The appearance of intermediate states in atomic and nuclear reactions is considered as the most interesting physical phenomenon since the development of quantum theory. The development of femtosecond lasers has tremendously enhanced the experimental identification of these states in chemical and physical reactions. Theoretical investigation on positron– atom or proton– atom scattering has been carried out by many authors. Most theoretical treatments of positron– atom or proton– atom scattering are based on calculations of differential, partial, and total cross sections as functions of incident energy using different approximations. Abdel-Raouf^[1] investigated the collision of positrons by lithium and sodium atoms using the coupled static approximation. Elkilany^{[2– 5]} used the variational static approximation or iterative Green's function partial wave expansion method to obtain partial and total cross sections of elastic scattering of positron or electron by inert gas atoms. Surdutovitch et al.^[6] reported measurements of Ps formation cross sections for positron collision by lithium and sodium atoms. Jun et al.^[7] discussed positronium formation cross sections for positron collision by potassium at low energies using an optical-model method. Roy and Ho^[8] calculated the P-partial wave resonance in e⁺ – Li scattering below the Ps threshold using the stabilization method. Gilmore et al.^[9] obtained results for electron– or positron– positronium scattering in low energies using a coupled pseudo-state approximation. Ratnavelu and Ng^[10] used the coupled-channel optical method to study positron collision by lithium atom. Abdel-Raouf and Elkilany^[11] applied the iterative Green’ s function partial wave expansion method to calculate partial and total cross sections of the collision of positron by helium, neon, argon, and krypton atoms. Abdel-Raouf and Elkilany^[12] discussed the appearance of intermediate states (resonant states) in positron– lithium collision. Elkilany^[13] computed the partial and total cross sections of excited positronium formation in the scattering of positrons by positronium using the coupled static approximation. Jun et al.^[14] calculated the Ps formation cross section in positron collision by lithium atom using the coupled channel optical method. Fang et al.^[15] used a coupled-channel optical method to calculate positron collision with lithium atom. Banyard and Shirtcliffe^[16] used the continuum-distorted wave method to calculate cross sections for protons scattered from atomic lithium. Ferrante et al.^[17] calculated total cross sections for hydrogen formation in protons scattering with alkali atoms using the OBK approximation. Daniele et al.^[18] calculated total cross-sections for high-energy proton scattering by alkali atom. Ferrante and Fiordilino^[19] investigated high-energy proton collision with alkali atom using the Eikonal approximation. Fritsch and Lin^[20] discussed the hydrogen formation in proton scattering by lithium atoms using a modified two-center atomic orbital expansion. Choudhury and Sural^[21] calculated the hydrogen formation in the collision of proton with alkali-metal atoms using the wave formulation of impulse approximation. Tiwiri^[22] reported the differential and total cross sections in the hydrogen formation in the collision of proton by lithium and sodium atoms using the coulomb projected Born approximation. Elkilany and Abdel-Raouf^[23] investigated the effect of polarization potentials on antiproton scattered by positronium using coupled static approximation. Elkilany^[24] used the coupled static approximation and Green's function iterative method to calculate partial and total cross sections of anti-hydrogen formation in antiproton positronium collision.

For the first time, we investigate the inelastic scattering of protons by sodium atoms as a two-channel problem in which the elastic and hydrogen (2s state), H(2s), formation channels are open using the coupled-static approximation within the framework of the frozen-core picture of Na atom. The considered method has been applied successfully in the inelastic scattering of proton by lithium atom by Elkilany.^[25]

The two-channel scattering problems under investigation can be sketched by

The one valence electron model of an alkali target (Na) is described (in Rydberg units) by^[1]

where r is the position vector of the valence electron with respect to the origin of the scattering system, in which the nucleus of the target is infinitely heavy. V_c(r) is a screened potential defined by

where V_cCoul(r) and V_cex(r) are the Coulomb and exchange parts of the core potential.

Following Clementi and Roetti, ^[26] the wave function of the i-th electron in the orbital j of the target is expanded by

where | χ _jp(r_i )〉 is a Slater-type wave function given by

The k_jp are integers or zero, A_jp are normalization factors determined by

and the spherical harmonics Y_jp are normalized by 〈 Y_jp(ϑ _i, φ _i)| Y_jp(ϑ _i, φ _i )〉 = 1. The constants c_jp and α _jp are adjusted within the framework of the Hatree– Fock– Roothaan approach. Substituting Eq.  (5) into Eq.  (4) and introducing the notation , we obtain

where m_j is the number of basis functions characteristic to each orbital. Usually, all the orbitals of the same type have the same m_j.

