Bohm velocity

In order to be consistent with the Bohm criterion for ions, the sheath edge is defined as the point where the ions have been accelerated (presumably by the presheath electric field. Fig. 5) to the Bohm velocity, i.e. the presheath is included as part of the bulk plasma. The Bohm flux also provides a boundary condition (applied at the wall because of the thinness of the sheath) for the positive ion continuity equation. The negative ion density is assumed zero at the walls. 

The critical parameter that controls ion modulation in collisionless RF sheaths is WT+, where co is the frequency of the applied field, and is the ion transit time through the sheath. The ion transit time can be estimated by assuming a collisionless Child-Langmuir sheath (Eq. 54), and neglecting the velocity of ions entering at the sheath edge (Bohm velocity). The result is [181]... 

Solution of (3-234) can exist only if its right-hand side is positive. Expanding (3-234) to the second order in a Taylor series shows that the sheath can exist only if the initial ion velocity exceeds the critical one, known as the Bohm velocity, mb ... 

In Equation 4.56, the real quantities p, v, and j are the charge density, velocity field, and current density, respectively. The above equations provide the basis for the fluid dynamical approach to quantum mechanics. In this approach, the time evolution of a quantum system in any state can be completely interpreted in terms of a continuous, flowing fluid of charge density p(r,t) and the current density j(r,t), subjected to forces arising from not only the classical potential V(r, t) but also from an additional potential VqU(r, t), called the quantum or Bohm potential the latter arises from the kinetic energy and depends on the density as well as its gradients. The current... 

If the velocity U of an electron within the beam is constant outside the solenoid, the variation of the vector potential A as a function of time in the medium, and thus also in the solenoid, will induce a modification of the phase, as indicated by the equations written above. This will produce a modification of the boundary conditions on the boundary of the solenoid for the quantities a and b. We must also stress that the modification of the vector potential outside the solenoid is generated by either an external or an internal source feeding the solenoid. This can explain the existence of the Aharonov-Bohm effect for toroidal, permanent magnets. The interpretation of the Aharonov-Bohm effect is therefore classic, but the observation of this effect requires the principle of interference of quantum mechanics, which enables a phase effect to be measured. 

The causal interpretation of quantum theory as proposed by De Broglie and Bohm is an extension of the hydrodynamic model originally proposed by Madelung and further developed by Takabayasi [36]. In Madelung s original proposal R2 was interpreted as the density p(x) of a continuous fluid with stream velocity v= VS/rri. Equation (5) then expresses conservation of fluid, while (6) determines changes of the velocity potential S in terms of the classical potential V, and the quantum potential... 

The first is the Bohm condition, affecting the radial velocity of the ions, at the sheath of the anode the second... 

With a Bohm ion velocity at the wall, the condition of zero total current is written as follows, still without secondary electron emission ... 

With a Bohm ion velocity [BOH 49] and a secondary electron emission rate at the wall cr, we have ... 

This is the Bohm-Stavev relation [4.70]. Since Ep = igVp/2, where is the electron mass and Vp the Fermi velocity, we can write... 

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