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

Abstract

The inelastic scattering of proton with a potassium atom is treated for the first time as a three-channel problem within the framework of the improved coupled static approximation by assuming that the ground (1s state) and the excited (2s state) hydrogen formation channels are open for seven values of total angular momentum, ℓ (0≤ℓ≤6) at energies between 50 and 500 keV. The Lipmann–Swinger equation and the Green’s function iterative numerical method are used to calculate iterative partial and total cross-sections. This can be done by calculating the reactance matrix at different values of the considered incident energies to obtain the transition matrix that gives partial and total cross-sections. Present results are in reasonable agreement with previous results.

Appendices

Appendix A: The reactance matrix R:

Let us consider the partial wave expansions of the scattering wave functions |ψ ₁〉, |ψ ₂〉 and |ψ ₃〉, that is,

$$ \psi_{1} =\frac{1}{x}\sum (2\ell +1)i^{\ell }f_{\ell} (x)Y_{\ell }^{0} (\hat{{x}}), $$

(40)

$$ \psi_{2} =\frac{1}{\sigma }\sum (2\ell +1)i^{\ell }g_{\ell} (\sigma )Y_{\ell}^{0} (\hat{{\sigma}}) $$

(41)

and

$$ \psi_{3} =\frac{1}{\sigma^{\prime}}\sum (2\ell +1)i^{\ell }h_{\ell} (\sigma^{\prime})Y_{\ell}^{0} (\hat{{\sigma }}^{\prime}), $$

(42)

where f _ℓ (x), g _ℓ (σ) and h _ℓ (σ ^′) are the radial partial wave functions corresponding to the total angular momentum ℓ, of the first, second and third channels, respectively, $Y_{\ell }^{0} (\hat {{x}})$, $Y_{\ell }^{0} (\hat {{\sigma }})$ and $Y_{\ell }^{0} (\hat {{\sigma }}^{\prime })$ are the related spherical harmonics. $\hat {{x}}$, $\hat {{\sigma }}$ and $\hat {{\sigma }}^{\prime }$ are the angles between the vectors $\bar {{x}}$, $\bar {{\sigma }}$ and $\bar {{\sigma }}^{\prime }$, and the z-axis.

Substituting eqs (40)–(42) into eqs (23)–(25), for each value of ℓ, we obtain the following coupled integrodifferential equations:

$$\begin{array}{@{}rcl@{}} &&{\kern-7.5pt} \left[{\frac{{\mathrm{d}}^{2}}{{{\mathrm{d}x}}^{2}}-\frac{\ell (\ell +1)}{x^{2}}+k_{1}^{2}}\right]f_{\ell} \left( x \right)\\ &&\quad = 2\mu_{\mathrm{M}} U_{st}^{\left( 1\right)} (x){} f_{\ell}(x)\\ &&\qquad+{\int}_{0}^{\infty} {K_{12}} ({\kern-.5pt}{x,\sigma}{\kern-.5pt}) g_{\ell} ({\kern-.5pt}\sigma{\kern-.5pt})\mathrm{d}\sigma\\ &&\qquad + {\int}_{0}^{\infty}\!{K_{13}}({x,\sigma^{\prime}}) h_{\ell} (\sigma^{\prime}) \mathrm{d}\sigma^{\prime}, \end{array} $$

(43)

$$\begin{array}{@{}rcl@{}} &&{\kern-7.5pt} \left[ {\frac{{\mathrm{d}}^{2}}{{{\mathrm{d}\sigma}}^{2}}-\frac{\ell (\ell +1)}{\sigma^{2}}+k_{2}^{2}}\right]g_{\ell} (\sigma)\\ &&\quad = 2\mu_{\mathrm{M}^{\prime}} U_{st}^{\left( 2\right)} (\sigma) g_{\ell} (\sigma)\\ &&\qquad + {\int}_{0}^{\infty} {K_{21}} ({\sigma,x}) f_{\ell} (x)\mathrm{d}x\\ &&\qquad +{\int}_{0}^{\infty} \!\!{K_{23}} ({\sigma,\sigma^{\prime}}) h_{\ell} (\sigma^{\prime})\mathrm{d}\sigma^{\prime} \end{array} $$

(44)

and

$$\begin{array}{@{}rcl@{}} &&{\kern-7.5pt}\left[{\frac{{\mathrm{d}}^{2}}{{{\mathrm{d}\sigma^{\prime}}}^{2}}-\frac{\ell (\ell +1)}{\sigma^{\prime2}}+k_{3}^{2}}\right]h_{\ell}(\sigma^{\prime})\\ &&\quad = 2\mu_{\mathrm{M}^{\prime\prime}} U_{st}^{\left( 3\right)} (\sigma^{\prime}) h_{\ell} (\sigma^{\prime})\\ &&\qquad + {\int}_{0}^{\infty} {K_{31}} ({\sigma^{\prime},x}) f_{\ell} (x) \mathrm{d}x \\ &&\qquad + {\int}_{0}^{\infty} {K_{32}} ({\sigma^{\prime},\sigma}) g_{\ell} ({\sigma})\mathrm{d}\sigma, \end{array} $$

(45)

where the kernels K ₁₂, K ₁₃, K ₂₁ and K ₃₁ are expanded by

$$\begin{array}{@{}rcl@{}} K_{12} ({x,\sigma})&=&2\mu_{\mathrm{M}} \text{(}8x\sigma \text{)} \iint { {\Phi}_{\text{K(}4s\text{)}}} (r) {\Phi}_{\text{H(}1s\text{)}}(\rho)\\ &&\times\left[{-\frac{1}{2\mu_{\mathrm{M}^{\prime}}}({\nabla_{\sigma}^{2} + k_{2}^{2}})+ V_{\text{int}}^{\left( 2\right)}}\right] \\ &&\times\, Y_{\ell}^{o} \left( {\hat{{x}}}\right) Y_{\ell}^{o} \left( {\hat{{\sigma}}} \right)\mathrm{d}\hat{{x}}\mathrm{d}\hat{{\sigma}}, \end{array} $$

