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
UCLA undergraduate Lab
Plasma is a collection of charged particles
Why to study single particle motion?
• A large part of phenomena can be understood (even
quantitatively) by single particle motion
•Single particle description valid under some conditions:
•
- e.g. high energy particles in low density plasma where
collisions are infrequent
- fields induced by motion of particles negligible when
compared with applied fields
Task: solve the Eq. of motion of a charged particle:
Lorentz Force
non-EM forces acting on
the particle (usually
gravitation)
macroscopic field:
sum of the externally applied field and the
fields generated by the plasma particles
collectively
1) E = const and B = 0
Eq. of motion
i.e. linear acceleration in the direction of E
2) E = 0 and B = const
Eq. of motion:
kinetic energy and the speed remain
constant
4.2
0
c
t
cyclotron frequency
gyro frequency
Larmor frequency
particle motion divided in two components:
1) motion along B at constant speed
2) circular motion perpendicular to B
https://www.youtube.com/watch?v=a2_wUDBl-g8
radius of the circle:
(Larmor radius)
;
the larger the B, the more tightly particles are ‘bound’ to B
electrons more tightly bound than ions
gyro period (cyclotron period, Larmor time) is:
guiding centre (GC)
• guiding centre system (GCS): frame of reference where v|| = 0 :
• guiding centre approximation:division to GC and gyro motion
The pitch angle ( ) of the helical path: angle between the
velocity vector of the particle and the magnetic field
B
v
90
0
v
v
v
v ||
• uniform magnetic field:
constant
• non-uniform magnetic field:
changes as the ratio between v and v|| changes
• frequently used concept e.g., when studying the magnetosphere
• In GCS charge gives rise to a microscopic current
with associated magnetic moment:
in vector form
Magnetic moment vector always opposite to the ambient field (rL ~ sign of q)
charged particles weaken the external magnetic
plasma can be considered a diamagnetic medium
4.2
+ ion
electron
ExB-drift
Inside ~4 Earth radii
forces plasma to rotate
with the planet (called
plasmasphere)
plasmasphere
ROSETTA
2014: 67P/ChuryumovGerasimenkoa
Co-rotational electric field
Measurements of the plasma flow
velocities in space and laboratory
plasma
Electric field : from Langmuir probe
force capable of accelerating/decelerating particles gyrating about B
drifts to B and the force
perpendicular eq. of motion:
transform
In GCS the last two terms must sum to 0
Note: F charge independent
drift opposite direction for electrons and ions
separates charges
currents B
F = mg gives the gravitational drift
charge separation
caused by the
gravitational drift
gravitational drift is horizontal not vertical!
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
ExB drift
Slowly varying E
Exercise!
polarization drift
Corresponding current: polarization current:
Carried primarily by the more massive ions!
Dipoles in a plasma are formed by ions and electrons
-
steady E
no polarization field P (charges move around and
maintain quasi-neutrality)
- E oscillates
polarization current results from the lag of ion inertia.
periodic motion
conserved quantity
Hamiltonian: q & p are canonical variables
(generalized momentum and coordinate)
almost periodic motion
is constant, called adiabatic invariant
Example: time dependent harmonic oscillator
A=amplitude of the oscillation
total energy
magnetic moment is the first adiabatic invariant!
4.4
conservation can be shown by direct calculation in the case of static B and E=0
4.5
magnetic flux stays also constant
What if the magnetic field varies slowly in time?
4.6 …
again remains constant
also the flux enclosed by the gyro
orbit is constant
B
E
Electric field induced by slowly
varying B
charges gain energy when
moving along E
Recall:
mirror point
mirror point:
°, W||
0
magnetic mirror
pushed back by parallel component of the gradient force (mirror-force):
magnetic bottle
two magnetic 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 escapes at the end of the bottle
trapped
particles
loss cone
v||
= 0°
The dipole field of the Earth is a large magnetic bottle
http://www.youtube.com/watch?v=6CpNOu4l1dM
GC drifts across field lines
motion is not exactly periodic
BUT
If field does not change much in one bounce period, the motion is
nearly periodic
Constant of motion:
J
mv|| ds
ds = element of path length
along the field line
bounce motion in a
magnetic bottle
b
J
p || d s
second adabatic invariant
a
bounce period
sm'
b
ds
2
sm v|| ( s )
4.7
require
b
>>
L=2
/
c
sm'
ds
2
v sm 1 B( s) / Bm
1/ 2
(weaker invariant than )
What happens if magnetic bottle shrinks?
