Chapter 4 SINGLE PARTICLE MOTIONS

Chapter 4 SINGLE PARTICLE MOTIONS 4.1 Introduction We wish now to consider the effects of magnetic fields on plasma behaviour. Especially in high temperature plasma, where collisions are rare, it is important to study the single particle motions as governed by the Lorentz force in order to understand particle confinement. Unfortunately, only for the simplest geometries can exact solutions for the force equation be obtained. For example, in a constant and uniform magnetic field we find that a charged particle spirals in a helix about the line of force. This helix, however, defines a fundamental time unit – the cyclotron frequency ωc and a fundamental distance scale – the Larmor radius rL. For inhomogeneous and time varying fields whose length L and time ω scales are large compared with ωc and rL it is often possible to expand the orbit equations in rL/L and ω/ωc. In this “drift”, guiding centre or “adiabatic” approximation, the motion is decomposed into the local helical gyration together with an equation of motion for the instantaneous centre of this gyration (the guiding centre). It is found that certain adiabatic invariants of the motion greatly facilitate understanding of the motion in complex spatio-temporal fields. We commence this chapter with an analysis of particle motions in uniform and time-invariant fields. This is followed by an analysis of time-varying electric and magnetic fields and finally inhomogeneous fields. 4.2 Constant and Uniform Fields The equation of motion is the Lorentz equation dv F = m = q(E + v×B) (4.1) dt 88 4.2.1 Electric field only In this case the particle velocity increases linearly with time (i.e. accelerates) in the direction of E 4.2.2 Magnetic field only ˆ It is customary to take the coordinate system oriented so that k is in the direction ôf B (i.e. B = Bk). Then Eq. (4.1) gives ꢀꢀ ꢀꢀ<ul style="display: flex;"><li style="flex:1">ˆ</li><li style="flex:1">ˆ</li><li style="flex:1">ˆ</li></ul><ul style="display: flex;"><li style="flex:1">i</li><li style="flex:1">j</li><li style="flex:1">k</li></ul><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢀ</li></ul><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢀ</li></ul>ꢀmv˙ = q ꢀ (4.2) (4.3) vx vy vz 0ꢀꢀ ꢀꢀ0Band the separate component equations are <ul style="display: flex;"><li style="flex:1">mv˙x = qBvy </li><li style="flex:1">mv˙y = −qBvx </li><li style="flex:1">mv˙z = 0. </li></ul>The magnetic field acts perpendicularly to the particle velocity so that there is no force in the z direction and we write vz = v = constant. It is clear that ꢀthe x and y motions are closely coupled. Taking the time derivative allows the equations to be decoupled. For vx we obtain qB mq2B2 m2 v¨x = v˙y = − vx (4.4) (4.5) and similarly for vy vÿ = −ωc2vy where we have introduced the cyclotron frequency |q| B ωc = .(4.6) mFor B = 1 Tesla we find ωce = 28 GHz and ωci = 15.2 MHz (proton). Ions gyrate much more slowly due to their greater mass. The solution to Eq. (4.4) can be written as <ul style="display: flex;"><li style="flex:1">vx = v exp (iωct) </li><li style="flex:1">(4.7) </li></ul>⊥with the convention that we take the real part (vx = v cos ωct). Substituting ⊥Eq. (4.7) into Eq. (4.3) gives an expression for vy <ul style="display: flex;"><li style="flex:1">m</li><li style="flex:1">imωc </li></ul>vy = <ul style="display: flex;"><li style="flex:1">v˙x = </li><li style="flex:1">v exp (iωct) = iv exp (iωct) </li><li style="flex:1">(4.8) </li></ul><ul style="display: flex;"><li style="flex:1">⊥</li><li style="flex:1">⊥</li></ul><ul style="display: flex;"><li style="flex:1">qB </li><li style="flex:1">qB </li></ul><ul style="display: flex;"><li style="flex:1">4.2 Constant and Uniform Fields </li><li style="flex:1">89 </li></ul>where in the last step we have substituted q = e for ions and electrons and the plus sign for vy is for protons and the minus for electrons. Taking the real part gives vy = ∓v sin (ωct) ⊥and the resultant speed in the transverse x–y plane is (vx2 + vy2)1/2 = v . The ⊥transverse velocity v can be regarded as an initial condition in the solution to ⊥Eq. (4.3). We can integrate the equations once more to obtain the particle trajectory. For this, it is convenient to use the complex forms. Integrating from t = 0 to t gives iv ⊥x − x0 = − exp (iωct) ωc v⊥y − y0 <ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">exp (iωct) </li><li style="flex:1">(4.9) </li></ul>ωc where (x0, y0) are constants of integration. Taking real parts gives x − x0 = rL sin (ωct) y − y0 <ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">rL cos (ωct) </li><li style="flex:1">(4.10) </li></ul>with (x − x0)2 + (y − y0)2 = rL2 and we have introduced the Larmor radius <ul style="display: flex;"><li style="flex:1">v</li><li style="flex:1">mv </li></ul><ul style="display: flex;"><li style="flex:1">⊥</li><li style="flex:1">⊥</li></ul>rL = =.