Collisions

Collisions between charged particles in a plasma differ fundamentally from those between molecules in a neutral gas because of the long range of the Coulomb force. In fact, the discussion in Section 1.6 implies that binary collision processes can only be defined for weakly coupled plasmas. However, binary collisions in weakly coupled plasmas are still modified by collective effects, because the many-particle process of Debye shielding enters in a crucial manner. (See Chapter 3.) Nevertheless, for large ${\mit\Lambda}$ we can speak of binary collisions, and therefore of a collision frequency, denoted by $\nu_{ss'}$. Here, $\nu_{ss'}$ measures the rate at which particles of species $s$ are scattered by those of species $s'$. When specifying only a single subscript, one is generally referring to the total collision rate for that species, including impacts with all other species. Very roughly,

$$\nu_s \simeq \sum_{s'} \nu_{ss'}.$$ (1.21)

The species designations are generally important. For instance, the relatively small electron mass implies that, for unit ionic charge and comparable species temperatures [see Equation (1.27)],

$$\nu_e \sim \left(\frac{m_i}{m_e}\right)^{1/2}\!\nu_i.$$ (1.22)

The collision frequency, $\nu$, measures the frequency with which a particle trajectory undergoes a major angular change due to Coulomb interactions with other particles. Coulomb collisions are, in fact, predominately small angle scattering events, so the collision frequency is not the inverse of the typical time between collisions. (See Chapter 3.) Instead, it is the inverse of the typical time needed for enough collisions to occur that the particle trajectory is deviated through $90^\circ$. For this reason, the collision frequency is sometimes termed the $90^\circ$ scattering rate.

It is conventional to define the mean-free-path,

$$\lambda_{\rm mfp} \equiv \frac{v_t}{\nu}.$$ (1.23)

Clearly, the mean-free-path measures the typical distance a particle travels between “collisions” (i.e., $90^\circ$ scattering events). A collision-dominated, or collisional, plasma is simply one in which

$$\lambda_{\rm mfp} \ll L,$$ (1.24)

where $L$ is the observation lengthscale. The opposite limit of long mean-free-path is said to correspond to a collisionless plasma. Collisions greatly simplify plasma behavior by driving the system toward statistical equilibrium, characterized by Maxwellian distribution functions. (See Section 3.11.) Furthermore, short mean-free-paths generally ensure that plasma transport is local (i.e., diffusive) in nature, which is a considerable simplification.

The typical magnitude of the collision frequency is (see Section 3.14)

$$\nu \sim \frac{\ln{\mit\Lambda}}{{\mit\Lambda}}\,{\mit\Pi}.$$ (1.25)

Note that $\nu\ll {\mit\Pi}$ in a weakly coupled plasma. It follows that collisions do not seriously interfere with plasma oscillations in such systems. On the other hand, Equation (1.25) implies that $\nu\gg {\mit\Pi}$ in a strongly coupled plasma, suggesting that collisions effectively prevent plasma oscillations in such systems. This accords well with our basic picture of a strongly coupled plasma as a system, dominated by Coulomb interactions, that does not exhibit conventional plasma dynamics.

Equations (1.7), (1.23), and (1.25) imply that the ratio of the mean-free-path to the Debye length can be written

$$\frac{\lambda_{\rm mfp}}{\lambda_D} \sim \frac{{\mit\Lambda}}{\ln{\mit\Lambda}}.$$ (1.26)

It follows that the mean-free-path is much larger than the Debye length in a weakly coupled plasma. This is a significant result because the effective range of the inter-particle force (i.e., the Coulomb force) in a plasma is of approximately the same magnitude as the Debye length. We conclude that the mean-free-path is much larger than the effective range of the inter-particle force in a weakly coupled plasma.

Equations (1.5) and (1.20) yield

$$\nu \sim \frac{3\,e^4\,\ln{\mit\Lambda}}{4\pi\,\epsilon_0^{2}\, m^{1/2}} \frac{n}{T^{3/2}}.$$ (1.27)

Thus, diffuse, high temperature plasmas tend to be collisionless, whereas dense, low temperature plasmas are more likely to be collisional.

While collisions are crucial to the confinement and dynamics of neutral gases, they play a far less important role in plasmas. In fact, in many plasmas the magnetic field effectively plays the role that collisions play in a neutral gas. In such plasmas, charged particles are constrained from moving perpendicular to the field by their small Larmor orbits, rather than by collisions. Confinement along the field-lines is more difficult to achieve, unless the field-lines form closed loops (or closed surfaces). Thus, it makes sense to talk about a “collisionless plasma,” whereas it makes little sense to talk about a “collisionless neutral gas.” Many plasmas are collisionless to a very good approximation, especially those encountered in astrophysics and space plasma physics contexts.