  Motivated by the recent discovery of electron vortex beams carrying orbital
angular momentum (AM), we construct exact Bessel-beam solutions of the Dirac
equation. They describe relativistic and nonparaxial corrections to the scalar
electron beams. We describe the spin and orbital AM of the electron with
Berry-phase corrections and predict the intrinsic spin-orbit coupling in free
space. This can be observed as a spin-dependent probability distribution of the
focused electron vortex beams. Moreover, the magnetic moment is calculated,
which shows different $g$-factors for spin and orbital AM and also contains the
Berry-phase correction.
