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 importantto study the single particle motions as governed by the Lorentz force in order tounderstand particle confinement.

Unfortunately, only for the simplest geometries can exact solutions for theforce equation be obtained. For example, in a constant and uniform magneticfield 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 frequencyωc and a fundamental distance scale – the Larmor radius rL. For inhomogeneousand time varying fields whose length L and time ω scales are large comparedwith ωc and rL it is often possible to expand the orbit equations in rL/L andω/ωc. In this “drift”, guiding centre or “adiabatic” approximation, the motion isdecomposed into the local helical gyration together with an equation of motionfor the instantaneous centre of this gyration (the guiding centre). It is found thatcertain adiabatic invariants of the motion greatly facilitate understanding of themotion in complex spatio-temporal fields.

We commence this chapter with an analysis of particle motions in uniformand time-invariant fields. This is followed by an analysis of time-varying electricand magnetic fields and finally inhomogeneous fields.

4.2 Constant and Uniform Fields

The equation of motion is the Lorentz equation

F = mdv

dt= q(E + v×B) (4.1)

88

4.2.1 Electric field only

In this case the particle velocity increases linearly with time (i.e. accelerates) inthe direction of E

4.2.2 Magnetic field only

It is customary to take the coordinate system oriented so that k is in the directionof B (i.e. B = Bk). Then Eq. (4.1) gives

mv = q

∣∣∣∣∣∣∣i j kvx vy vz

0 0 B

∣∣∣∣∣∣∣ (4.2)

and the separate component equations are

mvx = qBvy mvy = −qBvx mvz = 0. (4.3)

The magnetic field acts perpendicularly to the particle velocity so that there isno force in the z direction and we write vz = v‖ = constant. It is clear thatthe x and y motions are closely coupled. Taking the time derivative allows theequations to be decoupled. For vx we obtain

vx =qB

mvy = −q2B2

m2vx (4.4)

and similarly for vy

vy = −ω2cvy (4.5)

where we have introduced the cyclotron frequency

ωc =|q | B

m. (4.6)

For B = 1 Tesla we find ωce = 28 GHz and ωci = 15.2 MHz (proton). Ions gyratemuch 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). SubstitutingEq. (4.7) into Eq. (4.3) gives an expression for vy

vy =m

qBvx =

imωc

qBv⊥ exp (iωct) = ±iv⊥ exp (iωct) (4.8)

4.2 Constant and Uniform Fields 89

where in the last step we have substituted q = ±e for ions and electrons and theplus sign for vy is for protons and the minus for electrons. Taking the real partgives

vy = ∓v⊥ sin (ωct)

and the resultant speed in the transverse x–y plane is (v2x + v2

y)1/2 = v⊥. The

transverse velocity v⊥ can be regarded as an initial condition in the solution toEq. (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 tgives

x − x0 = − iv⊥ωc

exp (iωct)

y − y0 = ±v⊥ωc

exp (iωct) (4.9)

where (x0, y0) are constants of integration. Taking real parts gives

x − x0 = rL sin (ωct)

y − y0 = ±rL cos (ωct) (4.10)

with

(x − x0)2 + (y − y0)

2 = r2L

and we have introduced the Larmor radius

rL =v⊥ωc

=mv⊥|q | B

. (4.11)

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 theelectrons are right-handed (see Fig. 4.1). Charged particles follow the lines offorce provided there are no electric fields (unless E is parallel to B) and that theB-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. 4.1it is clear that inside the orbit, the induction is into the page, i.e. opposite thedirection of B. The same is true for the electrons - opposite v, opposite q. Thecurrent flowing in the loop is I = q(ωc/2π) and the loop area is A = πr2

