Appleton–Hartree equation

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

Full Equation

The equation is typically given as follows:[3]

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

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

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

Definition of Terms

${\displaystyle n}$ = complex refractive index

${\displaystyle i}$ = ${\displaystyle {\sqrt {-1}}}$

${\displaystyle X={\frac {\omega _{0}^{2}}{\omega ^{2}}}}$

${\displaystyle Y={\frac {\omega _{H}}{\omega }}}$

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

${\displaystyle \nu }$ = electron collision frequency

${\displaystyle \omega =2\pi f}$ (radial frequency)

${\displaystyle f}$ = wave frequency (cycles per second, or Hertz)

${\displaystyle \omega _{0}=2\pi f_{0}={\sqrt {\frac {Ne^{2}}{\epsilon _{0}m}}}}$ = electron plasma frequency

${\displaystyle \omega _{H}=2\pi f_{H}={\frac {B_{0}|e|}{m}}}$ = electron gyro frequency

${\displaystyle \epsilon _{0}}$ = permittivity of free space

${\displaystyle B_{0}}$ = ambient magnetic field strength

${\displaystyle e}$ = electron charge

${\displaystyle m}$ = electron mass

${\displaystyle \theta }$ = angle between the ambient magnetic field vector and the wave vector

Modes of propagation

The presence of the ${\displaystyle \pm }$ sign in the Appleton–Hartree equation gives two separate solutions for the refractive index.[5] For propagation perpendicular to the magnetic field, i.e., ${\displaystyle {\mathbf {k}}\perp {\mathbf {B}}_{0}}$, the '+' sign represents the "ordinary mode," and the '−' sign represents the "extraordinary mode." For propagation parallel to the magnetic field, i.e., ${\displaystyle {\mathbf {k}}\parallel {\mathbf {B}}_{0}}$, 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.

${\displaystyle {\mathbf {k}}}$ is the vector of the propagation plane.

Reduced Forms

Propagation in a collisionless plasma

If the electron collision frequency ${\displaystyle \nu }$ is negligible compared to the wave frequency of interest ${\displaystyle \omega }$, the plasma can be said to be "collisionless." That is, given the condition

${\displaystyle \nu \ll \omega }$,

we have

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

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

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

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., ${\displaystyle \theta \approx 0}$, we can neglect the ${\displaystyle Y^{4}\sin ^{4}\theta }$ term above. Thus, for quasi-longitudinal propagation in a cold, collisionless plasma, the Appleton–Hartree equation becomes,

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