## 2. Electric field generation by a neutral beam in an unresponsive plasma## 2.1. Infinite homogeneous beam caseIn this case the background plasma only exerts a collisional drag on the beam particles so that electrodynamic effects are confined to the beam. Since the system is infinite and homogeneous and the beam is purely , any arising must be uniform in space so that so that no magnetic field exists if at . Even then, this purely temporal problem is complicated by the large range of timescales involved - from the electron plasma period up to the proton collision time. Since the proton collision time is long compared to the electron collision time and extremely long compared to the electron plasma period, we first simplify the problem by considering the proton speed to remain constant and the electron speed to vary by only a small fraction of the initial beam (and proton) speed so that the electron collision term remains constant. That is, we start from the equations (in a mean particle description of plasma processes): where Based on our previous remarks, these equations simplify to: where is the constant proton speed. These equations can be rewritten into one for the (oscillatory)
motion of the relative proton-electron velocity the solution of which is where is the angular beam electron plasma frequency with and since . The corresponding electric field can be expressed as It follows that the electrons here are fully dragged by the protons
due to the mean electric field where Here is the Coulomb logarithm,
is the density of the background gas, A more complete numerical solution of the full analytic mean
particle equations, including both and
and their dependences on velocity, shows
exactly the same behaviour with the electrons oscillating about the
protons as they decelerate together due to the combined dragged force
, the electric field increasing as the
velocities fall and the ## 2.2. Semi-infinite beam caseWe now consider the more realistic case of injection of a semi-infinite neutral beam rather than an infinite one `launched' homogeneously. When such a neutral beam is injected and penetrates into a dense gas then, due to collisions, its electron component is again decelerated first. This deceleration force is equivalent to that of an electric field of magnitude given by The beam electron velocity initially (before where is the neutral beam density initially
equal to electron and proton beam densities everywhere, and the
thickness of the charged layers is . This
electric field continues to grow, with the charge layer thickness
## 2.3. Acceleration of electronsThough the charge separation induced by differential collisional
drag is very small, the resulting In Simnett and Haines (1990), a calculation was performed to find the maximum possible energy of an accelerated runaway electron. The first step was to find , the potential due to the assumed double layer. This step is questionable given their subsequent analysis since, as we shall show, this potential cannot remain unaffected if significant runaway occurs. A second less fundamental objection is that the in the denominator of the collision term in their Eq. (8) should actually be . The consequence of this last error is to reduce the maximum possible electron energy to 75% of the proton energy rather than 100%, as they stated. Runaway electrons with high energies are in principle possible in
an unresponsive plasma, if a particle has a speed greater than the
average beam speed. The key issue now is how many electrons can be
involved in this. Till now we considered only one test electron, or
more generally a number sufficiently small not to affect the
electrodynamic equations we used to find the neutral beam generated
Any runaway electrons accelerated in this field would effectively result in a double layer of charges with associated electric field where is the density of runaway electrons,
and is the length of the runaway electron
pulse. In order for the runaways not to modify the where is the beam `plasma parameter' (number
of beam electrons in a Debye sphere). Since We conclude, therefore, that runaway acceleration of electrons,
whatever their origin, in the © European Southern Observatory (ESO) 1998 Online publication: March 3, 1998 |