Appleton–Hartree equation

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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 German radio physicist H. K. Lassen.[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]


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[edit]

The equation is typically given as follows:[3]

or, alternatively, with damping term Z = 0 and rearranging terms:[4]

Definition of Terms[edit]

= complex refractive index


= electron collision frequency

(radial frequency)

= wave frequency (cycles per second, or Hertz)

= electron plasma frequency

= electron gyro frequency

= permittivity of free space

= ambient magnetic field strength

= electron charge

= electron mass

= angle between the ambient magnetic field vector and the wave vector

Modes of propagation[edit]

The presence of the sign in the Appleton–Hartree equation gives two separate solutions for the refractive index.[5] For propagation perpendicular to the magnetic field, i.e., , the '+' sign represents the "ordinary mode," and the '−' sign represents the "extraordinary mode." For propagation parallel 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.

is the vector of the propagation plane.

Reduced Forms[edit]

Propagation in a collisionless plasma[edit]

If the electron collision frequency is negligible compared to the wave frequency of interest , the plasma can be said to be "collisionless." That is, given the condition


we have


so we can neglect the terms in the equation. The Appleton–Hartree equation for a cold, collisionless plasma is therefore,

Quasi-Longitudinal Propagation in a Collisionless Plasma[edit]

If we further assume that the wave propagation is primarily in the direction of the magnetic field, i.e., , we can neglect the term above. Thus, for quasi-longitudinal propagation in a cold, collisionless plasma, the Appleton–Hartree equation becomes,

See also[edit]


Citations and notes
  1. ^ Lassen, H., I. Zeitschrift für Hochfrequenztechnik, 1926. Volume 28, pp. 109–113
  2. ^ C. Altman, K. Suchy. Reciprocity, Spatial Mapping and Time Reversal in Electromagnetics – Developments in Electromagnetic Theory and Application. Pp 13–15. Kluwer Academic Publishers, 1991. Also available online, Google Books Scan
  3. ^ Helliwell, Robert (2006), Whistlers and Related Ionospheric Phenomena (2nd ed.), Mineola, NY: Dover, pp. 23–24 
  4. ^ Hutchinson, I.H. (2005), Principles of Plasma Diagnostics (2nd ed.), New York, NY: Cambridge University Press, p. 109 
  5. ^ Bittencourt, J.A. (2004), Fundamentals of Plasma Physics (3rd ed.), New York, NY: Springer-Verlag, pp. 419–429