Probes in the Orbital Motion-Limited (OML) Regime

Probes in the Orbital Motion-Limited (OML) Regime Ref. Laframboise, J. G., and L. W. Parker. "Probe Design for Orbit-limited Current Collection." Physics of Fluids 16, no.5 (1973): 629-36. For 2-D or 3-D objects collecting ions or electrons, the perturbation to the background plasma dies away over distance comparable to the object’s size, and beyond that one can assume equilibrium conditions (this not so for plasma, 1-D, conditions). In equilibrium,  at ∞ : f (w  ) = n∞ m 2πkT∞ 3/2  E∞ − kT e E∞ ∞  1 2 2 2 = m(wx + wy + wz ) 2 (1) If there are no collisions (mf p >> object size), then Vlasov’s equation applies, and it states that f does not change following a particle’s trajectory. Hence f is a function of any of the constants of motion, and in particular, any function of the energy E satisfies Vlasov’s equation. The particular function  f (w  ) = n∞ m 2πkT∞ 3/2 E e− kT∞ (2) is one such, and so, if a trajectory connects to ∞, where (1) applies and serves as initial condition, this is the solution. Since E = 12 mw2 + qφ, qφ − kT f (w) = n∞ e  ∞ m 2πkT∞ 3/2 mw2 e− 2kT∞ (3)  ’s to occur. This This is valid for any w  which actually occurs, but there is no need for all w may happen for two reasons: (a) For some w  ’s (at a given r) the energy may be negative, in which case the trajectory cannot come from ∞ (where E = E∞ = 12 mw2 + qφ). (b) Even if E > 0, some trajectories still do not connect to ∞, because they begin and end on the body surface. This is very dependent on geometry and field distribution. By definition, the OML regime is that for which (b) does not happen on the body’s surface, i.e., any energetically possible trajectory reaching the body originates at ∞. In that case, it is very easy to calculate the collected current for any shape of the collector. The only distinction to be made is whether the collector is 2-D or 3-D; this is because in 2-D the w  component parallel to the axis (w ) is separately conserved, and so we need to impose E⊥ > 0 instead of E > 0 (if we imposed only E > 0, E⊥ + E > 0, E⊥ > −E , we would be 2 admitting some trajectories with 0 > (w⊥ )∞ > −(w2 )∞ , which cannot be). 1. Repelling Bodies 1 Use (3) on the body surface, whenever ⎧ 1 ⎪ ⎪ mw2 + qφp > 0 ⎪ ⎪ ⎨2 (3 − D) (4) ⎪ ⎪ ⎪ 1 ⎪ 2 ⎩ mw⊥ + qφp > 0 2 (2 − D) For the repulsive case q and φp have the same sign, and (4) or (5) is automatic, so there is no restriction. Hence, in this case, shape does not matter at all. We have an arriving flux ˚ J = |wn |f (w)d3 w (5) Using a local polar coordinate system, and counting all one-sided arriving velocities J = n∞ e qφ ∞ − kT p  m 2πkT∞ 3/2 ˆ∞ dwe w = qφp n∞ − kT J = e ∞ 2π 2kT∞ m  1/2 2πkT∞ m 2 2kT∞ m π  2 3 0  mw2 = x 2kT∞ ˆπ/2 ∞w cosθsinθ2πdθ mw − 2kT 0   ∞ ´ 0 2π 12 =π  x3/2 e−x 12 x−1/2 dx=π 2 2kT∞ m  2  1 ∞ Γ(2)= 2π1 2πkT 2 m 1 x1/2 1/2 qφp n∞ c̄∞ − kT e ∞ J = 4  c̄∞ = 8 kT∞ π m  (6) Notice this gives the same current density at any point on the (equipotential) body, for any body shape (in the OML regime). 2. 3-D Attracting Body (electrons to φp > 0, ions to φp < 0) 1 mw2 + qφp > 0 2 Impose and since φp < 0, now, w > J = n∞ e qφ ∞ − kT p  2|qφp | m m 2πkT∞ (7) and repeat calculation 3/2 ˆ∞ π  2  1 2 2kT∞ m 2 − ∞ ´ mw2 2qφp 3 − 2kT ∞ dw w e m   −qφp >0 kT∞ xe−x dx= 12 |qφ | ∞ 1+ kT p qφp e kT∞ −qφp mw2 > 2kT∞ kT∞ x = ˆ∞ −x −xe−x |∞ a xe dx = a ˆ∞ + −a e−x dx = −e−x (x + 1)|∞ a = (a + 1)e a   |qφp | n∞ c̄∞ 1+ J = 4 kT∞ (8) 3. 2-D Attracting Body 1 2 mw⊥ + qφp > 0 , 2 and now use a local cylindrical system at the surface: Impose ˆ∞ 3/2 J = ˆπ/2 − n∞ (..) w⊥ = q e 2 −2|qφ | mw⊥ p 2kTinf ty w⊥ > 2|qφp | m ˆ∞ (w⊥ cosθ)(w⊥ dw⊥ dθ) θ=− π2 2|qφp | m (9) mw2  dw ∞ − 2kT e −∞   1/2 2πkT∞ m  J = 2n∞ 2πkT∞ m −1  ˆ∞ dw⊥ q   2kT∞ m ˆ∞ 2 −x2 xe a 1 1 2 dx = − xe−x |∞ a + 2 2  ˆ∞ −x2 e 2|qφp m 3/2 ´∞ r |qφp | kT∞ 2 x2 e−x dx √ 1 −a2 1 π dx = ae erf c(a) + 2 2 2 a     √   |qφ | 2 2kT∞ π |qφp | qφp  − kT p  J = n∞ e  kT∞ e ∞ + 2 erf c kT∞ 2π m    √ |qφ | p π kT |qφp | n∞ c̄∞ 2 |qφp | √ e ∞ erf c J = + 4 2 kT∞ π kT∞ |qφp | kT∞ (10) For strong potentials (|qφp | >> kT∞ ), √ π x2 2 e erf c(x) = ex 2 ˆ∞ −t2 e x x2 ˆ∞ dt = e   −t2  ˆ∞ dt e 1 1 2 2 −t x ∞ −t2 − 2 |x + e − d(e ) = e − → 2t 2t 2t 2x x x 3 and so the second term (∼ J 1 ) 2x is negligible with regard to the first (x):  n∞ c̄∞ 4 J n∞ π |qφ∞ | 2 n √ = √∞ kT∞ π 2 π 2|qφp | m  8 kT∞ π m  |qφp | kT ∞  (|qφp | >> KT∞ ) (11) (again, anywhere on the 2-D attracting body) Examples: 1. Spherical probe in OML     ⎧ |e|φe nex |e|φp 2 ⎪ ⎪ − c̄i e kTi ⎨ 4πR 4 − c̄e 1 + kT e     I = |eφe | ⎪ nex eφp ⎪ 2 ⎩ 4πR − c̄e e kTe − c̄i 1 + 4 kTi 2. Cylindrical probe in OML (with |eφp | >> kT ) ⎧   ⎪ 2(eφp ) ⎪ ⎪ −∼0 ⎪ ⎨  2RLnex me I =   ⎪ ⎪ 2(eφp ) ⎪ ⎪ −∼0 ⎩  2RLnex mi 4 (φp > 0) (φp < 0) (φp > 0) (φp < 0)