# Minimum number of interacting particles to form a plasma

After reading this question https://physics.stackexchange.com/questions/82486/what-is-the-minimum-number-of-electrons-necessary-to-propagate-a-plasmon I began to wonder what is the minimum number of particles needed to have a plasma? I figure that you need at least two to have density oscillations, but I would imagine you would need quite a bit more to have anything that resembles debye screening. I know that the solar wind plasmas have a number density on the order of unity so I dont think its an issue of $n_e$. Thanks!

It actually turns out that the number density is an important aspect of defining a plasma. The general definition is

a plasma is a quasi-neutral gas of charged and neutral particles which exhibit collective motions

The three conditions for exhibiting collective motions is temperature and (number) density dependent, stemming from the Saha equation, are

1. the time scale of oscillatory motion ought to be less than collisional time scales ($\tau \ll \tau_n\propto1/n_n\sqrt{T}$ with $n_n$ the number of neutrals)
2. the length scale of plasma dynamics needs to be much larger than the mean-free-path ($\lambda\gg\lambda_D\propto\sqrt{T/n}$ with $n$ the number of charged particles)
3. the number of particles in the Debye sphere needs to be significant ($N_D\propto\sqrt{T^3/n}\gg1$)

Pictorially, one can see it from the following image (which comes from one of Hans Goedbloed's lectures, specifically MHD1.pdf at the bottom of the linked page). A slightly different version of this image appears in his book Principles of Magnetohydrodynamics (co-authored by Stephan Poedts).

The caption in the book reads,

Conditions for collective plasma behaviour, in terms of the density $n\equiv n_e\approx Zn_i$ and temperature $T\sim T_e\sim T_i$, are satisfied in the shaded1 area for time scales $\tau<\tau_n=1\,{\rm s}$ and length scales $\lambda>\lambda_D=1\,{\rm m}$, where $N_D\gg1$ is also satisfied. The restrictions on the upper time limit of low density astrophysical plasmas quickly approach the age of the Universe, whereas the restrictions on the lower length limit for high density laboratory fusion experiments approach microscopic dimensions.

It seems, then, that the "minimum" density that satisfies this is a $T\sim10^{3.5}\,{\rm K}$ plasma of $n\sim10^5\,{\rm m^{-3}}$ charged particles (corresponding to $n_n\sim10^{-20}\,{\rm m^{-3}}$ neutrals).

1 The shaded region in the book is the white region above

• Thanks, apparently I have forgotten some of my basics. One concern remains though, In the solar wind plasma the number density is somewhere between .1-10, so it would seem that this region is not covered by the graph. Does some assumption about the collision time scale break down? – Anode Dec 3 '13 at 15:23
• No, your units are the problem :D. The number density of solar wind you cite is in CGS units, the number density in the above graph is in SI. $1\,{\rm cm}^{-3} = 10^6\,{\rm m}^{-3}$, so it fits neatly on the graph. – Kyle Kanos Dec 3 '13 at 15:34