Download NRE4610-notes - RTF Technologies

NRE 4610 Intro to plasma physics and fusion engineering
Fall 2005
By Andrew Seltzman
Basic plasma properties
A plasma must meet 3 criteria to be considered a plasma instead of an ionized gas.
(1) A plasma is a quasi-neutral collection of electrons and ions that exhibits collective
behavior. By quasi-neutral it is meant that there are an equal number of free electrons and
ions in any given region such that the net charge of that area is 0.
The most relevant effect of quasi-neutrality is that plasma can polarize to screen out an
electric field.
Let n0 = the average electron or ion density.
ne (r ) = the actual local electron density
Then the electron density satisfies poisons equation on the macroscopic scale.
1
 2 (r )   (ne  n0 )
0
The local electron density is then given by
 e 

e 
ne (r )  n0 
  n0 1 

 kTe 
 kTe 
Combining the results we obtain
 n e2 
1
2 (r )   0   (r )  2  (r )
d
  0 kTe 
We now define d as the Debye length or distance at which the remaining plasma
experiences an inconsequential amount of interaction with the electric field generated by
a test charge immersed in the plasma. Any area outside of the Debye sphere is not
affected by the test charge.
1/ 2
  kT 
d   0 2e 
 n0 e 
In order to properly screen a test charge, the plasma must have sufficient ionization
density. In this case the number of electrons within the Debye sphere must be sufficient
to screen the test charge.
nd = electron density
 T 3/ 2 
4
nd  d3n0  1.38E 6  1/ 2 
3
 n0 
(2) For a classical plasma (not a weakly ionized gas) the volume must exceed the Debye
length. L d
(3) The number of collisions with neutral particles must be minimal, eg the majority of
particles must be ionized.
Plasma frequency
A plasma will exhibit oscillations at a given frequency.
Take a Plasma and displace all the electrons from X1<X<X0 to X1
The excess charge at X1 is given by   n0e( x0  x1 ) generating an electric field
ne
E ( x1 )    0 ( x0  x1 ) which will generate a force on the charge distribution at X1,
0
attempting to restore equilibrium.
The equation of motion for this system is given by
n e2
me x1  eE ( x1 )    0 ( x0  x1 )
0
  x0  x1
 n0 e 2 
  0
m

 e 0
 
 (t )  Ae
i pet
 Be
 i pet
Which will oscillate at  pe for electrons and  pi
1/ 2
 e2 n 
 pe   0 
 me 0 
 pi 
me
1
 pe   pe
mi
50
For fusion plasma  pe is about 1E11 rad/s
Coulomb scattering
We now divide the plasma into 2 parts; inside and outside the Debye sphere. Inside the
Debye sphere two charged particles can interact and scatter of each other. As one particle
approaches another, it experiences a force due to coulomb repulsion. This will cause the
particles to deflect and scatter off one another.
Often we like to work this problem in the center of mass reference frame (right) instead
of the lab reference frame (left).
Let X= impact parameter (distance of closest approach)
Converting between lab and COM reference frame.
m1r1  m2 r2  (m1  m2 )rcm
F12 
e1e2 (r1  r2 )
4 0 r1  r2
3
mr  m1r1  m2 r2  r  r1  r2
mrcm 
m1m2
m1m2
r m
m1  m2
m1  m2
(m1  m2 )rcom  m1r1  m2 r2
r  r1  r2
r  r1  r2
F12 
e1e2 (r1  r2 )
4 0 r1  r2
m1r1 
r1 
3

e1e2 rˆ
4 0 r1  r2
2
e1e2
rˆ  m2 r2
4 0 r 2
1 e1e2
1 e1e2
rˆ, r2 
rˆ
2
m1 4 0 r
m2 4 0 r 2
 m  m2 
r  1
 F12
 m1m2 
1/ 2
   e e 2  
 1   1 22   
   xV   
yr  

x2






cos( c   ) 
X= impact parameter
c = angle in COM reference frame
 =phase
V r
e1e2
mrV 2 x 2
e1e2
 
tan  c  
r
 2  4 0 mr xv
m 
cot( L )   1  csc  c  cot  c
 m2 
1
c  2
xv
Some important limits of this are
m1 m2   L   c
1
m1  m2   L   c
2
m
m1 m2   L  1 sin  c
m2
Cross sectional scattering area
d   2 sin  c d c
d  2 xdx
 (c ) 
d
2 xdx

d  2 sin  c d c
Rutherford scattering formula =
 e1e2 
2

2
2  c  
 8 0 mr v sin  2  
 

2
90 degree scattering
Occurs by either of two methods; single or multiple collisions.
 c  90 
c 

