Experimental measurements of the $m=1$ unstable diocotron mode

The first experimental observation of an exponentially growing $m=1$ unstable diocotron mode on a trapped, magnetized, partially hollow electron column was in 1990\footnote{C. F. Driscoll, {\it Phys. Rev. Lett.} {\bf 64}, 645 (1990).}, and subsequent measurements examined the effect of end shape curvature\footnote{A. A. Kabantsev and C. F. Driscoll, Non-Neutral Plasmas III, 208 (1999).}. We present new experimental measurements of the $\mu$ (initial column hollowness parameter) and $\kappa$ (end shape curvature parameter) dependences of growth rates of the $m=1$ instability. Measurements of the perturbed longitudinal temperatures of the electron column have been incorporated into the present experiments. We have experimentally established a $\mu^{4/3}$ scaling of the growth rates on the column hollowness $\mu$ for $\mu<2.2$. Our results of growth-rate scaling on $\kappa/\mu$ for relatively large $\mu$ and $\kappa$ are in agreement with theoretical predictions for the instability near onset $(\kappa,\mu)\rightarrow 0$ by Finn et al.\footnote{J. M. Finn, D. del-Castillo-Negrete and D. C. Barnes, {\it Phys. Plasmas} {\bf 6}, 3744 (1999).}.