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
F =m
dv
= q(E + v×B)
dt
(4.1)
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 k̂ is in the direction
of B (i.e. B = B k̂). Then Eq. (4.1) gives
mv̇ =
q î ĵ k̂ vx vy vz 0 0 B (4.2)
and the separate component equations are
mv̇y = −qBvx
mv̇x = qBvy
mv̇z = 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
v̈x =
and similarly for vy
qB
q2B2
v̇y = − 2 vx
m
m
(4.4)
v̈y = −ωc2 vy
(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 gyrate
much more slowly due to their greater mass.
The solution to Eq. (4.4) can be written as
vx = v⊥ exp (iωc t)
(4.7)
with the convention that we take the real part (vx = v⊥ cos ωc t). Substituting
Eq. (4.7) into Eq. (4.3) gives an expression for vy
vy =
m
imωc
v̇x =
v⊥ exp (iωc t) = ±iv⊥ exp (iωc t)
qB
qB
(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 (ωc t)
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⊥
exp (iωc t)
ωc
v⊥
= ± exp (iωc t)
ωc
x − x0 = −
y − y0
(4.9)
where (x0 , y0) are constants of integration. Taking real parts gives
x − x0 = rL sin (ωc t)
y − y0 = ±rL cos (ωc t)
(4.10)
with
(x − x0 )2 + (y − y0 )2 = rL2
and we have introduced the Larmor radius
rL =
v⊥
mv⊥
.
=
ωc
|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 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
B
X
-
+
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
2
qωc πv⊥
=
2π ωc2
2
mv⊥
=
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
4.2 Constant and Uniform Fields
the
the
the
the
91
radius decreases. The result is that the ion executes a cycloidal motion with
guiding centre drifting in the direction perpendicular to both E and B. For
electrons, the cycloidal orbits are smaller (smaller mass). However, we note
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
mv̇ = qE
(4.13)
describing a free acceleration along B. The perpendicular motion is described by
mv̇ ⊥ = q(E ⊥ + v ⊥ ×B).
(4.14)
Anticipating the result, we make a transformation into the reference frame moving
with drift velocity v E such that v = v E + v c and Eq. (4.14) becomes
mv̇ c = q(E ⊥ + v E ×B) + qv c ×B.
(4.15)
In the drifting frame the velocity v c is just the cyclotron motion so that we can
set
E ⊥ + v E ×B = 0.
(4.16)
92
This can be solved for v E as follows:
E ⊥ ×B = −(v E ×B)×B
= v E B 2 − B(v E .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
.
(4.19)
vE =
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
v = v k̂ (along B) + v E (perpendicular drift) + v c (Larmor gyration). (4.20)
In this case, v E 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
.
(4.21)
vF =
q B2
An example is the gravitational drift F = mg which gives
m g×B
.
(4.22)
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:
j g = qe ne v e + qi ni v i
g×B
= n(mi + me ) 2
B
The magnitude of j g is usually negligible. However, curved lines of force ⇒
effective gravitational (centrifugal) force ⇒ curvature drift (more below).
4.3 Time Varying Fields
Figure 4.3: The orbit in 3-D for a charged particle in uniform electric and magnetic fields.
4.3
4.3.1
Time Varying Fields
