Download Motion of a charged particle in an EM field

Department of physics
Seminar - 4th year
Motion of a charged particle in an EM field
Author: Lojze Gačnik
Mentor: Assoc. Prof. Tomaž Gyergyek
Ljubljana, November 2011
Abstract
In this paper I will present various types of single particle motion in externally applied
electro-magnetic fields, where they arise in nature and engineering, and discuss the applicability of such a model for the representation of plasmas.
Contents
1 Introduction
3
2 Uniform B field
2.1 E = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Constant E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Additional force F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
4
6
3 Non-uniform B field
3.1 ∇B drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Curvature drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Magnetic mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
6
8
11
4 Time-varying E field
4.1 Polarization drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Cyclotron resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
12
13
5 Conclusion
14
References
15
2
1
Introduction
Roughly 99% of matter in the universe is in the plasma state - a gas composed of neutral
and electrically charged particles. Stars, nebulae, and most of the interstellar hydrogen are all
plasma. And yet we rarely encounter naturally-occuring plasma in our everyday lives - almost
exclusively in the form of lightning bolts or the Aurora Borealis. This is explained by the
Saha equation, which relates the amount of ionization in a gas at thermal equilibrium to its
temperature
ni
T 3/2 −Ui /KT
≈ 2.4 × 1021
e
nn
ni
where ni and nn are the number densities of ionized and neutral atoms, Ui is the ionization energy
of the gas, T is the temperature and K is the Boltzmann constant. For air at room temperature,
this gives an ni /nn ration of 10−122 , explaining why we so seldom encounter plasma [2].
To understand the behavior of a plasma it is necessary to first understand the motion of
a single charged particle in an externally imposed electro-magnetic (EM) field. While such an
approach does not explicitly touch upon the collective behavior exhibited by plasmas, it is a
useful first approximation. It can reveal many faults in magnetic confinement devices, such as
those used for nuclear fusion, and tells us about the behavior of space plasmas, such as the
Van-Allen radiation belts in the Earth’s magnetosphere, which is relevant to artificial satellites,
since the highly energetic plasma trapped in the belts can cause damage to both electronics and
organic life.
Throughout this paper I will take E to be the electric field, B the magnetic field, v the
velocity, m will be mass, and q the electric charge. Vectors are written as boldface, such as B,
and we take the same non-boldface variable to mean the absolute value, so that B = kBk. We
will also adopt the convention that A · B = AB.
2
2.1
Uniform B field
E=0
In this case a particle has a simple cyclotron gyration. The equation of motion is
v̇ =
q
v×B
m
We can assume B = (0, 0, B), which yields
qB
vy
m
qB
v̇y = −
vx
m
v̇z = 0
v̇x =
(2.1)
qB 2
qB
v̇y = −
vx
m
m
qB 2
qB
v̈y = −
vy
v̇x = −
m
m
v̈x =
(2.2)
With the exception of describing velocity, not position, these equations are otherwise identical
to those of a harmonic oscillator, with a frequency of
ωc =
3
qB
m
also known as the angular cyclotron frequency [3]. Typical cyclotron frequencies inside a tokamak
are a few tens of GHz for electrons and a few tens of MHz for ions. The solution will be of the
form
vx = a cos(ωc t + δx )
(2.3)
vy = b cos(ωc t + δy )
(2.4)
Inserting eq. 2.3 into eq. 2.1, we find that a = b = v⊥ and δx = δ = δy + π2 . Setting t to
0, we divide eq. 2.4 by eq. 2.3, which yields vy0 /vx0 = − tan δ, where vx0 = vx (t = 0) and
vy0 = vy (t = 0). This gives us
vy0
δ = − arctan
vx0
vx = v⊥ cos(ωc t + δ)
vy = −v⊥ sin(ωc t + δ)
vz = vz0
2 , it follows that v is the velocity in the plane perpendicular to B. To find
Since vx2 + vy2 = v⊥
⊥
the position, we simply integrate the above equations
x = rL sin(ωc t + δ) + xc
y = rL cos(ωc t + δ) + yc
Where xc and yc are integrating constants derived from δ and position at t = 0, and
v⊥
rL =
ωc
is the Larmor radius. These equations produce a guiding center motion around the point
(xc , yc , z), and a constant velocity parallel to B. Particles thus follow a helical trajectory. The
direction of gyration is always such that the magnetic field produced by the particle is opposite
the external field. Plasmas are therefore diamagnetic [2].
