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(circul ar) 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, th e total 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 e quation 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 ab ove) , th e 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 g uiding 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) i s a nonli near differenti al equati on 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 (2 (218).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 0000 sintt cosee sin2 t B (13) yyxyy zero-order rr Sub. (13) into (2.20) and average over t . 01 orbit
vB2 r x qq00 0 first-order m vvB 11cc0 ey (14) tt 2y orbit 0 0 vB2 BvvBBe(14)BB2 ( ) 00 0 0000011tt2y x 22 11vB00 v 0 veBB1 t x 2 00 B [grad- drift velocity] (2.28) 2By0 2B0
Question : Does BeB (y )z satisfy B 0 ? 13
2.4 Curvature Drifts By turning a charged particle around to perform gyrational motion, the magnetic field tends to resist a par ticle's motion across its field lines, while the particle is completely free to move along the field line. This results in a helical orbit for the particle, with its guiding center foll owi ng a fi eld li ne. H owever, wh en th e field li ne curves, th e part cil e will feel an outward centrifugal force given by 2 mv ˆ Fc eBB,,[ [ e is R B in Nicholson] (()2.29) RB where RB is the radius of curvature of the magnetic field line, and eB is an outward unit vector along the radius of curvature and is to B0 . SbSub. FF for in th e FB d diftfrift formul a: magnetic field line c e0 cFB v vFB 0 [ drift velocity] e (2.8) d qB2 R B 0 B we obta in the curva ture dr ift vel ocitity: 22 cmv v veBeeed 2 BB000 : unit vector alon g B 0 (2.31) qRB B0 RB 14 2.4 Curvature Drifts (continued) A field gradient is usually associated with curved field lines. Consider, for example , the magnetic field due to a straight , thin wire 2I e carrying current IBrBr : Be00 ( ) with 0 ( )cr B (15) 2I BB00 We have from (()15): B0 2 eeerr B , ()(16) cr r RB which results in a drift velocity given by the RB 2 1 v0 formula in (2.28): vBBd 2 00B , [grad- drift velocity] (2.28) 2B0
where vv00 is the particle velocity B , which we denote below by
because the particle also has a v . Sub. B0 from (16) into (2.28), 22 vv we obtain vBd 2 00B eeB 0 (17) 2B0 2RB magnetic field line Combining (2.31) and (17), we get the e0 v direction of total drift velocity in the curved magnetic e B RB eB and B0 vv221 tot 2 field of (16): vd eB e0 (2.32) RB 15
2.5 Polarization Drift
cEB 0 Rewrite the E B drift velocity : vd 2 (2.2) B0 ctEB() 0 Assume that Ev varies with t , then d 2 (18) B0 FiliitFor simplicity, we assume EE() ()(t (t)(fibl)Th) B0 (see figure below). The acceleration of vvdd requires a force m , which cannot come from E because Ev d . Hence, a polarization drift velocity vp in the direction of E (t ) is developed to provide a magnetic force equal to mv d : q mcEB() t 0 y c vBp 0 m v d 2 (19) B0 0 0 22 EE()tt & () B vvBB() BtEEBB() [ () t ] 000000 pp B x 0 q mcBE00[() t B ] BBvB000(19) c (()p ) 2 B0 z mc2EE() t c () t v p 2 [polarization drift velocity] (2.41) qB0 B0 16 2.5 Polarization Drift (continued) Polarization Current : mc2EE() t c () t RitRewrite: v p 2 (2.41) qB0 B0
Note : Since v p is proportional to m , the polarization drift velocity is much fas ter for the ion than for the electron.
Question : Why is v p proportional to m ?
