Near-threshold behavior of positronium-antihydrogen scattering cross sections

We have performed a comprehensive numerical investigation of the scattering of antihydrogen ($\overline{\mathrm{H}}$) and positronium (Ps) atoms, at collision energies just above the threshold for the rearrangement reaction resulting in ${\overline{\mathrm{H}}}^{+}$ ions. Formation of ${\overline{\mathrm{H}}}^{+}$ in $\overline{\mathrm{H}} +\mathrm{Ps}$ collisions is of vivid interest for experiments with $\overline{\mathrm{H}}$ at CERN. We treat all open channels below the threshold for the rearrangement into $\overline{\mathrm{H}} +\mathrm{Ps}$ (which includes elastic scattering, excitation and deexcitation of positronium to and from the excited states with $n=2,3$, polarization and depolarization of $\mathrm{Ps}(n,l)$, rearrangement into ${\overline{\mathrm{H}}}^{+}+e^{-}$, and formation of Ps and $\overline{\mathrm{H}}$ in the reverse reaction). Our method is a combination of the variational calculation with the coupled rearrangement channel method. The four-body solutions of the Schrödinger equation, obtained via diagonalization of the full Hamiltonian, are combined with the channel functions that satisfy the correct asymptotic boundary conditions, in a procedure that delivers the scattering matrix $S$. The complete set of $S$-matrix elements for all allowed scattering processes is calculated (both in forward and reverse directions, without assumption of the detailed balance). The scattering cross sections are obtained from the $S$-matrix elements calculated for the total angular momenta $J=0–6$, and for the positronic spin $S=0$, required for the formation of ${\overline{\mathrm{H}}}^{+}$. The numerical accuracy of the calculation is checked using the criterion of unitarity of the $S$ matrix. Convergence of the cross section is examined with respect to the partial wave contributions, under the constraint of conservation of total angular momentum and parity. We show that the energy behavior of the cross sections near thresholds is consistent with the Wigner laws. The cross sections for processes that compete with ${\overline{\mathrm{H}}}^{+}$ formation are presented.

Figure 1

Threshold energies of the $\mathrm{Ps}\overline{\mathrm{H}}$ system. The thresholds for $\mathrm{Ps}(n=3)+\overline{\mathrm{H}}\left(1s\right)$ and $e^{-}+{\overline{\mathrm{H}}}^{+}$ are almost degenerate; the difference between them is only $6\times {10}^{-5}$ a.u. (see the inset)

Figure 2

(a) Interparticle vectors. We denote two positrons as $e_1^{+}$ and $e_2^{+}$. (b–f) Jacobi coordinate systems for Ps-$\overline{\mathrm{H}}$ scattering.

Figure 3

Jacobi arrangements for intermediate couplings during the collision. $\overline{p}, e^{+}$, and $e^{-}$ are illustrated by the black, red, and blue circles. The labels beside the vectors indicate the angular momenta along the Jacobi coordinates $\{{\mathbf{r}}_c,{\mathbf{R}}_c,{\mathit{\rho }}_c\}$. ${\lambda }_c$ is chosen so that the vector sum of ${\lambda }_c$ and ${\mathrm{\Lambda }}_c$ configures $J$. The large grey circles suggest the clusterings of the system that are suitably described by the various sets of Jacobi coordinates. In group 1, the intermediate coupling ${\mathrm{\Lambda }}_c$ is uniquely determined in each arrangement. In group 2, we select ${\mathrm{\Lambda }}_c=0$ so that the basis functions describe the ${\overline{\mathrm{H}}}^{+}$ subsystem with the electron in the ${\lambda }_c$ wave. In group 3, ${\mathrm{\Lambda }}_c$ is chosen from the possible vector sums of $l_c$ and $L_c$. In group 4, we select ${\mathrm{\Lambda }}_c=0$ again so that the basis functions describe the ${\mathrm{Ps}}^{+}$ subsystem that would be formed virtually during the collision. In group 5, ${\mathrm{\Lambda }}_c=1$ is selected so that the basis functions of this group describe deformation of the ${\overline{\mathrm{H}}}^{+}$ subsystem, whose ground-state configuration is described by group 2. Naturally, the groups are nonexcluding and they all collectively participate in the description of a given state of the system.

Figure 4

(a) Eigenvalues $E_{\upsilon }$ of the four-body eigenvalue problem as a function of $\upsilon$. The horizontal lines denote $\overline{\mathrm{H}}\left(1s\right)+\mathrm{Ps}\left(nl\right)$ threshold energies (light blue) and $\overline{\mathrm{H}}(n\ge 2)+\mathrm{Ps}\left(1s\right)$ threshold energies (blue). Convergence properties of $J=0$ scattering at the four-body energy $E=-0.524727$ a.u. are shown in (b)–(j). (b) Convergence of the unitarity condition, $|U_{{\alpha }_{\mathrm{i}}}-1|$. For each ${\alpha }_i, U_{{\alpha }_i}$ is a sum over all opened channels $\alpha$ [see Eq. (28)]; e.g., for $J=0$ there are seven open channels at the energy chosen for this graph. The conservation of scattering flux, illustrated by $|U_{{\alpha }_i}-1|$, is seen to depend on the choice of initial state ${\alpha }_i$. (c) Convergence of the symmetric condition, $\mathrm{\Delta }{S}^{\left(J\right)}$. (d)–(j) Convergence of cross sections ${\sigma }_{\alpha {\alpha }_{\mathrm{i}}}^{\left(J\right)}$ against ${\upsilon }_{\max }$. Each panel (d)–(j) corresponds to a different initial state; the final states are indicated by different colors. The error bars denote $\mathrm{\Delta }{\sigma }_{\alpha {\alpha }_{\mathrm{i}}}$.

