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)^2Y^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., $\bold k\perp \bold B_0$ , the '+' sign represents the "ordinary mode," and the '−' sign represents the "extraordinary mode." For propagation parallel to the magnetic field, i.e., $\bold k\parallel \bold B_0$ , 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.

$\bold k$ 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