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.
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