1629373397-425d4de58b7aea127ffc7c337418ea8d (Introduction to Plasma Physics and Controlled Fusion Francis F. Chen), страница 7

The electrons remain Maxwellian and move to form a Debye shield, but the ions are stationary duringthe time frame of the experiment.(a) Assuming eϕ/KTe 1, write Poisson’s equation for this problem interms of λD.(b) Show that the equation is satisfied by a function of the form ekr/r.Determine k and derive an expression for ϕ(r) in terms of a, ϕ0, and λD.181 Introduction1.11.

A field-effect transistor (FET) is basically an electron valve that operates on afinite-Debye-length effect. Conduction electrons flow from the source S tothe drain D through a semiconducting material when a potential is appliedbetween them. When a negative potential is applied to the insulated gate G,no current can flow through G, but the applied potential leaks into thesemiconductor and repels electrons. The channel width is narrowed and theelectron flow impeded in proportion to the gate potential.

If the thickness ofthe device is too large, Debye shielding prevents the gate voltage frompenetrating far enough. Estimate the maximum thickness of the conductionlayer of an n-channel FET if it has doping level (plasma density) of 1022 m3,is at room temperature, and is to be no more than 10 Debye lengths thick.(See Fig. P1.11.)Fig. P1.111.12. (Advanced problem) Ionization is caused by electrons in the tail of a Maxwellian distribution which have energies exceeding the ionization potential.For instance, this potential is Eioniz ¼ 15.8 eV in argon.

Consider aone-dimensional plasma with electron velocities u in the x direction only.What fraction of the electrons can ionize for given KTe in argon? (Give ananalytic answer in terms of error functions.)Chapter 2Single-Particle Motions2.1IntroductionWhat makes plasmas particularly difficult to analyze is the fact that the densitiesfall in an intermediate range. Fluids like water are so dense that the motions ofindividual molecules do not have to be considered.

Collisions dominate, and thesimple equations of ordinary fluid dynamics suffice. At the other extreme in verylow-density devices like the alternating-gradient synchrotron, only single-particletrajectories need be considered; collective effects are often unimportant. Plasmasbehave sometimes like fluids, and sometimes like a collection of individual particles. The first step in learning how to deal with this schizophrenic personality is tounderstand how single particles behave in electric and magnetic fields. This chapterdiffers from succeeding ones in that the E and B fields are assumed to be prescribedand not affected by the charged particles.2.22.2.1Uniform E and B FieldsE¼0In this case, a charged particle has a simple cyclotron gyration.

The equation ofmotion ismdv¼ qv Bdtð2:1ÞTaking z^ to be the direction of B (B ¼ B z^ ), we haveThe original version of this chapter was revised. An erratum to this chapter can be found athttps://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_219202Single-Particle Motionsmv_ x ¼ qBv ymv_ y ¼ qBvx mv_ z ¼ 0 2qBqB€vx ¼vxv_ y ¼ mm 2qBqB€v y ¼ v_ x ¼ vymmð2:2ÞThis describes a simple harmonic oscillator at the cyclotron frequency, which wedefine to beωc jqjBmð2:3ÞBy the convention we have chosen, ωc is always nonnegative. B is measured intesla, or webers/m2, a unit equal to 104 G. The solution of Eq. (2.2) is thenvx, y ¼ v⊥ exp iωc t þ iδx, ythe denoting the sign of q.

We may choose the phase δ so thatvx ¼ v⊥ eiωc t ¼ x_ð2:4aÞwhere v⊥ is a positive constant denoting the speed in the plane perpendicular to B.Thenvy ¼m1v_ x ¼ v_ x ¼ iv⊥ eiωc t ¼ y_qBωcð2:4bÞIntegrating once again, we havex x0 ¼ iv⊥ iωc teωcy y0 ¼ v⊥ iωc teωcð2:5ÞWe define the Larmor radius to berL v⊥ mv⊥¼ωc jqjBð2:6ÞTaking the real part of Eq. (2.5), we havex x0 ¼ r L sin ωc ty y0 ¼ r L cos ωc tð2:7ÞThis describes a circular orbit around a guiding center (x0, y0) which is fixed(Fig. 2.1). The direction of the gyration is always such that the magnetic field2.2 Uniform E and B Fields21Fig.

2.1 Larmor orbitsin a magnetic fieldgenerated by the charged particle is opposite to the externally imposed field. Plasmaparticles, therefore, tend to reduce the magnetic field, and plasmas are diamagnetic.In Fig. 2.1, the right-hand rule with the thumb pointed in the B direction would giveions a clockwise gyration. Ions gyrate counterclockwise to generate an opposing B,thus lowering the energy of the system. In addition to this motion, there is anarbitrary velocity vz along B which is not affected by B. The trajectory of a chargedparticle in space is, in general, a helix.2.2.2Finite EIf now we allow an electric field to be present, the motion will be found to be thesum of two motions: the usual circular Larmor gyration plus a drift of the guidingcenter.

