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# 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 K. Lassen.

## Equation

### Full Equation

The equation is typically given as follows [1]:

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

### Definition of Terms

n = complex refractive index

i = $\sqrt{-1}$

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

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

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

ν = electron collision frequency

ω = 2πf

f = wave frequency

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

$\omega_H = 2\pi f_H = \frac{B_0 |e|}{m}$ = electron gyro frequency

ε0 = permittivity of free space

μ0 = permeability of free space

B0 = ambient magnetic field strength

e = electron charge

θ = 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 [2]. For propagation parallel to the magnetic field, i.e., $k\parallel B_0$, the '+' sign represents the "ordinary mode," and the '-' sign represents the "extraordinary mode." For propagation perpendicular to the magnetic field, i.e., $k\perp B_0$, 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.

## Reduced Forms

### Propagation in a Collisionless Plasma

If the wave frequency of interest ω is much smaller than the electron collision frequency ν, 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 Y4sin4θ 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
1. ^ Helliwell, Robert (2006), (2nd ed.), Mineola, NY: Dover, pp. 23-24
2. ^ Bittencourt, J.A. (2004), (3rd ed.), New York, NY: Springer-Verlag, pp. 419-429