Appleton–Hartree equation

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]

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 $n^{2}=\left({\frac {ck}{\omega }}\right)^{2}$ .

Full Equation

The equation is typically given as follows:^[3]

n^{2}=1-{\frac {X}{1-iZ-{\frac {{\frac {1}{2}}Y^{2}\sin ^{2}\theta }{1-X-iZ}}\pm {\frac {1}{1-X-iZ}}\left({\frac {1}{4}}Y^{4}\sin ^{4}\theta +Y^{2}\cos ^{2}\theta \left(1-X-iZ\right)^{2}\right)^{1/2}}}

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

n^{2}=1-{\frac {X\left(1-X\right)}{1-X-{{\frac {1}{2}}Y^{2}\sin ^{2}\theta }\pm \left(\left({\frac {1}{2}}Y^{2}\sin ^{2}\theta \right)^{2}+\left(1-X\right)^{2}Y^{2}\cos ^{2}\theta \right)^{1/2}}}

Definition of Terms

$n$ = complex refractive index

$i$ = ${\sqrt {-1}}$

$X={\frac {\omega _{0}^{2}}{\omega ^{2}}}$

$Y={\frac {\omega _{H}}{\omega }}$

$Z={\frac {\nu }{\omega }}$

$\nu$ = electron collision frequency

$\omega =2\pi f$ (radial frequency)

$f$ = wave frequency (cycles per second, or Hertz)

$\omega _{0}=2\pi f_{0}={\sqrt {\frac {Ne^{2}}{\epsilon _{0}m}}}$ = electron plasma frequency

$\omega _{H}=2\pi f_{H}={\frac {B_{0}|e|}{m}}$ = electron gyro frequency

$\epsilon _{0}$ = permittivity of free space

$B_{0}$ = ambient magnetic field strength

$e$ = electron charge

$m$ = electron mass

$\theta$ = angle between the ambient magnetic field vector and the wave vector

Modes of propagation

The presence of the $\pm$ sign in the Appleton–Hartree equation gives two separate solutions for the refractive index.^[5] For propagation perpendicular to the magnetic field, i.e., ${\mathbf {k}}\perp {\mathbf {B}}_{0}$ , the '+' sign represents the "ordinary mode," and the '−' sign represents the "extraordinary mode." For propagation parallel to the magnetic field, i.e., ${\mathbf {k}}\parallel {\mathbf {B}}_{0}$ , 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.

${\mathbf {k}}$ is the vector of the propagation plane.

Reduced Forms

Propagation in a collisionless plasma

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

$\nu \ll \omega$ ,

we have

$Z={\frac {\nu }{\omega }}\ll 1$ ,

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

n^{2}=1-{\frac {X}{1-{\frac {{\frac {1}{2}}Y^{2}\sin ^{2}\theta }{1-X}}\pm {\frac {1}{1-X}}\left({\frac {1}{4}}Y^{4}\sin ^{4}\theta +Y^{2}\cos ^{2}\theta \left(1-X\right)^{2}\right)^{1/2}}}

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., $\theta \approx 0$ , we can neglect the $Y^{4}\sin ^{4}\theta$ term above. Thus, for quasi-longitudinal propagation in a cold, collisionless plasma, the Appleton–Hartree equation becomes,

n^{2}=1-{\frac {X}{1-{\frac {{\frac {1}{2}}Y^{2}\sin ^{2}\theta }{1-X}}\pm Y\cos \theta }}

References

Citations and notes

↑ Lassen, H., I. Zeitschrift für Hochfrequenztechnik, 1926. Volume 28, pp. 109–113
↑ 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
↑ Helliwell, Robert (2006), Whistlers and Related Ionospheric Phenomena (2nd ed.), Mineola, NY: Dover, pp. 23–24
↑ Hutchinson, I.H. (2005), Principles of Plasma Diagnostics (2nd ed.), New York, NY: Cambridge University Press, p. 109
↑ Bittencourt, J.A. (2004), Fundamentals of Plasma Physics (3rd ed.), New York, NY: Springer-Verlag, pp. 419–429

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