2
x90 

2
1
xv 2

, x  x90
e1e2
2
 
4 0 mr v 2 tan  
4
2

1  e1e2
 ( c  )   x  

2
  4 0 mr v 2 
2 e1e2
   d 
4 0 mr xv 2

2
2
90
Mean square derivation
  
2
 n0 L
xmax
   
2
d  xmin  x90 , xmax  d
xmin
  
xmax

xmin
2
2


e1e2
 n0 L  
2 xdx
2 
2

m
v
x
0 r

xmin 
xmax
1/ 2

 ( 0 kT )3  
xmax
dx
 ln(
)  ln()  ln 12 
 

x
xmin
n2 e24e12  



2
  
2
  
2
n L ee 
 0  1 2 2  ln( )
2   0 mr v 
1
ln()  e1e2 
1



0 L
2   0 mr v 2 
2
 90
 8ln 
 ( c  90)
Particles therefore are more likely to scatter through multiple small collisions.
for a fusion plasma ln  =10-15
Characteristic times
Time for 90 deg deflection in COM reference frame by small angle scattering. This
characteristic time is the time it takes for a plasma to loose organized momentum, or for
all parts of the plasma temperature to equalize.
 90 
L90
2 02 mr2v3

v
n0 (e1e2 ) 2 ln()
2 0 m1/r 2 (3kT )3/ 2
 90 
n0 (e1e2 ) 2 ln()
In COM frame
electron->electron
 90ee 
6 6 02 me1/ 2 (kT )3/ 2
ne e4 ln()
about 1E-4s
ion->ion
 90ii 
6 6 02 mi1/ 2 (kT )3/ 2
ni ei4 ln( )
about 1E-2s
electron->ion
6 3 02 me1/ 2 (kT )3/ 2
 
ni e 2ei2 ln( )
about 1E-4s
ion->electron
 90ie 
ei
90
6 3 02 me1/ 2 (kT )3/ 2
ne e 2 ei2 ln( )
In lab reference frame
m1<<m2
l  c
1
l   c
m1=m2
2
m
l  2 sin  c
m1>>m2
m1
Most of the time
 90L   90CM
m1  m2 => ee,ii,ei
 90ee :  90ii :  90ei :  90ie :1:
mi
m
:1: i
me
me
about 1s
Single particle motion
Resistively: Collisions destroy ordered momentum
F  ma  FE  Fsc  eE 
me ve
 lee
 me ve
J  ene ve
me d
J  E  RJ
e2 ne dt
E  JR
R
ei e me ln()
6 3 02 (kT )3/ 2
The hotter the plasma, the lower the resistance
Gyromotion
An ionized particle will travel in a helix around a magnetic field.
Consider
B  B0 zˆ
FL  eV  B  V  V  V
2
vx 
eB0
 eB 
vx , vx    0  vx
m
 m 
2
eB
 eB 
vy   0 vy , vy    0  vy
m
 m 
x   2 x
vx  v cos(t   ), v y  v sin(t   )
eB0
m
A particle orbits about the magnetic field axis at the Lamor Radius

rl 
v mv mvth


 e B0 e B0
Te
, T  keV , B  T
B
AT
rli  4.57 E  3 c e , Ac  AMU
zi B
rle  1.066 E  4
Drifts
Uniform B and E
B  B0 zˆ
E  E  E  Ez zˆ  Ex xˆ
F  mv  e( E  V  B )
eE
eB
vz  z  vz  v  t
m
m
e
vx  Ex  v y , vx   2v y
m
E
E
2
2 Ex
v y  v y , v y   (  v y )  2 (v y  x )   2 (v y  x )
B0
t
B0
B0
vx  v cos(t   )
v y  v sin(t   ) 
Ex
B
v  v  vE B
mv  mv  ev  B
mvE B  e( E  VEB  B )  0
EB
B2
1 FB
vF 
e B2
vE  B 
Non-uniform B field
mv 2
Fc  
nc
R
Where nc is the normal vector to the radius of curvature of the applied magnetic field.
0
mv 2
nc  evc  B
R
mv 2 B  nc
vc 
eR B 2
1 mv2
F 
B
2 B
mv 2 B B
vB  
2e
B3
Magneto Hydrodynamics (MHD)
Plasmas as fluids, E and B fields are not fixed.
To analyze plasmas as fluids, we must break up the approximately 1E20 particles per
cubic meter into fluid elements in order to simplify calculations.
Particles in a plasma interact with each other due to magnetic fields and Maxwell’s
equations.


