1、Chapter 1 Introduction 1.1 Definition of plasma A plasma is a quasineutral gas of charged and neutral particles which exhibits collective behavior. 1.2 Characterization of plasma 1.2.1 Plasma density e n ei nn Electron density Ion density Quasineutral condition i n 1.2.2 The temperature of plasma 32
2、 2 ( )()exp() 2 2 m mv fn kT kT v 2 13 22 mvkT Maxwellian distribution at thermal equilibrium The mean kinetic energy e T i T g T Electron temperature Ion temperature Gas temperature 2 3 ee kT 2 1 2 eee m 1eV e kT The electron temperature at nonequilibrium is defined as , 1eV = 11600 K . It is calle
3、d as kinetic temperature. The temperature in units of energy eV for example 1.3 Classification of plasma 1.3.1 Cold plasma The plasmas can be generated by a direct current glow discharge, a high frequency or a microwave discharges at low pressures. The plasma is called a nonequilibrium plasma. Plasm
4、a in a processing reactor (computer model, by M. Kushner) 1.3.2 Thermal plasma The plasmas are generated by the arc discharges with the temperature eig TTT :p : e n 163 10 cm : e T The plasma is at thermal equilibrium. over 100 Torr 2000K50000K ,: ei T T The plasma used for controlled thermonuclear
5、fusion 104 eV (108K) Plasmas are also classified as low temperature plasma and high temperature plasma. 1.4 Debye shielding and Debye length Let us introduce a negative charge into a plasma having equilibrium densities . We assume immobile ions, such that 0ei nnn 0i nn Poissons equation after the ch
6、arge is introduced. 2 0 ( )() ie e rnn (1-1) From the Boltzmann relation, we set 0 exp() e e e nn kT (1-2) i e Substituting for ne and ni in Eq(1-1), we have 2 0 0 ( )(1) e e kT en re (1-3) exp() e e kT e ekT Expanding in a Taylor series for, Eq(1-3) becomes 2 2 0 2 0 1 () e ndde r r drdrkT (1-4) (
7、) DD rr AB ree rr ,0r 0 0, 4 q r r The general solution is (1-5) 0 ( ) 4 D r q re r 1 0 2 D 2 0 () e kT n e the solution becomes (1-7) (1-6) where The quantity D D is called the Debye length. Using the condition 3 7.0 10 cm D D We find for glow discharge n0 =1010cm-3 and kTe =1 eV that The Debye len
8、gth serves as a characteristic scale to shield the Coulomb potential in plasma. D D It is also the characteristic scale of the quasi-neutrality region in plasma. Imagine that the neutrality is disturbed in some volume of the plasma. We assume for simplicity that this disturbance is due to the displa
9、cement of a plane layer of electrons. 1.5 Plasma oscillations Fig.2-1 e n ex e o n ex E The surface density of the charge on the plates is (1-8) The electric field E is determined (1-9) The electron motion equation has the form 22 2 e e o n e xd x m dt (1-10) It describes harmonic oscillations with
10、a frequency of 2 12 () e pe oe n e m (1-11) The frequency is accordingly called the plasma frequency or electron plasma frequency. Therefore the time scale of charge separation in a plasma is determined by 12 2 1 () oe D pee m n e (1-12) 8980 2 pe pee fn Hz The frequency in practical units can be wr
11、itten (ne in cm-3 ) (1-13) Plasma frequencies for discharges are typically in the microwave region (1-10GHz). If the assumption of infinite mass ions is not made, then the ions also move slightly, we obtain the plasma frequency 1 22 2 () ppepi 2 12 () o pi o e n M (1-14) where (1-15) is the ion plas
12、ma frequency. For Mm, ppe . 1.6 Criteria for plasma D L D 1 3 0D n The dimensions L of a system are much larger than The three conditions a plasma must satisfy are: 1. 2. The picture of Debye shielding is valid only if there are enough particles in a “Debye sphere”. If there are only one or two part
13、icles in the region, Debye shielding would not be a statistically valid concept. 3. pc We require the third condition for the gas to behave like a plasma rather than a neutral gas. exp() j jo j q nn kT 1 2 2 () oei D eio kTT TT n e D Show that the shielding distance is then given approximately by an
