On the Breakup Reaction in Three-Particle Coulomb Systems with Application to the Description of Dissociative Recombination and Charge-Exchange Processes in the Antiproton Physics

Abstract

A new approach is proposed to the description of the breakup reaction in systems of three charged quantum particles in the presence of Coulomb pair attraction potentials. The analytic asymptotic form of the wavefunction at infinity in the configuration space is assumed to be known. For the first time, a simplified asymptotic form associated with the separation of the main contributions generated by an infinite set of pair asymptotic scattering channel is proposed. The approach proposed here appears as fundamental for describing the breakup reaction in multiparticle systems. The possibility of application of the approach developed here to the description of dissociative recombination and charge-exchange processes in experiments on accumulation of antiatoms is considered.

APPENDIX

1.1 Determination of Kernel φ_n(x, ${\mathbf{\hat {k}}}$) of Generating Integral

Let us consider eigenfunction ${{\tilde {\psi }}_{c}}$(x, ${\mathbf{\hat {k}}}$) of absolutely continuous spectrum of the Coulomb two-particle Schrödinger operator, which has been determined in standard manner to within normalization ((normalization factor is assumed to be equal to unity),

$${{\tilde {\psi }}_{c}}({\mathbf{x}},{\mathbf{k}}) = {{e}^{{i\langle {\mathbf{k}},{\mathbf{x}}\rangle }}}\Phi ( - i\gamma ,1,ikx(1 - \langle {\mathbf{\hat {k}}},{\mathbf{\hat {x}}}\rangle )),$$

(60)

and perform its partial analysis (decomposition in spherical functions). In the case when the function is invariant to rotation of the coordinate system (in this case, the angular dependence is contained in scalar product 〈${\mathbf{\hat {k}}}$, ${\mathbf{\hat {x}}}$〉), such analysis is equivalent to the expansion in the Legendre polynomials [35].

To perform such an expansion, we can use Kummer transform (9.212.1) [24]

$$\Phi (a,c,z) = {{e}^{z}}\Phi (c - a,c, - z),$$

which allows us to write expression (60) in form

$${{\tilde {\psi }}_{c}}({\mathbf{x}},{\mathbf{k}}) = {{e}^{{ikx}}}\Phi (1 + i\gamma ,1, - ikx(1 - t)),$$

(61)

where t = 〈${\mathbf{\hat {k}}}$, ${\mathbf{\hat {x}}}$〉. The angular dependence is now concentrated only in the confluent hypergeometric function. Let us expand it into a series in orthogonal Legendre polynomials P_l. In accordance, for example, with relation (8.904) [24], this expansion takes form

$$\Phi (1 + i\gamma ,1, - ikx(1 - t))\, = \,\sum\limits_{l = 0}^\infty {(2l + 1){{\Phi }_{l}}(k,x){{P}_{l}}(t).} $$

(62)

Here, Φ_l(k, x) are the particle components of function Φ(1 + iγ, 1, –ikx(1 – t)), which are calculated as follows in accordance with the orthogonality conditions for the Legendre polynomials:

$${{\Phi }_{l}}(k,x) = \frac{1}{2}\int\limits_{ - 1}^1 {dt\Phi (1 + i\gamma ,1, - ikx(1 - t)){{P}_{l}}(t).} $$

(63)

For evaluating the integral in this expression, we can use the hypergeometrical expansion

$$\begin{gathered} \Phi (a,1,b(1 - t)) = \sum\limits_{j = 0}^\infty {\frac{{{{{(a)}}_{j}}}}{{{{{(j!)}}^{2}}}}{{b}^{j}}{{{(1 - t)}}^{j}}} , \\ a = 1 + i\gamma ,\quad b = - ikx,\quad {{(a)}_{j}} = \frac{{\Gamma (a + j)}}{{\Gamma (a)}}, \\ \end{gathered} $$

(64)

in terms of which

$${{\Phi }_{l}}(k,x) = \frac{1}{2}\sum\limits_{j = 0}^\infty {\frac{{{{{(a)}}_{j}}}}{{{{{(j!)}}^{2}}}}{{b}^{j}}\int\limits_{ - 1}^1 {dt{{{(1 - t)}}^{j}}{{P}_{l}}(t).} } $$