In a plasma with neinn0 , the polarization drifts lead to a polarization current density J p given by 2 nc0 Jppipeiene0 ()() v v2 m m E B0 2 mc 2 E (2.43) B0 where m nm00 (ie m ) nm i is the mass density. Applications of th e EB drift and polarization drift can be found in Sec. 6.12 on the interpretation of the Alfv en wave and the magnetosonic wave. 17
2.6 Magnetic Moment We have thus far considered drift motion across the B -field. In this section, we consider particle motio n along the B-field . Magnetic Moment of a Gyrating Particle : A current loop of surface area AI and current produces a mgnetic (dipole) moment : See, flklilfor example, Jackson, "Classical IA n (2.44) c Electrodynamics," Eq. (5.57). where n is a unit vector surface area A and pointing in the direction determined by the right-hand rule. For a gyrating particle in a B -field, we have qq I 221 2 q vqv mv W c 2 2 (2.46) v2 c2 2 2c BB Ar 2 L 2 ion where W is the perpendicular energy of th e particle . rL B has a direction opposite to B independent of the sign of q . the plasma is diamagnetic. 18 2.6 Magnetic Moment (continued) Conservation Laws in a Slowly--Varying B Field : Expression of near - axis magnetic field : Before considering the motion of a charged particle in a static, axisymmetric, nonuniform B -field, we first express the field in cylilind ri cal coordi nat es as
B(rz , ) Brrzz ( rz , ) e B ( rz , ) e (20) B 0()01 rB Bz z r rzr (rB ) r Bz (21) r r z
On or near the zBB-axis , we have r z , i.e. the B-field lines have a negligible angular divergence. Hence, the rB -dependence of z can be neglected and we may approximate BBz by . Integrating (21) over r , we obtain B rrBBz (22) r 22zz
Note : The z -axis here is the x -axis in Nicholson Sec. 2.6 19
2.6 Magnetic Moment (continued) Forces on a gyating particle in a slowly -varying static B -field : Consider the motion of a gyrating particle with its guiding center on the z -axis (see figure). The Larmor radius will vary as the particle moves axially into a stronger or weaker field, which implies a radial moti()fhion ()v for the parti ilcle. We assume B 1, ()(23) r B
where BB is the change in seen by the particle in particle orbi one gyro-period.
Under (()23), we ma y ne glect vr
and write vevv zz e (24) Then, with BeBBrr zz e [(20)], we can write down the 3 components of the magnetic force on the particle and identify the function of each. (: Note qv B 0) increase (decrease) v z qq FvB (vB e vB e vBe ) (25) cczr z r rz centripetal force for gyrations decrease (increase) v z 20 2.6 Magnetic Moment (continued) Conservation of the magnetic moment - adiabatic invariant : qq Rewrite (25): FvBcc () (vBzr e vB z r e vB rz e ) (25) qq qv qv2 mv2 FvBvBr BBB zrrcc c22zzzcB 2 B v 0 r B v , r B (22) rr L (26) z r 2 z where Fqz is independent of the sign of . In a static B -field, the total kinetic energy (W ) of the particle is a constant. Hence, ddWWmv (1 2 ) 0 dt dt 2 z W ddWvmvvvFvvm 2 d BB (27) dt 2 dtzzzzzzz dt zzB d WdW (26) d ( ) 11 W dB dt dtBB dt B2 dt (27) Wv11BB W v 0 (()2.55) zzBB22zz W valid under assumption (23); const (2.56) B is called an adiabatic invariant 21
2.6 Magnetic Moment (continued) Conservation of the magnetic flux : W Rewrite (2.56): const (2.56) B 1mv2 (2.56) 2 const B 2 v B2 const B z v2 B const 2 r2 B const 2 L BrL const The magnetic flux enclosed by the particle orbit is conserved.