The Coulomb part of the core potential is defined by

where M is the number of orbitals, and N_j is the number of electrons occupying the orbital j. The prime on the sum sign means that the term − 2/r is repeated for each j. The exchange part of the core potential is defined by

where the subscript Na(3s) is employed for distinguishing the wave function for the valence electron of the target atom.

According to Eq.  (2), the binding energy of the valence electron of the target is determined by

The total Hamiltonian of the first channel, elastic channel, has the following form (in Rydberg units and frozen core approximation):

where μ _M is the reduced mass of the first channel. In Fig.  1, and are the position vectors of the projectile proton and the valence electron of the target with respect to the center of mass of target, is the position vector of the projectile proton with respect to the valence electron of the target , is the position vector of the center of mass of H(2s) from target, M_T is the mass of the nucleus of target, and denotes the interaction between the incident proton and the target atom, i.e.,

where

and the corresponding total energy in the first channel is determined by

The second term on the right is the kinetic energy of incident proton relative to the target nucleus.

The total Hamiltonian of the second channel, H(2s) formation, is expressed (in Rydberg units and frozen core approximation) as

where μ _M′ is the reduced mass of the second channel, represents the interaction between the particles of H(2s) and the rest of the target atom, i.e.,

and the total energy of the second channel is determined by

The second term on the right is the kinetic energy of center of mass of H(2s) with respect to the nucleus of the target. This is related to the energy of the incident proton by

where E_H(2s) = − 0.25  Ry is the first excited 2s-state energy of H(2s) and means that H(2s) channel is open, otherwise it is closed. Thus, H(2s) formation is only possible if .

Fig.  1.

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	Fig.  1. Configuration space of p– Na scattering.

The coupled static approximation states that the solution of the two-channel scattering problem under consideration is subjected to the following conditions:^[25]

where | Ψ 〉 is the total wave function describing each scattering process, i.e.,

where

is the ground-state wave function of H(2s), Ψ ₁(x) is the wave function describing scattered protons, and Ψ ₂(x) is the scattering wave function of the second channel. Substituting Eqs.  (14) and (21) into Eq.  (19), we obtain

Substituting Eqs.  (17) and (21) into Eq.  (20), we obtain

where Schrö dinger’ s equations of the target and H(2s) are employed. The potentials and are defined by

Using the partial wave expansions of the scattering wave functions | Ψ ₁〉 and | Ψ ₂〉 in Eqs.  (23) and (24), solutions are given (formally) by the Lippmann– Schwinger equation (see appendix)

where G₀ is the Green operator (ε − H₀)^{− 1} and | ξ ₀〉 is the solution of the homogeneous equation

The partial wave expansions of Green operators corresponding to operators in the two differential equations enable us to write their solutions in integral form that can be solved by iterative numerical technique. Then we obtain the reactance matrix R^ν which is related to the transition matrix T^ν

where ν is the order of iteration, I is a 2 × 2 unit matrix, and . Partial cross sections obtained in an iterative way are determined (in units) by

Finally, the total cross sections (in units) are expressed (in the ν -th iteration) by

We begin with testing the variation of static potentials of two channels, and , with increase of x and σ , respectively. Values of x and σ have been chosen such that x = σ = 1/16, 2/16, 3/16, … , 512/16 , where h = 1/16 is the mesh size (or Simpson’ s step) employed to calculate integrals appearing in integral equations using Simpson’ s rule. Calculation of cross sections of proton– sodium (p– Na) scattering has been proceeded by investigating the variation of the elements of R^ν with the increase of integration range (IR) and the number of iterations. We set h to be 1/16 and use 512 mesh points (i.e., IR = 32a₀). It is found that excellent convergence can be obtained with υ = 50. This demonstrates the stability of our iterative method. Final calculations have been carried out for seven partial waves corresponding to 0 ≤ ℓ ≤ 6 at values of representing the kinetic energy region .

Table  1 shows the partial and total elastic cross sections for all incident values of energy between 10 and 1000  keV. We can conclude the following points: (i) the main contributions to the total elastic cross sections, σ ₁₁’ s, are due to the S and P partial cross sections; (ii) the total elastic cross sections, σ ₁₁’ s, decrease steadily with ; (iii) the seven partial waves employed are quite satisfactory to calculate the total elastic cross sections σ ₁₁’ s to a higher degree of accuracy within the framework of the coupled-static approximation.

