Test particle Field particle m m f v v f e Ze and assume m fi m f and (1=2)mv 2 fl (1=2)m f vf 2 (i.e., v fl v f ). We begin with a single Coulomb col

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1 Ph777, week 2* Part 1. Particle Kinetics of Thermal Plasmas 1.1 Debye Shielding, Collective Behavior of Plasma, Plasma and Cyclotron Oscillations Debye length: Debye number: h kt i T [104 K] D = 690cm 4ßne 2 n[cm 3 ] 1=2 N D = n 4ß 3 3 D =1:38 10 (T K]) 9 [104 3=2 (n[cm 3 ]) 1=2 Since N D fl 1, we expect collective behaviors. Plasma frequency: s 4ße! p = 2 n =5: Hz(n[cm 3 ]) 1=2 Plasma n [cm 3 ]! p Interstellar Medium khz Earth Ionosphere MHz (radio) Fusion experiment GHz (mm) Sun's core Hz (SX) Cyclotron frequency:! c = eb c =1:76 HzB[gauss] 107 Plasma B [gauss]! c Interstellar Medium ο 10 6 ο 100 khz Earth/Sun magnetosphere ο 1 ο 10 MHz (radio) Neutron star magnetosphere ο GHz (HX) Fusion experiment 10 6 ο Hz (IR) 1.2 Coulomb Collisions of a Test" Particles with Field" Particles It is essential to understand basic physics of Coulomb scattering (or collisions) in this study as Coulomb scattering will be the dominant physical process in a plasma. For an order-ofmagnitude estimate, we take * Note from the lecturer (JL): I suppose Ph 777 is primary for solar MHD study. During its first three weeks we will review some fundamental aspects of the plasma physics as appropriate for the rest of the course. Last week, we learned Debye shielding, and plasma/cyclotron frequencies as well as the motions of individual particles moving in a largescale electromagnetic fields. Here I continue some related issues that we can do with thermal (or just mean, if not Maxwellian,) velocity distribution. In the next (week 3), I will intend to introduce the kinetic theory for warm plasmas. Most of materials here are derived from the lecture note of Ph136 at Caltech which I attended in The credit of this material goes to Dr. Roger Blandford. 1

2 Test particle Field particle m m f v v f e Ze and assume m fi m f and (1=2)mv 2 fl (1=2)m f vf 2 (i.e., v fl v f ). We begin with a single Coulomb collision as shown below: There is a critical impact parameter (which gives (1=2)mv 2 = Ze 2 =b 0 ) b 0 2Ze2 mv 2 by which we distinguish two regimes of the deflection angle. For b fl b 0, deflected through a very small angle in which case we use mv D = R Ze 2 b=(b 2 + v 2 t 2 ) 3=2 dt =2Ze 2 =vb. Thus D = b b 0 : For b» b 0, the deflection angle is of order ß=2. D ο ß 2 : The test particle's energy loss 4E equals to the energy gained by the field particle 1 m 2 f (4v f ) 2. Also the momentum conserves, mv D = m f 4v f,so 4E = 1 2 m f(4v f ) 2 = m b0 2E: m f b Next we turn from an individual Coulomb collisions to the net, statistically averaged effect of many collisions, and consider the mean time t D required for the orbit of the field particles to get deflected by ß=2 from initial direction. If single large-angle scattering is dominant source of this deflection, t D would be: t D = 1 n f ffv = 1 n f ßb 2 0 v (single large angle scattering) However, the cumulative, random walk effects of many single scattering actually produces a net ß=2 deflection in a time scale shorter than this. In this case the mean deflection angle will vanish, but the mean square deflection angle will be: 2

