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Chapter 2: Single-particle Motions 1. Introduction 2. Uniform B, E fields 3. Gravitational drift 4. Nonuniform B, E fields 5. Time varying E, B fields 6. Adiabatic invariants 2.1 Introduction For very low density devices, collective effects are often unimportant. Single particle motions is the first step toward understanding plasma dynamics. Equation of motion dv m q( E v B) mg dt (2.1) Lorenz force and gravity The fields include self-generated or externally imposed 2.2 Uniform B, E fields • E=0, dv m q( E v B) mg dt g=0 • Equation of motion: Assume Then dv m qv B dt B Bez mvx qBv y mv y qBv x mvz 0 which means vz v// const. 2.2 Uniform B, E fields 2 vx qBv y / m c vx 2 vy qBvx / m c v y (2.2) This describes a simple harmonic oscillator at the cyclotron frequency: c qB m The solution of eq.(2.2) is vx v cos( c t ) v y v sin( c t ) v x x By Integrating, we can get x xg y yg v c v c sin(c t ) ( x xg ) 2 ( y y g ) 2 ( v c ) 2 rL cos(c t ) Gyroradius, or Larmor radius rL v c mv qB The direction of the gyration is always such that the magnetic field generated by the particle is opposite to the externally imposed field. 2 Plasma particles, therefore, tend to reduce the magnetic field, and plasma are diamagnetic. In addition to this motion, there is an arbitrary velocity. guiding center slides along magnetic field line. The motion along B which is not affected by B. The trajectory of a charged particle in space is a helix. Using magnetic field, we can confine plasma. Gyro-orbit is a current loop magnetic moment IrL 2 q c mv rL 2 2 2B 2 E cons. 0 dv m q( E v B) dt The sum of two motions: circular Larmor gyration plus a drift of the guiding center. • We take Ey 0 In z direction dvz q E , z dt m vz q E z t vz 0 m This is a straightforward acceleration along B. In transverse direction: x c 2 v x v E y c ( x v y ) v B 2 q Ex cv y m v y c v x vx d 2 Ex 2 Ex ( v ) vy ) y c ( 2 dt B B vx v cos( ct ) v y v sin( ct ) x xg y yg v c Ex B sin( c t ) Ex v t cos(c t ) B c The trajectory of particle is Ex 2 v 2 2 ( x xg ) ( y y g t) ( ) rL B c 2 The Larmor motion is the same as before, but there is superimposed a drift of the guiding center in the –y direction. To obtain a general formula for drift velocity, we can use vector methods as following. dv m At first, we write E in two components B E E// a B B dt EB E// B Then we determine a B E B E// B (a B) B B(a B) B 2 a B EB a B ( ) B 2 B EB dv B m q( E v B) q[ E// (v ) B] dt B B2 q( E v B) EB dv B m q( E v B) q[ E// (v ) B] 2 dt B B parallel direction: m dv// qE// , dt Perpendicular direction: VE B EB B2 v// du m qu B, dt EB v u u VE B 2 B only gyromotion no drift as if E=0 qE// t v// 0 m EB u v B2 radius of gyro-orbit is larger on the right half of gyro-orbit. Ion and electron drift at the same direction and speed so that no net current , only mass flow appear. Guiding center drift is perpendicular to E ! The three-dimensional orbit in space is a slanted helix with changing pitch. 2.3 Gravitational drift The foregoing result can be applied to other forces by replacing qE in the equation of motion by a general force F. dv m F qv B dt The guiding center drift caused by F is then VE B FB Vf qB 2 EB B2 2.3 Gravitational drift For Gravitational Force F mg , mg B Vg qB 2 gravitational drift is horizontal Ion and electron drift at different direction and speed: produce net current, resulting horizontal electric field moves plasma downward 2.4 Nonuniform B, E field As soon as we introduce inhomogeneity, the problem becomes too complicated to solve exactly. • To get an approximate answer, it is customary expand in the small ratio rL / L Where L is the scale length of the inhomogeneity. This type of theory called as orbit theory. 