14、d that is determined by the temperature of the colder species. Problem 1-1 In a strictly steady state situation, both the ions and the electrons will follow the Boltzmann relation. Chapter 2 Single particle motions 2.1 Uniform magnetic field d mq dt v vB In this case, the equation of motion for a pa
15、rticle is (2-1) z B()BzB Taking to be the direction of , we have 0 x y d mqB dt 0 y x d mqB dt 0 z d m dt (2-2a) (2-2b) (2-2c) Differentiating (2-2a) and (2-2b), we obtain 2 2 00 2 y x x d dqBqB dtmdtm 22 00 2 y x y d qB dqB dtmdtm (2-3a) (2-3b) 2.1.1 Particle motion with E=0 00 cos xct 00 sin yct 0
16、zz where Solving (2-3) and (2-2), we find (2-4a) (2-4b) (2-4c) 0 c qB m (2-5) o 0 B 0 is the cyclotron frequency, and is the speed perpendicular to , and is an arbitrary phase. 00 sin o c c xtx 00 cos o c c yty /0 ztz Integrating (2-4) yields the particle position (2-6c) (2-6a) (2-6b) 22 2 00c xxyyr
17、 0 oo c c m r q B Using (2-6a) and (2-6b), we have (2-7) where (2-8) is defined the Lamar radius. The direction of the gyration is always such that the magnetic field generated by the charged particle is opposite to the externally imposed field. Plasmas are diamagnetic. Fig.2-1 2.1.2 Finite E EB d m
18、q dt v EvB The equation of motion in uniform and (2-11) (2-9) EEE BWe take. We obtain a uniform acceleration along. d mqq dt v vBE c D vvv The equation for the transverse motion is (2-10) We let Putting (2-11) into (2-10), we have d mqqq dt c cD v vBvBE (2-12) c v 0qq D vBE Since is the velocity of
19、gyration, we obtain (2-13) 2 B DDD EBBvBvB vB Taking the cross product with B, we have (2-14) 2 B D EB v The electric drift velocity is (2-15) D v EB is perpendicular to both and is independent of the mass and charge of the particles. It is important to note that and , Fig.2-2 2.1.3 Gravitational fi
20、eld qE FF The foregoing result can be applied to other forces by replacing in the equation (2-15) by a general force is then . The drift caused by 2 1 qB f FB v (2-16) F mg 2 m qB g gB v In particular, if is the force of gravity , there is a drift (2-17) g vB The drift is perpendicular to both the f
21、orce and but it changes sign with the particle charge. , There is a net current density in the plasma given by 2 n Mm B gB j (2-18) Fig.2-3 2.2 Non-uniform magnetic field 2.2.1 Gradient drift ( ) d mq dt v vB r ( )( )( ) 0c0 B rB rrB r The equation of motion for a particle in the non-uniform magneti
22、c field is (2-19) The magnetic field in the neighborhood of the guiding center is expanded as (2-20) 00 /1 c rBB cD B vvv We let (2-21) with the approximation (2-22) c v D B v where is the velocity of gyration,is the drift velocity. Substituting (2-20) and (2-22) in (2-19), we obtain d mqqq dt c c0D
23、 B0cD Bc0 v vBvBvvrB (2-23) 0qq D B0cc0 vBvrB Averaging over a gyro-period, the rapidly rotating terms average to zero, so that (2-24) where denotes an average over a gyro-period. 0 B D B0 vB Cross multiplying (2-24) by , we obtainand considering 2 0 1 B D Bcc00 vvrBB （2-25） sin,cos,0 cc cc tt c r c
24、os,sin,0 cc tt c v For the motion of gyration, we have (2-26a) (2-26b) where the magnetic field B is along z. 0,0,B yB B By y Assuming the magnetic field and substituting (2-26) in (2-25), we obtain and 2 0 0 1 2 c B x By D B v(2-27) It can be generalized to 3 w B qB D B vB (2-28) B w B The drift ve
25、locity is perpendicular to both Using the expression of magnetic moment we can write (2-28) as and grad-B and depends on the sign of the charge. , 22 B qBqB BG D B BFB v (2-29) where B BG F is the equivalent force on the particle. Fig.2-4 2.2.2 Curvature drift Here we assume the lines of force to be