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
averaged parallel energy <W||>
changes according to
W||
1
W||
2
l12
l22
mirror points move toward each other
Fermi acceleration
W|| increases
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
(second order) Fermi acceleration
Third adiabatic invariant
• associated with azimuthal drift motion
• exists only for fields with a well defined axial symmetry
A trapped particle performs a drift motion in
axially symmetric magnetic field
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
Summary of adiabatic invariants
Invariant
Velocity
Time scale
magnetic
moment
gyro motion
gyro period
longitudinal
invariant
longitudinal
velocity of GC
bounce period
flux invariant
perpendicular
velocity of GC
drift period
Validity
Examples of violation of adiabatic invariant
First invariant
• magnetic pumping: collosion freq. larger than magnetic pumping freq
collisions transfer perpendicular to parallel energy (heating)
• cyclotron heating: B oscillates at
•
c
c
constant acceleration
0 (e.g. magnetic cusp, close to current sheet)
Second invariant
• any resonance with bounce motion
• changes in magnetic field configuration (no longer trapped)
Third invariant
• Solar wind interaction changing E and B fields in magnetosphere
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
Kristian Birkeland and terella in 1909
Carl Stormer photographing auroras in
northern Norway in 1910s
The dipole field familiar from basic electrodynamics.
We use the notations of geomagnetism:
M
-longitude
increases towards east
Earth’s dipole
moment points
southward (–z)!
components of B:
magnitude of B:
(azimuthal symmetry)
line elements
Eq for magnetic field lines (spherical coord.)
length of line element:
important geometrical factor
field lines of
different L
L-parameter
RE : Earth radius
For given L, at what latitude the field line crosses the Earth’s surface?
Lat. where field line crosses the
Earth surface: e=arccosL-1/2
m
>
e
:particle hits the Earth’s surface before mirroring
leaks out of the bottle
What is the minimum pitch angle that a particle can have at the equator (at given L
value) and still be trapped?
4.8
Insert now
m=
loss cone
e
The particle leaks out if its
(loss cone)
Up to what energies J stays still invariant?
bounce period in dipole field:
Use:
for pitch-angles
In the Earth’s dipole field 1 keV electrons
bounce inseconds,1 keV protons in minutes. Magnetosphere can change
considerably in minutes
Invariance of J questionable for > keV protons
Particles circle the Earth due to gradient and curvature drifts
angular drift speed around the Earth:
where
for
For relativistic particles this is
Finally the average
drift time is:
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
where
for
For relativistic particles this is
Finally the average
drift time is:
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
drift time for 1 keV particles
Van Allen belts (or radiation belts)
- Inner belt protons: 0.1 MeV – 100 MeV
beta decay of neutrons which are emitted from the Earth's
atmosphere as it is bombarded by cosmic rays
- Outer belt electrons: 0.1 – 10 MeV
injection from the outer magnetosphere
http://www.youtube.com/watch?v=QwNNh4c9bR0
Example of adiabatic heating of magnetospheric particles
Insert this to:
i.e. if L2 < L1 particle
gains energy
inward drift
L decreases
Adiabatic heating creates the 10-100 keV ring current ions from the 1-10
keV plasma sheet ions
Current sheets are important in plasma physics
- separate different plasma domains
- sites of the most important energy release process, magnetic reconnection
Bn << B0
Current flowing along positive y-axis in
the region where Bx changes sign
E.g. magnetotail
Harris current sheet:
; Bn and B0 are constant and
L gives the thickness of the current sheet
According to Ampère’s law
1-D
Bn =0
2-D
Motion in the current sheet