Appleton-hartree Equation

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Appleton-hartree Equation

EQUATION

Full Equation

The equation is typically given as follows :

:

n^2 = 1 - rac{X}{1 - iZ - rac{rac{1}{2}Y^2\sin^2	heta}{1 - X - iZ} \pm rac{1}{1 - X - iZ}\left(rac{1}{4}Y^4\sin^4	heta + Y^2\cos^2	heta\left(1 - X - iZ
ight)^2
ight)^{1/2}}

Definition of Terms

n

= complex refractive index

i

=

\sqrt{-1}

X = rac{\omega_0^2}{\omega^2}

Y = rac{\omega_H}{\omega}

Z = rac{
u}{\omega}

u

= electron collision frequency

\omega = 2\pi f

f

= wave frequency

\omega_0 = 2\pi f_0 = \sqrt{rac{Ne^2}{\epsilon_0 m}}

= electron Plasma Frequency

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.,

heta \approx 0

, we can neglect the

Y^4\sin^4	heta

term above. Thus, for quasi-longitudinal propagation in a cold, collisionless plasma, the Appleton-Hartree equation becomes,

:

n^2 = 1 - rac{X}{1 - rac{rac{1}{2}Y^2\sin^2	heta}{1 - X} \pm Y\cos	heta}

REFERENCES

;Citations and notes