Fundamentals of Plasma Physics

12 Chapter 1. Basic concepts

assignment 1, this chapter) that the scattering angleθis given by

tan

(

θ

2

)

=

qTqF

4 πε 0 bμv 02

∼

Coulombinteractionenergy

kineticenergy

(1.12)

whereμ−^1 =m−T^1 +m−F^1 is the reduced mass,bis the impact parameter, andv 0 is the
initial relative velocity. It is useful to separate scattering events (i.e., collisions) into two
approximate categories, namely (1) large angle collisions whereπ/ 2 ≤θ≤πand (2)
small angle (grazing) collisions whereθ <<π/ 2.
Let us denotebπ/ 2 as the impact parameter for 90 degree collisions;from Eq.(1.12) this
is

bπ/ 2 =

qTqF

4 πε 0 μv^20

(1.13)

and is the radius of the inner (small) shaded circle in Fig.1.3. Large angle scatterings will
occur if the test particle is incident anywhere within this circle andso the total cross section
foralllarge angle collisions is

σlarge ≈ πb^2 π/ 2

= π

(

qTqF

4 πε 0 μv^20

) 2

. (1.14)

Grazing (small angle) collisions occur when the test particle impingesoutsidethe shaded
circle and sooccur much more frequentlythan large angle collisions. Although each graz-
ing collision does not scatter the test particle by much, there are far more grazing collisions
than large angle collisions and so it is important to compare thecumulativeeffect of graz-
ing collisions with thecumulativeeffect of large angle collisions.
To make matters even more complicated, the effective cross-section of grazing colli-
sions depends on impact parameter, since the largerbis,the smaller the scattering. To take
this weighting of impact parameters into account, the area outside the shaded circle is sub-
divided into a set of concentric annuli, called differential cross-sections. If the test particle
impinges on the differential cross-section having radii betweenbandb+ db,then the test
particle will be scattered by an angle lying betweenθ(b)andθ(b+ db)as determined by
Eq.(1.12). The area of the differential cross-section is 2 πbdbwhich is therefore the effec-
tive cross-section for scattering betweenθ(b)andθ(b+ db).Because the azimuthal angle
about the direction of incidence is random, the simple average ofNsmall angle scatterings
vanishes, i.e.,N−^1

∑N

i=1θi= 0whereθiis the scattering due to thei

thcollision andN

is a large number.
Random walk statistics must therefore be used to describe the cumulativeeffect of
small angle scatterings and so we will use thesquareof the scattering angle, i.e.θ^2 i,as the
quantity for comparing the cumulative effects of small (grazing) and largeangle collisions.
Thus, scattering is a diffusive process.
To compare the respective cumulative effects of grazing and large angle collisions we
calculate how many small angle scatterings must occur to be equivalentto a single large
angle scattering (i.e.θ^2 large≈ 1 );here we pick the nominal value of the large angle scat-

tering to be 1 radian. In other words, we ask what mustNbe in order to have

∑N

i=1θ

2
i≈^1
where eachθirepresents an individual small angle scattering event. Equivalently,we may