Collisional Sheath Law

For a high-pressurehigh-voltage,collisional sheath, the ion drift velocity can be written as v=4;E where u4; = e/Mvmiis the constant ion mobility, with vmi a constant ion-neutral momentum trans-fer frequency.

(a) Using particle conservation and Poisson's equationderive the high-pressure, collisional Child law for ions (6.5.8).

(b) For an argon discharge withN; = (330p)-1 cm, with the pressure p in Torr and p = 10 Torr, calculate the sheath thickness s for n,= 10cm3 at thesheath edge, T=2V,T;=0.026 V, and Vo =100 V across the sheath.Assume a constant vmi= uB/;.Compare this s to that obtained for the same discharge parameters from the collisionless Child law.

6.4. Collisional Sheath Law For a high-pressure, high-voltage, collisional sheath, the ion drift velocity can be written as o = /; E, where M; = e/Mymi is the constant ion mobility, with vm; a constant ion-neutral momentum trans- fer frequency. (a) Using particle conservation and Poisson's equation, derive the high- pressure, collisional Child law for ions (6.5.8). (b) For an argon discharge with A; = (330p)" cm, with the pressure p in Torr and p = 10 Torr, calculate the sheath thickness s for n, = 10" cm at the sheath edge, T. = 2 V, T; = 0.026 V, and Vo = 100 V across the sheath. Assume a constant Vmi = #B/Aj. Compare this s to that obtained for the same discharge parameters from the collisionless Child law.6.6 ELECTROSTATIC PROBE DIAGNOSTICS 185 position x = s, we obtain 3/2 1/2 3/2 Jo = EO (6.5.7) TIM Equation (6.5.7) gives a collisional form of the Child law for the regime in which Ai is independent of ion velocity. We note that the current scales the same with voltage, but differently with sheath spacing, than for the collisionless case. For a fixed Jo and Vo. the sheath width scales as soc A, and therefore weakly decreases as the gas pressure is increased. Alternatively to our relation (6.5.7), we could equally well have chosen the higher pressure regime to make the calculation, taking Vm and hence , independent of vel- ocity. In this case, a similar integration procedure leads to the result (Problem 6.4) Jo = EOMi (6.5.8) We note here that the scalings of Jo with both Vo and s in (6.5.8) are different from (6.5.7). More detailed use of these various relations will be given in Chapter 11. where we use sheath physics in a complete description of capacitive discharges