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 It is customary to take the coordinate system oriented so that kˆ is in the direction of B (i.e. B = Bkˆ). Then Eq. (4.1) gives ˆi jˆ kˆ m q v v v v˙ = x y z (4.2) 00B
and the separate component equations are
mv˙x = qBvy mv˙y = −qBvx mv˙z =0. (4.3)
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 q2B2 v¨ = v˙ = − v (4.4) x m y m2 x
and similarly for vy 2 v¨y = −ωc vy (4.5) where we have introduced the cyclotron frequency |q | B ω . c = m (4.6)
For B = 1 Tesla we find ωce =28GHzandωci =15.2 MHz (proton). Ions gyrate much more slowly due to their greater mass. The solution to Eq. (4.4) can be written as
vx = v⊥ exp (iωct) (4.7)
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 m mω v v i c v ω t ± v ω t y = qB ˙x = qB ⊥ exp (i c )= i ⊥ exp (i c ) (4.8) 4.2 Constant and Uniform Fields 89 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) x y v2 v2 1/2 v and the resultant speed in the transverse – plane is ( x + y) = ⊥.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 =0tot gives
iv⊥ x − x0 = − exp (iωct) ωc v⊥ y − y0 = ± exp (iωct) (4.9) ωc where (x0,y0) are constants of integration. Taking real parts gives
x − x0 = rL sin (ωct)
y − y0 = ±rL cos (ωct) (4.10) with 2 2 2 (x − x0) +(y − y0) = rL and we have introduced the Larmor radius
v⊥ mv⊥ 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 mag- netic 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,oppositeq.The 2 current flowing in the loop is I = q(ωc/2π) and the loop area is A = πrL so that the magnetic dipole moment IA (proportional to the excluded magnetic 90 B X
-
+ Guiding centre
Figure 4.1: Electrons and ions spiral about the lines of force. The ions are left- handed and electrons right. The magnetic field is taken out of the page
flux ∆BA)is
µ = IA magnetic moment 2 qωc πv⊥ = 2 2π ωc mv2 = ⊥ (4.12) 2B 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 4.2 Constant and Uniform Fields 91 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 oppo- site 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
mv˙ = qE (4.13) describing a free acceleration along B. The perpendicular motion is described by
mv˙ ⊥ = q(E⊥ + v⊥×B). (4.14)
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 2 = vEB − B(vE.B) (4.17) where we have used the vector identity (A×B)×C = B(C.A) − A(C.B). (4.18) 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 v . E = B2 (4.19) 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 ˆ v = 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 im- mersed 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 v . F = q B2 (4.21) An example is the gravitational drift F = mg which gives m g×B v = . (4.22) g 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: q n q n jg = e eve + i ivi 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). 4.3 Time Varying Fields 93
Figure 4.3: The orbit in 3-D for a charged particle in uniform electric and mag- netic 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 2 (ω ωc) and transform to the frame moving with velocity vE =(E×B)/B 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 4.3 Time Varying Fields 95 analogously by translating to the frame moving with velocity m v˙ ×B v = − E P q B2 1 E˙ = ± (4.24) ωc B to give (show this) q ¨ d m − q − × − E . [ (vc vP )] = (vc vP ) B 2 (4.25) dt ωc
2 The explicit E dependent term is now O(ω/ωc) 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 v = 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 ne − jP = (vP i vP e) ne dE = (mi + me) eB2 dt ρ dE = (4.26) B2 dt
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 dis- placement 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 ∂ ε ε E . jP = r 0 ∂t (4.28) Comparing with Eq. (4.26) we obtain an expression for the low-frequency plasma electric susceptibility ρ µ ρ ε 0 r = 2 = 2 ε0B ε0 µ0B c2 ≡ v2 (4.29) A 96
where B vA = (4.30) µ0ρ
is the Alfv´en wave speed. Typically, vA c so εr 1.