22 2.6 Magnetic Moment (continued) Problem : Show that is conserved in a B -field which varies slowly with time . Solution : For simplicity, we assume that B is constant in space, while varying with time. Assume also that the guiding center is stationary. If the variation of the field (hence the Larmor radius) is sufficiently slow so that the orbit almost closes on itself in one
revolution, then the change of W in one revolution is q WdqdSE= ( Ea ) [ : surface spanning the orbit] 4 S 2 ion d qqqB 2 BBv ccc dra L 2 S ttt r q v2 L B mc B1 mv2 1 2 B c qBtt2 B da W B [ B 2 B chfilti]hange of B in on revolution] WB t
(WB / ) 0 is conserved. (28) 23
2.6 Magnetic Moment (continued) Magnetic Mirror - An Example of Adiabatic Motion : The simplest magnetic mirror field is pr oduced by 2 coils with equal currents (I0 ) flowing in the same direction (see figure). If condition (23) is satisfied, we have Bmax mv2 const . B (2.56) 2B min z vv0 , z0 At BBmin , let v v0 I I I0 2 0 0 mv0 and vvzz0 . Then, (29) 2Bmin Since FB B [(26)], as the particle moves away from , z z min vvz decreases (hence increases). Define a reflection field Brefl by mv2 m(vv22 ) i.e. at BB , we have the relation: 0 0 z0 refl 2B 2B 222 2 min refl vvvvvzz0and0 and 0 0 22 vv0 z0 or BBrefl 2 min (30) v0 24 2.6 Magnetic Moment (continued) 22 vv0 z0 Bmax mirror Rewrite (30): BBrefl 2 min and define R (31) v0 Bmin ratio 2 v0 1 Case (i): BBmax refl or 2 2 B vv0 z0 R max
The p artic le w ill p ass thr ough Bmin vv, z the B point. 0 z0 max I 2 I0 I0 0 v0 1 Case (ii): BBmax refl or 2 2 vv0 z0 R The particle will stop at the Bmax point. 2 v0 1 Case (iii): BBmax refl or 2 2 vv0 z0 R The particle will be reflected before reaching the Bmax point. 00 v0 Thus, if we define a "loss cone" in the (vv , z ) space bby si n,1 1 th e parti cl e will b e L R loss 00 cone lost if its vv and z lie in the loss cone. L vz0 25
2.6 Magnetic Moment (continued)
z z
(These two figures are taken from G. Schmidt,"Physics of High Temperature Plasmas".)
Discussion : For simppy,licity, we have considered particle motion with the guiding center on the z -axis (left figure). The off-axis motion (right figure) is complicated by the grad-B and curvature drifts, which cause the guiding center to rotate slowly around the z-axis. However, the same results (conservation of and condition for
reflection, etc.) still hold true. 26 2.6 Magnetic Moment (continued) Magnetic Cusp - An Example of Non - adiabatic Motion : If the two coils for the mirror field have opposite currents, a cusp field will be generated. In the figure below, we show without derivation a particle orbit in the cusp field. At the far left, the field varies slowly and the motion is adiabati c. But as the partcile moves into the region of rapidly varying field, the motion becomes non- adiabatic and the orbit eventually encircles the z -axis. ThecuspfieldisoftenusedtogenThe cusp field is often used to generate an axis -encircling electron beam, in which case the Larmor radius in the adiabatic portion of the orbit is made negligibly small. non-adiabatic motion adiabatic motion
z
(figure taken from G. Schmidt,"Physics of High Temperature Plasmas") 27
2.8 Ponderomotive Force The ponderomotive force is a single-particle effect occurring in spatially-varying, high-frequency electric fields, with or without a BB-field. We assume no -field in the following analysis.
