Introduction to Plasma Physics PDF

Chap 1 : Introduction to plasma and basic concepts WHAT IS PLASMA? In essence a plasma is an ionized gas. However, the behavior of ionized gases is sufficiently different from their nonionized cousins that it is meaningful to talk of plasma as a fourth state of matter. (The other three states being...

Chap 1 : Introduction to plasma and basic concepts WHAT IS PLASMA? In essence a plasma is an ionized gas. However, the behavior of ionized gases is sufficiently different from their nonionized cousins that it is meaningful to talk of plasma as a fourth state of matter. (The other three states being solid, liquid, and gas.) As is well known, a liquid is produced when a crystalline solid is heated sufficiently that the thermal motions of its constituent atoms disrupt its interatomic bonds. Likewise, a neutral gas is produced when a liquid is heated sufficiently that atoms evaporate from its surface at a faster rate than they recondense. Finally, a plasma is produced when a neutral gas is heated until interatomic collisions become sufficiently violent that they detach electrons from colliding atoms. Heating a plasma does not, however, produce a fifth state of matter. Plasmas resulting from ionization of neutral gases consist of positive and negative charge carriers. In this situation, the oppositely charged particles, which are coupled electrostatically, tend to electrically neutralize one another on macroscopic lengthscales. Such plasmas are termed quasi-neutral (“quasi” because the small deviations from exact neutrality can have important consequences). FUNDAMENTAL PARAMETERS Consider an idealized plasma consisting of an equal number of electrons, with mass me and charge −e (here, e denotes the magnitude of the electron charge), and ions, with mass mi and charge +e. We shall employ the symbol 1 Ts ≡ ms vs2 , (1) 3 to denote a kinetic temperature measured in units of energy. Here, v is a particle speed, and the angular brackets denote an ensemble average (Reif 1965). The kinetic temperature of species ”s” is a measure of the mean kinetic energy of particles of that species. (Here, s represents either e for electrons, or i for ions.) In plasma physics, kinetic temperature is invariably measured in electron-volts (1 joule is equivalent to 6.24 × 1018 eV). Quasi-neutrality demands that ne ≃ ni ≡ n. (2) where ns is the particle number density (that is, the number of particles per cubic meter) of species ”s”. Assuming that both ions and electrons are characterized by the same temperature, T (which is, by no means, always the case in plasmas), we can estimate typical particle speeds in terms of the so-called thermal speed,  1/2 2T vts ≡. (3) ms Incidentally, the ion thermal speed is usually far smaller than the electron thermal speed. In fact,  1/2 me vti ∼ vte. (4) mi Of course, n and T are generally functions of position in a plasma. 1 PLASMA FREQUENCY The plasma frequency, 1/2 4πns0 q 2  ωs = , (5) ms which reads, in MKSA units, as, 1/2 ns0 q 2  ωs = , (6) ϵ0 ms is the most fundamental timescale in plasma physics. There is a different plasma frequency for each species, and we have 1/2 1/2 4πni0 e2 4πne0 e2   ωpi = < ωpe =. (7) mi me However, the relatively large electron frequency is, by far, the most important of these, and references to “the plasma frequency” in textbooks generally mean the electron plasma frequency. Plasma frequency corresponds to the typical electrostatic oscillation frequency of a given species in re- sponse to a small charge separation. For instance, let’s take a simple model of a plasma made up of ions, with charge e, assumed to be infinitely heavy, i.e. at rest, and mobile electrons with charge −e. At equilibrium, ion density and electron density are homogeneous and equal to n0. Let’s consider a one-dimensional pertur- bation, along the x−axis, such that all electrons at the position x0 at equilibrium are displaced in the plane x = x0 + ξ(x0 ; t) at time t, the thermal movement of the electrons being neglected in front of the movement due to the disturbance. All electrons initially located between the x = x0 plane and the x = x0 + ξ (x0 , t) plane are moved to coordinates x > x0 + ξ (see Fig. (1)). Figure 1: Electronic perturbaion The spatial perturbation ξ of the electron density translates the creation of an electric field E. Thus, Newton’s second law of motion applied to an individual particle inside the slab yields ∂2ξ me = −eE. (8) ∂t2 2 Furthermore, from Gauss’s theorem, the electric field can be given by   n0 e S ξ (x0 , t) E S = 4πn0 e S ξ (x0 , t) = (in SI) , (9) ϵ0 where S is any surface parallel to the yOz plane. Thus, we have ∂2ξ 4πn0 e2 2 + ξ = 0, (10) ∂t me giving 1/2 4πn0 e2  ξ (x0 , t) ≡ cos (ωpe t) , with ωpe = (11) me Plasma oscillations are observed only when the plasma system is studied over time periods, τ , longer than the plasma period, τp ≡ 1/ω , and when external influences modify the system at a rate no faster than. In the opposite case, the system cannot usefully be considered to be a plasma. Similarly, observations over length scales L shorter than the distance (∼ τp vt ) traveled by a typical plasma particle during a plasma period will also not detect plasma behavior. 