L sothat the magnetic dipole moment IA (proportional to the excluded magnetic

90

+

-

XB

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

=qωc

2π

πv2⊥

ω2c

=mv2

⊥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 tothe total perpendicular charged particle kinetic energy. The greater the plasmathermal energy, the more it excludes the magnetic field. This results in a balancebetween the thermal and magnetic pressures as we shall see later. A loop externalto the plasma and encircling it will measure the flux excluded by the plasma asthe particles are heated. This is a very fundamental way to measure the plasmastored 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 radiusof 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 withthe guiding centre drifting in the direction perpendicular to both E and B. Forthe electrons, the cycloidal orbits are smaller (smaller mass). However, we notethe 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 percycle 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 intoits 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 movingwith drift velocity vE such that v = vE + vc and Eq. (4.14) becomes

mvc = q(E⊥ + vE×B) + qvc×B. (4.15)

In the drifting frame the velocity vc is just the cyclotron motion so that we canset

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)

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, requiringthat the drift velocity must be perpendicular to B. We then obtain an expressionfor the drift velocity that is independent of the species charge and mass

vE =E×B

B2. (4.19)

Equation (4.15) describes the residual cyclotron motion of the particle aboutthe field lines at angular frequency ωc and radius rL = vc/ωc. The total particlemotion is composed of three parts

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 theLarmor 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 azimuthaldirection 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

vF =1

q

F×B

B2. (4.21)

An example is the gravitational drift F = mg which gives

vg =m

q

g×B

B2. (4.22)

This changes sign with q and is different for different masses. This will give riseto a net current flow in a plasma:

jg = qeneve + qinivi

= n(mi + me)g×B

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 withtime, and unlike the static case, a polarization current can flow. The origin ofthe drift is illustrated in Fig. 4.5

We assume that the electric field is uniform and perpendicular to B. Theparallel 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)]:

mvc = −mvE + qvc×B (4.23)

where the first term on the right side is O(ω/ωc) and so is small compared withthe left hand side. The equation has the form of Eq. (4.14) and so can treated


analogously by translating to the frame moving with velocity

vP = −m

q

vE×B

B2

= ± 1

ωc

E

B(4.24)

to give (show this)

d

dt[m(vc − vP )] = q(vc − vP )×B − qE

ω2c

. (4.25)

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 〈v〉 = vE +vP . The new polarization drift vP given by Eq. (4.24) (correctto first order in ω/ωc) is charge dependent and points in the direction of E. Thepolarization current flow that results is given by

jP = ne(vP i − vP e)

=ne

eB2(mi + me)

dE

dt

=ρ

B2

dE

dt(4.26)

where ρ = n(mi + me) is the plasma mass density. The polarization currentvanishes 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 andεr is the electric susceptibility. When the electric field varies with time, it drivesthe polarization current

jP = εrε0∂E

∂t. (4.28)

Comparing with Eq. (4.26) we obtain an expression for the low-frequency plasmaelectric susceptibility

εr =ρ

ε0B2=

µ0

ε0

ρ

µ0B2

≡ c2

v2A

(4.29)

96

where

vA =B

µ0ρ(4.30)

is the Alfven 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 acomposition of Fourier components

E(t) =∫ ∞

−∞E(ω) exp (−iωt)

dω

2π. (4.32)

We decompose the solution to the Lorentz equation into the sum of a magneticallydriven term vc (the Larmor motion) and the harmonic polarization term vP =vP exp (−iωt). Substituting into Eq. (4.1) gives

m

(dvc

dt− iωvP exp (−iωt)

)= q [E exp (−iωt) + vc×B + vP ×B exp (−iωt)] .

(4.33)This equation can be separated:

mdvc

dt= qvc×B Cyclotron motion (4.34)

−iωmvP = q(E + vP ×B) Polarization drift (4.35)

To solve Eq. (4.35) we break it into its components parallel and perpendiclarto B. Then

vP = vP‖ + vP⊥ (4.36)

vP‖ = − 1

iω

q

mE‖ (4.37)

B∗vP⊥ =q

mE⊥. (4.38)

where B∗ is the complex conjugate of the vector operator

B =(iω +

q

mB×

). (4.39)

Because the natural motion in the plane perpendicular to B is circular, it wouldseem that a reasonable simplification could be obtained by expressing the driving


field E⊥ as the sum of left and right hand circularly fields:

