Plasma Parameter

Let us define the average distance between particles,

$$r_d \equiv n^{-1/3}.$$ (1.16)

We can also define a mean distance of closest approach,

$$r_c\equiv \frac{e^2}{4\pi\,\epsilon_0\, T},$$ (1.17)

by balancing the one-dimensional thermal energy of a particle against the repulsive electrostatic potential of a binary pair. In other words,

$$\frac{1}{2}\, m\,v_t^{2} = \frac{e^2}{4\pi\,\epsilon_0 \,r_c}.$$ (1.18)

The significance of the ratio $r_d/r_c$ is readily understood. If this ratio is small then 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, if the ratio is large then 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 modified Boltzmann equation (in other words, the same type of equation that is conventionally used to describe a neutral gas). (See Chapter 3.) Understanding the strongly coupled limit is far more difficult, and will not be attempted in this book. (Interested readers are directed to Fortov, Iakubov, and Khrapak 2007.) Actually, a strongly coupled plasma has more in common with a liquid than a conventional weakly coupled plasma.

Let us define the plasma parameter,

$${\mit\Lambda} = \frac{4\pi}{3}\, n\, \lambda_D^{3}.$$ (1.19)

This dimensionless parameter is obviously equal to the typical number of particles contained in a Debye sphere. However, Equations (1.8), (1.16), (1.17), and (1.19) can be combined to give

$${\mit\Lambda} = \frac{\lambda_D}{3\,r_c}=\frac{1}{3\!\sqrt{4\pi}}... ...\right)^{3/2} = \frac{4\pi \,\epsilon_0^{3/2}}{3\,e^3} \frac{T^{3/2}}{n^{1/2}}.$$ (1.20)

It can be seen that the case ${\mit\Lambda}\ll 1$, in which the Debye sphere is sparsely populated, corresponds to a strongly coupled plasma. Likewise, the case ${\mit\Lambda}\gg 1$, in which the Debye sphere is densely populated, corresponds to a weakly coupled plasma. It can also be appreciated, from Equation (1.20), that strongly coupled plasmas tend to be cold and dense, whereas weakly coupled plasmas tend to be 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 that 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 1.1 lists the key parameters for some typical weakly coupled plasmas. In conclusion, characteristic plasma behavior is only observed on timescales longer than the plasma period, and on lengthscales larger than the Debye length. The statistical character of this behavior is controlled by the plasma parameter. Although ${\mit\Pi}$, $\lambda_D$, and ${\mit\Lambda}$ are the three most fundamental plasma parameters, there are a number of other parameters that are worth mentioning.

Table 1.1: Key parameters for some typical weakly coupled plasmas.

Plasma	$n({\rm m}^{-3})$	$T({\rm eV})$	${\mit\Pi}({\rm sec}^{-1})$	$\lambda_D({\rm m})$	${\mit\Lambda}$
Solar wind (1AU)	$10^7$	$10$	$2\times 10^5$	$7\times 10^0$	$5\times 10^{10}$
Tokamak	$10^{20}$	$10^4$	$6\times 10^{11}$	$7\times 10^{-5}$	$4\times 10^8$
Interstellar medium	$10^6$	$10^{-2}$	$6\times 10^4$	$7\times 10^{-1}$	$4\times 10^6$
Ionosphere	$10^{12}$	$10^{-1}$	$6\times 10^7$	$2\times 10^{-3}$	$1\times 10^5$
Inertial confinement	$10^{28}$	$10^4$	$6\times 10^{15}$	$7\times 10^{-9}$	$5\times 10^4$
Solar chromosphere	$10^{18}$	$2$	$6\times 10^{10}$	$5\times 10^{-6}$	$2\times 10^3$
Arc discharge	$10^{20}$	$1$	$6\times 10^{11}$	$7\times 10^{-7}$	$5\times 10^2$