Single particle motion
Plasma is a collection of a very large number of charged particles moving and
giving rise to EM fields. Before going to the statistical descriptions, let us learn
about the motion of individual particles in given EM-fields:
• Guiding centre approximation
• Adiabatic invariants
• Motion in dipole field
• Motion in the field of a current sheet
Guiding centre approximation
The motion in a magnetic field of a charged particle can be treated as the
superposition of a relatively fast circular motion around a point called the guiding
center and a relatively slow drift of this point.
The task is to solve the equation of motion of a charged particle:
Lorentz Force
(assume, for the time being,
nonrelativistic motion; = 1 and p = mv)
macroscopic field: sum of the externally
applied field and the fields generated by
the plasma particles collectively
Fnon-EM contains all non-electromagnetic forces acting on the particle. In practise,
this usually means gravitation:
F n o n -E M
mg
Plasma is a collection of charged particles
Why to study single particle motion?
• A large part of phenomena we shall meet later can be understood (even
quantitatively) by the single particle motion
• Single particle description valid under some conditions:
- e.g. high energy particles in low density plasma where collisions are infrequent
- also it is required that fields induced by motion of particles are neglible
compared with the applied fields.
It is impossible to solve the Eq. of motion analytically in a general case, where the
fields depend on time and spatial coordinates
For slow temporal changes and smooth spatial dependencies of the fields we can
use perturbation theory, i.e. calculate the motion of particles starting from the
solution with constant fields B, E and g
This way the particle trajectory, as averaged over many gyro-periods, is obtained
The case of E = const and B = 0 (neglect the non-EM forces) is trivial:
Eq. of motion
, i.e., linear acceleration in the direction of E
Motion in a homogenous and static
magnetic field
Consider next the case E = 0 and B = const
The equation of motion is now
(static uniform magnetic field)
(*)
Since we chose electric field to be zero, Lorentz force is perpendicular to the
velocity the particle. Consequently, it can change only direction of the velocity ,
but the kinetic energy and the speed remain constant.
(*)
The orbit can be found in several ways, let’s do it very simply starting from the
components of the equation of motion (let B be in the z-direction)
(*)
Since v and vz are constants
Write: vx
v cos
and v y
v sin
v
v 2 vz2
vx2 v y2
is constant
And calculate time derivative of vx: vx
And inserting these to (*) gives:
mv sin
qv sin B
qB
m
0
cyclotron frequency
gyro frequency
Larmor frequency
c
c
t
The motion of the particle is
gyromotion in the xy-plane!
v sin
The solution describes helical motion. In a homogeneous magnetic field the
particle motion can, thus be divided in two components:
- linear motion along the field lines at constant speed (v|| = vz = constant)
- circular motion in the plane (xy plane) perpendicular to B
The radius of the circle is
(Larmor radius)
rL
B
1
;
The higher B, the smaller rL (i.e. particles bound
more tightly to the magnetic field when B has
large values)
rL
m
rLe
rLi
For a given value of the magnetic
field electrons are bound more tightly
to B than ions
The gyro period (cyclotron period, Larmor time) is:
The centre of the gyro motion is called guiding centre (GS)
The pitch angle ( ) of the helical path is defined as the angle
between the velocity vector of the particle and the magnetic field
B
v
90
0
v
v
v
v
In a non-uniform magnetic field:
- changes as the ratio between v and v|| changes
- pitch angle is a frequently used concept when studying the magnetosphere e.g in
determining whether a charged particle will be lost to the Earth's atmosphere or not.
The frame of reference where v|| = 0 : Guiding centre system (GCS)
Division of the motion of the guiding centre and
to the gyro motion is called the guiding centre approximation
• In the GCS we can formally address a microscopic current with the Larmour
motion of a charged particle: I = q / L.
• The magnetic field arising from this microcurrent is always opposite to the
applied magnetic field (rL depends on the sign of q) and thus plasma can be
considered a diamagnetic medium
• Placing a large number of charged particles to external magnetic
field reduces the field.