E⊥ = EL + ER (4.40)

EL =1

2

(E⊥ − iB×E⊥

)(4.41)

ER =1

2

(E⊥ + iB×E⊥

)(4.42)

where B ≡ k. The imaginary term is the orthogonal electric field componentretarded 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:

BB∗vP⊥ ≡(iω +

q

mB×

)(−iω +

q

mB×

)vP⊥

= ω2vP⊥ +q2

m2B×B×vP⊥

= (ω2 − ω2c )vP⊥. (4.43)

98

In obtaining this relation we have used the fact that B×B×vP⊥ = −B2vP⊥.Moreover, the left and right handed fields are eigenvectors of this operator:

BER =1

2

(iω +

q

mB×

)(E⊥ + iB×E⊥

)

=1

2

[iω(E⊥ + iB×E⊥) +

qB

m(B×E⊥ − iE⊥)

]

=1

2

[iω(E⊥ + iB×E⊥) ∓ iωc(E⊥ + iB×E⊥)

]= i(ω ∓ ωc)ER (4.44)

where we have used ωc =|q| B/m and the minus and plus signs are for ions andelectrons 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

(ω2 − ω2c)vP⊥ = i

q

m[(ω ∓ ωc)ER + (ω ± ωc)EL] . (4.46)

Finally, decomposing the perpendicular polarization velocity into left and righthand components vP⊥ = vL + vR allows the solution for the guiding centre driftin the oscillating electric field to be written

vR = iq

m

ER

(ω ± ωc)(4.47)

vL = iq

m

EL

(ω ∓ ωc). (4.48)

Note that for positive ions, there is a resonance between the ions and theleft handed wave as ω → ωci. The reverse is true for electrons. To obtain theresonance 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. It isconvenient to represent this combined response to the driving field in the form

vR

vL

vP‖

=

iq

mω

ω

ω ± ωc0 0

0ω

ω ∓ ωc0

0 0 1

ER

EL

E‖

(4.49)

orvP =

↔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.


The conductivity tensor for a collisionless magnetized plasma is obtained using

j = ne(ui − ue)

= ne(↔µ i − ↔

µe)E

=↔σ E (4.51)

where

↔σ =

↔σ i +

↔σe

↔σ i

e= ne

↔µ i

e. (4.52)

In Cartesian coordinates, we obtain

↔σ i

e=

ine2

mω

ω2

ω2 − ω2c

±iωcω

ω2 − ω2c

0

∓iωcω

ω2 − ω2c

ω2

ω2 − ω2c

0

0 0 1

. (4.53)

The factor i indicates that the current and the applied electric field are 90 degreesout of phase.

Synchrotoron emission

At ω = ωci or ω = ωce (resonance for ions or electrons) it can be shown that thesolution for the perpendicular component of the particle velocity is (for the ions)

v⊥ = vc +q

mELt exp (−iωcit). (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 electromagneticwaves at a rate [6]

dK

dt=

e2

6πε0mc3a2. (4.55)

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 timeconstant is

τR = 3πε0mc3/e2ω2c . (4.57)

Since τR scales as m3 radiation damping through cyclotron emission (or magneticbremstrahhlung) 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 bereabsorbed by the plasma. Indeed, the inverse process is used to provide resonantheating 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 inthe limit ω/ωc 1. In this limit, the velocity is expressed by

vx

vy

vz

=

±iq

mω

−ω2

ω2c

∓ iω

ωc0

± iω

ωc

−ω2

ω2c

0

0 0 1

.

Ex

00

exp (−iωt). (4.58)

This reduces to

vx =iq

mω

−ω2

ω2c

Ex exp (−iωt)

vxi =q

mω2c

∂E

∂t

= ± 1

ωcB

∂E

∂t(4.59)

which is the same as Eq. (4.24) with the plus sign for ions and the minus forelectrons. 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 particlepicture!) by following the procedure used in the low frequency case. The dielectrictensor is extremely important to an understanding of wave propagation in aplasma.