 E   B    E  0    B   B  0
t
t



  B  0 J   0 0 E     B  0   0    0 0  E
t
t
t
 E   / 0
 B0
In a media
D E
B  H
Plasma Approximation
The charges in a plasma are too complicated in motion to practically solve for so we use
the vacuum equations and approximated the plasma as a fluid consisting of bound charge
 and assign a current J based on their average motion.

 E   B
t

 H  J  D
t
 D 
 B0
Plasma is a non-linear magnetic medium
mV 2 1


2B
B
1
M   i
v i
Jb  D  M
M  H
Convective Derivative
A plasma is considered a fluid and contains electrons, ions, and neutrals.
Define average velocity for a group of particles as:
u V
The charged particles then satisfy the following equation of motion.
mV  q( E  V  B)
mn
d
u  qn( E  u  B)
dt
However this can not be practically evaluated at all points in space. We want an
expression for the change in a quantity at a fixed location in the lab reference frame.
G  G (r , t )
d

 x
G (r , t )  G (r , t )   G
dt
t
t
x x
d


G (r , t )  G   u x G
dt
t
x
x
d

G (r , t )  G  (u )G
dt
t
This equation is known as the convective derivative
d

G (r , t )  (  u )G
dt
t

is the change at a fixed point in time and u  is the change due to moving to
t
a new location.
Where
For a plasma the convective derivative is written as:
d
 du

mn u  mn   (u )u   qn( E  u  B)
dt
 dt

Thermal motion in the plasma causes pressure due to collisions, while electric and
magnetic fields allow the ions and electrons within the plasma to interact with each other.
Consider right moving particles in the diagram. The number of particles entering
boundary A is
A  nvx yz  nvx  vx  f (vx , v y , vz )dv y dvz
PAt   vx mvx2 yz   vx vx2 
1
vx vx2
2
1

PAt  yz  nmvx2 
2
 x0 x / 2
where P is the number of particles
1

PAt  yz  nmvx2 
2
 x0 x / 2
1 

PAt  PBt  yz m   nvx2 
  nvx2 





x

x
/
2
x

x
/
2
2 
0
0
 where P is the number of particles
1

PAt  PBt  yz m(x) (nvx2 )
2
x
Consider the case of left and right moving particles, the left moving particles carry
momentum in the –x direction, while the right moving particles carry momentum in the
+x direction.
More specifically particles carry momentum in the nvx2 direction.
The total change in momentum at x0 is


(nmu x )xyz  m (nvx2 )xyz
t
x
Now decompose particle velocity into average fluid velocity and random thermal
velocity.
vx  ux  vth
ux  vth
mvth2  3kt Thermal Velocity



kT
(nmu x )  m [n(u x2  2u x vth  vth2 )]  m [n(u x2  )]
t
x
x
m





nm (u x )  mu x (n)  mu x (nu x )  mnu x (u x )  (nkT )
t
t
x
x
x
The continuity equation now reads


n  (nu x )  0 rate of change in particle density = -flux through cube’s surface
t
x




u x  u x u x )   (nkT )   P where P is pressure
t
x
x
x
Likewise in 3 dimensions
mn(


mn  u  (u )u   qn( E  u  B )  P when taking scalar pressure into account
 t

To account for Y momentum transfer in the X direction we define the pressure tensor
Pij  mnvi v j and replace P with  P
Collisions with Neutrals
Momentum from the plasma is exchanged with that of the neutrals.
Let
u  plasma fluid velocity
un  neutral fluid velocity
Momentum exchange is proportional to the relative velocity u  un with a mean free time
between collisions  . Therefore the force between the neutrals and the plasma is
mn(u  un )

leading to:

mn(u  un )


mn  u  (u )u   qn( E  u  B)   P 
for plasmas with neutrals

 t

Continuity (Mass Conservation)
Total number of particles N enclosed in a volume bounded by surface S can change only
if there is a net flux through the surface.
N
n
  dV    nu dS    (nu )dV
t V t
S
V
Since volume V is arbitrary

n   (nu )  0 Continuity equation
t
Equation of State
From thermodynamics, we take P    where  
Cp
Cv
1
P  C   for constant C
P
n
# d .o. f .  2

thus
and  
and is valid when thermal conductivity is low
P
n
# d .o. f .
Complete set of Equations for MHD
 0 E  ni qi  ne qe
 E  