3 Z bmax h b0 i h bmax i h 2 i = n f vt2ßbdb = n f 2ßb 2 b min b 0vt ln : b min In this case, the deflection time (when h 2 iο1) is therefore t D = 1 n f 2ßb 2 0 v ln (multiple random scattering): This is shorter by (1=2) ln Λ than the t D for single large-angle scattering. In addition to t D,wewant toknow the mean time for the energy of the test particle to change significantly. Since by analogy, Therefore, 4E E = m m f b0 b 2 = m m f 2 D ; h 4E E i = m m f h D i 2 : t E = m f m t D: If we know Λ, we will be able to calculate t D, t E,... etc. It can immediately be seen that b max = D. The minimum impact parameter b min has different values depending on whether quantum mechanical (QM) wave packet spreading is important or not. Suppose the particle is in a QM wave packet with transverse size l 0 as it nears the field particle. Then it will have a QM spread 4p y ' μh=l 0 in its initial transverse momentum. After time fi the packet's size will be h 4py i l 2 (fi) =l m fi 2 h ' l 2 μhfi i 2 μhfi + 0 ml 0 m l r μhfi r μhb m ο mv where fi is a time for its encounter with the field particle. Since wave packet can be no smaller than p μhb=mv, the nearest distance that the test particle can come to the field particles is b nearest =μh=mv, the debroglie wavelength. If b nearest <b 0, the large-angle scatterings are 3

4 allowed and b min = b 0. If b nearest > b 0, the large-angle scatterings are prevented from occurring by wave packet spreading, b min = b nearest. In summary, b min = max [b 0 ; DeBroglie ] = max [ 2Ze2 mv ; μh 2 mv ] and Λ= D =b min Note that we can express the ratio of the two b min 's as μh=mv b 0 = v=c 2Zff where ff e 2 =μhc ' 1=137, the fine structure constant. This means that (for the field particles of either protons or electrons i.e. Z = 1) wave packet spreading is important, and b min = μh=mv at test particle speeds > 2ffc; scattering is unimportant and b min = b 0 at speeds < 2ffc. If the test particle is an electron (m = ) and has a velocity given by (1=2)mv 2 = (3=2)kT and if plasma is made of hydrogen (Z = 1), the switch of b min from μh=mv to b 0 occurs at 1 = 2 mv2 3kT mc 2 =2ff2 2, i.e., at kt ' (one Rydberg), i.e. T ' K. By contrast, if the test particle is proton then the switch occurs at kt ' (m p = ) (one Rydberg), i.e. T ' K. See the table (from Spitzer 1962) in HW1. In some literature (e.g., Melrose 1986) ln Λ C ' 16:0 1 2 ln n e ln T e T e» K ' 21:6 1 2 ln n e +lnt e T e K: 1.3 Thermal Equilibration Time Scales in a Plasma The time required for particle of one species with another achieving a Maxwellian distribution would be similar to the time for significant energy exchange, i.e., t eq ο t E. (1) t eq; e e time required for electrons to equilibrate with each other. Assume electrons begin with typical individual energies of order T e where T e is the temperature to which they are going to equilibrate, but their initial velocity distribution is non-maxwellian. Choose a typical electron as test particle and all other electrons as the field particles. m = ; m f = ; mv 2 ο m f v 2 ο 3kT e t eq; e e = 2e 2 2 2e 2 2 n f = n; b 2 = 0 = mv 2 3kT e mf m 1 n f 2ßb =2 = v ln Λ 8ß m 1=2 e (kt e ) 3=2 e 4 n ln Λ A more careful calculation based on the Fokker-Planck equation gives 3 3=2 =8ß = 0:21 instead of 3=4 p ß =0:42. 4

