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Astron. Astrophys. 325, 401-413 (1997)

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4. Coulomb collisions

Speaking about the Coulomb scattering one usually implies the lowest order approximation, which is called Moller scattering when referred to identical particles [FORMULA], and Bhabha scattering when referred to distinct particles [FORMULA]. The effect of bremsstrahlung in ee -collisions is strictly not separable from that of scattering, however, it is convenient and generally accepted to treat them separately. Expressions for Coulomb energy losses and dispersion have been obtained by Dermer (1985) and Dermer & Liang (1989). Here we describe briefly their results for the self-consistency of consideration.

The average LS energy change during a collision is (asterisk denotes CMS variables)

[EQUATION]

where [FORMULA] is the differential cross section, [FORMULA], [FORMULA] and [FORMULA] are the polar and azimuthal angles, respectively. The LS energy change expressed in these variables is

[EQUATION]

[EQUATION]

where [FORMULA] is the CMS velocity,

[EQUATION]

are the Lorentz factor and momentum of a particle in the CMS prior to scattering, [FORMULA] and [FORMULA] are those after scattering, and [FORMULA] is a kinematic angle

[EQUATION]

Energy losses of a particle due to elastic Coulomb scattering are given by Eq. (16) with the cross section weighted by [FORMULA]. Using azimuthal symmetry of the cross section, Dermer (1985) obtains

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

since [FORMULA]. In the case of elastic scattering [FORMULA] and [FORMULA], that gives

[EQUATION]

[EQUATION]

[EQUATION]

where

[EQUATION]

The value of [FORMULA] can be assigned from geometrical consideration: [FORMULA] for distinct particles and [FORMULA] for identical particles. The minimum scattering angle [FORMULA] can be related to the excitation of a plasmon of energy [FORMULA]. The correction for double counting in the case of identical particles appears now as the above condition for [FORMULA].

Integration of Eq. (23) with Moller ([FORMULA]) and Bhabha ([FORMULA]) scattering cross sections (Jauch & Rohrlich 1976) gives

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The term [FORMULA] appearing in Eqs. (24)-(25) is the Coulomb logarithm. It is a slowly varying function of [FORMULA], and often can be approximated by a constant. In the Born regime for the cold plasma limit, the Coulomb logarithm is given by Dermer (1985) [FORMULA], [FORMULA]. Where the plasma frequency [FORMULA] can be obtained from the usual expression by replacing the electron rest mass with an average inertia per gas particle [FORMULA] (Gould 1981), [FORMULA].

Substitution of the Rutherford cross section yields the cold plasma limit

[EQUATION]

The energy dispersion coefficients [FORMULA] can be obtained from Eq. (3). Another way is to square Eq. (18) and to follow the above-described method. For Moller scattering of an electron by a thermal electron distribution the correct form of the coefficient has been obtained by Dermer & Liang (1989)

[EQUATION]

[EQUATION]

[EQUATION]

where

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

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© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998

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