chapter 5 diffusion and resistivity 5.1 diffusion and mobility in weakly ionized gases 5.2 decay of...
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Chapter 5 Diffusion and resistivity
5.1 Diffusion and mobility in weakly ionized gases5.2 Decay of a plasma by diffusion5.3 Steady state solutions5.4 Recombination5.5 Diffusion across a magnetic field5.6 Collisions in Fully ionized plasma5.7 The Single-fluid MHD equations5.8 Diffusion in fully ionized plasmas5.9 Solutions of the diffusion equation5.10 Bohm diffusion and neoclassical diffusion
Coulomb collisions
Cross section for scattering of an electron by a neutral atom:
Mean free path:
Collision frequency:
20~ an
1)( nnmfp n
n n nn v
Collision with charged particles
b
b0
2tan
20
2
0 4 mv
zeb
In a plasma, most encounter : small angle deflection.
Consider an electron with initial velocity v, suppose that it undergoes a large number of small angle scattering events.
Each deflection gives a small perpendicular velocity component
1g
0bb
20
0
)/(1
)/(2sin
bb
bbvvv
0 v2)( v Increase with time
Because of Debye shielding, electron cannot fell the electric field of an ion at a distance
integration bound
Energy conservation
Db
],0[ D
ln,/
2
ln
])/(1[
)/(8)(
0
220
42
220
2032
b
vm
ezn
bb
bdbbbvnv
dt
d
D
ii
Coulomb logarithm
2)( v //v
High temperature plasma is collisionless!
Neutral particle diffusion:
Diffusion coefficient D is proportional to temperature, mean free path ….
0n
t
D n
2 0n
D nt
5.1 Diffusion and mobility in weakly ionized gases
Any realistic plasma will have a density gradient. The central problem in controlled thermonuclear reaction
is to impede the rate of diffusion by using magnetic field. It is called as weakly ionized gas when the collisions with
neutral atoms are dominant.
• Collision Parameters
Scattering cross section
The fraction of the slab blocked by atom is
Flux:
dxnAAdxn nn /
mn xxn
n
n
ee
n
dxdxd
dxn
/00
'
'
/)(/
)1(
Mean free path:
Mean time between collision
Mean frequency of collision
Collision frequency
nm n
1
vm /
mv /1
nn vν
Diffusion Parameters The fluid equation of motion including collision is
Considering a steady state, and assuming sufficiently small. Then
Mobility
Diffusion coefficient
vmnpEenvvt
vmn
dt
vdmn
])([
v
n
n
m
KTE
m
ev
m
q
m
KTD
Einstein relation:
the flux of the jth species can be written
If mobility is zero, the above equation change into Fick’s law
KT
Dq
nDEnvn jjj
nD j
5.2 Decay of a plasma by diffusion
Ambipolar diffusion
Continuity equation:
It is clear that if and were not equal, a serious charge imbalance would soon arise, an electric field is set up of such a polarity as to retard the imbalance. The required E field is found by setting
0
jj t
nvn
t
n nDEnvn jjj
i
e
ei
nDEnnDEn eeii
n
nDDE
ei
ei
If , then
For ,
nDnDD
ei
eiie
nDt
n 2
ie
ei
eiie DDD
ii
ei
e
eii D
T
TD
DDD
KT
Dq
ie TT iDD 2
• Diffusion in a slab:
Separation of Variables:
)()(),( rStTtrn
nDt
n 2
)()()(
)( 2 rStTDdt
tdTrS
1
)()(
)(
)(
1 2 rSrS
D
dt
tdT
tT
SD
S
eTT t
12
/0
In slab geometry,
Boundary conditions S=0 at
2/12/1
2
2
)(sin
)(cos
1
D
xB
D
xAS
SDdx
Sd
Lx
L
xenn
DL
t
2cos
2
/0
2
In general,
2
/ /0
1
( 1/ 2)
( 1/ 2)( cos sin )l m
l
t tl m
l m
L
l D
l x m xn n a e b e
L L
Diffusion in a cylinder
))(
(
011
2/10
2
2
D
rJS
SDsr
dS
rdr
Sd
5.3 Steady state solutions
)(2 rQnDt
n
In steady state, we have
• For constant Ionization function , Q=Zn
The solution is Cosine or Bessel function.