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 v E = (E×B)/B 2 to
93
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 + qv c ×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 v̇ E ×B
q B2
1 Ė
= ±
ωc B
vP = −
(4.24)
to give (show this)
d
q Ë
[m(v c − v P )] = q(v c − v P )×B − 2 .
dt
ωc
(4.25)
The explicit E dependent term is now O(ω/ωc)2 and can be neglected. The
residual equation for v c − v P describes the Larmor motion.
Averaging the total motion over a gyro-period gives the overall guiding centre
drift as v = v E +vP . The new polarization drift v P 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
j P = ne(v P i − v P e )
ne
dE
=
(mi + me )
2
eB
dt
ρ dE
=
B 2 dt
(4.26)
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 = ε0 E + P ≡ εr ε0 E
(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
.
(4.28)
j P = εr ε0
∂t
Comparing with Eq. (4.26) we obtain an expression for the low-frequency plasma
electric susceptibility
ρ
µ0 ρ
=
2
ε0 B
ε0 µ0 B 2
c2
≡ 2
vA
εr =
(4.29)
96
where
vA =
B
µ0 ρ
(4.30)
is the Alfvén 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
E(t) =
∞
−∞
E(ω) exp (−iωt)
dω
.
2π
(4.32)
We decompose the solution to the Lorentz equation into the sum of a magnetically
driven term v c (the Larmor motion) and the harmonic polarization term v P =
v P exp (−iωt). Substituting into Eq. (4.1) gives
dv c
m
− iωv P exp (−iωt) = q [E exp (−iωt) + v c ×B + v P ×B exp (−iωt)] .
dt
(4.33)
This equation can be separated:
dv c
Cyclotron motion
= qv c ×B
dt
Polarization drift
−iωmv P = q(E + v P ×B)
m
(4.34)
(4.35)
To solve Eq. (4.35) we break it into its components parallel and perpendiclar
to B. Then
vP = vP + vP ⊥
1 q
E
vP = −
iω m
q
E⊥.
B∗ v P ⊥ =
m
(4.36)
(4.37)
(4.38)
where B∗ is the complex conjugate of the vector operator
q
B = iω + B× .
m
(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
1
E ⊥ − iB̂×E ⊥
EL =
2
1
ER =
E ⊥ + iB̂×E ⊥
2
(4.40)
(4.41)
(4.42)
where B̂ ≡ k̂. 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:
∗
BB v P ⊥
q
q
≡ iω + B× −iω + B× v P ⊥
m
m
2
q
= ω 2 v P ⊥ + 2 B×B×v P ⊥
m
= (ω 2 − ωc2 )v P ⊥ .
(4.43)
98
In obtaining this relation we have used the fact that B×B×v P ⊥ = −B 2 v P ⊥ .
Moreover, the left and right handed fields are eigenvectors of this operator:
1
q
iω + B× E ⊥ + iB̂×E ⊥
2
m
1
qB
(B̂×E ⊥ − iE ⊥ )
=
iω(E ⊥ + iB̂×E ⊥ ) +
2
m
1
=
iω(E ⊥ + iB̂×E ⊥ ) ∓ iωc (E ⊥ + iB̂×E ⊥ )
2
= i(ω ∓ ωc )E R
BE R =
(4.44)
where we have used ωc =|q| B/m and the minus and plus signs are for ions and
electrons respectively. Similarly for the left hand field we have
BE L = i(ω ± ωc )E L .
(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
(ω 2 − ωc2)v P ⊥ = i
q
[(ω ∓ ωc )E R + (ω ± ωc )E L ] .
m
(4.46)
Finally, decomposing the perpendicular polarization velocity into left and right
hand components v P ⊥ = v L + v R allows the solution for the guiding centre drift
in the oscillating electric field to be written
ER
q
m (ω ± ωc )
EL
q
vL = i
.
m (ω ∓ ωc )
vR = i
(4.47)
(4.48)
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 v c , v P , v R and v L . It is
convenient to represent this combined response to the driving field in the form



