Plasma Physics Exam 2024

Plasma Physics Exam 2024 1.5 Plasma Frequency The plasma frequency, 2 = ne2 , (1.5) ωp ǫ0 m is the most fundamental time-scale in plasma physics. Clearly, there is a different plasma frequency for each species. However, the relatively fast electron frequency is, by far, the most important, and references to “the plasma frequency” in text-books invariably mean the electron plasma frequency. It is easily seen that ωp corresponds to the typical electrostatic oscillation frequency of a given species in response to a small charge separation. For instance, consider a onedimensional situation in which a slab consisting entirely of one charge species is displaced from its quasi-neutral position by an infinitesimal distance δx. The resulting charge density which develops on the leading face of the slab is σ = enδx. An equal and opposite charge density develops on the opposite face. The x-directed electric field generated inside the slab is of magnitude Ex = −σ/ǫ0 = −enδx/ǫ0. Thus, Newton’s law applied to an individual particle inside the slab yields m d2δx2 = eEx 2 (1.6) = −mωp δx, dt giving δx = (δx)0 cos (ωp t). Note that plasma oscillations will only be observed if the plasma system is studied over time periods τ longer than the plasma period τp ≡ 1/ωp, and if external actions change the system at a rate no faster than ωp. In the opposite case, one is clearly studying something other than plasma physics (e.g., nuclear reactions), and the system cannot not usefully be considered to be a plasma. Likewise, observations over length-scales L shorter than the distance vt τp traveled by a typical plasma particle during a plasma period will also not detect plasma behaviour. In this case, particles will exit the system before completing a plasma oscillation. This distance, which is the spatial equivalent to τp, is called the Debye length, and takes the form . (1.7) Note that (1.8) is independent of mass, and therefore generally comparable for different species. Clearly, our idealized system can only usefully be considered to be a plasma provided that λD 1, (1.9) L ≪ and (1.10) Here, τ and L represent the typical time-scale and length-scale of the process under investigation. It should be noted that, despite the conventional requirement (1.9), plasma physics is capable of considering structures on the Debye scale. The most important example of this is the Debye sheath: i.e., the boundary layer which surrounds a plasma confined by a material surface. 1.6 Debye Shielding Plasmas generally do not contain strong electric fields in their rest frames. The shielding of an external electric field from the interior of a plasma can be viewed as a result of high plasma conductivity: i.e., plasma current generally flows freely enough to short out interior electric fields. However, it is more useful to consider the shielding as a dielectric phenomena: i.e., it is the polarization of the plasma medium, and the associated redistribution of space charge, which prevents penetration by an external electric field. Not surprisingly, the length-scale associated with such shielding is the Debye length. Let us consider the simplest possible example. Suppose that a quasi-neutral plasma is sufficiently close to thermal equilibrium that its particle densities are distributed according to the Maxwell-Boltzmann law, ns = n0 e−es Φ/T, (1.11) where Φ(r) is the electrostatic potential, and n0 and T are constant. From ei = −ee = e, it is clear that quasi-neutrality requires the equilibrium potential to be a constant. Suppose that this equilibrium potential is perturbed, by an amount δΦ, by a small, localized charge density δρext. The total perturbed charge density is written δρ = δρext + e (δni − δne) = δρext − 2e2 n0 δΦ/T. Thus, Poisson’s equation yields (1.12) ∇2 !, (1.13) which reduces to ext . (1.14) If the perturbing charge density actually consists of a point charge q, located at the origin, so that δρext = qδ(r), then the solution to the above equation is written q δΦ(r) = e . (1.15) 4πǫ0 r Clearly, the Coulomb potential of the perturbing point charge q is shielded on distance scales longer than the Debye length by a shielding cloud of approximate radius λD consisting of charge of the opposite sign. Note that the above argument, by treating n as a continuous function, implicitly assumes that there are many particles in the shielding cloud. Actually, Debye shielding remains statistically significant, and physical, in the opposite limit in which the cloud is barely populated. In the latter case, it is the probability of observing charged particles within a Debye length of the perturbing charge which is modified. 1.7 Plasma Parameter Let us define the average distance between particles, rd ≡ n−1/3, and the distance of closest approach, e2 (1.16) rc ≡ . (1.17) 4πǫ0 T Recall that rc is the distance at which the Coulomb energy 1 e2 U(r,v) = mv2 − (1.18) 2 4πǫ0 r of one charged particle in the electrostatic field of another vanishes. Thus, U(rc,vt) = 0. The significance of the ratio rd/rc 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 Λ = 4πnλD3. (1.19) This dimensionless parameter is obviously equal to the typical number of particles contained in a Debye sphere. However, Eqs. (1.8), (1.16), (1.17), and (1.19) can be combined to give 3/2 . (1.20) 1/2 It can be seen that the case Λ ≪ 1, in which the Debye sphere is sparsely populated, corresponds to a strongly coupled plasma. Likewise, the case Λ ≫ 1, in which the Debye sphere is densely populated, corresponds to a weakly coupled plasma. It can also be appreciated, from Eq. (1.20), that strongly coupled plasmas tend to be cold and dense, whereas weakly coupled plasmas are diffuse and hot. Examples of strongly coupled plasmas include soliddensity 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