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P4.34 Schematic of a pulsed HCN laser4.21The CMA Diagram139(c) Show that the plasma is a focusing lens for the whistler mode.(d) Can one use the whistler mode and therefore go to much higherdensities?4.35. Use Maxwell’s equations and the electron equation of motion to derive thedispersion relation for light waves propagating through a uniform,unmagnetized, collisionless, isothermal plasma with density n and finiteelectron temperature Te. (Ignore ion motions.)4.36.
Prove that transverse waves are unaffected by the ∇ p term wheneverk B0 ¼ 0, even if ion motion is included.4.37. Consider the damping of an ordinary wave caused by a constant collisionfrequency ν between electrons and ions.(a) Show that the dispersion relation isω2pc2 k 2¼1ωðω þ ivÞω2(b) For waves damped in time (k real) when v/ω 1, show that the dampingrate γ Im (ω) is approximatelyγ¼ω2p vω2 2(c) For waves damped in space (ω real) when v/ω 1, show that theattenuation distance δ (Im k)1 is approximately2c ω2δ¼v ω2pω2p1 2ω!1=24.38. It has been proposed to build a solar power station in space with huge panelsof solar cells collecting sunlight 24 h a day.
The power is transmitted to earthin a 30-cm-wavelength microwave beam. We wish to estimate how much ofthe power is lost in heating up the ionosphere. Treating the latter as a weaklyionized gas with constant electron-neutral collision frequency, what fractionof the beam power is lost in traversing 100 km of plasma with ne ¼ 1011 m3,nn ¼ 1016 m3, and σv ¼ 1014 m3 = sec ?4.39. The Appleton–Hartree dispersion relation for high-frequency electromagnetic waves propagating at an angle θ to the magnetic field isc2 k 2¼1ω22ω2p 1 ω2p =ω21=22222222422222ω 1 ω p =ω ωc sin θ ωc ωc sin θ þ 4ω 1 ω p =ωcos θDiscuss the cutoffs and resonances of this equation.
Which are independentof θ?1404 Waves in Plasmas4.40. Microwaves with free-space wavelength λ0 equal to 1 cm are sent through aplasma slab 10 cm thick in which the density and magnetic field are uniformand given by n0 ¼ 2.8 1018 m3 and B0 ¼ 1.07 T. Calculate the number ofwavelengths inside the slab if (see Fig. P4.40)(a) the waveguide is oriented so that E1 is in the ^z direction;(b) the waveguide is oriented so that E1 is in the ^y direction.Fig.
P4.404.41. A cold plasma is composed of positive ions of charge Ze and mass M+ andnegative ions of charge e and mass M. In the equilibrium state, there is nomagnetic or electric field and no velocity; and the respective densities are n0+and n0 ¼ Zn0+. Derive the dispersion relation for plane electromagneticwaves.4.42. Ion waves are generated in a gas-discharge plasma in a mixture of argon andhelium gases. The plasma has the following constituents:(a) Electrons of density n0 and temperature KTe;(b) Argon ions of density nA, mass MA, charge + Ze, and temperature 0; and(c) He ions of density nH, mass MH, charge + e, and temperature 0.Derive an expression for the phase velocity of the waves using a linearized,one-dimensional theory with the plasma approximation and the Boltzmannrelation for electrons.4.43.
In a remote part of the universe, there exists a plasma consisting of positronsand fully stripped antifermium nuclei of charge Ze, where Z ¼ 100. Fromthe equations of motion, continuity, and Poisson, derive a dispersion relationfor plasma oscillations in this plasma, including ion motions.
Define theplasma frequencies. You may assume KT ¼ 0, B0 ¼ 0, and all other simplifying initial conditions.4.21The CMA Diagram1414.44. Intelligent life on a planet in the Crab nebula tries to communicate with usprimitive creatures on the earth. We receive radio signals in the 108–109 Hzrange, but the spectrum stops abruptly at 120 MHz. From optical measurements, it is possible to place an upper limit of 36 G on the magnetic field inthe vicinity of the parent star.