Conversely, we can solve the above equations using vector analysis. We can write v = v⊥ +vk ,
where v⊥ is perpendicular, and vk is parallel, to B. The equation of motion is thus split in two:
q
v̇k = (vk × B) = 0
m
q
v̇⊥ = (v⊥ × B)
m
The first equation tells us that vk is constant. The second equation tells us that v̇⊥ is always
perpendicular to v⊥ . Thus v⊥ , and consequently |v⊥ × B|, are constant. This means that
acceleration is constant in magnitude and always perpendicular to B and v⊥ , and therefore
describes circular motion. [4]
2.2
Constant E
We introduce a constant external field E = (Ex , 0, Ez ). The equation of motion is now
mv̇ = q(v × B + E)
Or separately for each component
q
E x + ωc v y
m
v̇y = −ωc vx
q
v̇z = Ez
m
v̇x =
4
(2.5)
(2.6)
Figure 1: Guiding-center motion of a positive ion [3].
The solution for the z-component is simple acceleration
vz =
qEz
t + vz0
m
Taking the time derivative of 2.5 and 2.6 we get
v̈x = ωc v̇y = −ωc2 vx
q
E x + ωc v y
v̈y = −ωc v̇x = −ωc
m
2 Ex
= −ωc
+ vy
B
The second equation is equivalent to
d2 Ex Ex 2
v
+
=
−ω
v
+
y
y
c
dt2
B
B
If we replace vy + Ex /B with vy , the above expression becomes identical to eq. 2.2. Thus the
solution is
vx = v⊥ cos(ωc t + δ)
Ex
vy = −v⊥ sin(ωc t + δ) −
B
In addition to the previous motion, we now have an additional drift parallel to ŷ. To find the
direction of this drift in general, let us recall that B = (0, 0, B) and E = (Ex , 0, Ez ), meaning
that this drift is perpendicular to both B and E, meaning it is parallel to E × B = (0, −Ex B, 0).
If we merely divide this by B 2 , we obtain
vE×B =
Ex
E×B
= (0, − , 0)
2
B
B
(2.7)
which we shall call the E × B drift [2], and is exactly the additional velocity introduced by E
in the direction perpendicular to B. Since we can transform any reference frame into one where
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Figure 2: E × B of a positive ion, with additional acceleration parallel to B [2].
Bkẑ and E ⊥ ŷ, it holds that the vector form of the above equation is always applicable (for
uniform and constant B and E).
Thus the motion of a charged particle in this case is that of a slanted spiral - the particle
circles a guiding center, which is accelerated in the ẑ direction by E, and moves at a velocity of
vE in the ŷ direction. Note that vE does not depend on particle mass, or even charge, meaning
both positively and negatively charged particles will drift in the same direction.
2.3
Additional force F
If we introduce an additional force F of arbitrary origin, the equation of motion now reads
mv̇ = q(v × B) + F = q(v × B +
F
)
q
Thus the additional drift caused by F can be obtained by replacing E with F/q in eq. 2.7, which
yields
F×B
vF =
(2.8)
qB 2
Note that unlike vE , vF depends upon the charge q, and thus points in opposite directions for
positively and negatively charged particles, which results in a net current and subsequent charge
separation. F can, among other things, represent a systemic force in a non-inertial reference
frame.
If it represents gravity it is called the gravitational drift [1], in which case its magnitude
is on the order of 10−11 m/s in typical fusion setups, and 10−5 m/s in the Earth’s Van-Allen
radiation belts, and is in both cases negligible compared to the curvature and gradient drifts.
