Elastic scattering of photons byK-shell electrons at high energies

The matrix element for Rayleigh scattering by atomic $K$-shell electrons is evaluated in the limit of high photon energies at finite momentum transfers $\Delta$. The limiting form of the matrix element $\overline{M}$ is derived from its relativistic expression at finite energies $\mathfrak{M}$ in which the one-electron Green's function $G$ is replaced by its nonrelativistic approximation $G_0$ with adequately modified parameters. The expression obtained for $\overline{M}$ is exact in $\Delta$ and the atomic number $Z$, and is equivalent to the one found by Goldberger and Low. The evaluation of the matrix element is carried out in momentum space for the case of a Coulomb atomic field. Exact integral representations are used for $G_0$ and the ground-state eigenspinors. The integration of $\mathfrak{M}$ is carried out analytically as far as possible and at one stage the high-energy limit is taken. For one $K$-shell electron, when electron spin-flip is possible, $\overline{M}$ is expressed in terms of three real amplitudes $H$, $K$, and $L$, whereas the matrix element for the closed $K$ shell, ${\overline{M}}_K$, depends only on $H$. The amplitudes are obtained as one-dimensional integrals over Appell functions $F_1$. They simplify considerably in the case of forward scattering, and the connection of their imaginary parts with absorption cross sections is discussed. Expansions of the forward amplitudes in powers of $a=\alpha Z$ are also given. Further, a connection between $H$ and the nonrelativistic form factor is established, and the asymptotic behavior of $H$ with respect to $\Delta$ is derived. Then, a description is given of the numerical methods used. $H$ is computed for $0\le \frac{\Delta }{am}\le 15$ and $1\le Z\le {\alpha }^{-1\mathrm{}}$. Of the electron spin-flip amplitudes only $K$ is computed for forward scattering ($L$ vanishes in this case). The numerical results are discussed and comparison is made with other works. The validity of the high-energy result ${\overline{M}}_K$ at lower photon energies is considered. Finally, the magnitude of the Rayleigh matrix element ${\overline{M}}_K$ is compared with the one for Delbrück scattering.