(65)

Performing the substitution of variable s² = (1 – t))/2 in the integral, we can evaluate it explicitly and obtain a new hypergeometrical expansion that leads to the following limiting relation for the partial component:

$$\begin{gathered} {{\Phi }_{l}}(k,x) = \frac{{{{{( - 1)}}^{l}}}}{{\Gamma (l + 2)}}\mathop {\lim }\limits_{u \to - m} \frac{1}{{\Gamma (u)}} \\ \times {{\,}_{2}}{{F}_{2}}(a,1;u,l + 2;2b),\quad m = l - 1, \\ \end{gathered} $$

(66)

where ₂F₂ is the hypergeometrical function. Let us now use relation

$$\begin{gathered} \mathop {\lim }\limits_{u \to - m} {{\frac{1}{{\Gamma (u)}}}_{2}}{{F}_{2}}(A,B;u,D;z) \\ = \frac{{{{{(A)}}_{{m + 1}}}{{{(B)}}_{{m + 1}}}}}{{{{{(D)}}_{{m + 1}}}}}\frac{{{{z}^{{m + 1}}}}}{{(m + 1)!}} \\ \times {{\,}_{2}}{{F}_{2}}(A + m + 1,B + m + 1;m + 2,D + m + 1;z), \\ \end{gathered} $$

(67)

where

$$A = a,\quad B = 1,\quad m = l - 1,\quad D = l + 2,\quad z = 2b.$$

Substituting relation (67) into (66), we obtain the final expression for the partial component:

$$\begin{gathered} {{\Phi }_{l}}(k,x) \\ = \frac{{\Gamma (i\gamma + l + 1)}}{{\Gamma (i\gamma + 1)\Gamma (2l + 2)}}{{(2ikx)}^{l}}\Phi (i\gamma \, + \,l\, + \,1,2l\, + \,2, - 2ikx). \\ \end{gathered} $$

(68)

Substituting the resulting expression into expansion (61), (62) for the continuous spectrum eigenfunction of the Coulomb two-particle Schrödinger operator, we get

$$\begin{gathered} {{{\tilde {\psi }}}_{c}}({\mathbf{x}},{\mathbf{k}}) = {{e}^{{ikx}}}\sum\limits_{l = 0}^\infty {(2l + 1)\frac{{\Gamma (i\gamma + l + 1)}}{{\Gamma (i\gamma + 1)\Gamma (2l + 2)}}{{{(2ikx)}}^{l}}} \\ \, \times \Phi (i\gamma + l + 1,2l + 2, - 2ikx){{P}_{l}}(t). \\ \end{gathered} $$

(69)

Let us now consider the analytical continuation of the function constructed here to the upper half-plane of complex plane k for

$$k = {{k}_{n}} = i\frac{{{\text{|}}\alpha {\text{|}}}}{{2n}},\quad \alpha < 0,$$

$$i\gamma = i{{\left. {\frac{\alpha }{{2k}}} \right|}_{{k = {{k}_{n}}}}} = i\frac{\alpha }{{2{{k}_{n}}}} = - n,\quad n = 1,2,3,...$$

Here, $k_{n}^{2}$ = –|α|/4n² is the energy of the two-particle bound state corresponding to principal quantum number n. In this case, expression (69) takes form

$$\begin{gathered} {{{\tilde {\psi }}}_{c}}({\mathbf{x}},{{{\mathbf{k}}}_{n}}) = \exp \left( { - \frac{{{\text{|}}\alpha {\text{|}}}}{{2n}}x} \right) \\ \times {{\sum\limits_{l = 0}^\infty {{{\beta }_{{nl}}}\frac{{2l + 1}}{{\Gamma (2l + 2)}}\left( { - \frac{{{\text{|}}\alpha {\text{|}}}}{n}x} \right)} }^{l}} \\ \times \,\Phi \left( { - n + l + 1,2l + 2,\frac{{{\text{|}}\alpha {\text{|}}}}{n}x} \right){{P}_{l}}(t). \\ \end{gathered} $$