3
3.1
Non-uniform B field
∇B drift
To calculate the motion in a non-uniform magnetic field, it is necessary to make some approximations. We shall assume that the inhomogeneity of our field is small on scales of rL , so that
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kB(r0 + r1 ) − B(r0 )k B(r0 ) for any r1 ≤ rL . We write
r = r0 + r1
v = v0 + v1
v0 = ṙ0 , v1 = ṙ1
where r0 and v0 describe simple guiding center motion in a homogenous magnetic field, and r1
and v1 are small perturbations due to inhomogeneity. Next we split B into two parts
B = B0 + B1
Where B0 = (0, 0, B0z ) is the main part, and B1 is a small disturbance - the source of our
inhomogeneity. The equation of motion becomes
mv̇ = q(v × B)
= q((v0 + v1 ) × (B0 + B1 ))
= q(v0 × B0 ) + q(v0 × B1 ) + q(v1 × B0 ) + q(v1 × B1 )
Since r1 r0 and v1 v0 , the right-most term in the above equation is a second-order correction
and can be discarded. Joining the 1st and 3rd term then yields
mv̇ ≈ q(v0 × B0 ) + q(v0 × B1 ) + q(v1 × B0 )
≈ q(v × B0 ) + q(v0 × B1 )
Moving into a stationary reference frame where the guiding center is at ~0 at time 0, and setting
B0 to be the total field at ~0 means B(~0) = B0 and B1 (~0) = ~0, thus we may write
B1 (r) ≈ (r∇)B
We introduced the notation B(~0) = B. This gives the equation
mv̇ ≈ q(v × B0 ) + q(v0 × (r∇)B)
≈ q(v × B0 ) + F(r, v0 )
Which is very similar to 2.8, except that now F is a function of position and velocity. It is
possible to further simplify the expression for F:
F = q(v0 × (r∇)B)
= q(v0 × ((r0 + r1 )∇)B)
= q(v0 × ((r0 ∇)B) + q(v0 × ((r1 ∇)B)
≈ q(v0 × ((r0 ∇)B)
In the last step we dropped the final term, since it is of second-order. Thus F becomes a function
of only r0 and v0 . To use equation 2.8, we will average F over a cyclotron gyration. First we
define r0 and v0 :
r0 = (x0 , y0 , z0 ) v0 = ṙ0
x0 = rL sin (ωc t + ϕ)
y0 = rL cos (ωc t + ϕ)
z0 = vz0 t
7
Figure 3: ∇B-drift of a positive and negative particle [2].
We used the total field at ~0, B, to calculate ωc . Next we average over a cyclotron gyration


Z t0
−∂x Bz v⊥ − 2∂z Bz vz0 cos ϕ
2
mv⊥ 
1

∂y Bz v⊥ − 2∂z Bz vz0 sin ϕ
Fpavg =
F dt =
t0 − t 0
2B
2
−∂z Bz v⊥ + 2∂z Bx vz0 cos ϕ − 2∂z By vz0 sin ϕ
Where t0 = 2π/ωc , and we used ∇B = 0 to replace ∂x Bx + ∂y By with −∂z Bz . This has given
us a force that is dependent upon the arbitrary angle ϕ, which is unexpected. Since a particle’s
initial angle ϕ is equally likely to have any value between 0 and 2π we again average over ϕ:


Z π
∂ B
1
W⊥  x z 
∂ y Bz
Favg =
Favg1 dϕ = −
2π −π
B
∂ z Bz
2 /2. Immediately we find it produces a constant acceleration parallel to B.
Where W⊥ = mv⊥
Let’s try to write this expression using general vector notation:
(∂x Bz , ∂y Bz , ∂z Bz ) = ∇Bz = ∇(Bb)
B
= ∇(B ) = ∇B
B
Where b is a unit vector parallel to B. Thus
Favg = −W⊥
∇B
B
Inserting Favg into 2.8 gives the drift velocity
v∇B =
W⊥ B × ∇B
q
B3
This is called the gradient B drift [3]. We notice that it depends on q, giving opposite values
for positively and negatively charged particles and produces a net current and charge separation
in a plasma. In addition to its dependence on W⊥ , the magnitude of the gradient drift falls as
1/B, meaning it is smaller in stronger magnetic fields. Typical values inside a tokamak at fusion
temperatures are on the order of 400 m/s, while electrons with energies of 0.1 MeV in the outer
Van-Allen radiation belt have a drift speed on the order of 3 km/s.
3.2
Curvature drift
However, this result is incomplete. We assumed the guiding center velocity is parallel to B, even
though the direction of B changes as the particle moves. We will deal with this by viewing the
8
particle in an accelerated frame that keeps the direction of B at the particle’s position constant.
This introduces a systemic centrifugal force Fcf , while keeping our current equations of motion
accurate. We can write
Fcf × B W⊥ B × ∇B
vdrif t =
+
qB 2
q
B3
All that is left now is to express Fcf as a function of v and B. Since we are following the
direction of B with speed vk , the particle is affected by a centrifugal force
Fcf = m
vk2 Rc
Rc Rc
where Rc is the local curvature radius vector of B.