Table 1.

Table 1.

Table 1. Present partial and total elastic cross sections (σ 11 in ) of p– Na scattering. Integration range is equal to 32a0, and the number of iterations υ = 50. k2/keV ℓ = 0 ℓ = 1 ℓ = 2 ℓ = 3 ℓ = 4 ℓ = 5 ℓ = 6 σ 12 10 3.5672× 10− 2 9.5125× 10− 3 1.9025× 10− 3 3.8050× 10− 4 7.6100× 10− 5 1.5220× 10− 5 3.0440× 10− 6 4.7562× 10− 2 20 2.7869× 10− 2 7.4317× 10− 3 1.4863× 10− 3 2.9727× 10− 4 5.9454× 10− 5 1.1891× 10− 5 2.3781× 10− 6 3.7158× 10− 2 30 2.1772× 10− 2 5.8059× 10− 3 1.1612× 10− 3 2.3224× 10− 4 4.6447× 10− 5 9.2894× 10− 6 1.8579× 10− 6 2.9029× 10− 2 40 1.7010× 10− 2 4.5359× 10− 3 9.0718× 10− 4 1.8144× 10− 4 3.6287× 10− 5 7.2574× 10− 6 1.4515× 10− 6 2.2679× 10− 2 50 1.3289× 10− 2 3.5436× 10− 3 7.0873× 10− 4 1.4175× 10− 4 2.8349× 10− 5 5.6698× 10− 6 1.1340× 10− 6 1.7718× 10− 2 60 9.5680× 10− 3 2.5515× 10− 3 5.1029× 10− 4 1.0206× 10− 4 2.0412× 10− 5 4.0823× 10− 6 8.1647× 10− 7 1.2757× 10− 2 70 6.8891× 10− 3 1.8371× 10− 3 3.6742× 10− 4 7.3484× 10− 5 1.4697× 10− 5 2.9393× 10− 6 5.8787× 10− 7 9.1853× 10− 3 80 4.9600× 10− 3 1.3227× 10− 3 2.6453× 10− 4 5.2907× 10− 5 1.0581× 10− 5 2.1163× 10− 6 4.2325× 10− 7 6.6132× 10− 3 90 3.5712× 10− 3 9.5233× 10− 4 1.9047× 10− 4 3.8093× 10− 5 7.6187× 10− 6 1.5237× 10− 6 3.0475× 10− 7 4.7616× 10− 3 100 2.5713× 10− 3 6.8568× 10− 4 1.3714× 10− 4 2.7427× 10− 5 5.4854× 10− 6 1.0971× 10− 6 2.1942× 10− 7 3.4283× 10− 3 110 1.8513× 10− 3 4.9369× 10− 4 9.8738× 10− 5 1.9748× 10− 5 3.9495× 10− 6 7.8990× 10− 7 1.5798× 10− 7 2.4684× 10− 3 120 1.3330× 10− 3 3.5546× 10− 4 7.1091× 10− 5 1.4218× 10− 5 2.8436× 10− 6 5.6873× 10− 7 1.1375× 10− 7 1.7772× 10− 3 130 9.5970× 10− 4 2.5592× 10− 4 5.1184× 10− 5 1.0237× 10− 5 2.0474× 10− 6 4.0947× 10− 7 8.1895× 10− 8 1.2796× 10− 3 140 6.9100× 10− 4 1.8427× 10− 4 3.6854× 10− 5 7.3707× 10− 6 1.4741× 10− 6 2.9483× 10− 7 5.8966× 10− 8 9.2133× 10− 4 150 4.9752× 10− 4 1.3267× 10− 4 2.6534× 10− 5 5.3069× 10− 6 1.0614× 10− 6 2.1228× 10− 7 4.2455× 10− 8 6.6335× 10− 4 160 3.5821× 10− 4 9.5524× 10− 5 1.9105× 10− 5 3.8209× 10− 6 7.6419× 10− 7 1.5284× 10− 7 3.0568× 10− 8 4.7761× 10− 4 170 2.5791× 10− 4 6.8777× 10− 5 1.3755× 10− 5 2.7511× 10− 6 5.5022× 10− 7 1.1004× 10− 7 2.2009× 10− 8 3.4388× 10− 4 180 1.8570× 10− 4 4.9519× 10− 5 9.9038× 10− 6 1.9808× 10− 6 3.9615× 10− 7 7.9231× 10− 8 1.5846× 10− 8 2.4759× 10− 4 190 1.3370× 10− 4 3.5654× 10− 5 7.1308× 10− 6 1.4262× 10− 6 2.8523× 10− 7 5.7046× 10− 8 1.1409× 10− 8 1.7827× 10− 4 200 9.6266× 10− 5 2.5671× 10− 5 5.1342× 10− 6 1.0268× 10− 6 2.0537× 10− 7 4.1073× 10− 8 8.2147× 10− 9 1.2835× 10− 4 210 6.9311× 10− 5 1.8483× 10− 5 3.6966× 10− 6 7.3931× 10− 7 1.4786× 10− 7 2.9572× 10− 8 5.9145× 10− 9 9.2413× 10− 5 220 4.9905× 10− 5 1.3308× 10− 5 2.6616× 10− 6 5.3232× 10− 7 1.0646× 10− 7 2.1293× 10− 8 4.2585× 10− 9 6.6539× 10− 5 230 3.5931× 10− 5 9.5816× 10− 6 1.9163× 10− 6 3.8327× 10− 7 7.6653× 10− 8 1.5331× 10− 8 3.0661× 10− 9 4.7907× 10− 5 240 2.5870× 10− 5 6.8988× 10− 6 1.3798× 10− 6 2.7595× 10− 7 5.5190× 10− 8 1.1038× 10− 8 2.2076× 10− 9 3.4493× 10− 5 250 1.8626× 10− 5 4.9671× 10− 6 9.9341× 10− 7 1.9868× 10− 7 3.9737× 10− 8 7.9473× 10− 9 1.5895× 10− 9 2.4835× 10− 5 300 1.0245× 10− 5 2.7319× 10− 6 5.4638× 10− 7 1.0928× 10− 7 2.1855× 10− 8 4.3710× 10− 9 8.7421× 10− 10 1.3659× 10− 5 500 7.1715× 10− 6 1.9124× 10− 6 3.8248× 10− 7 7.6496× 10− 8 1.5299× 10− 8 3.0598× 10− 9 6.1197× 10− 10 9.5618× 10− 6 800 5.0199× 10− 6 1.3386× 10− 6 2.6773× 10− 7 5.3546× 10− 8 1.0709× 10− 8 2.1418× 10− 9 4.2837× 10− 10 6.6931× 10− 6 1000 3.5139× 10− 6 9.3705× 10− 7 1.8741× 10− 7 3.7482× 10− 8 7.4964× 10− 9 1.4993× 10− 9 2.9986× 10− 10 4.6852× 10− 6

