Transmission of an Electron Density Disturbance Along a Positive Column in a Longitudinal Field

The distribution of ions and electrons in the cross section of a uniform positive column is maintained by the radial motions of these particles. This distribution is designated as "normal." A disturbance of this distribution at some point in the column is followed on the anode side by an asymptotic approach to the normal. In the absence of a longitudinal magnetic field the recovery of a normal distribution occurs within a very short distance, but a longitudinal magnetic field slows down the readjustment by decreasing the radial mobility of the electrons. By making certain simplifying assumptions a theory for the approach of a disturbed column back to normal when the disturbance is cylindrically symmetrical has been worked out. The distribution is developed in a series of zero order Bessel functions, and it is found that the first term, which corresponds to the normal distribution, approaches a constant amplitude along the column whereas successive terms have successively greater space decrements. The decrement of the $i$th term approaches the constant value $\frac{({x_i}^2-{x_1}^2)\alpha (T_e+T_p)}{(1-\alpha )T_e{a}^2(-\epsilon )},$ where $x_i$ is the $i$th root of $J_0\mathrm{}\left(x\right)=0\mathrm{}$, $\alpha$ is the factor giving the reduction in transverse electron mobility, $T_e$ and $T_p$ are the electron and positive ion temperatures, $a$ is the radius of the column and $\epsilon$ is $\left(\frac{e}{k{T}_e}\right)\left(\frac{\partial V}{\partial z}\right)$. When compared with the experimental results of Cummings and Tonks, the theory calls for a reduction of the second term of the series expansion to 50 percent in 12.5 cm of arc length whereas experiment gave 33 percent.