PLASMA DYNAMICSX. PLASMAS AND CONTROLLED NUCLEAR FUSIONA. Waves and RadiationAcademic and Research StaffProf. G. Bekefi Prof. W. M. Manheimer J. J. McCarthyProf. S. C. Brown Prof. B. L. Wright W. J. MulliganGraduate StudentsR. J. Becker L. P. Mix, Jr. E. N. SpithasA. J. Cohen L. D. Pleasance D. W. SwainL. Litzenberger G. L. Rogoff J. H. VellengaC. E. Speck1. THEORY OF ION ACOUSTIC WAVESThe propagation characteristics of ion acoustic waves have been studied in highlyionized collisionless helium and argon plasmas. The study indicates that the wavesare dispersionless and obey the classical dispersion relation for ion acoustic wavespropagating in an infinite medium along a magnetic field. The waves, however,are heavily damped. In an attempt to explain this heavy damping, two differentapproaches have been used in the derivation of the ion acoustic wave theory. Onetheory, based on the Vlasov equation, ignores collisions and predicts the dampingto be independent of frequency and a function only of the ratio of electron to iontemperatures for plasma parameters typically observed in PF-1. The other theoryis based on the macroscopic transport equations and predicts a damping that has astrong frequency dependence and is a function of electron and ion temperature anddensity. In this report these theories and their predicted damping curves are pre-sented.The strong magnetic field used to constrain the plasma to the axis of the system sug-gests that only one dimension need be considered for these longitudinal waves. The goodagreement between the observed dispersion curves and the classical curves, which arederived under the assumption of an infinite geometry, hints that boundary effects maybe unimportant for this simple longitudinal mode of the plasma. Therefore, both the-ories will be derived, under the assumption of an infinite homogeneous plasma in onedimension and neglecting the magnetic field. Drift of the plasma has been observedexperimentally; therefore, the equations will be derived to include the effects of drifton the dispersion relations. The effect of a non-Maxwellian electron distribution is alsoconsidered.This work was supported by the U. S. Atomic Energy Commission (ContractAT(30-1)-3980).QPR No. 94(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)Collisionless TheoryThe basis for this theory will be the collisionless Boltzmann equation or Vlasovequation. If we assume the electrostatic approximation in addition to the approxima-tions stated above, the following well-known dispersion relation for longitudinal wavesis obtained.2S C Fs (u)D(k,l) = 1 - s _ = 0. (1)14k2 1 2 -ccu-k 2 -o0 u-s kHere F is the one-dimensional distribution function, k is the wavevector, and L isSo Psthe plasma frequency of the species. Under the assumption that the species are in adrifted Maxwellian distribution with a streaming velocity Us , the equation furtherreduces to the following form2D(k,) = 1- k , (2)2 2 kcsas swhere Z' is the first derivative of the plasma dispersion function,2 and a = 2T /mis the thermal speed of the species. Making the usual assumption ae > >> ai , we mayuse the power-series expansion and the asymptotic expansion for the plasma dispersionfunction. Keeping two terms in the ion expansion and one in the electron expansion forthe dispersion function, we obtain for one of the two sets of roots2 2e1 + Deem.2where D = e is the electron Debye length. If we further assume that kk D << 1, whiche Pe eis a very good approximation for the plasma produced in PF 1, Eq. 3 becomes= V +U.,k s iwhereT + 3T.e 1 (4)s m.1QPR No. 94(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)If instead, two terms in the electron expansion are retained, and only the leading termin the ion expansion is retained, one of the two sets of roots that is obtained is2 T 1 mk Ui m 2 2 4m. (5)1 k D 1 U i1+ e2If the frequency is assumed to have a small imaginary part, we can expand the dispersionYfunction in a power series of the real part, that is, if w = w + iy and -<< 1, thenr2D(k, W) = D(k,wr) + iyD' (k, Or) - 2! D" (k, r +... (6)Assuming y = 0, we obtain to first orderIm (D(k, wr))y Z - (7)Re (D' (k, wr))where D' (k, r) is the first derivative of Eq. 2 with respect to w and evaluated at w = wrBecause y is assumed small, the power-series expansion of the plasma dispersion func-tion for the ions will be used in which it is assumed y equal to zero. Putting into Eq.7 thevalues obtained from the first-order expansion, the following equation for y is obtainedT T m3 / 2y =k i3 m e exp 2 . (8)In Eq. 8, the ions and electrons are assumed to have equal streaming velocities.Although these forms of the solution are relatively simple and easy to use, for ourexperiments the approximations are not strictly valid. To avoid this difficulty, Eq. 2was solved on the computer, using a modified Newton-Raphson method. 3 The exact solu-tion of the dispersion equation obtained in this manner indicates that Eq. 4 gives thereal part of the dispersion relation within an accuracy of 3% for values of Te/Ti from1 to 40. Equation 8 is found to be in poor agreement with the exact computer solution.The quantity,De -o e _ (9)X 2ny 'rather than y, is the quantity that is measured experimentally; therefore this quantity,QPR No. 9424202416120 I--- Upstream WaveSDownstream Wave16 D e 12DIe 12Ix 884-4-0 a I10 20 30 4010 20 30 40T /T.T T.e(b)100 10.0_80 8.0> (VFig. X-1.a >D D D60 I 6.0 Theoretical values of Landaue IDownstream Wave e ratio of electron-to-ion tem-8 h x perature in a helium plasma.40 -4.0 (a) No electron or ion drifts.I (b) Ion drift velocity 0. 3 Vselectron drift velocity 0. 0.2020(c) Ion drift velocity 0. 3 Vs20 2.0 electron drift velocity Vs010 20 30 40T eT.(C)QPR No. 94(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)rather than y, will be discussed in the rest of this report. If the electron and ion driftvelocities are equal, the value of the damping term D /X is found to vary from the valueobtained with no drift, (De/ ) o , in the following wayD e D .o U1e__s Q )1 + (10)Because of the good agreement (less than 3% variation) between the exact solution andthis equation, the theoretical curve of (D e/) as a function of T /T i is shown inFig. X-la. These values of (De/\)o are found to be independent of ion temperature, atleast for values between 0. 1 and 0. 4 eV, and a function only of the temperature ratioTe/Ti, For comparison, Fig. X-lb and X-lc gives the values of De/\ as a function ofTe/Ti for upstream and downstream propagation with the electron and ion drift velocitiesunequal.The pronounced breaks observed in some of the log I vs V curves that were usedto obtain the electron temperature suggests that the plasma might consist of a two-temperature Maxwellian electron distribution.4 The fact that the probe curves generallysuggested a slightly higher temperature than that calculated from the wavelength of theion acoustic wave gives additional support to the idea that the body of the electron dis-tribution may not be Maxwellian. It has also been found that if the ion saturation currentis extrapolated by using a more horizontal line, i. e. , extrapolating that part of the ionsaturation curve which appears linear rather than using a tangent to the curve at a pointwhich gives the most linear log I vs V plot, pronounced breaks are observed even nearthe axis of the column. The temperature obtained by this more horizontal extrapolationindicates a high energy tail typically 9 or 10 eV.Because of the long mean-free paths for the electrons in this system, it is notunreasonable to assume that the electrons are not in a Maxwellian distribution. Thegrid farthest from the plasma-generating structure is observed to act as an electro-static reflector, causing the mirror magnetic field to have no observable effect oneither the plasma or the wave propagation. The reflecting efficiency of electrostaticreflectors is greatest when the velocity of the reflected particles normal to thereflector is the smallest. Therefore, because of the observed discrepancies betweenmeasurements of the electron temperature and the definite possibility that the gridscould cause an overpopulation of the low parallel energy electrons relative to aMaxwellian distribution, the effects of a two-temperature Maxwellian will be con-sidered.The effect of a colder component of the electron distribution on the dispersion rela-tion is simply to add an additional term to the sum over the species in Eq. 1. Thisadditional term causes Eq. 2 to take the following form:QPR No. 94(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)T W- U T22 e T k eD(k, w) = k X Z'D T.. Te 1 1 eU'k e6Z'aeke- EZ' kaewhere 6 and E are the fractional concentrations of the cold and hottions, and the primed variables denote the parameters of the colderFor simplicity, assume U. = U = U' = 0, and a i << << a' < a1 e e 1 k e epriate series expansions for Z' (a), the following dispersion relationthe leading terms of the expansions are kept:2 T T'e eek mi(6T +T')or2 T- eff2 m. 'k 1whereT T'Tef f = eeT' + 6(T -T')e eeelectron distribu-electron species.Using the appro-is obtained if only(12)e + 6 = 1.The imaginary part of the dispersion relation is computed by using (7)terms in the expansions of Z' (a). The following equation is obtained.