Formation of ground and excited hydrogen atoms in proton potassium inelastic scattering

Transcription

1 Pramana J. Phys. (6) 87: 78 DOI.7/s4-6-8-y c Indian Academy of Sciences Formation of ground excited hydrogen atoms in proton potassium ineastic scattering S A ELKILANY, Department of Mathematics, Facuty of Science, University of Dammam, Dammam, Kingdom of Saudi Arabia Department of Mathematics, Facuty of Science, Kafresheikh University, Kafresheikh, Egypt E-mai: sabbekiany@yahoo.com MS received December 4; revised February 6; accepted 6 February 6; pubished onine 7 October 6 Abstract. The ineastic scattering of proton with a potassium atom is treated for the first time as a threechanne probem within the framework of the improved couped static approximation by assuming that the ground (s state) the excited (s state) hydrogen formation channes are open for seven vaues of tota anguar momentum, ( 6) at energies between kev. The Lipmann Swinger equation the Green s function iterative numerica method are used to cacuate iterative partia tota cross-sections. This can be done by cacuating the reactance matrix at different vaues of the considered incident energies to obtain the transition matrix that gives partia tota cross-sections. Present resuts are in reasonabe agreement with previous resuts. Keywords. PACS Nos Ineastic scattering; proton potassium; hydrogen formation. 4.8.Dp; 4.8.Gs; 4.8.Ht. Introduction The appearance of ermediate states in atomic nucear reactions is considered as the most eresting physica phenomenon since the deveopment of the quantum theory. The deveopment of the femtosecond aser has tremendousy enhanced the experimenta identification of these states in chemica physica reactions. Most theoretica treatments of eectron, positron proton atom eractions are based on the cacuations of differentia, partia tota crosssections as functions of incident energies using different approximations. It is we known that hydrogen formation pays an important roe in the scattering of proton from akai atoms at ow ermediate impact energies. Theoretica investigation on eectron, positron proton atom scattering has been carried out by many researchers. Abde-Raouf [ appied the couped-static frozen core approximations to positron ithium positron sodium scattering as a two-channe probem. E-Bakry Abde-Raouf [ appied couped static approximation to the production of positronium in positron ithium scattering as a three-channe probem. Ekiany Abde-Raouf [ investigated the effect of poarization potentias on antiproton scattered by positronium using the couped static mode. Ekiany [4, treated in detai the effect of poarization potentia on ineastic scattering of proton ithium or proton sodium as a two-channe probem, using the improved couped-static approximation. Banyard Shirtciffe [6 used the continuumdistorted wave method to cacuate cross-sections for protons scattered from atomic ithium. Ferrante et a [7 cacuated tota cross-sections for hydrogen formation in proton scattering with akai atoms using the Oppenheimer Brinkman Kramer s (OBK) approximation. Daniee et a [8 cacuated tota cross-sections of proton scattering by akai atoms using eikona approximation treatment, deveoped previousy within the framework of the time-dependent impact parameter method for high-energy charge transfer. Ferrante Fiordiino [9 investigated high-energy proton akai atom coision using Eikona approximation. Fritsch Lin [ discussed the hydrogen formation in proton scattering by ithium atoms using a modified two-centre atomic orbita expansion. Choudhury Sura [ cacuated the hydrogen formation crosssections in the coision of protons with akai meta atoms using the wave formuation of impuse approximation. Tiwari [ reported the differentia tota