If the star is located in an HII region (one whichcontains ionized hydrogen), and if the radio signals are affected by some sortof cutoff in the plasma there, what is a reasonable lower limit to the plasmadensity? (1 G ¼ 104 T.)4.45. A space ship is moving through the ionosphere of Jupiter at a speed of100 km/s, parallel to the 105-T magnetic field.
If the motion is supersonic(v > vs), ion acoustic shock waves would be generated. If, in addition, themotion is super-Alfvénic (v > vA), magnetic shock waves would also beexcited. Instruments on board indicate the former but not the latter. Findlimits to the plasma density and electron temperature and indicate whetherthese are upper or lower limits. Assume that the atmosphere of Jupitercontains cold, singly charged molecular ions of H2, He, CH4, CO2, andNH4 with an average atomic weight of 10.4.46. An extraordinary wave with frequency ω is incident on a plasma from theoutside. The variation of the right-hand cutoff frequency ωR and the upperhybrid resonance frequency ωh with radius are as shown. There is an evanescent layer in which the wave cannot propagate. If the density gradient at thepoint where ω ’ ωh is given by |∂n/∂r| ’ n/r0, show that the distanced between the ω ¼ ωR and ωh points is approximately d ¼ (ωc/ω)r0.Fig.
P4.461424 Waves in Plasmas4.47. By introducing a gradient in B0, it is possible to make the upper hybridresonance accessible to an X wave sent in from the outside of the plasma(cf. preceding problem).(a) Draw on an ωc/ω vs. ω2p /ω2 diagram the path taken by the wave, showinghow the ωR cutoff is avoided.(b) Show that the required change in B0 between the plasma surface and theupper hybrid layer isΔB0 ¼ B0 ω2p =2ω2c4.48.
A certain plasma wave has the dispersion relationω2pc2 k 2¼12 ðω Ω Þ2ccω2ω2 ωc Ωc þ ωω2 ω2 þω Ωcpcwhere ω2 ω2p þ Ω2p : Write explicit expressions for the resonance and cutofffrequencies (or for the squares thereof), when ε m/M 1.4.49. The extraordinary wave with ion motions included has the following dispersion relation:ω2pΩ2pc2 k 2¼1ω2 ω2c ω2 Ω2cω22ωc ω pω ω2 ω2c1Ω2p Ωωc ω2 Ω2ω2pω2 ω2cc2Ω2pω2 Ω2c(a) Show that this is identical to the equation in the previous problem.(Warning: this problem may be hazardous to your mental health.)(b) If ωl and ωL are the lower hybrid and left-hand cutoff frequencies of thiswave, show that the ordering Ωc ωl ωL is always obeyed.(c) Using these results and the known phase velocity in the ω ! 0 limit,draw a qualitative v2ϕ /c2 vs.
(ω plot showing the regions of propagationand evanescence).4.50. We wish to do lower-hybrid heating of a hydrogen plasma column withωp ¼ 0 at r ¼ a and ωp ¼ ½ωc at the center, in a uniform magnetic field.The antenna launches an X wave with k║ ¼ 0.(a) Draw a qualitative plot of ωc, Ωc, ωL, and ωl vs. radius. This graphshould not be to scale, but it should show correctly the relative magnitudes of these frequencies at the edge and center of the plasma.(b) Estimate the thickness of the evanescent layer between ωl and ωL (cf. theprevious problem) if the rf frequency ω is set equal to ωl at the center.(c) Repeat (a) and (b) for ωp (max) ¼ 2ωc, and draw a conclusion about thisantenna design.4.21The CMA Diagram1434.51.