(13)and the leadingT- ff)3/Z8. exp -_ 2 f) +mme T____3/m.1The full equation, Eq. 11, was solved exactly for several values of Teff and Te, with zerodrift assumed. The real part of the dispersion relation agrees with Eq. 15 within 3%:2 T + 3T.k eff ik2 m.k 1 (15)The imaginary part of the dispersion relation is plotted as a function of T' in Fig. X-2,efor values of Teff typically obtained from wavelength measurements, and values of Tetypical of those measured by using the more horizontal ion saturation current extrapola-tion. From Eq. 13 5 and E were obtained.Collisional TheoryThe high densities that are obtainable with this plasma source suggested that viscousdamping might be playing an important role in the damping of the waves. Since, inQPR No. 94(11)/2+ E(14)12.010.08.06.0Teff 3.5 eVNo DriftT. 0.15 eVSi ' ,I , ll ITemperature of low energy component of electron distribution (eV)(a)18.0 i-16.014.012.010.08.06.0Teff 4.5 eVNo DriftT. 0.15 eV, j u lI I0.4 0.8 1.2Temperature of low energy component of electron distribution (eV)(b)Fig. X-2. Theoretical values of the Landau damping for a helium plasmahaving a two-temperature Maxwellian electron distribution.The temperature of the high-energy component of the electrondistribution is 9 eV for curve (a) and 11 eV for curve (b).QPR No. 94 95(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)general, the mean-free path of the ions is less than the wavelength of the waves that areobserved, a fluid description is valid, at least for the ions. Since the magnetic fieldused to contain the plasma is sufficiently large to satisfy the following inequalities:Qe >> 0i > Vi' Ve'where Qi and 0e are the ion and electron-cyclotron frequencies, and vi and ve are theself-collision frequencies of the ions and electrons, many simplifying assumptions maybe made. Braginskii 5 has considered these assumptions in some detail; therefore, thetransport equations derived by him will be used. The variables in the transport equationwill be assumed to consist of a steady-state plus a small perturbation having a harmonicvariation of the form, exp(i(wt+kz)). This is essentially equivalent to Fourier-Laplacetransforming the transport equations and solving them for times sufficiently long thatthe initial-value terms introduced by the Laplace transforms can be neglected and onlythe natural modes of the system considered. Absolute neutrality will also be assumed.With these assumptions, the mass conservation equation for the ions may be linear-ized to give(W+Uik) = -knU., (16)where n is the density and the tilde denotes a small perturbation to the steady state.Defining w 1 = kUe, the electron-mass conservation equation becomes(W+wll)n + nkUe = 0. (17)The momentum conservation equations may be linearized to yield the following relationsif we neglect electron inertia; that is, we assume the frequency of the variation of theelectron velocity to be much greater than the frequency of the collective modes of inter-est.0 = -k(iT +nT ) (18)nmi(w+kUi ) U. = -k(nT.+nT.). (19)Adding Eqs. 18 and 19, we obtainnm.i(w+kU.i) U. = -k(e(T +T.)+n(T.+T )). (20)Proceeding to the energy balance equations for the ions and electrons, we haveT. k a.(w+kUi) A- ix (w+kUi) -n (21)T. - i } n)11QPR No. 9450 100Frequency (kHz)Ti (eV) Te(eV) Vx 10- 40.1 4.0 3.9790.1 2.0 3.9790.2 4.0 1.5700.2 2.0 1.570(a)50 100FrequencyTi(eV) Te(eV)0.1 4.00.1 2.00.2 4.00.2 2.0(kHz)i x 10- 51.3891.3893.4673.467(b)Fig. X-3. Theoretical values of the viscous damping in a helium plasmaas a function of frequency. The density for curves (a) and (b)is 4 X 109/cm 3 and 4 X 1010/cm3QPR No. 94500 1000Sx 10- 42.1785.7192.1755.719500 1000CurveABCDCurveACD/e x 10- 52.0055.2292.0055.229(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)S 3 2 2T a e - (w (22)e (+I) - iXe 0 (22)e eThe mass conservation equations have been used to obtain these forms; Xi and Xe arenumerical factors equal to 1. 95 and 1. 58 for singularly ionized ions and electrons,respectively. Combining Eqs. 20, 21, and 22, we arrive at the following expression:w+kU.N T. T-nm U (T +T) + + e (23)i3 i 3 e3 TXi 3 _iXe2 w + kU i 2 o + 0wherek 2 25X, = Xs vs22In the limit k2a e>> e(C+ l) (long electron mean-free paths), this equation takes the forme e3 + *Xi-nm(w+kUi) U. i( ) + kUk = n Te 1 + + T + . (24)k e 12Xe _9+ Xi4 2(w+kU.)1Using Eq. 16, we arrive at the dispersion relation2 (+kU(w+kU.) T. +2 T + X + iXe T (25)1 e 1Equation 5 may be put into the form of a fourth-order polynomial in , and solved25)by a number of methods. Muller's method23 was used to solve this equation for various4 2 4 2(w+kU.) (w+kU.)1 iwhere 5 = T /m.s eiEquation 25 may be put into the form of a fourth-order polynomial in , and solvedby a number of methods. Muller's method 3 was used to solve this equation for variousvalues of the wavevector k, electron temperature, ion temperature, and density. Thereciprocals of the self-collision times from Spitzer were used for the collision fre-quencies. The theoretical curves relating De/X to the frequency, f = w/2Tr, are shownfor helium in Fig. X-3 for typical ion and electron temperatures and densities. Thenumerical factors, Xi and Xe' were set equal to one for these calculations.L. P. Mix, Jr., G. BekefiQPR No. 94(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)1. L. P. Mix, Jr., L. Litzenberger, and G. Bekefi, Quarterly Progress Report No. 93,Research Laboratory of Electronics, M. I. T., April 15, 1969, pp. 69-84.2. B. D. Fried and S. D. Conte, The Plasma Dispersion Function (Academic Press,New York, 1961).3. C. E. Froberg, Introduction to Numerical Analysis (Addison-Wesley Publishing Co.,Reading, Mass., 1965).4. L. P. Mix, Jr., E. W. Fitzgerald, and G. Bekefi, Quarterly Progress ReportNo. 92, Research Laboratory of Electronics, M. I. T., January 15, 1969, p. 227.5. S. I. Braginskii, Reviews of Plasma Physics, M. A. Leontovich (ed.) (ConsultantsBureau Enterprises, Inc. , New York, 1965), Vol. 1, pp. 205-311.6. L. Spitzer, Jr., Physics of Fully Ionized Gases (Interscience Publishers, New York,1962).QPR No. 94(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)2. THREE-WAVE COUPLING IN CLYNDRICAL PLASMASWe consider a plasma cylindrical column in the presence of a uniform magnetic field,directed along the axis of the cylinder. Our description of the system is simple: (a) Theelectrons are assumed to behave like a charged conducting fluid. Their thermal motionis neglected. (b) The ions constitute a uniform neutralizing background. (c) The exter-nal magnetic field is of such strength that the motion of electrons perpendicular to it isneglected.We are interested in plasma waves whose phase velocity is much smaller than thevelocity of light, so that we may use the quasi-static approximation (neglect of AC mag-netic field).The equations that describe the system areav e _ 1 8 2(v ) (1)at - 8z 2 8zman 8 v a8S+ N (nv) (2)t 8z az72 = 4wren. (3)We have taken the external magnetic field in the z-direction (along the axis of the plasmacolumn); v is the component of velocity perturbation in the z-direction; m and (-e) arethe mass and charge of an electron; N is the number density of the ions, n + N the num-ber density of electrons; 4 is the perturbation potential.Linear SolutionThe linear solution is found by neglecting products of first-order quantities, andassuming solutions in the form of plane waves in the axial direction. The well-knownresult is14(rt) = AJm(pr) ei(kz-wt+mO) (4)e kv(rt)- N (rt) (5)2n(rt)- e N (rt) (6)2 2p -- k E. (7)Here, E is the cold-plasma dielectric function, and A is a complex constant.QPR No. 94 100(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)In the case of a perfectly conducting plasma-filled waveguide, the potential vanishesat the wall, r = R. This gives the dispersion relation2xmnR2where x is the nthmnphase velocity isRvph = pP Xmn2-P (8)root of the mth-order Bessel function. For low frequencies, the(9)The group velocity is easily found from Eq. 8,a= = E(.)Vg- ak- (W) 1 Equation 8 shows that the waves cut off at the plasma frequency.Coupled-Mode EquationsWe combine Eqs. 1-3 to obtain2 2 2\ a2 4TeN 8 2 2S+ p + (v ) - 4e (uv).at 2 t z 8z(10)(11)Here, we have separated the Laplacian operator into its axial and transverse com-ponents.The nonlinear terms are assumed sufficiently small, so that we may use the linearexpressions for the number density and the velocity. We first expand n and v in the formi(kz-wo t+m6)mnv(rt ) = Vm(r) emni(kz-omnt+m)n(rt) = nmn(r) e mn (12)mnWe substitute these relations in the right-hand side of Eq. 11, denoted by R, andobtain in simplified notation,R - 4reN (k'+k")(2 vvG - e (k'+k")(w'+w")[n'v"+n"v' ]G, (13)QPR No. 94 101(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)whereG - exp[i(k'+k")z-i(wo'+w")t+i(m'+m")6].We use (5) and (6) to relate the eigenfunctions for the density and velocity to theeigenfunction for the potential. We substitute these relations in (13) to get, after somereductions,e 2 k' k" (k '+ k ') k' k"Sp +(k +k")(w'+w")+ + ' "G. (14)These are the driving terms in (11). If they are set equal to zero, the last equation hasthe linear solutions already discussed. Since the radial and azimuthal modes form acomplete set, at given z and t we can expand the potential in (11) as a sum of suchmodes. If we set4(rt) = 4mn(zt) Jm(pmnr) exp[i(kz- mnt+mO)], (15)mnthen 4(zt) will be a slowly varying function of z and t if the nonlinearity is weak.Substituting the expansion (15) in the left-hand side of (11), denoted by L, andneglecting second derivatives in z and t, we obtain- 