0t
n
)(12 rQD
n
nD
Zn 2
Plane source
Line source
)0()( 0QrQ
)1(
)0(
0
02
2
L
xnn
D
Q
dx
nd
)/ln(
0)(1
0 rannr
nr
rr
5.4 Recombination
Recombination need the third body, Because of the conservation of momentum.
• Radiative recombination
emitted photon
• Three-body recombination
with a particle
The loss of plasma by recombination will be proportional to .2nnn ei
the continuity equation without diffusion is
is called recombination coefficient.
This equation is a nonlinear equation.
Its solution is
2nt
n
trntrn
)(
1
),(
1
0
5.5 Diffusion across a magnetic field
The rate of plasma loss by diffusion can be decreased by a magnetic field. This is the problem of confinement in controlled fusion research.
Charge particle will
move along B by
diffusion and mobility
z
nDnEzz
If there are no collisions, particle will not diffuse at all in the perpendicular direction.
Particle will drift across B because of electric fields or gradients in B.
When there are collisions, particle migrate across B along the gradients.
Diffusion across B can be slowed down by decreasing
Larmor radius; that is by increasing B
Fluid equation of motion:
0)( vmnnKTBvEen
dt
dvmn
Bebvy
nKTenEvmn
Bebvx
nKTenEvmn
xyy
yxx
xc
yy
yc
xx
vy
n
n
DEv
vx
n
n
DEv
1
x
n
neB
KT
B
E
y
n
n
DEv
y
n
neB
KT
B
E
x
n
n
DEv
cx
cyy
cy
cxx
1
1
2222
2222
)/(1 22c
DE VV
n
nDEv
where
x
n
neB
KTV
y
n
neB
KTV
B
EV
B
EV
DyDx
xEy
yEx
11
2222 11
cc
DD
When , the magnetic field significantly retards
the rate of diffusion across B.
122 c
Lmmc rvm
Bq// vm /
mv
BqrL
22222
1
ccc m
KT
m
KTDD
/~~
/~~
22
22
2
22
Lth
Lth
c
mth
rV
rV
m
KTD
Vm
KTD
• Ambipolar diffusion across B
5.6 Collisions in Fully ionized plasma
• Collisions between like particles
• Collisions between unlike
particles
• Collsions between like particles
give rise to very little diffusion.
• Unlike particle collisions
give rise to diffusion.
Plasma Resistivity
The fluid equation of motion including the effects of charged-particle collisions may be written as
eieeee
ieiiii
PpBvEendt
vdmn
PpBvEendt
vdMn
)(
)(
eiie PP
)(
)(22
ei
eieiei
VVne
VVmnP
m
neei
2
The constant is the specific resistivity of the
plasma.
• Mechanics of Coulomb Collisions
90
20
2
0
02
00
2
4
4)(
mV
er
V
r
r
etFmVmV
Vn
r
ei
2
0
2/320
2/12
320
2
2
)()4(
16
e
ei
KT
me
mV
e
ne
m
If considering the small angle collisions,
0
2/320
2/12
/
ln)()4(
r
KT
me
D
e
Physical Meaning of
Let us suppose that an electric field E exists in a plasma and that the current that it drives is all carried by the electrons. Let B=0 and KTe=0. Then in steady state, the electron equation of motion reduces to
( )ee e e ei
dvmn en E v B p P
dt
eiPEen
jenVVneP eiei
)(22
)( ei VVenj
jE
This is simply Ohm’s Law. is the specific resistivity
is independent of density
In weakly ionized plasma,
the current is proportional to the plasma density
is proportional to . As a plasma is heated,
the coulomb cross section decreases, and the resistivity
drops rather rapidly.
The plasma becomes such a good conductor at
temperatures above 1kev that ohmic heating is a very slow
process in that range.
ln)()4( 2/32
0
2/12
eKT
me
EneVenj ee
2/3)( eKT
The fast electrons in the tail of the velocity distribution make very few collisions.
The current is therefore carried mainly by these electrons rather than by the bulk of the electrons in the main body of the distributions.
If an electric field is suddenly applied to a plasma, a phenomenon known as electron runaway can occur.