(46)

$$\begin{array}{@{}rcl@{}} K_{13} (x,{\sigma}^{\prime})\thinspace &=&2\mu_{\mathrm{M}} \text{(}8x{\sigma}^{\prime}\text{)}\iint { {\Phi}_{\text{K(}4s\text{)}}} (r) {\Phi}_{\text{H(}2s\text{)}} ({\rho}^{\prime})\\ &&\times\left[{-\frac{1}{2\mu_{\mathrm{M}^{\prime\prime}}}({\nabla_{\sigma^{\prime}}^{2} + k_{3}^{2}}) + V_{\text{int}}^{\left( 3\right)}} \right] \\ &&\times\, Y_{\ell}^{o} (\hat{{x}}) Y_{\ell}^{o} \text{(}{\hat{{\sigma}}}^{\prime}\text{)d}\hat{{x}}\mathrm{d}{\hat{{\sigma}}}^{\prime}, \end{array} $$

(47)

$$\begin{array}{@{}rcl@{}} K_{21} (\sigma,x) &=&2\mu_{\mathrm{M}^{\prime}} \text{(}8\sigma x\text{)}\iint{{{\Phi}_{\text{H(}1s\text{)}}(\rho ){\Phi} }_{\text{K(}4s\text{)}}} (r)\\ &&\times\left[{-\frac{1}{2\mu_{\mathrm{M}}}({\nabla {_{x}^{2}} + k{_{1}^{2}}} )+ V_{\text{int}}^{\left( 1\right)}}\right] \\ &&\times\, Y_{\ell}^{o} ({\hat{{x}}}) Y_{\ell}^{o} ({\hat{{\sigma}}})\mathrm{d}\hat{{x}}\mathrm{d}\hat{{\sigma}} \end{array} $$

(48)

and

$$\begin{array}{@{}rcl@{}} K_{31} ({\sigma}^{\prime},x)\thinspace &=&2\mu_{\mathrm{M}^{\prime\prime}} \text{(}8{\sigma}^{\prime}x\text{)}\iint{{{\Phi}_{\text{H(}2s\text{)}} ({\rho}^{\prime}){\Phi} }_{\text{K(}4s\text{)}}} (r)\\ &&\times\left[{-\frac{1}{2\mu_{\mathrm{M}}}({{\nabla_{x}^{2}} + k{_{1}^{2}}})+ V_{\text{int}}^{\left( 1\right)}}\right] \\ &&\times\, Y_{\ell}^{o} (\hat{{x}}) Y_{\ell}^{o} ({\hat{{\sigma}}}^{\prime})\mathrm{d}\hat{{x}}\mathrm{d}{\hat{{\sigma}}}^{\prime} \end{array} $$

(49)

and the kernels K ₂₃(σ,σ ^′) and K ₃₂(σ ^′,σ) are equal to zero due to the orthogonality of the wave functions of the ground and excited states of hydrogen. (The number eight appearing in the preceding equations refers to the Jacobians of the transformations $\int {\mathrm {d}r\to 8\int {\mathrm {d}\sigma }}$ and $\int {\mathrm {d}\rho \to 8\int {\mathrm {d}x}}$.

Let us now rewrite eqs (43)–(45) as

$$\begin{array}{@{}rcl@{}} {\kern-.2pc}\left[{\frac{ {\mathrm{d}}^{2}}{ {{\mathrm{d}x}}^{2}}-\frac{\ell (\ell +1)}{ x^{2}}{}+{} k{_{1}^{2}}}\right]{} f_{\ell} (x)&=&{}2\mu_{\mathrm{M}} U_{st}^{\left( 1\right)} (x){} f_{\ell} (x)\\&&+\,Q_{1} (x)+Q_{3} (x),\\ \end{array} $$

(50)

$$\begin{array}{@{}rcl@{}} {\kern-.2pc}\left[{}{\frac{ {\mathrm{d}}^{2}}{ {{\mathrm{d}\sigma }}^{2}}{}-{}\frac{\ell (\ell{} +{}1)}{ \sigma^{2}}+ k{_{2}^{2}}}\right]{} g_{\ell} (\sigma){}&=&{}2\mu_{\mathrm{M}^{\prime}} U_{st}^{\left( 2\right)} (\sigma) g_{\ell} (\sigma)\\&&+\,Q_{2} (\sigma ) \end{array} $$

(51)

and

$$\begin{array}{@{}rcl@{}} \left[{\frac{ {\mathrm{d}}^{\text{2}}}{ {{\mathrm{d}\sigma^{\prime}}}^{2}}-\frac{\ell (\ell +1)}{ {\sigma^{\prime}}^{2}}+ k{_{3}^{2}}}\right] h_{\ell} (\sigma^{\prime})\!&=&\!2\mu_{\mathrm{M}^{\prime\prime}} U_{st}^{(3)} ({\sigma^{\prime}})\\ &&\!\times\, h_{\ell} {\kern-.5pt}(\sigma^{\prime}{\kern-.5pt}) {}+{} Q_{5} {\kern-.5pt}({\kern-.5pt}\sigma^{\prime}{\kern-.5pt}){\kern-.5pt},\\ \end{array} $$

(52)

where

$$\begin{array}{@{}rcl@{}} Q_{1} (x)={\int}_{0}^{\infty} {K_{12}} ({x,\sigma}) g_{\ell} (\sigma)\mathrm{d}\sigma, \end{array} $$

(53)

$$\begin{array}{@{}rcl@{}} Q_{3} (x)={\int}_{0}^{\infty} {K_{13}} ({x,\sigma^{\prime}})h_{\ell} (\sigma^{\prime})\mathrm{d}\sigma^{\prime}, \end{array} $$