Figure 4: Illustration of the curvature vector Rc [2]
We express Rc with B using db/ds = −Rc /Rc2 , where s measures length along the field line,
so that b = B/B. Since d/ds is the derivative along the b direction, we can write
−
Rc
= (b∇)b
Rc2
Giving
Fcf = −mvk2 (b∇)b
Thus the drift equals
vcurv =
2Wk B × [(b∇)b]
q
B2
(3.1)
This is known as the curvature drift [2]. Note that since Fcf kRc ⊥ B, the only effect of this force
is a drift, with no acceleration parallel to B. Also note that once again the drift velocity depends
upon q, and will thus drive a current and produce charge separation. Typical magnitudes for a
fusion plasma in a tokamak are on the order of 400 m/s, and 800 m/s for electrons in the outer
Van-Allen radiation belt.
Finally we can write the total guiding center drift as
vdrif t = v∇B + vcurv + vF
W⊥ B × ∇B 2Wk B × [(b∇)b] F × B
+
+
=
q
B3
q
B2
qB 2
(3.2)
And the guiding center acceleration as
v̇gc = −W⊥
∇k B
1
+ Fk = v̇k
B
m
9
(3.3)
Figure 5: ∇B and curvature drifts inside a tokamak. [3]
Where ∇k B denotes the rate of change of B along field lines, that is, ∇k B = b((b∇)B), and
any force caused by E is included in F. Note that the acceleration is parallel to B.
We encounter such drifts when working with tokamaks - toroidal fusion devices. A tokamak
is, in its most basic form, a simple magnetic coil shaped into a torus, creating a toroidal magnetic
field. This is intended to trap charged particles without the losses that occur in a magnetic mirror
(covered later). In a toroidal coil B is larger closer to the inner wall, which means both ∇B
and the curvature vector Rc point towards the inner wall, creating a vertical current due to the
∇B and curvature drifts, until the E field caused by charge separation balances the two drifts.
However, this field gives rise to an E × B drift that drives the plasma towards the outer wall.
[3]
Figure 6: Combination of poloidal and toroidal B field in a tokamak. [3]
We can avoid this by adding a poloidal B field, either by driving a current through the
plasma, as is the case with tokamaks, or with additional or more complex coils, as is the case
with stellarators. This causes particles to follow a helical path, mixing those on the top and
those on the bottom, and thus undoing any charge separation that might occur [3].
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3.3
Magnetic mirror
We will show that particles can be reflected when traveling into a region with higher B. First
we find the magnetic moment caused by our particle, since its gyration forms a current loop
2
mv⊥
2B
Let us assume that F = 0, E = 0, so that the only field acting upon our particle is B. Writing
the guiding center acceleration, we get
µ=
∇k B
mv 2
d
vk = −W⊥
= − ⊥ ∇k B
dt
B
2B
Multiplying both sides by b yields
dvk
∂B
= −µ
m
dt
∂s
ds
Where B(s) runs parallel to B. Since dt = vk we may multiply the left side by vk and the right
by ds
dt .
dvk
d 1
∂B ds
dB
mvk
= ( mvk2 ) = −µ
= −µ
(3.4)
dt
dt 2
∂s dt
dt
dB
dt is the variation of B as seen by the particle, while B itself is not time-dependent. Since the
only field is B, the energy of the particle must be conserved
m
1
d 1
d 1
( mv 2 + mv 2 ) = − ( mvk2 + µB)
dt 2 k 2 ⊥
dt 2
2 = µB. Combining this with eq. 3.4 yields
Where we used the relation 12 mv⊥
−µ
d
dB
+ (µB) = 0
dt
dt
Which finally gives
dµ
=0
(3.5)
dt
Note that this only holds if B, as seen by the particle, changes slowly compared to ωc . This
will serve as the basis for constructing a magnetic mirror. We can arrange two magnetic coils
symetrically, so that they form a field B0 in their center, and a stronger field Bmax nearer each
coil. Suppose a particle is in the central B0 region with vk = vk0 and v⊥ = v⊥0 . It will be
0 and B = B 0 . The invariance of µ yields
reflected when vk = 0, v⊥ = v⊥
µ0 =
2
02
mv⊥0
mv⊥
= µ0 =
2B0
2B 0
And the conservation of energy yields
2
2
02
v⊥
= v⊥0
+ vk0
= v02
Combining the above equations gives
2
2
v⊥0
v⊥0
B0
=
= sin2 θ
=
02
B0
v⊥
v02
B0
sin2 θ
Where θ is the angle between B and v in the weak-field region. This means that when the
field reaches B 0 , a particle will be reflected, and if B 0 > Bmax , a particle will not be reflected at
all. This is the source of particle losses in a magnetic bottle made from two magnetic mirrors
[2].
B0 =
11
Figure 7: A simple magnetic mirror trap [1].
Figure 8: A banana orbit between two points on the high-field side of a tokamak [3].
