Langmuir Probes

Figure 4.5: Current-voltage curve of a Langmuir probe.

A Langmuir probe is a device used to determine the electron temperature and electron number density of a plasma. It works by inserting an electrode that is biased with respect to the vacuum vessel into the plasma. Provided that the bias voltage is not too positive, we would expect the probe current to vary as

$$I = A\,e\,n_s\,\left(\frac{T_e}{m_i}\right)^{1/2}\left[\left(\frac{m_i}{2\pi\,m_e}\right)^{1/2}\exp\left(\frac{e\,V}{T_e}\right)-1\right],$$ (4.268)

where $A$ is the surface area of the probe, and $V$ its bias with respect to the vacuum vessel. [See Equation (4.261).] This current-voltage relation is illustrated in Figure 4.5 For strongly negative biases, the probe current saturates in the ion (negative) direction. The characteristic current that flows in this situation is called the ion saturation current, and is of magnitude

$$I_s = A\,e\,n_s\,\left(\frac{T_e}{m_i}\right)^{1/2}.$$ (4.269)

For less negative biases, the current-voltage relation of the probe has the general form

$$\ln\left(\frac{I}{I_s}+1\right) = C + \frac{e\,V}{T_e},$$ (4.270)

where $C$ is a constant. Thus, a plot of $\ln (I/I_s+1)$ versus $V$ gives a straight-line from whose slope the electron temperature can be deduced. Note, however, that if the bias voltage becomes too positive then electrons cease to be effectively repelled from the probe surface, and the current-voltage relation (4.268) breaks down. Given the electron temperature, a measurement of the ion saturation current allows the electron number density at the sheath edge, $n_s$, to be calculated from Equation (4.269). Now, in order to accelerate ions to the Bohm velocity, the potential drop across the pre-sheath needs to be $e\,({\mit\Phi}_p-{\mit\Phi}_s) =T_e/2$, where ${\mit\Phi}_p$ is the electric potential in the interior of the plasma. It follows from Equation (4.249) that the relationship between the electron number density at the sheath boundary, $n_s$, and the number density in the interior of the plasma, $n_p$, is

$$n_s = n_p\,{\rm e}^{-0.5} \simeq 0.61\,n_p.$$ (4.271)

Thus, $n_p$ can also be determined from the probe.