Fundamentals of Plasma Simulation (I) 核融合基礎学（プラズマ・核融合基礎学） 李継全（准教授）/岸本泰明（教授）/今寺賢志（D1） 2007.4.9 — 2007.7.13 Lecture One (2007.4.) Part one: Basic concepts & theories of plasma physics ➣ Plasma & plasma fluctuations Definitions & basic properties of plasma Basic parameters describing the plasma Plasma oscillation & fluctuations Reference books: S Ichimaru, Basic principles of plasma physics, Chpt.1 1 …… What is a plasma? Plasma is quasi-neutral ionized gas containing enough free charges to make collective electromagnetic effects important for its physical Basic properties— quasi-neutrality & collective behaviour (motion of charged particle may produce electric and magnetic fields, then influence other particles) A plasma is regarded as the fourth state of matter in addition to the solid, liquid, and gaseous states. It is remarked that 99% of universe consists of PLASMA. State of matter: solid → fluid → gas → plasma By heating a gas (to a temperature of 105– 106 K) one can make a plasma. (collisions → ionization) 2 Where is a plasma? In nature: Sun & solar corrona & solar wind in space; Aurora & lighting on earth In nature: Sun & solar corrona & solar wind in space; Aurora & lighting on earth In man-made devices for applications: Fluorescent and neon lights; Plasma TV; Magnetic fusion (tokamak, stellarator, Magnetic mirror ……) Inertial fusion (laser plasma ) 3 Where is a plasma? Plasma in nature corona Magnetic reconnection And solar physics Vortex structures in non-neutral plasma Lightning and discharge physics 4 Aurora physics Where is a plasma? Plasma in man-made devices Plasma lamp Fluorescent & neon lamps Plasma TV Magnetic fusion Inertial fusion 5 Basic properties of plasma – Debye shielding In plasma, binary Coulomb scattering CANNOT correctly describe the behavior of charged particles interacting. Remarkable difference from neutral gas for plasmas is COLLECTIVE behavior – Debye shielding Binary Coulomb interaction +e -e +q +q Consider a positive point charge +q at the origin, Ti=Te, ni=ne. Now think about what the positive charge does. It will attract the electrons and repel the ions, making a cloud of net negative charge around itself, reducing (shielding) the electric field the point charge makes. 6 Basic properties of plasma – Debye shielding (cont.) In plasma -e +q -e +q +q But the electrons can’t just collapse onto the point charge to completely neutralize it because they have too much thermal energy. If we wait for inter-particle collisions to allow this competition between Coulomb attraction and thermal motion to come to equilibrium, we have the situation first studied by Peter Debye and called Debye shielding. 7 What is Debye Length? Coulomb potential of a test charge +q at origin is: q 4 0 r In thermal equilibrium of a plasma with temperature T, the probability of a particle being in a state with energy ε is proportional to the Boltzmann factor, f e T Since probability and number density are proportional to each other in a plasma and since the energy of a particle is simply ε = qU by potential U, we may write the electron and ion densities as ni Be eU T ne Ae eU T When U approaches to zero, no electric field to disturb the thermal equilibrium, so A=B=n0. Therefore the potential equation can be determined by Poisson’s equation 2U n0e 0 e eU T e eU T 8 What is Debye Length? (Cont.) Consider the case that the potential energy of particle in the electric field is much smaller than its kinetic energy, then, using the Taylor expansion to get 2 2 n e 2 0 U U 0T We define a characteristic length (namely, Debye length) D 0T 2n0e 2 the potential equation becomes U 2 U 2 D 1 2 U U r U r 2 r r 2D We can solve this equation as qe r D U (r ) 4 0 r Debye potential (Yukawa potential) 9 Physical meaning of Debye length The electric field tends to 0 much faster, or in other words, the electric field from the test charge is effectively shielded at distances larger than the Debye length. U→0 λD It is the “screening” distance, or the distance over which the usual Coulomb 1/r field is killed off exponentially by the polarization of the plasma. This is the most important length in plasma physics. If you have a gas with equal numbers of charged particles in which this length is larger than the size of the gas, you don’t have to do plasma physics. But if the Debye length is smaller than the size of the gas, then you have to consider the fact that electric fields applied to such plasmas don’t penetrate into them any deeper than a few Debye lengths. 