Formation of 2s-State Hydrogen Atom in Proton-Lithium Inelastic Scattering

Abstract

Elaborate coupled static formalism is employed for treatment of proton-lithium collisions at wide range of incident energies between 10 and 1000 Kev. Coupled static and frozen core approximations are employed for calculating partial and total cross sections. Only elastic and formation of excited hydrogen, H(2s), channels are considered. Total cross sections are calculated using seven partial waves Green’s function expansion technique of total angular momentum ℓ(0≤ℓ≤6). Proposed iterative approach allows for reliable representation of the core potentials using elaborate variational calculation of target orbitals. Polarization potential of lithium atom is taken into consideration in calculating corresponding total cross sections. Quite interesting reliable results were obtained in comparison with other theoretical approaches.

1. Introduction

There has been a growing interest in the investigation of electron capture from alkali-metal atom. As an example, charge-transfer process with Li has been suggested to occur in plasma diagnostic probes [1]. Besides, alkali-metal atoms are many-electron systems that can be simplified to be one-electron systems due to a single valence electron. Theoretical calculations as well as experimental measurements have been carried out for ionization of and electron capture from alkali-metal atoms by proton impact [1, 2]. Investigation of elastic scattering of positrons from noble gases using an iterative Green’s function partial wave expansion formalism and a model static and polarization potentials are derived for describing the interactions of positrons with closed shell atoms [3, 4]. The formation of excited hydrogen atom (H(2s)) in the process H⁺+Li→H(2s)+Li⁺ has been studied only in low energy ranges. The electron capture cross sections from Li by high energy (for heavy particle collision on atoms, the ratio of the projectile velocity to the orbital electron velocity is much larger than 1) incident protons in energy range between 200 and 10000 Kev have been investigated by using the continuum distorted wave (CDW) approximation [5]. The formation of excited hydrogen atom (H(2s)) in Oppenheimer Brinkman and Kramer (OBK) approximation is investigated [6]. The differential and total cross sections have been investigated in the formation of H-atom in the 2s-excited state of proton-lithium scattering by using the Coulomb projectile Born (CPB) approximation in the energy range from 50 to 10000 Kev [7]. Proton-alkali atom (Na, K, Rb, and Cs) collision has also studied in the wave formulation of impulse approximation in the energies ranging from 50 to 500 Kev [8]. The polarized orbital method (POM) has been applied to study elastic scattering of positrons by large atoms, finding that this method is extremely successful in the treatment of elastic scattering of positrons by noble gases [9]. The POM yields a considerable improvement in static phase shifts which were always shifted towards the variational ones [10]. The improvement was due to the effect of the polarization potentials which play an important role in the real physical picture of the considered collision process.

In this work, an elaborate coupled static formalism is employed [11] for the treatment of proton-lithium collisions at wide range of incident energies from 10 to 1000 Kev. The proposed iterative approach allows for reliable representation of the core potentials using elaborate variational calculation of the target orbitals. The effect of “switching on” polarization potential of lithium atom on their coupled static cross sections is investigated. We assume that the elasticity and the formation of H(2s) channels are open and that all other channels of target are closed. Quite interesting reliable results were obtained in comparison with other theoretical approaches.

2. Theoretical Formalism

The two-channel scattering problems under investigation can be sketched by [11] {}={)H{}𝑝+Li2s𝑝+Li2s}(elasticchannel2s+Li⁺(H(2s)formationchannel). (1)