(54)

$$\begin{array}{@{}rcl@{}} Q_{2} (\sigma )={\int}_{0}^{\infty} {K_{21}} ({\sigma,x}) f_{\ell} (x)\mathrm{d}x, \end{array} $$

(55)

and

$$\begin{array}{@{}rcl@{}} Q_{5} (\sigma^{\prime})={\int}_{0}^{\infty} {K_{31}} ({\sigma^{\prime},x})f_{\ell} (x)\mathrm{d}x. \end{array} $$

(56)

The three coupled integrodifferential (scattering) equations (50)–(52) are identical with the inhomogeneous equation

$$ [\varepsilon -H_{0} (u)]\left|\xi (u) \right\rangle =\left|\zeta(u) \right\rangle\!, $$

(57)

where u = x implies:

$$\begin{array}{@{}rcl@{}} \varepsilon ={k_{1}^{2}}, \quad H_{0} (x)=-\frac{\mathrm{d}}{\mathrm{d}x^{2}}+\frac{\ell (\ell +1)}{x^{2}}, \end{array} $$

$$\begin{array}{@{}rcl@{}} \left|\xi (x) \right\rangle &=&f_{\ell} (x),\\ \left|\zeta(x) \right\rangle &=&2\mu_{\mathrm{M}} U_{st}^{\left( 1 \right)} (x) f_{\ell} (x)+Q_{1} (x)+Q_{3} (x), \end{array} $$

u = σ implies:

$$\begin{array}{@{}rcl@{}} \varepsilon ={k_{2}^{2}}, \quad H_{0} (\sigma )=-\frac{\mathrm{d}}{\mathrm{d}\sigma^{2}}+\frac{\ell (\ell +1)}{\sigma^{2}}, \end{array} $$

$$\begin{array}{@{}rcl@{}} \left| \xi(\sigma)\right\rangle &=&g_{\ell} (\sigma ),\\ \left|\zeta(\sigma) \right\rangle &=&2\mu_{\mathrm{M}^{\prime}} U_{st}^{\left( 2 \right)} (\sigma) g_{\ell} (\sigma)+Q_{2} (\sigma ) \end{array} $$

and u = σ ^′ implies:

$$\begin{array}{@{}rcl@{}} \varepsilon ={k_{3}^{2}}, \quad H_{0} (\sigma^{\prime})=-\frac{\mathrm{d}}{\mathrm{d}\sigma^{\prime2}}+\frac{\ell (\ell +1)}{\sigma^{\prime2}}, \end{array} $$

$$\begin{array}{@{}rcl@{}} |\xi(\sigma^{\prime})\rangle &=&h_{\ell} (\sigma^{\prime}),\\ |\zeta(\sigma^{\prime})\rangle &=& 2 \mu_{\mathrm{M}^{\prime\prime}} U_{st}^{(3)} (\sigma^{\prime}) h_{\ell} (\sigma^{\prime})+Q_{5} (\sigma^{\prime}). \end{array} $$

The formal solutions of eq. (57) is given by the Lippmann–Schwinger equation

$$ \left|\xi \right\rangle =\left|\xi_{0} \right\rangle +G_{0}{\kern1.5pt}|\zeta\rangle, $$

(58)

where G ₀ is the Green operator (ε−H ₀)⁻¹ and |ξ ₀〉 is the solution of the homogeneous equation

$$ (\varepsilon-H_{0})~|\xi_{0}\rangle=|0\rangle. $$

(59)

The partial-wave expansion of G ₀ in the three channels enables us to write the formal solution of (50)–(52) as

$$\begin{array}{@{}rcl@{}} f_{{}\ell}^{(i)}{}({}x{}){}&=&{}\left\{{}\delta_{i{}1} {}+{}\frac{1}{k_{{}1}}{}{\int}_{{}0}^{\infty} {} \tilde{g}_{\ell} (k_{{}1} {x}^{\prime})[2\mu_{\mathrm{M}}{} U_{st}^{({}1)}{} ({x}^{\prime}) f_{\ell}^{(i)} ({x}^{\prime}){\vphantom{{\int}_{0}^{\infty}}}\right.\\ && \left.{\vphantom{{\int}_{0}^{\infty}}}{} + Q_{{}1}^{(i)}({x}^{\prime})+Q_{3}^{(i)} (x^{\prime}) ]\mathrm{d}{x}^{\prime} \right\}_{1} \tilde{f}_{\ell}(k_{1}x)\\ \end{array} $$

$$\begin{array}{@{}rcl@{}} &&{\kern2pt}+{}\left\{{}-{}\frac{1}{k_{1}}{}{\int}_{{}0}^{\infty}{} \tilde{f}_{\ell}(k_{1} {x}^{\prime})[ 2\mu_{\mathrm{M}} U_{st}^{({}1)} ({x}^{\prime}) f_{\ell}^{(i)}({x}^{\prime}){\vphantom{{\int}_{0}^{\infty}}}\right.\\ && \left.{\vphantom{{\int}_{0}^{\infty}}}{} {\kern5pt}+\,{} Q_{{}1}^{(i)}({x}^{\prime})+Q_{3}^{(i)} (x^{\prime})]\mathrm{d}{x}^{\prime}{} \right\}_{{}2} \tilde{g}_{\ell} (k_{1} x),\\ && {\kern2pt}i{}={}1,{}2,{}3 \end{array} $$