B
t
 B0
 

  B  0  ni qi ui  ne qeue   0 E 
t 



mi ni  ui  (ui )ui   qi ni ( E  ui  B )  Pi
 t


ni   (ni ui )  0
t
Pi  Ci ni
Fluid Drifts
In the fluid elements there exists a P term not present in the particle drift equations.
Fluid elements are made of particles and are expected to drift perpendicular to B.
Neglecting the (u )u term the diamagnetic drift is
FD  B
1 P  B

2
B
qn B 2
And is valid for (u )u  0 (this happens when E  0 and E    P ), otherwise
(u )u enters the expression.
P
n

When the plasma is isothermal (   1 ), P  (nkT )  kT n , generally
so
P
n
P   kT n
Therefore
 kT zˆ  n
in a plasma cylinder
uD 
qB
n
Drift current is defined as
B n
J D  ne (uDi  uDe )  (kTi  kTe )
B2
uD 
Other Fluid Drifts
ExB drift for fluids
uE B 
EB
B2
Curvature drift as the plasma follows a curve in the B field
uc  
nkT r  B
qRc B 2
B Drift does not exist for a fluid.
Parallel Drift occurs parallel to B. For B  Bzˆ we have



mn  u z  (u )u z   qnEz  P Ignoring the convective term u  since the potential
z
 t

is nearly uniform

q
 kT 
u z  Ez 
n fluid is accelerated along B
t
m
mn t
When this is isothermal and can be integrated,
Boltzmann Relation (balance of electrostatic and plasma pressure)
 kTe 

e 
n   1
z
n z
e  kTe ln(n)  B
e
n  n0 e kTe
Drift Summary
Drift
Particle
Fluid
ExB
EB
B2
mv 2 B  nc
vc  
eRc B 2
EB
B2
nkT r  B
uc  
qRc B 2
none
Curvature
B
P
vE  B 
vB 
none
mv2 B B
2e
B3
uE B 
uP  
1 vp  
 kT zˆ  vn
or uP  
eB 
en 
Charge
Separation
No
Yes
Yes
Yes
Plasma Waves
Equilibrium and Transport
Plasma equilibrium and momentum balance are determined by the Lorentz force which is
generates a magnetic field in a plasma column carrying current.
0 j   B
This magnetic field balances out the pressure gradient in the confined plasma column.
0  p  j  B
j gp  j g( j  B)  j  ( j  B )
Bgp  Bg( j  B)  B  ( j  B)
j gp  0
Bgp  0
Solving for the last two equations, we find that there exist constant pressure surfaces, or
Isobaric surfaces (Flux surfaces) within a plasma column.
An ion in the plasma is therefore constrained to move along an isobaric surface. The
particle can be transported along the plasma columns length and around the column at a
constant radius. In reality the particle gyro rotates along a magnetic field line that lies on
the isobaric surface, exchanging field lines with other particles due to collisions.
In a Tokamak a plasma column is subjected to a toroidal magnetic field which is used to
prevent pinch and kink instabilities. In a column approximation, an axial magnetic field
travels along the length of the column, causing the ions and electrons to rotate about the
center along a corresponding isobaric surface, generating the diamagnetic current.
The net current is determined by the density of ions and electrons.
J 
B  p
B2
B  n
J   (kTi  kTe )
B2
Diamagnetic current
In a plasma a pressure balance exists.
B2
1
p
 cons 
( Bg) B This is usually small and approximately 0 for straight lines.
2 0
0
Where kinetic pressure p   nkT
We now define the  factor.

pKenetic
 nkT
 2
pMagnetic B / 20
Transport
In a magnetically confined plasma, ions and electrons will gyro rotate about the magnetic
field line that they are traveling on. When two particles collide, they are knocked to
separate field lines. The result of this is particle transport across isobaric surfaces. The net
displacement generated by this process is a random walk given by.
 x 
D

2

rL2

    L90 where rL is the lamor radius.
 rl 2  n
The radial transport across a flux surface is nvr  