The magnetic moment associated to the circular loop is
rL
Note:
in vector form
W
average
• For n particles per unit volume the magnetization M
and the magnetization current
JM
Relation of applied and induced fields:
(remember: we required that Bind << B)
M
1
0
Bind
B
n
Bind
0
n W
B2
induced field
E x B drift
Let’s move on by introducing a constant, uniform electric field: E = const
Eq. of motion is now:
dp
dt
q (E
v
B)
The electric field can be resolved into components:
- in the direction of B: EII
- perpendicular to B: E
1. Components parallel to B
Only E can exert force along B. This force causes a constant acceleration of ions
in the direction parallel to B and moves electrons in the opposite direction
charge separation field that rapidly cancels out large-scale E|| in plasma!
Eq. of motion along B becomes:
m
dv
dt
qE
0
2. Components perpendicular to B
Let’s assume again that
B || eˆ z
and that
B
E || eˆ x
This yields
Solution same as for E=0, but modified by the term Ex/B
Substitution
leads again to gyro motion but now
the GC drifts in the y-direction ( both to B and E) with speed Ex /B
In vector form:
The drift speed is consistent with the Lorentz transformation of the E to GCS:
E
'
E
v
E
B
by demanding E’=0, we can solve for vE
+q accelerates when
moving with E
(rL increases)
+q slows when
moving opposite to E
( rL deccreases)
vE independent of both m and q, W
+ ion
electron
- same for all particles
- drift to the same direction ( E and B)
no electric current!
negative electron gyrates in the
opposite direction, but also gains
energy in the opposite direction
Co-rotational electric field
E (
r) B
: Earth’s angular velocity
Inside ~4 Earth radii corotational electric field is the
electric field for magnetic field
lines
forces plasma to rotate
with the planet (called
plasmasphere)
plasmasphere
ROSETTA
Measurements of the plasma flow velocities
in space and laboratory plasma
Electric field can be measured using a conducting, so
called Langmuir probe
E
(
2
1
)/L
Other non-magnetic drifts
Any force that is capable of accelerating/decelerating particles as they gyrate about
B will result in drifts perpendicular to both B and the force
Write the perpendicular eq. of motion in the form
Assume that F gives rise to a drift vD and transform
In GCS the last two terms must sum to 0
( )
This requires F/qB << c. If F > qcB , the GC approximation cannot be used!
Note: If F is charge independent
separates charges
currents
drift opposite direction for electrons and ions
B
Inserting F = qE into v D
F B
qB 2
we get the ExB-drift
F = mg gives the gravitational drift
Note that the gravitational drift is horizontal not vertical!
charge separation caused
by the gravitational drift
However, downward acceleration occurs due
to the E B-drift:
Charge separation caused by the gravitational drift
causes E B-drift in the direction of the gravitational
force
downward acceleration
ExB drift
Slow time variations in E
polarization drift
Directed along electric field, but oppositely for electrons and ions
The corresponding current is called the polarization current:
Carried primarily by the more massive ions!
Compare with the polarization in solid dielectric, D= 0E+P
Dipoles in a plasma are ions and electrons separated by a distance r.
However, in plasma application of a steady E does not result in a polarization field
P since charges can move around to maintain quasi-neutrality. If E oscillates,
polarization current results from the lag of ion inertia.
Nonuniform magnetic field drifts
When B has a transverse gradient or is curved, the GC tends to move perpendicular
to B because of variations in the instantaneous radius of curvature
recall: rL
mv
|q|B
1
B
Particle orbit is nearly
a circle, but it no
longer closes on itself
B
r
1. Gradien drift
Assume static but inhomogeneous magnetic field. Consider that the field is only
weakly inhomogenous, where changes are small during one Larmor gyration,
i.e. GC approximation is useful if the gradients of B are small as compared to the
gyro motion
Expand the field as a Taylor series about the GC:
B(r ) B0
r r0
B
0
...