4.3.2 Electric field with arbitrary time variation As before we consider fields uniform in space but that are now harmonic in time
E ≡ E exp (−iωt). (4.31)
Since the equation of motion is linear, any time variation can be expressed as a composition of Fourier components ∞ dω E(t)= E(ω)exp(−iωt) . (4.32) −∞ 2π We decompose the solution to the Lorentz equation into the sum of a magnetically driven term vc (the Larmor motion) and the harmonic polarization term vP = vP exp (−iωt). Substituting into Eq. (4.1) gives dv m c − iωv exp (−iωt) = q [E exp (−iωt)+v ×B + v ×B exp (−iωt)] . dt P c P (4.33) This equation can be separated: dv m c = qv ×B Cyclotron motion (4.34) dt c −iωmvP = q(E + vP ×B) Polarization drift (4.35)
To solve Eq. (4.35) we break it into its components parallel and perpendiclar to B.Then
vP = vP + vP ⊥ (4.36) 1 q v = − E (4.37) P iω m q B∗ . vP ⊥ = mE⊥ (4.38) where B∗ is the complex conjugate of the vector operator q B ω × . = i + mB (4.39)
Because the natural motion in the plane perpendicular to B is circular, it would seem that a reasonable simplification could be obtained by expressing the driving 4.3 Time Varying Fields 97
field E⊥ as the sum of left and right hand circularly fields:
E⊥ = EL + ER (4.40) 1 E = E⊥ − iBˆ ×E⊥ (4.41) L 2 1 E = E⊥ +iBˆ ×E⊥ (4.42) R 2 where Bˆ ≡ kˆ. The imaginary term is the orthogonal electric field component ◦ retarded or advanced in phase by 90 compared with E⊥ as shown in Fig. 4.6. The linearly polarized field E⊥ is equivalent to the sum of left and right circularly
Figure 4.6: The decomposition of E⊥ into left and right handed components. polarized fields. To solve Eq. (4.38) we first note the result that BB∗ is a scalar operator: q q BB∗ ≡ ω × − ω × vP ⊥ i + mB i + mB vP ⊥ 2 2 q ω v ⊥ B×B×v ⊥ = P + m2 P 2 2 =(ω − ωc )vP ⊥. (4.43) 98
2 In obtaining this relation we have used the fact that B×B×vP ⊥ = −B vP ⊥. Moreover, the left and right handed fields are eigenvectors of this operator: q B 1 ω × ˆ × ER = i + mB E⊥ +iB E⊥ 2 1 qB = iω(E⊥ +iBˆ ×E⊥)+ (Bˆ ×E⊥ − iE⊥) 2 m 1 = iω(E⊥ +iBˆ ×E⊥) ∓ iωc(E⊥ +iBˆ ×E⊥) 2 =i(ω ∓ ωc)ER (4.44)
wherewehaveusedωc =|q| B/m and the minus and plus signs are for ions and electrons respectively. Similarly for the left hand field we have
BEL =i(ω ± ωc)EL. (4.45) Operating on the left of Eq. (4.38) with the operator B and using Eq. (4.40) together with results (4.43), (4.44) and (4.45) gives q ω2 − ω2 ω ∓ ω ω ± ω . ( c )vP ⊥ =im [( c)ER +( c)EL] (4.46) Finally, decomposing the perpendicular polarization velocity into left and right hand components vP ⊥ = vL + vR allows the solution for the guiding centre drift in the oscillating electric field to be written
q ER vR =i (4.47) m (ω ± ωc)
q EL vL =i . (4.48) m (ω ∓ ωc)
Note that for positive ions, there is a resonance between the ions and the left handed wave as ω → ωci. The reverse is true for electrons. To obtain the resonance behaviour, we must start with the Lorentz equation and set ω = ωc. The total particle motion is obtained by combining vc, vP , vR and vL.Itis convenient to represent this combined response to the driving field in the form ω 00 vR q ω ± ωc ER i ω vL = EL (4.49) mω 0 ω ∓ ω 0 v c E P 001 or ↔ vP =µ E (4.50) ↔ where µ is the mobility tensor. This should be compared with the scalar mobility in the absence of a B-field µ =|q | /mν. 4.3 Time Varying Fields 99
The conductivity tensor for a collisionless magnetized plasma is obtained using
j = ne(ui − ue) ↔ ↔ = ne(µi − µe)E ↔ = σ E (4.51) where ↔ ↔ ↔ σ = σ i + σe ↔ ↔ σ i ne µ i . e = e (4.52) In Cartesian coordinates, we obtain 2 ω ±iωcω ω2 − ω2 ω2 − ω2 0 ne2 c c ↔ i 2 σ i ∓iωcω ω . e= (4.53) mω 2 2 2 2 0 ω − ωc ω − ωc 001 The factor i indicates that the current and the applied electric field are 90 degrees out of phase.
