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44 Chapter 3 Most of the charged particles in electric thrusters have a Maxwellian velocity distribution, which is the most probable distribution of velocities for a group of particles in thermal equilibrium. In one dimension, the Maxwellian velocity distribution function is m 1/2 mv2 f ( v ) = 2 k T e x p 2 k T , ( 3 . 4 - 1 ) where m is the mass of the particle, k is Boltzmann’s constant, and the width of the distribution is characterized by the temperature T. The average kinetic energy of a particle in the Maxwellian distribution in one dimension is 1mv2 f(v)dv Eave = – 2 . (3.4-2) f(v)dv – By inserting in Eq. (3.4-1) and integrating by parts, the average energy per particle in each dimension is Eave = 12 kT . (3.4-3) If the distribution function is generalized into three dimensions, Eq. (3.2-8) becomes where u, v, and w represent the velocity components in the three coordinate axes. The average energy in three dimensions is found by inserting Eq. (3.4-2) in Eq. (3.4-4) and performing the triple integration to give m 3/2 m f(u,v,w)= 2 kT exp 2kT u +v +w , (3.4-4) Eave = 23 kT . The density of the particles is found from n = + nf (v) dv – (3.4-5) (3.4-6) (222) + m3/2 (222) mu+v+w = – n 2 kT exp 2kT dudvdw.PDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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