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The Appleton-Hartree equation, sometimes also referred to as the Appleton-Lassen equation is a mathematical expression that describes the refractive index for electromagnetic wave propagation in a cold magnetized plasma. The Appleton-Hartree equation was developed independently by several different scientists, including Edward Victor Appleton, Douglas Hartree and K. Lassen.
The equation is typically given as follows :
Definition of Terms
n = complex refractive index
ν = electron collision frequency
ω = 2πf
f = wave frequency
= electron plasma frequency
= electron gyro frequency
ε0 = permittivity of free space
μ0 = permeability of free space
B0 = ambient magnetic field strength
e = electron charge
m = electron mass
θ = angle between the ambient magnetic field vector and the wave vector
Modes of Propagation
The presence of the sign in the Appleton-Hartree equation gives two separate solutions for the refractive index . For propagation parallel to the magnetic field, i.e., , the '+' sign represents the "ordinary mode," and the '-' sign represents the "extraordinary mode." For propagation perpendicular to the magnetic field, i.e., , the '+' sign represents a left-hand circularly polarized mode, and the '-' sign represents a right-hand circularly polarized mode. See the article on electromagnetic electron waves for more detail.
Propagation in a Collisionless Plasma
If the wave frequency of interest ω is much smaller than the electron collision frequency ν, the plasma can be said to be "collisionless." That is, given the condition
so we can neglect the Z terms in the equation. The Appleton-Hartree equation for a cold, collisionless plasma is therefore,
Quasi-Longitudinal Propagation in a Collisionless Plasma
If we further assume that the wave propagation is primarily in the direction of the magnetic field, i.e., , we can neglect the Y4sin4θ term above. Thus, for quasi-longitudinal propagation in a cold, collisionless plasma, the Appleton-Hartree equation becomes,
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Appleton-Hartree_equation". A list of authors is available in Wikipedia.|