2 exp[i(kz-m t+mO)]mn mumn22imk + ' -k~X V12 + 21E k + 2mn Vk mn )t mn(zt)SJm(Pmnr). (16)Substituting (15) in (14) and setting the resulting expression equal to (16), we obtainthe desired nonlinear equation. We can eliminate the dependence on the transversecoordinates by making use of the orthogonality relation2O i(m+m')O R R2 J2de eim+m rdr J m(pmnr) Jm, (Pmn' r) = 6 6 n, J' (p R)and the fact that7(V+pm2 ) Jm(pmnr) = 0.m\ m mnQPR No. 94 102(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)We finally obtaina + I a t ( z t) IvI W it)z-+ Umn t mn(zt) = - k V( )mn mn nn"n' n"X tm'n tm" n" a(n' n")X exp[-iAkz+iAw mnt], (17)where_ e i 2 1 k" (wk'+k") k' k ilV(W 0' w") = 2 -(k'+k")(w'++") + +2 m p W 2 W " ('+) W1 Ra(n'n") = 2 r dr Jm mnr)Jm(Pm' n' r) Jm" (Pm"n" r)J' (P mnR)S mn mnk mn mnu-mn= (2 +k2) E m nandmn mn (m' n' +m"n")Ak = k - (k'+k") (18)0 = m - (m' +m")Equation 17 is the final relation. It describes the coupling of the (m, n)th mode toall other modes that satisfy the relations (18). Note that there is no such relation forthe radial (quantum) number, n. Also notice that the velocity u is the group velocityth mnof the (m, n)th mode. For appreciable energy exchange either Acw = Ak = 0 (resonance),or these quantities are small compared with any of the frequencies or wavevectors(quasi-resonance).For three-wave coupling, we drop the summation in (17), and introduce the simplifiednotation(m, n) - (1)(m', n') - (2)(m", n") - (3).QPR No. 94 103(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)We also put in the expression for the coupling coefficientsk 1 = k 2 + k 3 ,' W = 2 + W 3'which is exact in the case of resonance, and holds approximately in the case of quasi-resonance.With this notation, we getaz.V.= + VS kj(-Eij)1 (zt) = +(2) VI 24 3 exp[-iAkl z+iAwI t]02 (zt) = +() V 2p 41 exp[+iAk 2 z-i w 2 t]3 (zt)= +( V 3 1 2 exp[+iAk 3 z-iAw3 t],(j = 1, 2, 3)e 2 klk 2 k3 k1 k2 k3V= --+ +2m p olo 2 3 1 2 ;3Equations 19 is a system of partial differential equations. Since they are very com-plicated in general, we deal only with two specific cases.Spatial Variation in the Mode AmplitudesIf we put in Eqs. 19 at = 0, we find that the resulting equations are in exactly the2same form as in a previous report. If we treat them in exactly the same way, we findthe solutionsN 1 (z) = Nla + (Nlb-Nla) y 2 [z]y 2[z] = sn2 (Nc-N 1/ 2 z (20)where sn(u) denotes the elliptic integral of the first kind of modulus A,0 < 2 N lb -Nla<=N -Nlalc laQPR No. 94u tu 8tu 3where(19)104(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)H = [VV 2 V 3 ] 1/2N -j V. (j = 1, 2, 3),and A is the amplitude of mode j.4K(A 2 ) = 4 s '0The period of the elliptic function is 3dt (1-t2)-1/2 (-A2 y2 -1/2so that the period in z is given by4K(A 2 ]1/2L = H/2 [N1 c-N 1 a-/2Temporal Variation in the Mode Amplit/2Temporal Variation in the Mode Amplitudes8We now put in Eqs. 19 aZ 0. Thenwe find(21)again treating the resulting equations as before,N1(t) = Nla + (Nlb-N1a) y2 [t]y 2 [t] = sn (N 1 c-N la/2) t-tland the period in time is4K(A 2 )T =H/2[N1 cN 1 ]-1/2(22)(23)We see from Eqs. 19 thatV. = u.V.A 1N.= N.j uj JH= (ulu 2 u 3 ) 1/2Using (24) and comparing (21) and (23), we getQPR No. 94(24)105(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)T = (u 2 u 3 ) -1/ Lor, put another way,y = (uu 3 ) 1/2 ;that is, the time growth rate, y, is equal to the space growth rate, F, multiplied by thegeometric mean of the two group velocities.E. N. Spithas, W. M. ManheimerReferences1. A. W. Trivelpiece, Slow-Wave Propagation in Plasma Waveguides (San FranciscoPress, 1967).2. L. N. Litzenberger and G. Bekefi, "Nonlinear Coupling of Three Ion AcousticWaves," Quarterly Progress Report No. 93, Research Laboratory of Electronics,M.I.T., April 15, 1969, pp. 72-84.3. Handbook of Mathematical Functions (U.S. Department of Commerce, NationalBureau of Standards, 1965, pp. 569-570.3. LANGMUIR PROBE IN THE ORBITAL MOTION REGIMEThe purpose of this report is to correct a formula used for determining plasma prop-erties from Langmuir probe curves. The corrected formula is important practically,as well as conceptually, as it allows the plasma density to be computed independentlyof the electron temperature. Thus the density determination will not be subject to errorsin the temperature measurement.This formula is applicable in the regime in which Langmuir's orbital motion theoryapplies, namely rp << D, where rp is the probe radius, and XD is the electron Debyelength. Chen has written the expression for the ion current to a cylindrical probe as2 1/2I = eJ (1+ p) (1)rpwhere J = 27r p(kTi/2mdi) 1/2 fn is the total random ion flux, and p = -eV p/kT isr p o p p ethe normalized probe potential, withr = probe radius T = electron temperaturep ef = probe length m i = ion massT i = ion temperature no = ion densityV = potential with respect to plasma potential.The proper normalized probe potential, however, should be qp = -eV /kT. WithQPR No. 94 106(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)this qp , as T.i approaches zero, the ion current properly remains nonzero.Consider the orbit of an ion in an attractive central force field. Let the initial veloc-ity be v 0 and the impact parameter be "b". At the point of closest approach to the poten-tial center let the velocity be va and the radius be "a". Then1 2 1 2-- my = mvy + eV (conservation of energy)o aby = av (conservation of angular momentum)ba( 2eV 1/2b = a 1 e a- (2)2where V must be referenced to zero (plasma potential) and mvy /2 = kT.. Then b =a 0 1a(l+rlp)1/2 , where qp is defined as -eVp/kT i .If "a" is the probe radius r , then any ion with "b" smaller than r (1+71) will2 1 / 2be collected. Therefore the effective probe radius is r/ (+r ) , and the correctformula iseV 1/22pn =2 J 1 kT (3)r kTim (A 2 1/2m e /AI2'o 3/2 ) AV (4)(2e) (r f )where all values are in MKS units, and n is independent of T. It is then easy to plot12 against V from the measured probe characteristic and use the slope of the resulting3straight line in this formula to determine n o .The densities derived from this formula are equal to those derived from the exactcomputer calculations of Laframboise 4 for cylindrical probes with r p/ D < 2. 5.A. J. CohenReferences1. F. F. Chen, C. Etievant, and D. Mosher, Phys. Fluids 11, 811 (1968); see Eqs. (1)and (2). This correction similarly holds for spherical probes in the orbital regime.2. J. E. Allen, R. L. F. Boyd, and P. Reynolds, Proc. Phys. Society (London),Vol. LXX, No. 3-B, p. 297, 1957.3. For electron current in the electron attracting regime this same formula will hold.See F. F. Chen in Plasma Diagnostic Techniques by R. H. Huddlestone and S. L.Leonard (Academic Press, 1965), Sec. 3. 1. 2.4. James G. Laframboise, UTIAS Report No. 100, University of Toronto Institute forAerospace Studies, June 1966; see Fig. 43 and Eq. (9. 10).QPR No. 94 107(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)4. NONLINEAR HARMONIC GENERATION AT PLASMARESONANCESThe subject of this report is the generation of harmonics of an RF signal applied toan antenna immersed in the center of a low-density (n ~ 109 electrons/cm 3 ), low-pressure (p ~ 0.4 p.), DC discharge in Argon.This harmonic generation was reported in two previous reports. 1,2 Various changeshave been made in the system, which have resulted in more precise data of the kindreported previously and in new interesting data.The experiment is basically the same as that reported earlier. The monopole spher-ical antenna has been replaced by the split-sphere dipole reported on by Waletzko andCohen previously.1,3,4 This was done to increase the Q of the resonance that naturallyresults in better definition of the observed effects. Since the dipole field is more local-ized than the monopole field, the dipole field is more intense for a given applied voltageto the antenna. Therefore the harmonic generation is more intense. The spectrum ana-lyzer used earlier as a detector can now be replaced by an RF voltmeter so as to get adirect voltage reading for the harmonics. The coaxial cable used earlier to feed thedipole, RG174, was shown to be significantly lossy at the harmonic frequencies. It hasbeen replaced by precision rigid and semirigid cables. The glass vacuum chamber wassurrounded with a cylindrical screen closed top and bottom. This prevented RF leakagefrom the antenna from going directly into the detecting apparatus. It also prevented spu-rious pickup of radio and television stations that had been thought to be Tonks-Dattnerresonances. Since the shield formed a cylindrical cavity resonant in the range of the har-monics, the RF absorber had to be placed strategically to damp out the cavity resonance.A 20-dB directional coupler was mounted to sample the power incident to the dipole. Thispower was kept constant during each run of data. The dipole could not be DC-biasedbecause no double DC blocks were available at the frequencies of operation. The antennais coated with a thin layer of dielectric, Insl-X, so it can remain DC floating throughoutthe experiment. A movable cylindrical Langmuir probe operating in the Langmuirorbital regime (see Sec. X-A. 3) was used to obtain radial density profiles. A diagramof the experiment is shown in Fig. X-4.Most RF data were taken by sending 0. 