1. Grad-B drift B B Assume B = B(y) local gyromotion radius is larger at the bottom of the orbit than at the top. This should lead to a drift, in opposite direction for ions and electrons, perpendicular to both B and B The drift velocity should obviously be proportional to rL / L and v Consider the Lorenz force Clearly F qv B Fx 0 vx v cos(c t ) v y v sin( c t ) So we only need to calculate Fy B Fy qvx Bz ( y ) qv cos c t ( B0 rL cos c t ) y cos 2 c t 1/ 2 1 B Fy qv rL 2 y Guiding center drift B B 1 B B VB v r L qB 2 2 B2 stands for the sign of the charge IrL 2 q c mv rL 2 2 2B 2 1 mv2 B 2 Potential Force FB B FB Vf qB 2 It is in opposite directions for ions and electrons and causes a current transverse to B 2. Curved B: Curvature drift Assume curved magnetic field line with a constant radius of curvature Centrifugal force Curvature Drift mv//2 Fc Rc 2 Rc 2 F B mv// Rc B VR c 2 qB qB 2 Rc2 We must compute the grad-B drift Vacuum field B 0 In the cylindrical coordinates: 1 1 e e ez z B B ( )e 1 B ( B )ez 0 1 B B 1 , Rc B Rc 2 B Rc rL v c 1 B B 1 m 2 Rc B VB v rL v 2 B2 2 q Rc2 B 2 2 2 m(v// v / 2) m Rc B 1 2 2 VR VB (v// v ) B B 2 2 3 q Rc B 2 qB mv qB 3. Magnetic Mirrors: B // B Axisymmetry B 0, B 0. B 0, 1 Bz B (rBr ) 0 r r z 1 r Bz 1 Bz Br r dr r r 0 z 2 z r 0 The components of the Lorentz force are Fr qv Bz F q (v z Br vr Bz ) 1 Bz Fz qv Br qv r 2 z r 0 • For simplicity, consider a particle whose guiding center lines on the axis, then v v , r rL . The average force is 1 Bz 1 mv Fz qv rL 2 z 2 B 2 Bz z 1 Bz 1 mv Fz qv rL 2 z 2 B 2 Bz z Define the magnetic moment of the gyrating particle mv 2B Fz 2 Bz z which means the gyrating particle is a diamagnetic particle. the general form can be written as F// // B dv// m F// // B dt B s F// // B // B As the particle moves into regions of stronger or weaker B, its larmor radius changes, but magnetic moment remains invariant. Conservation of magnetic moment For general nonuniform magnetic field is magnetic moment conserved? B B 0 B B0ez B m dV V B q dt Order: 0 1 Vz c (Vx 0 ,V y 0 ) gyromotion m dVz1 V0 ( B) q dt m dVz1 V x 0 ( B ) y V y 0 ( B ) x q dt B y B y B B Vx 0 ( x ' y' ) V y 0 ( x' x y ' x ) x y x y gyro-average: c 2 2 0 c dt vx v cos( c t ) Vx 0 x' V y 0 y ' 0 1 V2 Vx 0 y ' V y 0 x' 2 c v y v sin( c t ) x xg y yg m dVz1 1 V2 By Bx Bz ( ) q dt 2 c y x q z d mV// dVz1 B dz dB mVz1 dt 2 dt z dt dt 2 v sin( c t ) v cos( c t ) c c Energy conservation by Lorentz force qv B d mV//2 ( B) 0 dt 2 dB d ( B ) 0 dt dt d 0 dt Magnetic Mirror Make magnetic field stronger at the end. Paralle velocity : mV//2 W B 2 when particle moves toward higher field region If B is high enough, V// This particle is reflected. V// decrease. become zero at some point. The condition for barely trapped particle is mV//2 ( x 0) Bmin Bmax 2 mV//2 2 mV 2 ( x 0) ( Bmax Bmin ) Bmax 2 trapping condition : the particle with V// B (1 min )1/ 2 V Bmax V// B (1 min )1/ 2 V Bmax escapes the mirror. The magnetic mirror was first proposed by Enrico Fermi as a mechanism for the acceleration of cosmic rays. Nonuniform E field For Simplicity, we assume E to be in the x direction and to vary sinusoidally in the x direction: E E0 cos kxex In practice, such a charge distribution can arise in a plasma during a wave motion. dv m q ( E ( x) v B) dt vx q qB E x ( x) vy m m v y qB vx m E x 2 vx c c vx B 2 vy c Ex 2 c vy B Here E(x) is the electric field at the position of the particle. To evaluate this, we need to know the particle’s orbit, which we are trying to solve for in the first place. We assume the electric field is weak, then we can use the undisturbed orbit to evaluate E(x). x x0 rL sin ct 2 2 vy c v y c Ex E0 cos k ( x0 rL sin ct ) E0 cos k ( x0 rL sin ct ) B 2 2 vy 0 c v y c Assume E0 cos k ( x0 rL sin c t ) B krL 1 cos k ( x0 rL sin ct ) (cos kx0 )(1 1 k 2 rL sin 2 ct ) (sin kx0 )krL sin ct 2 2 vy E0 E (x ) 1 1 2 2 (cos kx0 )(1 k 2 rL ) x 0 (1 k 2 rL ) B 4 B 4 Thus the usual EB drift is modified by the inhomogeneity as EB 1 2 2 vE ( 1 k rL ) B2 4 For an arbitrary variation of E, the drift is 1 2 2 EB vE (1 rL ) 2 4 B The second term is called the finite Larmor-radius effect. Since Larmor-radius is much larger for ions than for electrons, drift velocity is dependent of species. Which will lead to drift instability. 