My watch list

Home
Encyclopedia
Appleton-Hartree_equation

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.

Additional recommended knowledge

How to ensure accurate weighing results every day?

Recognize and detect the effects of electrostatic charges on your balance

How to quickly check pipettes?

1 Equation
2 Reduced Forms
- 2.1 Propagation in a Collisionless Plasma
- 2.2 Quasi-Longitudinal Propagation in a Collisionless Plasma
3 References
4 See also

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

$ω = 2π f$

$f$ = 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

$ε 0$ = permittivity of free space

$μ 0$ = permeability of free space

$B 0$ = ambient magnetic field strength

$e$ = electron charge

$m$ = electron mass

$θ$ = 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 $Y 4 sin 4 θ$ 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

^ Helliwell, Robert (2006), (2nd ed.), Mineola, NY: Dover, pp. 23-24
^ Bittencourt, J.A. (2004), (3rd ed.), New York, NY: Springer-Verlag, pp. 419-429