2 78 Page of Pramana J. Phys. (6) 87: 78 cross-sections in the hydrogen formation in the coision of protons with ithium sodium atoms using the Couomb-projected Born approximation. In this paper, we modify the couped static approximation used in refs [ to investigate the ineastic scattering of a proton by a potassium atom as a threechanne probem. The formation of ground excited hydrogen are investigated in proton potassium (p K) ineastic scattering as a three-channe probem in which the eastic hydrogen (in s s states) formation channes are open for seven vaues of the tota anguar momentum ( 6) at energies between kev. The target is described using Roothan Hartree Fock atomic wave functions within the framework of the one-vaence-eectron mode [. The iterative Green s function partia-wave expansion technique is used. The next section contains a brief presentation of our mathematica formaism, is devoted to the discussions of our resuts a concusion on the present resuts is given in 4. The appendices invove a short treatment of the methods that are used in cacuating the reactance matrix (Appendix A) the numerica iterative method (Appendix B), which are used in cacuating the egras of our probem.. Theoretica formaism The three-channe scattering probem under investigation can be sketched by (see figure ) pk(4s) Eastic channe (first channe) H(s)K p K(4s)= H(s) formation () (second channe) H(s)K H(s) formation channe (third channe) The one-vaence-eectron mode of the target, K, is described (in Rydberg units, Ry) by the Hamitonian [: H T = r μ m r V c(r), () where r is the position vector of the vaence eectron with respect to the origin of the scattering system in which the nuceus of the target is infinitey heavy, μ m is the reduced mass of the target V c (r) is a screened potentia defined by V c (r) = V ccou (r) V cex (r), () where V ccou (r) V cex (r) are the Couomb exchange parts of the core potentia, respectivey. e M T r σ ρ Figure. Configuration space of p K(4s) scattering: x r are the position vectors of the projectie proton the vaence eectron of the target with respect to the centre of mass of the target, ρ is the position vector of the projectie proton with respect to the vaence eectron of the target, σ is the position vector of the centre of mass of H from the target, M T is the mass of the nuceus of the target. Using Cementi Roetti Tabes [, the wave function of the ith eectron in the orbita j of the target is exped by m j j (r i ) = c jp χ jp (r i ), (4) p= where χ jp (r i ) is a Sater-type wave function given by χ jp (r i ) = A jp r k jp i e α jpr i Y jp (ϑ i,ϕ i ). () The k jp are egers or zero A jp are normaization factors determined by A jp =[(α jp ) k jp /(k jp )! / (6) the spherica harmonics Y jp are normaized by Y jp (ϑ i,ϕ i ) Y jp (ϑ i,ϕ i ) =. The constants c jp α jp are adjusted within the framework of the Hartree Fock Roothaan approach. Substituting eq. () o eq. (4) roducing the notation c jp = c jp A jp, we obtain j (r i ) = m j p= x c jp r k jp i e α jpr i Y jp (ϑ i,ϕ i ), (7) where m j in eqs (4) (7) is the number of basis functions characteristic to each orbita (usuay, a orbitas of the same type have the same m j ). The Couomb part of the core potentia is defined by V ccou (r) = M N j j (r i ) r r i r j (r i ), (8) j= where M is the number of orbitas N j is the number of eectrons occupying the orbita j. The prime on the summation sign means that the term /r is repeated p

3 Pramana J. Phys. (6) 87: 78 Page of 78 for each j. The exchange part of the core potentia is defined by V cex (r) = M K(4s) (r i ) r r i j (r i ), (9) j= where the subscript K(4s) is empoyed for distinguishing the wave function for the vaence eectron of the target atom. According to eq. (), the binding energy of the vaence eectron of the target is determined by E K(4s) = K(4s) (r) H T K(4s) (r). () The tota Hamitonian of the first channe, eastic channe (in Rydberg units frozen core approximation) has the foowing form: H = H () = H T x μ V (), () M where μ M is the reduced mass of the first channe V () denotes the eraction between the incident proton the target atom. That is, V () = x x r V ccou(x), () where V ccou (x) = V ccou (r) r=x, () the corresponding tota energy in the first channe is determined by E = E () = E K(4s) k μ, (4) M where (/μ M )k is the kinetic energy of the incident proton reative to the target nuceus. The tota Hamitonian of the second channe, H(s), formation is expressed (in Rydberg units frozen core approximation) as H = H () = H H(s) σ μ V (), () M where μ M is the reduced mass of the second channe V () represents the eraction between the partices of H(s) the rest of the target atom. That is, V () = x r V