We start with Maxwell’s equation

∇×B = µ0

(j + ε0

∂E

∂t

). (4.60)

Considering the plasma as a dielectric, we write this as

∇×B = µ0∂D

∂t(4.61)


with

D = ε0E − 1

iωj

= ε0E − 1

iω

↔σ E

= ε0↔εr E

≡ ↔ε E (4.62)

where↔ε= ε0

(↔I +

i

ε0ω

↔σ)

(4.63)

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:

∇×E = −∂B

∂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 particlemight 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:

∇×E = −∂B

∂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 theunperturbed particle trajectory. Now work done equals change in kinetic energy,so

δ(mv2⊥/2) =

∫F .dl

= q∮

E.dl

= q∫

S∇×E.dS

= −q∫

S

∂B

∂t.dS

= |q | ∂B

∂tπr2

L (4.66)

102

+

B

ds

B.ds >0 electrons

B.ds <0 ions

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 ofopposite sign for ions and electrons as seen in Fig. 4.7.

The change in B that occurs over one orbit is

δB =∂B

∂tδt =

∂B

∂t

2π

ωc

so that

δ(mv2⊥/2) = |q | πr2

L

ωc

2πδB

=mv2

⊥2B

δB

= |µ | δB (4.67)

where

|µ |≡ µ =mv2

⊥2B

(4.68)

is the magnitude of the orbital magnetic dipole moment of the charged particleencountered earlier [see Eq. (4.12)].

Note that the left side gives δ(mv2⊥/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) forslowly changing fields. Now

δµ = 0

⇒ mv2⊥/B = constant

⇒ Br2L ∝ Φ = constant

4.4 Inhomogeneous Fields 103

where Φ is the magnetic flux linked by the particle orbit. Thus, if the magneticfield increases (decreases) slowly compared with ωc, the orbit radius decreases(increases) in such a way that the particle always encircles the same number ofmagnetic “lines of force”.

4.4 Inhomogeneous Fields

4.4.1 Nonuniform magnetic field

Grad B drift

z B

B|Β|

∆

x

y

+

-

Figure 4.8: The grad B drift is caused by the spatial inhomogeneity of B. It is

in opposite directions for electrons and ions but of same magnitude.

In this case we consider E = 0. As alluded in the introduction, we Taylorexpand the variation of B, B = bk, assuming that the variation in B across aLarmor orbit is small. This obtains

B = B0 + y∂B

∂y+ . . . (4.70)

where we have assumed that B varies only in the y-direction and that the firstorder term is small. Since we consider variation in y of order the Larmor radiusrL, we require

y <∼ rL B/(∂B

∂y) ∼ L

where L is the scale length for variation of B. Substituting into Eq. (4.3) andusing Eq. (4.10) we obtain

mvy = −qvxB

= −qv⊥ cos (ωct)

[B0 ± rL cos (ωct)

∂B

∂y

]. (4.71)

104

Since the B-field is time invariant, we can average over a cyclotron period

〈Fy〉 = 〈mvy〉 = ∓qv⊥rL∂B

∂y〈cos2(ωct)〉 (4.72)

so that there is a residual y-force (but no x directed force - show this). Theresulting drift is given by Eq. (4.21)

v∇B =1

q

F×B

B2

=1

q

〈Fy〉BB2

i

= ∓v⊥rL

2B

∂B

∂yi. (4.73)

Alternatively, this can be expressed in vector form

B×∇B =

∣∣∣∣∣∣∣∣∣

i j k0 0 Bz

0∂B

∂y0

∣∣∣∣∣∣∣∣∣(4.74)

where

∇B ≡ ∇ |B|= i∂ |B|∂x

+ j∂ |B|∂y

+ k∂ |B|∂z

.

