\def\path{Education/ExperimentMenu/ProbeMeasurements/figs}
%\newcommand\Excerpt[3]{\expandafter\newcommand\csname #1\endcsname{\def\topic{#2}\def\content{#3}}}

\Excerpt{DiagnosticsBasicsAtProbeMeasurements}{Basics of the diagnostic}{
Langmuir probes can be considered as the simplest plasma diagnostic tool. It basically consists of a short and picked piece of metal which is usually biased to some voltage. By a Langmuir probe measurement we mean the measurement of the electric current flowing through the probe - this current can eventually be zero. The simplicity of the hardware can be somewhat misleading since it does not imply the simplicity of the theoretical interpretation of the measurement.        

It is a general fact that any piece of material  inserted into a quasi-neutral plasma in thermal equilibrium ($T_e \approx T_i$, $n_e \approx n_i$) gets negatively charged due to the large electron mobility. Around this piece of material (let's call it \emph{probe}) a potential barrier (Debye-sheath) is formed which repels electrons and drags ions. The width of this sheath is in the order of Debye-length  $\lambda_D = \sqrt{\frac{\epsilon_0T_e}{e^2n_{\infty}}}$. Formation of such Debye-sheath implies that the potential of the probe drops from the plasma potential until the electron current is totally compensated by the ion current. The probe potential that corresponds to zero total current is called \emph{floating potential} ($V_{fl}$).

Figure \ref{fig:IV-characteristic} shows the current-voltage characteristics of a Langmuir-probe. Let us start the qualitative interpretation of the I-V curve by imagining  that we apply a voltage biasing to our probe equal to the plasma potential  ($V_p$). In this case the probe will collect mainly electrons with flux $\Gamma_e = \frac{1}{4}n_e\cdot <v_e>$. Increasing the biasing potential above $V_p$, the probe current will not increase without limit since we collect all the available electrons per unit area and unit time - this current is called \emph{electron saturation current} ($I_{es}$). We have to note that the probe usually cannot be operated in this regime due to high heat load. From the practical point of view of the real measurements more important is the other limit when the biasing potential is lower than the surrounding plasma potential $V_p$. In this case a potential barrier builds up for electrons, therefore the current flowing throughout the probe dramatically drops until becomes zero at the 
floating potential ($V_{fl}$). Decreasing further the biasing potential the current changes sign indicating the dominance of ion current. This ion current will also saturate but at much lower level than   $I_{es}$, this is called \emph{ion saturation current} ($I_{\mathrm{is}}$). The ion saturation current can be calculated solving in self-consistent way the Poisson equation together with the equation of motion. The I-V curve in the regime of $V_{bias}\lesssim V_p$ reads:

\begin{equation}\label{eq.I_V_char}
I(V)= I_{\mathrm{is}}\left[1-e^{\frac{e(V-V_{fl})}{T_e}}\right],
\end{equation}
 where 
$$
I_{\mathrm{is}} = - Aen_\infty e^{-1/2}\sqrt{\frac{T_e}{m_i}}.
$$

Equation (\ref{eq.I_V_char}) consists the basis of the Langmuir probe measurements. The I-V characteristic of a given Langmuir probe can be measured using voltage-sweep measurements. If the measurement data are reliable fitted by the Eq. (\ref{eq.I_V_char}), important plasma parameters can be determined as the electron density ($I_{\mathrm sat}\propto n_\infty$), the electron temperature ($T_e$) and the plasma potential  ($V_f\propto V_p$).