ccou(x) V ccou (r) V cex (r), (6) the tota energy of the second channe is determined by E = E () = E H(s) k μ, (7) M where (/μ M )k is the kinetic energy of the centre of mass of H(s) with respect to the nuceus of the target. This is reated to the energy of the incident proton by k ( = E K(4s) ) k μ M μ E H(s), (8) M where E H(s) =. Ry is the ground state energy of H(s)(k /μ M )>means that the H(s) channe is open, otherwise it is cosed. Thus, H(s) formation is ony possibe if k > μ M(E H(s) E K(4s) ). The tota Hamitonian of the third channe, H(s)formation, is expressed (in Rydberg units frozen core approximation) as H = H () = H H(s) σ μ M V (), (9) where μ M is the reduced mass of the third channe V () represents the eraction between the partices of H(s) the rest of the target atom. That is, V () = x r V ccou(x ) V ccou (r ) V cex (r ), () the tota energy of third channe is determined by E = E () = E H(s) k μ, () M where (/μ M )k is the kinetic energy of the centre of mass of H(s) with respect to the nuceus of the target. This is reated to the energy of the incident proton by k ( = E K(4s) ) k μ M μ E H(s), () M where E H(s) =. Ry is the first excited state energy of the excited H(s)(k /μ M )>means that H(s) channe is open, otherwise it is cosed. Thus, H(s) formation is ony possibe if k > μ M(E H(s) E K(4s) ). The couped static approximation states that the soution of the three-channe scattering probem under consideration is subjected to the foowing conditions: K(4s) H E =, () H(s) H E =, (4) H(s) H E =, () where is the tota wave function describing each scattering process, that is, = K(4s) (r) ψ (x) H(s) (ρ) ψ (σ ) H(s) (ρ ) ψ (σ ), (6)

4 78 Page 4 of Pramana J. Phys. (6) 87: 78 H(s) = π exp( ρ), (7) H(s) = π ( ρ ) exp( ρ /), (8) where H(s) is the ground-state wave function of H(s), H(s) is the first excited state wave function of H(s), ψ (x) is the wave function describing scattered protons, ψ (σ ) is the scattering wave function of the second channe ψ (σ ) is the scattering wave function of the third channe. Substituting eqs (4) (6) in eq. (), we obtain μ M ( x k ) ψ = U () st (x) ψ K(4s) H () E () H(s) ψ K(4s) H () E () H(s) ψ. (9) Substituting eqs (7) (6) in eq. (4), we obtain ( σ μ k ) ψ = U st () (σ ) ψ H(s) M H () E () K(4s) ψ H(s) H () E () H(s) ψ. () Substituting eqs () (6) in eq. (), we obtain ( σ μ M k ) ψ = U st () (σ ) ψ H(s) H () E () K(4s) ψ H(s) H () E () H(s) ψ, () where Schrödinger s equations of the target, H(s) H(s) are empoyed. The potentias U st () (x), U st () (σ ) U st () (σ ) are defined by U st () (x) = K(4s) (r) V () K(4s)(r), () Tabe. Present partia tota hydrogen formation, H(s), cross-sections (in πa ) of p K scattering. Tota k cross- (kev) = = = = = 4 = = 6 section.46e E-.844E-.697E-6 7.8E-7.476E-7.96E E E E-.9E-.9E-6.8E-7.4E-7.47E-8.E E E- 8.76E-6.7E-6.46E E-8.7E-8.44E E-.646E-.94E-6.8E-6.7E-7 4.4E E-9.E-4 9.8E-.4E-.874E-6.749E-7.48E-7.964E E E-.9E- 6.94E-6.89E E E-8 9.9E-9.98E-9.97E-.849E- 4.98E E-7.97E E E-9.778E-9.46E-.49E-.879E E-7.487E-7.97E-8 6.9E-9.9E-9.99E-.E-.9467E-6.894E-7.787E-7.74E E E-.47E E-6.66E-6 4.9E E-8.7E-8.466E E-.8E-.78E-6.6E-6.E E-8.9E-8.447E E- 7.6E E-6.E-6.66E-7.4E-8.649E-8.97E E- 6.6E E-6.49E-6.88E E-8 9.4E-9.887E-9.67E-.74E E-6 9.E-7.86E-7.7E E-9.49E-9.984E- 4.66E E E-7.46E-7.9E-8.846E-9.69E-9.8E-.68E-6.989E-6.877E-7.7E-7.E-8 4.E E-.69E-.648E-6.99E-6.E-7.E-7.E-8 4.E-9 8.E-.64E-.6E-6.847E E-7 9.8E-8.974E-8.948E E-.76E-.46E-6.777E E E-8.89E-8.79E E-.64E-.69E E-6 4.8E E-8.8E-8.6E E-.4E-.69E-6.647E E E-8.74E-8.8E-9 7.7E-.44E-.94E-6.9E-6.4E-7 7.9E-8.48E-8.86E-9.674E-.4E-.776E-6.E-6.699E-7.99E-8.798E-8.96E-9 4.9E- 8.68E-.497E E-7.84E E-8 7.7E E-9.949E-.8986E- 9.64E E E-8.96E-8.99E E-.74E-.447E- 4.96E-7 4.4E-8.747E-8.49E E E-.8796E-.79E-.87E-8

5 Pramana J. Phys. (6) 87: 78 Page of 78 E-4 =, =, =, =, =4, =, =6, Tota H(s) Partia tota hydrogen formation cross-section(π a ) E- E-6 E-7 E-8 E-9 E Incident energy k (kev) Figure. Present partia tota hydrogen formation, H(s), cross-sections (in πa ) of p K scattering.. E- =, =, =, =, =4, =, =6, Tota Eastic Partia tota eastic cross-section (π a ) E-4 E- E-6 E-7 E-8 E Incident energy k (kev) Figure. Present partia tota eastic cross-sections (in πa ) of p K scattering.

