In the Bennett-Nordsieck self-pinched regime of high power REB propagation in a neutral atmosphere, radial expansion is generally associated with transverse entropy generation caused by elastic electron-neutral multiple scattering: LN ∝ 1/s′⊥ elast, where LN is the Nordsieck length, the distance for one e-folding of beam radius, and where s′⊥ elast is the elastic collision space rate of transverse mean entropy per particle.

For ultrarelativistic beams (γ ≳ 100), the bremsstrahlung, which is the dominant energy loss process, also plays an essential rôle in the radial expansion.

A general treatment could be based on the proper time evolution equation of the beam electron pressure 4-tensor pλμ (λ, μ = 0, 1, 2, 3) where source terms linked to elastic, inelastic and bremsstrahlung collisions are introduced, as is also a closure relation. This approach is currently being studied at LPPG.

When the various implied scale lengths have clearly different orders of magnitude, a much simpler approximate description may be given.

In the λmbrems < z < λstrbrems propagation distance range, where λmbrems is the depth threshold beyond which bremsstrahlung scattering becomes multiple, and λstrbrems a characteristic distance for bremsstrahlung straggling, the rôle of bremsstrahlung in radial expansion is similar to that of elastic multiple scattering. The calculated s′⊥ brems/s′⊥ elast increases rapidly with both propagation distance and beam electron energy. For γ ≫ 103, the bremsstrahlung transverse entropy source term s′⊥ brems is no more negligible before s′⊥ elast.

In the z > λstrbrems propagation distance range, where bremsstrahlung straggling is dominant, an evaluation of its effect is deduced by applying the Haftel–Lampe–Aviles criterion to a statistical study of this straggling. A completely different estimation, based on an oversimplified version of the above-cited general thermodynamic method, gives a result which is in rather good agreement.