5 t eq; e e = 3 4 p ß m 1=2 e (kt e ) 3=2 e 4 n ln Λ = h 1:6 109 s (ln Λ)=10 ih (kte =1keV) i 3=2 n=1cm 3 (2) t eq; i i time required for ions to equilibrate with each other. We replace by m p and T e by T i t eq; i i = h mp i 1=2h Ti T e i 3=2 teq; e e ß 43t eq; e e = h s (ln Λ)=10 ih (kte =1keV) i 3=2 n=1cm 3 (3) t eq; i i time required for electrons to equilibrate with ions. Set m =, m f = m p and T e = T i. Correspondingly mv 2 = m f v 2 f = 3kT e. We then find t eq; e i = (m p = )t eq; e e, but a more careful calculation based on Fokker-Planck equation gives an additional factor, 1=2 3=2 =0:35: t eq; e i = 1 2 3=2 h mp i t eq; e e ß 650t eq; e e = h s (ln Λ)=10 ih (kte =1keV) i 3=2 n=1cm 3 Note that the energy equilibration times t eq are almost always enormously long compared to 1=! p,1=! c, and other dynamical time scales of a plasma. As a result, it is common that the electron and ion velocity distributions are not Maxwellian and the non-maxwellian nature plays an important role in the dynamics and evolution of the plasma. In analyzing the dynamics, one must use the phase space distribution function N(x; v; t) for electrons and ions, respectively. 1.4 Thermoelectric Transport Coefficients Electrons are much more mobile than ions, and are mainly responsible for the transport of heat and charge through a plasma. But there typically are two impediments to an electron's motion: Coulomb collisions and the Lorentz force v B due to magnetic field B. If t D;e fi 1=! c then magnetic field has little effect on thermal and electrical conductivity. But much more common is t D;e fl 1=! c in which an electron typically makes many circuits around the magnetic field line before Coulomb scattering significantly affects its orbit. In this case the electrons have great difficulty traveling distances larger than their Lamor radii s r L ν ' 3kTc 2 kt[kev] = 131 cm! c eb B[gauss] in directions transverse to the field. This suppression, strictly speaking, makes MHD approach in valid. However, when MHD predicts negligible current flow across field lines, it recovers its validity. Here we shall derive the expression for the electrical conductivity» e and thermal conductivity» T in regimes where the principal impediment to electron mobility is Coulomb scattering: In the presence of a temperature gradient rt, not only will a flow of heat, Q, appear, but an electric current J will also flow. Similarly an electric produces a flow of heat. In the absence of magnetic fields, 5

6 Q =» T rt fie; J =» e E + ffrt where ff and fi are called thermoelectric transport coefficients. The derivation can be found in some books (e.g., Kittel (1958) Elementary Statistical Physics"). Here we will use the above result to do an order of magnitude estimate. (a) In a situation where rt =0we will show» e ο ne2 t D;e ; fi ο kt e» e: Calculate J = nev. To estimate v from Fffit = v with ffit taken as the mean diffusion time, t D;e, and Thus Also v = ee t D;e J = ne2 t D;e E: hence» e = ne2 t D;e Q = nve = n ee t D;e (kt)= kt e fi ο kt e» e: ne2 t D;e E = kt e» ee (b) In a situation where E =0,we will show» T ο kn kt e t D;e ; ff ο e kt e» T Consider the temperature and density gradient exist such that a region with (T; n; v) is separated from a region with (T + ffit; n + ffin; v 0 )by a distance l = e, the mean free path of electrons. Then r kt Q = n e v(kt) (n + ffin)v 0 k(t + ffit) =nk k ffit if we use ffit = l (dt=dx) and v t D;e l we have Q = nk kt t D;e rt Also, we have J = nev = nev 2 v 1 ο ne(kt= )(t D;e =l). Since J = ffrt, ff ο (e=kt e )» T. (c) According to the thermodynamics these coefficients are not independent but satisfy a specific relation (the Onsager relation): 6