6 78 Page 6 of Pramana J. Phys. (6) 87: 78 =, =, Partia tota hydrogen H(s) formation cross-section E- E-4 E- E-6 E-7 E-8 =, =, =4, =, =6, Tota H(s) E Incident energy k (kev) Figure 4. Present partia tota hydrogen formation, H(s), cross-sections (in πa ) of p K scattering. Tabe. Present tota cross-sections (in πa ) of p K scattering with the compared resuts [7,8,. σ [7 [7 [8 σ k (kev) H(s) OBK FBA Eikona [ H(s) [.E-.9E- 4.64E-4 6.8E E-.E-4 7.9E E- 4.66E-.48E-.44E E-4.499E-.E-.9E-.E E E-.7E-.489E- 7.76E E-4.499E-.E-.86E-.E-4.97E- 4.4E-.79E-4.48E- 9.4E-4 8.8E-4.46E-.997E-4 6.4E-4 4.E-4 6.4E-4.99E-.E-4.496E-4.89E-4.8E-4.47E E-4.996E-4.9E-4 4.6E-4.8E-.4E-4.E-4 7.6E-6.6E- 6.9E-4 6.6E E-.74E E- 4.66E E-.98E-4.68E-6 4.4E-.E-.648E-6.68E E-.6E-6 4.E-.46E-6 4.E-.69E E-.69E-6.78E-.94E-6.467E-.776E-6.89E-.497E E- 9.64E E E-7 8.9E-7 9.E-7.87E-8 8.E-8

7 Pramana J. Phys. (6) 87: 78 Page 7 of 78 Ref. [ Ref. [8 Ref. [7 Ref. [7 Figure. Present tota hydrogen, H(s), formation cross-sections (σ in πa ) of p K scattering in comparison with theoretica resuts [7,8,. U st () (σ ) = H(s) (ρ) V () H(s)(ρ) () U st () (σ ) = H(s) (ρ ) V () H(s)(ρ ). (4) Using partia wave expansions of the scattering wave functions ψ, ψ ψ in eqs () (), soutions are given (formay) by Lippmann Schwinger equation (see Appendix A): ξ = ξ G ζ, () where G is the Green operator (ε H ) ξ is the soution of the homogeneous equation: (ε H ) ξ =. (6) The partia wave expansions of the Green operators corresponding to operators in the three differentia equations enabe us to write their soutions in egra form, which can be soved using the iterative numerica technique. Then we obtain reactance matrix R ν that is reated to transition matrix T ν by T ν = R ν (I ĩr), (7) where ν is the order of iteration, I isa unit matrix ĩ =. Partia cross-sections obtained in an iterative way are determined (in πa units) by σ (,ν) ij = 4( ) k T ν ij, i,j =,,. (8) Finay, the tota cross-sections (in πa expressed (in νth iteration) by σ ν ij = = units) are σ (,ν) ij, i,j =,,, ν>. (9). Resuts discussion We begin by testing the variation of static potentias of three channes, U st () (x), U st () (σ ) U st () (σ ), with increasing x, σ σ, respectivey. Vaues of x, σ σ have been chosen such that x = σ = σ = 6, 6,,..., 6 6 where h = /6 is the mesh size (or Simpson s step) empoyed to cacuate egras appearing in egra

8 78 Page 8 of Pramana J. Phys. (6) 87: 78 equations using Simpson s rue. Cacuation of crosssections of p K scattering has been proceeded by investigating variation of the eements of R ν with the increase of egration range (IR) the number of iterations. We set h to be /6 use mesh (that is, IR = a, see Appendix B). It is found that iterations are enough for the convergence of the reactance matrix eements the partia cross-sections vaues; this demonstrates stabiity of our iterative method. Fina cacuations were carried out for seven partia waves corresponding to 6, which are quite satisfactory for cacuating the tota eastic hydrogen formation, H(s) H(s) cross-sections (because the main contributions of those vaues are due to S, P D partia cross-sections at higher we have very sma vaues of the partia cross-sections) at vaues of k, representing the kinetic energy region ( k kev). In tabe, we present the partia tota H(s) formation cross-sections, σ (channe ) at a incident vaues of energy between kev. These vaues are aso dispayed in figure. We can concude the foowing pos: () The main contributions to the tota H(s) formation cross-sections, σ, are due to the S, P D partia cross-sections. () The partia cross-sections of H(s) formation decrease steadiy with the anguar momentum. () The tota H(s) formation cross-sections, σ, decrease steadiy with k. (4) The seven partia waves empoyed are quite satisfactory for cacuating the tota H(s) formation cross-sections, σ, to a higher degree of accuracy within the framework of the improved couped static approximation. The common features of the obtained vaues of the partia tota hydrogen formation, H(s) (channe ), are repeated for the partia tota eastic cross-sections (channe ) the partia tota H(s)formation cross-sections (channe ) at a incident energies (see figures 4). The most eresting resuts are accumuated in tabe figures 6. In tabe figure, we find a comparison between the tota H(s) formation cross-sections, σ,h(s) formation cross-sections, σ (in πa ) the avaiabe resuts determined by Ferrante et a [7 using the Oppenheimer Brinkman Kramer s (OBK) approximation, Daniee Figure 6. Present tota hydrogen, H(s), formation cross-sections (σ in πa ) of p K scattering in comparison with theoretica resuts [.

