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Note for Plasma Physics - PP by Abhishek Apoorv

  • Plasma Physics - PP
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Text from page-3

Figure 1.3: Charged slab This results in a force on the charges tending to expel whichever species is in excess. That is, if ni > ne , the E field causes ni to decrease, ne to increase tending to reduce the charge. This restoring force is enormous! Example Consider Te = 1eV , ne = 1019 m−3 (a modest plasma; c.f. density of atmosphere nmolecules ∼ 3 × 1025 m−3 ). Suppose there is a small difference in ion and electron densities Δn = (ni − ne ) so ρ = Δn e (1.6) Then the force per unit volume at distance x is Fe = ρE = ρ2 x x = (Δn e)2 �0 �0 (1.7) Take Δn/ne = 1% , x = 0.10m. Fe = (1017 × 1.6 × 10−19 )2 0.1/8.8 × 10−12 = 3 × 106 N.m−3 (1.8) Compare with this the pressure force per unit volume ∼ p/x : p ∼ ne Te (+ni Ti ) Fp ∼ 1019 × 1.6 × 10−19 /0.1 = 16N m−3 (1.9) Electrostatic force >> Kinetic Pressure Force. This is one aspect of the fact that, because of being ionized, plasmas exhibit all sorts of col­ lective behavior, different from neutral gases, mediated by the long distance electromagnetic forces E, B. Another example (related) is that of longitudinal waves. In a normal gas, sound waves are propagated via the intermolecular action of collisions. In a plasma, waves can propagate when collisions are negligible because of the coulomb interaction of the particles. 8

Text from page-4

1.2 1.2.1 Plasma Shielding Elementary Derivation of the Boltzmann Distribution Basic principle of Statistical Mechanics: Thermal Equilibrium ↔ Most Probable State i.e. State with large number of possible ar­ rangements of micro­states. Figure 1.4: Statistical Systems in Thermal Contact Consider two weakly coupled systems S1 , S2 with energies E1 , E2 . Let g1 , g2 be the number of microscopic states which give rise to these energies, for each system. Then the total number of micro­states of the combined system is (assuming states are independent) g = g1 g2 (1.10) If the total energy of combined system is fixed E1 + E2 = Et then this can be written as a function of E1 : and g = g1 (E1 )g2 (Et − E1 ) dg dg1 dg2 g2 − g1 . = dE dE dE1 The most probable state is that for which dg dE1 (1.11) (1.12) = 0 i.e. 1 dg1 1 dg2 d d = or ln g1 = ln g2 g1 dE g2 dE dE dE Thus, in equilibrium, states in thermal contact have equal values of (1.13) d dE ln g. One defines σ ≡ ln g as the Entropy. d And [ dE ln g]−1 = T the Temperature. Now suppose that we want to know the relative probability of 2 micro­states of system 1 in equilibrium. There are, in all, g1 of these states, for each specific E1 but we want to know how many states of the combined system correspond to a single microstate of S1 . Obviously that is just equal to the number of states of system 2. So, denoting the two values of the energies of S1 for the two microstates we are comparing by EA , EB the ratio of the number of combined system states for S1A and S1B is g2 (Et − EA ) = exp[σ(Et − EA ) − σ(Et − EB )] g2 (Et − EB ) 9 (1.14)

Text from page-5

Now we suppose that system S2 is large compared with S1 so that EA and EB represent very small changes in S2 ’s energy, and we can Taylor expand � g2 (Et − EA ) dσ dσ � exp −EA + EB g2 (Et − EA ) dE dE � (1.15) Thus we have shown that the ratio of the probability of a system (S1 ) being in any two micro­states A, B is simply � � −(EA − EB ) , (1.16) exp T when in equilibrium with a (large) thermal “reservoir”. This is the well­known “Boltzmann factor”. You may notice that Boltzmann’s constant is absent from this formula. That is because of using natural thermodynamic units for entropy (dimensionless) and temperature (energy). Boltzmann’s constant is simply a conversion factor between the natural units of temperature (energy, e.g. Joules) and (e.g.) degrees Kelvin. Kelvins are based on ◦ C which arbitrarily choose melting and boiling points of water and divide into 100. Plasma physics is done almost always using energy units for temperature. Because Joules are very large, usually electron­volts (eV) are used. 1eV = 11600K = 1.6 × 10−19 Joules. (1.17) One consequence of our Botzmann factor is that a gas of moving particles whose energy is 2 1 mv 2 adopts the Maxwell­Boltzmann (Maxwellian) distribution of velocities ∝ exp[− mv ]. 2 2T 1.2.2 Plasma Density in Electrostatic Potential When there is a varying potential, φ, the densities of electrons (and ions) is affected by it. If electrons are in thermal equilibrium, they will adopt a Boltzmann distribution of density ne ∝ exp( eφ ) . Te (1.18) This is because each electron, regardless of velocity possesses a potential energy −eφ. Consequence is that (fig 1.5) a self­consistent loop of dependencies occurs. This is one elementary example of the general principle of plasmas requiring a self­consistent solution of Maxwell’s equations of electrodynamics plus the particle dynamics of the plasma. 1.2.3 Debye Shielding A slightly different approach to discussing quasi­neutrality leads to the important quantity called the Debye Length. Suppose we put a plane grid into a plasma, held at a certain potential, φg . 10

Text from page-6

Figure 1.5: Self­consistent loop of dependencies Figure 1.6: Shielding of fields from a 1­D grid. Then, unlike the vacuum case, the perturbation to the potential falls off rather rapidly into the plasma. We can show this as follows. The important equations are: Poisson� s Equation Electron Density d2 φ e = (ni − ne ) − dx2 �0 ne = n∞ exp(eφ/Te ). �2 φ = (1.19) (1.20) [This is a Boltzmann factor; it assumes that electrons are in thermal equilibrium. n∞ is density far from the grid (where we take φ = 0).] Ion Density ni = n∞ . (1.21) [Applies far from grid by quasineutrality; we just assume, for the sake of this illustrative calculation that ion density is not perturbed by φ­perturbation.] Substitute: d2 φ en∞ eφ = exp −1 . dx2 �0 Te � � � � (1.22) This is a nasty nonlinear equation, but far from the grid |eφ/Te | << 1 so we can use a Taylor expression: exp eφ � 1 + eφ . So Te Te d2 φ e2 n∞ en∞ e φ = φ = dx2 �0 Te �0 Te 11 (1.23)

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