Dielectric permittivity tensor with 2 ions

Induced magnetic field

Electron density

wave frequency

Ion 1 Mass

Ion 1 charge

Ion 2 Mass

Ion 2 charge

Ion 2 ratio

Angle btw n and B

Electron Temperature

Plot frequency from

Plot frequency To

Ion 1 density `n_(i1)`

`n_e=n_(i1) Z_1+n_(i2) Z_2`

Ion 2 density `n_(i2)`

`n_(i2)=r n_(i1)`

Electro plasma frequency 

\(f_{pe}=\frac{1}{2\pi}\sqrt{\frac{n_e e^2}{\epsilon_0 m}}\)

Electro cyclone frequency  

\(f_{ce}=\frac{e Bt}{m}\frac{1}{2\pi}\)

Ion plasma frequency 

\(f_{pi}=\frac{1}{2\pi}\sqrt{\frac{n_i (Z e)^2}{\epsilon_0 m_i}}\)

Ion 1 cyclone frequency `f_(ci1)`

\(f_{ci}=-\frac{Z e B0}{m_i}\frac{1}{2\pi}\)

Ion 2 plasma frequency 

`f_(p i2)=1/(2pi) sqrt((n_(i2)(Z_2e)^2)/(epsilon_0 m_(i2))`

Ion 2 cyclone frequency `f_(ci2)`

`f_(ci2)=-(Z_2 e B)/(2 pi m_(i2))`

Right cutoff  

\(\omega_R=\sqrt{\omega_{ce}^2/4+\omega_{pe}^2+\omega_{ce}\omega_{ci}}+\omega_{ce}/2 \)

Left Cutoff 

\(\omega_L=\sqrt{\omega_{ce}^2/4+\omega_{pe}^2+\omega_{ce}\omega_{ci}}-\omega_{ce}/2 \)

\(K_{\bot}\) 

`K_(bot)=1-Sigma(omega_(pk)^2/(omega^2-omega_(ck)^2))`

\(K_{\times}\) 

`K_x=-Sigma(omega_(pk)^2/(omega^2-omega_(ck)^2) omega_(ck)/omega)`

\(K_{\parallel}\) 

`K_(||)=1-Sigma(omega_(pk)^2/omega^2)`

\(\gamma\) 

\(\gamma=\frac{\mu_0(n_im_i+n_em_e)c^2}{B_0^2} \,\,\,\, \omega_{pi}^2=\gamma\omega_{ci}^2 \,\,\,\, v_A^2=\frac{c^2}{\gamma}\)

Norm B square 

\(\frac{\omega_{ce}^2}{\omega^2}\)

Norm density 

\(\frac{\omega_{pe}^2+\omega_{ce}^2}{\omega^2}\)

R 

\(R=K_{\bot}+K_{\times}\)

L 

\(L=K_{\bot}-K_{\times}\)

electro ratio 

\(\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2}\)

ion 1 ratio 

\(\frac{\omega_{pi}^2}{\omega^2-\omega_{ci}^2}\)

ion 2 ratio 

\(\frac{\omega_{pi2}^2}{\omega^2-\omega_{ci2}^2}\)

Alfven velocity 

\(V_A=\frac{B_0}{\sqrt{\mu_0 \rho}}\)

mass density 

\(\rho=n_i m_i\)

upper hybrid frequency 

\( \omega_{UH}=\sqrt{\omega_{pe}^2+\omega_{ce}^2}\)

lower hybrid frequency 

\(\omega_{LH}=\sqrt{\frac{\omega_{ce}^2\omega_{ci}^2+\omega_{pi}^2\omega_{ce}^2}{\omega_{pe}^2+\omega_{ce}^2}}\)

permittivity 

\( \mathbf K=\begin{bmatrix} K_{\bot} & -i K_{\times} & 0 \\ i K_{\times} & K_{\bot} & 0 \\ 0 & 0 & k_{\parallel} \end{bmatrix}\)

eigenmode equation 

\(\begin{bmatrix} S-n^2\cos^2\theta & -i D & n^2\cos\theta\sin\theta \\ i D & S-n^2 & 0 \\ n^2\sin\theta\cos\theta & 0 & P-n^2\sin^2\theta \end{bmatrix}\begin{bmatrix} E_x \\ E_y \\ E_z \end{bmatrix}=0 \)

A 

\(K_{\bot}\sin^2 \theta+K_{\parallel} \cos^2 \theta \)

B 

\(RL\sin^2\theta+K_{\parallel}K_{\bot}(1+\cos^2\theta)\)

C 

\( K_{\parallel}RL\)

n1 wave 

\(An^4-Bn^2+C=0\,\,\,\, n_2=\sqrt{\frac{B+\sqrt{B^2-4AC}}{2A}}\)

n1 polarization 

\(\frac{i E_x}{E_y}=\frac{n^2-S}{D} \,\,\,\, left: -1\)

n2 wave 

\(n_2=\sqrt{\frac{B-\sqrt{B^2-4AC}}{2A}}\)

n2 polarization 

\(\frac{i E_x}{E_y}=\frac{n^2-S}{D}\,\,\,\, right: 1\)

\(\epsilon_0 \omega\) 

\(-\epsilon_0 \omega\)

conductivity 

\(\mathbf K=\mathbf I+\frac{i \mathbf{\sigma}}{\epsilon_0 \omega} \)

thermal velocity  

\(\frac{1}{2}mv_{th}^2=KT\)

reciprocal of n 

\(\lambda=\frac{\lambda_0}{n}\,\,\,\, v_{\phi}=\frac{c}{n}\)