Charged particle motion in external fields

Chapter 2 Charged particle motion in external fields A (fully ionized) plasma contains a very large number of particles. In general, their motion can only be studied statistically, taking appropriate averages. Each particle motion is a ected by the local electric and magnetic fields, due to the charges and current due to the other plasma particles as well as externally applied fields. However, to get insight into the relevant physical processes, it useful to study, sometimes in a simplified way, aspects of the motion of individual particles. This approach will be taken in this Chapter and in the next one. In this chapter we will discuss the motion of individual charged particles in given electric and magnetic fields. In the next chapter we will consider the collisions of a charged particle with the surrounding particles. A more general statistical treatment requires the study of the self-consistent evolution of the particle velocity distribution function, ruled by the Boltzmann equation (see Chapter 6), and the Maxwell equations. Such an approach is in most cases extremely complex and solutions of the relevant equations are possible, even numerically, only resorting to rather drastic simplifications. In many cases, however, more a ordable fluid (actually, magneto-fluid, or magnetohydrodynamic, MHD) approaches can used, as discussed in Chapters 6 and 7. In the next sections we will consider the motion of a single particle in externally applied, time-independent magnetic (and electric) fields. We will start from the case of a uniform magnetic field, and then, in sequence, uniform magnetic and electric fields, and a few cases of spatially nonuniform magnetic fields. 2.1 Uniform magnetic field Lorentz force on a particle with charge q and velocity v in a magnetic field B F = qv B (2.1) 15

16 CHAPTER 2. MOTION IN EXTERNAL FIELDS Figure 2.1: Normal acceleration in the plane orthogonal to the magnetic field, due to Lorentz force. Lorentz Force is normal to both magnetic field and velocity. Particle motion is the sum of uniform motion, at constant velocity v k in direction parallel to B. Uniform circular motion, with velocity v? in a plane orthogonal to B In the plane orthogonal to B, i.e. circular motion with Larmor radius F = qv? B = ma n = m v2? r L, (2.2) r L = mv? q B (2.3) In a thermal plasma at temperature T, r 1 2kT 2 mv2? ' kt ) v? ' m (2.4) hence average radius r L = p 2mkT q B. (2.5) Gyration occurs with cyclotron angular frequency! c = v = q B r L m and cyclotron frequency f c =! c 2 Note that often angular frequencies are just referred to as frequencies. (2.6) (2.7)

2.1. UNIFORM MAGNETIC FIELD 17 Numerical evaluations: r L,e =3.3 10 6 p T [ev] B [T] [m] where index e refers to electrons. For T = 1000 ev and B = 5 T we have electron Larmor radius r L,e 20 µm, and proton Larmor radius r L,p 1mm. Figure 2.2: Electrons and ions rotate in opposite direction. Electron Larmor radius is typically much smaller than ion Larmor radius For the cyclotron frequency! c, e =1.7 10 11 B [T] [s] 1 ; f c,e = 28 B [T] [GHz] and f c,p =1.5 B [T] [MHz], where index p refers to protons.

18 CHAPTER 2. MOTION IN EXTERNAL FIELDS 2.2 Gyromotion, guiding center, drift The magnetic field can be uniform only in a limited spatial region. In general, it is not spatially uniform and may change with time. In many cases, the distance over which the magnetic field changes significantly (also called magnetic field gradient scale-length) is much larger than the relevant Larmor radii, and the time over which it changes is much longer than the reciprocal of the plasma frequency. In such cases, the motion of a charged can be viewed as the sum of the fast gyration discussed in the previous section, and a slow motion of the center of the orbit. We then have a gyration around a moving guiding center, and a drift of the guiding centre. In summary, when no electric fields are present, particle trajectories are a sort of helix: particles move at constant speed in the direction parallel to the field, rotate in the plane orthogonal to the field, and drift. The presence of an electric field also a ects charged particle motion. The electric field component parallel to the magnetic field accelerates particles along its direction, i.e. along the magnetic field. In addition, the component of the electric field orthogonal to the magnetic field causes a drift of the guiding center, orthogonal to both electric and magnetic field. A few, simple and important cases of guiding center drifts, caused by timeindependent fields, are discussed in the following sections.

