MEMORANDUM-4. n id (n 2 + n2 S) 0 n n 0. Det[ɛ(ω, k)]=0 gives the Dispersion relation for waves in a cold magnetized plasma: ω 2 pα ω 2 cα ω2, ω 2
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1 Fundamental dispersion relation MEMORANDUM-4 n S) id n n id n + n S) 0 } n n 0 {{ n P } ɛω,k) E x E y E z = 0 Det[ɛω, k)]=0 gives the Dispersion relation for waves in a cold magnetized plasma: n P ) [ n S)n + n S) D ] n n n + n S) = 0 where S = 1 + α ω pα ω cα ω, D = α ω pα ω cα ω ω cα ω, α = e, i Waves in plasmas B 0 = 0) Electromagnetic waves P = 1 + α. ω pα ω Dispersion relation ω = ω p + k c in plasma) ω = kc in vacuum) v g v φ = c Skin-depth δ = c ω p ω ) 1/ Waves in plasmas B 0 0) Electrostatic waves Upper hybrid frequency ω h = ω p + ω c ) 1/, ωc = eb 0 /m Lower hybrid frequency ω l = ω c Ω c ) 1/ = ω c m/m) 1/, Ω c = eb 0 /M Electrostatic ion cyclotron wave ω = Ω c + k v S, v S =ion sound speed 1
2 Electromagnetic waves k B 0 E 1 B 0 O - wave): ω = ω p + k c E 1 B 0 X - wave): ñ c v φ k c ω = 1 ω p ω ω ω p ω ω h Resonance ñ ) : ñ =refractive index ω = ω h Cut-off ñ 0 ): ω R = 1 ω L = 1 k B 0 R - wave): ñ = k c ω = 1 ω p ωω ω c ) L - wave): ñ = k c ω = 1 ω p ωω+ω c) [ ω ) ] c + 4ωp 1/ + ωc right-hand) [ ω ) ] c + 4ωp 1/ ωc left-hand)
3 Problem 4-9 A space capsule making a reentry into the earth s atmosphere suffers a communications blackout because a plasma is generated by the shock wave in front of the capsule. If the radio operates at a frequency of 300MHz, what is the minimum plasma density during the blackout? Dispersion relation: where, ω = ω p + k c 1) If Cut-off takes place when ) e 1/ n 0 ω p = ) mε 0 ω p ω ω < ωp k < 0 k = i c { Ex, t) = E 0 exp {ikx ωt)} = E 0 exp { iωt} exp ω = ω p min = ω p n 0 min ) ω p ω } x } c {{} damping ω = πf = ) e 1/ n 0 min ε 0 m n 0 min = ε 0m e πf) n 0 min = ) ) = m 3 3
4 Problem 4-13 An 8-mm microwave interferometer is used on an infinite plane-parallel plasma slab 8 cm thick see Fig P4-13 in Chen). a) If the plasma density is uniform, and a phase shift of 1/10 fringe is observed, what is the density? Note: One fringe corresponds to a 360 degrees phase shift). b) Show that if the phase shift is small, it is proportional to the density. Vacuum: ω = k 0 c, φ = π 10 k 0 = π λ 0 1) Plasma: ω = ω p + k pc ) E exp {ikx ωt)} Let us choose some fixed moment, t and direct x-axis along wave propagation with the origin point aligned with left wall of the chamber. φ 0 t=t,x=0 = φ p t=t,x=0 φ 0 t=t,x=l = k 0L ωt 3) φ p t=t,x=l = k pl ωt 4) 5) : φ = k 0 k p )L 5) k p = k 0 1 φ ) k 0 L 6) 4
5 Relation 6) into dispersion relation ): ωp = ω k0c 1 φ ) = ω 1 1 φ )) = ω φ 1 1 k 0 L k 0 L k 0 L )) φ k 0 L As far as, e n 0 ω p = mε 0 ) 1/ n 0 = ε 0m e c π) λ 0 λ 0 πl φ 1 1 φ π )) λ 0 L n 0 = 4π ε 0mc e λ 0 L φ 1 1 φ π )) λ 0 L 7) As φ << π, λ 0 << L the last term in the brackets ) φ λ 0 = << 1 π L n 0 = 4π ε 0mc φ 8) e λ 0 L For given λ 0, L and φ from 8): 5) φ n 0 n 0 = = m 3 5