∇ |B| often simplifies to ∇Bz because Bz Br, Bθ. The general result is

v∇B = ±1

2v⊥rL

B×∇B

B2. (4.75)

The drift is in opposite directions for electrons and ions (see Fig. 4.8) but of thesame 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 centrifugalforce proportional to the radius of curvature Rc (see Fig. 4.9). The force felt is

F c =mv2

‖Rc

r =mv2

‖Rc

R2c

(4.76)

and the resulting drift can be written as

vR =mv2

‖qB2

Rc×B

R2c

(4.77)


B

R c

r

θ

Fc

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:

B = Bθ = (B0/r)θ

so

∇B = r∂B0/r

∂r= −r(B0/r

2) = −rBθ/r = −r(Bθ/r2)

where we have used ∇×B = 0 in vacuum and

(∇×B)z =1

r

∂rBθ

∂r⇒ Bθ ∼ 1

r

Using Eq. (4.75) we have

v∇B = ±1

2v⊥rL

B×∇B

B2

= ±1

2

v2⊥

ωc

B×(−Rc |B|)B2R2

c

=1

2

mv2⊥

q

Rc×B

R2cB

2(4.78)

where we have used B/ωc = m/ |q|. Combining with the curvature drift we find

vT = v∇B + vR =m

q

(Rc×B

R2cB

2

)(v2‖ +

1

2v2⊥

). (4.79)

106

Figure 4.10: The grad B drift for a cylindrical field.

Note that the two contributions add with similar magnitude because 〈v2‖〉 ∼

kBT/m and 12〈v2

⊥〉 ∼ kBT/m.

Magnetic mirrors — ∇B ‖ B

We have looked at particle drifts when ∇B is at an angle to B. What happenswhen the gradient is aligned with B? This situation is encountered in magneticmirrors where the magnetic field strength increases along the direction of thelines 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 canbe trapped under certain circumstances. Let’s describe mathematically the fieldstructure. The field must be divergence free (no sources or sinks): ∇.B = 0. In


cylindrical geometry this gives

1

r

∂rBr

∂r+

∂Bz

∂z= 0. (4.80)

Provided ∂Bz/∂z does not vary much with r we have

rBr = −∫ r

0r∂Bz

∂zdr ≈ −r2

2

∂Bz

∂z(4.81)

or

Br = −r

2

∂Bz

∂z. (4.82)

Any radial inhomogeneity of Br gives an azimuthal drift Bzk×∇Brr about theaxis 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?

F = qv×B =

∣∣∣∣∣∣∣r θ zvr vθ vz

Br 0 Bz

∣∣∣∣∣∣∣= r(qvθBz) − θq(vrBz − vzBr) − z(qvθBr). (4.83)

For simplicity, consider a particle spiralling along the axis (r = rL) so that wecan ignore grad B drifts. The logitudinal (axial) force is

Fz = vθrLq

2

∂Bz

∂z

= ∓v⊥rLq

2

∂Bz

∂zions and electrons

= ∓qv2⊥

2ωc

∂Bz

∂z

= −mv2⊥

2B∇‖B

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 thismotion provided the rate of change is slow. Since µ is a constant of the motion,then

v2⊥0

B0=

v2⊥m

Bm(4.85)

108

where the subscript 0 refers to the low field conditions and subscript m is forthe high field “mirror” region. Thus if Bm > B0 then v⊥m > v⊥0. However, theB-field does no work so that the total particle kinetic energy remains unchanged:K = m(v2

‖0+ v2

⊥0)/2 is constant. Therefore, we must have v‖m < v‖0 and theaxial particle velocity decreases as the particle moves into the high field region.The axial velocity is given by

1

2mv2

‖ = K − 1

2mv2

⊥

= K − µB

⇒ v‖ =[

2

m(K − µB)

]1/2

. (4.86)

If B is high enough, the particle can be stopped and F‖ forces the particle backinto 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 conditionEq. (4.85) requires that

Bm

B0=

v2⊥m

v2⊥0

=v2‖0 + v2

⊥0

v2⊥0

=1

sin2 θm

(4.87)

≡ Rm Mirror ratio (4.88)

where we have used the fact that kinetic energy K is conserved. The angle θm

is shown in Fig. 4.13. Particles having θ < θm (i.e. high v‖) penetrate the mirrorand 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 scatteredby collisions into this region of velocity space. Since the electron collision rate ishigher, however, they are preferentially lost and the plasma acquires a positivepotential.

