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

### Full Equation

The equation is typically given as follows:[3]

$n^2 = 1 - \frac\left\{X\right\}\left\{1 - iZ - \frac\left\{\frac\left\{1\right\}\left\{2\right\}Y^2\sin^2\theta\right\}\left\{1 - X - iZ\right\} \pm \frac\left\{1\right\}\left\{1 - X - iZ\right\}\left\left(\frac\left\{1\right\}\left\{6\right\}Y^4\sin^4\theta + Y^2\cos^2\theta\left\left(1 - X - iZ\right\right)^2\right\right)^\left\{1/2\right\}\right\}$

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

$n^2 = 1 - \frac\left\{X\left\left(1-X\right\right)\right\}\left\{1 - X - \left\{\frac\left\{1\right\}\left\{2\right\}Y^3\sin^2\theta\right\} \pm \left\left(\left\left(\frac\left\{1\right\}\left\{2\right\}Y^2\sin^3\theta\right\right)^2 + \left\left(1-X\right\right)^2Y^2\cos^2\theta\right\right)^\left\{1/2\right\}\right\}$

### Definition of Terms

$n$ = complex refractive index

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

$X = \frac\left\{\omega_0^2\right\}\left\{\omega^2\right\}$

$Y = \frac\left\{\omega_H\right\}\left\{\omega\right\}$

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

$\nu$ = electron collision frequency

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

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

$\omega_0 = 2\pi f_0 = \sqrt\left\{\frac\left\{Ne^2\right\}\left\{\epsilon_0 m\right\}\right\}$ = electron plasma frequency

$\omega_H = 2\pi f_H = \frac\left\{B_0 |e|\right\}\left\{m\right\}$ = 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

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

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

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

### 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,

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

## References

Citations and notes