  A new Bateman-Hillion solution to the Dirac equation for a relativistic
Gaussian electron beam taking explicit account of the $4$-position of the beam
waist is presented. This solution has a pure Gaussian form in the paraxial
limit but beyond it contains higher order Laguerre-Gaussian components
attributable to the tighter focusing. One implication of the mixed mode nature
of strongly diffracting beams is that the expectation values for spin and
orbital angular momenta are fractional and are interrelated to each other by
\textit{intrinsic spin-orbit coupling}. Our results for these properties align
with earlier work on Bessel beams [Bliokh \textit{et al.} Phys. Rev. Lett.
\textbf{107}, 174802 (2011)] and show that fractional angular momenta can be
expressed by means of a Berry phase. The most significant difference arises,
though, due to the fact that Laguerre-Gaussian beams naturally contain Gouy
phase, while Bessel beams do not. We show that Gouy phase is also related to
Berry phase and that Gouy phase fronts that are flat in the paraxial limit
become curved beyond it.
