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.
[1] 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.
[2] Further, regarding the derivation by Appleton, it was noted in the historical study by Gillmor that Wilhelm Altar (while working with Appleton) first calculated the dispersion relation in 1926.
[3] 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: The full equation is typically given as follows:[4] or, alternatively, with damping term
sign in the Appleton–Hartree equation gives two separate solutions for the refractive index.
, the '+' sign represents a left-hand circularly polarized mode, and the '−' sign represents a right-hand circularly polarized mode.
The Appleton–Hartree equation for a cold, collisionless plasma is therefore, If we further assume that the wave propagation is primarily in the direction of the magnetic field, i.e.,
Thus, for quasi-longitudinal propagation in a cold, collisionless plasma, the Appleton–Hartree equation becomes,