content_image/354935.jpg content_image/354934.jpg content_image/354936.jpg content_image/354937.jpg content_image/354938.jpg content_image/354939.jpg
equation is derived from Maxwell’s equations and from the quantum fluid equations for the two electrons species, 

and, 

where − q < 0 and m are the electron charge and mass respectively. The subscripts “+” and “-” stand here for the electron population going to the positive and negative directions respectively. These equations are now linearized perturbing every quantity with a term proportional to exp( i k · r − iωt ), where k = ( k$_{x}$, 0 , k$_{z}$ ). The linearized mass conservation equation yields straightforwardly, 

while the linearized quantum Euler equation reads, 

<img src='content_image/354935.jpg'>

<img src='content_image/354934.jpg'>

<img src='content_image/354936.jpg'>

<img src='content_image/354937.jpg'>

While flow aligned unstable modes ( k$_{x}$ = 0) are exactly electrostatic with B$_{1}$ = 0, such is not the case for a general orientation of the wave vector. This is why the exact calculation requires a Lorentz force term. Although some approximations will be used in the sequel to analyze the problem, we first outline the exact one. 

The linearized equation (3–4) allow to express the perturbed velocities v ± 1 in terms of the perturbed fields E$_{1}$ and B$_{1}$ . The first order magnetic field is then eliminated through the Maxwell-Faraday equation B$_{1}$ = ( c/ω ) k × E$_{1}$ so that the first order current J$_{1}$ = − q ∑ $_{±}$( n ± $_{0}$v ± 1 + n ± $_{1}$v ± 0 ), can eventually be expressed in terms of E$_{1}$ only. Combining now Maxwell-Faraday and Maxwell-Amp` ere equations gives, 

from which the formal dispersion relation is finally derived, 

<img src='content_image/354938.jpg'>

<img src='content_image/354939.jpg'>

where ϵ ( k , ω ) = 1 + 4 iπ ω J$_{1}$ is the dielectric tensor. The dispersion equation derived from Eq. (6) is very involved and will not be reproduced here. Instead, we now introduce the 