9 Pramana J. Phys. (6) 87: 78 Page 9 of 78 et a [8 using Eikona-approximation treatment those by Choudhury Sura [ using the wave formuation of impuse approximation. Our cacuations show that the tota cross-sections are consideraby ower than those of the compared resuts at the considered vaues of the incident energies except at k = 4 kev. We find that our vaues are higher than those of Ferrante et a (FBA) [7. We can aso concude that our resuts are in good agreement with those of Choudhury Sura [ at a considered vaues of the incident energies between kev. In tabe figure 6, we aso find our vaues of the tota H(s) formation cross-sections, σ (in πa ) with those of Choudhury Sura [ using the wave formuation of impuse approximation. Our resuts of the tota H(s) formation cross-sections have the same trend of the compared resuts are in reasonabe agreement with those of Choudhury Sura [ at a considered vaues of the incident energies. The reason for the difference ies in the different approximations used for the scattering wave function in negecting poarization potentia effects of the H(s) formation H(s) formation in our cacuations. Before eaving this section, it is important to mention that further improvements on the present tota coisiona cross-sections can aso be obtained by considering the effect of poarization potentia of H(s) formation H(s) formation atoms by considering higher excitation channes of H formation in our cacuations. 4. Concusions Proton potassium ineastic scattering was studied for the first time using the modified couped static approximation (three-channe probem). Our erest is focussed on H(s) formation H(s) formation atoms in p K ineastic scattering as a three-channe probem. The present cacuations show that the excited hydrogen channe is needed to investigate the coision of the proton with potassium atom. Our vaues are, in genera, ower than the compared resuts, we expect that our coisiona cross-sections wi increase by taking the effect of poarization potentias in our cacuations, either by empoying a poarized orbita technique or by incuding pseudopoarization potentias in our cacuations. We can aso improve our resuts by considering higher excitation channes of hydrogen formation atom (which are now under consideration by the author). Appendix A: The reactance matrix R Let us consider the partia wave expansions of the scattering wave functions ψ, ψ ψ, that is, ψ = ( )i f (x)y ( ˆx), (A.) x ψ = ( )i g (σ )Y ( ˆσ) (A.) σ ψ = ( )i σ h (σ )Y ( ˆσ ), (A.) where f (x), g (σ ) h (σ ) are the radia partia wave functions corresponding to the tota anguar momentum, of the first, second third channes, respectivey, Y ( ˆx), Y ( ˆσ) Y ( ˆσ ) are the reated spherica harmonics. ˆx, ˆσ ˆσ are the anges between the vectors x, σ σ,thez-axis. Substituting eqs (A.) (A.) o eqs () (), for each vaue of, we obtain the foowing couped egrodifferentia equations: [ d ( ) dx x k = μ M U st () (x)f (x) f (x) K (x, σ)g (σ)dσ K (x, σ )h (σ )dσ, [ d ( ) dσ σ k g (σ ) = μ M U st () (σ )g (σ ) K (σ, x)f (x)dx K (σ, σ )h (σ )dσ [ d ( ) dσ σ k h (σ ) = μ M U st () (σ )h (σ ) K (σ,x)f (x)dx K (σ,σ)g (σ )dσ, (A.4) (A.) (A.6) where the kernes K, K, K K are exped by K (x, σ ) = μ M (8xσ) K(4s) (r) H(s) (ρ) [ ( σ μ k ) V () M Y o ( ) ( ) ˆx Y o ˆσ d ˆxd ˆσ, (A.7)