where 0 means the quantity evaluated at the GC position r=ro(t). Thus,
B (r )
B
0
r
B
0
A straightforward (but tedious) calculation shows that the inhomogeneous
magnetic field causes a force
(averaged over a gyro period)
magnetic moment
Force component parallel to the magnetic field causes an acceleration:
The change in v|| must be accompanied by a change in v to keep v constant
The component perpendicular to the magnetic field F
causes a drift at speed:
vG
F
B
qB 2
qB 2
B
B
B
W
B
qB 3
B
This is called a gradient drift
A magnetic field gradient leads to a gradient drift perpendicular to both the magnetic
field and the gradient
Note: vG~q-1
gradient drift associated with a current
ion
B
|B|
electron
More energetic particles drift faster, since
they have larger gyroradius and experience
more of the inhomogenity of the field
2. Curvature drift
Let’s now assume that the magnetic field lines are curved with a constant radius
of curvature RC (positive inwards). GC drift now arises from the centrifugal force
felt by the particles as they move along the field lines:
w|| is the velocity of the GC along B
(not exactly v|| but w|| v|| is for us a good enough
approximation)
Applying this force as before we get the curvature drift
exercise
If there are no local currents (
B = 0),
and vG and vC can be combined to
unit vectors
n || RC, t || B
arccos
v||
v
(pitch angle)
v GC
kinetic energy
gradient and curvature drifts tend to control ‘hot’ (high energy) particles
while E B-drift tends to control “cold” low energy particles
vG & vC
q -1
give rise to currents
Example of gradient and curvature drift: Radiation belt
In a dipole field vG and vC
are in the same direction. Ions drift west
and electrons east producing westward
current called a ring current
Example of gradient and curvature drift: Tokamak
Gradient and curvature drifts are both in the same
direction. Electrons and ions drift to opposite
directions and finally hit the walls of the machine
polarization charge
electric field causing ExB
drift. Particles get lost when they hit the walls of
the machine.
Tokamak
Solution: toroidal magnetic field given a twist (e.g.
by applying axial current through the plasma)
same direction!
polarization charge
E
B, R
Adiabatic invariants
periodic motion
conserved quantity
H
action integral:
I
pdq
L
i
Hamiltonian formulation:
Let q & p be canonical variables
qi pi
L
pi
1 2
mv
2
H
qi
qA v q
qi
H
pi
Assume that the motion almost periodical
is constant, called adiabatic invariant
Adiabatic invariants help us to obtain simpler solution in many cases that involve
complicated motion
Example:
Consider a charged particle in Larmor motion.
Assume that the B does not change much during within one circle.
The canonical coordinate is rL and the canonical momentum
charge!
I
p drL
mv drL
q
A dS
S
2 rL
mv dl q B dS
0
S
2
L
2 mv rL | q | B r
2 m
|q|
i.e., the magnetic moment is an
adiabatic invariant
magnetic moment in terms of magnetic flux:
1
2
C
2
L
|q|r
1 qB
| q | rL2
2 m
1 q2
2m
remains constant, irrespective whether the change is due to time variation or
the motion in an inhomogenous magnetic field.
B
B
Example: Lorentz-Einstein pendulum
length of the pendulum is
changed slowly (as compared to
)
changes slowly
l
Now the energy per mass unit
is not conserved.
Lorentz asked Einstein at a conference in 1911: What is the conserved
quantity in this case?
Einstein: W / .
This was a relevant question while quantum mechanics was being
developed. Recall that
Relation to the conservation of magnetic moment:
First adiabatic invariant
In plasma physics
is called the first adiabatic invariant.
Let’s show its conservation by direct calculation in the case of static B and E=0
i.e. total energy is conserved:
1. Change in perpendicular energy
W
B
dW
dt
dB
dt
d
B
dt
,where
dB
dt
ds dB
dt ds
v||
dB
ds
along the
path of GC
2. Change in parallel energy:
Long calculation
multiply LHS by v|| and RHS by ds/dt
and thus
i.e.,
is constant in the GC approximation
Let’s then study the motion of a charged particle in a time varying magnetic field
Assume, field is varying slowly compared with the cyclotron period
Work gained during one gyroperiod:
E
Since field changes slowly:
B
Stokes
w
q E dl
q
C
E dS
q
S
For small changes:
W
Faraday
1
q
2
B
t
2
r
C L B
S
B
C
L
2
B
B
dS
t
B
are of orbit ~ rL2
On the other hand we can write:
must again remain constant
Note that also the flux enclosed by the gyro orbit
is constant
E
B
t
Magnetic mirror
W
B
m v 2 s in 2
2B
In GC approximation both W and
can change.
are conserved. Thus, when B changes, only
and B are related through:
When a particle moves into regions of stronger B its
W increases and W|| decreases
mirror point:
°, W||
increases
mirror point
0
particle is pushed back by the parallel
component of the gradient force, called
mirror-force:
B0,
The strength of mirror field Bm depends on particle’s
(smallest
0
Bm, =90°
in some reference field B0.
to confine a particle)
Magnetic bottle
A simple magnetic bottle consists of two mirrors facing each other.