Synchrotoron emission
At ω = ωci or ω = ωce (resonance for ions or electrons) it can be shown that the solution for the perpendicular component of the particle velocity is (for the ions) q t − ω t . v⊥ = vc + mEL exp ( i ci ) (4.54) The first term represents the usual cyclotron motion. The second term is a con- stant acceleration which causes the Larmor radius to increase linearly with time. However, an accelerating charge radiates energy in the form of electromagnetic waves at a rate [6] K e2 d a2. = 3 (4.55) dt 6πε0mc This non-relativistic expression can be integrated to show that
K⊥ = K⊥0 exp (−t/τR) (4.56) where K⊥ is the energy of gyration of the particle and the radiative decay time constant is 3 2 2 τR =3πε0mc /e ωc . (4.57) 3 Since τR scales as m radiation damping through cyclotron emission (or magnetic bremstrahhlung) can be important only for electrons. For fusion relevant condi- tions, this time constant is in the range 1 to 10 s and is thus considerable larger 100
than other plasma characteristic times such as the energy and particle confine- ment times. The radiative loss is also overestimated, since the radiation can be reabsorbed by the plasma. Indeed, the inverse process is used to provide resonant heating of the plasma as indicated by Eq. (4.54).
Low frequency limit It is instructive to show that the low-frequency polarization drift is recovered in the limit ω/ωc 1. In this limit, the velocity is expressed by ω2 iω − ∓ v ω2 ω 0 E x ± q c c x i 2 v = iω ω . 0 exp (−iωt). (4.58) y mω ± − v 2 0 z ωc ωc 0 001
This reduces to q −ω2 v i E − ωt x = 2 x exp ( i ) mω ωc q ∂ v ˆ E xi = 2 mωc ∂t 1 ∂E = ± (4.59) ωcB ∂t
which is the same as Eq. (4.24) with the plus sign for ions and the minus for electrons. What is the interpretation of the non-zero vy response to the field Ex?
Plasma dielectric tensor (no collisions) ↔ We may also now derive an expression for the plasma dielectric tensor ε valid at all frequencies (but without the effects of collisions - this is a single particle picture!) by following the procedure used in the low frequency case. The dielectric tensor is extremely important to an understanding of wave propagation in a plasma. We start with Maxwell’s equation ∂ ∇× µ ε E . B = 0 j + 0 ∂t (4.60)
Considering the plasma as a dielectric, we write this as ∂ ∇× µ D B = 0 ∂t (4.61) 4.3 Time Varying Fields 101 with 1 D = ε0E − j iω 1 ↔ = ε0E − σ E iω ↔ = ε0 εr E ↔ ≡ ε E (4.62) where ↔ ↔ i ↔ ε= ε0 I + σ (4.63) ε0ω ↔ is the dielectric tensor, I is the unit tensor and the conductivity tensor is given by Eq. (4.53). We can now derive the wave equation in a plasma: ∂ ∇× − B E = ∂t ↔ ∇× ⇒∇×∇×E = −µ0 ε E¨ . (4.64)
The solution is examined in later chapters.