2.3. UNIFORM E AND B FIELDS: E B DRIFT 19 2.3 Uniform E and B fields: E B drift Let us consider the case of magnetic field B and electric field E orthogonal to each other. m dv = q (E + v B) (2.8) dt We write the velocity as v w + E B B 2, (2.9) where the second term on the right hand side is a constant, depending only on the fields. Inserting this expression into Eq. (2.8) we obtain an equation for w: m dw dt + m d apple E B dt B {z 2 = q E + w + E B B } 2 B (2.10) =0 Using (E B) B = B (E B) = E (B B)+B (B E) = {z } EB 2 (2.11) =0 we then have m dw dt = qe + qw B + q EB 2 B 2, (2.12) m dw = qw B dt (2.13) This is just the equation for gyromotion in a uniform magnetic field. Since v = w +(E B/B 2 ), the resulting motion is gyromotion summed to drift of the guiding centre with velocity v E = E B B 2, (2.14) orthogonal to both electric and magnetic field. Notice that the direction of the drift does not depend on the sign of the particle charge (see Fig. 2.3). Qualitative explanation. Refer to the ion motion (red trajectory in the figure). When the ion is in the lower part of the orbit it has higher velocity (due to the acceleration caused by the electric field) and then larger Larmor radius. The opposite occurs in the upper portion of the orbit. This leads to the drift towards the right hand side. The expression of the E B drift can be generalized, by replacing the electric force qe with any force F, v F = F B qb 2. (2.15)

20 CHAPTER 2. MOTION IN EXTERNAL FIELDS Figure 2.3: E B drift.

2.4. NON-UNIFORM MAGNETIC FIELD 21 2.4 Non-uniform magnetic field We now we consider non-uniform magnetic fields, and no electric fields. shall consider a few cases separately. We 2.4.1 r B?B Figure 2.4: Left: r B?B; right: unperturbed gyromotion in the x (Electron and ion orbits not to scale.) y plane. We assume that the magnetic field gradient scale-length is much larger than the Larmor radius r L, so that the particle orbit is unperturbed, with radius r L and frequency f c =! c /2. Unperturbed orbit: x x 0 = r L sin! c t, (2.16) y y 0 = ±r L cos! c t, (2.17) where the positive sign refers to ions and the negative sign to electrons. Hence Force on the particle 0 F = q @ v x = r L! c cos! c t = v? cos! c t (2.18) v y = v? sin! c t (2.19) î ĵ ˆk v x v y v z 0 0 B z 1 A = qv y B z î qv x B z ĵ. (2.20) This force changes as the particle moves because the magnetic field is not uniform, B z = B z (y), with y = y 0 ± r L cos! c t. However, since the change of the field over a Larmor radius is small, we can compute it along the particle orbit by using a Taylor expansion limited to the first term. We then write " # F y = qv x B z (y) ' qv? cos! c t B z0 +(y y 0 ) @B z, (2.21) @y y=y0

22 CHAPTER 2. MOTION IN EXTERNAL FIELDS and then F y = qv? cos! c t " B z0 + @B z @y y=y0 (±r L cos! c ) #, (2.22) where B z0 is the field at the center of the orbit, i.e. B z0 = B z (y = y 0 ). This force changes in time. Since gyrations are fast (with respect to drift) we can average the force over an orbit, to obtain F y =0 r L qv? @B z @y Proceeding in the same way we find cos 2! c t y0 {z } 1 2 (2.23) F x =0. (2.24) In conclusion, the force (2.23) normal to the magnetic field acts on the particle. According to Eq. 2.15 this force causes a drift, with velocity v = F B qb 2 = 1 2 v?r L r B B B 2 (2.25) Notice that the direction of this drift depends on the sign of the charge (see Fig. 2.5) Figure 2.5: r B?B drift.

