By Robert Motley (Auth.)
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31]. They have solved the Poisson equation for azimuthally symmetric electrostatic waves, satisfying the wave equation y 2 j/_£ 2 e T/ = o in situations where the dielectric constant e is given by kinetic theory e=\-(G>v2/k2v2)Z'(Gj/kv) (3-7) where Z is the plasma dispersion relation. The dispersion curves obtained with a parabolic profile [n = n0(l —r2/R2)] are shown in Fig. 3-3. The curves are sensitive to the parameter GJpR/v or the plasma radius in Debye lengths. The most precise values of density can be deduced for large columns.
The growth rate of the instability may be obtained from the real part of Eq. 2/4ri ^4"14j One readily notes that the growth rate depends critically on the finite cyclotron radius parameter b. The driving term of Eq. (4-14), which is linear in b, is derived from the inertia term in the ion momentum equation. The (negative) damping term, proportional to b2, comes from the ion viscosity term. Numerical calculations, shown in Fig. 4-2, demonstrate a characteristic feature of the instability—maximum growth occurs for an intermediate range of parallel wavelengths.
Plasma is normally allowed to pass through access holes cut in the end walls of the cavity, if the hole size is not small compared to the cavity diameter Dc, power loss through the holes may degrade the Q of the cavity. Sleeves brazed to the ends of the cavity will improve the Q, provided the sleeve diameter Ds is less than the cutoff wavelength of the cylindrical waveguide formed by the sleeves (Xc = 2nDj2A05). Fringe fields generated by the access holes and the sleeves may lead to error in the calculation of the plasma density in terms of the frequency shift of the resonator.