Consider an isolated system of two ions of charge qa and qb, and possessing corresponding masses rria and гпь. The magnitude of the Coulomb force is given

by

1 ЧаЧь

4rt£o r2

where r is the distance of separation at any instant; evidently, as the charges move, this distance and also the direction of the force varies with time. The resultant trajectory of the particles a and b is suggested in Fig.3.1 for charged particle repulsion and attraction. In order to clearly specify a cross section for this process, it is useful to introduce the impact parameter r0 and the scattering angle 0S with respect to the initial and final asymptotic particle trajectories. In Fig. 3.1 the two colliding particles are shown in antiparallel motion because such a simplified situation always applies if the collision is analyzed in the centre of mass reference frame. Thus, the deflection angle 0S here indicated is identical to the scattering angle in the centre of mass system 0C, which is associated with the directional change of the relative velocity vr, to be subsequently introduced. Intuition based on physical grounds suggests a rigid relationship between the three parameters r0, 0C and vr.

For the case of azimuthally symmetric scattering, Fig. 3.2, and which here applies because of the specific form of Fc, every ion of mass mb and charge qb moving through the ring of area

dA = 2Tir0dr0 (3.2)

will scatter off particle а-which has a charge of the same sign-through an angle 0C into the conical solid angle element d£2* associated with the shadowed ring of Fig. 3.2; this conical solid angle element is evidently

dQ* =2Ksm(ec)dec. (3.3)

The number of ions which scatter into a solid angle element can change substantially with r0 and vr; it is therefore necessary to look for an angle dependent cross section 0(Q) which here, due to azimuthal symmetry, is only a function of 0c and yields a total scattering cross section CTS according to

4к к

Cs = ^Gs(ec)d2Q = ^(js(ec)d&

That is, as(0c) is a function of the conical solid angle £2* corresponding to a specified 0C and possesses units of bams per steradian (b/sr). Since das represents the differential cross sectional target area dA of Fig.3.2, and all ions entering this

area will depart through the solid angle element dQ*, we may write

N2tu r0dr0 = Nas (вс )dQ* (3.6)

with N denoting a particle number per unit area, which, upon insertion of Eq.(3.3) yields specifically

crs(ec) =

where 0 < 0C < K. Here, the standard absolute-value notation for the Jacobian of a transformation has been incorporated. The process discussed above refers to the so-called Rutherford scattering and CTs(0c) is known as the corresponding differential scattering cross section.

Case of Repulsion

Particle

Case of Attraction

where the parameters introduced are as follows:

0C = scattering angle for particle b off of particle a in the COM (Centre of Mass) system which relates to 0L in the LAB (Laboratory) system according to

cot( eL) = — csc( вс) + cot( вс) ; ma

mr = reduced mass of the two body system = (ma mb ) / (ma + mb ) ;

vr = relative speed of the two particles of interest

= I ve — vj. (3.9c)

The meaning of the impact parameter r0 is unchanged.

Equation (3.8) provides a useful connection between the impact parameter r0, the scattering angle 0C in the COM system, and the relative particle speed vr. For

Charged Particle Scattering simplicity, we may write

d6,=-dr0 r„

jsec2

(3.11)

and hence

V z J dr0 2 _ Го 2 sec і® 1 ° d вс 2K K2) Го Y 2

(3.12)

Then, substitution of this expression in Eq.(3.7) gives

(3.14)

(3.15)

This explicit algebraic form is frequently called the Rutherford scattering cross section.

The more significant information about this charged particle interaction cross section is suggested in Fig. 3.3 and illustrates the following important result: the cross section approaches GS(0C) —> K2/4 for "head-on" collisions (i. e. r0 —» 0) and becomes arbitrarily large for increasingly smaller "glancing" angles (i. e. r0 —» °o).

An informative conclusion about the singularity as 0C —» 0 is that, for example, if there were to exist only two nonstationary charged particles in the universe then some deflection would occur with certainty regardless of how far apart they were; that is, the Coulomb 1/r2 force dependence may become infinitesimally small at sufficient separations but, in principle, it never vanishes. Identifying a "reasonable" length beyond which charge effects can be considered to be unimportant, or even be ignored, is provided by the Debye length concept to be discussed next.

Fig. 3.3: Depiction of the Rutherford cross section as a function of scattering angle. Note GS(0C) —» as 0C —H).