(60)

$$\begin{array}{@{}rcl@{}} &&{}g_{\ell}^{(i)} (\sigma)\\ &&{}\quad=\left\{\delta_{i2} +\frac{1}{k_{2}}{\int}_{0}^{\infty} \tilde{g}_{\ell} (k_{2} \sigma^{\prime})[ {2\mu_{\mathrm{M}^{\prime}} U}_{st}^{(2)} (\sigma^{\prime}) g_{\ell}^{(i)} {}(\sigma^{\prime}) {\vphantom{{\int}_{0}^{\infty}}}\right.\\ &&{}\qquad \left.{\vphantom{{\int}_{0}^{\infty}}} {}+ Q_{2}^{(i)} (\sigma^{\prime})]\mathrm{d}\sigma^{\prime} {\kern-.5pt}\right\}_{3} \tilde{f}_{\ell} (k_{2}\sigma)\\ &&{}\qquad +\left\{-\frac{1}{k_{2}}{\int}_{0}^{\infty} \tilde{f}_{\ell} (k_{2} \sigma^{\prime})[ {2\mu_{\mathrm{M}^{\prime}} U}_{st}^{(2)} (\sigma^{\prime}) g_{\ell}^{(i)}(\sigma^{\prime}){\vphantom{{\int}_{0}^{\infty}}}\right.\\ &&{}\qquad \left. {\vphantom{{\int}_{0}^{\infty}}}{} +\,{} Q_{2}^{(i)} (\sigma^{\prime})]\mathrm{d}\sigma^{\prime} {\kern-.5pt}\right\}_{4} \tilde{g}_{\ell}(k_{2}\sigma),\quad i{\kern-.5pt}={\kern-.5pt}1{\kern-.5pt},{\kern-.5pt}2{\kern-.5pt},{\kern-.5pt}3 \end{array} $$

(61)

and

$$\begin{array}{@{}rcl@{}} &&{} h_{\ell}^{(i)} (\sigma^{\prime})\\ &&{}\quad =\left\{\delta_{i3} + \frac{1}{k_{3}} {\int}_{0}^{\infty} \tilde{g}_{\ell} (k_{2} \sigma^{\prime\prime})[{\vphantom{{\int}_{0}^{\infty}}} {2\mu_{\mathrm{M}^{\prime\prime}} U}_{st}^{(3)}(\sigma^{\prime\prime}) h_{\ell}^{(i)} (\sigma^{\prime\prime})\right.\\ &&{}\qquad + \left. {\vphantom{{\int}_{0}^{\infty}}}\!\! Q_{5}^{(i)} (\sigma^{\prime\prime})]\mathrm{d}\sigma^{\prime\prime}\right\}_{5} \tilde{f}_{\ell} (k_{3} \sigma^{\prime})\\ &&{}\qquad +\left\{-\frac{1}{k_{3}}{\int}_{0}^{\infty} \tilde{f}_{\ell} (k_{2} \sigma^{\prime\prime})[{\vphantom{{\int}_{0}^{\infty}}}{2\mu_{\mathrm{M}^{\prime\prime}} U}_{st}^{(3)} (\sigma^{\prime\prime}) h_{\ell}^{(i)} (\sigma^{\prime\prime})\right.\\ &&{}\qquad +\! \left. {\vphantom{{\int}_{0}^{\infty}}} Q_{5}^{(i)} (\sigma^{\prime\prime})]\mathrm{d}\sigma^{\prime\prime} \right\}_{6} \tilde{g}_{\ell} (k_{3} \sigma^{\prime}),\quad i=1,2,3,\\ \end{array} $$

(62)

where the delta functions δ _{i j} , i,j=1,2,3, specify two independent forms of solutions for each of f _ℓ (x), g _ℓ (σ) and h _ℓ (σ ^′) in channels i=1,2,3 according to the channel considered. Thus, if i=1, the first element in the {} bracket of f _ℓ (x), for example, will be 1 defining the first form of solution. For i=2,3 this element will be zero defining the second form. The functions ${\tilde {{f}}}_{\ell }(\eta )$ and ${\tilde {g}}_{\ell } (\eta )$, η = k ₁ x,k ₂ σ or k ₃ σ ^′, are related to the Bessel functions of the first and second channels, that is, j _ℓ (η) and y _ℓ (η), respectively, by the relations ${\tilde {{f}}}_{\ell }(\eta )=\eta j_{\ell }(\eta )$ and ${\tilde {g}}_{\ell }(\mu )=-\eta y_{\ell } (\eta )$.

It is obvious from eqs (60)–(62) that the solutions can be only found iteratively and the νth iterations are calculated by

$$\begin{array}{@{}rcl@{}} &&{} f_{\ell}^{(i,\nu)}(x)=\left\{\delta_{i1} +\frac{1}{k_{1}}{{\int}_{0}^{X}}{\tilde{g}}_{\ell} ({k_{1} {x}^{\prime}})[{\vphantom{{\int}_{0}^{\infty}}} 2\mu_{\mathrm{M}} U_{st}^{(1)} ({x}^{\prime})\right.\\ &&{} \left.\left. {\vphantom{{\int}_{0}^{\infty}}} \quad\times f_{\ell}^{(i,\nu)} ({x}^{\prime}) {}+{} Q_{1}^{(i,\nu-1)} ({x}^{\prime})\right.\right.\\ &&{\kern1pt}\left.+Q_{3}^{(i,\nu -1)} (x^{\prime})]\mathrm{d}{x}^{\prime} \right\}_{{}1} {} \tilde{f}_{\ell} ({k_{1} x})\\ \end{array} $$

$$\begin{array}{@{}rcl@{}} &&{}\quad +\left\{-\frac{1}{k_{1}}{{\int}_{0}^{X}}{\tilde{{f}}}_{\ell} ({k_{1} {x}^{\prime}}) {} [{\vphantom{{\int}_{0}^{\infty}}}{2\mu_{\mathrm{M}} U}_{st}^{(1)} ({{x}^{\prime}}) f_{\ell}^{(i,\nu)} ({x}^{\prime})\right.\\ &&{}\quad \left. {\vphantom{{\int}_{0}^{\infty}}} + Q_{1}^{(i,\nu -1)} ({x}^{\prime})+Q_{3}^{(i,\nu -1)} (x^{\prime})]\mathrm{d}{x}^{\prime} \right\}_{2} \tilde{g}_{\ell} ({k_{1} x}),\\ &&{}\quad i=1,2,3;\quad\nu \ge 1, \end{array} $$

