Chapter 2: Single Particle Motion 2.1 Introduction A plasma moves in self-consistent electric and magnetic fields, i.e. a superposition of the self fields produced by the plasma under study and the prescribed fields from external sources (if any).The plasma motion and the fields are governed by a set of coupled dynamic and field equations [e.g. Eq. (3) of Ch. 1]. Here in Ch. 2,,gp we consider single-particle ((,g)mass m, charge q) motion in external fields (EB00 , ) with self-fields neglected. The 1 particle obeys the equation of motion: mqvE (00c vB ) (2.3) with EB00 and prescribed in a way to satisfy the Maxwell equaions . BeBB(0)(1z ) 0 (incorrect) 00L z Example: B(0) Be rB(0)(1z ) eB 0 (correct) 002L rzL Understanding of single-particle behavior in prescribed fields is an important first step toward the understanding of plasma behavior. 1 2.2 E×B Drifts A Physical Picture : B0 ion electron The EB drift motion is the most common form of single-particle behavior in a plasma. Refer to the figure above. In a static and uniform magnetic fi eld B0 , an i on or el ect ron perf orms (i(circu lar ) cycltlotron motion with a Larmor radius given by (see Sec. 1.5) v r , (1.28) L qB0 where mc cyclotron frequency (1.26) Note that here and in subsequent equations, q (hence ) carries the sign of the charge. 2 2.2 E×B Drifts (continued) B0 ion electron smaller r L larger rL B0 smaller r E0 larger rL ion electron L If now a static electric field EE00 is added,,pp and is perpendicular to B0 and points downward (see lower figure), the ion will then be decelerated when it moves upward whereas the electron will be acce- lerated in its upward motion. This will result in a smaller (larger) instantaneous Larmor radius for the ion (electron). The opposite takes place in downward motions. As shown in the lower figure, the overall effect will be a drift motion in the EB di rection for both the ion and the electron. With a qualitative knowledge of the EB drift motion, we peoceed with quantitative analyses by different methods. 3 2.2 E×B Drifts (continued) A Simple Derivation of the E B Drift Velocity : 1 RitthtiftiRewrite the equation of motion: mqv () (EBE00 c vB ) (2.3) If EB00, are static and uniform, and E 0 B 0, then the particle will be continuously turned around by B0 while being accelerated by Ev0. The time-averaged acceleration t will thus be 0. Hence, 1 balance of electric force mqvEvB 0 [00 c ] (2.1) ttand averaggge magnetic force 112 BEBBvBvBvB00000000(2.1)cc (ttt ) [B ( )] Assume 2-D motion on the planeBvB00t 0. Then, cEB Question: If EB , we have vv00, 00 (2.2) t d B2 0 vcEBcd 00/ . What is wrong? where vEBvvEBdd is called the drift velocity. v is independent of q and m, i.e. electrons and ions drift in the same direction with the same speed. Hence, EB drifts do not result in a net current in the plasma. 4 2.2 E×B Drifts (continued) A More Detailed Analysis : 1 Rewrite the equation of motion: mqvE () (00c vB ) (2.3) Let Be0 be along z and again assume 2-D motion on the xy- plane. osc We let vvd v (2.4) 2 oscillatory velocity B vEBd cB000/ [(2.2)] 0 Sub. (2.4) into (2.3), we obtain q mqv osc()(25)() E11 vB osc vB vB osc (2.5) 000cccd 0 11 22()EB00 B 0 BEB 000 () EE00 BB00 (2.5) is in the form of (1.24) in Ch. 1, where we have obtained the osc solution: veevt(sinxy cos t ) (2.6) cEB00 Thus, t he tota l sol uti on i s vee2 vt ( si ncos x ty )(27)), (2.7) B0 where vv is a constant. For 0, the particle moves in a straight line. 5 2.2 E×B Drifts (continued) A Thorough Analysis : In obtaining (2.7), we have ignored the possibility of a contant velocity along B0 , because it is completely decoupled from the drift motion. Also, to get the general form of the solution, initial conditions for determining the amplitude and phase angle of the oscillatory motion have bee n ignored. For a complete solution, we start from the equation of motion: E0ey mqd vE(1 vB ) e (2.3) dt 00c x B e Ee E 0 z initial with 00y and veee(tvvv 0) (1) xyz000xyzcondition Be00 B z (()2.3) is a linear differential eq uation in v. Let ve vvvxx yy e zz e, we