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. It is named after Edward Victor Appleton and Douglas Hartree.

History

The Appleton–Hartree equation was developed independently by several different scientists, including H. K. Lassen (1926), Edward Victor Appleton (1927,1931), Douglas Hartree (1929, 1931). Lassen's work, that predates the work of Appleton and 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. Further, regarding the derivation by Appleton, it was noted in the historical study by C. Stewart Gillmor that Wilhelm Altar (while working with Appleton) first calculated the dispersion relation in 1926.

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

=

(

c

k

ω

)

2

.

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

The full equation is typically given as follows:

n

2

=

1

−

X

1

−

i

Z

−

1

2

Y

2

sin

2

⁡

θ

1

−

X

−

i

Z

±

1

1

−

X

−

i

Z

(

1

4

Y

4

sin

4

⁡

θ

+

Y

2

cos

2

⁡

θ

(

1

−

X

−

i

Z

)

2

)

1

/

2

{\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

{\displaystyle Z=0}

and rearranging terms:

n

2

=

1

−

X

(

1

−

X

)

1

−

X

−

1

2

Y

2

sin

2

⁡

θ

±

(

(

1

2

Y

2

sin

2

⁡

θ

)

2

+

(

1

−

X

)

2

Y

2

cos

2

⁡

θ

)

1

/

2

{\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:

n

{\displaystyle n}

: complex refractive index

i

=

−

1

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

: imaginary unit

X

=

ω

0

2

ω

2

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

Y

=

ω

H

ω

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

Z

=

ν

ω

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

ν

{\displaystyle \nu }

: electron collision frequency

ω

=

2

π

f

{\displaystyle \omega =2\pi f}

: angular frequency

f

{\displaystyle f}

: ordinary frequency (cycles per second, or Hertz)

ω

0

=

2

π

f

0

=

N

e

2

ϵ

0

m

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

: electron plasma frequency

ω

H

=

2

π

f

H

=

B

0

|

e

|

m

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

: electron gyro frequency

ϵ

0

{\displaystyle \epsilon _{0}}

: permittivity of free space

B

0

{\displaystyle B_{0}}

: ambient magnetic field strength

e

{\displaystyle e}

: electron charge

m

{\displaystyle m}

: electron mass

θ

{\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. For propagation perpendicular to the magnetic field, i.e.,

k

⊥

B

0

{\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.,

k

∥

B

0

{\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.

k

{\displaystyle \mathbf {k} }

is the vector of the propagation plane.