Positronium-atom scattering at low energies

A pseudopotential for positronium-atom interaction, based on electron-atom and positron-atom phase shifts, is constructed, and the phase shifts for Ps-Kr and Ps-Ar scattering are calculated. This approach allows us to extend the Ps-atom cross sections, obtained previously in the impulse approximation [I. I. Fabrikant and G. F. Gribakin, Phys. Rev. Lett. 112, 243201 (2014)], to energies below the Ps ionization threshold. Although experimental data are not available in this low-energy region, our results describe well the tendency of the measured cross sections to drop with decreasing velocity at $v<1$ a.u. Our results show that the effect of the Ps-atom van der Waals interaction is weak compared to the polarization interaction in electron-atom and positron-atom scattering. As a result, the Ps scattering length for both Ar and Kr is positive, and the Ramsauer-Townsend minimum is not observed for Ps scattering from these targets. This makes Ps scattering quite different from electron scattering in the low-energy region, in contrast to the intermediate energy range from the Ps ionization threshold up to $v\sim 2$ a.u., where the two are similar.

Figure 1

Positron scattering phase shifts for Ar obtained using the static potential of the Hartree-Fock atom.

Figure 2

Electron scattering phase shifts (modulo $\pi$) for Ar obtained using the static-exchange (i.e., Hartee-Fock) atomic potential (solid black lines), and the pseudopotential (3) (dashed red lines), with parameters listed in Table 1.

Figure 3

Ps-Kr scattering phase shifts for $L=0$, 1 and 2. For each partial wave, the lower curve (solid black) is for the static pseudopotential and the upper curve (dashed red) includes the effect of the van der Waals interaction with $C_6=152$ a.u. and $R_c=3.0$ a.u.

Figure 4

Ps-Kr scattering phase shifts for $L=2$. Shown are the results obtained with the full pseudopotential (i.e., with nonlocal effects) and its local approximations $v_{\mathrm{loc}}\left(r\right)=v_l\left(r\right)$ for $l=1$ and 2. The van der Waals interaction is included in all cases. Solid black lines: $R_c=3.0$ a.u.; red dashed line: $R_c=3.5$ a.u.

Figure 5

Dependence of the scattering length, given by Eq. (20), for the van der Waals potential (17) on the radius of the repulsive core. The three curves correspond to $C_6=104.5$, 152, and 234 a.u., for Ps-Ar, Ps-Kr, and Ps-Xe, respectively. Open circles show the estimates of $A$ from Table 2.

Figure 6

Ps-Kr elastic scattering cross sections. Solid black curve “IA” is the result of the impulse approximation [4]; solid black curve “static” is the present static-field calculation (i.e., $C_6=0$); dashed red curve is the present calculation with the static and van der Waals interaction ($C_6=152$ a.u., $R_c=3.0$ a.u.); dotted blue curve is the same for $R_c=3.5$ a.u.; dot-dashed magenta curve is the static-exchange calculations of Blackwood et al. [5].

Figure 7

Ps-Kr total scattering cross sections. Solid black curve is the total cross section in the impulse approximation [4]; dashed red curves present elastic and total cross sections obtained with $C_6=152$ a.u., $R_c=3.0$ a.u. Solid squares are experimental data from Ref. [1].

Figure 8

Ps-Ar scattering phase shifts. For each $L$, the lower curve (solid black) was calculated using the static-field pseudopotential, while the upper (dashed red) also includes the effect of the van der Waals interaction with $C_6=104.52$ a.u. and $R_c=3.0$ a.u.

Figure 9

Ps-Ar elastic scattering cross sections. Solid black curve “IA” is the result of the impulse approximation [4]; solid black curve “static” is the present static calculation $\left(C_6=0\right)$; dashed red curve is the calculation with the static and van der Waals potential ($C_6=104.5$ a.u., $R_c=2.5$ a.u.); dotted blue curve is the same for $R_c=3.0$ a.u.; dot-dashed magenta curve is close-coupling calculations of Blackwood et al. [5].

Figure 10

Ps-Ar total scattering cross sections. Solid black curve “IA” is the result of the impulse approximation [4]; dashed red curve is the present elastic cross section obtained with $C_6=104.5$ a.u., $R_c=2.5$ a.u., augmented by the ionization cross sections of Starrett et al. [22]; dotted blue curve is the same for $R_c=3.0$ a.u. Open circles: experiment [23]; solid squares: experiment [1].