Velocity Distribution and Temperature f(v,t) dv is the number of particles per unit volume in the velocity space interval v, v + dv. n(t) = ∫ f(v,t) dv (1.1.1-1) For thermal equilibrium, this distribution is the so-called Maxwellian distribution f(v) = A.exp(-1/2mv2/(kBT)) (1.1.1-2) where kB is Boltzmann’s constant = 1.38x10-23 JK-1. Integrating as in (1.1.1-1) A = n(m/(2kBT)0.5 (1.1.1-3) so the Maxwellian distribution is f(v) = n(m/(2kBT)1.5.exp(-1/2mv2/(kBT)) (1.1.1-4) Now the average kinetic energy of the particles in the distribution is Eav = ∫ f(v) (1/2mv2) dv/∫ f(v) dv = 3kBT/2 (1.1.1-5) This identifies T as the temperature, so that the average energy is kBT/2 per degree of freedom. In plasma physics, it is customary to quote temperatures in energy units corresponding to kBT e.g. for kBT = 1 eV = 1.6x10-19 J. Hence T = 1.6x10-19/(1.38x10-23) = 11,600 K.

Debye Shielding 1/2 Poisson’s equation in one dimension is (Z = 1 i.e. singly charged ions): 0d2/dx2 = -e(ni-ne) (1.1.2-1) If the density far away from the grid is n then ni = n. (1.1.2-2) In the presence of the potential the electrons have a velocity distribution f(u) = A.exp(-(1/2mu2 +q)/(kBTe)) (1.1.2-3) This is intuitively clear as this means that there are fewer electrons where the potential energy (q) is large, since not all electrons have the energy to get there. Integrating this distribution over u and setting q = -e, and noting that ne  n. as x  , ne = n.exp(e)/(kBTe)) (1.1.2-4) Substituting for ni and ne into (1), we have; 0d2/dx2 = e n.(exp(e)/(kBTe)) – 1) (1.1.2-5)

(x) = 0exp( -x/D) (1.1.2-7) Debye Shielding 2/2 Expanding the exponential in a Taylor series and keeping only the linear term (which can be justified only a posteriori because very near the grid (e)/(kBT) can be large. (the contribution of this region is small as the potential falls rapidly there), and defining a length D (the Debye Length) as; D = (0 kBTe/ (nee2))0.5 (1.1.2-6) then the solution of the potential (1.1.2-5) is (x) = 0exp( -x/D) (1.1.2-7) Thus the characteristic dimension of the sheath is the Debye length. We are now in a position to define “quasi-neutrality”. This obtains if the characteristic dimensions L of a system are much larger than the Debye length, D << L (1.1.2-8)

Collective behaviour requires ND >> 1 (1.1.3-2) Criteria for Plasmas Quasineutrality D << L (1.1.2-8) Collective behaviour requires ND >> 1 (1.1.3-2) where ND = n4/3D3 (1.1.3-1) Electromagnetic effects dominate collisions with neutrals  >> 1 (1.1.3-3) where  is the mean time for collisions of charged particles with neutrals and  is the characteristic frequency of typical plasma oscillations