Figure 5

Convergence properties of $J=0$ scattering at the four-body energy $E=-0.524727$ a.u. against the matching radius ${\rho }_{\mathrm{match}}$.

Figure 6

Cumulative plots of the cross sections of elastic (“ela”), Ps excitation (“ex”), Ps deexcitation (“dex”), (de)polarization (“pl”), and rearrangement (“rearr”) scattering. Each row is devoted to a different initial state of Ps. The principal quantum number noted after the “ex” and “dex” indicates the final state of Ps, and the contributions from various possible final $l$ states are added together. Similarly, if multiple final $l$ states are possible for (de)polarization, the corresponding cross sections are added up.

Figure 7

Partial cross sections for various processes just above $e^{-}+{\overline{\mathrm{H}}}^{+}$ threshold energy are shown against the momentum in the initial and final channels, $k_{\mathrm{i}/\mathrm{f}}$. The shades indicate the trends expected from Wigner's law. The orbital angular momentum between the fragments, ${\lambda }_{\mathrm{i}/\mathrm{f}}$, is annotated by symbols. (a) Ps excitation cross sections of $\mathrm{Ps}(n\le 2$) $+ \overline{\mathrm{H}}\left(1s\right) \to \mathrm{Ps}(n=3)+\overline{\mathrm{H}}\left(1s\right)$. Shades are in proportion to $k_{\mathrm{f}}^{2{\lambda }_{\mathrm{f}}+1}$. (b) Ps deexcitation cross sections of $\mathrm{Ps}(n=3)+\overline{\mathrm{H}}\left(1s\right) \to \mathrm{Ps}(n\le 2$) $+ \overline{\mathrm{H}}\left(1s\right)$. Shades are in proportion to $k_{\mathrm{i}}^{2{\lambda }_{\mathrm{i}}-1}$. (c) ${\overline{\mathrm{H}}}^{+}$ production cross sections of $\mathrm{Ps}(n\le 2$) $+ \overline{\mathrm{H}}\left(1s\right) \to {\overline{\mathrm{H}}}^{+}+e^{-}$. The gray shade indicates no dependency on the momentum $k_{\mathrm{f}}$. (d) ${\overline{\mathrm{H}}}^{+}$ production cross sections of $\mathrm{Ps}(n=3)+\overline{\mathrm{H}}\left(1s\right) \to {\overline{\mathrm{H}}}^{+}+e^{-}$. (e) The cross sections for recombination $e^{-}+{\overline{\mathrm{H}}}^{+}\to \mathrm{Ps}(n=1,2$) $+ \overline{\mathrm{H}}\left(1s\right)$. The shade is in proportion to $k_{\mathrm{i}}^{-2}$. (f) The cross sections for recombination $e^{-}+{\overline{\mathrm{H}}}^{+}\to \mathrm{Ps}(n=3)+\overline{\mathrm{H}}\left(1s\right)$. (g) Elastic scattering cross sections for $\mathrm{Ps}(n=3)+\overline{\mathrm{H}}\left(1s\right)$ collision. (h) Ps (de)polarization cross sections of Ps($3l$) $+ \overline{\mathrm{H}}\left(1s\right) \to \mathrm{Ps}\left(3l^{\prime }\right)+\overline{\mathrm{H}}\left(1s\right)$ where $l\ne l^{\prime }$. Shades are in proportion to $k_{\mathrm{i}}^{2{\lambda }_{\mathrm{i}}+2{\lambda }_{\mathrm{f}}}$.

Figure 8

Cross sections for formation of $\mathrm{Ps}\left(nl\right)$ in ${\overline{\mathrm{H}}}^{+}+e^{-}$ collisions, ${\sigma }_{\mathrm{Ps}}\left(nl\right)={\sigma }_{\mathcal{F}{\mathcal{F}}_i}$ (${\mathcal{F}}_i=7, \mathcal{F}=1,2,...,6$), as a function of collision energy. The solid lines denote the cross sections calculated by summing up ${\sigma }_{\alpha {\alpha }_{\mathrm{i}}}$. The shades represent the accumulated errors of Eq. (27). Due to the smallness of these errors, only the purple shade is visible.

Figure 9

Partial wave convergence of the cross sections for $\mathrm{Ps}(n\le 2$) formation in ${\overline{\mathrm{H}}}^{+}+e^{-}$ collisions, as a function of collision energy. The cross sections are plotted cumulatively.

Figure 10

Various scattering cross sections for $\mathrm{Ps}\left(nl\right)+\overline{\mathrm{H}}\left(1s\right)$ collisions are compared in the same energy range above the ${\overline{\mathrm{H}}}^{+}$ production threshold. The three panels are for different initial states of $\mathrm{Ps}\left(nl\right)$: (a) $n=1$, (b) $n=2$, and (c) $n=3$. The quantum numbers in the parentheses denote the Ps state in the initial channel, $\mathrm{Ps}\left(nl\right)+\overline{\mathrm{H}}\left(1s\right)$. The label “ela$+$pol” denotes the sum of elastic and the (de)polarization scattering cross sections, “ex” denotes excitation of Ps, “dex” denotes deexcitation of Ps, and “rearr” denotes ${\overline{\mathrm{H}}}^{+}$ production cross sections.

Figure 11

${\overline{\mathrm{H}}}^{+}$ production cross section (red lines, this work) are compared with previous results indicated by black lines: CDW-FS (using a Le Sech function) [44], distorted-wave Born approximation (DWBA) and FBA [37], and the CPA [40]. The red arrows indicate the threshold energy of ${\overline{\mathrm{H}}}^{+}$ production in the present calculation, and the blue arrows indicate that reproduced in the CDW-FS calculation.