Charged particle motion in external...

Chapter 2

Charged particle motion inexternal fields

A (fully ionized) plasma contains a very large number of particles. In general,their motion can only be studied statistically, taking appropriate averages. Eachparticle motion is a↵ected by the local electric and magnetic fields, due to thecharges and current due to the other plasma particles as well as externallyapplied fields.

However, to get insight into the relevant physical processes, it useful to study,sometimes in a simplified way, aspects of the motion of individual particles. Thisapproach will be taken in this Chapter and in the next one. In this chapter wewill discuss the motion of individual charged particles in given electric andmagnetic fields. In the next chapter we will consider the collisions of a chargedparticle with the surrounding particles.

A more general statistical treatment requires the study of the self-consistentevolution of the particle velocity distribution function, ruled by the Boltzmannequation (see Chapter 6), and the Maxwell equations. Such an approach is inmost cases extremely complex and solutions of the relevant equations are possi-ble, even numerically, only resorting to rather drastic simplifications. In manycases, however, more a↵ordable fluid (actually, magneto-fluid, or magnetohy-drodynamic, MHD) approaches can used, as discussed in Chapters 6 and 7.

In the next sections we will consider the motion of a single particle in ex-ternally applied, time-independent magnetic (and electric) fields. We will startfrom the case of a uniform magnetic field, and then, in sequence, uniform mag-netic and electric fields, and a few cases of spatially nonuniform magnetic fields.

2.1 Uniform magnetic field

Lorentz force on a particle with charge q and velocity v in a magnetic field B

F = qv ⇥B (2.1)

15

16 CHAPTER 2. MOTION IN EXTERNAL FIELDS

Figure 2.1: Normal acceleration in the plane orthogonal to the magnetic field,due to Lorentz force.

Lorentz Force is normal to both magnetic field and velocity. Particle motionis the sum of

• uniform motion, at constant velocity vk in direction parallel to B.

• Uniform circular motion, with velocity v? in a plane orthogonal to B

In the plane orthogonal to B,

|F | = qv?B = man

= mv2?rL

, (2.2)

i.e. circular motion with Larmor radius

rL

=mv?|q|B (2.3)

In a thermal plasma at temperature T ,

1

2mv2? ' kT ) v? '

r2kT

m(2.4)

hence average radius

rL

=

p2mkT

|q|B . (2.5)

Gyration occurs with cyclotron angular frequency

!c

=v

rL

=|q|Bm

(2.6)

and cyclotron frequency

fc

=!c

2⇡(2.7)

Note that often angular frequencies are just referred to as frequencies.

2.1. UNIFORM MAGNETIC FIELD 17

Numerical evaluations:

rL,e

= 3.3⇥ 10�6

pT [eV]

B [T][m]

where index e refers to electrons.

For T = 1000 eV and B = 5 T we have electron Larmor radius rL,e

⇡ 20 µm,and proton Larmor radius r

L,p

⇡ 1 mm.

Figure 2.2: Electrons and ions rotate in opposite direction. Electron Larmorradius is typically much smaller than ion Larmor radius

For the cyclotron frequency

!c, e

= 1.7⇥ 1011 B [T] [s]�1;

fc,e

= 28 B [T] [GHz]

andfc,p

= 1.5 B [T] [MHz] ,

where index p refers to protons.


2.2 Gyromotion, guiding center, drift

The magnetic field can be uniform only in a limited spatial region. In general, itis not spatially uniform and may change with time. In many cases, the distanceover which the magnetic field changes significantly (also called magnetic fieldgradient scale-length) is much larger than the relevant Larmor radii, and thetime over which it changes is much longer than the reciprocal of the plasmafrequency. In such cases, the motion of a charged can be viewed as the sumof the fast gyration discussed in the previous section, and a slow motion of thecenter of the orbit. We then have a gyration around a moving guiding center,and a drift of the guiding centre. In summary, when no electric fields are present,particle trajectories are a sort of helix: particles move at constant speed in thedirection parallel to the field, rotate in the plane orthogonal to the field, anddrift.

The presence of an electric field also a↵ects charged particle motion. Theelectric field component parallel to the magnetic field accelerates particles alongits direction, i.e. along the magnetic field. In addition, the component of theelectric field orthogonal to the magnetic field causes a drift of the guiding center,orthogonal to both electric and magnetic field.

A few, simple and important cases of guiding center drifts, caused by time-independent fields, are discussed in the following sections.