(63)

$$\begin{array}{@{}rcl@{}} {}g_{\ell}^{(i,\nu)} (\sigma)&=&\left\{{\vphantom{\frac{1}{k_{3}} {\int}_{0}^{{\Sigma}^{\prime}}}}{}\delta_{i2} +\frac{1}{k_{2}}{\int}_{0}^{\Sigma} {\tilde{g}}_{\ell} ({k_{2} {\sigma}^{\prime}})\right.\\ &&\times[{\vphantom{{\int}_{0}^{\Sigma}}}{2\mu_{\mathrm{M}^{\prime}} U}_{st}^{(2)} ({{\sigma}^{\prime}}) g_{\ell}^{(i,\nu)} ({{\sigma}^{\prime}})\\ && + \left. \! Q_{2}^{(i,\nu)} ({{\sigma}^{\prime}}){\vphantom{{\vphantom{\frac{1}{k_{3}} {\int}_{0}^{{\Sigma}^{\prime}}}}}}]\mathrm{d}{\sigma}^{\prime} \right\}_{3} {\tilde{{f}}}_{\ell} ({k_{2}\sigma})\\ &&+\left\{{\vphantom{\frac{1}{k_{3}} {\int}_{0}^{{\Sigma}^{\prime}}}}{}-\frac{1}{ k_{2}}{\int}_{0}^{\Sigma}{\tilde{{f}}}_{\ell} ({k_{2}{\sigma}^{\prime}})\right.\\ && \times [{2\mu_{\mathrm{M}^{\prime}} U}_{st}^{(2)} ({{\sigma}^{\prime}}) g_{\ell}^{(i,\nu)} ({{\sigma}^{\prime}})\\ && + \left. Q_{2}^{(i,\nu)}({{\sigma}^{\prime}})]\mathrm{d}{\sigma}^{\prime} {\vphantom{{\vphantom{\frac{1}{k_{3}} {\int}_{0}^{{\Sigma}^{\prime}}}}}}\right\}_{4} {\tilde{g}}_{\ell} ({k_{2}\sigma}),\\ &&i=1,2,3;\quad\nu \ge 0 \end{array} $$

(64)

and

$$\begin{array}{@{}rcl@{}} h_{\ell}^{(i,\nu)} (\sigma^{\prime})&=&\left\{\delta_{i3} + \frac{1}{k_{3}} {\int}_{0}^{{\Sigma}^{\prime}} {\tilde{g}}_{\ell} ({k_{3} \sigma^{\prime\prime}})\right.\\ && \times [{2\mu_{\mathrm{M}^{\prime\prime}} U}_{st}^{(3)} (\sigma^{\prime\prime}) h_{\ell}^{(i,\nu)} (\sigma^{\prime\prime})\\ && \left. +\, Q_{5}^{(i,\nu)} (\sigma^{\prime\prime})]\mathrm{d}\sigma^{\prime\prime} {\vphantom{-\frac{1}{ k_{2}}{\int}_{0}^{\Sigma}}}\right\}_{5} {\tilde{f}}_{\ell} ({k_{3} \sigma^{\prime}})\\ &&+\left\{ -\frac{1}{k_{3}}{\int}_{0}^{{\Sigma}^{\prime}} {\tilde{{f}}}_{\ell} ({k_{3} \sigma^{\prime\prime}})\right.\\ &&\times [{2\mu_{\mathrm{M}^{\prime\prime}}U}_{st}^{(3)} ({{\sigma}^{\prime}}) h_{\ell}^{(i,\nu)} ({{\sigma}^{\prime}}) \\ && \left. + \,Q_{5}^{(i,\nu)} (\sigma^{\prime\prime}) ]\mathrm{d}\sigma^{\prime\prime} {\vphantom{{\vphantom{\frac{1}{k_{3}} {\int}_{0}^{{\Sigma}^{\prime}}}}}}\right\}_{6} {\tilde{g}}_{\ell} ({k_{3} \sigma^{\prime}}),\\ && i=1,2,3;\quad \nu \ge 0. \end{array} $$

(65)

The zeroth iteration of $f_{\ell }^{(i,0)}(x)$ is obtained by

$$\begin{array}{@{}rcl@{}} f_{\ell}^{(i,0)} (x)\!&=&\!\left\{\delta_{i1} +\frac{1}{k_{1}}{{\int}_{0}^{X}}{\tilde{g}}_{\ell} ({k_{1} {x}^{\prime}})\right.\\ &&\left.\! \times\, [{2\mu_{\mathrm{M}} U_{st}^{(1)} ({x}^{\prime}) f_{\ell}^{(i,0)} ({x}^{\prime})}]\mathrm{d}{x}^{\prime}{\vphantom{{\int}_{0}^{X}}}\right\}_{1}\tilde{f}_{\ell} ({k_{1} x})\\ &&\!+{}\left\{{} - \frac{1}{k_{1}}{{\int}_{0}^{X}} \tilde{f}_{\ell} ({k_{1}{x}^{\prime}})\right.\\ &&\!\left. \times {}[{{2\mu_{\mathrm{M}}U}_{st}^{(1)} ({x}^{\prime}) f_{\ell}^{(i,0)} ({x}^{\prime})}]\mathrm{d}{x}^{\prime}{} {\vphantom{{\int}_{0}^{X}}}\right\}_{2} \tilde{g}_{\ell} ({k_{1}x}),\thinspace \thinspace\\ &&i=1,2,3, \end{array} $$

