Transcript PlasmaIntro002

Chapter 2: Single-particle Motions
1.
Introduction
2.
Uniform B, E fields
3.
Gravitational drift
4.
Nonuniform B, E fields
5.
Time varying E, B fields
6.
Adiabatic invariants
2.1
Introduction
 For very low density devices, collective effects are often
unimportant.
 Single particle motions is the first step toward
understanding plasma dynamics.

Equation of motion

  

dv
m
 q( E  v  B)  mg
dt
(2.1)
Lorenz force and gravity
The fields include self-generated or externally imposed
2.2 Uniform B, E fields
• E=0,

  

dv
m
 q( E  v  B)  mg
dt
g=0
• Equation of motion:
Assume
Then

 
dv
m
 qv  B
dt


B  Bez
mvx  qBv y
mv y  qBv x
mvz  0
which means
vz  v//  const.
2.2 Uniform B, E fields
2
vx  qBv y / m   c vx
2
vy  qBvx / m   c v y
(2.2)
This describes a simple harmonic oscillator
at the cyclotron frequency:
c 
qB
m
The solution of eq.(2.2) is
vx  v cos( ct   )
v y  v sin(  c t   )
v x  x
By Integrating, we can get
x  xg 
y  yg 
v
c
v
c
sin(c t   )
( x  xg ) 2  ( y  y g ) 2  (
v
c
) 2  rL
cos(c t   )
Gyroradius, or Larmor radius
rL 
v
c

mv
qB
The direction of the gyration is always such that the
magnetic field generated by the particle is opposite to the
externally imposed field.
2

Plasma particles, therefore, tend to reduce the magnetic
field, and plasma are diamagnetic.
 In addition to this motion, there is an arbitrary velocity.
guiding center slides along magnetic field line.
 The motion along B which is not affected by B. The
trajectory of a charged particle in space is a helix.
 Using magnetic field, we can confine plasma.

Gyro-orbit is a current loop
magnetic moment
  IrL
2
q c
mv

rL 2 
2
2B
2
E  cons.  0

  
dv
m
 q( E  v  B)
dt
The sum of two motions:
circular Larmor gyration plus a drift of the guiding center.
• We take
Ey  0
In z direction dvz  q E ,
z
dt
m
vz 
q
E z t  vz 0
m
This is a straightforward acceleration along B.
In transverse direction:
x   c 2 v x
v
E
y   c ( x  v y )
v
B
2
q
Ex   cv y
m
v y   c v x
v x 
d 2 Ex
2 Ex
(

v
)



 vy )
y
c (
2
dt
B
B
vx  v cos( ct   )
v y  v sin(  ct   ) 
x  xg 
y  yg 
v
c
Ex
B
sin( c t   )
Ex
v
t   cos(c t   )
B
c
 The trajectory of particle is
Ex 2
v 2
2
( x  xg )  ( y  y g 
t)  (
)  rL
B
c
2
The Larmor motion is the same as before, but there is
superimposed a drift of the guiding center in the –y direction.
To obtain a general formula for drift velocity, we can

use vector methods as following.
 
dv
m
At first, we write E in two components


B  
E  E//  a  B
B
dt

 q( E  v  B)
 
EB
E// 
B

Then we determine a

 
   
 

B 
E  B  E//  B  (a  B)  B  B(a  B)  B 2 a
B
 



EB
a  B  (
) B
2
B

 





 EB
dv
B
m
 q( E  v  B)  q[ E//  (v 
)  B]
dt
B
B2

 





 EB
dv
B
m
 q( E  v  B)  q[ E//  (v 
)  B]
2
dt
B
B
parallel direction:
m
dv//
 qE// ,
dt
Perpendicular direction:

VE B
 
EB

B2



du
m
 qu  B,
dt
 



EB 
v  u 
 u  VE B
2
B
only gyromotion no drift
as if E=0
v// 
qE//
t  v// 0
m
 


EB
u  v 
B2

radius of gyro-orbit is larger on the right half of gyro-orbit.

Ion and electron drift at the same direction and speed so
that no net current , only mass flow appear.
 Guiding center drift is perpendicular to E !

