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Guiding Center Motion GUIDING CENTER MOTION H.J. de Blank FOM Institute DIFFER – Dutch Institute for Fundamental Energy Research, Association EURATOM-FOM, P.O. Box 1207, 3430 BE Nieuwegein, The Netherlands, www.differ.nl. ABSTRACT Secondly, the Coulomb force is a long range interaction. In a well-ionized plasma, particles rarely suffer large-angle The motion of charged particles in slowly varying electro- deflections in two-particle collisions. Rather, their orbits are magnetic fields is analyzed. The strength of the magnetic deflected through weak interactions with many particles si- field is such that the gyro-period and the gyro-radius of the multaneously. Hence, the effects of collisions can be best particle motion around field lines are the shortest time and described statistically, in terms of distributions of particles. length scales of the system. The particle motion is described The kinetic equation for the particle distribution function as the sum of a fast gyro-motion and a slow drift velocity. will be discussed at the end of this chapter, with emphasis on the role of the particle orbits, not the collisions. The equations of motion of a particle with mass m and I. INTRODUCTION charge q in electromagnetic fields E(x, t) and B(x, t) are, The interparticle forces in ordinary gases are short-ranged, q x˙ = v, v˙ = (E + v B), (1) so that the constituent particles follow straight lines between m × collisions. At low densities where collisions become rare, N the gas molecules bounce up and down between the walls of where the dot denotes the time derivative. Each of the the containing vessel before experiencing a collision. plasma particles satisfies such equations. The solutions to the 6N equations are the particle trajectories. These trajec- High-temperature plasmas, however, cannot be con- tories determine the local charge and current density which tained by a material vessel, but only by magnetic fields. The are the sources in Maxwell’s equations and which determine Lorentz forces that act on the particles tie them to the mag- the electromagnetic fields E and B. In turn, these fields de- netic field and forcethem to follow the field lines. In orderto termine the particle trajectories. This self-consistent picture confine the particles in a bounded volume, the magnetic field is extremely complex. must be curved and inhomogeneous. In addition, it must be However, as illustrated above, in a weakly collisional strong. So strong, that the Lorentz force dominates all other plasma one can first study the behaviour of test particles in forces. Therefore, charged particles do not follow straight given fields E(x, t) and B(x, t). The role of the particles lines between collisions but follow strongly curved orbits as sources of charge density and current in Maxwell’s equa- under the influence of the magnetic field. In fact, many tions is disregarded. The fields E and B of course obey the properties of a magnetically confined plasma are dominated subset of Maxwell’s equations, by the motion of the particles subject to the Lorentz force qv B B . Here is the macroscopic field, i.e., the sum E = ∂B/∂t, B = 0. (2) of externally× applied field and the fields generated by the ∇ × − ∇· plasma particles collectively, but excluding the microscopic variations of the fields due to the individual particles. II. GYRATION AND DRIFT The particle motion in the macroscopic field is the sub- ject of this lecture. The microscopic fields, i.e., the interac- A. Motion in a Constant Magnetic Field tions between individual particles (“collisions”), cause devi- Let us first consider the motion of a charged particle in the ations from these particle orbits. Collisions in a plasma are presence of a constant magnetic field B, caused by Coulomb interactions between the particles, with properties that are very different from collisions in a gas. mv˙ = q(v B). Firstly, the cross-section of Coulomb collisions is a × strongly decreasing function of the energies of the interact- The kinetic particle energy remains constant because the ing particles. Hence, the mean free paths of chargedparticles Lorentz force is always perpendicularto the velocity and can in high-temperaturefusion devices are very long and the par- thus change only its direction, but not its magnitude. The ticles will trace out their trajectories over distances that can particle velocity can be decomposed into components paral- be comparable to or even larger than the size of the device lel and perpendicular to