Abstract

Calculations of the energy levels of atoms and ions with Z ≤ 10 are carried out in this paper using the complex rotation method. Using a new wave function with two terms including a new method to calculate the correlation factor taking into account spherical harmonics through hypergeometric functions to calculate excited 2sns^+1,3S^e, 2snp^+1,3P⁰, 2pnp^+1,3D^e, 3d4d ^+1,3G^e states. The results obtained show quantitatively the great significance of electron correlation effects in the doubly excited states. These results are in compliance with some experimental and theoretical data.

1. Introduction

The study of atomic resonances in plasma environment has gained tremendous attention in the recent years and is one of the interesting topics of current research ^{1, 2, 3, 4}. The complex rotation method was used in atomic physics by Doolen ⁵ to determine the resonance parameters in the system e- + H and by Lonnie W. Manning and Frank C. Sanders ⁶ to study the resonance parameters (¹P⁰ and ³P⁰) of He-like systems. Subsequently other calculations were made; using the variational wave functions built on a Hylleraas basis, by Ho ^{7, 8, 9, 10}, Gning et al ¹¹ used the Hylleraas wave functions to compute resonance parameters. Some of his work ⁷ focused on the determination of the energies and widths of the first two resonances ¹S^e and ^1,3P⁰ of helium and the H^- and Li⁺ ions under the excitation threshold n = 4. Similarly the complex rotation method allows calculating positions and widths of resonances in atomic systems ^{12, 13, 14}. To mention just a few applications, the method has been used in nonrelativistic quantum-mechanical calculations of resonance parameters of doubly excited states ^{15, 16} or Stark resonances ¹⁷ in such systems as He, H^-, or Ps^-. The above mentioned applications of the complex rotation procedure neglected relativistic effects. Recent developments of the experimental technique perfected experimental data to a degree where relativistic effects become observable.

A number of methods based on close-coupling approximation ¹⁸ and the Feshbach projection operator formalism ^{19, 20} have been used to calculate positions and widths of doubly excited states. In the latter method, resonance positions and widths are calculated separately and therefore need not be of the same accuracy.

In this work we report a resonance parameters calculation for some lower-lying doubly excited states for a He-like systems below the n = 2 hydrogenic thresholds. Hylleraas-type wave functions are used and 2sns^+1,3S^e, 2snp^+1,3P⁰, 2pnp^+1,3D^e, 3d4d ^+1,3G^e states are calculated.

2. Wave Functions and Calculations

The existence of this term makes impossible to solve exact the fundamental equation of quantum mechanics (Schrödinger equation):

(1)

In this equation, is the wave function of the quantum state of the system (core-electrons) and E is the total energy of the same quantum state. The exact solution of the last equation is impossible because of the correlation term involved in the expression of total Hamiltonian. Hence it is necessary to implement a method of approximate calculation using a correlated wave function, which is given by:

(2)

Where j+k+m ≤ Ω, Ω is a positive integer number which determines the expansion length N; α is a nonlinear parameter and S the total spin of atomic system, r₁ and r₂ are the coordinates of electrons with respect to the nucleus; N is the principal number ; ℓ₁ and ℓ₂ are orbital angular, Z the nuclear charge. C_jkm are parameters to be determined, r₀ is Bohr radius.

Where

(3)

represent the spherical harmonics of electron 1 or 2.

Which

(4)

and i= 1, 2.

The resonance parameters are determined by finding a rate of change that is stable with respect to the nonlinear parameters α, and the complex value of the energy.

(5)

where E_r gives the resonance position and gives the width.

And the Hamiltonian can be written as:

(6)

With

(7)

(8)

(9)

In equation (6), T is the kinetic energy; C is the Coulomb interaction between the atomic nucleus and the two electrons and W is the Coulomb interaction between electrons. In the study of atomic resonances, the application of this theorem consists in the transformation of real Hamiltonian system H(r) to a complex one H(r,θ) where r is the inter-particle radial coordinate and θ is a positive real number denoting the rotation angle. This processing is executed by a change of the real variable r to complex variable r exp (iθ).