Table 1. Present partial and total elastic cross sections (σ 11 in ) of p– Na scattering. Integration range is equal to 32a0, and the number of iterations υ = 50.

In Table  2 we present the partial and total H(2s) formation cross sections, σ ₁₂’ s, at all considered energies. The values indicate that the results shown in Table  1 have been repeated for the values of partial and total H(2s) formation cross sections, σ ₁₂’ s. The most interesting results are accumulated in Table  3, where we find a comparison between total H(2s) formation cross sections (in ) determined by different authors using different approaches. In column 2, we find the present total H(2s) formation cross sections. Column 3 involves the values of H(2s) formation cross sections obtained by Choudhury and Sural^[21] using the wave formulation of impulse approximation. The next column contains the values of total H(2s) formation cross sections of Tiwari^[22] in protons scattering with alkali atoms using the coulomb projected Born approximation. In Fig.  2 we also show the present results together with those of Choudhury and Sural^[21] and Tiwari^[22] in the energy range of . The results show that our values of total H(2s) formation cross sections are in good agreement with those of Choudhury and Sural^[21] for incident energy range of , specially between 150 and 200  keV. The present values of the total H(2s) formation cross sections are larger than those of Choudhury and Sural^[21] and smaller than those of Tiwari.^[22] The reason for the difference lies in the different approximation used for the scattering wave function and in neglecting the polarization potentials of the target and H(2s) in our calculations. Before leaving this section it is important to mention that further improvements upon the present total collisional cross sections can be also obtained by taking the polarization effect of the hydrogen formation atom in our calculations. We can also consider a three-channel problem by opening H(2p) formation channel.

Table 2.

Table 2.

