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

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

Collision with charged particles

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

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!

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

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:

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

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

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

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

If , then For ,

Diffusion in a slab: Separation of Variables:

In slab geometry, Boundary conditions S=0 at

In general,

Diffusion in a cylinder

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

Plane source Line source

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 .

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

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

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:

where

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

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

The constant is the specific resistivity of the plasma.

Mechanics of Coulomb Collisions

If considering the small angle collisions,

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

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.

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.

numerical values of Spitzer resistivity For KTe= 100eV,

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

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

Generalized Ohm’s Law

For slow motion, m/M 0. This is the generalized Ohm’s Law. The last term often is small, can be neglected.

The set of MHD equations is 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.

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

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.

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

Time dependence separation of variables:

Time-independent solutions recombination For 1-dimension:

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.

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.

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.

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