The total Hamiltonian of the first channel, elastic channel (in Rydberg units and frozen core approximation), has the following form: 𝐻=𝐻⁽¹⁾=𝐻_Li(2s)−12𝜇_𝑀∇²_𝑥+𝑉⁽¹⁾_int1=−2𝜇_𝑚∇²_𝑟−2𝑟−12𝜇_𝑀∇²_𝑥+𝑉⁽¹⁾_int, (2) where 𝜇_𝑚,𝜇_𝑀 are reduced masses of the target and first channel, respectively. 𝑥  and  𝑟 are the position vectors of projectile proton and valence electron of target with respect to center of mass of target, 𝜌 is the position vector of projectile proton with respect to valence electron of target, and 𝜎 is the position vector of center of mass of H(2s) from target nucleus (see Figure 1). 𝑀_𝐴 is mass of the nucleus of target and 𝑉⁽¹⁾_int denotes the interaction between the incident proton and target atom; that is, 𝑉⁽¹⁾_int=2𝑥−2|𝑥−𝑟|+𝑉_𝑐Coul(𝑥), (3) where 𝑉_𝑐Coul(𝑥)=−𝑉_𝑐Coul(𝑟)_𝑟=𝑥,𝑉_𝑐Coul(𝑟)=Φ_Li(1s)𝑟_𝑖2𝑟−𝑟_𝑖−2𝑟Φ_Li(1s)𝑟_𝑖, (4) and the corresponding total energy in first channel is determined by 𝐸=𝐸⁽¹⁾=𝐸_Li(2s)+12𝜇_𝑀𝑘²₁, (5) where (1/2𝜇_𝑀)𝑘²₁ is the kinetic energy of incident proton relative to target nucleus. |Φ_Li(1s)(𝑟_𝑖)⟩,𝑖=1,2, is the wave function of 𝑖th electron in 1s-orbital and |Φ_Li(2s)(𝑟)⟩ is the wave function of valence electron. They are expanded following Clementi and Roetti [12] and adjusted such that the binding energy of the valence electron 𝐸_Li(2s) is minimum; that is, 𝐸_Li(2s)=Φ_Li(2s)(𝑟)𝐻_Li(2s)Φ_Li(2s)(𝑟). (6)

Figure 1 

Configuration space of 𝑝-Li scattering.

The total Hamiltonian of the second channel, H(2s) formation, is expressed (in Rydberg units and frozen core approximation) as 𝐻=𝐻⁽²⁾=𝐻_H(2s)−12𝜇_𝑀′∇²_𝜎+𝑉⁽²⁾_int1=−2𝜇_𝑚′∇²_𝜌−2𝜌−12𝜇_𝑀′∇²_𝜎+𝑉⁽²⁾_int, (7) where 𝜇_𝑚′,𝜇_𝑀′ are reduced masses of H(2s) and the second channel, respectively. 𝑉⁽²⁾_int represents the interaction between the particles of H(2s) and rest of target atom; that is, 𝑉⁽²⁾_int=2𝑥−2𝑟+𝑉_𝑐Coul(𝑥)+𝑉_𝑐Coul(𝑟)+𝑉_𝑐𝑒𝑥(𝑟), (8) where 𝑉_𝑐𝑒𝑥(𝑟)=Φ_Li(2s)𝑟_𝑖2𝑟−𝑟_𝑖−2𝑟Φ_Li(2s)𝑟_𝑖, (9) and the total energy of second channel is determined by 𝐸=𝐸⁽²⁾=𝐸_H(2s)+12𝜇_𝑀′𝑘²₂, (10) where (1/2𝜇_𝑀′)𝑘²₂ is the kinetic energy of center of mass of H(2s) with respect to nucleus of target. It is related to the energy of the incident proton by 𝑘²₂2𝜇_𝑀′=𝐸_Li(2s)+12𝜇_𝑀𝑘²₁−𝐸_H(2s), (11) where 𝐸_H(2s)=−0.25Ry is the ground state energy of H(2s) and 𝑘²₂/2𝜇_𝑀′>0 means that H(2s) channel is open; otherwise, it is closed. Thus, H(2s) formation is only possible if 𝑘²₁>2𝜇_𝑀(𝐸_H(2s)−𝐸_Li(2s)).

The coupled static approximation states that the solution of the two-channel scattering problem under consideration is subjected to the following conditions [11]: Φ_Li(2s)||ΨΦ𝐻−𝐸=0,_H(2s)|𝐻−𝐸|Ψ=0, (12) where |Ψ⟩ is the total wave function describing each scattering process; that is, |Ψ⟩=Φ_Li(2s)(𝑟)𝜓₁(𝑥)+Φ_H(2s)𝜌𝜓₂(𝜎), (13) where Φ_H(2s)=(1/√32𝜋)(2−𝜌)exp(−𝜌/2) is the ground state wave function of H(2s), 𝜓₁(𝑥) is the wave function describing scattered protons, and 𝜓₂(𝑥) is the scattering wave function of second channel.