4.3.3 Slowly time varying magnetic field Generally speaking, the magnetic field acts perpendicularly to the particle ve- locity and no work is done so that the change in kinetic energy of the particle might be expected to be zero when the field strength changes. However, when ∂B/∂t = 0 there is an associated induced emf which acts on the particle orbit: ∂ ∇× − B . E = ∂t (4.65)
Assuming the rate of change of B is small compared with ωc [i.e.(1/B)(∂B/∂t) ωc] then the work done on the particle during a cycle can be evaluated over the unperturbed particle trajectory. Now work done equals change in kinetic energy, so 2 δ(mv⊥/2) = F.dl = q E.dl = q ∇×E.dS S ∂B = −q .dS S ∂t ∂B |q | πr2 = ∂t L (4.66) 102
B.ds >0 electrons B.ds <0 ions B
+ ds
Figure 4.7: When the magnetic field changes in time, the induced electric field does work on the cyclotron orbit.
where we take the absolute value of the charge because the flux B.dS is of opposite sign for ions and electrons as seen in Fig. 4.7. The change in B that occurs over one orbit is ∂B ∂B 2π δB = δt = ∂t ∂t ωc so that ωc δ(mv2 /2) = |q | πr2 δB ⊥ L 2π mv2 = ⊥ δB 2B = |µ| δB (4.67) where mv2 |µ|≡ µ = ⊥ (4.68) 2B is the magnitude of the orbital magnetic dipole moment of the charged particle encountered earlier [see Eq. (4.12)]. 2 Note that the left side gives δ(mv⊥/2) = δ(µB)=µδB + Bδµ. Comparing with the right side of Eq. (4.67) shows that, for slowly varying magnetic fields, δµ =0. (4.69) In other words, the magnetic moment is invariant (a conserved quantity) for slowly changing fields. Now δµ =0 2 ⇒ mv⊥/B = constant 2 ⇒ BrL ∝ Φ= constant 4.4 Inhomogeneous Fields 103 where Φ is the magnetic flux linked by the particle orbit. Thus, if the magnetic field increases (decreases) slowly compared with ωc, the orbit radius decreases (increases) in such a way that the particle always encircles the same number of magnetic “lines of force”.
4.4 Inhomogeneous Fields
4.4.1 Nonuniform magnetic field
Grad B drift
y ∆ |Β| B + x - z B
Figure 4.8: The grad B drift is caused by the spatial inhomogeneity of B.Itis in opposite directions for electrons and ions but of same magnitude.
In this case we consider E = 0. As alluded in the introduction, we Taylor expand the variation of B, B = bkˆ, assuming that the variation in B across a Larmor orbit is small. This obtains ∂B B B y ... = 0 + ∂y + (4.70) wherewehaveassumedthatB varies only in the y-direction and that the first order term is small. Since we consider variation in y of order the Larmor radius rL,werequire ∂B y < r B/ ∼ L ∼ L ( ∂y ) where L is the scale length for variation of B. Substituting into Eq. (4.3) and using Eq. (4.10) we obtain
mv˙ = −qv B y x ∂B −qv ω t B ± r ω t . = ⊥ cos ( c ) 0 L cos ( c ) ∂y (4.71) 104
Since the B-field is time invariant, we can average over a cyclotron period ∂B F mv ∓qv r 2 ω t y = ˙y = ⊥ L ∂y cos ( c ) (4.72)
so that there is a residual y-force (but no x directed force - show this). The resulting drift is given by Eq. (4.21) 1 F ×B v∇ = B q B2 1 F B = y ˆi q B2
v⊥rL ∂B = ∓ ˆi. (4.73) 2B ∂y Alternatively, this can be expressed in vector form ˆi jˆ kˆ B ×∇B 00 z B = ∂B (4.74) 0 ∂y 0
where ∂ | | ∂ | | ∂ | | ∇B ≡∇| | ˆ B ˆ B ˆ B . B = i ∂x + j ∂y + k ∂z
∇|B| often simplifies to ∇Bz because Bz Br,Bθ. The general result is 1 B×∇B v∇ = ± v⊥rL . (4.75) B 2 B2 The drift is in opposite directions for electrons and ions (see Fig. 4.8) but of the same magnitude. The drift therefore results in a net current across B.
