Compton scattering from positronium and validity of the impulse approximation

The cross sections for Compton scattering from positronium are calculated in the range from 1 to 100 keV incident photon energy. The calculations are based on the $A^2$ term of the photon-electron or photon-positron interaction. Unlike in hydrogen, the scattering occurs from two centers and the interference effect plays an important role for energies below 8 keV. Because of the interference, the criterion for validity of the impulse approximation for positronium is more restrictive compared to that for hydrogen.

Figure 1

Total cross section for Compton scattering on positronium Ps (squares) and hydrogen H ($\times$) in $A^2$ approximation. Also shown is the magnitude $\left|\mathbb{I}\right|$ of the interference term contribution (triangles) for scattering on positronium, as well as the Klein-Nishina cross sections ${\sigma }_{\mathrm{KN}}$ (dashes) and $2{\sigma }_{\mathrm{KN}}$ (dots).

Figure 2

Total cross sections in $A^2$ approximation for Compton scattering on positronium and hydrogen in the low-energy range. The symbols are the same as in Fig. 1.

Figure 3

Ratios of positronium to hydrogen total cross sections $R={\sigma }_{\mathrm{Ps}}/{\sigma }_{\mathrm{H}}$, and $R_D={\sigma }_{\mathrm{Ps}}^D/{\sigma }_{\mathrm{H}}$, the ratio when the interference term in positronium scattering is omitted (direct terms only).

Figure 4

DDCS for Compton scattering of photons with energy ${\omega }_1=10\hspace{0.5em}\mathrm{keV}$ (left panel) and ${\omega }_1=5\hspace{0.5em}\mathrm{keV}$ (right panel) from positronium Ps (squares) and from hydrogen H ($\times$) for the scattering angle $\theta ={123}^{°}$. Also shown is the absolute value of the interference term $\left|\mathbb{I}\right|$ (triangles).

Figure 5

Same as Fig. 4 for incident photon energy ${\omega }_1=2\hspace{0.5em}\mathrm{keV}$.

Figure 6

SDCS for scattering of 5 keV photons (upper panel) and 2 keV photons (lower panel) from positronium Ps (squares) and hydrogen H ($\times$) as a function of the scattering angle ${\Theta }_2$. Also shown is the absolute value of the interference term $\left|\mathbb{I}\right|$ (triangles) and the cross section obtained by using only direct terms in the scattering $D$ (line).

Figure 7

DDCS from positronium Ps and hydrogen H in the $A^2$ approximation and IA for the scattering angle $\theta ={123}^{°}$ and the incident photon energy ${\omega }_1=5\hspace{0.5em}\mathrm{keV}$ (a) and ${\omega }_1=2\hspace{0.5em}\mathrm{keV}$ (b). In the legend, $\left|\mathbb{I}\right|$ is the absolute value of the interference contribution to the DDCS obtained in $A^2$ approximation and ${\mathbb{I}}_{\mathrm{IA}}$ is the interference contribution to the DDCS obtained in IA.

Figure 8

Total cross sections for positronium and hydrogen in the $A^2$ calculation and the IA calculation (a). In (b), the lower-energy region is displayed in more detail. Symbols are the same as in Fig. 7.