  We explore the opportunities of using electron scattering by screened Coulomb
potential as a tool to retrieve properties of the relativistic vortex beams of
electrons, such as their transverse momentum and orbital angular momentum
(OAM). We focus on relativistic and ultra-relativistic regimes of the electron
energies of at least several MeV and higher, in which the transverse beam
momentum is typically much smaller than its longitudinal momentum. Different
scattering scenarios for the incident electron beam are considered. In
particular, the scattering by a very wide target can be used to probe the
electron transverse momentum when its values are larger than 10 keV. The
scattering by a target of a width comparable to that of the incident beam
allows one to obtain information about the electron OAM. Varying target sizes
in the range from couple to hundreds of nanometers, one can in principle
distinguish OAM values from several units of $\hbar$ up to thousands and more.