Curvature drift If the magnetic lines of force are curved, the charged particles feel a centrifugal force proportional to the radius of curvature Rc (see Fig. 4.9). The force felt is
2 2 mv mv Rc F = rˆ = (4.76) c R R2 c c and the resulting drift can be written as
2 mv R ×B v c R = qB2 R2 (4.77) c 4.4 Inhomogeneous Fields 105
r^ F c
θ^
B
R c
Figure 4.9: The curvature drift arises due to the bending of lines of force. Again this force depends on the sign of the charge.
Combined grad B and curvature drifts Consider the ∇B drift that accompanies curvature in a cylindrical geometry: ˆ B = Bθ =(B0/r)θ so ∂B /r ∇B 0 − B /r2 − B /r − B /r2 = rˆ ∂r = rˆ( 0 )= rˆ θ = r( θ ) wherewehaveused∇×B = 0 in vacuum and ∂rB ∇× 1 θ ⇒ B ∼ 1 ( B)z = r ∂r θ r Using Eq. (4.75) we have 1 B×∇B v∇ = ± v⊥rL B 2 B2 v2 × − | | ±1 ⊥ B ( Rc B ) = ω B2R2 2 c c 1 mv2 R ×B = ⊥ c (4.78) q R2B2 2 c wherewehaveusedB/ωc = m/ |q|. Combining with the curvature drift we find m Rc×B 2 1 2 v = v∇ + v = v + v . (4.79) T B R q R2B2 ⊥ c 2 106
Figure 4.10: The grad B drift for a cylindrical field.
2 Note that the two contributions add with similar magnitude because v ∼ k T/m 1 v2 ∼k T/m B and 2 ⊥ B .
Magnetic mirrors — ∇B B We have looked at particle drifts when ∇B is at an angle to B. What happens when the gradient is aligned with B? This situation is encountered in magnetic mirrors where the magnetic field strength increases along the direction of the lines of force as shown in Fig. 4.11.
Figure 4.11: Schematic diagram showing lines of force in a magnetic mirror device.
We shall show that a charged particle inside such a magnetic topology can be trapped under certain circumstances. Let’s describe mathematically the field structure. The field must be divergence free (no sources or sinks): ∇.B =0.In 4.4 Inhomogeneous Fields 107 cylindrical geometry this gives ∂rB ∂B 1 r z . r ∂r + ∂z =0 (4.80)
Provided ∂Bz/∂z does not vary much with r we have r 2 ∂Bz r ∂Bz rBr = − r dr ≈− (4.81) 0 ∂z 2 ∂z or r ∂B B = − z . (4.82) r 2 ∂z ˆ Any radial inhomogeneity of Br gives an azimuthal drift Bzk×∇Brrˆ about the axis of symmetry [see Fig. 4.11] but there is no radial drift (why?). What is the effect of the Lorentz force in the cylindrical field? rˆ θˆ zˆ q × v v v F = v B = r θ z Br 0 Bz ˆ = rˆ(qvθBz) − θq(vrBz − vzBr) − zˆ(qvθBr). (4.83)
For simplicity, consider a particle spiralling along the axis (r = rL)sothatwe can ignore grad B drifts. The logitudinal (axial) force is
q ∂Bz F = v rL z θ 2 ∂z q ∂Bz = ∓v⊥rL ions and electrons 2 ∂z qv2 ∂B = ∓ ⊥ z 2ωc ∂z 2 mv⊥ = − ∇ B 2B where v⊥ is the cyclotron speed. Expressed in terms of the magnetic moment,
F = −µ∇ B. (4.84)
This force is away from increasing B and is equal for particles of equal energy (independent of charge). A particle moving from a weak field region to a strong field sees a time chang- ing magnetic field. However, the magnetic moment stays constant during this motion provided the rate of change is slow. Since µ is a constant of the motion, then v2 v2 ⊥0 = ⊥m (4.85) B0 Bm 108
