Fundamentals of Plasma Physics

484 Chapter 17. Dusty plasmas

Debye

radius

ion trajectory

b rd

dust

grain

lmfp

Figure 17.1: Electrons and ions may be considered as being collisionless inside asphere of
diameterlmfpsurrounding a dust grain. The trajectory an ion making a grazing collision
is shown;this ion has an impact parameterb.

17.2 Electron and ion currentflow to a dust grain

Because an individual dust grain is much heavier than an electron or an ion, itcan be

approximated as an infinitely massive scattering center for the electrons and ions colliding

with it. In typical dusty plasma laboratory experiments the collisional mean free pathlmfp

greatly exceeds the Debye length. This situation is sketched in Fig.17.1 where a dust grain

with radiusrdis surrounded by an imaginary sphere with diameterlmfpand the Debye

radius is much smaller thanlmfp. Electrons and ions are collisionless inside thelmfp

sphere and so have Keplerian orbits associated with the electrostaticcentral force produced

by the charge on the dust grain. Because the dust grain is shielded by other particles,

the electrons and ions only experience the central force when inside a sphere having the

nominal Debye radius.

Dust grain charging involves calculating effective collision cross-sections for the elec-

trons and ions colliding with the dust grain. A useful benchmark reference for thiscalcu-

lation is the cross-section of charged particles colliding with a neutral dust grain. Because

charged particles incident on a neutral dust grain have straight line trajectories, the cross-

section of a neutral dust grain is just the geometrical area it projectsonto a plane, i.e.,

σgeometric=πr^2 d. The cross-section of achargeddust grain differs fromσgeometricbe-

cause incident electrons and ions are deflected from straight line trajectories by the electro-

static central force produced by the charge on the dust grain. The respectivecross-sections

for ions and electrons will now be calculated and related toσgeometric.

Fig.17.1 shows the impact parameterband trajectory of an ion colliding with a nega-