2.4. NON-UNIFORM MAGNETIC FIELD 23 2.4.2 Curvature drift Let us now consider a circular B-field line, with curvature radius R c ; see Fig. 2.6. We indicate with ˆr = R c /R c the versor (unit vector) directed along the radius. Figure 2.6: Particle motion around a curved magnetic field line. In the guiding centre reference frame, a centrifugal force F c = mv2 k R c ˆr (2.26) acts on the particle; here v k is the component of the velocity parallel to B. This force causes a curvature drift with velocity v R = F c B qb 2 and then v R = mv2 k qb 2 R c B R 2 c (2.27) Referring to the figure, the drift is normal to the plane of the figure, and its sign depends on the sign of the charge. However, if the magnetic field lines are curved, then we also have a finite r B that causes an additional drift. We evaluate this drift as follows. We refer to the cylindrical coordinate system in Fig. 2.7, so that B r = 0 (2.28) B z = 0 (2.29) B = B (r) (2.30) If in the region we are considering there are no currents, then, for Ampere law,

24 CHAPTER 2. MOTION IN EXTERNAL FIELDS Figure 2.7: Circular field line and cylindrical coordinate system. r B = 0. The z component of the curl operator, (r B) z = 1 r apple @ @r (rb ) @B r @, (2.31) in our case reads 1 @ r @r (rb ) (2.32) and must be identically zero for the Ampere law. We then find that B / 1 r. (2.33) It follows that (r B ) r=rc = r B ˆ = cost r=r c Rc 2 ˆr, (2.34) and then r B = cost B r=r c Rc 2 and the relevant drift velocity is R c cost ˆr = ˆr R c = R c Rc 2, (2.35) v r B = 1 2 v?r L r B B B 2 = ± 1 2 v?r L R c B B R 2 cb 2. (2.36) Finally, by using the definition of Larmor radius, r L = mv? / q B = ±mv? /qb, we can write v r B = m v? 2 R c B q 2 RcB 2 2, (2.37)

2.4. NON-UNIFORM MAGNETIC FIELD 25 which di ers from the curvature drift [Eq. (2.27)] only for the term v? 2 /2, replacing vk 2. The total drift velocity, sum of Eq. (2.27) and (2.37) is then v = v R + v r B = m q vk 2 + 1 Rc B 2 v2? RcB 2 2 (2.38) Figure 2.8: Drift in a purely toroidal field. An important consequence of curvature drift concerns plasma confinement in toroidal geometry. Let us consider a toroidal (i.e. donut shaped) configuration, with a purely toroidal magnetic field, i.e. a sort of toroidal solenoid (see Fig. 2.8). According to Biot-Savart s law, B(r) / 1/r and then the field gradient is directed towards the torus major axis and is at any point orthogonal to the field. The combined curvature and gradient drift pushes electron and ions vertically, in opposite directions, thus producing charge separation, and a vertical electric field, orthogonal to the magnetic field. This leads to an E B drift, pushing particles of both signs out of the torus. Confinement in toroidal geometry then requires more complex field topologies, as discussed in a later Chapter. Excercise Compute electron thermal velocity, ion thermal velocity and (curvature plus gradient) drift velocity in a toroidal device with major radius R c = 1 m, magnetic field of 5 T, and plasma temperature of 1 kev. Verify that the drift velocity is much smaller than the thermal velocities.

26 CHAPTER 2. MOTION IN EXTERNAL FIELDS 2.4.3 r B kb; Magnetic mirror Figure 2.9: Cylindrically symmetric magnetic field and cylindrical coordinate system. We now consider the case of magnetic field gradient (nearly) parallel to the magnetic field B. We refer, for simplicity, to a configuration with cylindrical symmetry, as illustrated in Fig. 2.9. The magnetic field B is nearly axial and weakly varying (i.e. the gradient scale-length is much larger than the Larmor radius). In summary B = (B r, 0,B z ); B r << B z ; (2.39) @B z @z B z /r L. (2.40) We use Maxwell s equation r B = 0 to determine the relation between the radial and axial component of the magnetic field. Using the expression of the divergence operator in cylindrical coordinates, 1 r @ @r (rb r)+ @B z @z + 1 r @B @ =0, (2.41) we obtain @ @r (rb r)= @B z r. (2.42) @z (the third term in Eq. (2.41) vanishes identically, because B = 0). We integrate along the radius between the axis and a radius r Z r 0 @ @r 0 (r0 B r ) dr 0 = Z r 0 @B z @z r0 dr 0. (2.43) Since @B z /@z varies weakly with r, we can take it as a constant (or take an average value), and then we have B r = r @B z 2 @z. (2.44)

