(1) dσc(ε,Θ')/dΩ' = Z2.e4 / [16.ε2.sin4(Θ'/2) ] ,where Z is the nuclear charge number, e the elementary charge, ε the center of mass energy of the two charges and Θ' the scattering angle in the center of mass system (note that this expression is for cgs-units; in SI-units it requires a further numerical factor 8.08.1019 (the square of the 'Coulomb constant')). The second aspect entering into the problem is the dependence of the energy transfer function on the scattering angle. This can be determined by considering the energy transferred to a surface element dS of a spherical target in an elastic collision with a homogeneous particle beam. The surface element may be hereby defined by the angles η and η+dη (Fig.1).
(2) η = (π-Θ')/2 .The amount of energy transferred to the surface element dS is determined by two factors : a) the amount of energy transferred by a single particle hitting the surface element dS and b) the number of particles (or probability of the single particle) hitting dS.
(3) Δ(Θ') = 4.m1.m2.sin2(Θ'/2) / (m1+m2)2 ,where Δ(Θ') designates the relative energy change as a function of the scattering angle Θ' (related to the surface element by Eq.(2)). m1 and m2 are the particle masses, where m2 is assumed at rest in the laboratory system before the collision (at this stage of the derivation m1 and m2 are arbitrary; when evaluating the total cross section, the use of the center of mass variable Θ' implies m1<<m2 which does however not restrict the general validity of the so obtained qualitative results).
(4) f(η) = 1/π.cos(η) ,where f(η) is normalized to half the sphere surface which is exposed to the incident particle beam, i.e.
(5) (2π)∫dΩH f(η) = 1 .(This normalization is of importance only for the definition of the average energy transfer Δav . For the evaluation of the total cross section a different normalization is necessary (see Eq.(8))).
(6) f(Θ') = 1/π.sin(Θ'/2) .Eq.(6) maps a homogeneous distribution of impact parameters into Θ'-space, i.e. it provides the geometrical connection between the monodirectional incident particle beam and the spherical scattering surface. This connection is ignored in the literature throughout, which invalidates in these cases the interpretation of the scattering angle Θ' as an independent integration variable in Eqs.(11) and (12) (see below). The identification of the scattering angle Θ' with the polar angle of a coordinate system fixed in the target center is only correct if the exact geometrical relations in the scattering process on a microscopical scale are of no importance. This is however not the case when considering the energy transfer function, and the neglection of this aspect is the reason for the well known logarithmic divergence of the total cross section for Coulomb scattering (Coulomb Logarithm) in other treatments of this problem. The total amount of energy transferred to the surface element is thus given by the product of Eqs.(3) and (6), i.e. the corresponding normalized energy transfer function is of the form
(7) F(Θ') = cF.sin3(Θ'/2) ,where cF is a normalization constant which is obtained from the condition
(8) (4π)∫dΩ F(Θ') = 1as
(9) cF = 5/(8π) ,i.e.
(10) F(Θ') = 5/(8π) .sin3(Θ'/2) ,Eq.(10) describes the distribution of the energy transfer over scattering angle. The normalization (Eq.(8)) is such that it corresponds to the average fractional energy loss
(11) Δav = 1/4π.(4π)∫dΩ' Δ(Θ').f(Θ') = 8/(5π) .m1.m2/(m1+m2)2 =
= 8/(5π) .m1/m2 if m1<<m2 (see remarks below Eq.(12)).
(12) σc(ε) = (4π)∫dΩ' F(Θ') .dσc(ε,Θ')/dΩ'Since the integrations in Eqs.(11) and (12) are performed in the center of mass system, whereas the cross section shall constitute a quantity in the laboratory system, it is clear that these definitions are strictly (i.e. quantitatively) valid only for m1<<m2, in which case the scattering angles in those two frames of reference nearly coincide.
(13) σc(ε) = 5/16 .Z2.e4 / ε2 .(this expression is for cgs-units; in SI-units it requires a further numerical factor 8.08.1019 (the square of the 'Coulomb constant')).