# Debye length

In a quasi-neutral plasma, the mobile electrons and ions respond to an electric field in such a way to reduce that electric field. This means the plasma is a dielectric. For example, a positive charge placed in the center of a plasma will attract more electrons towards it on average, and push more ions away from it. The electrons will increase the negative charge near the positive charge, so that from far away it appears that there is no charge there. This is known as Debye shielding. The length scale on which it happens is the Debye length. The most commonly used length is given only by the electron shielding, which happens on a faster timescale than the ion shielding:

$\lambda_D=\sqrt{\frac{kT}{4\pi n e^2}}$

If ion shielding is included, and we allow separate temperatures, we find:

$\lambda_D=\sqrt{\frac{kT_e}{4\pi n e^2 \left(1+ZT_e/T_i\right)}}$

We assume that the electron density takes on the boltzmann response $n_i=n_0 e^{-Ze\phi/kT_i} /Z$ (quasi-neutrality gives us the same background charge density). We then use the Poisson equation :

$\frac{d^2\phi}{dx^2}=4\pi e\left(n_e-Z n_i\right)=4\pi en_0 \left(e^{-e\phi/kT_e} -e^{-Ze\phi/kT_i}\right)$

Expanding around small eφ / kT:

$\frac{d^2\phi}{dx^2}=4\pi en_0 \left(\frac{e\phi}{kT_e} + \frac{Ze\phi}{kT_i}\right)=\frac{\pi e^2n_0}{kT_e} \left(1 + \frac{Z T_e}{T_i}\right)\phi$

Which we can solve easily:

$\phi=e^{-x/\lambda_D}$

with:

$\lambda_D=\sqrt{\frac{kT_e}{4\pi n e^2 \left(1+ZT_e/T_i\right)}}$

This page was recovered in October 2009 from the Plasmagicians page on Debye_length dated 17:46, 6 June 2006.