In all corresponding treatments in the literature, the total cross section for Coulomb scattering, as derived over the differential cross section, is obtained as a divergent expression unless an arbitrary cut-off value for the impact parameter (or scattering angle) is introduced (this leads to the well known Coulomb Logarithm factor). The following analysis shows that a finite effective total cross is in fact obtained if the exact scattering geometry is properly taken into account with regard to the energy transfer function.
As usual, the starting point of the derivation is the well known Rutherford-formula for the differential cross section
^{.}10^{19} (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).

η=0..π/2 designates in general the angle between the velocity vector of the incident particle before the collision and the velocity vector of the target particle (initially assumed at rest) after the collision. The scattering angle Θ' is hereby related to η by the general expression for elastic collisions

The first factor (a) is well known to be given by the formula (see for instance Landau and Lifschitz, Mechanics)_{1} and m_{2} are the particle masses, where m_{2} is assumed at rest in the laboratory system before the collision (at this stage of the derivation m_{1} and m_{2} are arbitrary; when evaluating the total cross section, the use of the center of mass variable Θ' implies m_{1}<<m_{2} which does however not restrict the general validity of the so obtained qualitative results).

The second factor (b) is obviously proportional to cosη (for the case of a homogeneous particle beam, i.e. a homogeneous probability distribution of impact parameters). One can thus define a corresponding distribution function_{av} . For the evaluation of the total cross section a different normalization is necessary (see Eq.(8))).

By means of Eq.(2), one obtains f as a function of the scattering angle (center of mass) i.e._{F} is a normalization constant which is obtained from the condition
_{1}<<m_{2}, in which case the scattering angles in those two frames of reference nearly coincide.

With this restriction, one gets from Eqs.(1),(10) and (12)^{.}10^{19} (the square of the 'Coulomb constant')).

One should note again that ε is the centre of mass energy of both particles, which for m_{1}>>m_{2} differs from the energy of the incident mass m_{1} in the rest frame of m_{2} by a factor (m_{1}+m_{2})/m_{2}, i.e. if ε is taken as the energy of m_{1}, this results then in a further factor ((m_{1}+m_{2})/m_{2})^{2} for the cross section. On the other hand, the energy transferred in a single collision is only a factor 8/(5π)^{.}m_{1}^{.}m_{2}/(m_{1}+m_{2})^{2} (see Eq.(11)), so as far as the effective cross section for a total energy loss of the incident particle is concerned, Eq.(13) has to be multiplied by 8/(5π) ^{.}m_{1}/m_{2} in case of the incident projectile mass m_{1}>>m_{2} (with ε then understood as the energy of m_{1} in the rest frame of m_{2}) (this explains for instance the much shorter mean free path of high energy ions in matter compared to electrons of the same energy, namely because their collision cross section with the atomic electrons is much larger).
In any case, with Eq.(13) a finite expression for the total cross section has been obtained, contrary to the usual but erroneous treatment of the problem where neglection of the additional factor Eq.(6) leads to the well known logarithmic divergence (Coulomb Logarithm) of the total cross section. Taking the correct energy transfer function (Eq.(10)) into account, the sin^{-4}(Θ'/2) behaviour of the Rutherford formula is exactly cancelled by F(Θ') since additionally there adds a further factor sin (Θ'/2) ) from the differential dΩ' in Eq.(12). Obviously, the 1/r - potential is the most slowly decreasing potential function for which a finite total cross section can be obtained , because for geometrical reasons no potentials decreasing asymptotically slower than 1/r are thinkable. The notion of 'infinite-range' potentials must therefore be abandoned , since the effective total cross section (as a physical quantity) always has a finite value independent of macroscopic parameters of the associated plasma.

(1) dσ_{c}(ε,Θ')/dΩ' = Z^{2}^{.}e^{4} / [16^{.}ε^{2}^{.}sin^{4}(Θ'/2) ] ,

η=0..π/2 designates in general the angle between the velocity vector of the incident particle before the collision and the velocity vector of the target particle (initially assumed at rest) after the collision. The scattering angle Θ' is hereby related to η by the general expression for elastic collisions

(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.The first factor (a) is well known to be given by the formula (see for instance Landau and Lifschitz, Mechanics)

(3) Δ(Θ') = 4^{.}m_{1}^{.}m_{2}^{.}sin^{2}(Θ'/2) / (m_{1}+m_{2})^{2} ,

The second factor (b) is obviously proportional to cosη (for the case of a homogeneous particle beam, i.e. a homogeneous probability distribution of impact parameters). One can thus define a corresponding distribution function

(4) f(η) = 1/π^{.}cos(η) ,

(5) _{(2π)}∫dΩ_{H} f(η) = 1 .

By means of Eq.(2), one obtains f as a function of the scattering angle (center of mass) i.e.

(6) f(Θ') = 1/π^{.}sin(Θ'/2) .

(7) F(Θ') = c_{F}^{.}sin^{3}(Θ'/2) ,

(8) _{(4π)}∫dΩ F(Θ') = 1

(9) c_{F} = 5/(8π) ,

(10) F(Θ') = 5/(8π) ^{.}sin^{3}(Θ'/2) ,

(11) Δ_{av} = 1/4π^{.}_{(4π)}∫dΩ' Δ(Θ')^{.}f(Θ') = 8/(5π) ^{.}m_{1}^{.}m_{2}/(m_{1}+m_{2})^{2} =

= 8/(5π) ^{.}m_{1}/m_{2} if m_{1}<<m_{2} (see remarks below Eq.(12)).

(12) σ_{c}(ε) = _{(4π)}∫dΩ' F(Θ') ^{.}dσ_{c}(ε,Θ')/dΩ'

With this restriction, one gets from Eqs.(1),(10) and (12)

(13) σ_{c}(ε) = 5/16 ^{.}Z^{2}^{.}e^{4} / ε^{2} .

One should note again that ε is the centre of mass energy of both particles, which for m

Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)

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