(66)

where X, Σ and Σ^′ specify the integration range away from the nucleus over which the integrals at eqs (63)–(66) are calculated using Simpson’s expansions. The functions $Q_{1}^{({i,\nu -1})} ({x}^{\prime })$, $Q_{3}^{(i,\nu -1)} ({x}^{\prime })$, $Q_{2}^{(i,\nu )} (\sigma )$ and $Q_{5}^{(i,\nu )} (\sigma ^{\prime })$, in eqs (63)–(65), are now defined by

$$ Q_{1}^{(i,\nu -1)} (x^{\prime}) = {\int}_{0}^{\Sigma} {K_{12}} ({x^{\prime},\sigma}) g_{\ell}^{(i,\nu -1)} (\sigma) \mathrm{d}\sigma, $$

(67)

$$ Q_{3}^{(i,\nu -1)} (x^{\prime})={\int}_{0}^{{\Sigma}^{\prime}} {K_{13}} ({x^{\prime},\sigma^{\prime}}) h_{\ell}^{(i,\nu -1)} (\sigma^{\prime})\mathrm{d}\sigma^{\prime}, $$

(68)

$$ Q_{2}^{(i,\nu)} (\sigma^{\prime}) = {{\int}_{0}^{X}} { K_{21}} ({\sigma^{\prime},x}) f_{\ell}^{(i,\nu)} (x)\mathrm{d}x $$

(69)

and

$$ Q_{5}^{(i,\nu)} (\sigma^{\prime\prime})={{\int}_{0}^{X}} { K_{31}} ({\sigma^{\prime\prime},x}) f_{\ell}^{(i,\nu)} (x)\mathrm{d}x. $$

(70)

To find the starting value of $f^{(i,0)}_{\ell }(x)$, we consider the Taylor expansion of $U^{(1)}_{st}(x),\tilde {f}_{\ell }(k_{1}x)$ and $\tilde {g}_{\ell }(k_{1}x)$ around the origin (see Appendix B).

Equations (63)–(65) can be abbreviated to

$$\begin{array}{@{}rcl@{}} &&{} f_{\ell}^{({i,\upsilon})} (x)= a_{1}^{({i,\upsilon})} {\tilde{{f}}}_{\ell} ({k_{1} x})+ b_{1}^{({i,\upsilon})} {\tilde{g}}_{\ell} ({k_{1} x}),\\ &&{\kern3.4pc} i=1,2,3, \end{array} $$

(71)

$$\begin{array}{@{}rcl@{}} &&{} g_{\ell}^{({i,\upsilon})} (\sigma)= a_{2}^{({i,\upsilon})} {\tilde{{f}}}_{\ell} ({k_{2} \sigma})+ b_{2}^{({i,\upsilon})} {\tilde{g}}_{\ell} ({k_{2} \sigma}),\\ &&{\kern3.4pc} i=1,2,3 \end{array} $$

(72)

and

$$\begin{array}{@{}rcl@{}} &&{} h_{\ell}^{({i,\upsilon})} (\sigma^{\prime}) = a_{3}^{({i,\upsilon})} {\tilde{{f}}}_{\ell} ({k_{3} \sigma^{\prime}}) + b_{3}^{({i,\upsilon})} {\tilde{g}}_{\ell} ({k_{3} \sigma^{\prime}}), \\ &&{\kern3.4pc} i=1,2,3, \end{array} $$

(73)

where

$$\begin{array}{@{}rcl@{}} \begin{array}{l} a_{1}^{\left( {i,\upsilon}\right)} = {\left\{~\right\}}_{1}, \text{\thinspace \thinspace} b_{1}^{\left( {i,\upsilon}\right)} = {\left\{~\right\}}_{2}, \text{\thinspace} a_{2}^{\left( {i,\upsilon}\right)} = {\left\{~\right\}}_{3},\\ b_{2}^{\left( {i,\upsilon}\right)} = {\left\{~\right\}}_{4}, {\text{\thinspace}a}_{3}^{\left( {i,\upsilon}\right)} = {\left\{~\right\}}_{5}, \text{\thinspace} b_{3}^{\left( {i,\upsilon}\right)} = {\left\{~\right\}}_{6}, \end{array} \end{array} $$

and the preceding six brackets are the coefficients of eqs (63)–(65).

The coefficients at eqs (71)–(73) are elements of the following matrices:

$$\begin{array}{@{}rcl@{}} &&{} a^{\upsilon} =\\ &&{} \left( {\begin{array}{l} {}\sqrt {2\mu_{\mathrm{M}}/k_{1}} a_{1}^{\left( {1,\upsilon}\right)} \sqrt {2\mu_{\mathrm{M}}/k_{1}} a_{1}^{\left( {2,\upsilon}\right)} \sqrt {2\mu_{\mathrm{M}}/k_{1}} a_{1}^{\left( {3,\upsilon}\right)}\\ {}\sqrt {2\mu_{\mathrm{M}^{\prime}}/k_{2}} a_{2}^{\left( {1,\upsilon}\right)} \sqrt{2\mu_{\mathrm{M}^{\prime}}/k_{2}} a_{2}^{\left( {2,\upsilon}\right)} \sqrt{2\mu_{\mathrm{M}^{\prime}}/k_{2}} a_{2}^{\left( {3,\upsilon}\right)}\\ {}\sqrt {2\mu_{\mathrm{M}^{\prime\prime}}/k_{3}} a_{3}^{\left( {1,\upsilon}\right)} \sqrt {2\mu_{\mathrm{M}^{\prime\prime}}/k_{3}} a_{3}^{\left( {2,\upsilon}\right)} \sqrt {2\mu_{\mathrm{M}^{\prime\prime}}/k_{3}} a_{3}^{\left( {3,\upsilon}\right)} \end{array}}{}\right)\\ \end{array} $$