2.5 Time-varying E, B field Let us now take E and B to be uniform in space, but varying in time. it it E E0e E0e ex E x i Ex 2 2 vx c c vx c (vx ) B c B 2 vy c Ex 2 c vy B vp i E x c B vE Ex B 2 vx c (vx v p ) 2 vy c (v y vE ) (2.63) we try a solution which is the sum of a drift and a gyratory motion v x v e i c t v p (2.64) v y iv e i c t vE Derivative respect to time, we get x c 2v eict 2v p c 2vx ( c 2 2 )v p v y c 2iv eict 2vE c 2v y ( c 2 2 )vE v This is not the same as above equation, unless If we assume that E varies slowly, so that 2 c 2 2 c 2 Eq.(2.64) is the approximate solution of Eq.(2.63) we define polarization drift as i E 1 dE vp c B c B dt since polarization drift is in opposite directions for ions and electrons, there is a polarization current. For Z=1 ne dE dE j p ne(vip vep ) 2 ( M m) 2 eB dt B dt Time-varying B field Consider a spatially uniform, but slowly varying B-field: 1 B c B t An electric field associated with time-varying B field is given as E B Assume B 0 t dv m q ( E ( x) v B) dt qv E 0 Everywhere on the gyro-orbit is accelerated. change of perpendicular energy averaged over one gyro period: 2 d 1 2 ( mv ) q E v q c dt 2 2 0 c v Edt c c q E dl q E dS 2 2 c c B B q d S q dS 2 t 2 t B d 1 B ( mv2 ) q c 2 dt 2 2 t t By definition of magnetic moment d 1 d d dB ( mv2 ) B B dt 2 dt dt dt d 0 dt c • magnetic moment is a constant of motion if LB rL magnetic flux encolsed in a gyro-orbit is also conserved. v2 B m 2v2 2m B const. c2 q2B q2 2 Plasma volume shrinks as B increase Plasma and field line move together: Frozen-in-line Adiabatic compression A plasma is injected into the region between the mirrors A and B. Coils A and B are then pulsed to increase B and hence v 2 . The heated plasma can then be transferred to the region C-D by a further pulse in A; increasing the mirror ratio there. The coils C and D are then pulsed to further compress and heat the plasma. Summary of Guiding center Drift General force F: Electric field: Gravitational field Nonuniform E field Nonuniform B field FB Vf qB 2 mg B Vg qB 2 1 2 EB vE (1 rL 2 ) 2 4 B Grad-B drift Curvature drift Curved vacuum field Polarization drift VE B EB B2 1 B B VB v rL 2 B2 2 F B mv// Rc B VR c 2 qB qB 2 Rc2 m Rc B 1 2 2 VR VB ( v v ) // q Rc2 B 2 2 1 dE vp c B dt 2.6 Adiabatic invariants E ~ 1 is a constant to the order of E is called adiabatic invariant. In Hamiltonian system, if the motion is periodic in q, the action integral J pdq taken over a period is a constant of the motion. ~ 1 under adiabatic change: although the change is small in any one period, over a long interval of time the property of motion may undergo large quantitative change. Canonical perturbation theory shows that J is an adiabatic invariant. 1 J c O( 2 ) J Adiabatic invariants play an important role in plasma physics. The First Adiabatic Invariant , If we take p to be the angular momentum mv r and dq to be the angular coordinate pdq mv rL d 2 mv rL 2 mv c 2 4 m q d mv 2B 2 The condition that the invariance of is not violated is c when above condition is violated, for example, Cyclotron Heating, is not conserved, and the plasma can be heated. The Second Adiabatic Invariant, J Consider a particle trapped between two magnetic mirrors: It bounces between them and therefore has a periodic motion. A constant of motion is given by mv // ds However, since the guiding center drifts across field lines, the motion is not exactly periodic, and the constant of the motion becomes an adiabatic invariant. This is called the longitudinal invariant j b J v// ds a The Third Adiabatic Invariant,