26、 curved with a constant radius of curvature R. The centrifugal force felt by the particles as they move along the field lines is 2 / 2 m R BG FR(2-30) 2 / 2222 2mw qB RqB R DBC vRBRB According to (2-16), this gives rise to a drift (2-31) DBC v The drift Such a field does not obey Maxwells is called
27、the curvature drift. Consider a curved field line with a constant radius of curvature R. equations in a vacuum for the uniform field. component and 0B 0BB B In a vacuum, we have . In the cylindrical coordinates of Fig.2-5, has only a z component, and since has only a only an r component. We have 1 0
28、 z rB rr B 1 B R 2 B BR R , (2-32) Thus (2-33) Using (2-28), we have 322 ww B qBq R B D B RB vB (2-34) / 22 1 2ww qR B DBD BDBC vvvRB The total drift in a curved vacuum field is (2-35) Fig.2-5 2.3 Non-uniform electric field E xy Now we let the magnetic field be uniform and the electric field be no u
29、niform. to be in the we have direction and to vary sinusoidally in the Assuming direction, 0 cosEky xE (2-36) This field distribution has a wavelength 2 k . The equation of motion is d mqy dt v EvB (2-37) xcy q E y m ycx whose transverse components are (2-38b) (2-38a) xy Vi c q ViVE y m By defining
30、, we have (2-39) E y 0 0 coscos cc c qE ViVk yt m If the electric field is weak, we may use the undisturbed orbit to evaluate . That is (2-40) 0 coscos c c k yt 00 coscoscossinsincos cc cc kk kytkyt Expanding the cosine, we have (2-41) 2 1 cos1 2 sin 1 c c kv kr Using the Taylor expansions for the s
31、mall Lamar radius case and , we can write 22 00 coscoscos() 1 4 c c c k r k ytky 22 00 cos()cos 2sin()cos 4 c cc c k rk kytkyt (2-42) 22 0 0 cos1 4 c c qEk r ViVky m 22 00 cos()cos 2sin()cos 4 c cc c k rk kytkyt Substituting (2-42) in (2-40), we have (2-43) 22 0 0 cossincos1 4 c cc c qEk r Vtitiky m
32、 + Cosine and Sine terms The solution of (2-43) is (2-44) xy Vi 22 0 0 cos1 4 c c qEk r iky m (2-45) Averaging over time, we obtain 0 x 22 0 0 cos1 4 c DEy c qEk r ky m 22 1 4 c k rE B The drift velocity in a non-uniform electric field is (2-47) Equation (2-45) gives (2-46) where 00 cosEEkyis the el
33、ectric field at the guiding center. 22 2 1 4 c k r B DE EB v Eik 22 2 1 1 4 c r B DE EB v For an arbitrary variation of (2-48) , we need only replace by and write (2-47) as c r DE v The second term is called the finite-Lamar-radius effect. Since is much larger for ions than for electrons, is no long
34、er independent of species. 2.4 Time varying electric field EBE x Let us now take axis: and to be uniform in space. Consider that varies sinusoidally in time and lie along the 0 sinEtxE ( ) d mqt dt v EvB (2-49) The equation of motion is (2-50) Assuming B to be in the z direction, we can write the tr
35、ansverse components as c xcy E t B ycx (2-51b) (2-51a) xy Vi c c ViVE t B Defining , we have (2-52) 22 c c it c c dE iE t dt Ve B 22 / cos c xc c dE dt t B 2 22 sin c yc c E t B The solution of (2-52) is (2-53) We can find (2-54a) (2-54b) E 22 c Assuming that varies slowly, so that, we can obtain th
36、e approximate expression 2 cos xc m dE t qBdt sin yc E t B (2-55b) (2-55a) /E B EB The first terms in (2-55) express the motion of gyration and the second terms are the drift. is the drift. 2 m d qBdt DP E v We define the polarization drift as (2-56) DP v is in opposite directions for ions and elect
37、rons, Since there is a polarization current 22 ndd nemM BdtBdt pDPiDPe EE jvv is the mass density. (2-57) where B 2.5 Adiabatic constancy of the magnetic moment 2.5.1 Time varyingfield B t B E We consider the magnetic field to vary in time. The electric field associated with is given by (2-58) ( ) d
38、 mqt dt v EvB The equation of motion for a charged particle is (2-59) v 2 1 2 dd mqq dtdt l E vE Taking the dot product of the equation of motion with the transverse velocity (2-60) we have , where is the element of path along a particle trajectory. ddt lv The change of kinetic energy in one gyratio