Substituting (3)–(5) in (2), we obtain 12𝜇_𝑀∇²_𝑥+𝑘²₁𝜓₁=𝑈⁽¹⁾_𝑠𝑡(𝑥)𝜓₁+Φ_Li(2s)𝐻⁽²⁾−𝐸⁽²⁾Φ_H(2s)𝜓₂. (14)

Substituting (4) and (8)–(10) in (7), we obtain 12𝜇_𝑀′∇²_𝜎+𝑘²₂𝜓₂=𝑈⁽²⁾_𝑠𝑡(𝜎)𝜓₂+Φ_H(2s)𝐻⁽¹⁾−𝐸⁽¹⁾Φ_Li(2s)𝜓₁, (15) where Schrödinger’s equations of the target and H(2s) are employed. The potentials 𝑈⁽¹⁾_𝑠𝑡(𝑥) and 𝑈⁽²⁾_𝑠𝑡(𝜎) are defined by 𝑈⁽¹⁾_𝑠𝑡(𝑥)=Φ_Li(2s)(𝑟)𝑉⁽¹⁾_intΦ_Li(2s)(𝑟,𝑈_(2)𝑠𝑡)(𝜎)=Φ_H(2s)(𝑟)𝑉⁽²⁾_intΦ_H(2s)(𝑟). (16)

In order to test the effect of polarization of lithium, we switch on the polarization potential of lithium atom, that is, 𝑉^Li(2s)_pol(𝜎). This can be done by replacing 𝑈⁽²⁾_𝑠𝑡(𝑥) by a potential 𝑈⁽²⁾(𝑥) such that 𝑈⁽²⁾(𝑥)=𝑈⁽²⁾_𝑠𝑡(𝜎)+𝜆₁𝑉^Li(2s)_pol(𝜎), (17) where 𝜆₁=1 if the polarization is switched on (Model II) and 𝜆₁=0 otherwise (Model I). We choose 𝑉^Li(2s)_pol(𝜎) in the Peach form [13],𝑉^Li(2s)_pol(𝑥)𝛼=−𝑥⁴1.0−1.0+𝛾𝑥+𝛾𝑥²2𝑒^−𝛾𝑥, (18) where 𝛼=0.19 and 𝛾=3.91.

Using partial wave expansions of the scattering wavefunctions |𝜓₁⟩ and |𝜓₂⟩ in (14) and (15) that solutions are given (formally) by Lippmann-Schwinger equation [14] 𝜉=𝜉₀+𝐺₀𝜁, (19) where 𝐺₀ is Green operator (𝐸−𝐻₀)⁻¹ and |𝜉₀⟩ is solution of the homogeneous equation: 𝐸−𝐻₀𝜉₀=|0⟩. (20)

The partial wave expansions of Green operators corresponding to operators on the two differential equations enable us to write their solutions in integral form that can be solved by iterative numerical technique; then, we obtain reactance matrix 𝑅^𝜈 that is related to transition matrix 𝑇^𝜈 by 𝑇^𝜈=𝑅^𝜈𝐼−𝑖𝑅̃⁻¹, (21) where 𝜈 is order of iteration, 𝐼 is a 2 × 2 unit matrix, and 𝑖̃=√−1. Partial cross sections obtained in an iterative way are determined (in 𝜋𝑎²₀ units) using the relation 𝜎^(ℓ,𝜈)_𝑖𝑗=4(2ℓ+1)𝑘²_𝑖𝑇^𝜈_𝑖𝑗²,𝑖=1,2. (22)

Finally, the total cross sections (in 𝜋𝑎²₀ units) are expressed (in 𝜈th iteration) by 𝜎^𝑣_𝑖𝑗=^∞_ℓ=0𝜎^(ℓ,𝑣)_𝑖𝑗,𝑖,𝑗=1,2,𝜈>0. (23)

3. Results and Discussion

We start our investigation by testing the variation of static potentials of two channels, 𝑈⁽¹⁾_𝑠𝑡(𝑥) and 𝑈⁽²⁾_𝑠𝑡(𝜎), and the polarization potential 𝑉^Li(2s)_pol(𝜎) of lithium atom, with increase of 𝜎 and 𝑥. Values of 𝜎 and 𝑥 have been chosen such that 𝜎=𝑥=1/16,2/16,3/16,…,512/16 where ℎ=1/16 is the mesh size (or Simpson’s step) employed for calculating integrals appearing in integral equations using Simpson’s rule.

Calculation of cross sections of proton-lithium (𝑝-Li) scattering has been proceeded by investigating variation of elements of 𝑅^𝜈 with increase of integration range (IR) and the number of iterations. We fix ℎ at 1/16  and have obtained all the results presented below with 512 mesh points (i.e., IR = 32𝑎₀). It is found that excellent convergence can be obtained with 𝜈=50, and this demonstrates stability of our iterative method. Final calculations have been carried out for seven partial waves corresponding to 0≤ℓ≤6 at values of 𝑘²₁ representing the kinetic energy region (10≤𝑘²₁≤1000Kev).

