Asymptotic cross section and scaling law: positronium formation in Rydberg states in positron–hydrogen collisions

Abstract

Positronium formation in Rydberg states from the ground state of the hydrogen atom by positron impact has been studied within the framework of a distorted wave theory which includes static dipole polarization potential. The distorted wave scattering amplitude has been obtained in a closed form. A detailed investigation has been made on the differential and total cross sections in the energy range 25–300 eV of incident positron. It has been found that asymptotic cross sections for the positronium formation into different angular momentum states obey a simple law.

Appendix

Evaluation of the $1s \rightarrow nlm$ capture amplitude $g_{B}(\vec {k}_f, \vec {k}_i).$

The wave function of Ps atom in nlm state is given by

$$ \begin{aligned} \eta _{{nlm}} (\vec{r}_{1} ) & = R_{{nl}} (r_{1} )Y_{{lm}} (\theta _{1} ,\phi _{1} ) \\ & = \left\{ { - \left[ {\left( {\frac{2}{{na_{0} }}} \right)^{3} \frac{{(n - l - 1)!}}{{2n[(n + l)!]^{3} }}} \right]^{{1/2}} e^{{ - \frac{{r_{1} }}{{na_{0} }}}} \left( {\frac{2}{{na_{0} }}r_{1} } \right)^{l} L_{{n + l}}^{{2l + 1}} \left( {\frac{2}{{na_{0} }}r_{1} } \right)} \right\}Y_{{lm}} (\theta _{1} ,\phi _{1} ), \\ \end{aligned} $$

(7)

where $Y_{lm}$ denote the spherical harmonics, and $L_{n+l}^{2l+1}$ denote the associated Laguerre polynomial of degree $(n+l)$ and order $(2l+1).$

$1s \rightarrow nlm$ capture amplitude $g_{B}(\vec {k}_f, \vec {k}_i)$ is given by

$$ \begin{aligned} g_{B} (\vec{k}_{f} ,\vec{k}_{i} ) & = \left( { - \frac{{m_{f} }}{{2\pi }}} \right)\left\langle {\Phi _{f} |V_{f} |\Phi _{i} } \right\rangle \\ & = \left( { - \frac{{m_{f} }}{{2\pi }}} \right)\int {e^{{i[\vec{A}.\vec{r}_{2} - \vec{B}.\vec{r}_{1} ]}} } \eta _{{nlm}}^{*} (\vec{r}_{1} )\left[ {\frac{1}{{R_{{12}} }} - \frac{1}{{r_{2} }}} \right]\phi _{{1s}} (\vec{r}_{2} )d\vec{r}_{1} d\vec{r}_{2} \\ & = \left( { - \frac{{m_{f} }}{{2\pi }}} \right)\left[ {I_{1} - I_{2} } \right], \\ \end{aligned} $$

(8)

where

$$ \vec{A} = \frac{{m_{p} }}{{m_{p} + 1}}\vec{k}_{i} - \vec{k}_{f} ,\quad \vec{B} = \vec{k}_{i} - \frac{{\vec{k}_{f} }}{2} $$

$$ I_{1} = \int {e^{{i[\vec{A} \cdot \vec{r}_{2} - \vec{B} \cdot \vec{r}_{1} ]}} } \eta _{{nlm}}^{*} (\vec{r}_{1} )\frac{1}{{R_{{12}} }}\phi _{{1s}} (\vec{r}_{2} )d\vec{r}_{1} d\vec{r}_{2} , $$

(9)

$$ I_{2} = \int {e^{{i[\vec{A} \cdot \vec{r}_{2} - \vec{B} \cdot \vec{r}_{1} ]}} } \eta _{{nlm}}^{*} (\vec{r}_{1} )\frac{1}{{r_{2} }}\phi _{{1s}} (\vec{r}_{2} )d\vec{r}_{1} d\vec{r}_{2} $$

(10)

We first consider the integral $I_1$ which contains positron–proton interaction and hence rather difficult to evaluate. Taking Fourier transforms of the function $exp(-\lambda r)/r$ and then utilizing the $\delta -$function properties $\vec {r}_2$ integration can be carried our easily and we obtain

$$I_1 = -2\left( \frac{\gamma _{1}}{\pi } \right) ^{3/2} \frac{\partial }{\partial \gamma _{1}} \int \frac{e^{i(\vec {s}-\vec {B}).\vec {r}_1}}{(|\vec {A}-\vec {s}|^2 + \gamma _1^2)s^2}\eta _{nlm}^*(\vec {r}_1)~ d \vec {r}_1 d \vec {s}.$$