39、n is obtained by integrating over one period 2/ 0 c d wqdt dt l E (2-61) If the field changes slowly, we can replace the time integral by a line integral over the unperturbed orbit. ss wqdqdqd t B ElEss (2-62) S 0dBs Here for ions and 0 for electrons. Then equation (2-62) become is the surface enclo
40、sed by the Lamar orbit and has a direction given by the right-hand rule. Since the plasma is diamagnetic, we have 2 c dB wqr dt (2-63) 2 c cc qrdwwdBdB dtTTdtdt (2-64) The change rate of kinetic energy with time is just the average change during one period of gyration where 2/ cc T . wB dwdBd B dtdt
41、dt Using , we have (2-65) 0 d dt const or Comparing (2-64) with (2-65), we have the desired result (2-66) The magnetic moment is invariant in slowly varying magnetic field. 2.5.2 Space varying fieldB Now we consider a magnetic field which is pointed primarily in z direction and whose magnitude varie
42、s in the z direction. The field is axisymmetric as shown in Fig.2-6. Fig.2-6 The Lorentz force has a component along z given by zr FqB (2-67) 0 0 We have for ions and for electrons. Let so we have , zr FqB (2-68) r B 0B We can obtain from 1 0 z r B rB rrz (2-69) r BrThis yields upon integrating with
43、 respect to 2 0 1 2 c r zz crc BB r Brdrr zz 2 c r rB B z （2-70） 1 1 z c z B r Bz From the averaging procedure it is seen to be valid only for (2-71) 2 1 22 zc mBB Fqr zBz z wBB F Bzz Substituting (2-70) in (2-68), we have (2-73) (2-72) or Because the magnetic field does no work, the total kinetic e
44、nergy of the particle is conserved z wwconst (2-74) 2 1 2 zz wmdzwhere . If the particle moves a distance, then zz w dwF dzdB B (2-75) z dwdw dwdB wB Differentiating (2-74) yields , hence (2-75) becomes (2-76) Integrating (2-76), we obtain w const B (2-77) The magnetic moment is an adiabatic invaria
45、nt that is approximately conserved if the magnetic field changes slowly. 2.6 Magnetic mirrors As a particle moves from a weak-field region to a strong-field region, it sees an constant.B / B / / F must increase in order to keep Since its total energy must remain constant, is high enough in the “thro
46、at” of the mirror, which causes the reflection. increasing, and therefore its must necessarily decrease. If eventually becomes zero, and the particle is reflected back to the weak-field region. It is, of course, the force , which particle will escape? Assume that a particle with 0 B m B 0 /0 m / 0 m
47、 B For given and andat the midplane will have and at the point where the field reaches its maximum value . Then the invariance of yields 22 0 0 11 22 m m mm BB 2222 0/00m (2-78) Conservation of energy requires (2-79) 22 2 000 22 0 sin m mm B B 20 1 sin m mm B BR Combining (2-78) with (2-79), we have
48、 (2-81) (2-80) or m m R is the mirror ratio. where is the smallest pitch angle of a confined particle, Eq.(2-81) defines the boundary of a region in velocity space in the shape of a cone, called a loss cone. 0 11 2 sin1 cos1 2 m m m m R Pd R 1 m R 11 1/21/2 mm PRR Assuming that the velocity distribu
49、tion of particles is isotropic and the collisions of particles can be neglected, we can evaluate the loss probability for (2-83) (2-82) The magnetic moment of a charged particle gyrating in a magnetic field is defined as the product of the current generated by the rotating particle times the area en
50、closed by the rotation. Show that this is equal to wB . Problems 2-1 Consider a uniform magnetic field and a transverse electric field that varies slowly with time. Then the electric drift velocity also varies slowly with time. Therefore there is an inertial force . Show that the Problems 2-2 /mddt