4.4.2 Particle drifts in a toroidal field

The ramifications of field curvature and inhomogeneity are clearly evident fortoroidal magnetic fields. For the toroidal field, the radius of curvature vector isnormal to the magnetic field line Rc.B = 0. For a typical charged particle wetake v2

‖ = 12v2⊥ so that Eq. (4.79) gives

vT =mv2

⊥qRcB

∼ rL

Rcvth (4.89)


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

vT

vth∼ ±rL

Rc≡ κ. (4.90)

For H-1NF, κ ≈ 1 × 10−3 (drift angle to field line) so that the toroidal traveldistance for a particle to drift out of the magnetic volume is dT = 0.1 m/κ = 100m which is about 16 toroidal orbits.

As already noted, the electrons and ions drift in opposite directions. Thisgenerates a vertical electric field as shown in Fig. 4.14. The resulting E×B driftpushes the plasma to the wall and the plasma is not confined.

This problem can be remedied by twisting the field lines (by introducinga toroidal current). Particles moving freely along B will then short out the

110

vy

vx

vz

v⊥

vθm

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 ofthe 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 therotational 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. Tosee this, let us apply Ampere’s law around a closed circular loop threading thetorus. We assume this imaginary loop encloses N toroidal field coils each carryingcurrent I in the same direction. We then have

∮B.dl = 2πRB = µ0NI


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.

B =µ0NI

2πR. (4.91)

112

To first order in ε = r/R0, where R0 is the radius of the magnetic axis, the fieldstrength at some point in the torus is (show this)

B = B0(1 − ε 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 systemused here).

If we now follow an electron along a helical field line, then if it starts at theoutside of the torus and moves towards the torus axis (inside), the magnetic fieldincreases. 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 ontothe R–z plane shows a banana-shaped orbit. Between the reflection points thereis an upward drift due to curvature and ∇B. Typically, the banana width fora tokamak <∼ 0.1a where a is the minor radius. Particles with sufficiently highparallel velocity will penetrate the magnetic well and complete a toroidal circuit.These are called passing particles.


Passing orbits

As the particle moves freely toroidally, its orbit projected onto the R-z (poloidal)plane is given by

dx

dt= −Ωz

dz

dt= Ωx + vz (4.93)

where we have taken x = R − R0 and where the vertical z drift velocity is givenby

vz =m

2qBφR(2v2

‖ + v2⊥). (4.94)

In Eq. (4.93) Ω = dθ/dt is the angular velocity of the particle orbit projectedonto the poloidal plane (imagine the torus straightened into a cylinder and lookingalong the axis of the helical magnetic field line). The rotation in this plane arisesfrom the helicity of the magnetic field line. The rate of spiralling of the field lineis given by

rdθ

Bθ

=Rdφ

B0

and is characterised by the so-called winding number or safety factor q(r) whichis defined by

q(r) =dφ

dθ=

rB0

RBθ= ε

B0

Bθ. (4.95)

In tokamaks, q typically lies in the range 0.7 < q(r) < 3 or 4. In terms of q, thepoloidal rotation frequency is given by

Ω =dθ

dt=

1

q(r)

dφ

dt=

vφ

q(r)R(4.96)

with vφ = v.φ.Aside: In stellarator studies (as opposed to tokamaks), it is more usual to talkof the rotational transform ι/2π = 1/q of the field line, defined as the number ofturns poloidally (the short way) per turn toroidally (the long way) (or “twist perturn”). Typically ι <∼ 1.

The projected particle orbit (ignoring variations in the vertical drift velocityvz) is

(x + vz/Ω)2 + z2 = constant.