or

vR
iq
vL 
=
mω
vP 





ω
ω ± ωc
0
0
↔
vP = µ E
↔

0
0   E 
R
 
ω
  EL 

0 

ω ∓ ωc
E
0
1
(4.49)
(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( µ i − µ e )E
↔
= σE
(4.51)
where
↔
↔
↔
σ = σi + σe
↔
↔
σ i = ne µ i .
e
(4.52)
e
In Cartesian coordinates, we obtain

ine2
σ i=
e
mω
↔







ω2
ω 2 − ωc2
∓iωc ω
ω 2 − ωc2
0

±iωc ω
0 

ω 2 − ωc2

2
.
ω

0


ω 2 − ωc2
0
1
(4.53)
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)
v⊥ = vc +
q
E L t exp (−iωci t).
m
(4.54)
The first term represents the usual cyclotron motion. The second term is a constant 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]
dK
e2
a2 .
(4.55)
=
3
dt
6πε0 mc
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
τR = 3πε0 mc3 /e2 ωc2.
(4.57)
Since τR scales as m3 radiation damping through cyclotron emission (or magnetic
bremstrahhlung) can be important only for electrons. For fusion relevant conditions, 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 confinement 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





vx
±iq 



 vy  =
mω 

vz


ω2
iω

− 2 ∓
0  

E
ωc
ωc
x
 
.  0 
ω2
iω
 exp (−iωt).
− 2 0 
±

0

ωc
ωc
0
0
1
(4.58)
This reduces to
iq −ω 2
Ex exp (−iωt)
vx =
mω ωc2
q ∂E
vx î =
mωc2 ∂t
1 ∂E
= ±
ωc B ∂t
(4.59)
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
∇×B = µ0
∂E
j + ε0
.
∂t
(4.60)
Considering the plasma as a dielectric, we write this as
∇×B = µ0
∂D
∂t
(4.61)
4.3 Time Varying Fields
101
with
1
j
iω
1 ↔
σE
= ε0 E −
iω
↔
= ε0 εr E
↔
≡ ε E
D = ε0 E −
where
↔
ε = ε0
↔
i ↔
I +
σ
ε0 ω
(4.62)
(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:
∂B
∂t
↔
∇×∇×E = −µ0 ε Ë.
∇×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 velocity 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
.
∂t
∇×E = −
(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) =
= q
= q
F .dl
E.dl
S
∇×E.dS
∂B
.dS
S ∂t
∂B 2
= |q |
πr
∂t L
= −q
(4.66)
102
B.ds >0 electrons
B
B.ds <0 ions
+
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
δB =
∂B 2π
∂B
δt =
∂t
∂t ωc
so that
2
/2) = | q | πrL2
δ(mv⊥
ωc
δB
2π
2
mv⊥
δB
2B
= | µ | δB
=
(4.67)
where
2
mv⊥
(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,
| µ |≡ µ =
δµ = 0.
(4.69)
In other words, the magnetic moment is invariant (a conserved quantity) for
slowly changing fields. Now
⇒
⇒
δµ = 0
2
/B = constant
mv⊥
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
4.4.1
Inhomogeneous Fields
Nonuniform magnetic field
Grad B drift
∆
B
y
|Β|
+
x
z
B
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 Taylor
expand the variation of B, B = bk̂, assuming that the variation in B across a
Larmor 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 first
order term is small. Since we consider variation in y of order the Larmor radius
rL , we require
∂B
y<
∼ rL B/( ∂y ) ∼ L
where L is the scale length for variation of B. Substituting into Eq. (4.3) and
using Eq. (4.10) we obtain
mv̇y = −qvx B
∂B
.
= −qv⊥ cos (ωc t) B0 ± rL cos (ωc t)
∂y
(4.71)
104
Since the B-field is time invariant, we can average over a cyclotron period
Fy = mv̇y = ∓qv⊥ rL
∂B
cos2 (ωc t)
∂y
(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
q B2
1 Fy B
=
î
q B2
v⊥ rL ∂B
î.
= ∓
2B ∂y
v ∇B =
(4.73)
Alternatively, this can be expressed in vector form
î ĵ
k̂ B×∇B = 0 0 Bz ∂B
0
0 ∂y
where
∇B ≡ ∇ |B|= î
(4.74)
∂ |B|
∂ |B|
∂ |B|
+ ĵ
+ k̂
.
∂x
∂y
∂z
∇ |B| often simplifies to ∇Bz because Bz Br , Bθ . The general result is
B×∇B
1
v ∇B = ± v⊥ rL
.
2
B2
(4.75)
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
mv2 Rc
mv2
r̂ =
Fc =
Rc
Rc2
(4.76)
and the resulting drift can be written as
mv2 Rc ×B
vR =
qB 2 Rc2
(4.77)
4.4 Inhomogeneous Fields
105
r^
F
c
θ^
B
Rc
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 = B θ = (B0 /r)θ̂
so
∂B0 /r
= −r̂(B0 /r 2 ) = −r̂Bθ /r = −r(Bθ /r 2 )
∂r
where we have used ∇×B = 0 in vacuum and
∇B = r̂
(∇×B)z =
1 ∂rBθ
r ∂r
⇒
Bθ ∼
1
r
Using Eq. (4.75) we have
B×∇B
1
v ∇B = ± v⊥ rL
2
B2
2
1 v B×(−Rc |B|)
= ± ⊥
2 ωc
B 2 Rc2
2
Rc ×B
1 mv⊥
=
2 q Rc2 B 2
(4.78)
where we have used B/ωc = m/ |q|. Combining with the curvature drift we find
m
v T = v ∇B + v R =
q
Rc ×B
Rc2 B 2
v2
1 2
+ v⊥
.
2
(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 ∼
2
kB T /m and 12 v⊥
∼ kB T /m.
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
1 ∂rBr ∂Bz
+
= 0.
r ∂r
∂z
(4.80)
Provided ∂Bz /∂z does not vary much with r we have
rBr = −
r
0
r
r 2 ∂Bz
∂Bz
dr ≈ −
∂z
2 ∂z
or
Br = −
r ∂Bz
.
2 ∂z
(4.81)
(4.82)