% Miután általánosságban megtátgyaltuk a Langmuir szondák mûködési elvét, most röviden ismertetjük azt a szondarendszert, amelynek segítségével a  jelen értekezés néhány fontos kísérleti eredménye született.
%\begin{figure}
%	\centering
%	\includegraphics[width=6cm]{Langmuire_electronics.eps}
%	\includegraphics[width=6cm]{radial_lang.eps}
%% Langmuire_electronics.eps: 300dpi, width=3.37cm, height=3.92cm, bb=14 14 412 477
%	\caption{A mérsek során alkalmazott Langmuir-szondák kapcsolási sémája (bal oldali ábra). A jobb oldali ábrán látható a zonális áramlások kimutatására és vizsgálatára szolgáló Langmuir-szonákból álló szondasor (\emph{rake probe}).\label{fig:lang_electronics}}
%\end{figure}


\begin{figure}
\center
\GWincludegraphics{scale=1}{\path/characteristic_langmuir-new-eps-converted-to.pdf} 
\caption{The I-V characteristic of a Langmuir probe immersed in hot plasma.}
\end{figure}\label{fig:IV-characteristic}
}

\Excerpt{GolemLangmuirProbesAtProbeMeasurements}{Langmuir probe array used on GOLEM}{

GOLEM tokamak is equipped with an array of 16 Langmuir-probe tips measuring signals at different radial positions of the tokamak vessel. This probe array, called the \emph{rake probe}, is inserted into the tokamak from bottom, as it is seen in Fig. \ref{fig:arrangement}.

\begin{figure}[!]
\center
\GWincludegraphics{width=10cm}{\path/lang_arrangement-eps-converted-to.pdf} 
\caption{Present measurement arrangement at GOLEM.}
\end{figure}\label{fig:arrangement}

The probe head can be moved in vertical direction on the shot-to shot basis. It is also viewed by the fast camera through the corresponding vertical port. Such arrangement allows the observation of probe-plasma  interaction at temporal resolution of 0.8 ms.
The probe head is composed of 16 tips spaced radially by 2.5 mm. Picture in Fig. \ref{fig:arrangement} shows the probe head used in the measurement.

The tips are made of Molybdenum wire of thickness 0.7 mm. The length of the tip is 2 mm. The orientation of tips with respect to the magnetic field line can be changed in between shots. In the current experiments, the tips are perpendicular to the magnetic field lines and turned towards the low-field side of the torus. Signals collected by  the probe tips are digitized with sampling frequency of 1 MHz. Presently, only 11 ADC channels are available in the DAS system. The 150 kHz bandwidth measuring circuit is schematically shown in Fig. \ref{fig:circuit}.

\begin{figure}[h!]
\center
\GWincludegraphics{scale=1}{\path/lang_circuit.png} 
\caption{Measurement circuit of a given Langmuir probe.}
\end{figure}\label{fig:circuit}
}


\Excerpt{MeasurementtasksAtProbeMeasurements}{Measurement tasks}{
\begin{enumerate}
\item Measurement of the radial electric field profile in different GOLEM discharges: density scan, comparison of hydrogen and helium plasmas. Calculation of the poloidal component of the $\mathbb{E\times B}$ rotation as a function of radial position. \underline{Hint}: let's assume that the plasma potential can be estimated by the floating potential. To what extent can this approximation be correct? 
\item Measurement of the floating potential fluctuations. Calculate the amplitude distribution and first 4 statistical moments of the floating potential fluctuation and compare them with the same quantities in the radial electric field fluctuations. Examine the gaussianity of the signal. Determine the auto-correlation time for $\tilde{V}_{fl}$ and $\tilde{E}_{r}$. Calculate the distribution of cross-correlations along the radial direction, estimate the radial correlation length of the structures.
\item Measurement of plasma ion saturation current (not yet ready for measurement). Applying appropriate biasing voltage (between -100 and -300 V) to the probe it can be driven to ion saturation current regime. Estimate the radial electron density profile. Calculate the amplitude distribution and first 4 statistical moments of the $\tilde{I}_{\mathrm{is}}$ fluctuations. Examine the gaussianity of the signal. Determine the auto-correlation times and the distribution of cross-correlations along the radial direction, estimate the radial correlation length of the structures.
\item Estimate the statistical properties (PDF, moments, correlations) of the radial turbulent particle flux $\Gamma_r = \left\langle  \tilde{n}_e\cdot \tilde{v}_r\right\rangle $.


\end{enumerate}
}