Excerpt from Zaveryaev et al

The theory of probes in general is discussed in Refs [4.74–4.76]. In accordance with this theory, the usual method to determinate the values of electron temperature and density with a probe is to register the current–voltage characteristics. Figure 4.3 shows typical current–voltage characteristics of different kinds of electrical probes in a plasma consisting of electron and singly charged ions, where both particle species have a Maxwellian energy distribution. A conventional analysis of such characteristics involves fitting the data up to the voltage at which electron current saturates with the standard Langmuir formula:

\[I=I_{is}(1-e^{(V-V_f)/T_e})\]

where \(I\) is the current drawn by the probe at applied voltage \(V\), \(I_{is}\) is the ion saturation current, \(T_e\) is the electron temperature and \(V_f\) is the floating potential of the probe.

Continuation at the special page

#Excerpt from Fusion Physics M.Kikuchi, K.Lackner, M. Quang Tran (Ed), IAEA, 2012, pp. 369-371}, reprinted with kind permission to reproduce IAEA materials

The characteristics are asymmetric, since the ion saturation current is much smaller than the electron saturation current. The floating potential \(V_f\) of such a probe is more negative than the plasma potential \(V_p\). The reason for the strong discrepancy between the currents lies in the fact that the electrons have a much smaller mass than the ions and therefore a much higher mean velocity and average flux than those of the ions. In the absence of a magnetic field, the ratio between the ion and electron saturation currents should be around 50 but in a tokamaks it is found to be much lower [4.77]. If the electron temperature \(T_e\) is known, it is possible to measure the floating potential of a , and to calculate the plasma potential using the well known relation between the floating potential of a cold probe and the plasma potential in the case of a conventional Maxwellian plasma:

\[V_p=V_f+kT_e\]

FIG. 4.3. Typical current–voltage characteristics of a cold probe (solid line) and of an emissive probe (dashed line), in a plasma with Maxwellian velocity distributions for the electrons and ions. Here Vprobe is the variable potential of the probe.

where \(k = 1.9–2.8\) under different plasma conditions [4.78]. However, it is not so easy to measure \(T_e\) with sufficient accuracy. Also \(T_e\) can fluctuate during the measurement and there can be temperature gradients in the region of investigation. The electron emissive probe is a kind of a Langmuir probe heated sufficiently to have a high electron emission current [4.79]. When the emission current just compensates the electron saturation current, the floating potential of such a probe equals the plasma potential (the dashed line in Fig. 4.3). This equilibrium is largely self-establishing, and the established floating potential is little sensitive to the probe temperature. So, it can be obtained directly without \(T_e\) measurements. When the electron temperature \(T_e\) is known it is also possible to calculate the electron density \(n_e\) from the standard probe formula: \[I_{is}=n_eS_{pr}ec_s\] where \(c_s=(k(T_e+T_i)/m_i)^{1/2}\) is the ion sound speed. Assuming \(T_i\approx T_e\) which is valid in many cases, we have a relation for electron density (\(S_{pr}\) is the probe surface): \[n_e[10^{19} m^{–3}]=1.12I_{is} [mA]/S_{pr}[mm^2](T_e[eV] /m_i[at.u.])^{1/2}\] A disadvantage of the determination of the plasma parameters from the current/voltage characteristics of a single Langmuir probe is its low temporal resolution, which naturally is limited by the frequency with which the characteristic can be scanned. Also \(T_e\) can fluctuate during the measurement. The advantages of the measurements with a single Langmuir probe are high locality of the measurements (one of the best amongst ) due to the small size of the probes and high signal to noise value due to the usual high level of the probe signal.