10 78 Page of Pramana J. Phys. (6) 87: 78 K (x, σ ) = μ M (8xσ ) K(4s) (r) H(s) (ρ ) [ ( σ μ M k ) V () Q (σ ) = K (σ,x)f (x)dx. (A.7) Y o ( ˆx)Yo ( ˆσ )d ˆxd ˆσ The three couped egrodifferentia (scattering), (A.8) equations (A.) (A.) are identica with the inhomogeneous equation K (σ, x) = μ M (8σx) H(s) (ρ) K(4s) (r) [ [ε H (u) ξ(u) = ζ(u), (A.8) ( x μ k ) V () M where Y o ( ˆx)Yo ( ˆσ)dˆxd ˆσ (A.9) u = x impies: ε = k K (σ,x) = μ M (8σ x) H(s) (ρ, H (x) = d ( ) dx x, ) K(4s) (r) [ ξ(x) = f (x), ( x μ k ) V () M ζ(x) = μ M U st () (x)f (x) Q (x) Q (x), Y o ( ˆx)Yo ( ˆσ )d ˆxd ˆσ (A.) the kernes K (σ, σ ) K (σ,σ)are equa to zero due to the orthogonaity of the wave functions of the ground excited states of hydrogen. (The number eight appearing in the preceding equations refers to the Jacobians of the transformations dr 8 dσ dρ 8 dx. Let us now rewrite eqs (A.4) (A.6) as [ d ( ) dx x k f (x) = μ M U st () (x)f (x) Q (x) Q (x), (A.) [ d ( ) dσ σ k [ d dσ where Q (x) = Q (x) = Q (σ ) = ( ) σ k g (σ ) = μ M U st () (σ )g (σ ) Q (σ ) (A.) K (x, σ )g (σ )dσ, K (x, σ )h (σ )dσ, K (σ, x)f (x)dx, h (σ ) = μ M U () st (σ ) h (σ ) Q (σ ), (A.) (A.4) (A.) (A.6) u = σ impies: ε = k, H (σ ) = d ( ) dσ σ, ξ(σ) = g (σ ), ζ(σ) = μ M U () st (σ )g (σ ) Q (σ ) u = σ impies: ε = k, H (σ ) = d ( ) dσ σ, ξ(σ ) =h (σ ), ζ(σ ) =μ M U () st (σ )h (σ ) Q (σ ). The forma soutions of eq. (A.8) is given by the Lippmann Schwinger equation ξ = ξ G ζ, (A.9) where G is the Green operator (ε H ) ξ is the soution of the homogeneous equation (ε H ) ξ =. (A.) The partia-wave expansion of G in the three channes enabes us to write the forma soution of (A.) (A.) as f (i) (x) = { δ i k Q (i) (x ) Q (i) (x )dx g (k x )[μ M U () st (x )f (i) (x ) (k x)

11 Pramana J. Phys. (6) 87: 78 Page of 78 g (i) { k (k x )[μ M U st () (x )f (i) (x ) Q (i) (x ) Q (i) (x )dx g (k x), i =,, (σ ) = {δ i k h (i) Q (i) (σ )dσ (A.) g (k σ )[μ M U () st (σ )g (i) (σ ) (k σ) { k (k σ )[μ M U () st (σ )g (i) (σ ) Q (i) (σ )dσ g (k σ), i =,, (A.) (σ ) = {δ i k 4 g (k σ )[μ M U () st (σ )h (i) (σ ) Q (i) (σ )dσ (k σ ) { k (k σ )[μ M U () st (σ )h (i) (σ ) Q (i) (σ )dσ g (k σ ), i =,,, 6 (A.) where the deta functions δ ij, i, j =,,, specify two independent forms of soutions for each of f (x), g (σ ) h (σ ) in channes i =,, according to the channe considered. Thus, if i =, the first eement in the { bracket of f (x), for exampe, wi be defining the first form of soution. For i =, this eement wi be zero defining the second form. The functions (η) g (η), η = k x,k σ or k σ,are reated to the Besse functions of the first second channes, that is, j (η) y (η), respectivey, by the reations (η) = ηj (η) g (μ) = ηy (η). It is obvious from eqs (A.) (A.) that the soutions can be ony found iterativey the νth iterations are cacuated by f (i,ν) (x) = {δ i k X g (k x )[μ M U st () (x ) f (i,ν) (x ) Q (i,ν ) (x ) (x )dx (k x) Q (i,ν ) { k X (k x )[μ M U () st (x )f (i,ν) (x ) Q (i,ν ) (x ) Q (i,ν ) (x )dx g (k x), i =,, ; ν, (A.4) { g (i,ν) (σ ) = δ i g (k σ ) k [μ M U () st (σ )g (i,ν) (σ ) Q (i,ν) (σ )dσ { k f (k σ ) f (k σ) [μ M U () st (σ )g (i,ν) (σ ) Q (i,ν) (σ )dσ g (k σ), i =,, ; ν (A.) h (i,ν) (σ ) = {δ i k g (k σ ) [μ M U () st (σ )h (i,ν) (σ ) Q (i,ν) (σ )dσ (k σ ) { k (k σ ) [μ M U () st (σ )h (i,ν) (σ ) Q (i,ν) (σ )dσ g (k σ ), 6 i =,, ; ν. (A.6) The zeroth iteration of f (i,) (x) is obtained by f (i,) (x) = {δ i k X g (k x ) [μ M U st () (x )f (i,) (x )dx (k x) { k X (k x ) [μ M U () st (x )f (i,) (x )dx g (k x), i =,,, (A.7) where X, specify the egration range away from the nuceus over which the egras at eqs (A.4) (A.7) are cacuated using Simpson s expansions. The functions Q (i,ν ) (x ), Q (i,ν ) (x ), 4

12 78 Page of Pramana J. Phys. (6) 87: 78 Q (i,ν) (σ ) Q (i,ν) (σ ), in eqs (A.4) (A.6), are now defined by Q (i,ν ) (x ) = K (x,σ)g (i,ν ) (σ )dσ, (A.8) Q (i,ν ) (x ) = K (x,σ )h (i,ν ) (σ )dσ, (A.9) Q (i,ν) (σ ) = Q (i,ν) (σ ) = X X K (σ,x)f (i,ν) (x)dx (A.) K (σ,x)f (i,ν) (x)dx. (A.) To find the starting vaue of f (i,) (x), we consider the Tayor expansion of U st () (x), (k x) g (k x) around the origin (see Appendix B). Equations (A.4) (A.6) can be abbreviated to f (i,υ) g (i,υ) (x) = a (i,υ) f (k x) b (i,υ) g (k x), i =,,, (σ ) = a (i,υ) f (k σ) b (i,υ) g (k σ), i =,, (A.) (A.) h (i,υ) (σ ) = a (i,υ) f (k σ ) b (i,υ) g (k σ ), i =,,, (A.4) where a (i,υ) = {, b (i,υ) = {,a (i,υ) = {, b (i,υ) = { 4, a (i,υ) = {,b (i,υ) = { 6, the preceding six brackets are the coefficients of eqs (A.4) (A.6). The coefficients at eqs (A.) (A.4) are eements of the foowing matrices: a υ = μm /k a (,υ) μm /k a (,υ) μm /k a (,υ) μm /k a (,υ) μm /k a (,υ) μm /k a (,υ) μm /k a (,υ) μm /k a (,υ) μm /k a (,υ) (A.) b υ = μm /k b (,υ) μm /k b (,υ) μm /k b (,υ) μm /k b (,υ) μm /k b (,υ) μm /k b (,υ) μm /k b (,υ) μm /k b (,υ) μm /k b (,υ) (A.6) which are connected with the reactance matrix, R υ, through the reation {R v ij ={b v (a v ) ij. (A.7) Appendix B: The numerica iterative method This appendix contains a brief discussion of the iteration procedure used for cacuating the eements of the reactance matrix. This has been achieved using eqs (A.4) (A.6) where X, represent the ranges of egrations over x, σ σ, respectivey. Physicay, X (X = nh,wherenis the number of mesh pos h is the Simpson step, or mesh size), represents the distance at which we assume that the scattered protons are not affected by the potassium atoms, (= nh) is the distance at which the ground-state hydrogen atom the proton of the target are totay separated (=nh) is the distance at which the excited hydrogen atom the proton of the target are totay separated. To cacuate the egras in eqs (A.4) (A.) we use Simpson s rue. Thus, we exp Q (i,ν ) (x), Q (i,ν ) (x), Q (i,ν) (σ ) Q (i,ν) (σ ) at po q of the configuration space as foows: Q (i,ν ) (x q ) = Q (i,ν ) (x q ) = Q (i,ν) (σ q ) = Q (i,ν) n {ω p () K() p= (σ p,x q ) ω p () K() (σ p,x q ) ω p () K() (σ p,x q )g (i,ν ) (σ p ), ν, (B.) n {ω p () K() p= (σ p,x q) ω p () K() (σ p,x q) ω p () K() (σ p,x q)h (i,ν ) (σ p ), ν, (B.) n (σ q ) = n {ω p () K() p= (σ q,x p ) ω p () K() (σ q,x p ) ω p () K() (σ q,x p )f (i,ν) (x p ), ν, {ω p () K() p= (σ q,x p) ω p () K() (σ q,x p) ω p () K() (σ q,x p)f (i,ν) (x p ), ν, (B.) (B.4)