(4.11) ωc |q| B In the frame of reference moving at velocity v the orbit is a circle of radius rL ꢀand guiding centre (x0, y0). The ions gyrate in the left-handed sense and the electrons are right-handed (see Fig. 4.1). Charged particles follow the lines of force provided there are no electric fields (unless E is parallel to B) and that the B-field is homogeneous. Diamagnetism The spiralling particles are themselves current loops and generate their own magnetic induction. Consider that generated by the ions. With reference to Fig. 4.1 it is clear that inside the orbit, the induction is into the page, i.e. opposite the direction of B. The same is true for the electrons - opposite v, opposite q. The current flowing in the loop is I = q(ωc/2π) and the loop area is A = πrL2 so that the magnetic dipole moment IA (proportional to the excluded magnetic 90 BX-Guiding centre +Figure 4.1: Electrons and ions spiral about the lines of force. The ions are lefthanded and electrons right. The magnetic field is taken out of the page flux ∆BA) is µ = IA magnetic moment qωc πv⊥2 ==2π ωc2 mv⊥2 2B (4.12) which is proportional to the perpendicular kinetic energy over the field strength. The important point is that plasmas are “diamagnetic” – all particle generated fluxes add to reduce the ambient field. The total change in B is proportional to the total perpendicular charged particle kinetic energy. The greater the plasma thermal energy, the more it excludes the magnetic field. This results in a balance between the thermal and magnetic pressures as we shall see later. A loop external to the plasma and encircling it will measure the flux excluded by the plasma as the particles are heated. This is a very fundamental way to measure the plasma stored perpendicular thermal energy. 4.2.3 Electric and magnetic fields Let’s consider the particular case where E is perpendicular to B as shown in Fig. 4.2. When the ion moves in the direction of E it is accelerated and the radius of its orbit increases (rL = v/ωc). However, when the ion moves against the field <ul style="display: flex;"><li style="flex:1">4.2 Constant and Uniform Fields </li><li style="flex:1">91 </li></ul>the radius decreases. The result is that the ion executes a cycloidal motion with the guiding centre drifting in the direction perpendicular to both E and B. For the electrons, the cycloidal orbits are smaller (smaller mass). However, we note the following important features: (i) Electrons and ions drift in the same direction E×B: the electron has opposite charge, but also gyrates in the opposite sense to the ions. (ii) The drift velocity for electrons and ions is the same: electrons drift less per cycle but execute more cycles per second. Figure 4.2: When immersed in orthogonal electric and magnetic fields, electrons and ions drift in the same direction and at the same velocity. We can generalize the treatment to arbitrary fields by decomposing E into its components parallel and perpendicular to B. The parallel motion is given by <ul style="display: flex;"><li style="flex:1">mv˙ = qE </li><li style="flex:1">(4.13) </li></ul>describing a free acceleration along B. The perpendicular motion is described by mv˙ = q(E + v ×B). (4.14) <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢀ</li></ul><ul style="display: flex;"><li style="flex:1">⊥</li><li style="flex:1">⊥</li><li style="flex:1">⊥</li></ul>Anticipating the result, we make a transformation into the reference frame moving with drift velocity vE such that v = vE + vc and Eq. (4.14) becomes mv˙ c = q(E + vE×B) + qvc×B. (4.15) ⊥In the drifting frame the velocity vc is just the cyclotron motion so that we can set E + vE×B = 0. (4.16) ⊥92 This can be solved for vE as follows: E ×B = −(vE×B)×B ⊥= vEB2 − B(vE.B) (4.17) (4.18) where we have used the vector identity (A×B)×C = B(C.A) − A(C.B). Since the left side is perpendicular to B the second term must vanish, requiring that the drift velocity must be perpendicular to B. We then obtain an expression for the drift velocity that is independent of the species charge and mass E×B vE = .