 ( 90 ) L  r
Diffusion
Given a volume of particles, area A and width dx subjected to an incoming particle flux.
# of atoms in slab nn Adx
Fraction of area blocked
 nn Adx
A
  nn dx
Incoming flux i
Outgoing flux o  i (1  nn dx)
d
 nn
dx
  i e  nnx  i e x / m
Where average velocities and times are:
v  nn  v
 1 
v
m
 nn v
Mean free path m 
Time  
m
v
1
 nn
Re deriving the radial transport equation
mn
d

v  mn[ v  (u g)v]   ni ei E  p  mi ni v
dt
t
Momentum exchange Ri between particle species i and all others j
nm
Ri   iij i (vi  v j )
i  j ( 90 ) L
Ficks Law
1
e
kt n
vi 
(enE  k ni )  
E
mi ni vi
mv
mv n
n
n
i  ni v   i ni E  Din
vi    E  D
   Di n
 

 pi  ni ei
  ni ei  vi Rz  viz  0 
r 
 r
1  pi
 
ni vi 
 ne 

eB  r
r 
2
 m(mvth )  n
rL2 n
mvr    2 2 90    90
 L r
 e Bz  i  r
ie 1
mi ( 90
) p
2 2
e Bz r
for i  e
ei
ie
ne me ( 90
)  ni mi ( 90
)
ni vir  
Radial current jr vanishes for a plasma column
jr  eni vir  (e)never  0
 n z 2  mi
viz   z 
vie
 ne  me
1/ 50
Confinement Concepts
Simple Magnetic Mirror
A simple mirror consists of 2 magnetic coils a set distance apart that trap plasma with
B drifts.
Since the stationary magnetic field does no work, conservation of KE requires that
1
m v 2 ( s )  v^2 ( s )   KE  const Where v^ is the gyro motion velocity.
2
Likewise conservation of angular momentum requires that
mv 2 ( s)
p  mrL v^  L
 const
B( s )
this is conventionally written as
1 mv^2 (s)

 const
2 B( s )
Combining the two relations
1 2
mv ( s )  KE   B( s )
2
In order for a particle to be trapped,
KE

 B( s ) for Bmin  B(s)  Bmax that is
v^  0 before the particle reaches the end of the mirror and falls through the coil.
Evaluating the expression at s  s0
1 2
1
1
1
B( s)
mv ( s )  ( mv 2 ( s0 )  mv^2 ( s0 ))  mv^2 ( s0 )
2
2
2
2
Bmin
Loss cone
The boundary velocity of a particle trapped in a mirror is given by
1/ 2
B

v^ ( s0 )    max  1 v ( s0 )
 Bmin

The slope of this equation generates the loss cone of the confinement system.
Particles initially below a certain
v^ ( s0 )
will be immediately lost as well as those
v ( s0 )
scattered into the loss cone.
The confinement time of a given mirror system is given by
B 
 p  ( 90 ) L log10  max 
 Bmin 
 Bmax 

