1. Chapter 5 Diffusion and resistivity
5.1 Diffusion and mobility in weakly ionized gases
5.2 Decay of a plasma by diffusion
5.3 Steady state solutions
5.4 Recombination
5.5 Diffusion across a magnetic field
5.6 Collisions in Fully ionized plasma
5.7 The Single-fluid MHD equations
5.8 Diffusion in fully ionized plasmas
5.9 Solutions of the diffusion equation
5.10 Bohm diffusion and neoclassical diffusion

2. Coulomb collisions
Cross section for scattering of an electron by a neutral atom:
Mean free path:
Collision frequency:

3. Collision with charged particles

4. 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
Increase with time

5. Because of Debye shielding, electron cannot fell the electric field of an ion at a distance
integration bound
Energy conservation
Coulomb logarithm
High temperature plasma is collisionless!

6. Neutral particle diffusion:
Diffusion coefficient D is proportional to temperature, mean free path ….

7. 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.

8. Collision Parameters
Scattering cross section
The fraction of the slab blocked by atom is
Flux:

9. Mean free path:
Mean time between collision
Mean frequency of collision
Collision frequency

10. Diffusion Parameters
The fluid equation of motion including collision is
Considering a steady state, and assuming sufficiently small. Then
Mobility
Diffusion coefficient

11. Einstein relation:
the flux of the jth species can be written
If mobility is zero, the above equation change into Fick’s law

12. 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

13. If , then
For ,

14. Diffusion in a slab:
Separation of Variables:

15. In slab geometry,
Boundary conditions S=0 at

16. In general,

17. Diffusion in a cylinder

18. 5.3 Steady state solutions
In steady state, we have
For constant Ionization function , Q=Zn
The solution is Cosine or Bessel function.

19. Plane source
Line source

20. 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

21. the continuity equation without diffusion is
is called recombination coefficient.
This equation is a nonlinear equation.
Its solution is

22. 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

23. 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

24. Fluid equation of motion:

25. where

26. When , the magnetic field significantly retards the rate of diffusion across B.

27. Ambipolar diffusion across B

28. 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.

30. Unlike particle collisions give rise to diffusion.

31. Plasma Resistivity
The fluid equation of motion including the effects of charged-particle collisions may be written as

32. The constant is the specific resistivity of the plasma.

33. Mechanics of Coulomb Collisions

34. If considering the small angle collisions,

35. 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
This is simply Ohm’s Law is the specific resistivity

36. 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.

37. 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.

38. numerical values of
Spitzer resistivity
For KTe= 100eV,

39. 5.7 The Single-fluid MHD equations
The equation of magnetohydrodynamics (MHD)
Mass density
Mass velocity
Current density

40. The motion equation of ion and electron :
The single fluid equation of motion.

41. Generalized Ohm’s Law

42. For slow motion, m/M
This is the generalized Ohm’s Law. The last term often
is small, can be neglected.

43. The set of MHD equations is
Together with Maxwell’s equations is often used to described the equilibrium state of the plasma.

44. 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.

45. For The perpendicular component is
The first term is just the drift velocity.
The second term is the diffusion velocity.
The diffusion coefficient is
For weakly ionized gas

46. Diffusion comparing with weakly ionized plasma
Both is proportional to
One is proportional to n, another is independent to n
Decreases with temperature increasing
opposite in weakly ionized plasma.

47. 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

48. Time dependence
separation of variables:

49. Time-independent solutions
recombination
For 1-dimension:

50. 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.

52. 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.

53. Explanations:
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.
There is the possibility of asymmetric electric fields.
There is the possibility of oscillating electric field arising from unstable plasma waves.

54. 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