In Table 1, we find the partial and total elastic cross sections 𝜎₁₁ of 𝑝-Li scattering (Model I). The table demonstrates that the S-wave scattering cross sections contribute with main part of total cross sections. The table shows also that the P- and D-waves are the most important partial waves after S-wave. This table emphasizes that the total elastic cross section decreases smoothly with the increase of the incident energy.

Table 1 

Partial and total elastic cross sections (𝜎₁₁ in 𝜋𝑎²₀) of proton-lithium scattering by coupled static approximation. Integration range (IR = 32𝑎_𝑜), number of iterations (𝜐 = 50) (Model I).

Table 2 contains the partial and total excited hydrogen, H(2s), formation cross sections 𝜎₁₂ (Model I). The table illustrates that the main contribution to 𝜎₁₂ is due to the S-wave scattering cross sections. The remaining contributions to 𝜎₁₂ are due to P- and D-waves. The total excited hydrogen, H(2s), formation cross section decreases also smoothly with the increase of 𝑘²₁.

Table 2 

Partial and total excited hydrogen formation cross sections (𝜎₁₂ in 𝜋𝑎²₀) of proton-lithium scattering by coupled static approximation. Integration range (IR = 32𝑎_𝑜), number of iterations (𝜐=50) (Model I).

Table 3 presents the partial and total elastic cross sections 𝜎₁₁ of Model II. The main contributions to 𝜎₁₁ are to the S- and P-waves. The seven partial waves employed are quite satisfactory for calculating 𝜎₁₁ in a high degree of accuracy within the framework of coupled static approximation.

Table 3 

Partial and total elastic cross sections (𝜎₁₁ in 𝜋𝑎²₀) of proton-lithium scattering by coupled static approximation. Integration range (IR = 32𝑎_𝑜), number of iterations (𝜐=50) (Model II).

Table 4 shows partial and total excited hydrogen, H(2s), cross sections 𝜎₁₂ of 𝑝-Li scattering of Model II. On considering the variation of 𝜎₁₂, with respect to incident energy, we notice that it has its maximum value at 𝑘²₁=10Kev and decreases smoothly thereafter. The values of total excited hydrogen, H(2s), cross sections 𝜎₁₂ in table give larger values than of Model I.

Table 4 

Partial and total excited hydrogen formation cross sections (𝜎₁₂ in 𝜋𝑎²₀) of proton-lithium scattering by coupled static approximation. Integration range (IR = 32𝑎_𝑜), number of iterations (𝜐=50) (Model II).

The total elastic cross sections of proton-lithium scattering for the two models are presented in Figure 2. In Figure 3, the present total hydrogen formation cross sections of proton-lithium scattering for Model I are displayed with that of Model II. The present results keep decreasing with the increasing of energy and its values of Model II are higher than that of Model I.

Figure 2 

Total elastic cross sections (𝜎11 in 𝜋𝑎20) of proton-lithium scattering by coupled static approximation for the two models.

Figure 3 

Total excited hydrogen (H(2s)) formation cross sections (𝜎12 in 𝜋𝑎20) of proton-lithium scattering by coupled static approximation for the two models.

Comparison between total excited hydrogen, H(2s), cross sections 𝜎₁₂ of 𝑝-Li scattering of the two Models (I and II) and those determined by Banyard and Shirtclliffe [5], Ferrante et al. [6], and Tiwari [7] are tabulated in Table 5. Our results of Model II are found in reasonably good agreement with the above mentioned theoretical results specially with Tiwari calculations [7].

Table 5 

Present total cross sections (𝜎₁₂ of Model I and Model II in 𝜋𝑎²₀) for excited hydrogen formation (H(2s)) with different theoretical results [5–7].

4. Conclusions

Proton-lithium scattering is studied using the coupled static approximation. Our interest is focused on the influence of polarization potential of the lithium atom. The switching on 𝑉^Li(2s)_pol(𝜎) in Model II has positive influence upon total excited hydrogen formation cross sections 𝜎₁₂. The present calculations for excited hydrogen formation cross sections, using Model II, show reasonable agreement with available theoretical calculations in wide energy region.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.