where the subscript 0 refers to the low field conditions and subscript m is for the high field “mirror” region. Thus if Bm >B0 then v⊥m >v⊥0. However, the B-field does no work so that the total particle kinetic energy remains unchanged: K m v2 v2 / v If B is high enough, the particle can be stopped and F forces the particle back into the plasma body in the low field region (see Fig. 4.12). Not all particles are trapped by the mirror. At the mirror point, the condition Eq. (4.85) requires that B v2 m ⊥m = 2 B0 v⊥0 2 2 v 0 + v⊥0 = 2 v⊥0 1 = 2 (4.87) sin θm ≡ Rm Mirror ratio (4.88) wherewehaveusedthefactthatkineticenergyK is conserved. The angle θm is shown in Fig. 4.13. Particles having θ<θm (i.e. high v ) penetrate the mirror and are lost. Particles not in the loss cone, i.e. having θ>θm are confined (electrons and ions equally). After loss cones are depleted, particles are scattered by collisions into this region of velocity space. Since the electron collision rate is higher, however, they are preferentially lost and the plasma acquires a positive potential. 4.4.2 Particle drifts in a toroidal field The ramifications of field curvature and inhomogeneity are clearly evident for toroidal magnetic fields. For the toroidal field, the radius of curvature vector is normal to the magnetic field line Rc.B = 0. For a typical charged particle we v2 1 v2 take = 2 ⊥ so that Eq. (4.79) gives 2 mv⊥ vT = qRcB rL ∼ vth (4.89) Rc 4.4 Inhomogeneous Fields 109 Figure 4.12: Top:The flux linked by the particle orbit remains constant as the particle moves into regions of higher field. The particle is reflected at the point where v = 0. Bottom: Showing plasma confined by magnetic mirror where the drift is up or down for electrons or ions (see Fig. 4.10). We thus obtain v ±rL T ∼ ≡ κ. (4.90) vth Rc For H-1NF, κ ≈ 1 × 10−3 (drift angle to field line) so that the toroidal travel distance for a particle to drift out of the magnetic volume is dT =0.1m/κ = 100 m which is about 16 toroidal orbits. As already noted, the electrons and ions drift in opposite directions. This generates a vertical electric field as shown in Fig. 4.14. The resulting E×B drift pushes the plasma to the wall and the plasma is not confined. This problem can be remedied by twisting the field lines (by introducing a toroidal current). Particles moving freely along B will then short out the 110 vz v θ m v⊥ vy vx Figure 4.13: Particles having velocities in the loss cone are preferentially lost potential difference. Another way of thinking of this is to note that at the top of the torus the particle is drifting out, while at the bottom, it is drifting inwards. These two drifts compensate. The helicity of the lines of force is called the rotational transform and is shown in Fig. 4.15 4.4.3 Particle orbits in a tokamak field The toroidal magnetic field in a tokamak varies inversely with major radius. To see this, let us apply Ampere’s law around a closed circular loop threading the torus. We assume this imaginary loop encloses N toroidal field coils each carrying current I in the same direction. We then have B.dl =2πRB = µ0NI 4.4 Inhomogeneous Fields 111 Figure 4.14: The grad B drift separates vertically the electrons and ions. The resulting electric field and E/B drift pushes the plasma outwards. Figure 4.15: A helical twist (rotational transform) of the toroidal lines of force is introduced with the induction of toroidal current in the tokamak. Electrons follow the magnetic lines toroidally and short out the charge separation caused by the grad B drift. µ0NI B = . (4.91) 2πR 112 To first order in = r/R0,whereR0 is the radius of the magnetic axis, the field strength at some point in the torus is (show this) B = B0(1 − cos θ) (4.92) where θ is the poloidal angle coordinate with respect to the magnetic axis and B0 is the magnetic field strength on axis. (See Fig. 4.16 for the coordinate