Table 2. Present partial and total H(2s) formation cross sections (σ 12 in ) of p– Na scattering. Integration range is equal to 32a0, and the number of iterations υ = 50. k2/keV ℓ = 0 ℓ = 1 ℓ = 2 ℓ = 3 ℓ = 4 ℓ = 5 ℓ = 6 σ 12 10 3.7434× 10− 3 9.9823× 10− 4 1.9965× 10− 4 3.9929× 10− 5 7.9858× 10− 6 1.5972× 10− 6 3.1943× 10− 7 4.9911× 10− 3 20 2.9245× 10− 3 7.7987× 10− 4 1.5597× 10− 4 3.1195× 10− 5 6.2390× 10− 6 1.2478× 10− 6 2.4956× 10− 7 3.8993× 10− 3 30 2.2847× 10− 3 6.0926× 10− 4 1.2185× 10− 4 2.4371× 10− 5 4.8741× 10− 6 9.7482× 10− 7 1.9496× 10− 7 3.0463× 10− 3 40 1.7849× 10− 3 4.7599× 10− 4 9.5197× 10− 5 1.9039× 10− 5 3.8079× 10− 6 7.6158× 10− 7 1.5232× 10− 7 2.3799× 10− 3 50 1.3945× 10− 3 3.7187× 10− 4 7.4375× 10− 5 1.4875× 10− 5 2.9750× 10− 6 5.9500× 10− 7 1.1900× 10− 7 1.8593× 10− 3 60 1.0041× 10− 3 2.6775× 10− 4 5.3550× 10− 5 1.0710× 10− 5 2.1420× 10− 6 4.2840× 10− 7 8.5680× 10− 8 1.3387× 10− 3 70 7.2290× 10− 4 1.9277× 10− 4 3.8554× 10− 5 7.7109× 10− 6 1.5422× 10− 6 3.0844× 10− 7 6.1687× 10− 8 9.6385× 10− 4 80 5.2050× 10− 4 1.3880× 10− 4 2.7760× 10− 5 5.5520× 10− 6 1.1104× 10− 6 2.2208× 10− 7 4.4416× 10− 8 6.9399× 10− 4 90 3.7476× 10− 4 9.9936× 10− 5 1.9987× 10− 5 3.9974× 10− 6 7.9948× 10− 7 1.5990× 10− 7 3.1979× 10− 8 4.9967× 10− 4 100 2.6983× 10− 4 7.1953× 10− 5 1.4391× 10− 5 2.8781× 10− 6 5.7563× 10− 7 1.1513× 10− 7 2.3025× 10− 8 3.5976× 10− 4 110 1.9427× 10− 4 5.1806× 10− 5 1.0361× 10− 5 2.0723× 10− 6 4.1445× 10− 7 8.2890× 10− 8 1.6578× 10− 8 2.5903× 10− 4 120 1.3988× 10− 4 3.7301× 10− 5 7.4602× 10− 6 1.4920× 10− 6 2.9841× 10− 7 5.9682× 10− 8 1.1936× 10− 8 1.8650× 10− 4 130 1.0071× 10− 4 2.6857× 10− 5 5.3713× 10− 6 1.0743× 10− 6 2.1485× 10− 7 4.2970× 10− 8 8.5941× 10− 9 1.3428× 10− 4 140 7.2512× 10− 5 1.9337× 10− 5 3.8673× 10− 6 7.7346× 10− 7 1.5469× 10− 7 3.0938× 10− 8 6.1877× 10− 9 9.6681× 10− 5 150 5.2209× 10− 5 1.3922× 10− 5 2.7845× 10− 6 5.5690× 10− 7 1.1138× 10− 7 2.2276× 10− 8 4.4552× 10− 9 6.9611× 10− 5 160 3.7590× 10− 5 1.0024× 10− 5 2.0048× 10− 6 4.0096× 10− 7 8.0193× 10− 8 1.6039× 10− 8 3.2077× 10− 9 5.0120× 10− 5 170 2.7065× 10− 5 7.2174× 10− 6 1.4435× 10− 6 2.8870× 10− 7 5.7739× 10− 8 1.1548× 10− 8 2.3096× 10− 9 3.6086× 10− 5 180 1.9487× 10− 5 5.1965× 10− 6 1.0393× 10− 6 2.0786× 10− 7 4.1572× 10− 8 8.3144× 10− 9 1.6629× 10− 9 2.5982× 10− 5 190 1.4031× 10− 5 3.7415× 10− 6 7.4830× 10− 7 1.4966× 10− 7 2.9932× 10− 8 5.9864× 10− 9 1.1973× 10− 9 1.8707× 10− 5 200 1.1632× 10− 5 3.1018× 10− 6 6.2037× 10− 7 1.2407× 10− 7 2.4815× 10− 8 4.9629× 10− 9 9.9259× 10− 10 1.5509× 10− 5 210 7.2734× 10− 6 1.9396× 10− 6 3.8792× 10− 7 7.7583× 10− 8 1.5517× 10− 8 3.1033× 10− 9 6.2067× 10− 10 9.6977× 10− 6 220 5.2369× 10− 6 1.3965× 10− 6 2.7930× 10− 7 5.5860× 10− 8 1.1172× 10− 8 2.2344× 10− 9 4.4688× 10− 10 6.9824× 10− 6 230 3.7705× 10− 6 1.0055× 10− 6 2.0110× 10− 7 4.0219× 10− 8 8.0438× 10− 9 1.6088× 10− 9 3.2175× 10− 10 5.0273× 10− 6 240 2.7148× 10− 6 7.2395× 10− 7 1.4479× 10− 7 2.8958× 10− 8 5.7916× 10− 9 1.1583× 10− 9 2.3166× 10− 10 3.6197× 10− 6 250 1.9547× 10− 6 5.2125× 10− 7 1.0425× 10− 7 2.0850× 10− 8 4.1700× 10− 9 8.3400× 10− 10 1.6680× 10− 10 2.6062× 10− 6 300 1.3436× 10− 6 3.5828× 10− 7 7.1656× 10− 8 1.4331× 10− 8 2.8663× 10− 9 5.7325× 10− 10 1.1465× 10− 10 1.7914× 10− 6 500 3.6069× 10− 7 9.6183× 10− 8 1.9237× 10− 8 3.8473× 10− 9 7.6947× 10− 10 1.5389× 10− 10 3.0779× 10− 11 4.8091× 10− 7 800 1.9164× 10− 7 5.1104× 10− 8 1.0221× 10− 8 2.0442× 10− 9 4.0883× 10− 10 8.1766× 10− 11 1.6353× 10− 11 2.5552× 10− 7 1000 1.7992× 10− 8 4.7978× 10− 9 9.5955× 10− 10 1.9191× 10− 10 3.8382× 10− 11 7.6764× 10− 12 1.5353× 10− 12 2.3988× 10− 8

