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title: Single Langmuir Probes
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# 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](/Theory/IAEAFusionPhysics/PlasmaDiagnostics/PassiveMethods/LangmuirProbes/SingleLangmuirProbes)


#Excerpt from [Fusion Physics](\href{http://www-pub.iaea.org/books/IAEABooks/8879/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 \href{../../../../../../Diagnostics/ParticleFlux/RakeProbe}{Langmuir probe}, 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:


$$V_p=V_f+kT_e$$
 
[<img src="http://golem.fjfi.cvut.cz/wiki/Theory/IAEAFusionPhysics/PlasmaDiagnostics/PassiveMethods/LangmuirProbes/SingleLangmuirProbes/figs-179.jpg" />](/Theory/IAEAFusionPhysics/PlasmaDiagnostics/PassiveMethods/LangmuirProbes/SingleLangmuirProbes/figs-179.jpg)

*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 \href{../../../../../../Diagnostics/diagnostics}{tokamak diagnostics}) due to the
small size of the probes and high signal to noise value due to the usual high level
of the probe signal.