2.3. UNIFORM E AND B FIELDS: E⇥B DRIFT 19

2.3 Uniform E and B fields: E⇥B drift

Let us consider the case of magnetic field B and electric field E orthogonalto each other.

mdv

dt= q (E+ v ⇥B) (2.8)

We write the velocity as

v ⌘ w +E⇥B

B2

, (2.9)

where the second term on the right hand side is a constant, depending only onthe fields. Inserting this expression into Eq. (2.8) we obtain an equation for w:

mdw

dt+m

d

dt

✓E⇥B

B2

◆

| {z }=0

= q

E+

✓w +

E⇥B

B2

◆⇥B

�(2.10)

Using

(E⇥B)⇥B = �B⇥ (E⇥B) = �E (B ·B) +B (B ·E)| {z }=0

= �EB2 (2.11)

we then have

mdw

dt= qE+ qw ⇥B+ q

��EB2

�

B2

, (2.12)

mdw

dt= qw ⇥B (2.13)

This is just the equation for gyromotion in a uniform magnetic field. Sincev = w + (E ⇥ B/B2), the resulting motion is gyromotion summed to drift ofthe guiding centre with velocity

vE

=E⇥B

B2

, (2.14)

orthogonal to both electric and magnetic field. Notice that the direction of thedrift does not depend on the sign of the particle charge (see Fig. 2.3).

Qualitative explanation. Refer to the ion motion (red trajectory in thefigure). When the ion is in the lower part of the orbit it has higher velocity (dueto the acceleration caused by the electric field) and then larger Larmor radius.The opposite occurs in the upper portion of the orbit. This leads to the drifttowards the right hand side.

The expression of the E⇥B drift can be generalized, by replacing the electricforce qE with any force F,

vF

=F⇥B

qB2

. (2.15)


Figure 2.3: E⇥B drift.

2.4. NON-UNIFORM MAGNETIC FIELD 21

2.4 Non-uniform magnetic field

We now we consider non-uniform magnetic fields, and no electric fields. Weshall consider a few cases separately.

2.4.1 r|B| ? B

Figure 2.4: Left: r|B| ? B; right: unperturbed gyromotion in the x� y plane.(Electron and ion orbits not to scale.)

We assume that the magnetic field gradient scale-length is much larger thanthe Larmor radius r

L

, so that the particle orbit is unperturbed, with radius rL

and frequency fc

= !c

/2⇡. Unperturbed orbit:

x� x0

= rL

sin!c

t, (2.16)

y � y0

= ±rL

cos!c

t, (2.17)

where the positive sign refers to ions and the negative sign to electrons. Hence

vx

= rL

!c

cos!c

t = v? cos!c

t (2.18)

vy

= ⌥v? sin!c

t (2.19)

Force on the particle

F = q

0

@i j kvx

vy

vz

0 0 Bz

1

A = qvy

Bz

i� qvx

Bz

j. (2.20)

This force changes as the particle moves because the magnetic field is not uni-form, B

z

= Bz

(y), with y = y0

± rL

cos!c

t. However, since the change of thefield over a Larmor radius is small, we can compute it along the particle orbitby using a Taylor expansion limited to the first term. We then write

Fy

= �qvx

Bz

(y) ' �qv? cos!c

t

"B

z0

+ (y � y0

)@B

z

@y

��y=y

0

#, (2.21)


and then

Fy

= �qv? cos!c

t

"B

z0

+@B

z

@y

��y=y

0

(±rL

cos!c

)

#, (2.22)

where Bz0

is the field at the center of the orbit, i.e. Bz0

= Bz

(y = y0

). Thisforce changes in time. Since gyrations are fast (with respect to drift) we canaverage the force over an orbit, to obtain

Fy

= 0⌥ rL

qv?@B

z

@y

��y

0

cos2 !c

t| {z }1

2

(2.23)

Proceeding in the same way we find

Fx

= 0. (2.24)

In conclusion, the force (2.23) normal to the magnetic field acts on the particle.According to Eq. 2.15 this force causes a drift, with velocity

v =F⇥B

qB2

= ⌥1

2v?rL

r |B|⇥B

B2

(2.25)

Notice that the direction of this drift depends on the sign of the charge (seeFig. 2.5)

Figure 2.5: r|B| ? B drift.


2.4.2 Curvature drift

Let us now consider a circularB-field line, with curvature radius Rc

; see Fig. 2.6.We indicate with r = R

c

/Rc

the versor (unit vector) directed along the radius.