10 Debye length in different plasmas (A Otto) 11 Plasma oscillation & plasma frequency The Debye length governs plasma behavior in equilibrium, but dynamics depends on another fundamental parameter called the plasma frequency. Simple model to understand plasma oscillation + + + + + + + + + ions electrons n0 - d Consider an infinite slab of electrons and ion with a width of d (in x ) and particle density of n0. Assume that the electrons are displaced by a small distance δx in the x direction. This creates two regions of nonzero charge density. The electric field is produced as U ene E x d 0 Homework: problem 1 derive this expression of electric field. 12 Plasma oscillation & plasma frequency (cont.) From the equation of particle motion ne e 2x d 2x eE 2 pex 2 dt me 0 me + + + + + + + + + ions electrons n0 - plasma oscillation frequency: 2 n e e 2pe 0 me Electron oscillation equation: d You may also find this relation: x x0 cos( pe t ) D pe 0T 2n0e 2 ne e 2 0 me T 2 me th 13 Physical meaning of plasma frequency Assuming the plasma is perturbed in some local place, how long time will the plasma respond to it? In other words, if the plasma may locally deviate from the quasi-neutrality due to some reason, how long time can it recover the charge neutrality? This is about the response time. From the oscillation equation of electron, the energy of oscillating electron is 2 about me ( x0 pe ) . If this energy is coming from the thermal energy, 2 me ( x0 pe )2 2 T 2 The amplitude of electron oscillation is approximately about the Debye length x0 ~ D . If the response time of plasma to the perturbation is defined as the time that a thermal electron needs to travel the distance of Debye length, D 1 tD th pe Then, inverse plasma frequency corresponds to the plasma response time to local perturbation. 14 Parameters describing a plasma Two important parameters: λD, and ωp, describe medium-like properties of plasma due to static and dynamic consequences of long-range Coulomb force. On the other hand, plasma consists of a large number of discrete particles. Hence, the interplay between medium-like character and individual particle-like behavior is one of the most interesting aspects of plasma physics. D 0T 2n0e 2 pe ne e 2 0 me 1 ( D ; p ) f ( m; e; ;T ) n Discreteness parameters for per particle: mass(m); electric charge(e); average volume occupied(1/n); average kinetic energy(κT) Fluid-like parameters: mass density (nm); charge density (ne); kinetic energy density (nT) 15 How to understand the fluid limit of plasma? Imaging a process to cut each particle into finer and finer pieces, the discreteness parameters all approach zero, but keep the fluidlike parameters ( nm; ne; nT) finite, regarding the discreteness parameters as infinitesimal quantities of the same order. This procedure is called fluid limit. 16 Plasma parameter Discreteness parameters (m,e,1/n,T) are very useful in plasma kinetic theory. They have finite physical dimensions. However, it is practical to conveniently use dimensionless parameter to treat with plasma. To construct a dimensionless parameter by using four discreteness parameters, write an equation [m x (1 / n) y T z e] 1 Notice: T here actually means κT, κ is Boltzmann constant. Dimensional analysis: The dimensions of a physical quantity are associated with mass; length and time, represented by symbols m, l, and t , each raised to rational powers. 