2.4. NON-UNIFORM MAGNETIC FIELD 27 Motion along z, magnetic moment as an adiabatic invariant We now study the axial component of the motion of a particle (actually, of its guiding centre) close to the axis. We can write m dv k dt = F z = q (v r B v B r )= qv B r (2.45) Now, writing v = v? = r L! c, approximating r ' r L, and using Eq. (2.44) for B r, we obtain m dv k r dt = q (±v @B z?) = 1 2 @z 2 q @B z r2 L! c @z = µ@b z (2.46) @z The quantity µ = 1 2 mv 2? B is called magnetic moment of the particle. (2.47) Exercise Show that this definition is just the same as that for the magnetic moment of a current carrying coil, µ = IS,whereI is the current and S is the surface area of the coil. We now show that the magnetic moment is an adiabatic invariant, i.e. it is a constant if the magnetic field changes weakly in space and time. Here we consider the case of a magnetic field which varies in space but is constant in time. We multiply both members of Eq. 2.46 by v k, which can also be written as d dt 1 2 mv2 k mv k dv k dt = µv @B z k @z, (2.48) @z @B z = µv k @t @z = µdb dt. (2.49) During motion, kinetic energy (equal to total energy, since there is no potential energy) is conserved, hence d 1 dt 2 mv2 = d 1 dt 2 mv2 k + d 1 dt 2 mv2? =0, (2.50) or, using the definition of magnetic moment, d 1 dt 2 mv2 k = d (µb). (2.51) dt

28 CHAPTER 2. MOTION IN EXTERNAL FIELDS By equating the right hand sides of Eq. (2.49) and (2.51) we obtain d dt (µb) = µdb dt (2.52) which is satisfied for constant values of µ only. (A full proof requires to show that µ is constant also in the case of magnetic field slowly varying in time. We omit such a proof here.) In conclusion, if r B is nearly parallel to B the magnetic moment remains constant as particles move. An important application of such a results concerns the magnetic mirror. Magnetic mirror Figure 2.10: Magnetic mirror: magnetic field topology We consider a cylindrically symmetric magnetic mirror, with field topology as shown on Fig 2.10. Two cylindrical coils, at some distance from each other generate a magnetic field, which is stronger close to the coils and weaker in the region between the coils. We now show that such a configuration allows for a (partial) confinement of plasma particles. Let us consider a particle close to the axis at position z = 0, and with velocity v 0 forming an angle 0 with the mirror axis (see Fig. 2.11). Due to the constancy of the magnetic moment v? 2 v?0 2 = B B 0. (2.53) Hence as the particle moves toward the region of higher magnetic field the component of the velocity orthogonal to the field, v? increases. At the same time, since kinetic energy is conserved, the parallel component decreases. When the orthogonal component becomes so large that the parallel component vanishes,

2.4. NON-UNIFORM MAGNETIC FIELD 29 Figure 2.11: Above: reflected particle guiding centre motion; below: velocity components in a position close to the origin of the coordinate system. Figure 2.12: Magnetic mirror: loss cone the particle is reflected, and therefore it is contained in the space between the two coils. However, particles with too small value of 0 cannot be reflected. To show this we proceed as it follows. Reflection occurs when This condition can also be written as or v 2 0 v 2 k = v2 0 v 2? =0. v0 2 v?0 2 B =0, (2.54) B 0 1 B sin 2 0 =0. (2.55) B 0 It follows that particles with central velocities forming an angle 0 with the

30 CHAPTER 2. MOTION IN EXTERNAL FIELDS mirror axis, such that r B0 0 < m = arcsin (2.56) B m cannot be confined, and are then lost by the mirror. This defines a loss cone; see Fig. 2.12. A word of caution: consideration of particle collisions and of kinetic e ects makes the study of mirrors substantially more complex.