Plasma Parameter

Let us define the average distance between particles,

$$r_d \equiv n^{-1/3},$$ (16)

and the distance of closest approach,

$$r_c\equiv \frac{e^2}{4\pi\epsilon_0\, T} .$$ (17)

Recall that $r_c$ is the distance at which the Coulomb energy

$$U(r,v) = \frac{1}{2}\, mv^2 - \frac{e^2}{4\pi\epsilon_0 \,r}$$ (18)

of one charged particle in the electrostatic field of another vanishes. Thus, $U(r_c,v_t) = 0$.

The significance of the ratio $r_d/r_c$ is readily understood. When this ratio is small, charged particles are dominated by one another's electrostatic influence more or less continuously, and their kinetic energies are small compared to the interaction potential energies. Such plasmas are termed strongly coupled. On the other hand, when the ratio is large, strong electrostatic interactions between individual particles are occasional and relatively rare events. A typical particle is electrostatically influenced by all of the other particles within its Debye sphere, but this interaction very rarely causes any sudden change in its motion. Such plasmas are termed weakly coupled. It is possible to describe a weakly coupled plasma using a standard Fokker-Planck equation (i.e., the same type of equation as is conventionally used to describe a neutral gas). Understanding the strongly coupled limit is far more difficult, and will not be attempted in this course. Actually, a strongly coupled plasma has more in common with a liquid than a conventional weakly coupled plasma.

Let us define the plasma parameter

$${\Lambda} = 4\pi\, n\, \lambda_D^{~3}.$$ (19)

This dimensionless parameter is obviously equal to the typical number of particles contained in a Debye sphere. However, Eqs. (8), (16), (17), and (19) can be combined to give

$${\Lambda} = \frac{\lambda_D}{r_c}=\frac{1}{\sqrt{4\pi}} \lef... ... \frac{4\pi \,\epsilon_0^{~3/2}}{e^3} \frac{T^{3/2}}{n^{1/2}}.$$ (20)

It can be seen that the case ${\Lambda}\ll 1$, in which the Debye sphere is sparsely populated, corresponds to a strongly coupled plasma. Likewise, the case ${\Lambda}\gg 1$, in which the Debye sphere is densely populated, corresponds to a weakly coupled plasma. It can also be appreciated, from Eq. (20), that strongly coupled plasmas tend to be cold and dense, whereas weakly coupled plasmas are diffuse and hot. Examples of strongly coupled plasmas include solid-density laser ablation plasmas, the very ``cold'' (i.e., with kinetic temperatures similar to the ionization energy) plasmas found in ``high pressure'' arc discharges, and the plasmas which constitute the atmospheres of collapsed objects such as white dwarfs and neutron stars. On the other hand, the hot diffuse plasmas typically encountered in ionospheric physics, astrophysics, nuclear fusion, and space plasma physics are invariably weakly coupled. Table 1 lists the key parameters for some typical weakly coupled plasmas.

Table 1: Key parameters for some typical weakly coupled plasmas.

	$n({\rm m}^{-3})$	$T({\rm eV})$	$\omega_p({\rm sec}^{-1})$	$\lambda_D({\rm m})$	${\Lambda}$
Interstellar	$10^6$	$10^{-2}$	$6\times 10^4$	$0.7$	$4\times 10^6$
Solar Chromosphere	$10^{18}$	$2$	$6\times 10^{10}$	$5\times 10^{-6}$	$2\times 10^3$
Solar Wind (1AU)	$10^7$	$10$	$2\times 10^5$	$7$	$5\times 10^{10}$
Ionosphere	$10^{12}$	$0.1$	$6\times 10^7$	$2\times 10^{-3}$	$1\times 10^5$
Arc discharge	$10^{20}$	$1$	$6\times 10^{11}$	$7\times 10^{-7}$	$5\times 10^2$
Tokamak	$10^{20}$	$10^4$	$6\times 10^{11}$	$7\times 10^{-5}$	$4\times 10^8$
Inertial Confinement	$10^{28}$	$10^4$	$6\times 10^{15}$	$7\times 10^{-9}$	$5\times 10^4$

In conclusion, characteristic collective plasma behaviour is only observed on time-scales longer than the plasma period, and on length-scales larger than the Debye length. The statistical character of this behaviour is controlled by the plasma parameter. Although $\omega_p$, $\lambda_D$, and ${\Lambda}$ are the three most fundamental plasma parameters, there are a number of other parameters which are worth mentioning.