 Bmin 
 p  ( 90ii ) L log10 
The simple mirror is unstable against flute instabilities. If the surface of the plasma is
deformed outward, the kinetic pressure will remain constant, however since the plasma is
now further from the magnetic coils, the restoring magnetic pressure is weaker, and the
plasma will continue to deform until confinement is lost.
Minimum B Mirror
Flute instabilities can be suppressed if the magnetic field increases farther from the center
of the plasma, applying a proportional correcting force to reestablish confinement.
A minimum B mirror consists of a pair of cresset shaped coils (similar to a baseball) that
confine plasma in a spherical region.
Tandem Mirror
To prevent loses out of the ends of the simple mirror, it is possible to form an
electrostatic plug at the ends of the magnetic coil by augmenting the design with
minimum B end plugs. In a mirror configuration, electrons escape confinement faster
then ions, so the mirror operates with a net positive charge. By creating a higher plasma
density at the ends of a simple mirror by use of minimum B end plugs, the imparted
potential difference repels ions back into the mirror, increasing confinement time.
Tokamaks
To compensate for confinement losses in the simple mirror, the magnetic field was bent
into a circular path, forcing the plasma column to form a torus. This type of closed
confinement concept is called the tokamak.
The Pappus--Guldinus theorem says that:
The area of the surface of revolution on a curve C is equal to the product of the length
of C and the length of the path traced by the centroid of C (which is 2 the distance from
this centroid to the axis of revolution).
The volume bounded by the surface of revolution on a simple closed curve C is equal to
the product of the area bounded by C and the length of the path traced by the centroid of
the area bounded by C.
Where the centroid in a given axis is:
 ida
Ci 
A
The area and volume of a tokamak (torus) is
A  4 2 Rr
V  2 2 Rr 2
The Tokamak traps a toroidal plasma column in several magnetic fields arranged to
confine and stabilize the plasma column.
Tokamak Confinement
A tokamak operates with the superposition of 3 independent magnetic fields; a toroidal
field ( ˆ ), a poloidal field ( ˆ ), and a vertical field with curvature about the ˆ direction.
The most obvious way to make a charged particle follow a toroidal path is to provide a
toroidal magnetic field, however confinement is not possible with a toroidal field alone
since the curvature and B drifts would cause charge separation in the plasma. The
ions would drift up, while the electrons drift downward, inducing a downward electric
field. This downward electric field, overlapped with the toroidal magnetic field
causes an outward radial drift ( E  B ) forcing the plasma into the reactor walls.
To compensate for this, a poloidal field generated by a toroidal current flow within the
plasma, confines the plasma column to a toroidal geometry and cancels out the E  B
drift.
Due to the toroidal curvature of the toroidal plasma geometry, the poloidal field is
stronger near the center of the torus then it is near the exterior. This provides an outward
force in the pressure balance equation since the magnetic pressure is weaker towards the
outer surface of the tokamak.
Magnetic Field Superposition
The poloidal field is generated by the toroidal current flowing within the plasma and
compresses the plasma to a toroidal column, confining the plasma. This however causes
pinch and kink instabilities.
The toroidal field is generated by toroidal field coils (wrapped around the plasma’s axis)
and stabilizes the plasma instabilities by forcing the plasma to follow a toroidal path.
The toroidal field within the tokamak is given by
B0
B0
B (r ,  ) 

(1  r / R0 ) cos  1   cos 
Due to the decrease in poloidal field strength with radius, the pressure balance attempts to
force the plasma into the outer wall, this is compensated by adding a vertical field.
The vertical field is generated by the central solenoid and prevents outward drift of the
plasma. The vertical field also possesses curvature which prevents vertical drift. As the
vertical field is generated by the central solenoid, the increasing magnetic flux induces a
toroidal electric field which produces the toroidal current.
Stability
In plasma columns confined by carrying current, two main types of instabilities exist;
pinch and kink instabilities.
The current carried by the plasma is constant, therefore as the dimensions of the plasma
column change, the current density increases and the total current is contained entirely
within the plasma’s surface. The poloidal magnetic field generated by the enclosed
current is then given by amperes law.
I
B (rp )  0
2 rp
Pinch Instability
If a plasma column experiences a perturbation in the form of a radial compression of
expansion, the magnetic field at the surface increases or decreases, increasing or
decreasing the magnetic pressure at the surface. The perturbation will therefore continue
to grow.
Kink Instability
If a plasma column experiences a perturbation that causes it to bend, the expanded
section will experience a weaker restoring magnetic pressure and continue to expand.
Likewise the inner section will continue to contract.
Safety Factor
To prevent the instabilities in plasma columns generated by purely poloidal magnetic
fields, toroidal fields are applied to prevent pinch and kink instabilities. The ratio of
poloidal and toroidal field magnitudes is given by the safety factor.
# of toroidal revolutions
q
1 poloidal revolution
The safety factor is geometrically defined as the number toroidal revolutions for every
poloidal revolution.
B 2 2 r 2
rB
q (r )  

0 I (r ) R0 R0 B (r )
Suppression of internal kink instabilities requires
q(0)  1
Suppression of surface kink instabilities (r=a) requires
q(a)
2 3
q(0)
By superposition of magnetic fields this specifies that for a tokamak, the toroidal field
must be greater then the poloidal field.
The safety factor constrains the maximum current that a plasma can carry wile
maintaining stability.
Plasma Heating
Ohmic Heating
Since a plasma is a resistive medium, it can be ohmically heated by a current flow.
The resistance of a plasma is:
R
ei e me ln()
6 3 02 (kT )3/ 2
1/ 2
 ( kT )3 
where   12  0 4 2 
 n2 e2 e1 
P  I 2 R
For a uniformly distributed current, this can also be written as
Z eff I 2 ( A)
3
6
2
15
P ( MW / m )  10  j  2.8 10
a 4 (m)Te3/ 2 (kev)
Using the definition of the safety factor and amperes law
aB
q(a) 
R0 B
0 I  2 aB
Ohmic heating can be written as
2
 1   B 
P ( MW / m )  7 10
 