(70)

Here, vector k_n is interpreted as a vector with the direction coinciding with the direction of initial vector k (${{{\mathbf{\hat {k}}}}_{n}}$ = ${\mathbf{\hat {k}}}$) and with the length assuming a purely imaginary value in the upper half-plane of complex plane k (k_n = i|α|/2n). Here, coefficient β_nl is defined as

$${{\beta }_{{nl}}} = (1 - n)(2 - n)...(l - n).$$

(71)

It should be noted that in accordance with this relation, coefficient β_nl vanishes for l ≥ n. Therefore, expression (70) turns out to be a finite sum of terms and can be written in form

$$\begin{gathered} {{{\tilde {\psi }}}_{c}}({\mathbf{x}},{{{\mathbf{k}}}_{n}}) = \exp \left( { - \frac{{{\text{|}}\alpha {\text{|}}}}{{2n}}x} \right) \\ \times \,\Phi \left( {1 - n,1,\frac{{{\text{|}}\alpha {\text{|}}}}{{2n}}x(1 - \langle {\mathbf{\hat {k}}},{\mathbf{\hat {x}}}\rangle )} \right) \\ = 4\pi \exp \left( { - \frac{{{\text{|}}\alpha {\text{|}}}}{{2n}}x} \right)\sum\limits_{l = 0}^{n - 1} {\sum\limits_{m = - l}^l {{{\beta }_{{nl}}}\frac{1}{{(2l + 1)!}}} } \\ \times {{\left( { - \frac{{{\text{|}}\alpha {\text{|}}}}{n}} \right)}^{l}}{{x}^{l}}\Phi \left( { - n + l + 1,2l + 2,\frac{{{\text{|}}\alpha {\text{|}}}}{n}x} \right)Y_{l}^{m}({\mathbf{\hat {x}}})Y_{l}^{m}{\text{*}}({\mathbf{\hat {k}}}), \\ \end{gathered} $$

(72)

where $Y_{l}^{m}$ are the corresponding spherical functions. This equality holds by virtue of the theorem of summation of spherical harmonics (5.17.9) [35],

$$\frac{{2l + 1}}{{4\pi }}{{P}_{l}}(\langle {\mathbf{\hat {k}}},{\mathbf{\hat {x}}}\rangle ) = \sum\limits_{m = - l}^l {Y_{l}^{m}({\mathbf{\hat {x}}})Y_{l}^{m}{\text{*}}({\mathbf{\hat {k}}})} .$$

Finally, we introduce function

$${{\varphi }_{n}}({\mathbf{x}},{\mathbf{\hat {k}}}) \equiv {{\tilde {\psi }}_{c}}({\mathbf{x}},{{{\mathbf{k}}}_{n}})$$

(73)

in accordance with expression (72). We have defined the function with partial components coinciding with the Coulomb radial functions of the discrete spectrum to within normalization [36],

$${{R}_{{nl}}} = {{\rho }^{l}}{{e}^{{ - \rho /2}}}\Phi ( - n + l + 1,2l + 2,\rho ),$$

$$l = 0,1,2,...,n - 1$$

for a fixed principal quantum number n.

Integrating function φ_n over d${\mathbf{\hat {k}}}$ on a unit sphere with a certain smooth function $\hat {a}$(${\mathbf{\hat {k}}}$), we obtain a standard expansion of form

$$\int\limits_{{{\mathbb{S}}^{2}}}^{} {{{\varphi }_{n}}({\mathbf{x}},{\mathbf{\hat {k}}})\hat {a}({\mathbf{\hat {k}}})d{\mathbf{\hat {k}}}} = \sum\limits_{l = 0}^{n - 1} {\sum\limits_{m = - l}^l {{{D}_{{nlm}}}{{R}_{{nl}}}(x)Y_{l}^{m}({\mathbf{\hat {x}}})} } $$

(74)

in the complete set of Coulomb paired states of the discrete spectrum, which correspond to a fixed quantum number n. In this case, the set of amplitudes D_nlm is completely determined by the analytic form of function $\hat {a}$(${\mathbf{\hat {k}}}$).