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Dr. Peter T. Gallagher
Astrophysics Research Group
Trinity College Dublin
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o  Gyrating particle constitutes an electric current loop with a dipole moment:
1/2mv"2
B
o  The dipole moment is conserved, i.e., is invariant. Called the first adiabatic
invariant.
µ=
!
o  µ = constant even if B varies spatially or temporally. If B varies, then vperp varies to
keep µ = constant => v|| also changes.
o  Gives rise to magnetic mirroring. Seen in planetary magnetospheres, magnetic
bottles, coronal loops, etc.
Bz
o  Right is geometry of mirror
Br
from Chen, Page 30.
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o  Consider B-field pointed primarily in z-direction and whose magnitude varies in zdirection. If field is axisymmetric, B! = 0 and d/d! = 0.
o  This has cylindrical symmetry, so write B = Br rˆ + Bz zˆ
o  How does this configuration give rise to a force that can trap a charged particle?
!
o  Can obtain Br from " # B = 0. In cylindrical polar coordinates:
"B
1"
(rBr ) + z = 0
r "r
"z
!
"B
"
=> (rBr ) = #r z
"r
"z
o  If "Bz /"z is given at r = 0 and does not vary much with r, then
!
!
$Bz
1 &$B )
dr % " r 2 ( z +
$z
2 ' $z *r=0
1 &$B )
Br = " r( z +
2 ' $z *r=0
rBr = " #0r r
(5.1)
!
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o  Now have Br in terms of BZ, which we can use to find Lorentz force on particle.
o  The components of Lorentz force are:
Fr = q(v" Bz # vz B" )
(1)
F" = q(#vr Bz + vz Br )
(2) 
(3)
Fz = q(vr B" # v" Br )
(4)
o  As B! = 0, two terms vanish and terms (1) and (2) give rise to Larmor gyration.
!
Term (3) vanishes on the axis and causes a drift in radial direction. Term (4) is
therefore the one of interest.
o  Substituting from Eqn. 5.1:
Fz = "qv# Br
=
!
qv# r $Bz
2 $z
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o  Averaging over one gyro-orbit, and putting v" = !v# and r = rL
qv"rL #Bz
2 #z
o  This is called the mirror force, where
! -/+ arises because particles of opposite charge
orbit the field in opposite directions.
Fz = !
!
o  Above is normally written:
1 v2 $Bz
Fz = ! q "
2 # c $z
=%
or
Fz = "µ
#Bz
where
#z !
µ"
1/2mv#2
B
o  In 3D, this can be generalised to:
!
1 mv"2 $Bz
2 B $z
F|| = "µ
is the magnetic moment.
dB
= "µ# ||B
ds
!
where F|| is the mirror force parallel to B and ds is a line element along B.
!
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o  As particle moves into regions of stronger or weaker B, Larmor radius changes, but
µ remains invariant.
o  To prove this, consider component of equation of motion along B:
dv||
dB
= "µ
dt
ds
Multiplying by v||:
dv||
dB
mv||
= "µv||
dt
ds
Then,
d "1 2 %
ds dB
!
$ mv|| ' = (µ
&
dt # 2
dt ds
!
dB
= (µ
(5.2)
dt
d #1 2 1 2 &
The particle’s energy must be conserved, so
% mv|| + mv" ( = 0
'
dt $ 2
2
1/2mv#2 !
Using µ "
%
d "1 2
B
$ mv|| + µB ' = 0
&
dt # 2
!
m
o 
o 
o 
o 
!
!
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o  Using Eqn 5.2,
dB d
+ (µB) = 0
dt dt
dB
dB
dµ
"µ
+µ
+B
=0
dt
dt
dt
"µ
=> B
! to 0, this implies that
o  As B is not equal
dµ
=0
dt
dµ
=0
dt
! (invariant).
o  That is, µ = constant in time
o 
! invariant of the particle orbit.
µ is known as the first adiabatic
o  As a particle moves from a weak-field to a strong-field region, it sees B increasing
and therefore vperp must increase in order to keep µ constant. Since total energy
must remain constant, v|| must decrease.
o  If B is high enough, v|| eventually -> 0 and particle is reflected back to weak-field.
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o  Consider B0 and B1 in the weak- and strong-field regions. Associated speeds are v0
and v1.
B0
o  The conservation of µ implies that
B1
µ=
mv02 mv12
=
2B0 2B1
o  So, as B increases, the perpendicular component of the particle velocity increases:
particle moves more and more perpendicular to B.
!
o  However, since we have E=0, the total particle energy cannot increase. Thus as v⊥
increases, v|| must decrease. The particle slows down in its motion along the field.
o  If field convergence is strong enough, at some point the particle may have v|| = 0.
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o  Say that at B1 we have v1,|| = 0. From conservation of energy:
o  Using µ =
!
mv02 mv12
=
we can write
2B0 2B1
B0
=
B1
v0,2"
=
v1,2"
v12 = v1,2"= v02
v
!
v0,2"
v02
v⊥
!
o  But sin(!) = v⊥/ v0 where ! is the pitch angle.
!
o  Therefore
v||
B
B0
= sin2 (" )
B1
o  Particles with smaller ! will mirror in regions of higher B. If ! is too small,
B1>> B0 and particle does not mirror.
!
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o  Mirror ratio is defined
Rm =
Bm
B0
!
o  So, the smallest ! of a confined particle is
sin2 (" ) =
B0
1
=
Bm Rm
o  This defines region in velocity space in the shape of a cone, called the loss cone.
!
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o  Particles are confined in a mirror with ratio
Rm if they have ! > !m.
o  Otherwise they escape the mirror. That
portion of velocity space occupied by
escaping particles is called the loss-cone.
o  The opening angle of the loss cone is not
dependent on mass or charge. Electrons and
protons are lost equally, if the plasma is
collisionless.
o  Coulomb collisions scatter charged particles, changing their pitch angle. Thus a
particle which originally lay outside the loss cone can be scattered into it.
o  Electrons are more readily Coulomb-scattered than ions; thus electrons will be
scattered out of the mirror trap more quickly.
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o  Early magnetic confinement machines involved designs using magnetic mirror. The
original idea was based on the fact that an electric current generates a magnetic
field, and that the currents flowing in the plasma will "pinch" the plasma,
containing it within its own magnetic field.
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o  A particle outside the loss-cone in a field structure which is convergent at both ends
(such as the Earth’s magnetic field) will be reflected by both mirrors and bounce
between them.
o  Superposed on this bounce will be drift motions. For example, particle orbits in the
Earth’s magnetic field are a combination of gyromotion, bounce motion, and grad-B
drift.
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o  Consider an initially Maxwellian distribution of particles inside a magnetic
o  mirror. In velocity space, the distribution is spherical.
o  As particles exit via the loss cone, the distribution becomes anisotropic
o  The distribution will try to relax back to an isotropic one (higher entropy); one of
the ways it does this is by radiating energy in the form of plasma waves. These
waves are somewhat more complex than cold plasma wave - they involve the
distribution functions of particles. They are generally called kinetic plasma waves,
driven by a kinetic instability.