Fundamentals of Plasma Physics

488 Chapter 17. Dusty plasmas

However, using Eq.(17.21) this becomes

(1+ψd)

√

meTi

miTe

exp(ψdTi/Te)=1−Pψd, (17.26)

a transcendental equation relatingψdandP.

Equation (17.26) shows that, for a given temperature ratioTi/Teand given mass ratio

me/mi,the normalized dust grain surface potentialψdis a function ofP,a dimension-

less parameter incorporating the geometrical information characterizing the dust grains in

the dusty plasma. Becauseψdappears non-algebraically in Eq.(17.26), it is not possible to

solve forψd(P)without resorting to numerical methods. However, solving the inverse rela-

tion, namelyP(ψd),is straightforward becausePoccurs only once in Eq.(17.26). Solving

forPgives (Havnes, Goertz, Morfill, Grun and Ip 1987)

P =

1

ψd

−

(

1+

1

ψd

)√

meTi

miTe

exp(ψdTi/Te). (17.27)

Figure 17.2: Plots oflogPandαv. logψdforTe= 100TiandTe=Ti. Logarithms are
base 10 and ion mass is 40 amu. SincePis proportional to the dust grain density the right
hand side of these curves, i.e., smallP,corresponds to the limit of low dust grain density.