Table 2. Present partial and total H(2s) formation cross sections (σ 12 in ) of p– Na scattering. Integration range is equal to 32a0, and the number of iterations υ = 50.

Table 3.

Table 3.

Table 3. Present total H(2s) cross-section formation (σ 12 in ) of p– Na scattering together with theoretical results (Refs.  [21] and [22]). k2/keV σ 12 Present Ref.  [21] Ref.  [22] 50 1.8593× 10− 3 1.16× 10− 3 5.269× 10− 3 60 1.3387× 10− 3 – – 70 9.6385× 10− 4 – – 80 6.9399× 10− 4 – – 100 3.5976× 10− 4 8.27× 10− 5 4.912× 10− 3 150 6.9612× 10− 5 2.41× 10− 5 1.827× 10− 3 200 1.5509× 10− 5 1.03× 10− 5 5.206× 10− 4 250 2.6062× 10− 6 – – 500 4.8092× 10− 7 2.91× 10− 7 6.022× 10− 7 800 2.5552× 10− 7 – 1.029× 10− 7 1000 2.3989× 10− 8 – 1.105× 10− 8

Table 3. Present total H(2s) cross-section formation (σ 12 in ) of p– Na scattering together with theoretical results (Refs.  [21] and [22]).

Fig.  2.

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	Fig.  2. Present total H(2s) formation cross sections (σ 12 in of p– Na scattering together with theoretical results (Refs.  [21] and [22]).

It is interesting to mention that all theoretical techniques applied to the collisions of electron with atoms can be extended to proton– atom collisions after allowing for hydrogen formation channel instead of recombination channel. Proton– sodium inelastic scattering is studied using the coupled static approximation. Our interest is focused on the influence of producing H(2s) atom in the second channel. The present calculations of the total H(2s) formation cross sections using the coupled static approximation show a reasonable agreement with available theoretical calculations in a wide region of incident energies.