Te3/ 2 (kev)  q(a)   R0 
3
2
Z eff
2
This implies that it becomes progressively more difficult to resistively heat high
temperature plasmas, and that resistive heating can achieve a higher temperature in
high field tokamaks.
Neutral Beam Injection
Plasma can be heated by kinetic energy exchange with high velocity beams of
deuterium fuel. Deuterium is ionized and accelerated electrostaticly across a potential
difference, however to be injected through the magnetic fields of the tokamak, it must
first be neutralized so there is no Lorenz force in the beam.
The ionized beam is guided through a neutralizer gas where it gains electrons from the
neutral deuterium. Any ionized portion of the beam is magnetically directed into a beam
dump and the remaining neutral beam is injected into the tokamak.
The mean free path of the neutral beam is
5.5 1017 Eb (kev)
 ( m) 
A(amu )n(m 3 ) Z eff
For effective heating the majority of the beam energy must be deposited at the center of
the plasma (   a / 4 ), so the beam energy must be greater then
EB (kev) 
n(m 3 )a (m) Z eff A(amu )
2.2 1018
RF Heating
An electromagnetic wave is launched into the plasma using a waveguide or antenna. As it
passes through the plasma the magnetic field varies with position, determining the
resonant frequency of the plasma at that point.
When the wave reaches an area of the plasma where its frequency matches the local
resonant frequency, it is absorbed and drives electron or ion oscillations in the plasma
thereby heating it.
For RF waves propagating parallel to the magnetic field (k B0 ) , the resonant frequencies
are given by the ion and electron cyclotron frequencies. This type of heating is called
ECRF and ICRF.
eB
 0
m
where e and m are either the ion or electron charges and masses
eB0
f  2
m
For RF waves propagating perpendicular to the magnetic field (k ^ B0 ) , the resonant
frequencies are given by upper and lower hybrid frequencies.
  ei
f  2 ei
Plasma Wall Interaction
Surface Erosion
Electrons or ions colliding with the surface of a material can remove atoms from the
surface by sputtering. A particle colliding with the surface of a material will transfer its
kinetic energy to the lattice of the material. If at any time the energy of a given lattice
atom at the surface of the material exceeds its surface binding energy, it will be sputtered
and escape the surface.
The rate of atomic sputtering scales directly with the mass of the incident particle,
further, below a certain threshold energy, sputtering will not occur. This threshold is
given by Et1 the minimum incident particle energy that will cause sputtering.
Et1 
(m1  m2 ) 2
U 02 where U 02 is the surface binding energy
4m1m2
Impurity Radiation
Impurities in a plasma will cause energy losses in the form of EM radiation by several
methods; Bremsstrahlung, line and recombination losses.
Bremsstrahlung
When a charged particle is accelerated, it will emit EM radiation. In the case of a
fusion plasma, this will be caused by electrons colliding with impurity.
The energy of the EM radiation given off by an accelerated charged particle is
2
2
1 e v
Wrad 
6  0 c3
The coulomb force of an impurity ion on an electron is
x
ze2
Taking place over time 
and distance x
F  mv 
2
v
4 0 x
and produces acceleration
ze2
v
4 0 me x 2
The total Bremsstrahlung power radiated is given by integrating over all interactions in
the plasma volume
 x 
W
 rad  ve  2 xdx
xmin
 
3
Pbrems ( MW / m ) 4.8  z 2 nz (m 3 )ne (m 3 )Te1/ 2 (kev)
Pbrems  nz ne ve
xmax
Line Radiation
Electrons in a plasma collide with partially ionized impurities and excite their outer
electrons to the next highest orbital. The electron will then decay back to its ground
state, emitting a photon at the atoms characteristic line frequency.
PL ( MW / m3 ) 1.8 1044
z 4 ne (m3 )nz (m3 )
Te1/ 2 (kev)
Recombination Radiation
Free electrons within a plasma can recombine with ionized impurities, emitting the
ionization energy difference as a photon.
3
PR ( MW / m )
4.110
46
z 6 ne (m3 )nz (m3 )
Te3/ 2 (kev)