50 V of frequency f onto a dipole,1 3/4 inches in diameter, and monitoring the reflected power both at f and at 2f, andby sending 2f onto the dipole and monitoring the reflected power at 2f. For a given fre-quency, the reflected signals were observed as a function of discharge current. Thenthe incident frequency was changed and the measurements were repeated. The RF admit-tance of the probe was monitored with a GR 1602B admittance meter. The value 0.50 Vwas chosen because it was large enough to give good results on the output detectors andsmall enough so as not to perturb the resonance significantly. The gas pressure couldQPR No. 94 108(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)V ,_ , IV i o-- LANGMUIR PROBETUNABLE MOTOR-DRIVENBANDPASS PROGRAMMINGFILTER RESISTOR / 10-2f, 3f r inchRADIUSVARIABLE I- inchCURRENT RADIUSDC POWERFUPPLY INDIRECTLYVOLTMETER CATHODESIGNAL - PUMPPROPORTIONALDC OUTPUT Y X TO DISCHARGEINLETX-Y RECORDERFig. X-4. Diagram of the experiment.also be varied from 0. 25 4 to 2 L by changing the flow rate. The measurements reportedhere are at 0. 25 i and 0. 4 t.Geometrical ResonanceIf a probe is inserted into a plasma, its RF admittance characteristic will show aresonance at a frequency somewhat below plasma frequency. This is attributed to aseries resonance between the capacitive sheath surrounding the probe and the inductiveplasma just beyond. (A plasma acts as an inductance below plasma frequency, since theconduction current exceeds the displacement current for w less than w ( VX H = J + D).)The variation of the resonance with electron density n has been calculated theoreti-cally from the formula3r 3 fp' (1)((11where R is the probe radius, f is electron plasma frequency, and s is the sheath thick-5 pness given by 4. 4 XD. For the conditions at 0. 4 p. at which the admittance runs weretaken, this formula can be written approximately as fr ~ n 3 0 for electron temperature,26 78T e a constant. Experimentally we observe a variation of fr I' but since n ~ I733from Langmuir probe data, we derive fr n 3 3 . This shows excellent agreement betweenfromLanguirprob dat, w derve rQPR No. 94 109(a)80(b)16080-- A200 A100 200 300 400(mA)(c)Fig. X-5. (a) Second harmonic as a function of density.(b) Fundamental power reflected as a functionof density.(c) Harmonic power reflected as a function ofdensity. (Zeros are displaced for clarity.)QPR No. 94 110(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)theory and experiment. The precise positions of the resonant frequencies calculatedfrom the same formula depend strongly on the value of density inserted for s and f .pType B PeaksIf we put frequency f onto the antenna and look at the generation of frequency 2f bythe plasma, or if we look at the reflected power from the plasma at f itself, bothmeasurements being made as a function of plasma density, we observe effects that wepreviously called "type B" peaks 2 (see Fig. X-5). These results can then be correlatedwith the measurements at frequency f with the admittance meter.a. Reflected Power from Plasma at Frequency fIn a simple series RLC circuit in which R, L, and C are independent of frequency W,the minimum in the power reflected from the circuit occurs at the resonant frequency,whereRG=1R + L 1is a maximum andB=R + wL )is zero. If, however, R, L, and C vary with frequency, then the minimum in reflectedpower does not have to occur at the resonance. This, in fact, is the case for the type Bpeaks observed in reflection at frequency f.For experimental simplicity, the frequency is kept constant and the plasma densityis varied instead. Since R, L, and C are functions of the density, the minimum inreflected power does not coincide with the conductance maximum (see Fig. X-6). Theobserved reflection curve at the fundamental f can be approximately generated from theobserved admittance at the fundamental by using the formulaV = /V = y L i (2)inc. o Lwhere p is the reflection coefficient, p = (Yo -YL)/(Y o +YL), Yo is the admittance of the50-0 cable, and YL is the measured admittance of the load that includes the insidecapacitance of the dipole (see Fig. X-7).QPR No. 94 111100 MHzo.- .60 MHz80 MHz- _ -- ....Yo CONDUCTANCEx SUSCEPTANCEp= 0.25 F .x100 200 300 400Id(mA)r-O-I0-20500Fig. X-6. (a) Admittance as a function of discharge current for 60, 80,and 100 MHz.(b) Absorbed power as a function of discharge current.(c) Harmonic generation as a function of discharge current.QPR No. 94 1123020-(a)(a)(b)-jzZ-zT(c)60 MHz80 MHz100 MHz p .25p =I 25p,---b~ ~z~~~ -7~~c~60 MHzp-0 254-o- THEORY ( FROM FIG. X-6o)0 5 -EXPERIMENT00 200 300 400Id (mA)(a)80 MHzp=0. O25,u.--- o---- THEORY (FROM FIG. X-60)EXPERIMENT', i [200100pp" 05200300 400Id (mA)(b)100 MHzp= 0. 2 5-o- THEORY (FROM FIG X-60)EXPERIMENT300 400Id (m A)Fig. X-7. Reflected power as a function of discharge current: (a) at60 MHz; (b) at 80 MHz; (c) at 100 MHz.QPR No. 94pp05[10'0 - -. L113(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)b. Generation of the Second HarmonicIf an RF signal is applied to a probe immersed in a plasma, then harmonics of thatapplied signal will be generated by the nonlinear exponential sheath. To explain thetype B peak observed in the second harmonic, the admittance characteristics at thefundamental must be considered.Since the harmonic is generated by rectification in the nonlinear exponential sheath,then the harmonic must peak when the voltage in the sheath is a maximum. The electroncurrent to a probe is= ne(kTe 1/2 -eV/kT e (3)e)Now, if V = VDC + V RF, where VRF Vsheath at frequency f, then(-eV RFR 1 VRFI DC +T 2 T 2e TWe are interested in the term in the expansion for I which varies at 2w. ForVRF/T small, the series converges rapidly, and the only term that contributes signif-icantly to I(2w) is the 1/2 (VRF/T)2 term. The validity of this approximation for theexperimental conditions is discussed in Appendix I. It is shown in Appendix II that I(2w)is given byIDC IVs(w) 2I(2w) =4 s 2 (5)TeIf the voltage across the terminals of the dipole could be kept constant, then thissheath voltage would be largest at the admittance peak. Unfortunately, only the incidentvoltage can be kept constant. The terminal voltage equals the incident voltage plus thereflected voltage.V V. + V (1+p)V 0 V. (6)T inc. refl = (l+p)Vinc. Y + Y L Vinc.Thus the terminal voltage drops as the resonance is approached. To determine thesheath voltage, a model more complex than the constant-voltage model used by othersmust be constructed.QPR No. 94 114(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)The system may be modeled as shown in Fig. X-8. Then2YV =IZ I =V Y V o Vs p s p VT p T Yo + YT inc.and sos YZ+ Y Vinc (7)A value for Vs can then be obtained from the measured values of YT and Y and from1 abPa calculation for Z which is j and C equals 4rE -, where a is the radius of theTILpVinc RpFig. X-8. Model for determining thePLASMA-SHEATH terminal voltage.SYSTEMCi=INSIDE CAPACITANCE OF DIPOLEYT = Yp + jCiT,, T2 = TERMINALS ACROSS WHICHVT IS DEVELOPEDantenna, s is the sheath size (4. 4 XD), and b = a + s. Since C s ~ 1/AD 1 /2 , the ratioof sheath voltages for two different densities at a fixed frequency isV Y +Y Y n )spo T1 p2 1where the various Y's are the values of admittances at frequency f. The ratioIV /Vs 2 12 calculated from formula (8), by using the experimental values for Y , YT'and n, can be compared with the harmonic generation at 2f. The calculated and experi-mental peaks are close to each other. The calculated widths, however, are significantlylarger than the observed widths.This model was used to calculate the relative amplitudes of the second harmonicpeaks for various frequencies, by using the experimentally determined values for theadmittances. The theory and the experiment do not agree.It should be noted that the comparisons above are valid only under the assumptionsthat the plasma admittance at the second harmonic is constant and that the only thingthat determines the height and width of the B peaks is the variation of plasma admittanceat the fundamental. These assumptions will be subject to further scrutiny.QPR No. 94--1 r115Id = 190 mAcoo CONDUCTANCEx SUSCEPTANCEo o p 4oo×xx x x 0x xxxXxIdoxp==260 mACONDUCTANCESUSCEPTANCEO 4 4xxxxXX50 60 70 80 90(b)(d)odId 370mA' CONDUCTANCEx SUSCEPTANCE op=O 4Lx o50 60 70 80 90 100(e)xFig. X-9. Admittance as a function of frequency.116Id=90 mAo CONDUCTANCEx SUSCEPTANCEpO 4A4Id= 140 mAo CONDUCTANCEx SUSCEPTANCEp=O 44QPR No. 94oox xx xxX x xxxx xxx[xxxx xx(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)Various Methods of Determining Series ResonanceOther investigators have measured resonance probe characteristics in three standardways. They have used admittance meters,5,6 observed transmitted or absorbed power,7'8and detected the extra rectified current to a biased probe when the radiofrequency isturned on.8, 9 This extra rectified current to the probe derives from the time averageof Eq. 4, where VRF is taken as V s sin wt. ThenIDC (swhich is just the time average of Eq. 5. The heights of the resonance curves have beenrelated to the damping by Buckley and others. 1 0 ,11 These methods have been comparedand the positions of the resonances have been found to be equivalent in the regimes inwhich the experiments have been performed. 