Editing Langmuir probe (section)

===Double probe===

An electrode can be biased relative to a second electrode, rather than to the ground. The theory is similar to that of a single probe, except that the current is limited to the ion saturation current for both positive and negative voltages. In particular, if <math>V_{bias}</math> is the voltage applied between two identical electrodes, the current is given by;

<math>
I 
= I_i^{max} \left( -1 + \,e^{q_e(V_2-V_{fl})/k_BT_e} \right)
= -I_i^{max} \left( -1 + \,e^{q_e(V_1-V_{fl})/k_BT_e} \right)
</math>,

which can be rewritten using <math>V_{bias}=V_2-V_1</math> as a [[Hyperbolic function|hyperbolic tangent]]:

<math>
I = I_i^{max} \tanh\left( \frac{1}{2}\,\frac{q_eV_{bias}}{k_BT_e} \right)
</math>.

One advantage of the double probe is that neither electrode is ever very far above floating, so the theoretical uncertainties at large electron currents are avoided. If it is desired to sample more of the exponential electron portion of the characteristic, an '''asymmetric double probe''' may be used, with one electrode larger than the other. If the ratio of the collection areas is larger than the square root of the ion to electron mass ratio, then this arrangement is equivalent to the single tip probe. If the ratio of collection areas is not that big, then the characteristic will be in-between the symmetric double tip configuration and the single-tip configuration. If <math>A_1</math> is the area of the larger tip then:

<math>
I = A_1 J_i^{max} \left[ \coth\left(\frac{q_eV_{bias}}{2k_BT_e}\right) + \frac{\left(\frac{A_1}{A_2}-1\right)\,e^{-q_eV_{bias}/2k_BT_e}}{2\sinh\left(\frac{q_eV_{bias}}{2k_BT_e}\right)} \right]^{-1}
</math>

Another advantage is that there is no reference to the vessel, so it is to some extent immune to the disturbances in a [[radio frequency]] plasma. On the other hand, it shares the limitations of a single probe concerning complicated electronics and poor time resolution. In addition, the second electrode not only complicates the system, but it makes it susceptible to disturbance by gradients in the plasma.