Compton Scattering by $K$-Shell Electrons. II. Nonrelativistic Dipole Approximation

The dipole approximation of the nonrelativistic cross section for Compton scattering by electrons bound in the ground state of a hydrogenlike atom, obtained by the present author, is computed and discussed. In this approximation the cross-section differential with respect to the angles and energy of the scattered photon is proportional to $C_1+C_2{cos}^2\theta$, where $\theta$ is the scattering angle of the photon. The coefficients $C_1$, $C_2$ are combinations of Appell functions $F_1$, depending on the atomic number $Z$ and the energies of the initial and final photons ${\kappa }_1$, ${\kappa }_2$ by means of the dimensionless variables $k_i=\frac{{\kappa }_i}{Z^2\mathcal{R}}$, $i=1, 2$, where $\mathcal{R}$ is the rydberg. In order to compute the Appell functions, these were first continued analytically and expressed in terms of other hypergeometric functions of two variables, which under our circumstances admit convergent series expansions. These were then summed numerically. The incident photon energies considered lie in the interval $1.05\le k_1\le 20$. For each value $k_1$ the coefficients $C_1$ and $C_2$ were computed for a number of values of $k_2$ ($0\le k_2\le k_1-1\mathrm{}$). Special attention was given to the end points $k_2=0\mathrm{}$ and $k_2=k_1$. $C_1$ and $C_2$ turn out to be monotonically decreasing functions of $k_2$ presenting the infrared-divergent behavior $\frac{1}{k_2}$ for $k_2\to 0$. The limitations of the result due to retardation corrections are considered.