We may choose E to lie in the x–z plane so that Ey ¼ 0. As before, thez component of velocity is unrelated to the transverse components and can betreated separately. The equation of motion is nowmdv¼ qð E þ v B Þdtð2:8Þwhose z component isdvz q¼ Ezmdtorvz ¼qEzt þ vz0mð2:9ÞThis is a straightforward acceleration along B. The transverse components ofEq.

(2.8) are222Single-Particle Motionsdvx q¼ E x ωc v ymdtdv y¼ 0 ωc v xdtð2:10ÞDifferentiating, we have (for constant E)€vx ¼ ω2c vxq2 Ex€v y ¼ ωc Ex ωc v y ¼ ωcþ vymBWe can write this asd2ExEx2vy þ¼ ωc v y þBBdt2ð2:11Þso that Eq. (2.11) is reduced to the previous case (Eq. (2.2)) if we replace vy byvy + (Ex/B). Equations (2.4a) and (2.4b) are therefore replaced byvx ¼ v⊥ eiωc tv y ¼ iv⊥ eiωc t ExBð2:12ÞThe Larmor motion is the same as before, but there is superimposed a drift vgc of theguiding center in the y direction (for Ex > 0) (Fig. 2.2).To obtain a general formula for vgc, we can solve Eq. (2.8) in vector form.We may omit the m dv/dt term in Eq.

(2.8), since this term gives only the circularmotion at ωc, which we already know about. Then Eq. (2.8) becomesE þ v B¼0Fig. 2.2 Particle drifts in crossed electric and magnetic fieldsð2:13Þ2.2 Uniform E and B Fields23Taking the cross product with B, we haveE B ¼ B ðv BÞ ¼ vB2 Bðv BÞð2:14ÞThe transverse components of this equation arev⊥gc ¼ E B=B2 vEð2:15ÞWe define this to be vE, the electric field drift of the guiding center. In magnitude,this drift isvE ¼EðV=mÞ mBðteslaÞ secð2:16ÞIt is important to note that vE is independent of q, m, and v⊥. The reason isobvious from the following physical picture. In the first half-cycle of the ion’s orbitin Fig. 2.2, it gains energy from the electric field and increases in v⊥ and, hence,in rL. In the second half-cycle, it loses energy and decreases in rL.

This differencein rL on the left and right sides of the orbit causes the drift vE. A negative electrongyrates in the opposite direction but also gains energy in the opposite direction; itends up drifting in the same direction as an ion. For particles of the same velocitybut different mass, the lighter one will have smaller rL and hence drift less percycle.

However, its gyration frequency is also larger, and the two effects exactlycancel. Two particles of the same mass but different energy would have the sameωc. The slower one will have smaller rL and hence gain less energy from E in a halfcycle. However, for less energetic particles the fractional change in rL for a givenchange in energy is larger, and these two effects cancel (Problem 2.4).The three-dimensional orbit in space is therefore a slanted helix with changingpitch (Fig. 2.3).Fig.

2.3 The actual orbit of a gyrating particle in space242.2.32Single-Particle MotionsGravitational FieldThe foregoing result can be applied to other forces by replacing qE in the equationof motion (2.8) by a general force F. The guiding center drift caused by F is thenvf ¼1F Bq B2ð2:17ÞIn particular, if F is the force of gravity mg, there is a driftvg ¼mg Bq B2ð2:18ÞThis is similar to the drift vE in that it is perpendicular to both the force and B, but itdiffers in one important respect. The drift vg changes sign with the particle’s charge.Under a gravitational force, ions and electrons drift in opposite directions, so thereis a net current density in the plasma given byj ¼ nðM þ mÞgBB2ð2:19ÞThe physical reason for this drift (Fig. 2.4) is again the change in Larmor radius asthe particle gains and loses energy in the gravitational field.

Now the electronsgyrate in the opposite sense to the ions, but the force on them is in the samedirection, so the drift is in the opposite direction. The magnitude of vg is usuallynegligible (Problem 2.6), but when the lines of force are curved, there is an effectivegravitational force due to centrifugal force. This force, which is not negligible, isindependent of mass; this is why we did not stress the m dependence of Eq. (2.18).Centrifugal force is the basis of a plasma instability called the “gravitational”instability, which has nothing to do with real gravity.Fig. 2.4 The drift of a gyrating particle in a gravitational field2.2 Uniform E and B Fields25Problems2.1. Compute rL for the following cases if vk is negligible:(a) A 10-keV electron in the earth’s magnetic field of 5 105 T.(b) A solar wind proton with streaming velocity 300 km/s, B ¼ 5 109 T.(c) A 1-keV He+ ion in the solar atmosphere near a sunspot, whereB ¼ 5 102 T.(d) A 3.5-MeV He++ ash particle in an 8-T DT fusion reactor.2.2.

In the TFTR (Tokamak Fusion Test Reactor) at Princeton, the plasma washeated by injection of 200-keV neutral deuterium atoms, which, after enteringthe magnetic field, are converted to 200-keV D ions (A ¼ 2) by chargeexchange. These ions are confined only if rL a, where a ¼ 0.6 m is theminor radius of the toroidal plasma. Compute the maximum Larmor radiusin a 5-T field to see if this is satisfied.2.3. An ion engine (see Fig.