A charged particle is trapped in the bottle if
= 90°
v
Otherwise it is said to be in the loss-cone
and escape at the end of the bottle
trapped
particles
loss cone
v||
The dipole field of the Earth is
a large magnetic bottle
= 0°
Adiabatic heating
Mirror effect is a consequence of the conservation of
magnetic field lines.
when particle moves along
Now let’s consider a particle drifting across the field lines from a region with a
magnetic field strength B1 into a region of increasing field strength B2
since B2 > B1
adiabatic heating
Second adiabatic invariant
Consider a particle bouncing between two magnetic mirrors
at the associated ”bounce frequency”
ds = element of path length along the
field line
mv|| ds
Constant of motion:
periodic motion
GC drifts across field lines, the motion is not exactly periodic
b
J
becomes second adabatic invariant
p || d s
a
sm'
bounce period
total energy:
2
b
W
ds
v ( s)
sm ||
W
W||
1
mv|| 2
2
B
v||
2W
1
m
B
W
'
sm
b
ds
2
v sm 1 B( s) / Bm
We need to require
b
1/ 2
>>
L=2
/
c
(weaker invariant than )
Fermi Acceleration
Second adiabatic invariant J
p|| ds
2ml
v||
l = total length of the field line between the mirror point
<v||> = average parallel velocity along the field line
Conservation of J implies that the averaged parallel
energy <W||> changes according to
Enrico Fermi
W||
1
W||
2
l12
l22
If mirror points move toward each other, W|| increases
(Fermi acceleration)
proposed by Enrico Fermi to explain acceleration of
galactic cosmic rays.
Cosmic rays striking the Earth’s
atmosphere and causing a high
energy particle shower
v||
s1 s2
s0
s1' s2'
s10
s2'
s1'
s1
J invariant
v||'
v||
s0
s0'
W||'
s2
s0
W||
s0'
v|| and v||’ parallel components of velocity at the mid-plane at t=0 and t=t’
W'
W ' W||'
m 2
v
2
s0
s0'
2
v||2
(assumed that B is constant at the mid-plane)
Third adiabatic invariant
• associated with azimuthal drift motion
• exists only for fields with a well defined axial symmetry
If the perpendicular drift of the GC is nearly periodic
(e.g. in a dipole field), the magnetic flux through the GC orbit
is conserved. This is the third adiabatic invariant.
Now
d
>>
b
>>
L
Summary of adiabatic invariants
There is a specific energization mechanism for each invariant
:
J:
W changed by changing the Larmor radius (i.e., |B|)
W|| changed by stretching or shortening the magnetic bottle
:
W changed by compressing or expanding the drift surface
Motion in a dipole filed
It is of great importance to understand the trajectories in the magnetosphere to
explain various phenomena, such as radiation belts and auroras.
Giant magnet: William Gilbert, English
physician: De Magnete 1600
Carl Stormer and his
assistant photographing
auroras in northern Norway
in 1910s. Stromer
calculated the motion of
charged particles in a
dipole magnetic field
supporting the work of K.
Birkeland’s
Kristian Birkeland in his laboratory with
his terella in 1909
Terella (“little Earth”): invented by William
Gilbert. Further developed by Kristian
Birkeland while studying particle motion in
the magnetosphere
Occurrence of aurora confined within the auroral
oval puzzled Birkeland and Stromer. Details
discussed in application of plasma physics!
The dipole field is familiar from basic electrodynamics.
We use the notations of geomagnetism:
- the latitude : zero at equator, increases towards north
- the longitude increases towards east
- instead of dipole moment ME, we use k0 = 0ME/4
M
Earth’s dipole
moment points
southward (–z)!
Dipole approximation: field generating currents compressed to a singular point;
elsewhere field is curl-free, it can be obtained as
, where
components of B:
magnitude of B:
(azimuthal symmetry)
Eq for magnetic field lines in spherical coord.
dr
Br
rd
B
r sin d
B
The length of line element is now
Note the important geometrical factor
Every field line is determined by two parameters:
(constant) longitude 0 and distance r0 where the line crosses the equator
Define the L-parameter: L = r0/RE , RE is the radius of Earth (or another planet)
For given L the field line crosses the surface at latitude
B on a given field line as a function of latitude:
for the Earth:
In calculations of particle orbits we need the curvature radius of B, RC .