Figure 2.6: Particle motion around a curved magnetic field line.

In the guiding centre reference frame, a centrifugal force

Fc

=mv2k

Rc

r (2.26)

acts on the particle; here vkis the component of the velocity parallel to B. Thisforce causes a curvature drift with velocity

vR

=F

c

⇥B

qB2

and then

vR

=mv2k

qB2

Rc

⇥B

R2

c

(2.27)

Referring to the figure, the drift is normal to the plane of the figure, and itssign depends on the sign of the charge.

However, if the magnetic field lines are curved, then we also have a finiter|B| that causes an additional drift. We evaluate this drift as follows. We referto the cylindrical coordinate system in Fig. 2.7, so that

Br

= 0 (2.28)

Bz

= 0 (2.29)

B✓

= B✓

(r) (2.30)

If in the region we are considering there are no currents, then, for Ampere law,


Figure 2.7: Circular field line and cylindrical coordinate system.

r⇥B = 0. The z component of the curl operator,

(r⇥B)z

=1

r

@

@r(rB

✓

)� @Br

@✓

�, (2.31)

in our case reads1

r

@

@r(rB

✓

) (2.32)

and must be identically zero for the Ampere law. We then find that

B✓

/ 1

r. (2.33)

It follows that

(r |B|)r=R

c

=⇣r

��B✓

✓��⌘

r=R

c

= �cost

R2

c

r, (2.34)

and then ✓r |B||B|

◆

r=R

c

= �cost

R2

c

Rc

cost· r = � r

Rc

= �Rc

R2

c

, (2.35)

and the relevant drift velocity is

vr|B| = ⌥1

2v?rL

r |B|⇥B

B2

= ±1

2v?rL

Rc

|B|⇥B

R2

c

B2

. (2.36)

Finally, by using the definition of Larmor radius, rL

= mv?/ |q|B = ±mv?/qB,we can write

vr|B| =m

q

v2?2

Rc

⇥B

R2

c

B2

, (2.37)


which di↵ers from the curvature drift [Eq. (2.27)] only for the term v2?/2, re-placing v2k. The total drift velocity, sum of Eq. (2.27) and (2.37) is then

v = vR

+ vr|B| =m

q

✓v2k +

1

2v2?

◆R

c

⇥B

R2

c

B2

(2.38)

Figure 2.8: Drift in a purely toroidal field.

An important consequence of curvature drift concerns plasma confinementin toroidal geometry. Let us consider a toroidal (i.e. donut shaped) configu-ration, with a purely toroidal magnetic field, i.e. a sort of toroidal solenoid(see Fig. 2.8). According to Biot-Savart’s law, B(r) / 1/r and then the fieldgradient is directed towards the torus major axis and is at any point orthogonalto the field. The combined curvature and gradient drift pushes electron andions vertically, in opposite directions, thus producing charge separation, and avertical electric field, orthogonal to the magnetic field. This leads to an E⇥Bdrift, pushing particles of both signs out of the torus. Confinement in toroidalgeometry then requires more complex field topologies, as discussed in a laterChapter.

ExcerciseCompute electron thermal velocity, ion thermal velocity and (curvature plusgradient) drift velocity in a toroidal device with major radius R

c

= 1 m, mag-netic field of 5 T, and plasma temperature of 1 keV. Verify that the drift velocityis much smaller than the thermal velocities.


2.4.3 r|B| k B; Magnetic mirror

Figure 2.9: Cylindrically symmetric magnetic field and cylindrical coordinatesystem.

We now consider the case of magnetic field gradient (nearly) parallel to themagnetic field B. We refer, for simplicity, to a configuration with cylindricalsymmetry, as illustrated in Fig. 2.9. The magnetic field B is nearly axial andweakly varying (i.e. the gradient scale-length is much larger than the Larmorradius). In summary

B = (Br

, 0, Bz

) ; |Br

| << |Bz

| ; (2.39)��@B

z

@z

�� ⌧ Bz

/rL

. (2.40)

We use Maxwell’s equation r ·B = 0 to determine the relation between theradial and axial component of the magnetic field. Using the expression of thedivergence operator in cylindrical coordinates,

1

r

@

@r(rB

r

) +@B

z

@z+

1

r

@B✓

@✓= 0, (2.41)

we obtain@

@r(rB

r

) = �@Bz

@zr. (2.42)