While not obvious at first glance, magnetic mirrors can appear in tokamaks as well. Since
the magnetic field lines, augmented by the poloidal field, now follow a helical trajectory in- and
out-of the inner, high-field side, a particle may bounce when it encounters the stronger B field.
The path such a particle follows is called a banana orbit [3].
The Van-Allen radiation belts, containing charged particles trapped in the Earth’s magnetosphere, are also a consequence of the mirror effect. As particles follow the field lines from the
equatorial region towards the polar regions, they encounter a stronger magnetic field. Those
that are reflected form the radiation belts, while those that are not produce the Aurora Borealis
when they encounter the atmosphere [2].
4
4.1
Time-varying E field
Polarization drift
We take E and B to be spatially uniform, where B is constant, while E varies in time with a
frequency much slower than ωc , and observe the effects such a setup gives. We recall the formula
12
Figure 9: Charged particle motion in the magnetosphere [2].
for E × B drift
vE×B =
E×B
B2
Since E now varies with time, so does vE×B
v̇E×B =
d E×B
dt B 2
In the guiding center frame this is equivalent to a force F = −mv̇E×B , which produces yet
another drift
m d
F×B
= − 4 (E × B) × B
vP =
2
qB
qB dt
m d
= − 4 ((EB)B − B 2 E)
qB dt
m d
m
=
(E − (Eb)b) =
Ė⊥
2
qB dt
qB 2
This is called the polarization drift, and is dependent upon charge, and thus in opposite directions
for positively and negatively charged particles, which produces a polarization current [4].
4.2
Cyclotron resonance
We shall now look at the effects of an E field that varies with the cyclotron frequency. We set
B = (0, 0, B) and E = (E cos(ωc t), 0, 0). The equations of motion become
v̇x = ωc vy + qE cos(ωc t)
v̇y = −ωc vx
(4.1)
v̇z = 0
v̈x = ωc v̇y − ωc qE sin(ωc t) = −ωc2 vx − ωc qE sin(ωc t)
v̈y = −ωc v̇x =
−ωc2 vy
− ωc qE cos(ωc t)
(4.2)
(4.3)
Lets guess that the solution will be of the form
vx = At cos(ωc t)
(4.4)
vy = At cos(ωc t + δ)
(4.5)
Inserting 4.4 and 4.5 into 4.1 gives δ = π/2, while inserting 4.4 into 4.2 yields
−2Aωc sin(ωc t) − Atωc2 cos(ωc t) = −Atωc2 cos(ωc t) − Eωc q sin(ωc t)
13
From which we can derive A to be
Thus the solution is
1
A = Eq
2
1
vx = Eqt cos(ωc t)
2
1
vy = − Eqt sin(ωc t)
2
We can express v⊥ and W⊥ as
q
1
vx2 + vy2 = Eqt
2
1
1
2
W⊥ = mv⊥
= mE 2 q 2 t2
2
8
v⊥ =
Figure 10: Spiral trajectory of a particle undergoing cyclotron heating.
We see that W⊥ increases as t2 ! This is the basis for cyclotron resonance heating - plasma
is irradiated with radio waves of frequency ωc , thus heating it. Since ions and electrons have
different ωc , we can choose which of them to heat directly. Furthermore, since B decreases with
distance from the inner wall of a tokamak, so does ωc , meaning we can control where the energy
of cyclotron heating is deposited by choosing an appropriate frequency. [3]
5
Conclusion
We have shown that charged particles follow complex paths in an EM-field, and some of the
effects that make containing charged particles difficult. However, missing from our analysis
are multi-particle effects, such as collisions with other particles, and collective behavior - the
interaction of a plasma with its own EM-fields, which further complicates the containment of
a hot plasma. Nonetheless such simple theory can often produce workable results, especially if
the particle density is low enough to be able to neglect collision effects, and the EM-field caused
by the plasma is small compared to the external field, as is the case in some space plasmas.
14
References
[1] R. J. Goldston, P. H. Rutherford, Introduction to Plasma Physics. Institute of Physics Publishing, Bristol and Philadelphia, 1995.
[2] F. Chen, Introduction to Plasma Physics and Controlled Fusion. 1974.
[3] A. A. Haarms, K. F. Schoepf, G. H. Miley, D. R. Kingdon, Principles of Fusion Energy:
An Introduction to Fusion Energy for Students of Science and Engineering. World Scientific
Publishing, Singapore, 2000.
[4] J. A. Bittencourt Fundamentals of Plasma Physics. Springer-Verlag, New York, 2004
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