The constant depends on the radius of the magnetic surface formed by the helicalfield lines on which the particle travels. The particle departs from this surface(due to inhomogeneity and curvature drifts) by

∆ = −vz

Ω

∼(

m

qB

v2th

R0

)(q(r)R0

vth

)

114

= q(r)(

vth

ωc

)= q(r)rL. (4.97)

This shift is shown in Fig. 4.17

Trapped particles

Let’s now assume that (v⊥/v‖)2 > 1/ε so that particles are trapped and bouncein the mirrors produced by the 1/R variation of the toroidal field.

We wish to find a relationship between the radial motion of trapped particlesand their parallel velocity. Geometry tells that the radial component of thevertical drift velocity is

vr = vz sin θ

=mv2

⊥2qB0R

sin θ (for v2‖ v2

⊥). (4.98)

The parallel force felt by the particle due to the increasing toroidal field is

mv‖ = −µ∂B

∂(4.99)

where ≈ Rφ is the distance coordinate along the magnetic field line. Thecoordinate is related to the poloidal angle θ through the safety factor q:

≈ Rφ =rB0

Bθ

θ

or

θ = κ with κ ≡ Bθ

rB0

where r is the radius of the field line with respect to the magnetic axis. The rightside of Eq. (4.99) can now be evaluated as

∂B

∂=

∂

∂[B0(1 − ε cos κ)]

= B0εκ sin κ

= B0r

R0

Bθ

rB0sin θ

=Bθ

R0sin θ (4.100)

where we have used Eq. (4.92). Equation (4.99) now gives

v‖ = −mv2⊥

2B0

Bθ

mR0sin θ

= −v2⊥2

Bθ

R0B0sin θ. (4.101)


Comparing with Eq. (4.98) we obtain

vr = − m

qBθv‖

which integrates to give

r − r0 = − m

qBθv‖ (4.102)

where r0 is the turning point at which v‖ = 0. Equation (4.102) describesthe poloidal projection of the banana orbit. To obtain the orbit width, we useEq. (4.86) with K ≈ mv2

⊥/2 to obtain for the change in v‖ around the orbit

δv‖ =[

2

m(K − µB)

]1/2

≈ v⊥ε1/2 (4.103)

where we have substituted from Eq. (4.92) for B. Also note that

ωcθ =qBθ

m≈ ε

q(r)

qB0

m=

εωc

q(r)(4.104)

so that Eq. (4.102) becomes

δr =δv‖ωcθ

≈ q(r)rL/ε1/2. (4.105)

The width of the banana orbit is bigger again by the factor ε−1/2 than thepassing particles. This has profound consequences, with neoclassical diffusionincreased again over the cylindrical value.

Problems

Problem 4.1 The polarization drift vP can also be derived from energy conserva-

tion. If E is oscillating, the E×B drift also oscillates, and there is an energy 12mv2

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 A 1keV proton with v‖ = 0 in 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⊥. The B 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.


Problem 4.3 Consider the magnetic mirror system shown below. Suppose that the

axial magnetic field is given by

B(z) = B0[1 + (z/a)2]

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

v‖(z) =(2µB0/m)[(zm/a)2 − (z/a0)

2]1/2

(b) The average force acting on the particle guiding centre along the z axis is given

by

〈F‖〉 = −µ∂B

∂zz

Show that the particle executes a simple harmonic motion between the mirroring

planes with a period given by

T = 2πa0[m/(2µB0)]1/2

.

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 = B0z, and an electric field that oscillates in time at the electron

cyclotron frequency ωc according to

E(t) = E0[x cos(ωct) + y 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):

d2vx

dt2+ ω2

cvx = 2(eE0/m)ωc sin(ωct)

d2vy

dt2+ ω2

cvy = −2(eE0/m)ωc cos(ωct)

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) = −(eE0/m)[(1/ω2c ) cos(ωct) + (t/ωc) sin(ωct) − 1/ω2

c ]

y(t) = −(eE0/m)[(1/ω2c ) sin(ωct) − (t/ωc) cos(ωct)]

single particle motions

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