The three-dimensional orbit in space is a slanted helix
with changing pitch.
2.3
Gravitational drift
The foregoing result can be applied to other forces by
replacing qE in the equation of motion by a general force F.


 
dv
m
 F  qv  B
dt
The guiding center drift caused by F is then

VE B
 

FB
Vf 
qB 2
 
EB

B2
2.3
Gravitational drift
For Gravitational Force


F  mg ,



 mg  B
Vg 
2
qB
gravitational drift is horizontal
 Ion and electron drift at different direction and speed:
produce net current, resulting horizontal electric field
moves plasma downward
2.4
Nonuniform B, E field
 As soon as we introduce inhomogeneity, the problem
becomes too complicated to solve exactly.
• To get an approximate answer, it is customary expand
in the small ratio rL / L
Where L is the scale length of the inhomogeneity.
This type of theory called as orbit theory.
1. Grad-B drift

B  B
Assume B = B(y)
local gyromotion radius is larger at the bottom of the orbit
than at the top. This should lead to a drift, in opposite
direction for ions and electrons, perpendicular to both B
and B
 The drift velocity should obviously be proportional to rL / L

and v
Consider the Lorenz force
Clearly

 
F  qv  B
Fx  0
v x  v cos(c t )
v y  v sin( c t )
So we only need to calculate Fy
B
Fy  qvx Bz ( y )  qv cos  c t ( B0  rL cos  c t
)
y
 cos 2 c t  1/ 2
1
B
Fy   qv rL
2
y
Guiding center drift



B  B
1
B  B
VB 


v
r
 L
qB 2
2
B2
stands for the sign of the charge
  IrL 2
q c
mv

rL 2 
2
2B
2
1
mv2  B
2
Potential
Force

FB  B
 

FB
Vf 
qB 2
It is in opposite directions for ions and electrons and causes a
current transverse to B
2. Curved B: Curvature drift
Assume curved magnetic
field line with a constant
radius of curvature
Centrifugal force
Curvature Drift

mv//2 
Fc 
Rc
2
Rc




2

F  B mv// Rc  B
VR  c 2 
qB
qB 2 Rc2
 We must compute the grad-B drift
Vacuum field

 B  0
In the cylindrical coordinates:

1  
1  
 
e 
e 
ez
 
 
z


B  B (  )e


1 
 B 
( B )ez  0
 
1
B 

B
1
,
Rc

B
Rc
 2
B
Rc
rL 
v
c




1
B  B 1 m 2 Rc  B
VB   v rL

v
2
B2
2 q
Rc2 B 2


2
2



m(v//  v / 2) 
m Rc  B
1 2
2
VR  VB 
(v//  v ) 
B  B
2
2
3
q Rc B
2
qB

mv
qB

3. Magnetic Mirrors: B // B
Axisymmetry
B  0,


B  0.

  B  0,
 1 
Bz
B 
(rBr ) 
0
r r
z
1 r Bz
1  Bz 
Br    r
dr   r 
r 0
z
2  z 
 r 0
 The components of the Lorentz force are
Fr  qv Bz
F  q (v z Br  vr Bz )
1
 Bz 
Fz   qv Br  qv r 
2
 z 
 r 0
• For simplicity, consider a particle whose guiding center
lines on the axis, then
v   v ,
r  rL .
The average force is
1
Bz
1 mv
Fz   qv rL

2
z
2 B
2
Bz
z
1
Bz
1 mv
Fz   qv rL

2
z
2 B
2
Bz
z
Define the magnetic moment of the gyrating particle
mv

2B
Fz   
2
Bz
z
which means the gyrating particle is a diamagnetic particle.
the general form can be written as
F//    // B
dv//
m
 F//    // B   
dt
B
s
F//   //  B    // B
 As the particle moves into regions of stronger or weaker B,
its larmor radius changes, but magnetic moment remains
invariant.
 Conservation of magnetic moment
For general nonuniform magnetic field is
magnetic moment conserved?