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 is defined  1/2 1 T λD ≡ , (12) ω m It follows that  1/2 T λD ≡. (13) 4πn0 e2 According to the preceding discussion, our idealized system can usefully be considered to be a plasma only if λD ≪ 1, (14) L and τp ≪ 1. (15) τ Here, and L represent the typical timescale and length scale of the process under investigation. PLASMA PARAMETER Let us define the average distance between particles, rd ≡ n−1/3. (16) where n is the density of particles at equilibrium. Now, by balancing the one-dimensional thermal energy of a particle against the repulsive electrostatic potential of a binary pair, 1 e2 mvt2 = , (in SI) (17) 2 4πϵ0 rc 3 we can also define a mean distance of closest approach, also known as "Landau length", which the closest that two particles with the elementary charge come to each other if they approach head-on and each has a velocity typical of the temperature, so that e2 rc ≡. (18) 4πϵ0 T In MKSA, the latter is given by e2 rc ≡. (19) T If the ratio rd /rc 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. Such plasmas are termed weakly coupled. Let us, now, define the plasma parameter, 4π 3 Λ= nλD. (20) 3 This dimensionless parameter is obviously equal to the typical number of particles contained in a Debye sphere. It can be written as (in MKSA) 3/2 3/2 4πϵ0 T 3/2  λD 1 rd Λ= = √ =. (21) 3rc 3 4π rc 3e3 n1/2 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 the latter equation, that strongly coupled plasmas tend to be cold and dense, whereas weakly coupled plasmas tend to be diffuse and hot. Examples Examples of strongly coupled plasmas include the very “cold” plasmas found in “high pressure” arc dis- charges1 , and the plasmas that constitute the atmospheres2 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. 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 the previous section 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. Nevertheless, for large Λ we can speak of binary collisions, 1 which is an electrical breakdown: a process that occurs when an electrically insulating material (a dielectric), subjected to a high enough voltage, suddenly becomes a conductor and current flows through it. 2 layer of gases that envelop an astronomical object 4  Table 1: Key parameters for some typical weakly coupled plasmas. and therefore of a collision frequency, denoted by. Here, 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, X νs ≃ νss′. (22) s′ The species designations are generally important. For instance, the relatively small electron mass implies that, for unit ionic charge and comparable species temperatures  1/2 mi νe ∼ νi. (23) me The collision frequency, ν, 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, predomi- nantly small angle scattering events, so the collision frequency is not the inverse of the typical time between collisions. Instead, it is the inverse of the typical time needed for enough collisions to occur that the particle trajectory is deviated through 90ž. For this reason, the collision frequency is sometimes termed the 90ž scattering rate. It is conventional to define the mean-free-path, vt λmfp ≡. (24) ν Clearly, the mean-free-path measures the typical distance a particle travels between “collisions” (i.e., 90ž scattering events). A collision-dominated, or collisional, plasma is simply one in which λmfp ≪ L, (25) where L is the observation length scale. 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 5  Figure 2: Coulomb Collision equilibrium. Furthermore, short mean-free-paths generally ensure that plasma transport3 is local in nature, which is a considerable simplification. The typical magnitude of the collision frequency is ln Λ ν∼ ω. (26) Λ Note that ν ≪ ω in a weakly coupled plasma. It follows that collisions do not seriously interfere with plasma oscillations in such systems. On the other hand, the latter equation implies that ν ≫ ω in a strongly coupled plasma, suggesting that collisions effectively prevent plasma oscillations in such systems. The ratio of the mean-free-path to the Debye length can be written λmfp Λ ∼. (27) λD ln Λ 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 (5) and (21) yield 3e4 ln Λ n ν∼. (28) 4πϵ20 m1/2 T 3/2 Thus, high temperature plasmas tend to be collisionless, whereas dense, low temperature plasmas are more likely to be collisional. Relaxation Time Relaxation time is the average time it takes for a perturbed system to return to equilibrium after a disturbance, particularly in the context of collision processes among particles in a plasma. This concept is crucial for understanding how particles lose their energy or momentum through interactions, which directly affects properties like conductivity and viscosity. The relaxation time can influence the mean free path of particles and is integral in predicting the behavior of plasmas under various conditions. 3 In plasma physics, plasma transport refers to the movement of particles, energy, and momentum within a plasma. This process is influenced by various factors, including electric and magnetic fields, collisions between particles, and pressure gradients. 6