Ee E0 cost x Ee E0 ()cosxt x x x spatially weaker stronger uniform E-field E-field E-field
t t A Physical Picture :In a uniform E -field ((g),pleft figure), a particle will oscillate with a constant amplitude. If the E -field is stronger to the right (right figure), the particle will be given a stronger stopping force, followed by a stronger pushing force, as it turns around on the right side. The reverse is true as it turns around on the left side. Thus,
the net result is a gradual acceleration to the left. 28 2.8 Ponderomotive Force (continued) A Quantitative Analysis : This is a one-dimensional problem w ith the following equation of motion for the particle:
mx qE0 ( x )cos t (2.71) xt()ttt x ()t x ()t 01 Let dE , (32) Ex() E x 0 001dx
xt0 (() ) is a slowly-varing component of xt() ( ), c alled the oscillation center. xt( ) is a rapidly-oscillating component of xt ( ). where 1 EE000 (treat ed as a const ant) i s () (xx ) eval uat ed at . dE dE() x 00 (treated as a constant) is evaluated at x . dx dx 0 Su b. (32) i nto (2 .71) , we obibtain
dE0 mx (01 x ) qE ( 01 x )cos t (2.72) dx 29
2.8 Ponderomotive Force (continued) dE Rewrite (2.72): mx ( x ) qE ( x0 )cos t , (2.72) 01 01dx where xx01 varies slowly while oscillates rapidly. Averaging (2.72) dE over one oscillation period gives mx q0 x cost (2.73) 01dx t dE Since xxEx , 0 , (2.72) gives mxqEt cos (2.74) 1001dx 1 0 qE with the solution: xt 0 cos (33) 1 m2 Sub. (33) into (2.73), we obtain q22 E dE q E dE mx 00 cos2 t 0 0 F , 0 m2 dxt 2m2 dx p q2 dE2 2 where F 0 m dv [ponderomotive force] (2.77) p 44m2 dx dx qE and we have defined vx ( )0 [quiver velocity]. 1max m 2 The pondermotive force Fp is related to (wave field amplitude) and hence is important for the study of nonlinear plasma behavior. 30 2.8 Ponderomotive Force (continued) A computational exercise : So far in this chapter , we have con sidered the following types of drift motion: EB drift, grad- B drift, curvature drift, and polarization drift. In addition, we have shown that the magnetic moment of a single particle is an "adiabatic invariant";na ; namely, it is a constant in a slowly varying magnetic field. We have aslo derived a nonlinear force, called the pondermotive force, in a spatially nonuniform electric field which variidlitiies rapidly in time. All these effects can be readily tested by computations. The student is encouraged to write a simple computer program to solve the single partilicle orbi ts numeri cally ll and compare thlihdiibhe results with predictions by the formulae developed in this chapter. It is also worthwhile to use the relativistic equation of motion [see Eq. (2) in Special Topic I] in this exercise. This will allow us to verify that the nonrelativistic EB drift theory breaks down when EB . 31
2.9 Diffusion (across the magnetic field) The guiding center motion gives a simple picture of collisional effects on particle diffusion across the B -field. For simplicity, we consider head-on (180 ) collisions, but the conclusion also applies to small-angle collisions. before after (1) head-on collisions between e e like particles with the same energy (fi)(upper figure) e no diffusion e
before after (2) head-on collisions between e unlike particles with slightly e different energies (lower figure)
significant diffusion e e
32 Appendix A. Taylor Expansion a translational operator, which translates a 1 n Define e a the argument of th e f uncti on it operat es on n0 n! to a distance a away from the argument. Taylor expansion of f (xa ) and A(xa ) about point x : a 1 n fef(xa ) () x n! a f () x n0 1 ff (xa ) ( x )2 a a f ( x ) (A.1) a 1 n Ax( a ) e Ax() n! a Ax () n0 1 AxaAxa ( ) ( )2 aAx ( ) (A.2) ()xa Similarly, operatingfe (xAx )at xa and ( )at xa with , we obtain the Taylor expansion of f (xAx ) and ( ) about point a : 1 ff(xaxaaxaxaa ) ( ) ( ) f ( ) 2 [( ) ][( ) ] f ( ) (A.3) Ax() Aa () ( x a ) Aa ()1 [( x a) ][(xa ) ] Aa ( ) (A.4) 2 33
Appendix A. Taylor Expansion (continued) In (A.1) and (A.2), we have [in Cartesian coordinates] 3 a aa12xxx a 3 ai x (A.5) 123i1 i 2 aa aai xxjjj aai j xx (A.6) ijij ij ij axfaff( ) i x ( xax ) ( ) (A.7) i i a A()xeaijjijjxx()(AAA a A ) e (A8)(A.8) ijii ji Example : axx [()]axxijjjjjx e a ea jii j ij For scalar functions with a scalar argument, (A.1) & (A.3) reduce to 1 2 fx( a ) fx () afx () 2 af () x (A.9) 1 2 fx ( ) fa ( ) ( xafa ) ( )2 ( xa ) fa ( ) (A.10) 34