13 Pramana J. Phys. (6) 87: 78 Page of 78 where ω p () s are the usua Simpson weights (h/, 4h/, h/,..., h/, 4h/,h/) ω p () s, ω p () s are modified weights used for avoiding the singuarities (see refs [4,). The variabes x p, σ p σ p are chosen such that x p = σ p = σ p = ph, p =,,..., n. An essentia po in the determination of the egras in eqs (A.4) (A.) is the cacuation of the starting vaues of the functions f (i,ν) (x), g (i,ν) (σ ) h (i,ν) (σ ), that is, their vaues at x p = σ p = σ p = h, respectivey, which is determined as foows (note that f (i,ν) () = g (i,ν) () = h (i,ν) () = ). Considering the Tayor expansion of U st () (r), (kr) g (kr) around the origin, we obtain ( ) U st () Z (x) x D D x, (B.) where Z =, D D are constants, (k x) = (k x) [ (k x) / ( )!!!( ) g (k x) = (k x) 4 /4!( )( ) [ ( )!! ( )(k x) (k x) /!( ) (k x) 4 /4!( )( ) (B.6). (B.7) Assume that f (i,) (x) behaves cose to the origin as f (i,) (x) = C x C x C x C 4 x 4. (B.8) Substituting eqs (B.) (B.8) o eq. (A.7) yieds C = k ( )!!, C = Z C, [ D C = Z ( )( ) k C ( ) C 4 ={ [C 4 D C Z D 4 C C 4 Z/ 4 C Z 4 [C D C Z C D C Z/( 4) C Z/( )/k, (B.9) where = k ( )!!, = k k ( )( )!!, = 8( )( )( )!! ( )!! 4 = ( )k, ( )!! = ( )( )k, ( )!! 6 = 8( )( )( )k 4. (B.) Introducing constants C, C, C C 4 o eq. (B.8) setting x = h (the erva of egration), we obtain a starting vaue f (i,) (x) (note that the first four terms of eq. (B.8) are enough for obtaining a good starting vaue speciay when h is reasonaby sma). The above procedure can aso be appied for cacuating the starting vaue of g (i,ν) (σ ) h (i,ν) (σ ) at different vaues of, i ν. The iteration process starts by roducing f (i,) (x), i =,, vaues o eqs (A.) (A.) to find Q (i,) (σ ) Q (i,) (σ ), which can be used in the right-h side of eqs (A.) (A.6) to obtain g (i,) (σ ) h (i,) (σ ). The vaues of the ast quantities can be roduced o eqs (A.8) (A.9) to cacuate Q (i,) (x ) Q (i,) (x ), which may be empoyed in eq. (A.4) for determining f (i,) (x). This iteration process can be repeated as many times as we need the judge of its quantity is the stationary variation of the eements of the reactance matrix Rij ν where ν increase. References [ M A Abde-Raouf, J. Phys. B: At. Mo. Opt. Phys., (988) [ S Y E-Bakry M A Abde-Raouf, Nuc. Instrum. Methods B 4, 49 (998) [ S A Ekiany M A Abde-Raouf, J.Phys.Conf.Ser.88, Artice ID 7 () [4 S A Ekiany, Chin. Phys. Lett. (9), 94 (4) [ S A Ekiany, Braz. J. Phys. 44(6), 69 (4) [6 K E Banyard G W Shirtciffe, J. Phys. B: At. Mo. Phys. (9), 47 (979) [7 G Ferrante, E Fiordiino M Zarcone, I Nuovo Cimento B (), (979) [8 R Daniee, G Ferrante E Fiordiino, I Nuovo Cimento B 4(), 8 (979) [9 G Ferrante E Fiordiino, I Nuovo Cimento B 7, (98) [ W Fritsch C D Lin, J. Phys. B: At. Mo. Phys. 6, 9 (98) [ K B Choudhury D P Sura, J. Phys. B: At. Mo. Phys., 8 (99) [ Y N Tiwari, Pramana J. Phys. 7(4), 7 (8) [ E Cementi C Roetti, At. Data Nuc. Data Tabes 4( 4), 77 (974) [4 P A Fraser, Proc. Phys. Soc. 78, 9 (96) [ Y F Chan P A Fraser, J. Phys. B 6, 4 (97)