n ve=- P (5) NOTE:This leaves the E field out. To include it, just replace Vne by Vn.+ ene - E Define the nondimensional factor ÷、eBc (Hall parameter (6) here o=eB is the cyclotron frequency for electrons Then n ve= 1+ (-D。vn-阝xDn Of these two terms, the second is perpendicular to both b and Vn,, and is called the"VP. xb drift". The main interest is on the first term which is along -Vn, as a regular diffusion We see that this"cross-field diffusion"is governed by n Ve =-D,Vne, witl D So, a high Hall parameter B can greatly reduce diffusion, compared to that in the absence of a magnetic field. High B means both, high B and/or low collision frequency In an ion engine, with T. 4ev =46400 K, the e-n and e-i cross-section are roughly Q。=10m2 Q。=4×1048m2, and 8 kTe e=1.34×10°m/ If also n=28×103m3, 16.522, Space P pessan Lecture Prof. Manuel martinez Page 9 of16.522, Space Propulsion Lecture 15 Prof. Manuel Martinez-Sanchez Page 9 of 11 e e e e 2 2 e e e e e e 2 e e kT ekT - n + n ×B m m nv = eB 1+ m ∇ ∇ ν ν ⎛ ⎞ ⎜ ⎟ ν ⎝ ⎠ JG JG (5) NOTE: This leaves the E G field out. To include it, just replace ∇ne by e e e en n+ E kT ∇ G Define the nondimensional factor c ee e eB β m ω ≡ ≡ ν ν (Hall parameter) (6) where c e eB = m ω is the cyclotron frequency for electrons. Then e ee ee e 2 ( ) 1 n v = -D n - ×D n 1+ ∇β ∇ β JG G (7) Of these two terms, the second is perpendicular to both, B JG and ∇ne , and is called the “ ∇P ×B e JG drift”. The main interest is on the first term, which is along e - n∇ , as a regular diffusion. We see that this “cross-field diffusion” is governed by e n v = -D n e e ⊥∇ JG , with e 2 D D = 1+ ⊥ β (8) So, a high Hall parameter β can greatly reduce diffusion, compared to that in the absence of a magnetic field. High β means both, high B and/or low collision frequency. In an ion engine, with T = 4eV = 46400 K, e the e-n and e-i cross-section are roughly -19 2 Q 10 m , en  -18 2 Q 4×10 m , ei  and e 6 e e 8 kT c = 1.34×10 m/s. π m  If also 17 -3 n 2.8×10 m , e