6 Problem 4-8 Find the dispersion relation for electrostatic electron waves propagating at an arbitrary angle θ relative to B 0. a) Show that the answer is ω ω ω h) + ω c ω p cos θ = 0. b) Write out the two solutions of this quadratic for ω, and analyze the solutions in the limits θ 0 and θ π/. Show that in these limits, one of the two solutions is a spurious root with no physical meaning. Assumptions: n i = n 0, T i = T e = 0, V 0 = E 0 = 0 Linearization: n e = n 0 + n 1, V e = V 1, E = E 1. Linearized system of equations: n 1 t + n 0 V 1 = 0 V 1 t = e m E 1 + V 1 B 0 ) ε 0 E 1 = en 1 All variables exp {ikr ωt)} and: k E 1, k = k x, 0, k z ), E 1 = E 1x, 0, E 1z ), B 0 = 0, 0, B 0 ) Separate into components and drop subscripts: iωn + in 0 k x V x + k z V z ) = 0 1) iωv x = e m E x e m V yb 0 ) iωv y = e m V xb 0 3) iωv z = e m E z 4) iε 0 k x E x + k z E z ) = en 5) 6
7 3) V y = i eb 0 mω V x = i ω c ω V x ω c = eb ) 0 m 6) ) V x = i ω e m E x + i eb ) 0 ω c m ω V x ω e V x = i ω ωc m E x 7) 4) V z = i e ωm E z 8) Substitution of expressions for V x, V z into Eq.1) yields: n = i n 0e m kx ω ω c E x + k ) z ω E z 9) Using, k x = k sin θ, k z = k cos θ, E x = E sin θ, E z = E cos θ, where k = k x + k z, E = E x + E z we arrive at: n = i n 0e sin m ke θ ω ωc ) + cos θ ω 10) Substituting expression for n into Eq.5), we find, 1 ω p ω ω c sin θ ω p ω cos θ = 0 11) Let take Eq.11) ω ω ω c ) and make some algebra; then ω 4 ω ω h + ω pω c cos θ = 0 1) 7
8 where ω h = ω p + ω c Solution of Eq.1) with respect to ω has the following form: ω = 1 ) ωh ± ωh 4 4ω pωc cos θ Consider limit cases with respect to θ: i) θ 0 cos θ 1 k z arbitrary k x fixed) : + ω ω = 1 p ωp > ωc ω p + ωc ± ωp ωc + ωc ) = ωp < ωc - ωc ωp > ωc - ωp ωp < ωc ii) θ π cos θ 0 k z 0 k x fixed) : ω = 1 ω p + ω c ± ω p + ω c )) = { + ω h - 0 Roots ω = ω c and ω = 0 are unphysical ones see Eq.11)). It comes from the fact that by obtaining the dispersion relation we multiplied Eq.7) with ω ω ω c ). But when ω ω c and ω 0 solution do exists and is described by lower curve. 8
9 Problem 4-5 A microwave interferometer employing the ordinary wave cannot be used above the cutoff density n c. To measure higher densities, one can use the extraordinary wave. a) Write an expression for the cutoff density n cx for the X-wave. b) On a ñ vs. ω p diagram show the branch of the X-wave dispersion relation on which such an interferometer would work. a) Dispersion relation for the O-wave ω = ω p + k c 1) Cut-off takes place at ω = ω p, or n co = mε 0ω e ) n < n co ). Dispersion relation for the X-wave: ñ k c ω = 1 ω p ω ω ω p ω ω h 3) ω h = ω p + ω c ) Cut-off takes place when k = 0 we can write the equation with respect to plasma frequency, Solution is Thus, ω 4 p ω ω p + ω ω ω c ) = 0 ω p = ωω ± ω c ) = Ω ±. n cx± = mε 0 e ωω ± ω c). 9
10 The plus sign corresponds to better performance. n cx = mε 0ω 1 + ω c e ω ) = n co 1 + ω c ω ) n cx > n co 10
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