(74)

and

$$\begin{array}{@{}rcl@{}} && {}b^{\upsilon} =\\ &&{} \left( {\begin{array}{l} {}\sqrt {2\mu_{\mathrm{M}}/k_{1}}b_{1}^{\left( {1,\upsilon}\right)} \sqrt {2\mu_{\mathrm{M}}/k_{1}}b_{1}^{\left( {2,\upsilon}\right)} \sqrt {2\mu_{\mathrm{M}}/k_{1}} b_{1}^{\left( {3,\upsilon}\right)}\\ {}\sqrt{2\mu_{\mathrm{M}^{\prime}}/k_{2}} b_{2}^{\left( {1,\upsilon}\right)} \sqrt{2\mu_{\mathrm{M}^{\prime}}/k_{2}} b_{2}^{\left( {2,\upsilon}\right)} \sqrt{2\mu_{\mathrm{M}^{\prime}}/k_{2}} b_{2}^{\left( {3,\upsilon}\right)}\\ {}\sqrt {2\mu_{\mathrm{M}^{\prime\prime}}/k_{3}} b_{3}^{\left( {1,\upsilon}\right)} \sqrt{2\mu_{\mathrm{M}^{\prime\prime}}/k_{3}} b_{3}^{\left( {2,\upsilon}\right)} \sqrt{2\mu_{\mathrm{M}^{\prime\prime}}/k_{3}} b_{3}^{\left( {3,\upsilon}\right)} \end{array}}{}\right)\\ \end{array} $$

(75)

which are connected with the reactance matrix, R ^υ, through the relation

$$ \{R^{v} \}_{ij} = \{b^{v} (a^{v})^{-1}\}_{ij}. $$

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Appendix B: The numerical iterative method:

This appendix contains a brief discussion of the iteration procedure used for calculating the elements of the reactance matrix. This has been achieved using eqs (63)–(65) where X, Σ and Σ^′ represent the ranges of integrations over x, σ and σ ^′, respectively. Physically, X (X = nh, where n is the number of mesh points and 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, Σ(= n h) is the distance at which the ground-state hydrogen atom and the proton of the target are totally separated and Σ^′ (= n h) is the distance at which the excited hydrogen atom and the proton of the target are totally separated.

To calculate the integrals in eqs (63)–(70) we use Simpson’s rule. Thus, we expand $Q_{1}^{(i,\nu -1)} (x)$, $Q_{3}^{(i,\nu -1)} (x)$, $Q_{2}^{(i,\nu )} (\sigma )$ and $Q_{5}^{(i,\nu )} (\sigma ^{\prime })$ at point q of the configuration space as follows:

$$\begin{array}{@{}rcl@{}} Q_{1}^{(i,\nu -1)} ({x_{q}}) &=& \sum\limits_{p=1}^{n} \{\omega_{p}^{(1)} K_{12}^{(1)} (\sigma_{p}, x_{q})\\ && +~\omega_{p}^{(2)} K_{12}^{(2)} (\sigma_{p}, x_{q})\\ && +~\omega_{p}^{(3)} K_{12}^{(3)} (\sigma_{p}, x_{q})\}g_{\ell}^{(i,\nu-1)} (\sigma_{p}),\\ &&\nu \ge 1, \end{array} $$

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$$\begin{array}{@{}rcl@{}} Q_{3}^{(i,\nu -1)} ({x_{q}}) &=& \sum\limits_{p=1}^{n} \{\omega_{p}^{(1)} K_{13}^{(1)} (\sigma_{p}^{\prime},x_{q})\\ &&+~\omega_{p}^{(2)} K_{13}^{(2)} (\sigma_{p}^{\prime},x_{q})\\ &&+~\omega_{p}^{(3)} K_{13}^{(3)} (\sigma_{p}^{\prime},x_{q})\}h_{\ell}^{(i,\nu-1)} (\sigma_{p}^{\prime}),\\ &&\nu \ge 1, \end{array} $$

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$$\begin{array}{@{}rcl@{}} Q_{2}^{(i,\nu)} ({\sigma_{q}}) &=& \sum\limits_{p=1}^{n} \{\omega_{p}^{(1)} K_{21}^{(1)} (\sigma_{q}, x_{p})\\ && +~\omega_{p}^{(2)} K_{21}^{(2)} (\sigma_{q}, x_{p})\\ && +~\omega_{p}^{(3)} K_{21}^{(3)} (\sigma_{q}, x_{p})\}f_{\ell}^{(i,\nu)} (x_{p}),\\ &&\nu \ge 0, \end{array} $$

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and

$$\begin{array}{@{}rcl@{}}Q_{5}^{(i,\nu)} ({\sigma_{q}^{\prime}}) &=& \sum\limits_{p=1}^{n} \{\omega_{p}^{(1)} K_{31}^{(1)} (\sigma_{q}^{\prime},x_{p})\\ && +~\omega_{p}^{(2)} K_{31}^{(2)} (\sigma_{q}^{\prime},x_{p})\\ &&+~\omega_{p}^{(3)} K_{31}^{(3)} (\sigma_{q}^{\prime},x_{p})\}f_{\ell}^{(i,\nu)} (x_{p}),\\ &&\nu \ge 0, \end{array} $$

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where $\omega _{p}^{(1)}$’s are the usual Simpson weights (h/3, 4h/3, 2h/3, ..., 2h/3, 4h/3,h/3) and $\omega _{p}^{(2)}$’s, $\omega _{p}^{(3)}$’s are modified weights used for avoiding the singularities (see refs [14,15]). The variables x _p , σ _p and $\sigma ^{\prime }_{p}$ are chosen such that $x_{p} =\sigma _{p} =\sigma _{p}^{\prime }=ph, p=1,2,...,n$.

An essential point in the determination of the integrals in eqs (63)–(70) is the calculation of the starting values of the functions $f_{\ell }^{(i,\nu )} (x)$, $g_{\ell }^{(i,\nu )} (\sigma )$ and $h_{\ell }^{(i,\nu )}(\sigma ^{\prime })$, that is, their values at $x_{p} =\sigma _{p} =\sigma _{p}^{\prime }=h$, respectively, which is determined as follows (note that $f_{\ell }^{(i,\nu )} (0)= g_{\ell }^{(i,\nu )} (0)= h_{\ell }^{(i,\nu )} (0)=0)$.