Any radial inhomogeneity of Br gives an azimuthal drift Bz k̂×∇Br r̂ 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?
F = qv×B =
r̂ θ̂ ẑ
vr vθ vz
Br 0 Bz
= r̂(qvθ Bz ) − θ̂q(vr Bz − vz Br ) − ẑ(qvθ Br ).
(4.83)
For simplicity, consider a particle spiralling along the axis (r = rL ) so that we
can ignore grad B drifts. The logitudinal (axial) force is
q ∂Bz
2 ∂z
q ∂Bz
= ∓v⊥ rL
2 ∂z
2
∂Bz
qv⊥
= ∓
2ωc ∂z
mv 2
= − ⊥ ∇ B
2B
Fz = vθ rL
ions and electrons
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 changing 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
2
v2
v⊥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:
2
)/2 is constant. Therefore, we must have vm < v0 and the
K = m(v20 + v⊥0
axial particle velocity decreases as the particle moves into the high field region.
The axial velocity is given by
1 2
1 2
mv = K − mv⊥
2
2
= K − µB
1/2
2
(K − µB)
.
(4.86)
⇒
v =
m
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
Bm
v2
= ⊥m
2
B0
v⊥0
2
2
v0
+ v⊥0
=
2
v⊥0
1
=
sin2 θm
Mirror ratio
≡ Rm
(4.87)
(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 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
2
take v2 = 12 v⊥
so that Eq. (4.79) gives
2
mv⊥
qRc B
rL
∼
vth
Rc
vT =
(4.89)
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
vT
±rL
∼
≡ κ.
vth
Rc
(4.90)
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 = 0.1 m/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
θm
v
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 = µ0 NI
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.
B =
µ0 NI
.
2πR
(4.91)
112
To first order in = r/R0 , where R0 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 + vz
(4.93)
dt
dt
where we have taken x = R − R0 and where the vertical z drift velocity is given
by
m
2
(2v2 + v⊥
).
(4.94)
vz =
2qBφ R
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
B0
dφ
rB0
= .
(4.95)
q(r) =
=
dθ
RBθ
Bθ
In tokamaks, q typically lies in the range 0.7 < q(r) < 3 or 4. In terms of q, the
poloidal rotation frequency is given by
Ω=
1 dφ
vφ
dθ
=
=
dt
q(r) dt
q(r)R
(4.96)
with vφ = v.φ̂.
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
(x + vz /Ω)2 + z 2 = 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
vz
Ω
2
m vth
q(r)R0
∼
qB R0
vth
∆ = −
114
vth
ωc
= q(r)rL .
= q(r)
(4.97)
This shift is shown in Fig. 4.17
Trapped particles
Let’s now assume that (v⊥ /v )2 > 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⊥
sin θ
=
2qB0 R
2
(for v2 v⊥
).
(4.98)
The parallel force felt by the particle due to the increasing toroidal field is
∂B
(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:
mv˙ = −µ
≈ Rφ =
rB0
θ
Bθ
or
Bθ
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
θ = κ
with κ ≡
∂B
∂
=
[B0 (1 − cos κ)]
∂
∂
= B0 κ sin κ
r Bθ
sin θ
= B0
R0 rB0
Bθ
=
sin θ
R0
(4.100)
where we have used Eq. (4.92). Equation (4.99) now gives
2
Bθ
mv⊥
sin θ
2B0 mR0
v 2 Bθ
sin θ.
= − ⊥
2 R0 B0
v˙ = −
(4.101)
4.4 Inhomogeneous Fields
115
Comparing with Eq. (4.98) we obtain
vr = −
m
v˙
qBθ
which integrates to give
m
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
r − r0 = −
δv
2
(K − µB)
=
m
≈ v⊥ 1/2
1/2
(4.103)
where we have substituted from Eq. (4.92) for B. Also note that
ωcθ =
qB0
ωc
qBθ
≈
=
m
q(r) m
q(r)
(4.104)
δv
≈ q(r)rL /1/2 .
ωcθ
(4.105)
so that Eq. (4.102) becomes
δr =
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 conservation. If E is oscillating, the E×B drift also oscillates, and there is an energy 12 mvE2
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 equating the rate of change of energy gain from v P .E, find the required value of vP .
HINT: v P 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 tokamak 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
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
1/2
v (z) = (2µB0 /m)[(zm /a)2 − (z/a0 )2 ]
(b) The average force acting on the particle guiding centre along the z axis is given
by
∂B
F = −µ
ẑ
∂z
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 magnetostatic field B = B0 ẑ, and an electric field that oscillates in time at the electron
cyclotron frequency ωc according to
E(t) = E0 [x̂ cos(ωc t) + ŷ sin(ωc t)]
(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):
d2 vx
+ ωc2 vx = 2(eE0 /m)ωc sin(ωc t)
2
dt
d2 vy
+ ωc2 vy = −2(eE0 /m)ωc cos(ωc t)
dt2
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(ωc t)
vy (t) = −(eE0 /m)t sin(ωc t)
and that its trajectory is given by
x(t) = −(eE0 /m)[(1/ωc2) cos(ωc t) + (t/ωc ) sin(ωc t) − 1/ωc2]
y(t) = −(eE0 /m)[(1/ωc2 ) sin(ωc t) − (t/ωc ) cos(ωc t)]