obtain from (2.3) a set of coupled linear differential equations: q mvdd Bv v v (2) dtxyc 0 dt xy q q mvd qEBv d vEv (3) dt yx00c dt yxm 0 mvd 0 d v 0 (4) dt z dt z 6 2.2 E×B Drifts (continued) d vv (2) dt xy q Rewrite: d vEv (3) dt yxm 0 d v 0 (4) dt z byy( (3) dd2 d q 2 cE0 (2)2 vvxy (m Ev0 x ) ( v x ) (5) dtdt dt B0 from (5) EB drift speed cE0 vvx sin( t ) B0 E0e y vvvt1 d cos( ) (6) yx dt ex B e vv 0 z z z0 constant speed along B0 where the constants v and are to be determined from the initial cE0 vtx (0) vx0 v sin conditions : B0 vt(0) v v cos y y0 7 2.2 E×B Drifts (continued) Extension to F B Drift : smaller r L smaller rL B0 larger r F larger rL ion electron L If the electric force qE0 in the previous model is replaced by a constant (in time and space) force FB perp endicular to 0 (e .g . the gravitational force), there will still be a drift motion. The new drift cEB00 velocity can be obtained from vEd 2 [(2.2)] by replacing 0 B0 with F /q . Thus, cFB 0 vFBd 2 [ drift velocity] (2.8) qB0 Note that, under the action of F , the electron and ion will drift in opposite directions (see figure above). 8 2.3 Grad-B Drift A Physical Picture : smaller rL y ion B larger rL smaller r B x L B=B()y ez z electron larger r L The grad-B drift ("grad" stands for "gradient") takes place in a static but nonuniform magnetic field. If the B-field increases in the thiidii(he positive y-direction (see figure a bove ), t he instantaneous Larmor radius of the charged particle will get smaller (larger) as it moves upward (downward). As shown the figure, the net effect will be a drift in the direction of qBB , the sign of which depends upon the sign of qB and . 9 2.3 Grad-B Drift (continued) A Quantitative Analysis: In a non-uniform B-field, the particle only "sees" the variation of the field over a distance of the particle's Larmor radius, which is usually much smaller than the scale length of the field. Thus we may expand the B-field about the guiding center of the particle using the formula for Taylor expansion [see (A.4) in Appendix A]: Ax() Aa () (xa)() Aa ( ()7) where, in Cartesian coordinates, ()xa (xa1212 ) ( xa ) ( xa 3 3 ) xxx123 (xaii )x (8) i i (xa ) Aa ( ) (xaii )[]x A j ( xe ) jxa iji {}(xaii )[]x A j (xe )xa j (9) ji i 10 2.3 Grad-B Drift (continued) Rewrite (7): Ax( ) Aa ( ) ( x a ) Aa ( ) (7) Let thfildbhhe vector field ABr be the magnetifildic field and dl let b e th e position vector of the particle: rexyzxyz e e (10) Then, we have from (7) Br( ) Ba ( ) ( r a ) Ba ( ) (11) Let the particle's guiding center be the origin of coordinetes. We expp()gand Bx() about the guidinggyg() center by setting a 0 in (11) and neglecting higher order terms, then B( r ) B (0) ( r ) B (0) (12) where r is now the vector distance from t he guiding center and (rB ) (0) {}rBij[]x ( r)er0 j . ji i For simplicity, we write (12) as BBB= BrB00(() r ) B (2.16) The equation of motion is thus qq q mvvB=vB cc 00c [vr ( ) B ] (2.17) 11 2.3 Grad-B Drift (continued) qq Rewrite (2.17): mv= cc v B00[ v ( r ) B ] (2.17) Un like (2. 3), (2 . 17) is a nonli near differenti a l equa tion of vari abl es r and v (r ). If the Lamor radius rL is much samller than the scale r length LBB of the magnetic field, then (rB ) ~L and the 2nd 000L term on the RHS of (2.17) is much smaller than the 1st term. We may then solve (2.17) by the perturbation method. rr()ttt () r () no free motion along B LtLet 01 0 (218)(2.18) vrv()tt ()01 () t v () t rv and on xy- plane. and assume rrvv1010, . Sub. (2.18) into (2.17) and equate the zero -order terms, we obtain [ v0 ()isafirstorderterm]()rB00 is a first-order term] q y mvvB 000c , (2.19) ion B0 which yields gyromotion given by r0 v Ω t r (ttt )0 ( cos , sin , 0) x (2.25) 0 zero-order v (tv ) (sin t , cos t , 0) (2.26) 00 orbit 12 2.3 Grad-B Drift (continued) Sub. (2.18) into (2.17) and equate the first-order terms, we obtain qq mvv BBB vr () B (2.20) 11cc0000 y B Let BeBy( ) , then ( rBe ) y 0 (2.23) zz000y B BBB vr() Bv y 00 e v y e x 00 000yxyyxy 00 z y vBvB22 0000sintt cosee sin2 t B (13) yyxyy zero-order rr Sub.