Langmuir probe

The above charging process is also the basic mechanism of an instrument designed to measure plasma parameters and called a Langmuir probe. The Langmuir's probe is a conductor for which the potential with respect to a ground (in general the whole satellite) and the collected current can be monitored.
 
 

In density mesearement mode, the potential $\Phi_p$ is fixed and the current I is measured. The plasma parameters can be retrieved if a theoretical model of I is available. In principle, a model of I must be determined by expressing each component of the current using the distribution function at the surface of the object,
 
 

$$J = q \int \int \int \mathbf{u}\cdot\mathbf{n} f(\phi,\mathbf{u})\cdotd^3u$$ (2)

where

                J is the current densities at the object surface.
                q is the charge of the particle.
                u is the speed of the particle.
                n is the normal to the surface.
                f is the distribution function.
                $\phi$
is the electric potential.

In general the determination of $f(\phi,\mathbf{u})$ requires to solve the Vlasov-Poisson system of equation,
 

$$\nabla^2 \phi = 4 \pi q (n_e+n_s - n_i)$$ (3)

$$\partial_t f+ \mathbf{u} \cdot \partial_{\mathbf{r}} f+ \frac{q......}(\mathbf{E} + \mathbf{u} \wedge \mathbf{B})\cdot \partial_{\mathbf{u}} f = 0$$ (4)

where

                E is the electric field.
                B is the magnetic field.
                m is the mass of the particle.

Fortunately, there are several possible simplifications of the problem leading to more tractable equations. For instance, it can be shown that in the case of a system with spherical symmetry and a radius much smaller than the plasma Debye-Length,
$\lambda_D = \left(\frac{\epsilon_0 k_B T_e}{n_e e^2}\right)^{\frac{1}{2}}$, the current due to attracted Maxwellian species can be written (Mott-Smith and Langmuir 1926),
 
 

$$I = 2 \pi r_p^2 n_e q \sqrt{\frac{k_BT_e}{2\pi m}}( 1 + \frac{q\phi_p}{k_BT_e})$$ (5)

where

                rp is the radius of the probe.
                ne is the electronic density.
                Te is the electronic temperature.
                $\phi_p$ is the potential of the probe.

1999-10-10