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Basic Plasma Physics 61 v = q E – kT n . (3.6-25) m m n The coefficients of the electric field and the density gradient terms in Eq. (3.6-25) are called the mobility, and the diffusion coefficient, μ = q [m2/V-s], m D = kT [m2/s]. m (3.6-26) (3.6-27) (3.6-28) These terms are related by what is called the Einstein relation: μ= q D. kT 3.6.2.1 Fick’s Law and the Diffusion Equation. The flux of diffusing particles in the simple case of Eq. (3.6-25) is = n v = μn E – D n . (3.6-29) A special case of this is called Fick’s law, in which the flux of particles for either the electric field or the mobility term being zero is given by = –D n . (3.6-30) The continuity equation, Eq. (3.5-10), without sink or source terms can be written as n + = 0 , ( 3 . 6 - 3 1 ) t where represents the flux of any species of interest. If the diffusion coefficient D is constant throughout the plasma, substituting Eq. (3.6-30) into Eq. (3.6-31) gives the well-known diffusion equation for a single species: n – D 2n = 0 . (3.6-32) t The solution to this equation can be obtained by separation of variables. The simplest example of this is for a slab geometry of finite width, where thePDF Image | Fundamentals of Electric Propulsion: Ion and Hall Thrusters
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