7 fi = fft e + 5 kt e 2 e» e Our order of magnitude estimate shows that fi» e ff» T ο kt e e kt = O(1) This is consistent with the relation, 1 fiff=» e» T =0:419 (see below). Also, fi ο (kt=e)» and ff ο kn(e= )t D;e ' k=e» e! fft ο (kt)=e» e. This is consistent with the Onsager relation. (d) If a temperature gradient persists for sufficiently long, the effective thermal conductivity reduces to about 42% of that in the absence of E. To see it, consider that after build-up of charge separation, Therefore q = J =0! E = ff» e rt: fiff» e» T rt =» T;eff rt which defines» T;eff =» T =1 fiff=» e» T. Spitzer showed that for Hydrogen plasma, fffi» e» T =0:581: 1.5 Anomalous Resistivity If Coulomb collision in a thermal plasma is the only process, electrical conductivity (called Spitzer value) will be ff = ne2 mν th with ν th = 4ßne4 ln Λ m 2 v 3 th where ν th is the thermal collision frequency. However, due to collective excitation, electromagnetic fields may build up, which is more effective than Coulomb scattering at deflecting the orbits of individual electrons and ions and exchanging energies with them. Correspondingly, the deflections by the excitation's fields cause the transport coefficients» e,» T, ff, and fi to be muchlower than predicted by Coulomb scattering. i.e. they produce larger resistivity to the flow of current and heat. The enhanced resistivities are called anomalous. Enhanced resistivity increases the collision rate and can make the plasma approach the equilibrium more quickly. For instance, low-frequency ion-sound turbulence leads to anomalous conductivity given as: ff Λ n ee 2 W with ν Λ ß! ν Λ pe : nkt 7

8 1.6 Dreicer Field In the context of electron acceleration by electric field, there is a concept of run-aways." The idea is based on the fact that the acceleration in an electric field is independent of the speed of the electron (only depends on charge and electric field E) whereas the slowing down due to collisions ο v 3 (see above and HW2). So there is a critical velocity (or energy) above which electrons will freely run away unhindered by collisions. To find the relation between the critical velocity v cr and the electric field, we start from the balance of the electric field force with thermal collisions: ee = mv d ν th where v d is a drift (away froquilibrium) velocity due to imposition of an external field E. The electric field is regarded weak" when v d fi v th so that the plasma is not greatly displaced en masse. If we increase the electric field to make v d = v th so that virtually all the electrons run away. This is called Dreicer field and thus given by E D = mv th n ν th = 4ße3 e m ln Λ vth 2 In other words, in a weak electric field (E <E D ) supra-thermal electrons can run away. Since E ο v 2,we can express v cr as v cr = v th p E=ED Note: 1. Due to runaways the tail of the Maxwellian distribution becomes depleted and collisions feed electrons into the depleted tail and there will be continuous supply of electrons. 2. You may express E D =4ße= 2 D if it seems more meaningful to you. 1.7 Charged Particle Motions in a Largescale Electromagnetic Fields done in the last class. References Blandford, R. Application of Classical Physics Jackson, J.D. (1962) Classical Electrodynamics Kittel, C. (1958) Elementary Statistical Physics Spitzer, L. Jr. (1962) Physics of Fully Ionized Gases Melrose, D. B. (1986) Instabilities in space and laboratory plasmas Reif, F. Fundamentals of Statistical and Thermal Physics 8

9 Homework (week 2) HW1 : Spitzer (1962) gives values of ln Λ as: Reproduce this table using the formulae derived in this class, noticing where the expression for b min switches. Take electron as test particle, and electron and positron as field particles. Compare the result with the expression given by Priest. Which value of ln Λ would be appropriate for the solar corona? HW2: Derive the expression for Spitzer's electrical conductivity ff and thermal collision frequency ν th when Coulomb collisions dominate (see sec. 1.5). Discuss why ν th / v 3 th. 9

Plasma Astrophysics Chapter 1: Basic Concepts of Plasma. Yosuke Mizuno Institute of Astronomy National Tsing-Hua University

Plasma Astrophysics Chapter 1: Basic Concepts of Plasma Yosuke Mizuno Institute of Astronomy National Tsing-Hua University What is a Plasma? A plasma is a quasi-neutral gas consisting of positive and negative