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] 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]

Equation

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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   and rearranging terms:[5]

 

Definition of terms:

 : complex refractive index
 : imaginary unit
 
 
 
 : electron collision frequency
 : angular frequency
 : ordinary 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

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The presence of the   sign in the Appleton–Hartree equation gives two separate solutions for the refractive index.[6] 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

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Propagation in a collisionless plasma

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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

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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

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References

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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. ^ C. Stewart Gillmor (1982), Proc. Am. Phil. S, Volume 126. pp. 395
  4. ^ Helliwell, Robert (2006), Whistlers and Related Ionospheric Phenomena (2nd ed.), Mineola, NY: Dover, pp. 23–24
  5. ^ Hutchinson, I.H. (2005), Principles of Plasma Diagnostics (2nd ed.), New York, NY: Cambridge University Press, p. 109
  6. ^ Bittencourt, J.A. (2004), Fundamentals of Plasma Physics (3rd ed.), New York, NY: Springer-Verlag, pp. 419–429