Bethe's bremsstrahlung spectrum revisited

Abstract

We derive an expression for the Born approximation cross section for bremsstrahlung integrated over final particle directions without making any high energy approximations. Using the momentum distribution which is the integrand of this “exact” cross section, we derive a high energy approximation for the momentum distribution that is uniformly valid in the region which contributes significantly to the cross section, viz., δ ⩽ q ≲ O( m), i.e. the errors in this high energy distribution are of order ( m 2 ϵ 2 ) ln( ϵ m ) for all momentum transfers, q, in this region. (Here, ϵ refers to either the initial or final electron energy, ϵ 1 or ϵ 2, m is the electron mass, and δ is the minimum kinematically allowed momentum transfer.) We make no assumptions with regard to the photon energy, k, which can take on any kinematically allowed value. Using our high energy momentum distribution, we analyze the errors in Bethe's bremsstrahlung spectrum, which we find to be less than indicated by his analysis. Our high energy expression for the integrated cross section, derived solely by neglecting terms of relative orders ( m 2 ϵ 2 ) ln 2( ϵ m ) and ( β 2 m 2 ) ln( m 2 β 2 ) , differs from that given by Bethe by only a single term of O( δ m ) , which does not affect the integral over momentum transfer. Thus, Bethe's spectrum is valid as well in the limit k → 0. Our expression for the bremsstrahlung spectrum has much smaller errors and is valid over a large range of photon energies. Thus, for example, even for initial electron energies as low as 50 MeV, the error is always less than 1% for 0⩽ k ϵ 1 ⩽ 0.9; for higher initial electron energies, the errors are even smaller.