  Classical radiation reaction is the effect of the electromagnetic field
emitted by an accelerated electric charge on the motion of the charge itself.
The self-consistent underlying classical equation of motion including
radiation-reaction effects, the Landau-Lifshitz equation, has never been tested
experimentally, in spite of the first theoretical treatments of radiation
reaction having been developed more than a century ago. Here we show that
classical radiation reaction effects, in particular those due to the near
electromagnetic field, as predicted by the Landau-Lifshitz equation, can be
measured in principle using presently available facilities, in the energy
emission spectrum of $30\text{-}\text{GeV}$ electrons crossing a
$0.55$-$\text{mm}$ thick diamond crystal in the axial channeling regime. Our
theoretical results indicate the feasibility of the suggested setup, e.g., at
the CERN Secondary Beam Areas (SBA) beamlines.