8, 6The various methods may, in fact, be equivalent only over a limited range of cur-rents, frequencies and pressures for which the Q of the resonance is not too high. Athigh Q' s, the differences between these methods begin to show up.As the resonant frequency is increased for a fixed pressure, the Q increases, asdoes the RF conductance as measured with the admittance meter (see Fig. X-9). It isseen, however, that the peak absorbed power at o and the peak harmonic generation at2w (which should be equivalent to the extra rectified current) attain peak values, and then2 4 x IO9--- ,-R E G IO N IE 20x109PROBESi9 RADIUSM 6 x10 875"zDISCHARGE CURRENT 300 mAI x IO 9 PRESSURE 041-FOR ANY CURRENTz 0 7 8o n I SHEATHREGIONne< ni6" 4" 2" 0DISTANCE FROM CENTER OF DIPOLE inches)Fig. X-10. Density profile as determined by a Langmuir probe.QPR No. 94 117finc = 95 MHzfreceived= 190 MHzp= 0.4/finc= 190 MHzfrefl.= 190 MHzp=0 4.ftronsmitied = 190 MHzp=0.4100 200 300 400Fig. X-ll1. (a)(b)(c)(d)(e)Excitation of peak A by the harmonic.Absorption of peak A.Transmission at peak A.Absorption at 160 MHz compared with admittance at 160 MHz.Absorption of RF signal as a function of discharge current forvarious frequencies.00(d)230 MHz0210 MHz0170 MHz120 MHz0 0 GEOMETRICALRESONANCEREGION100 200 300 400(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)decrease as a function of w (see Fig. X-5a and 5b and Fig. X-6b and 6c). Since the peakvalue of the RF conductance peak increases as expected for increasing resonance fre-quency and its position varies according to formula (1), then we expect that the RF admit-ance meter determination is the proper technique to use in a regime in which the variousmethods differ.Type A PeaksThe solutions to the RF probe in a plasma are of two types: the long-wavelengthsolution in which the phase velocity is comparable to the vacuum electromagnetic veloc-ity; and the short-wavelength solution in which the waves propagate at approximatelythe electron thermal velocity. The geometrical resonance and type B peaks were ofthe long-wavelength variety. We shall now consider the thermal waves.The effects called "type A" previously2 may result from a thermal standing wavethat propagates between the dipole antenna and a point in the plasma for which W = o (r).The wave equation 0 is W = c p(r) + 3V Tk2(r), where VT is the electron thermal velocity,and k is the wave number of the oscillation. The wave becomes evanescent; that is,k becomes imaginary for A < p(r). Standing waves of this type are known as Tonks-Dattner resonances, and have been discussed extensively by others.If we look at the radial density profile (Fig. X-10), measured with a thin cylindricalLangmuir probe operating in the orbital regime (see Sec. X-A. 3), we see that a regionin which the Tonks-Dattner resonances could occur does exist for this probe-plasmasystem (region 1 of Fig. X-10).A computer program is being prepared to predict the locations of the Tonks-Dattnerresonances from the measured density profiles and electron temperatures. Thesesolutions will be compared with the locations of the experimental type A peaks (seeFig. X-11).Appendix IIn the expansion of Eq. 4, only the terms in even powers of Vs contribute to thesecond-harmonic current.The term following V2/2T 2 is V4/24T 4 . If V/T is small, then higher order terms,n > 2, may be neglected.Let us see what Vs/T is for the case of maximum harmonic generation. If this issmall, then the approximation should be valid everywhere. See Fig. X-6 for 80 MHzand 155 mA.Y2Y pV o Vs Yo + YL Ys inc.'QPR No. 94 119(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)where Y = (13-3j) mmhos, and YL = (13+3j) mmhos.PY = jwCs s4TE abCs swhere s = 4 . 4XD, a = radius of the probe, and b = a + s.For T = 4. O eV and n s = n/4 = 4 X 108/cm2jwC s = 8. 5j mmhos.Vs = . 9 V.inc but V. = 0.5 V.inc.V s 1T -T, and the fourth-order contribution will beenegligible.Appendix IIThe applied signal is V. sin wt.inc. Then V across the sheath is given byV = o Y Z V.s Yo + YL p s inc.V = V sin wt + V. cos wtS T 12 2 2V = V sin wt +s r2V. cos. 2 1 - cos 2tsin t = 22wt + 2V V. sin wt cos wt,r i12 1 + cos 2wt, cos t = 22 sin wt cos wt = sin 2wtV 2 1 cos 20t - V 2s 2 1i r cos 2wt + 2V V. sin 2wt.r 1Then, from Eq. 4,I(2w) = IDC V r2 2 cos4T2 rbut we measure II(2w) I, soII(2w) I- V DC +V}4 T2 r2wt + 2V V. sinr 1IDC IVs!24 T 2A. J. CohenQPR No. 94but2t} ,120(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)References1. A. J. Cohen, Quarterly Progress Report No. 85, Research Laboratory of Elec-tronics, M.I.T., April 15, 1967, p. 103.2. A. J. Cohen, Quarterly Progress Report No. 90, op. cit., July 15, 1968, pp. 76-82.3. J. A. Waletzko, Quarterly Progress Report No. 79, op. cit., October 15, 1965,p. 95.4. J. A. Waletzko, Quarterly Progress Report No. 80, op. cit., January 15, 1966,p. 103.5. J. A. Waletzko and G. Bekefi, Radio Science (New Series), Vol. 2, No. 5, p. 489,May 1967.6. R. S. Harp and F. W. Crawford, J. Appl. Phys. 35, 3436 (1964).7. A. M. Messiaen and P. E. Vandenplas, J. Appl. Phys. 37, 1718 (1966).8. D. Lepechinsky, A. Messiaen, and P. Rolland, CEA (Paris)-R2945 (1964).9. H. Ikegami and K. Takayama, IPPJ-10 (Nagoya), 1963.10. R. Buckley, J. Plasma Phys. 1 (Pt. 2), 171 (1967).11. P. G. Davies, Proc. Phys. Soc. (London) 88, 1019 (1966).12. A. Dattner, Ericsson Technics 2, 310 (1957).13. F. C. Hoh, Phys. Rev. 133, A1016 (1964).14. H. J. Schmitt, Appl. Phys. Letters 4, 111 (1964).QPR No. 94 121(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)5. RADIATION TEMPERATURE OF EXTRAORDINARY WAVESObservations have been made in this laboratory of the effective temperature, T (w),1-4of microwave radiation emitted from weakly ionized gas discharges. In the presenceof an external magnetic field, B, resonant peaks in T (w) were observed at the electron-cyclotron frequency w = wc = eB/m (in MKS units.) A theoretical model for this behaviorwas obtained by treating the steady-state intensity of radiation as a balance between theemission and absorption (by orbiting electrons) of right circularly polarized (resonant)electromagnetic waves propagating in the direction of B. The resultant expression forthe effective temperature of the radiation is4S v m f(v) dvS(W- )OC + vkTr( =r 4 c m (1)m 1 af dv(co-c(W )2 + V2 m vc mwhere k is Boltzmann's constant, vm(v) is the electron-atom collision frequency formomentum transfer, and f(v) is the (isotropic) electron velocity distribution functionnormalized so that 4Tr f0 v 2 f dv = 1. Both the numerator and denominator (representingcoefficients of emission and absorption of right circular waves) exhibit resonant behaviorat w = wc and, if vm(v) is not constant and f(v) is not Maxwellian, their ratio, T r() is res-onant also. The model neglects warm plasma effects (for example, Doppler broadening)and assumes that electron collisions with neutral particles predominate. Because of itsmanifest simplicity, Eq. 1 has always been used - sometimes with remarkable success -in the analysis of resonant temperature data. 4 ',5 In the guided-wave systems normallyemployed in these experiments, however, the claim that right circularly polarized radia-tion is being monitored exclusively cannot be made. Furthermore, for waves propa-gating at an angle to the magnetic field, the wave resonant frequency (at which the wavephase velocity becomes quite large) is different from the electron-cyclotron frequency.Because of these potential influences, we have undertaken a study of the validity of(1) for other resonant polarizations.A more general expression for Tr (w), one which explicitly includes the wave polar-ization, may be based upon the theory of fluctuations. It is found that for weakly dampedmodes in a plasma confined by conducting walls (say, a waveguide) an effective radiationtemperature may be defined by 4E (r, w) . (r, w) E(r, w) d 3 rkT (O) = T, (2)E(r, o) H(r, O) E(r, ) drQPR No. 94 122(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)iwtwhere E(r, w) is the wave electric field (assumed to vary as e ), 4(r, w) is the micro-scopic current correlation dyadic, and H(r, w) is the Hermitian part of the conductivitytensor, ^. The two integrals are taken over the plasma volume, and it has been assumedhere that C and a reflect only local behavior (cold-plasma limit). The expression (2)represents the average energies of fluctuating modes in the system as a balance betweentheir rate of emission by random electron motion (the numerator) and their ohmic dis-sipation through the induced, in-phase current. In the case of a weakly ionized gas witha uniform magnetic field in the z-direction, all expressions are simplified by trans-formation to right and left circular coordinates:' - E =E iE, ,Z E =E +iE.r x y x yThis transformation diagonalizes the tensors 4 and o-, and for a uniform plasma wemay writeSE 2 Wr + I E 12 4 + Ez 12 zkTr( r) rz (3)Er 2Hr + E 2 GH + I Ez 2 aHzThe appropriate fQrm of the components of has been obtained by Bunkin42 00 v v4ne 2 v m f(v) dv (4)r, 3 0 (w T ) + vc m9 z = the same expression with c = 0.Here n is the electron number density. Similarly, for the components of the diago-nalized conductivity tensorS4Tnev 1 8f dv (5)r,f 3 i(: wc ) + my 8vaz = the same expression with w = 0.The components of the Hermitian tensor 0H are simply the real parts of (5). Combining(4) and (5) with (3), we recover the original result (1) for Tr(w) in the case of pure rightcircular polarization: E = Ez = 0.In the interpretation of temperature measurements of guided electromagnetic waves,the form of E(r, L) in the plasma column is not known. We can argue, however, that the2 2component