A few electrons which happen to be moving fast in the direction of –E when the field is applied will have gained so much energy that they can make only a glancing collision.
If E is large enough, runaway electrons never make a collision.
3220
4
20
16 Vm
neVn
r
ei
numerical values of
Spitzer resistivity
For KTe= 100eV,
mohmeVT
)(
ln102.5
2/35
//
mohm 7105
5.7 The Single-fluid MHD equations
The equation of magnetohydrodynamics (MHD)
Mass density
Mass velocity
Current density
)( mMnmnMn ei
mM
VmVMVmnVMnV ei
eeii
)(
1
)()( eieeii VVneVnVnej
The motion equation of ion and electron :
eieee
ieiii
PgmnpBvEendt
vdmn
PgMnpBvEendt
vdMn
)(
)(
ei
eiei
ppp
gmMnpBVVenVmVMt
n
)()()(
gpBjt
V
The single fluid equation
of motion.
Generalized Ohm’s Law
eieee
ieiii
PgmnpBvEendt
vdmn
PgMnpBvEendt
vdMn
)(
)(
eiei
eiei
PmMpMpm
BvMvmenEmMenvvt
Mmn
)(
)()()(
ne
jmMV
n
vvmvvMvmvMvMvm eiieeiei
)(
)()(
For slow motion, m/M 0.
jepMpm
BjmMBVeEen
j
te
Mmn
ei
)()(
]
)()([1
ei pMpm
BjmMn
j
te
Mmn
ejBVE
)(1
epBjen
jBVE
This is the generalized Ohm’s Law. The last term often is small, can be neglected.
The set of MHD equations is
jBVE
gpBjt
V
0)(
Vt
0
jt
Together with Maxwell’s equations is often used to described the equilibrium state of the plasma.
5.8 Diffusion in fully ionized plasmas
In the absence of gravity, MHD equation for a steady state plasma become
The parallel component of the latter equation is
this is a ordinary Ohm’s law.
pBj
jBVE
////// jE
For The perpendicular component is
The first term is just the drift velocity.
The second term is the diffusion velocity.
The diffusion coefficient is
pBB
BEV
pBVBE
pBjBBVBE
22
2
)(
KT
B
nD
2
2cm
KTD
For weakly ionized gas
Diffusion comparing with weakly ionized plasma
1. Both is proportional to
2. One is proportional to n, another is independent to n
3. Decreases with temperature increasing
opposite in weakly ionized plasma.
2
1
B
ln)()4( 2/32
0
2/12
eKT
me
5.9 Solutions of the diffusion equation
is not a constant in a fully ionized gas.
We define A which is a constant
For case:
The equation of continuity
KT
B
nD
2
D
2B
KTA
ei TT nAD 2
22
)2()(
nAt
n
nnAnDt
n
Time dependence
separation of variables: )()(),( rStTtrn
1
)()(
)(
)(
1 222
rSrS
A
dt
tdT
tT
t
TtT
0
1
)(
1
0)(1
)(22 rSA
rS
Time-independent solutions
222 )( nrnA recombination
22
22
nAx
n
For 1-dimension: ])(exp[ 2/120
2 xA
nn
5.10 Bohm diffusion and neoclassical diffusion
Bohm’s semi-empirical Formula
This formula was obeyed in a surprising number of
different experiments. Diffusion following this law is called
Bohm diffusion.
Be D
eB
KTD 16
1
In absolute magnitude, is also much larger than .
For example, For a 100-eV plasma in 1-T field,
If the density is .
The disagreement is 4 orders of magnitude.
BD D
.sec/25.6 2mDB
31910 m
.sec/1049.52 24
2m
B
nKTD
Explanations:
1. There is the possibility of magnetic field errors.
In a complicated geometries used in fusion research, it is not always clear the the lines of fore either close upon themselves or even stay within the chamber.
2. There is the possibility of asymmetric electric fields.
3. There is the possibility of oscillating electric field arising from unstable plasma waves.
Let the escape flux be proportional to the drift velocity:
Because of Debye shielding, the maximum potential in the plasma is
This leads to flux
BE
BnEnV /
eKTe max
eR
KT
RE e max
max
nDneB
KT
eB
KT
R
nB
ee
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