Diffusion across the magnetic field
The diffusion coefficient in a plasma can be influenced by the presence of a magnetic field. The behaviour of charged particles, such as electrons and ions, in a magnetic field is described by their gyration around magnetic field lines. This gyration affects the transport properties, including diffusion, in the plasma.
In the presence of a magnetic field, the diffusion of charged particles can be classified into two types: parallel diffusion and perpendicular diffusion.
Parallel diffusion: Parallel diffusion refers to the diffusion of particles along the magnetic field lines. In a uniform magnetic field, the gyration of particles keeps them confined along the field lines. However, there can be small variations and random motions that lead to a slow parallel diffusion of particles parallel to the magnetic field.
Perpendicular diffusion: Perpendicular diffusion occurs across the magnetic field lines. Charged particles can experience random displacements perpendicular to the field lines due to collisions with other particles or plasma turbulence. The strength of the magnetic field can affect the confinement and scattering of particles, thus influencing the perpendicular diffusion coefficient.
In general, the presence of a magnetic field can modify the diffusion coefficient by altering the paths and trajectories of particles in the plasma. The specific relationship between the diffusion coefficient and magnetic field strength depends on various factors, such as plasma conditions, magnetic field configuration, and the types of charged particles involved.
A magnetic field has the ability to slow the rate of plasma loss by diffusion. Consider a magnetically fielded weakly ionized plasma. The velocity direction is unaffected by the magnetic field B.
According to the equation for each species, the charged particle will migrate through diffusion and mobility in the direction of magnetic field B.
Until it collides, a charged particle in a magnetic field will oscillate along the same lines of force. Particles wouldn't diffuse at all in the perpendicular direction if there were no collisions. However, there are particles that naturally drift over B due to its electric field or gradient B.
Particles move randomly across B to the walls along the gradients when there are collisions. When an ion strikes a neutral atom, for example, the ion exits the collision traveling in several directions. It keeps rotating around the magnetic field, but the phase of that rotation is always changing.The radius may also change if the particles gets or loses the energy hence guiding centre shifts positions in a collision and undergoes a random walk. The particle not the 𝜆m but has magnitude of Larmer radius rL.
The diffusion across B can therefore be slowed down by decreasing rL ie by increasing B.
Now, writing the perpendicular components of the fluid equation of motion for either species as
T = isothermal temperature of plasma
n = density of plasma
𝛾 = collision frequency
If 𝛾 is large enough then dv/dt is negligible
Taking x and y component,
The last two terms of the equations contains E x B drift and diamagnetic drift as
Comparing this with equation 6 we can define perpendicular mobility and diffusion coefficient as
Equation 11 can be written in general form
This shows perpendicular velocity composed to two parts
1. V_E and V_D drifts perpendicular to gradient in potential and density. These drifts are slowed down by collisions as 𝛾 >> 1 with neutral atom and drag factor ( 1 + 𝛾²/w²) becomes unity if 𝛾-> 0
2. There are mobility and diffusion coefficients drifts parallel to gradient in potential and density. This drifts are same as in B=0 case but coefficients 𝜇 and D are reduces by the factor (1 + w²c 𝜏²).
The precise determination of the diffusion coefficient in a magnetic field requires careful experimental measurements and theoretical modeling specific to the particular plasma system of interest.
This note is a part of the Physics Repository.
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