E is probably comparable to Ed and E so that for (w-wc) > vc mIn our calculations we have also examined the resonant waves that propagate at anglesother than 0' and 90' relative to B. In all respects, they are intermediate between theright circular and extraordinary cases cited here. We also considered the nonresonantpolarizations which vary from the pure left circular wave at 0' to the "ordinary" waveQPR No. 94 127(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)at 900. For these cases the wave fields generally contain only a slight amount of E ,rand the radiation temperature does not vary dramatically.1.0 Rmin= 10.8Fig. X-14.0.6 Dependence on a2 and y of the minimum valueC12 0.4 5of R = Er12 r/I Eej 2 in the range from0.4 - 10 = 1 - y to p = 1 + y. Reduced collision fre-quency, y, assumed independent of electron20 velocity.0.2501000 0.2 0.4 0.6 0.8 1.0YOur initial concern has been with the radiation of guided waves in the neighborhoodof the electron-cyclotron frequency. These waves are not pure plane waves and the2 2suggestion that (1) may be correctly applied if 4w > v is probably still valid. As ac mpotential counterexample we have treated the extraordinary polarization and found thatTr(w), if properly measured, would be resonant at w = wc even though the rates ofemission and absorption are maximum elsewhere. Indeed (1) could be used with littleerror for moderate plasma densities. This is true in the collisionless limit y -* 0 (seeEq. 6), even though this implies Er - 0 at w = oc. A related question arises from studiesof the anomalous emission of microwaves from noble-gas discharges. Even in denseplasmas, the observed radiation is well tuned to the electron-cyclotron frequency. 9 Thephenomenon is thought to involve the negative absorption of extraordinary waves, and itseems apparent that as f(v) and v (v) are varied the denominator of (3) will first becomenegative at the particle resonance, w = uc, rather than at the wave resonance, w = WH.Computer programs are being written to pursue this and related problems.A. C. Reisz, B. L. WrightReferences1. H. Fields, G. Bekefi, and S. C. Brown, Phys. Rev. 129, 506 (1961).2. G. Bekefi, Radiation Processes in Plasmas (John Wiley and Sons, Inc., New York,1966).3. B. L. Wright, Quarterly Progress Report No. 80, Research Laboratory of Elec-tronics, M.I.T., January 15, 1966, p. 99.QPR No. 94. 128(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)4. B. L. Wright, "Measurement of the Time-Dependent Electron Velocity Distributionof a Plasma," Ph. D. Thesis, Department of Physics, M. I. T., June 1967.5. B. L. Wright, Quarterly Progress Report No. 83, Research Laboratory of Elec-tronics, M.I.T. , October 15, 1966, p. 59.6. F. V. Bunkin, Soviet Phys. - JETP 14, 206 (1961).7. B. L. Wright, Quarterly Progress Report No. 76, Research Laboratory of Elec-tronics, M.I.T., January 15, 1965, p. 81.8. W. P. Allis, S. J. Buchsbaum, and A. Bers, Waves in Anisotropic Plasmas (TheM. I. T. Press, Cambridge, Mass., 1963).9. C. Oddou, Quarterly Progress Report No. 93, Research Laboratory of Electronics,M.I.T., April 15, 1969, p. 93.QPR No. 94 129(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)6. DEVELOPMENT OF THE CASCADE IN LASER-PRODUCED PLASMASAccording to one physical picture of the process,I the breakdown of gases at thefocus of a high-intensity laser occurs in three stages:(a) A few initial free electrons appear, probably produced by low-order multiphotonionization of impurities.(b) A "cascade" of free electrons develops. In this stage, each free electron absorbsenergy from the photon field during collisions with the massive neutrals. The numberdensity of free electrons multiplies because those electrons that have gained enoughenergy to do so collide with and ionize the neutral atoms.(c) When the electron and ion densities become great enough, nonlinear processesbegin to dominate the cascade as follows:(i) Collisions of free electrons with ions. Because the cross section for this processis greater than that for collisions with neutrals, the electrons gain energy more rapidly,and the growth rate is enhanced.(ii) Electron-electron thermalization. This tends to fill out the high-energy tail ofthe electron distribution function, so that this process also increases the ionizationrate.(iii) Ambipolar diffusion. This decreases the effectiveness of one loss process.All of these act to enhance the growth of the cascade, while the competing nonlinearprocess of recombination remains negligible. Thus, as soon as the breakdown reachesthis third stage, it almost certainly goes to completion. (In support of the existence ofthis third stage, the data of Young and Hercher1 show a discontinuity in the amount ofcharge collected near the threshold laser-beam power.)A consequence of this physical picture is that it is stage (b) (linear growth of thecascade) that mainly determines the threshold for breakdown. Once stage (c) is reached,the cascade goes to completion; while the elimination of stage (a) does not seem to affectthe threshold. (Young and Hercher's data show that using a low-grade discharge to pro-vide, say, 100-1000 free electrons in the focal volume does not noticeably lower thethreshold.)Analogy with High-Frequency Microwave BreakdownThis report presents a calculation of gain and loss rates in stage (b) basedon the Boltzmann equation as analyzed by Allis.2 In his treatment, the distri-bution function, electric field intensity, and collision integrals are Fourier-analyzedin time and expanded in spherical harmonics in velocity space. For our pur-poses, the result may be writtenQPR No. 94 130(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)8F 1 a 2 p m 8Fa- vt 6v 2 8v m2 2 2 avmm 1 a Zv(M + m 2 av mmv avv2v2F[- (V)+ i(v)]F + g(v) F + 3v (1)mwhereF(v, r, t) = the spherically symmetric DC component of the electron velocity-space-time distribution function;v,e,m = the electron speed, charge, mass;M, T = the mass, temperature of the unperturbed neutral gas;gvm(v) = the electron-atom collision frequency for momentum transfer;v (v), .i(v) = the excitation and ionization collision frequencies for electrons incidenton atoms;S(v) F = the rate at which slow electrons reappear, because of inelastic collisionsof high-speed electrons;S= the peak electric field strength - a function of space and time;PW = the angular frequency of the incident radiation.This equation treats the photon field classically; it makes no provision for the discrete-ness of photon energies, which are approximately 5-10 per cent of gas ionization ener-gies.In extending this theory to the case of laser-produced plasmas, we must note, amongothers, the following conditions:1. The field is not uniform in space. This results in an uneven rate of growth forthe electron density. Since the rate of growth is fastest at the center, this condition mayenhance diffusion losses.2. The field is not uniform in time. In microwave terms, the field is pulsed, ratherthan cw. We must therefore compute and integrate the net gain over the duration of thepulse in order to predict the threshold.Also, we may expect this theory to be valid only provided that the mean-free path ofthe free electrons, and the amplitude of their oscillations, are both small compared withthe characteristic dimensions of the discharges.3For the present, these computations will be restricted to helium, for the followingreasons: (a) The collision frequency in helium is very nearly constant as a function ofelectron speed.3 This simplifies Eq. 1 considerably. (b) Fairly exact measurementsQPR No. 94(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)exist for the inelastic (excitation and ionization) cross sections of helium.4 (c) Next tohydrogen, helium is the simplest atom known. This means that oscillator strengths havebeen calculated in a number of different approximations,5 so that, if necessary, one mayguess appropriate radiative rates a little more confidently. (d) Helium is monatomic,so that there are no low-lying rotational or vibrational molecular states. In principle,however, other gases could also be treated by the present theory.As a first approximation to the solution, we may reduce Eq. 1 to an ordinary differ-ential equation in two steps.Step 1LetF(v,r,t) = e f t v(t)dt h(v, r, t). (2a)Then1 aF 1 ahF at (t)h atOur first approximation is to write1 8FF 8t = v(t); (2b)that is, we assume 8 In h/at to be small.Step 2Extending the same analysis to nonuniformity in space, we write quite generallyh(v, r, t) = f(r) g(v, r, t) (3a)-22 2 g gV2h Vf f Vg gh -+ 2 - + -h f f g gLetVf 1f 2f 2f A 2 (r)Then we make the second approximation:V2h 1h - (3b)h A 2(r)QPR No. 94 132(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)where the terms ignored are of order A- 1 IV In g . It is not obvious that these twoapproximations are valid; their validity must be checked against the solutions that theyyield. But Eq. 1 now becomes (locally and instantaneously) an ordinary differentialequation for g as a function of v. Furthermore, if we choose to normalize g(v) inde-pendently of the electron number density,Y50 g(v) v 2 dv = 14Tr'then the parameters v(t) and A(r) reflect the growth and spatial distribution of the num-ber density itself.We may simplify the appearance of (1) by introducing the following grouped variables:S e2 22 1 M PT m 2 2 6m 2 kT-M6D2 2312v = 3tv2 AD mm mSM+m M0=kf (v) = 1 (v) g(v)nm -LVmx, 1h (v) X,L - L m i( .At the same time, we take advantage offollowing ordinary differential equation1 d [ 2 2 dg + 3 v 22 v v vT dv gv Dthe function g(v) introduced by (2) to write thefor the reduced distribution function:hi (v)+ h(v) g + g - kfnm(v4 r IFor convenience, we normalizeYOf0 nm ~0 2 1g(v) v dv = -.For the sake of comparison with the theory of high-frequency microwave discharges, weQPR No. 94 133(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)note that v T corresponds roughly to the concept of an effective electric field, while vDis proportional to the "dimensionless" variable PA. 3The parameters k and 0 are still unknown. To solve for these self-consistently,we may impose further conditions on the parameters of (4) on physical grounds. Multi-plying by v2 and integrating, we obtainoo 2Tv dv 24w V 2 3 =(h) + Kh) + 0 - k + , (5)o vDwhere for an arbitrary function s(v)( s) = 4Y s(v) g(v) v 2 dv.0Now k is the (reduced) rate at which slow electrons appear following high-speed inelasticcollisions:k = (h > + 2(h' (6)Furthermore, the growth rate 0 should represent the difference between the ionizationrate and the diffusion loss rate:20 = (h 2 (7)vDFinally, we expect g to vanish rapidly at oo. All this is consistent with (5) if and onlyif2 dg + v3 = o. (8)Now, unless fnm(v) is singular at v = 0, the solution to Eq. 4 is either analytic at theorigin or has a simple pole there. But a simple pole at the origin violates condition (8),so we conclude thatg(v-0) = const. (9)It turns out that Eqs. 4, 6, and 7 yield a unique solution for any pair of valuesof (VT,VD); condition (9) is but an additional check on the correctness of the inte-gration procedure.QPR No. 94 134(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)Method of Numerical IntegrationFor a given set of values (vT, vD, k, 0), we may integrate (4) numerically, firstrewriting it as follows:2 2 dgdg I T dg v 3 - 2 g k2 2 v dv 2 2 v 2 nmdv v vT vD Torg" + A(v) g' + B(v) g = C(v).We divide v-space into discrete intervals Av = h, and label these intervals with anindex n:g+A g' + Bg =Cn nn nn nWe combine this equation with two integrations by the trapezoidal rulehgn gn- 1 2 gn gn- 1hgn = gn- 1 + -2 (gn+gn- 1 )and solve these three, step by step, as simultaneous equations in g , gn, and g."The values of k and 0 are obtained by iteration. Values of these two parametersare guessed initially, but with each integration of g, new values of k and E are computedfrom (6) and (7). Thus far, this iteration has in every case converged; moreover, thefinal form of g always satisfies condition (9).Preliminary ResultsThe computer program turns out values for 0, k, (hX+h i ), (hi ), and (v 2 ),as well as values of the distribution function and its derivatives at selected intervals.Variations of interval size, of the maximum velocity serving as cutoff for the inte-gration (in lieu of infinity), and of the normalization function f (v), indicate that theparameters obtained are accurate within 1%.The variation of E = (h ) with vT for infinite vD is shown in Fig. X-15, as is(hX+h l ). The corresponding variation of (v2 ) is shown in Fig. X-16.[I.1 i 22Figure X-17 shows the variation of (h ), k, and (h = v D with vD for fourvalues of vT. The point at which (h ) and (v 2 )/v D2 are equal is the value of vD for whichLwhichQPR No. 94 135)(h )0 2 3 4 5 6 70 I 2 3 4 5 6( VSDLOG,, o--sec)i (v2)(VD>e ~K=-Zz r ' 450 sec0 100 200 300 400 500t(psec)Fig. X-23. Decay of the energy density as determined from the plasmaBremsstrahlung measurements. The arrows indicate therange of the most probable time for the occurrence of theinstability.instability the hot-electron distribution is becoming highly peaked in energy in the range30-35 keV. This indicates that a nonequilibrium condition may exist but is not conclu-sive, since an instability theory requires the more detailed information contained in avelocity distribution. We are continuing the interpretation of these results in light ofprevious measurements in order to see if a consistent velocity distribution may beinferred.C. E. SpeckReferences1. C. E. Speck, "Experimental Study of Enhanced Cyclotron Radiation from an Electron-Cyclotron Resonance Discharge," Quarterly Progress Report No. 89, Research Lab-oratory of Electronics, M. I. T. , April 15, 1968, pp. 177-182.2. C. E. Speck and A. Bers, "Experimental Study of Enhanced Cyclotron Radiationfrom an Electron-Cyclotron Resonance Discharge," Quarterly Progress ReportNo. 90, Research Laboratory of Electronics, M. I. T. , July 15, 1968, pp. 135-144.3. T. J. Fessenden, "Pulsed Electron-Cyclotron Resonance Discharge Experiment,"Technical Report No. 442, Research Laboratory of Electronics, M. I. T. , Cam-bridge, Mass., March 15, 1966, pp. 70-72.4. D. W. Aitken, B. L. Beron, G. Yenicay, and H. R. Zulliger, "The FluorescentResponse of NaI(Tl), Cs I(T1), CsI(Na) and CaF2(Eu) to X-rays and Low EnergyGamma Rays," IEEE Trans., Vol. NS-14, No. 1, pp. 468-477, February 1967.5. C. E. Speck and A. Bers, op. cit., p. 143.QPR No. 94 145(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)8. INSTABILITIES DRIVEN BY HOT ELECTRONS IN AMAGNETIC MIRRORWe have continued our investigation of the possibility of negative dissipation effectsin a mirror-confined electron distribution in the presence of a high-frequency electricfield. Our previously reported calculations 1 were based on the approximation that theelectrons did not penetrate too deeply into the magnetic mirror. In this report we pre-sent additional calculations that remove this approximation.Subject to the experimental observations discussed in our previous report, we chooseto calculate the power absorbed (or emitted) by a distribution of electrons confinedby a mirror magnetic field with parabolic axial dependence, B(z, r = 0) = B (1 +z /L2),when they are acted upon by an RF electric field. We take the RF field to have an ejwtdependence (w = c + jwi, Wi << r ) and to be circularly polarized transverse to the mag-netic field. No variation of the electric field along z is assumed, and it is taken to beof infinite wavelength.2 (In our previous report we took the field to be linearly polarized,but have found that in the decomposition of this field into two oppositely rotating cir-cularly polarized fields only the component rotating with the electrons entered into theinteraction.) Taking the midplane (Z = 0) electron distribution in the absence of the elec-tric field to be of the formf = noFi(v2 ) F 1 (V 2 1 0 ), (1)where n0 is the electron density, we integrate the linearized Vlasov equation along thezero-order electron orbits predicted by adiabatic theoryz(t') = z cos [wm(t'-t)+4] (2)v (t') = o e 1 + z 2 /L e (t") dt'(3)= vx(t) + jv y(t),where z 0 = Lv 110/Vo is the maximum penetration depth into the mirror of an electronwhose midplane parallel and perpendicular velocity magnitudes are v 1 1 0 and viorespectively, = Vlo/L is the mirror frequency of the electron, w (t) = W (1+z 2 (t)/L 2 )is the instantaneous cyclotron frequency of the electron, is the phase of the electronin its longitudinal motion at t' = t and 9' is the phase of the transverse velocity att' = t. This integration yields the first-order distribution function for the electrons,fl(z' v110, v o' t), which is then used to calculate the first-order current density in theplasma. The time average power absorbed per unit transverse area by the electronscan then be determined from the expressionQPR No. 94 146(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)P = Re (E J) dz,where both El and J± are written in circular polarized variables. The results ofcalculation are(4)the2 22\T CEw io po! JS d(vo)2 5 d(zZ) S 2 3/2( v*2o mb3/(z) v1S o-00dT e j FFj F -FF )b(z +b( z')exp{-j[ c T+A sin 2(w T+) -Asin 2] },whereb(z) = 1 + z /Lz z cosT =t' - tz'= Z(T)wc co(1+z/ZL )co 0A= o z2 /(4 L2co o mpo =e 2 n/(Eom)and the primes on the distribution functions indicate derivatives with respect to theirarguments. In order to perform the integrations on T and 4, we introduce the real func-tions Mk(A' z2/L 2 ) defined byn o0-oon=_001+ coS2 Z(0 e-jA sin 6 Mk e-jn0n(6)which satisfy the relationdO ejnO - jA sin 021+ cosLQPR No. 94Pd = Re -E1Mk Trn Y(7)147F Fi1 bl/(z')(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)These integrations may then be performed to yield3d = -(E12 E 2 Re S d(v o Sd(z2o M3/2r(FF1 -F Fl )M-l/2 + F' M1/2z 1 (8)n 11 n j(W- - n(8)n=-oo j c-2nm) mThus it is clear that the "resonant" electrons are those whose average cyclotron fre-quency plus or minus even multiples of its mirror frequency equals the frequency of theRF field. In the limit of vanishing small negative imaginary part of c the integrationon z 2 is then performed to yield2co ccP d =2 3 -- - CoLIE I 112 dj Fl( ]) Qn ( ) ' (9)co n=-oowhereQ On) -- 13/2 M3/2 M/2 M--I/2 n + 1 7 + 1 F (1 - n3/2 3 1 3/2 -1/2 d 3/2 -1/2+ 3/2 Mn Mn 2 + M /2M / n FII Yn) I 1 (n ) (10)with2 Zn] I/Zn co co Lco2wL zk k co on n 41/2 L n4 1271= Vio.Of particular interest in the application of this theory to the explanation of the high-frequency instability present in our pulsed electron-cyclotron resonance discharge2 isthe nature of the power absorption at a frequency equal to the minimum cyclotron fre-quency along a field line. From Eq. 9, the sign of Pd for a sufficiently narrow perpen-dicular distribution about a particular 1 is determined by the sign of the sum of the QnnQPR