system used here). If we now follow an electron along a helical field line, then if it starts at the outside of the torus and moves towards the torus axis (inside), the magnetic field increases. As a result, some of the electrons will be reflected (at points P and Q Figure 4.16: Diagram showing toridal magnetic geometry in Fig. 4.17) by the mirror effect. The projection of the guiding centre drift onto the R–z plane shows a banana-shaped orbit. Between the reflection points there is an upward drift due to curvature and ∇B. Typically, the banana width for < a tokamak ∼ 0.1a where a is the minor radius. Particles with sufficiently high parallel velocity will penetrate the magnetic well and complete a toroidal circuit. These are called passing particles. 4.4 Inhomogeneous Fields 113 Passing orbits As the particle moves freely toroidally, its orbit projected onto the R-z (poloidal) plane is given by dx dz = −Ωz =Ωx + v (4.93) dt dt z wherewehavetakenx = R − R0 and where the vertical z drift velocity is given by m 2 2 vz = (2v + v⊥). (4.94) 2qBφR In Eq. (4.93) Ω = dθ/dt is the angular velocity of the particle orbit projected onto the poloidal plane (imagine the torus straightened into a cylinder and looking along the axis of the helical magnetic field line). The rotation in this plane arises from the helicity of the magnetic field line. The rate of spiralling of the field line is given by rdθ Rdφ = Bθ B0 and is characterised by the so-called winding number or safety factor q(r)which is defined by dφ rB0 B0 q(r)= = = . (4.95) dθ RBθ Bθ In tokamaks, q typically lies in the range 0.7 ˆ with vφ = v.φ. Aside: In stellarator studies (as opposed to tokamaks), it is more usual to talk of the rotational transform ι/2π =1/q of the field line, defined as the number of turns poloidally (the short way) per turn toroidally (the long way) (or “twist per < turn”). Typically ι ∼ 1. The projected particle orbit (ignoring variations in the vertical drift velocity vz)is 2 2 (x + vz/Ω) + z = constant. The constant depends on the radius of the magnetic surface formed by the helical field lines on which the particle travels. The particle departs from this surface (due to inhomogeneity and curvature drifts) by v ∆=− z Ω m v2 q r R ∼ th ( ) 0 qB R0 vth 114 vth = q(r) ωc = q(r)rL. (4.97) This shift is shown in Fig. 4.17 Trapped particles 2 Let’s now assume that (v⊥/v ) > 1/ so that particles are trapped and bounce in the mirrors produced by the 1/R variation of the toroidal field. We wish to find a relationship between the radial motion of trapped particles and their parallel velocity. Geometry tells that the radial component of the vertical drift velocity is vr = vz sin θ 2 mv⊥ 2 2 = sin θ (for v v⊥). (4.98) 2qB0R The parallel force felt by the particle due to the increasing toroidal field is ∂B mv −µ ˙ = ∂ (4.99) where ≈ Rφ is the distance coordinate along the magnetic field line. The coordinate is related to the poloidal angle θ through the safety factor q: rB0 ≈ Rφ = θ Bθ or B θ = κ with κ ≡ θ rB0 where r is the radius of the field line with respect to the magnetic axis. The right side of Eq. (4.99) can now be evaluated as ∂B ∂ B − κ ∂ = ∂ [ 0(1 cos )] = B0 κ sin κ r Bθ = B0 sin θ R0 rB0 B = θ sin θ (4.100) R0 where we have used Eq. (4.92). Equation (4.99) now gives 2 mv⊥ Bθ v˙ = − sin θ 2B0 mR0 v2 B = − ⊥ θ sin θ. (4.101) 2 R0B0 4.4 Inhomogeneous Fields 115 Comparing with Eq. (4.98) we obtain m vr = − v˙ qBθ which integrates