Diagnosing Energy and Angular Momentum Deposition Using Diocotron Mode Frequency Shifts

Monitoring frequency variations$ f_1 (t)$ of a small amplitude $m_\theta =1$ diocotron mode in pure electron plasmas determines the energy and angular momentum deposited by a variety of damped plasma waves. The finite length and temperature model\footnote{Fine, Driscoll, Phy. Plas {\bf 5} 601 (1997)} of $f_1$ determines the frequency variations due to plasma radius $R$ and temperature $T$, arising from thermal pressure on the ends. For energy $\Delta T$ and angular momentum input resulting in $\Delta R$, the model gives $\Delta f_1 / f_1 \approx 1.2 (R_w / L) [\Delta T / e^2 N - \Delta R / R ]$. Typical plasma and wall sizes give $R / R_w ~\sim 0.3$, $L / R \sim 30$, so $R_w / L \sim 0.1$. With accuracy $\Delta f_1 / f_1 \leq 10^{-4}$, we have confidently measured the energy deposits $( \Delta T )$ from Landau damped $m_\theta = 0$ plasma waves with $\Delta n / n \leq 10^{-2}$; as well as both the energy deposits $( \Delta T )$ and angular momentum $( \Delta R )$ deposits from collisionally damped $m_\theta = 1$ trapped-particle diocotron modes with $D / R_W \leq 10^{-2}$. In prior work, the $m=2$ frequency has been used to diagnose the plasma expansion, as $2 \dot{R} / R = - \dot{f}_2 / f_2$. Together, these two modes give a rather complete non-destructive diagnostic.