  A detection technique of ultra-high energy cosmic rays, complementary to the
fluorescence technique, would be the use of the molecular Bremsstrahlung
radiation emitted by low-energy electrons left after the passage of the showers
in the atmosphere. The emission mechanism is expected from quasi-elastic
collisions of electrons produced in the shower by the ionisation of the
molecules in the atmosphere. In this article, a detailed calculation of the
spectral intensity of photons at ground level originating from the transitions
between unquantised energy states of free ionisation electrons is presented. In
the absence of absorption of the emitted photons in the plasma, the obtained
spectral intensity is shown to be 5 10^{-26} W m^{-2}Hz^{-1} at 10 km from the
shower core for a vertical shower induced by a proton of 10^{17.5} eV.