(the third term in Eq. (2.41) vanishes identically, because B✓

= 0). We integratealong the radius between the axis and a radius r

Zr

0

@

@r0(r0B

r

) dr0 = �Z

r

0

@Bz

@zr0dr0. (2.43)

Since @Bz

/@z varies weakly with r, we can take it as a constant (or take anaverage value), and then we have

Br

⇠= �r

2

@Bz

@z. (2.44)


Motion along z, magnetic moment as an adiabatic invariant

We now study the axial component of the motion of a particle (actually, of itsguiding centre) close to the axis. We can write

mdvk

dt= F

z

= q (vr

B✓

� v✓

Br

) = �qv✓

Br

(2.45)

Now, writing v✓

= ⌥v? = �rL

!c

, approximating r ' rL

, and using Eq. (2.44)for B

r

, we obtain

mdvk

dt= �q · (±v?) ·

✓�r

2

@Bz

@z

◆= �1

2|q| r2

L

!c

@Bz

@z= �µ

@Bz

@z(2.46)

The quantity

µ =1

2

mv2?B

(2.47)

is called magnetic moment of the particle.

ExerciseShow that this definition is just the same as that for the magnetic moment of acurrent carrying coil, µ = IS, where I is the current and S is the surface areaof the coil.

We now show that the magnetic moment is an adiabatic invariant, i.e. itis a constant if the magnetic field changes weakly in space and time. Here weconsider the case of a magnetic field which varies in space but is constant intime. We multiply both members of Eq. 2.46 by vk,

mvkdvk

dt= �µvk

@Bz

@z, (2.48)

which can also be written as

d

dt

✓1

2mv2k

◆= �µvk

@z

@t

@Bz

@z= �µ

dB

dt. (2.49)

During motion, kinetic energy (equal to total energy, since there is no potentialenergy) is conserved, hence

d

dt

✓1

2mv2

◆=

d

dt

✓1

2mv2k

◆+

d

dt

✓1

2mv2?

◆= 0, (2.50)

or, using the definition of magnetic moment,

d

dt

✓1

2mv2k

◆= � d

dt(µB) . (2.51)


By equating the right hand sides of Eq. (2.49) and (2.51) we obtain

� d

dt(µB) = �µ

dB

dt(2.52)

which is satisfied for constant values of µ only. (A full proof requires to show thatµ is constant also in the case of magnetic field slowly varying in time. We omitsuch a proof here.) In conclusion, if r|B| is nearly parallel to B the magneticmoment remains constant as particles move. An important application of sucha results concerns the magnetic mirror.

Magnetic mirror

Figure 2.10: Magnetic mirror: magnetic field topology

We consider a cylindrically symmetric magnetic mirror, with field topologyas shown on Fig 2.10. Two cylindrical coils, at some distance from each othergenerate a magnetic field, which is stronger close to the coils and weaker in theregion between the coils. We now show that such a configuration allows for a(partial) confinement of plasma particles.

Let us consider a particle close to the axis at position z = 0, and withvelocity v0 forming an angle ✓

0

with the mirror axis (see Fig. 2.11). Due to theconstancy of the magnetic moment

v2?v2?0

=B

B0

. (2.53)

Hence as the particle moves toward the region of higher magnetic field the com-ponent of the velocity orthogonal to the field, v? increases. At the same time,since kinetic energy is conserved, the parallel component decreases. When theorthogonal component becomes so large that the parallel component vanishes,


Figure 2.11: Above: reflected particle guiding centre motion; below: velocitycomponents in a position close to the origin of the coordinate system.

Figure 2.12: Magnetic mirror: loss cone

the particle is reflected, and therefore it is contained in the space between thetwo coils. However, particles with too small value of ✓

0

cannot be reflected. Toshow this we proceed as it follows. Reflection occurs when

v2k = v20

� v2? = 0.

This condition can also be written as

v20

� v2?0

B

B0

= 0, (2.54)

or

v20

✓1� B

B0

sin2 ✓0

◆= 0. (2.55)

It follows that particles with central velocities forming an angle ✓0

with the


mirror axis, such that

✓0

< ✓m

= arcsin

rB

0

Bm

(2.56)

cannot be confined, and are then lost by the mirror. This defines a loss cone; seeFig. 2.12. A word of caution: consideration of particle collisions and of kinetice↵ects makes the study of mirrors substantially more complex.

Charged particle motion in external...

Documents

Transcript of Charged particle motion in external...