GC approximation is applicable in dipole field, if rL << RC ( )
This can be expressed in terms of the magnetic rigidity of the particle:
(quantity used widely in acceletrator physics and cosmic ray studies)
Centrifugal force and lorentz force must balance:
mv 2
RC
rL
B
B
qBv
1 rL
BRC
p
q
B
B
mv
| q | RC B
Condition reduces to
mv
| q | r0 B
mv
|q|
B0=k0/r03,
Magnetic field
conserved
Let
m
B( )
sin
2
( )
1/ 2
2
k0 1 3sin
r03
cos 6
B0
sin 2
B( )
0
cos6
0
1 3sin 2
1/ 2
sin 2
0
be the mirror latitude. The pitch-angle at equator is determined by:
The dipole field is a magnetic
bottle
If m > latitude where field line crosses the Earth surface ( e=arccosL-1/2) particle hits the
Earth’s surface before mirroring and leaks out of the bottle
The width of the loss cone at equator is given by
The particle leaks out if its
Consider the bounce motion in the dipole field:
0
arcsin
cos3
1 3sin 2
m
1/ 4
m
For pitch-angles
In the Earth’s dipole field 1 keV electrons bounce in seconds,
1 keV protons in minutes.
Magnetosphere can change considerably in minutes
questionable for > keV protons
Invariance of J
Particles circle the Earth due to gradient and curvature drifts
W
1 cos 2 ( )
qB( ) RC ( )
vGC ( )
B( ) RC ( )
B = 0)
2
1 3sin 2
k0
3r0 cos5
( now
2 cos 2
What we are interested is the average of the over the bouncing period <vgc>
The angular drift speed around the Earth is:
And averaged over the bounce period, gives:
4
b
b
/4
4
v b
dt
0
1
r0 f ( 0 )
m
0
m
0
ds d
d cos
vGC ( ) 1 3sin 2
cos2 cos ( )
4r0
f(
v
b
1/ 2
d
0
)
vGC
r cos
vGC
r0 cos 3
3 2
mv r0
2
g(
qk0
where
0
g(
)
0
)
1
f ( 0)
m
0
cos3 (1 sin 2 )(1 cos 2 )
d
2
3/ 2
(1 3sin ) cos
For
For particles staying on equator (
g( 0)=f( 0)=1
For relativistic particles this is
0
= /2)
Finally the average
drift time is given by
In case of the Earth: positive ions drift to the west, electrons to the east
westward net current (ring current)
At the ring current region (L ~ 2-7) drift times:
- 1 keV particles hundreds of hours
- 1 MeV few tens of minutes
Recall:
mp = 938 MeV/c2
me = 511 keV/c2
ions are nonrelativistic, the most energetic electrons are relativistic
Van Allen belts (or radiation belts)
Typical energies:
Inner belt protons: 0.1 MeV – 40 MeV
Outer belt electrons: keV – MeV
James Van Allen hoisting a model of Explorer
1 that was the first satellite launched by U.S.
in January 31, 1958. Explorer 1 carried only
one instrument (geiger counter) that detected
the inner belt and confirmed the existence of
the radiation belts.
Example of adiabatic heating of magnetospheric particles
• at equatorial plane, =0:
B(
0)
B0
L3
W
W
Insert this to:
2
1
L1
L2
3
i.e. if L2 < L1 particle gains energy
E.g. 1 keV particle, starts at L1 = 8 with pitch angle energies of 10 keV inside
L=3
Electrons and ions with equatorial pitch angles < 90° also undergo Fermi acceleration
W||2
W||1
L1
L2
2
for
2.5 for
eq
eq
0
90
Adiabatic heating creates the 10-100 keV ring current ions from the 1-10 keV
plasma sheet ions
Energy anisotropy:
For =2.25
AW 2
AW 1
W
W||
AW
L1
L2
AW 2
AW 1
B2 l22
B1 l12
0.75
The further one gets into the inner magnetosphere, the more there are
particles with large pitch angles (large W )