B  B  0




B  B0ez    B
m dV
V B
q dt
Order:
0
1
Vz  c
(Vx 0 ,Vy 0 )
gyromotion

 
m dVz1 
 V0  (   B)
q dt
 
 
m dVz1
 V x 0  (   B ) y  V y 0  (   B ) x
q dt
B y
B y
B
B
 Vx 0 ( x '
 y'
)  V y 0 ( x' x  y ' x )
x
y
x
y
gyro-average:

 c
2
2

0
c
dt
vx  v cos( ct   )
 Vx 0 x'  V y 0 y '  0
1 V2
 Vx 0 y '    V y 0 x'  
2 c
v y  v sin(  c t   )
x  xg 
y  yg 
m dVz1 1 V2 By Bx
 Bz

(

)
q dt
2  c y
x
q z
d mV//
dVz1
B dz
dB
 mVz1
 
 
dt 2
dt
z dt
dt
2
v
sin(  c t   )
v
cos( c t   )
c
c
Energy conservation by Lorentz force
 
qv  B
d mV//2
(
 B)  0
dt
2

dB d

( B )  0
dt
dt
d
0
dt
Magnetic Mirror
Make magnetic field stronger at the end.
Paralle velocity :
mV//2
 W  B
2
when particle moves toward higher field region
If B is high enough, V//
This particle is reflected.
V//
decrease.
become zero at some point.
 The condition for barely trapped particle is
mV//2
( x  0)  Bmin  Bmax
2
mV//2
2
mV 2
( x  0) 
 ( Bmax  Bmin )
Bmax
2
trapping condition :
the particle with
V//
B
 (1  min )1/ 2
V
Bmax
V//
B
 (1  min )1/ 2
V
Bmax
escapes the mirror.
The magnetic mirror was first proposed by Enrico Fermi as
a mechanism for the acceleration of cosmic rays.
Nonuniform E field
 For Simplicity, we assume E to be in the x direction and to
vary sinusoidally in the x direction:


E  E0 cos kxex
In practice, such a charge
distribution can arise in a
plasma during a wave
motion.


 
dv
m
 q ( E ( x)  v  B)
dt
vx 
q
qB
E x ( x) 
vy
m
m
v y  
qB
vx
m
E x
2
vx   c
  c vx
B
2
vy   c
Ex
2
 c vy
B
Here E(x) is the electric field at the position of the particle.
To evaluate this, we need to know the particle’s orbit,
which we are trying to solve for in the first place.
We assume the electric field is weak, then we can use the
undisturbed orbit to evaluate E(x).
x  x0  rL sin  ct
2
2
vy   c v y   c
Ex  E0 cos k ( x0  rL sin  ct )
E0
cos k ( x0  rL sin  ct )
B
2
2
vy  0   c v y   c
Assume
E0
 cos k ( x0  rL sin  c t ) 
B
krL  1
cos k ( x0  rL sin  ct )  (cos kx0 )(1  1 k 2 rL sin 2  ct )  (sin kx0 )krL sin  ct
2
2
vy  
E0
E (x )
1
1
2
2
(cos kx0 )(1  k 2 rL )   x 0 (1  k 2 rL )
B
4
B
4
Thus the usual
 
EB
drift is modified by the inhomogeneity as
 

EB
1 2 2
vE 
(
1

k rL )
B2
4
For an arbitrary variation of E, the drift is
 

1 2 2 EB
vE  (1  rL  ) 2
4
B
The second term is called the finite Larmor-radius effect.
Since Larmor-radius is much larger for ions than for
electrons, drift velocity is dependent of species. Which will
lead to drift instability.
2.5

Time-varying E, B field
Let us now take E and B to be uniform in space, but
varying in time.