Considering the Taylor expansion of $U_{st}^{(1)} (r)$, $\tilde {f}_{\ell } (kr)$ and $\tilde {g}_{\ell } (kr)$ around the origin, we obtain

$$ U_{st}^{(1)} (x)\approx 2\left( {\frac{Z}{x}+D_{0} +D_{1} x+\cdots} \right), $$

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where Z = 1, D ₀ and D ₁ are constants, and

$$\begin{array}{@{}rcl@{}} \tilde{f}_{\ell} (k_{1} x)&\cong& \frac{(k_{1} x)^{\ell +1}}{(2\ell +1)!!}\left[1-\frac{(k_{1} x)^{2}/2}{1!(2\ell -1)}\right.\\ && \left.+\, \frac{(k_{1}x)^{4}/4}{2!(2\ell +3)(2\ell +5)}+\cdots\right] \end{array} $$

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and

$$\begin{array}{@{}rcl@{}} \tilde{g}_{\ell} (k_{1} x) &\cong& \frac{(2\ell +1)!!}{(2\ell +1)(k_{1} x)^{\ell}}\left[1+\frac{(k_{1} x)^{2}/2}{1!(2\ell -1)}\right.\\ && \left. +\,\frac{(k_{1} x)^{4}/4}{2!(2\ell -1)(3\ell -2)}+\cdots\right]. \end{array} $$

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Assume that $f_{\ell }^{(i,0)} (x)$ behaves close to the origin as

$$\begin{array}{@{}rcl@{}} f_{\ell}^{(i,0)} (x)&\cong& C_{1} x^{\ell +1}+C_{2} x^{\ell +2}\\ &&+~C_{3} x^{\ell +3}+C_{4} x^{\ell +4}+\cdots. \end{array} $$

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Substituting eqs (81)–(84) into eq. (66) yields

$$\begin{array}{@{}rcl@{}} \left. \begin{array}{l} {\kern-.4pc}C_{1} \,=\,\frac{k^{\ell +1}}{(2\ell +1)!!},\text{\thinspace\thinspace\thinspace}C_{2} =\frac{Z}{\ell +1}C_{1}, \text{\thinspace\thinspace\thinspace}\\ {\kern-.4pc}C_{3} \,=\,\left[{\frac{D_{0}}{2\ell+3}+\frac{Z^{2}}{(\ell +1)(2\ell +3)}-\frac{k^{2}}{2(2\ell -1)}}\right]\!C_{1} \\ {\kern-.4pc}C_{4} \,=\,\{{\Omega}_{1} [C_{1} {\Omega}_{4}D_{1} +C_{1} {\Omega}_{5} Z+D_{0} {\Omega}_{4} C_{2}\\ {\kern2pc}+C_{3} {\Omega}_{4} Z]/3+{\Omega}_{2} {\Omega}_{4} C_{1}Z\\ {\kern2pc}-{\Omega}_{4}[C_{1}{\Omega}_{1} D_{1} +C_{1} {\Omega}_{2} Z+C_{2} {\Omega}_{1} D_{0}\\ {\kern2pc}+C_{3} {\Omega}_{1} Z]/(2\ell +4)\\ {\kern2pc}-{\Omega}_{1} {\Omega}_{5} C_{1} Z/(2\ell +2)\}/k, \end{array}\!\!\right]\\ \end{array} $$

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where

$$ \left. {\begin{array}{l} {\Omega}_{1} =\displaystyle\frac{k^{\ell +1}}{(2\ell +1)!!},\text{ \thinspace}{\Omega}_{2} =\frac{-k^{\ell +3}}{2(2\ell -1)(2\ell +1)!!}\text{,\thinspace \thinspace}\\ {\Omega}_{3} = \displaystyle\frac{k^{\ell +5}}{8(2\ell +3)(2\ell +5)(2\ell +1)!!} \\ {\Omega}_{4} = \displaystyle\frac{(2\ell +1)!!}{(2\ell +1)k^{\ell}},\text{\thinspace}{\Omega}_{5} =\frac{(2\ell +1)!!}{2(2\ell -1)(2\ell +1)k^{\ell -2}},\text{\thinspace\thinspace\thinspace}\\ {\Omega}_{6} = \displaystyle\frac{(2\ell +1)!!}{8(2\ell +1)(2\ell -1)(3\ell -2)k^{\ell -4}}.\end{array}}\right] $$

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Introducing constants C ₁, C ₂, C ₃ and C ₄ into eq. (84) and setting x = h (the interval of integration), we obtain a starting value $f_{\ell }^{(i,0)} (x)$ (note that the first four terms of eq. (84) are enough for obtaining a good starting value specially when h is reasonably small). The above procedure can also be applied for calculating the starting value of $g_{\ell }^{(i,\nu )} (\sigma )$ and $h_{\ell }^{(i,\nu )} (\sigma )$ at different values of ℓ, i and ν.

The iteration process starts by introducing $f_{\ell }^{(i,0)}$ (x), i = 1, 2, 3 values into eqs (69) and (70) to find $Q_{2}^{(i,0)} (\sigma ^{\prime })$ and $Q_{5}^{(i,0)}(\sigma ^{\prime \prime })$, which can be used in the right-hand side of eqs (64) and (65) to obtain $g_{\ell }^{(i,0)} (\sigma )$ and $h_{\ell }^{(i,0)} (\sigma ^{\prime })$. The values of the last quantities can be introduced into eqs (67) and (68) to calculate $Q_{1}^{(i,0)} ({x}^{\prime })$ and $Q_{3}^{(i,0)}({x}^{\prime })$, which may be employed in eq. (63) for determining $f^{(i,1)}_{\ell }(x)$. This iteration process can be repeated as many times as we need and the judge of its quantity is the stationary variation of the elements of the reactance matrix $R^{\nu }_{ij}$ where ν increase.