Relativistic total cross section and angular distribution for Rayleigh scattering by atomic hydrogen

We study the total cross section and angular distribution in Rayleigh scattering by hydrogen atom in the ground state, within the framework of Dirac relativistic equation and second-order perturbation theory. The relativistic states used for the calculations are obtained by making use of the finite basis-set method and expressed in terms of B splines and B polynomials. We pay particular attention to the effects that arise from higher (nondipole) terms in the expansion of the electron-photon interaction. It is shown that the angular distribution of scattered photons, while symmetric with respect to the scattering angle $\theta ={90}^{\circ }$ within the electric dipole approximation, becomes asymmetric when higher multipoles are taken into account. The analytical expression of the angular distribution is parametrized in terms of Legendre polynomials. Detailed calculations are performed for photons in the energy range 0.5 to 10 keV. When possible, results are compared with previous calculations.

Figure 1

The adopted geometry for the scattering process is displayed. The scattering angle $\theta$ uniquely defines the direction of the outgoing photon in the scattering ($xz$) plane. The hydrogen atom is placed at the origin of the coordinate axes.

Figure 2

Feynman diagrams which correspond to the first and second terms of the transition amplitude (1), respectively.

Figure 3

Total cross section in Rayleigh scattering by atomic hydrogen, as a function of the photon energy. Calculations obtained within the electric dipole approximation (red dashed line) are compared with those including all of the allowed multipoles (black solid line) and with those from Veigele [39] (blue dot-dashed line).

Figure 4

Angular distribution in Rayleigh scattering by atomic hydrogen in the ground state as a function of the scattering angle $\theta$. Results are calculated within the exact relativistic theory (solid black line) and within the electric dipole approximation (red dashed line), for three selected photon energies.

Figure 5

The anisotropy coefficients ${\beta }_i$ of Eq. (15) are plotted against the photon energy. Results obtained within the electric and magnetic dipole approximation (red dashed line) are compared with results obtained by taking into account all photon multipoles with $L_1⩽4$ and $L_2⩽4$ (black solid line), which are dipole, quadrupole, octopole, and hexadecapole moments of each photon field, both of magnetic and electric type.