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 German radio physicist H. K. Lassen. Lassen's work, completed two years prior to Appleton and five years prior to Hartree, included a more thorough treatment of collisional plasma; but, published only in German, it has not been widely read in the English speaking world of radio physics.
Contents
Equation
The dispersion relation can be written as an expression for the frequency (squared), but it is also common to write it as an expression for the index of refraction
Full Equation
The equation is typically given as follows:
or, alternatively, with damping term Z = 0 and rearranging terms:
Definition of Terms
Modes of propagation
The presence of the
Propagation in a collisionless plasma
If the electron collision frequency
we have
so we can neglect the
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.,