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Chapter 11: Plasma physics 61 For disturbances in the z -direction in a cold, homogeneous, magnetized plasma: ± B = B 0 ± e z + ± ˆ B e i ( kz - ω t ) and n = n 0 n e i ( kz - ω t ) (external E felds are screened) ±ollows, with the defnitions α = ω p / ω and β = Ω / ω and ω 2 p = ω 2 pi + ω 2 pe : ± J = ± ±σ ± E, with ± = i ε 0 ω ± s α 2 s 1 1 - β 2 s - i β s 1 - β 2 s 0 i β s 1 - β 2 s 1 1 - β 2 s 0 00 1 where the sum is taken over particle species s . The dielectric tensor E , with property: ± k · ( ± ± ± E )=0 is given by ± ± E = ± ± I - ± /i ε 0 ω . With the defnitions S =1 - ± s α 2 s 1 - β 2 s ,D = ± s α 2 s β s 1 - β 2 s ,P - ± s α 2 s ±ollows: ± ± E = S - iD 0 iD S 0 P The eigenvalues o± this hermitian matrix are R = S + D , L = S - D , λ 3 = P , with eigenvectors ± e r = 1 2 2(1 ,i, 0) , ± e l = 1 2 2(1 , - i, 0) and ± e 3 =(0 , 0 , 1) . ± e r is connected with a right rotating feld ±or which iE x /E y and ± e l is connected with a le±t rotating feld ±or which iE x /E y = - 1 . When k
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