 it
it 
E  E0e
 E0e ex
E x
i E x
2
2
vx   c
  c vx   c (vx 
)
B
c B
2
vy   c
Ex
2
 c vy
B
vp  
i E x
c B
vE  
Ex
B
2
vx   c (vx  v p )
2
vy   c (v y  vE )
(2.63)
 we try a solution which is the sum of a drift
and a gyratory motion
v x  v  e i c t  v p
(2.64)
v y  iv  e i c t  vE
Derivative respect to time, we get
x   c 2 v ei ct   2 v p   c 2 vx  ( c 2   2 )v p
v
y   c 2iv  ei ct   2 vE   c 2 v y  ( c 2   2 )vE
v
This is not the same as above equation, unless
If we assume that E varies slowly, so that
 2  c 2
 2  c 2
Eq.(2.64) is the approximate solution of Eq.(2.63)
we define polarization drift as



i E
1 dE
vp  

c B
 c B dt
 since polarization drift is in opposite directions for ions
and electrons, there is a polarization current. For Z=1



 
ne
dE  dE
j p  ne(vip  vep )  2 ( M  m)
 2
eB
dt B dt
Time-varying B field
Consider a spatially uniform, but slowly varying B-field:
1 B
    c
B t
An electric field associated with time-varying B field is


given as
  E  B
Assume
B
0
t


 
dv
m
 q ( E ( x)  v  B)
dt
qv  E  0
 Everywhere on the gyro-orbit is accelerated.
change of perpendicular energy averaged over one gyro
period:
 
 2   
d 1
2
( mv )  q  E  v  q c
dt 2
2

0
c
v  Edt
 
c  
c
q
E  dl  q
  E  dS
2 
2 

c
c
B 
B
 q

d
S

q
dS
2  t
2  t
 B
d 1
B
( mv2 )  q c
 2  
dt 2
2 t
t
By definition of magnetic moment
d 1
d
d
dB
( mv2 ) 
B  B

dt 2
dt
dt
dt
d

0
dt
   c
•
magnetic moment is a constant of motion if LB  rL

magnetic flux encolsed in a gyro-orbit is also conserved.
v2 B m 2 v2
2m
   B 



 const .
 c2
q2B
q2
2
 Plasma volume shrinks as B increase
 Plasma and field line move together: Frozen-in-line
 Adiabatic compression
A plasma is injected into
the region between the
mirrors A and B. Coils
A and B are then pulsed
to increase B and hence
v 2 . The heated plasma
can then be transferred
to the region C-D by a further pulse in A; increasing the
mirror ratio there. The coils C and D are then pulsed to
further compress and heat the plasma.
Summary of Guiding
center
Drift
 
 General force F:
 Electric field:
 Gravitational field
 Nonuniform E field
 Nonuniform B field

FB
Vf 
qB 2
 

mg  B
Vg 
qB 2
 

1 2
EB
vE  (1  rL  2 ) 2
4
B
 Grad-B drift
 Curvature drift
 Curved vacuum field
 Polarization drift

VE B
 
EB

B2


1
B  B
VB   v rL
2
B2




2

F  B mv// Rc  B
VR  c 2 
qB
qB 2 Rc2




m Rc  B
1 2
2
VR  VB 
(
v

v )
//
q Rc2 B 2
2


1 dE
vp  
 c B dt
2.6 Adiabatic invariants




E

~   1
is a constant to the order of 
E

is called adiabatic invariant.
In Hamiltonian system, if the motion is periodic in q,
the action integral J   pdq taken over a period is a
constant of the motion.

~   1
 under adiabatic change:

although the change is small in any one period, over a
long interval of time the property of motion may undergo
large quantitative change.

 Canonical perturbation theory shows that J is an
adiabatic invariant.
1 J
 c  O( 2 )
J
 Adiabatic invariants play an important role in
plasma physics.
 The First Adiabatic Invariant  ,
If we take p to be the angular momentum
mv r and dq to be the angular coordinate


pdq   mv rL d  2 mv rL  2
mv
c
2
 4
m

q
d
mv

2B
2
The condition that the invariance of  is not violated is
   c
when above condition is violated, for example, Cyclotron
Heating,  is not conserved, and the plasma can be
heated.

The Second Adiabatic Invariant, J
Consider a particle trapped between two magnetic mirrors:
It bounces between them and therefore has a periodic
motion. A constant of motion is given by
 mv
//
ds

However, since the guiding center drifts across field lines,
the motion is not exactly periodic, and the constant of the
motion becomes an adiabatic invariant. This is called the
longitudinal invariant j
b
J   v// ds
a
 The Third Adiabatic Invariant, 