(4.19) B2 Equation (4.15) describes the residual cyclotron motion of the particle about the field lines at angular frequency ωc and radius rL = vc/ωc. The total particle motion is composed of three parts ˆv = v k (along B) + vE (perpendicular drift) + vc (Larmor gyration). (4.20) ꢀIn this case, vE is the perpendicular drift velocity of the guiding centre of the Larmor orbit. When E is zero, the orbit about B is circular. When E is finite, ⊥the orbit is cycloidal. These motions are summarized in Fig. 4.3. Rotation of a cyclindrical plasma A radial electric field imposed between cyclindrical elecrodes across a plasma immersed in an axial magnetic field will cause the plasma to rotate in the azimuthal direction as shown in Fig. 4.4. 4.2.4 Generalized force We can replace qE in the Lorentz equation by a generalized force F then 1 F ×B vF = .(4.21) (4.22) q B2 An example is the gravitational drift F = mg which gives m g×B vg = .q B2 This changes sign with q and is different for different masses. This will give rise to a net current flow in a plasma: jg = qeneve + qinivi g×B = n(mi + me) B2 The magnitude of jg is usually negligible. However, curved lines of force ⇒ effective gravitational (centrifugal) force ⇒ curvature drift (more below). <ul style="display: flex;"><li style="flex:1">4.3 Time Varying Fields </li><li style="flex:1">93 </li></ul>Figure 4.3: The orbit in 3-D for a charged particle in uniform electric and magnetic fields. 4.3 Time Varying Fields 4.3.1 Slowly varying electric field When we later consider wave motions in plasma, the electric field will vary with time, and unlike the static case, a polarization current can flow. The origin of the drift is illustrated in Fig. 4.5 We assume that the electric field is uniform and perpendicular to B. The parallel component can be handled easily. We allow the field to vary slowly in time (ω ꢀ ωc) and transform to the frame moving with velocity vE = (E×B)/B2 to 94 Figure 4.4: The cylindrical plasma rotates azimuthally as a result of the radial electric and axial magnetic fields. Figure 4.5: When the electric field is changed at time t = 0, ions and electrons suffer an additional displacement as shown. The effect is opposite for each species. obtain [see Eq. (4.15)]: mv˙c = −mv˙E + qvc×B (4.23) where the first term on the right side is O(ω/ωc) and so is small compared with the left hand side. The equation has the form of Eq. (4.14) and so can treated <ul style="display: flex;"><li style="flex:1">4.3 Time Varying Fields </li><li style="flex:1">95 </li></ul>analogously by translating to the frame moving with velocity m v˙ E×B vP = − qB2 ˙1 E <ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">(4.24) </li></ul>(4.25) ωc B to give (show this) ¨qE ωc2 d[m(vc − vP )] = q(vc − vP )×B − .dt The explicit E dependent term is now O(ω/ωc)2 and can be neglected. The residual equation for vc − vP describes the Larmor motion. Averaging the total motion over a gyro-period gives the overall guiding centre drift as ꢁvꢂ = vE +vP . The new polarization drift vP given by Eq. (4.24) (correct to first order in ω/ωc) is charge dependent and points in the direction of E. The polarization current flow that results is given by jP = ne(vP i − vP e) ne eB2 dE <ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">(mi + me) </li></ul>dt ρ dE B2 dt <ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">(4.26) </li></ul>where ρ = n(mi + me) is the plasma mass density. The polarization current vanishes as ω/ωc → 0. Analogy with solid dielectric polarization For a solid dielectric immersed in an electric field we construct the electric displacement vector D = ε0E + P ≡ εrε0E (4.27) where P is the polarization vector due to the alignment of electric diploes and εr is the electric susceptibility. When the electric field varies with time, it drives the polarization current ∂E jP = εrε0 .(4.28) ∂t Comparing with Eq. (4.26) we obtain an expression for the low-frequency plasma electric susceptibility ρε0B2 c2 µ0 ε0 µ0B2 ρεr <ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">=</li></ul>≡(4.29) vA2

Chapter 4 SINGLE PARTICLE MOTIONS

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