17 Plasma parameter (cont.) Since the electric charge e has dimensions of [mass]1/2[length]3/2[time]-1, and κT has dimensions [mass]1[length]2[time]-2 , i.e., T m 1 l 2 t 2 e m 1 / 2 l 3 / 2 t 1 1 1 x 0; y ; z 6 2 For the defined dimensionless parameter, we have Defining a parameter with the same order of the discreteness parameters, x y z 1/ 6 3 [n T [m (1 / n) T e] Plasma parameter g 1 / 2 1 e] 8 3 / 2 n3D 3 1 n3D This parameter is also defined as the ratio of average (Coulomb) potential energy and the kinetic energy of particle, e 2 / 2 D 1 3 3T / 2 n D 18 Physical meaning of plasma parameter g 1 n3D Density n λD Particle number in a Debye sphere N n3D It implies that the number of particles in a Debye sphere N=4πnλD3/3 is much larger than unity. This is consistent with the shielding. A considerable shielding of individual charges can occur only on the Debye length if there are sufficient charges in the Debye sphere of each individual particle. For a plasma g 1 n 3 D 1 19 Collision frequency -- Role of binary Coulomb collision in plasma Coulomb collision frequency for momentum exchange 4ne 4 m ( ) nQm 2 3 ln m See: Ichimaru textbook Qm is Coulomb collision section For the particles with Maxwellian distribution, the average value of υ3 is, 1 3 8 Hence m 2 3/ 2 1/ 2 T m 3/ 2 ne 4 ln 1/ 2 3 / 2 m T 20 Collision frequency -- Role of binary Coulomb collision in plasma ( cont.) For the discrete parameters (m; 1/n; T; e) with the same order, νm is a quantity of the first order. Plasma frequency is of the zeroth order of the discreteness parameters. Hence, their ratio is m 1 ln 1 p 32 2 n3D In a plasma, binary collisions are less important than collective plasma effects!! Homework: problem 2 Calculate the average value lm of the Coulomb free path (nQm)-1 and show that λD/lm is of the same order in the plasma parameter as νm/ωp. 21 Collective vs particle behavior of density fluctuations (S Ichimaru, Basic principles of plasma physics, Chpt.1.4) In order to understand the essential features of the collective and individualparticle behaviors in a plasma, it is instructive to investigate the equation of motion for the density fluctuations of an electron gas. Assuming the point particles are treated, the density field of electrons is expressed n as (r ) (r r ) i i 1 The Fourier transformation of the density fluctuation is k (r ) exp( ik r )dr exp( ik ri ) i Differentiate this equation above twice with respect to time, 2 2 k t 2 [( k v ) ik vi )]exp( ik ri ) i The potential produced by all charged particles is 4e 1 (r ) 4e 2 exp[ ik ( r rj )] j r rj k ( 0) j k Here the summation does not include the component k=0 since it is cancelled by the contribution of background positive charge, i.e. (r ) e i k exp( ik r ) i 0 k k exp(ik r ) i k ( 0) 22 So the electric field is k E ( r ) i 4e 2 exp[ ik ( r rj )] k j k The acceleration of the ith electron is calculated from the force acting on it from all other particles(electrons and ions) 4e 2 v i m It can derive k 4e 2 exp[ ik ( ri r j )] i 2 k m j ( i )k ( 0 ) k k exp( ik ri ) 2 k k ( 0) 2 2k 4e 2 k q ( k v ) exp( ik ri ) k q q 2 t 2 m q i q( 0) The first term represents the influence of the translational motion of individual electrons; the second term comes from the mutual interaction, which can be separated as two terms with q=k and q≠k. the term for q=k is expressed by plasma frequency, i.e. 2 2k 4e 2 2 p ( k v ) exp( ik ri ) t 2 m i k q k q q 2 q q ( 0, k ) From here, it can be seen that the fluctuations in electron density oscillate at a plasma frequency if the last two terms can be ignored. 23 Now it will be analyzed that in what condition the last two terms may be less important. Assuming the velocity distribution is a Maxwellian, average the first term to get 2 2 2 T ( k v ) exp( i k r ) exp( i k r ) ( k v ) f ( ) d k k i i i i M m The second term is nonlinear term which involves a product of two density fluctuations. It is expected this term may be negligible in the first approximation for small perturbation. It can be seen that if the first term is small, the plasma (charged gas) is characterized by collective oscillation, i.e. k 2 k D2 with k D2 4ne 2 T Debye wave number Therefore, whether a plasma behaves collectively or like an assembly of individual particles depends on the wavelengths of the fluctuations. L D 24

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