The electromagnetic ion cyclotron wave (Stix wave) is sometimes used forradiofrequency heating of fusion plasmas. Derive the dispersion relation asfollows:(a) Derive a wave equation in the form of Eq. (4.118) but with displacementcurrent neglected.(b) Write the x and y components of this equation assumingkx ¼ 0, k2 ¼ k2y þ k2z , and k y kz Ez 0:(c) To evaluate j1 ¼ n0e(vi ve), derive the ion equivalent of Eq. (4.98) toobtain vi, and make a low-frequency approximation so that ve is simplythe E B drift.(d) Insert the result of (c) into (b) to obtain two simultaneous homogeneousequations for Ex and Ey, using the definition for Ωp in Eq.
(4.49).(e) Set the determinant to zero and solve to lowest order in Ω2p to obtain2ω ¼Ω2c"#1Ω2p 111þ 2 2þ 2c kz k4.52. Compute the damping rate of light waves in a plasma due to electroncollisions with ions and neutrals at frequency ν by adding a term mνve1to the equation of motion in Eq. (4.83). It is sufficient to show that, for smalldamping, Eq. (4.85) is replaced byω2 c2 k2 ¼ ω2p 1 iν=ω p :4.53. The two-ion hybrid frequency of S.J. Buchsbaum [Phys. Fluids 3, 418 (1960)]has the cold-plasma dispersion relationω2 ¼ Ωc1 Ωc2α1 Ωc2 þ α2 Ωc1α1 Ωc1 þ α2 Ωc2in the limit of perpendicular propagation (tan2θ ! 1). Here each ion specieshas cyclotron frequency Ωcj and fractional density αj.(a) Derive the equation by setting S ¼ 0 in Eq.
(B-29) in Appendix B.(b) In Fig. P4.53, which of (a) or (b) shows the guiding-center orbits inlower hybrid resonance, and which shows the orbits in two-ion hybridresonance?1444 Waves in PlasmasFig. P4.53(c) Which orbit in the lower hybrid case corresponds to the ions, and whichto the electrons?(d) Which orbit in the two-ion hybrid case corresponds to the majorityspecies, and which to the minority species? (This is non-trivial.)Chapter 5Diffusion and Resistivity5.1Diffusion and Mobility in Weakly Ionized GasesThe infinite, homogeneous plasmas assumed in the previous chapter for the equilibrium conditions are, of course, highly idealized.
Any realistic plasma will have adensity gradient, and the plasma will tend to diffuse toward regions of low density.The central problem in controlled thermonuclear reactions is to impede the rate ofdiffusion by using a magnetic field. Before tackling the magnetic field problem,however, we shall consider the case of diffusion in the absence of magnetic fields. Afurther simplification results if we assume that the plasma is weakly ionized, so thatcharge particles collide primarily with neutral atoms rather than with one another.The case of a fully ionized plasma is deferred to a later section, since it results in anonlinear equation for which there are few simple illustrative solutions. In any case,partially ionized gases are not rare: High-pressure arcs and ionospheric plasmas fallinto this category, and most of the early work on gas discharges involved fractionalionizations between 103 and 106, when collisions with neutral atoms are dominant.The picture, then, is of a nonuniform distribution of ions and electrons in a densebackground of neutrals (Fig.
5.1). As the plasma spreads out as a result of pressuregradient and electric field forces, the individual particles undergo a random walk,colliding frequently with the neutral atoms. We begin with a brief review ofdefinitions from atomic theory.5.1.1Collision ParametersWhen an electron, say, collides with a neutral atom, it may lose any fraction of itsinitial momentum, depending on the angle at which it rebounds. In a head-onThe original version of this chapter was revised.
An erratum to this chapter can be found at https://doi.org/10.1007/978-3-319-22309-4_11© Springer International Publishing Switzerland 2016F.F. Chen, Introduction to Plasma Physics and Controlled Fusion,DOI 10.1007/978-3-319-22309-4_51451465 Diffusion and ResistivityFig. 5.1 Diffusion of gasatoms by random collisionsFig.
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