to give m r − r0 = − v (4.102) qBθ where r0 is the turning point at which v = 0. Equation (4.102) describes the poloidal projection of the banana orbit. To obtain the orbit width, we use 2 Eq. (4.86) with K ≈ mv⊥/2 to obtain for the change in v around the orbit 1/2 δv 2 K − µB = m( ) 1/2 ≈ v⊥ (4.103) where we have substituted from Eq. (4.92) for B.Alsonotethat qB qB0 ωc ω = θ ≈ = (4.104) cθ m q(r) m q(r) so that Eq. (4.102) becomes δv 1/2 δr = ≈ q(r)rL/ . (4.105) ωcθ The width of the banana orbit is bigger again by the factor −1/2 than the passing particles. This has profound consequences, with neoclassical diffusion increased again over the cylindrical value. Problems Problem 4.1 The polarization drift vP can also be derived from energy conserva- × 1 mv2 tion. If E is oscillating, the E B drift also oscillates, and there is an energy 2 E associated with the guiding centre motion. Since energy can be gained from an E field only be motion along E, there must be a drift vP in the E direction. By equat- ing the rate of change of energy gain from vP .E, find the required value of vP . HINT: vP and E are in quadrature. Problem 4.2 A1keVprotonwithv =0in a uniform magnetic field B =0.1T is accelerated as B is slowly increased to 1T. It then makes an elastic collision with a heavy particle and changes direction so that v = v⊥.TheB field is then slowly decreased back to 0.1 T. What is the proton energy now? 116 ∆ Flux surface Passing Trapped Figure 4.17: Top: Schematic diagram of trajectory of “banana orbit” in a toka- mak field. Bottom: The projection of passing and banana-trapped orbits onto the poloidal plane. 4.4 Inhomogeneous Fields 117 Problem 4.3 Consider the magnetic mirror system shown below. Suppose that the axial magnetic field is given by 2 B(z)=B0[1 + (z/a) ] where B0 and a0 are positive constants and that the mirroring planes are at z = ±zm. (a) For a charged particle just trapped in this system, show that the z component of the particle velocity is given by 2 2 1/2 v (z)= (2µB0/m)[(zm/a) − (z/a0) ] (b) The average force acting on the particle guiding centre along the z axis is given by ∂B F −µ = ∂z zˆ Show that the particle executes a simple harmonic motion between the mirroring planes with a period given by 1/2 T =2πa0[m/(2µB0)] . Problem 4.4 A plasma with an isotropic velocity distribution is placed in a magnetic mirror trap with a mirror ratio Rm =4. There are no collisions so that the particles in the loss cone simply escape and the rest remain trapped. What fraction is trapped? Problem 4.5 Consider the motion of an electron in the presence of a uniform mag- netostatic field B = B0zˆ, and an electric field that oscillates in time at the electron cyclotron frequency ωc according to E(t)=E0[xˆ cos(ωct)+yˆ sin(ωct)] (a) What type of polarization has this electric field? (b) Obtain the following uncoupled differential equations satisfied by the velocity components vx(t) and vy(t): 2 d vx 2 + ω v =2(eE0/m)ω sin(ω t) dt2 c x c c 2 d vy 2 + ω v = −2(eE0/m)ω cos(ω t) dt2 c y c c 118 (c) Assume that at t =0, the electron is located at the origin of the coordinate system with zero velocity. Neglect the time varying par of B. Verify that the electron velocity is given by vx(t)=−(eE0/m)t cos(ωct) vy(t)=−(eE0/m)t sin(ωct) and that its trajectory is given by x t − eE /m /ω2 ω t t/ω ω t − /ω2 ( )= ( 0 )[(1 c )cos( c )+( c)sin( c ) 1 c ] y t − eE /m /ω2 ω t − t/ω ω t ( )= ( 0 )[(1 c )sin( c ) ( c)cos( c )]