orbitals from the previous step are used to derive the potential. Consequently, solving the eigenvalue problem using known potential is sufficient. As is standard, we convert Eq. (4) into a Schrodinger-like equation in two steps: eliminating the small component and performing the transformation [10]

We use the standard Numerov method to solve Eq. (6). However, it is customary to perform another transformation before seeking the solution

where 𝑡(𝑟) as a function of radial distance is suitably chosen so that a uniform grid can be used in the new variable 𝑡, and the corresponding transformation on the wavefunction is to bring the differential equation for 𝐺_𝑎(𝑡) to a Schrodinger-like form, i.e., without the first derivative term

However, for free orbitals with sufficiently high energy, solving Eq. (9) in a conventional way becomes impractical. We shall use a different approach for continuum states, namely, the phase amplitude method.

The minimum and maximum radial distances, 𝑟_{𝑚𝑖𝑛} and 𝑟_{𝑚𝑎𝑥}, in setting up the radial grid are chosen as.

where 𝑍_{𝑒𝑓𝑓} is the residual charge of the atomic ion that the electrons experience at large 𝑟. The low-𝑛 and high-𝑛 states are treated differently. The dividing 𝑛₀ is determined by the choice of 𝑟_{𝑚𝑎𝑥}, specifically, n0 = 0.5* /Z_effr_max For 𝑛 ≤ 𝑛₀, the orbitals were found by outward and inward integration of Eq. (9) with zero amplitudes at both ends, and matching at the outer classical turning point. Node counting is used to pick out the appropriate solution corresponding to the quantum numbers 𝑛 and 𝑙. The wavefunctions are then normalized by numerical integration. For 𝑛 > 𝑛₀, Eq. (9) is integrated outward until 𝑟 = 𝑟_{𝑐𝑜𝑟𝑒}, where the potential has reached its asymptotic Coulomb value. For 𝑟 > 𝑟_{𝑐𝑜𝑟𝑒}, the wavefunction is the exponentially decaying Whittaker function.

where v2 = -1/2 Z²_eff/ ·£, p = Z_effr and ðœ† = 𝑙 in the non-relativistic limit [8]. In the relativistic case, the asymptotic behavior of the effective potential is modified according to Eq. (7), and corresponds to

where 𝜇 is the quantum defect. The quantum defect of a Rydberg atom refers to a correction applied to the equations governing Rydberg atom behavior. For a non-hydrogen atom –alkali for example- the binding energy of the Rydberg states is e = R_y/( n- d )² where d  is the quantum defect and R_y is the Rydberg constant. For high 𝑛 states we are concerned with, (𝜈_𝑛) = 1 is a very good approximation.

Results and discussions

The calculation is performed up to n=5 which generate up to 49 levels. The 1s², 1snl (n=2-5) configurations are given in Table 1. We compare in this table our energy levels from FAC code with the level energies published by NIST [3]. The differences presented by percentage in the Table are very small. Our calculated energy levels agree within 1%, the only case where the difference is larger being for the 1s2p ³P₂ level. One can see that results from these two calculations match well for most of the levels and proves that our results are consistent. We can state with confidence that the results are in good agreement to the other published values for the energy levels of He I. New values of lifetimes are presented in the same Table 1.

Table 1: Energy levels [eV] calculated with FAC code and compared to data taken from NIST database. The last column represents the established lifetimes [s^-1] for 49 upper levels of neutral Helium.

Conclusion

We have presented in this paper results for energy levels among the lowest 49 levels, for He I. Based on the experimental published results in NIST database, our energy levels are accurate to better than 1%. Moreover, we presented the results for the lifetimes for almost levels of study. Good agreement between our calculated energy levels for He I and the available NIST data reflects the quality of calculation of the wavefunctions.  

As we don’t have other results of lifetimes, we expect that the present set of results will be highly useful for comparison with other future experimental work.

Acknowledgments

This work has been realised with the financial support of the Tunisian Ministry of Higher Education and Scientific Research.

References

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