# Special Plasmas

• Francis F. Chen

## Abstract

The concepts of plasma physics have so far been couched in terms of well-behaved, quasineutral plasmas, but there are other plasmas with special properties. Particle accelerators have a single species, but the kinematic effects are so large that the collective effects are not important. It is possible to generate single-species plasmas at low densities such that the electric fields are manageable. Ronald Davidson, Malmberg and O’Neil, Dan Dubin, and others have developed this interesting topic. Consider an infinite cylinder of electrons of uniform density n0 in a uniform coaxial magnetic field B (Fig. 9.1). A large electric field $$\mathbf{E}={E}_r(r)\widehat{\mathbf{r}}$$ will arise, (where Er is negative), and a typical fluid element of electrons will drift in a circular orbit, since everything is azimuthally symmetric. Those with small, off-axis orbits will drift around the axis in “diocotron” orbits, as shown at the right. We wish to calculate the rotation frequency ωr:
$${\omega}_r(r)\equiv {v}_{\theta }(r)/r.$$
In equilibrium, the inward and outward forces will balance:
$${\displaystyle \begin{array}{l}m{v}_{\theta}^2(r)/r=-e{E}_r(r)-e{v}_{\theta }(r)B\\ {} mr{\omega}_r^2=-e{E}_r-e{v}_{\theta }B=- eBr{\omega}_r-e{E}_r\end{array}}$$
The radial E-field is found from Poisson’s equation:
$${\displaystyle \begin{array}{l}\frac{1}{r}\frac{d}{dr}\left(r{E}_r\right)=-e{n}_e/{\varepsilon}_0\\ {}r{E}_r=\frac{-e{n}_e}{\varepsilon_0}{\int}_0^r{r}^{\prime }d{r}^{\prime }=\frac{-e{n}_e{r}^2}{2{\varepsilon}_0}\\ {}{E}_r=-\frac{r}{2}\frac{e{n}_e}{\varepsilon_0}=-\frac{r}{2}\frac{m}{e}{\omega}_p^2 r\le a\end{array}}$$
Equation (9.2) then becomes
$${\displaystyle \begin{array}{l}{\omega}_r^2+{\omega}_c{\omega}_r+\frac{e{E}_r}{mr}={\omega}_r^2+{\omega}_c{\omega}_r-\frac{1}{2}{\omega}_p^2=0\\ {}{\omega}_r=\frac{1}{2}{\omega}_c\left[-1\pm {\left(1-2{\omega}_p^2/{\omega}_c^2\right)}^{1/2}\right].\end{array}}$$
Since ωr is independent of radius, we see that such an pure electron plasma rotates as a solid body. When $${\omega}_p^2/{\omega}_c^2>\frac{1}{2}$$, there is no solution; otherwise, there are two solutions. For $$2{\omega}_p^2/{\omega}_c^2<<1$$, expanding the square root yields the frequencies $${\omega}_r\approx -{\omega}_c$$, and the lower frequency
$${\omega}_r\approx {\omega}_p^2/2{\omega}_c\equiv {\omega}_D.$$
Called the diocotron frequency, ωD has an orbit like the one at the right in Fig. 9.1.