The matrix element of the normalization constant is defined by:

(10)

Where is elementary volume element. The existence of the disturbance term (Equation 9) makes impossible the exact resolution of equation (1). This is how we used the complex rotation method by introducing a new method of calculating the correlation coefficient, which is one of the calculation techniques for providing the resonance parameters to two-electron atomic systems. According to the calculation of correlation coefficients, proposed by Varshalovich D.A et al. ³², we found a new formula that is proposed by Gning et al. ¹¹

(11)

With

(12)

the hypergeometric wave function defined by Samarskogo ³³

(13)

Replacing eqaution (13) with its value in equation (12)

(14)

So equation (14) becomes

(15)

And remplacing equation (15) in equation (11)

(16)

for 2sns^+1,3S^e states (ℓ=0;S=0),

(17)

so equation (2) becomes:

(18)

For 2snp^+1,3P⁰, 2pnp^+1,3D^e states corresponding to ,

(19)

by replacing the values of the spherical harmonics of e- 1or 2, we obtain:

(20)

For we have =1

(21)

For 3d4d ^+1,3Ge states

(22)

3. Results and Discussions

In our calculations, we set the dimension Ω to 4 for the 2s3s ^1,3S^e, 2s4s ^1,3S^e states and β=0.25 rad. Ω =3 for the 2s2p ^1,3P⁰, 2s3p ^1,3P⁰ states and β=0.2 rad. Similarly β=0.2 rad for the 2p3p ^1,3D^e, 2p4p ^1,3D^e, 3d4d ^1,3G^e states and Ω =4. After having fixed these parameters, we have varied the nonlinear parameter α. The minimum value of the energy is obtained following a stabilization of the latter with a variation of the nonlinear parameter. At each minimum value of the energy corresponds a minimum value of the given resonance width 10^-3. Based on the processing of the calculations, energy conversion of 1 a.u. = 2Ry = 27.211385 eV was used.

Table 1, Table 2, Table 3, Table 4 show the results of targeted energy 2s3s ^1,3S^e, 2s4s ^1,3S^e, 2s2p ^1,3P⁰, 2s3p ^1,3P⁰, 2p3p ^1,3D^e, 2p4p ^1,3D^e, 3d4d ^1,3G^e respectively.

Based on the processing of the calculations, energy conversion of 1 a.u. = 2Ry = 27.211385 eV was used. For the calculation of doubly excited states for Helium-like ions, several authors have used different methods. Thus, Ho ^{21, 29} used the method of complex rotation convoluted with a wave function of Hylleraas type, Dieng ²² used Special Forms of Hylleraas-Type Wave Functions, Ivanov and Safronova ²⁴ performed calculation using double sum method over the complete hydrogen spectrum, Sakho ^{23, 25} with the Screening Constant by Unit Nuclear Charge method, Ivanov and Safronova ²⁴ using computing double sum method, Bachau ³⁰ using the Fechbach projection formalism method.

In fact, a comparison is made in Table 5 based on the theoretical results from Ho ²¹, Dieng ²², Sakho et al. ^{23, 25} and Ivanov and Safronova ²⁴ for the 2s3s ^1,3S^e, 2s4s ^1,3S^e states. For these state, best consistency of the results was found and a little discrepancies regard Dieng results.

Table 6 compares results with those from Ho ^{8, 21}, Seminario and Sanders ²⁶, Drake et al ²⁷, Sakho et al. ^{23, 28}, Ivanov and Ho ²⁹, Bachau ³⁰ for the 2s2p ^1,3P⁰ , 2s3p ^1,3P⁰ For these states, we note generally a quite good agreement.

In Table 7, results of 2p3p^1,3D^e, 2p4p^1,3D^e states were compared with those of Ho ²¹, Sakho et al. ²⁵ and Dieng ²². For these states, we note generally a quite good agreement and little discrepancy with Dieng results.

We have also compared in Table 8, results with those from Ivanov and Ho ²⁹ and Bachau ³⁰ for the 3d4d¹G^e, 3d4d³G^e. We note good consistency.

In Table 8 our resonance width results of the same states above are listed, correspondingly, and also compared with those found in the literature Ho ^{8, 21}, Seminario and Sanders ²⁶, Manning and Sanders ³¹.

4. Conclusion

We have carried out calculations of the inter-shell nℓn′ℓ and nℓ′n′ℓ′ (with n ≤ 3, n′ ≤ 4 and ℓ= ℓ′ or ℓ ℓ′) singlet and triplet doubly excited states energies of helium-like ions by using the complex rotation method and new wave functions with two terms are used to calculate resonance parameters for intrashell states. The results of the resonance parameters reported here agree better with other theoretical calculations and experiment data for low Z than those with high Z. The results that we have obtained show that it is possible by using our theoretical approach, to describe the highly doubly excited two-electrons states. However, experimental techniques using synchrotron radiation are becoming more sophisticated and can provide very accurate values of the resonance parameters of the Rydberg series of multi-electron atomic systems. We can always use the method of complex rotation and with weak wave functions of terms to validate these experimental results. So this work also provides parameters for total widths for such resonances. Since the widths are related to the inverse of the autoionisation Jifetimes, the present results should be useful for future investigations of the underlying mechanism of autoionisation.

References