Thermodynamic Analogy for Electrons Interacting with a Magnetic Nozzle

The polytropic index of free electrons expanding in a magnetic nozzle of varying strength is experimentally investigated under a nearly zero electric field, allowing all the electrons to escape to the axial boundary and never return to the source. The measurements clearly demonstrate a continuous change in the polytropic index from adiabatic $5/3$ for a strong magnetic field to isothermal unity for a weak magnetic field, showing that the polytropic index depends on the magnetic field strength. It is shown that the cross-field diffusion and the resultant plasma loss out of the magnetic nozzle effectively reduce the polytropic index. The azimuthal current induced in the plasma is diamagnetic, does work on the magnetic nozzle, and contributes to the reduction of the electron internal energy during the expansion.

Figure 1

(a) Schematic diagram of the experimental setup. (b) Calculated magnetic field strength $B_z$ on axis for $I_B=1\mathrm{A}$.

Figure 2

(a) Typical EEPFs measured along the axis for $I_B=12\mathrm{A}$ together with the noise level (dashed line) and (b) same EEPFs data extracted for the 0–20 eV range. (c) Electron density $n_e$ and (d) effective electron temperature $T_{\mathrm{eff}}$ calculated over the EEPFs 0–20 eV energy range for $I_B=4\mathrm{A}$ (open circles), 8 A (filled triangles), and 12 A (open squares). The electron density $\widehat{n}$ and temperature $\widehat{T}$ are normalized at $z=15\mathrm{cm}$ and plotted in Figs. 2 and 2, respectively. The errors corresponding to the standard deviation in $n_e$, $T_{\mathrm{eff}}$, $\widehat{n}$, and $\widehat{T}$ are about $\pm 0.9\%$, $\pm 0.5\%$, $\pm 1.2\%$, and $\pm 0.7\%$, respectively.

Figure 3

(a) Radial profiles of $p_e$ measured at $z=20\mathrm{cm}$, together with their Gaussian fittings (solid lines). The error in $p_e$ is about $\pm 1\%$. (b) The HWHM obtained from the Gaussian fittings as a function of $I_B$. (c) Radial profiles of $V_p$ at $z=20\mathrm{cm}$. Typical error bar is obtained by analysis of the probability density function for $V_p$ in the Supplemental Material [27].

Figure 4

(a) Typical axial profiles of $\gamma$ calculated from the measured $\widehat{n}$ and $\widehat{T}$. The error bar is estimated according to the propagation of error formula. (b) The polytropic index $\overline{\gamma }$ (filled squares) averaged over the axial region of $z=15–45\mathrm{cm}$ as a function of $I_B$, together with the maximum (${\gamma }_{\max }$, open triangles) and minimum (${\gamma }_{\min }$, crosses) values. (c) Plasma-induced magnetic field $\mathrm{\Delta }{B}_z$ (open circles) measured at $z=20\mathrm{cm}$ as a function of $I_B$. The filled squares show the right-hand side term of Eq. (2).