(11)

where $\gamma _1 = 1/a_0.$ Using integral representation of Feynman

$$\frac{1}{ab} = \int \limits _{0}^{1} \frac{dx}{[ax+b(1-x)]^2}$$

and setting $\lambda ^2 = A^2x(1-x) + \gamma _{1}^2 x$ we obtain

$$ I_{1} = - 2\left( {\frac{{\gamma _{1} }}{\pi }} \right)^{{3/2}} \frac{\partial }{{\partial \gamma _{1} }}\int\limits_{0}^{1} d x~e^{{i(x\vec{A} - \vec{B}) \cdot \vec{r}_{1} }} \eta _{{nlm}}^{*} (\vec{r}_{1} )~d\vec{r}_{1} \int {\frac{{e^{{i(\vec{s} - x\vec{A}) \cdot \vec{r}_{1} }} }}{{[|s - x\vec{A}|^{2} + \lambda ^{2} ]^{2} }}} ~d\vec{s} $$

or,

$$\begin{aligned} I_{1} &= - 2\pi ^{2} \left( {\frac{{\gamma _{1} }}{\pi }} \right)^{{3/2}} \frac{\partial }{{\partial \gamma _{1} }}\int\limits_{0}^{1} {\frac{{e^{{i\vec{p} \cdot \vec{r}_{1} - \lambda r_{1} }} }}{\lambda }} \eta _{{nlm}}^{*} (\vec{r}_{1} )~d\vec{r}_{1} dx,\\ \vec{p} &= x\vec{A} - \vec{B} \end{aligned}$$

or,

$$ I_{1} = - 2\pi ^{2} \left( {\frac{{\gamma _{1} }}{\pi }} \right)^{{3/2}} \frac{\partial }{{\partial \lambda }}\int\limits_{0}^{1} {\frac{F}{\lambda }} \left( {\frac{{\partial \lambda }}{{\partial \gamma _{1} }}} \right)dx $$

or,

$$ I_{1} = {\text{ }}A_{1} \left[ {\int\limits_{0}^{1} {\frac{x}{{\lambda ^{2} }}} \frac{\partial }{{\partial \lambda }}(F)~dx - \int\limits_{0}^{1} {\frac{{xF}}{{\lambda ^{3} }}} ~dx} \right], $$

(12)

where

$$ A_{1} = - 2\pi ^{2} \gamma _{1} \left( {\frac{{\gamma _{1} }}{\pi }} \right)^{{3/2}} , $$

$$ F = \int {e^{{i\vec{p} \cdot \vec{r}_{1} - \lambda r_{1} }} } \eta _{{nlm}}^{*} (\vec{r}_{1} )~d\vec{r}_{1} . $$

(13)

Using the expansion formula of plane wave in terms of spherical Bessel’s function $J_L,$ such as $exp(i\vec {p}.\vec {r}_1) = 4\pi \sum _{L,M} i^L j_L(pr_1)Y_{LM}^*(\hat{p}) Y_{LM}(\hat{r}_1),$ and then utilizing the orthogonal property of spherical harmonics $Y_{LM}$ we obtain

$$F = 4\pi i^lY_{lm}^*(\hat{p})N_{lm}\int \limits _{0}^{\infty } j_l(pr_1) e^{-ar_1} L_{n+l}^{2l+1}(2\gamma _{n}r_1)r_1^{l+2}~dr_1, \quad a = \lambda + \gamma _{n} \;\mathrm{and}\;\gamma _{n} =\frac{1}{n a_0} .$$

(14)

Now using the expansion formula of the associated Laguerre polynomial and a typical integral involving spherical Bessel function, such as

$$ L_{{n + l}}^{{2l + 1}} (x) = \sum\limits_{{i = 0}}^{{n - l - 1}} {( - 1)^{{i + 1}} } \frac{{[(n + l)!]^{2} }}{{(n - l - 1 - i)!(2l + 1 + k)!}}\frac{{x^{i} }}{{i!}} $$

and

$$ \int\limits_{0}^{\infty } {e^{{ - ax}} } j_{l} (bx)x^{{l + 1}} dx = \frac{{(2b)^{l} l!}}{{(a^{2} + b^{2} )^{{l + 1}} }},~~({\text{Re}}(a) > |{\text{Im}}(b)|), $$

(15)

we finally obtain

$$F = C Y_{lm}^*(\hat{p}) \sum _{k=0}^{n-l-1} g(k)~D(p,a,l+1,k+1),$$

(16)

where

$$\begin{aligned}& D(x,y,l,m) = \frac{{d^{m} }}{{dy^{m} }}\left[ {\frac{1}{{(x^{2} + y^{2} )^{l} }}} \right]~\;{\text{and}}\\ & g(k) = \frac{{(2\gamma _{n} )^{k} }}{{(n - l - 1 - k)!(2l + 1 + k)!k!}}. \end{aligned}$$

Substituting (Eq. (16)) into (Eq. (12)) we obtain

$$I_1 = A_1C \sum _{k=0}^{n-l-1} g(k) \int \limits _{0}^{1} \frac{xY_{lm}^*(\hat{p})}{\lambda ^2} \left[ \frac{D(p,a,l+1,k+1)}{\lambda } - D(p,a,l+1,k+2) \right] (2p)^l~dx .$$

(17)

In the similar fashion the integral $I_2$ can be evaluated. The one-dimensional integration over [0, 1] appearing in integral (Eq. (17)) has been evaluated numerically by employing Gauss–Legendre quadrature formula. Note that the integral (Eq. (17)) has a fictitious singularity , which has been removed by taking a transformation of the form $x=z^2.$