the magnetic field, v = v b + v , k ⊥ before they are swept out of their orbits by collisions. where b B/B is the unit vector in the direction of ≡ 60 (a) (b) B, v|| B Fg e1 Figure 3: ion and elec- e v v ion φ tron drifts mg/qB in a ρρρ ρρ gravitational field. e2 electron B ρρρ x B. Drift due to an Additional Force R Ο If, in addition to the Lorentz force, a constant force F acts Figure 1: Definition of the gyro-angle φ (a) and guiding center (b). on the charged particle, the equation of motion is mv˙ = q (v B) + F . (4) × B. The Lorentz force does not affect the parallel motion: The motion of the particle due to F can be separated from v = constant. Only v interacts with B, leading to a k ⊥ the gyration due to B by using the guiding center as refer- circular motion perpendicular to B. The centrifugal force ence frame. Again the guiding center position R, the posi- mv2 /r balances the Lorentz force qv B for a gyration ra- ⊥ ⊥ tion of the particle x, and the gyration radius vector ρ are dius r equal to the “Larmor radius” related as in Eq. (3). The velocity of the guiding center can mv be obtained by differentiating the equation R = x ρ, ρ ⊥ . ≡ q B − | | ˙ 1 2 vg R = x˙ ρ˙ If we set 2 mv = kT for the two dimensional thermal mo- ≡ − m ⊥ 1/2 = v + v˙ B tion B, we obtain ρ = (2mkT ) / q B. In a typical 2 ⊥ | | qB × fusion plasma (kT = 10 keV, B = 5 T) the electrons have a 1 = v + (qv B + F ) B. gyroradius of 67 µm and deuterons 4.1 mm. qB2 × × The frequency of the gyration, called cyclotron fre- quency ωc, follows from v = ωc ρ, Using (v B) B = v B2 and v v = v b we obtain ⊥ × × − ⊥ − ⊥ k ω = qB/m. c F B vg = v b + × . In fusion experiments the electron cyclotron frequency is k qB2 of the same order of magnitude as the plasma frequency. Alhough the particle motion in a constant field is elemen- Thus, one sees that any force with a component perpendic- tary, the following notation will also serve more complicated ular to B causes a particle to drift perpendicular to both F and B. The basic mechanism for a drift in this direction is cases. Let e1, e2 be unit vectors perpendicular to each other and to b, and define co-rotating unit vectors (Fig. 1(a)): a periodic variation of the gyro-radius. When a particle ac- celerates in a force field, the gyroradius increases and when e (t) = e1 cos φ + e2 sin φ, ⊥ it slows down its gyroradius decreases, leading to the non- eρ(t) = e2 cos φ e1 sin φ, φ = φ0 ωct. closed trajectories shown in Fig. 3. The net effect is a drift − − perpendicular to the force and the magnetic field. As illustrated in Fig. 1(b), the particle position x can be de- A force parallel to B does not lead to a drift, but simply composed into a guiding center position R that moves with causes a parallel acceleration as can be seen from Eq. (4). velocity v b, and a rotating gyration radius vector ρ, k Summarizing, x = R + ρ, (3a) m F B dvg, F ρ = v B = ρ sgn(q) e , (3b) vg, = ⊥ × , k = k . (5) −qB2 × ρ ⊥ qB2 dt m v = ρ˙ = v e . (3c) ⊥ ⊥ ⊥ An example is the drift due to a constant gravitational The particle trajectory is a helix around the guiding center force Fg = mg perpendicular to the magnetic field. The re- magnetic field line (Fig. 2). sulting drift velocity, vg = mg/qB, is in opposite directions for electrons and ions (see Fig. 3). The net effect is a current v magnetic density. However, in laboratory plasmas g is far to small to 8 field be of importance(2 10− m/s in a magnetic field B = 5 T). × C. E B Drift × ions electrons A different situation arises in the presence of a constant elec- Figure 2: Orientation of the gyration orbits of electrons and ions in tric force qE. Since the electric force is in opposite direc- a magnetic field. The guiding center motion is also shown. tions for electrons and ions, the resulting drift velocity, 61 ion Figure 5: Inhomogeneous mag- Figure 4: E B drift grad B × netic field. Relation between the of ions and electrons. B E curvature radius and the field gra- B electron dient in a force-free magnetic field R ( B B). c ∇ × k E B vE = × , (6) B2 The other inhomogeneity that results in a drift is the does not depend on the sign of the charge or the particles. It transverse gradient of the magnetic field strength. The parti- is also independent of the particle mass and therefore iden- cle orbit has a smaller radius of curvature on that part of its tical for ions and electrons. Hence, this drift leads to a net orbit located in the stronger magnetic field. This leads to a flow of the plasma, not to a current.
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