Dispersion relation

The dispersion relation shows the response of the material to different wavenumbers by describing the frequency $ω (k)$

In Cold plasma

we can solve the Maxwell's equations in matter using the dielectric tensor of cold plasma , by considering that we are taking into account the linear response of the plasma to fluctuating electric fields and therefore the Electric Displacement Field can describe the currents generated by this motion.

Therefore Ampere Circuital Law becomes:

\nabla \times B = \frac{1}{c} \frac{\partial D}{\partial t}

and Maxwell-Faraday Law
#Question Why does this law remain unchanged? is D just quantity that should not be confused with the displacement field?

After fourier analysis these become:

i k \times B = \frac{- i ω}{c} ϵ E

where ε is the Dielectric Tensor
and:

i k \times E = \frac{i ω}{c} B

combining:

i k \frac{c}{i ω} \times i k \times E = - \frac{i ω}{c} ϵ E

k \times (k \times E) + \frac{ω^{2}}{c^{2}} ϵ E = 0

we can define a vector of the Refractive index

n = \frac{k c}{ω}

n \times (n \times E) + ϵ E = 0

From Cross-product properties:

n (n \cdot E) - E (n \cdot n) + ϵ E = 0

n^{2} E - (\begin{matrix} n_{x} (n_{x} E_{x} + n_{y} E_{y} + n_{z} E_{z}) \\ n_{y} (n_{x} E_{x} + n_{y} E_{y} + n_{z} E_{z}) \\ n_{z} (n_{x} E_{x} + n_{y} E_{y} + n_{z} E_{z}) \end{matrix}) + ϵ E = 0

((\begin{matrix} n^{2} & 0 & 0 \\ 0 & n^{2} & 0 \\ 0 & 0 & n^{2} \end{matrix}) - (\begin{matrix} n_{x} n_{x} & n_{x} n_{y} & n_{x} n_{z} \\ n_{y} n_{x} & n_{y} n_{y} & n_{y} n_{z} \\ n_{z} n_{x} & n_{z} n_{y} & n_{z} n_{z} \end{matrix}) + ϵ) E = 0

or symbolically often written as:

(n^{2} δ_{i j} - n_{i} n_{j} - ϵ_{i j}) E = 0, i, j = x, y, z

where $δ_{i j}$ the Kronecker delta Tensor @verkhoglyadovaPropertiesObliquelyPropagating2010a

We define $θ$ as the angle between the magnetic field $B_{0} = B_{0} \hat{z}$ and the direction of the propagation $\vec{n}$ and assume it to be in the x,z plane. Then:

n_{x} = n \sin θ

n_{z} = n \cos θ

((\begin{matrix} n^{2} - n^{3} \sin^{2} θ & 0 & - n^{2} \sin θ \cos θ \\ 0 & n^{2} & 0 \\ - n^{2} \sin θ \cos θ & 0 & n^{2} - n^{2} \cos θ \end{matrix}) - ϵ) (\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \end{matrix}) = ((\begin{matrix} n^{2} \cos^{2} θ & 0 & - n^{2} \sin θ \cos θ \\ 0 & n^{2} & 0 \\ - n^{2} \sin θ \cos θ & 0 & n^{2} \sin^{2} θ \end{matrix}) - ϵ) (\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \end{matrix})

where $ϵ$ Dielectric Tensor#Dielectric Tensor of cold plasma
The dispersion relation can then be found by solving the system

(\begin{matrix} n^{2} \cos^{2} θ - S & + i D & - n^{2} \sin θ \cos θ \\ - i D & n^{2} - S & 0 \\ - n^{2} \sin θ \cos θ & 0 & n^{2} \sin^{2} θ - P \end{matrix}) (\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \end{matrix}) = 0 $ $ F o r n o n - t r i v i a l s o l u t i o n t h e [[M a t h s / D e t e r m i n a n t ∥ D e t e r m i n a n t]] n e e d s t o b e z e r o :

D = (n^2 \cos^2\theta-S) (n^2-S)(n^2\sin^2\theta-P)+D^2(n^2 \sin^2\theta-P)+n^4\sin^2\theta \cos^2\theta=0

s o l v i n g t h i s g i v e s u s

\tan^2\theta= -\frac{P(n^2-R)(n^2-L)}

$ θ = 0 $

P=0 , n^2 =R, n^2 - L

$ θ = \frac{π}{2} $

n^2 = \frac{RL}{S},n^2 = P

@stixWavesPlasmas1992 <div class="transclusion internal-embed is-loaded"><div class="markdown-embed"> Title: Properties of obliquely propagating chorus Tags:#arctic,#Halocarbons,#iodineAuthors: Olga P. Verkhoglyadova, Bruce T. Tsurutani, Gurbax S. Lakhina Zotero link: [Full Text PDF](zotero://select/library/items/J473PWAX) > [!NOTE]- Abstract > We discuss chorus wave magnetic and electric field polarizations as functions on angle of propagation relative to the ambient magnetic field B0. For the first time, it is shown using a cold plasma approximation that the general whistler wave has circularly polarized magnetic fields for oblique propagation. This theoretical result is verified by observations. The electric field polarization plane is not orthogonal to the wave vector k and is in general highly elliptically polarized. Both the magnetic and the electric polarizations have important consequences for cyclotron resonant electron pitch angle scattering and for electron energization, respectively. A special case of the whistler wave called the Gendrin mode is discussed. # Questions > [!quote]+ Note ([Page ](zotero://open-pdf/library/items/J473PWAX?page=1&annotation=VI2HI7P4)) > In these coordinates the Hermitian tensor of dielectric permittivity for cold magnetized plasma#Disagreement> - **This seems like the susceptibility tensor not the permittivity** > > [!quote]+ Image ([Page 2](zotero://open-pdf/library/items/J473PWAX?page=2&annotation=PBYXYHVE)) > ![Litterature/verkhoglyadovaPropertiesObliquelyPropagating2010a/verkhoglyadovaPropertiesObliquelyPropagating2010a-2-x52-y155.png](/img/user/Litterature/verkhoglyadovaPropertiesObliquelyPropagating2010a/verkhoglyadovaPropertiesObliquelyPropagating2010a-2-x52-y155.png) > - **How?** > > [!quote]+ Highlight ([Page 2](zotero://open-pdf/library/items/J473PWAX?page=2&annotation=4Y3M5QXD)) > It is well known that chorus wave emission consists of separate “elements” rising or falling in frequency, which have fine structure of “wave packets” or “subelements.” These features were extensively studied by Santolik et al. [2003, 2004], Verkhoglyadova et al. [2009], and Tsurutani et al. [2009]. Each subelement is a quasi‐monochromatic wave modulated by a low‐frequency signal. We assume that our linear approach is applicable within a single subelement for a quasi‐periodic electromagnetic disturbance#Look-into> - **Chorus waves properties** > # Important points > [!quote]+ Note ([Page ](zotero://open-pdf/library/items/J473PWAX?page=1&annotation=FYSC6S7U)) > In the Earth’s nightside sector magnetosphere, chorus typically propagates almost parallel to the local background magnetic field B0 ( < 20°), where is the angle between the wave vector k and B0 [Tsurutani and Smith, 1977; Tsurutani et al., 2009]. However, observations show that dayside chorus can also propagate at highly oblique angles [Burton and Holzer, 1974; Goldstein and Tsurutani, 1984] > > [!quote]+ Note ([Page ](zotero://open-pdf/library/items/J473PWAX?page=1&annotation=TMNPJY5G)) > it is theoretically well known that whistler mode waves propagating parallel to B0 are circularly polarized in both magnetic and electric field components [Stix, 1992], > </div></div>