  The previously derived vortex atomic form factor, which is directly related
to a differential reaction cross section, is used to analyze the elastic
scattering of twisted vortex photons with a hydrogenic atomic target. The
vortex atomic form factor is expressed in a unified spherical basis and
implemented in a MatLab code that numerically evaluates it using globally
adaptive quadrature. The results of this code show the influence of variation
in the photon wavelength, Rayleigh range, and scattering angle on differential
reaction cross sections and the twist factor, which measures the impact of
introducing orbital angular momentum. The recently suggested double mirror
effect that accounts for a non-zero effect in the forward direction for twisted
photon interactions is numerically confirmed. Finally, it is shown that
differential reaction cross sections are greatly amplified when the Rayleigh
range and photon wavelength are brought close to the scale of an atom.
Experimental considerations and applications are briefly discussed, including
quantum information, in which the scattering of twisted photons on atomic
targets can be used to transfer information between light and matter.

Quantum Physics;Atomic Physics;Optics