No. 94 148(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)evaluated at r. Noting that for w = w none of the positive n values contribute andbecause the parallel distribution should decrease at large arguments, we expect that onlythe n = 0 and first few negative terms in the sum are necessary for the determination ofPd" Note also that to first order in yn the Mk(A, yn) functions defined by (7) are approx-imated by J (A ). We then find for w = wco that the n = 0 term is given by3 1/2F (0)(11)Qo 2 11and for the n = -1 term2 /3/61 2 4 +3/Z 3 1/Zz 43/2Q = J2(1) F 1 3 (1) F1 1 . (12)-1 L2 1 11 LwFrom (11), which in fact is exact, we see that the n = 0 term always predicts positivedissipation at w = wco. It vanishes only if F 1 1(0) vanishes. By combining (11) and (12),an approximate criterion for negative dissipation at w = wco may be derived to yield413/243/2 F11 L F (0)o >1+ 11 (13)F J (1) F1 1 Lw 1 11 LwIt can be shown that a Maxwellian parallel distribution cannot satisfy this criterion and,therefore, must result in positive dissipation at wco. In fact, numerical calculationsof the entire sum in (10) showed strong positive dissipation for w at and above W co andvery small negative dissipation in frequency bands below w co, when a Maxwellian F 1 1was used.The class of distribution functions that most easily satisfy (13) are those withF 1 1 (0) = 0. Thus we are led to consider peaked distributions of the form2 n 2 n2 n v1 1 0 v 1 1 0F11 v 1 0 11 exp- 2 (14)v11TF(1+1/2n) v11T 11T2 2which are observed to approach a delta function at v11 = v lT as n - o0. A detailednumerical study of Eq. 9 was undertaken, using the n = 4 member of the class of dis-tribution functions described by Eq. 14. The M k functions were evaluated by expandingthem in terms of an infinite sum of Bessel functions. Figure X-24 shows a characteristicQPR No. 94 149(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)/244 ( 4 4TIIF, I 250 1 1T2 4r( .125)VT5I \To -1 51-co L 10 -01S5.5 x 100 4v,, 5 x09 O i 3 o1.5,,o=9.5 x 1o9 -69.WCO CONTOURS OFS o -0. 10 -0. s05 0.05 0.10n-ocococoFig. X-24. Fig. X-25.Plot of the gain parameter Q -Z Qn of Contours of constant Q for the parallel dis-parameters in our experiment. Note that a wide region of negative absorption occursin the vicinity of w = w . Several such curves for different vio have been used to con-struct the plot of constant Q contours in v1 0 tribution co space shown in Fig. X-5.A region of negative absorption is observed for large values of vio and w - wco. We thusarrive numerically at the conclusion that for the relatively peaked parallel distributionunder consideration, the perpendicular velocity must also be peaked in order to achievenet negative dissipation (that is, Eq. 9 states that Q(v' ) is weighted by Fi(vlo) inooLodetermining the net dissipation).These calculations illustrate that net dissipation is possible for suitable distributionfunctions. We are continuing the study of this theory with particular interest in itsapplication to the experiment.C. E. Speck, R. J. BriggsReferences1. C. E. Speck and R. J. Briggs, "Instabilities Driven by Hot Electrons in a MagneticMirror," Quarterly Progress Report No. 93, Research Laboratory of Electronics,M.I.T., April 15, 1969, pp. 101-110.2. The results presented here can easily be generalized to include an arbitrarystanding-wave field, Elculationsf(z) illustrate tha. The only change is that in the expression for Mthe left-hand side of Eq. 6 will now be multiplied by f(withcos particular interest in itsn/).3. C. E. Speck, "Experimental Study of Enhanced Cyclotron Radiation from anElectron-Cyclotron Resonance Discharge," Quarterly Progress Report No. 89,Research Laboratory of Electronics, M.I.T., April 15, 1968, pp. 177-180.Research Laboratory of Electronics, M. I.T., April 15, 1968, pp. 177-180.QPR No. 94eI - --150X. PLASMAS AND CONTROLLED NUCLEAR FUSIONB. Diffusion and TurbulenceAcademic and Research StaffProf. T. H. DupreeProf. L. M. LidskyProf. W. M. ManheimerGraduate StudentsK. R. S. Chen H. R. Greenberg M. MurakamiH. Ching P. M. Margosian L. C. PittingerS. I. Friedman G. K. McCormick J. E. RobinsonR. L. McCrory, Jr.1. EXPERIMENTS ON WAVE-PARTICLE INTERACTIONThe construction of the linear nonadiabatic experimental device, which was intendedto provide a sensitive test of nonlinear theories of wave-particle interactions in1-4plasma, has been completed. This report describes a performance test in the deviceand preliminary results of an experiment on the interaction between electrons and wavesproduced by the electron beam itself.The experimental configuration, described in detail in a previous report,1 is shownschematically in Fig. X-26. An electron trap in a uniform magnetic field is producedby two negatively biased trapping electrodes. The pulse system enables us to injectthe electrons into the trap and to extract them for energy analysis after some trappingtime which is variable from 10- sec to 5 X10 sec. The interaction of the trapped elec-trons with the perturbation field located in the trap causes the electron velocity distri-bution to change from a delta function to some other broader one. The shape of thebroadened distribution as a function of the trapping time is the experimental informationthat we seek. The distribution of E 11 (energy parallel to the uniform magnetic field) isdetermined by a retarding potential technique, by using the Faraday cage current modu-lated by small AC voltage superposed on the DC retarding potential to monitor the energydistribution.To test the over-all performance of the device, the first experiment was done withno external perturbation field. The experimental data shown in Fig. X-27 indicate timeevolution of the distribution function for the 1000-eV electrons injected parallel to theuniform magnetic field of 100 G. The transit time of these electrons between the elec-trodes is 170 nsec. The data show that the electrons can be trapped without substantialchanges of their distribution function for more than 100 sec (600 transits). TheThis work was supported by the U. S. Atomic Energy Commission (ContractAT(30-1)-3980).QPR No. 94 151S4.0 m2.3 mECTOR MAIN FIELD INJECTION MAGNETSECTORBASIC PRESSURE 5 x 10 -5 x 107 TorrNOMINAL BEAM ENERGY 1600 eVMAIN MAGNETIC FIELD 100 GFig. X-26. Experimental configurationnonadiabatic experiment.used in the linear500.0250.05.0 -2.51.00.50.0r 600 800 1000 VPARALLEL ENERGYFig. X-27. Time evolution of the energy distribution functionin the weak-beam case.QPR No. 94100.050.0 INJECTION ENERGYMAGNETIC FIELDBEAM CURRENTBASIC PRESSURE1000 V100 G8 pA4x 107Torr152(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)appreciable broadening at 200 psec or later results from collisions with background neu-trals (the estimated total collision time is 200 lisec at 5 X 10- 7 Torr). A bump on eachdistribution function at 520 V corresponds to the electron population influenced by thetransient electric field when they are extracted, hence only the electrons whose parallelenergy is above this bump energy can be allowed to impinge on the energy analyzer.Besides the collisions, the following effects may result in distortion of the distribu-tion function. The first is nonadiabaticity caused by the trapping electrode potentials.The change in magnetic moment 5 is of the order of (rL/L)2 , where rL is the Larmorradius, and L is the characteristic length of the field - in this case, of the axial electricfield of the trapping electrodes. Since this nonadiabaticity limits the confinement time,bigger electrodes have been constructed. The second is instability caused by collectiveeffects, which will be discussed hereafter. In the actual experiments (corkscrew orconstant-pitch perturbation), the injected beam has a certain gyration radius; thus theelectron density is smaller than that in the present case, and hence the instability is lessimportant.Figure X-28 shows time evolution of the distribution function for an intense-beamcurrent, which is fifteen times as large as that in Fig. X-27. We can see that two peaksappear in the 2. 5-pLsec curve, while they merge at 10 1 sec. The resultant distributionfunctions are much broader than those in Fig. X-27. Again the collision effect becomesevident at 250 Lsec. Figure X-29 demonstrates evolution of the distribution function on500. 0250.0100.050.02INJECTION ENERGY 1000 VMAGNETIC FIELD 100 G10. 0 BEAM CURRENT 8 IA-7IBASIC PRESSURE 4 X 10 Torr2.50.50.0600 1000 VPARALLEL ENERGYFig. X-28. Time evolution of the energy distributionfunction in the intense-beam case.QPR No. 94 153(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)4.54.03.5 -3.0 -2.52.01.5 -1.00.50.0 -10.09.59.08.58.07.57.06.56.05.5Fig. X-29.600 800 o1000 VPARALLEL ENERGYTime evolutionfirst 10 1 sec in--,-- .. ...--+ - --+ :ib'i---- CI I'_" 12 ':-PARALLEL ENERGYof the energy distribution function during thethe same case as in Fig. X-28.a faster time scale during the first 10 psec. The distribution function has been broadenedat 0. 5 4sec, and in the 2. 0-i sec curve it splits into two peaks. Thereafter the lowerenergy peak becomes heavier and heavier, while the higher energy peak is decaying, andfinally they merge at ~6. 0 psec. In fact, the total particle energies calculated from thedistribution functions are smaller by approximately 1%0 than the initial particle energy.This fraction of the energy may be counted as the wave energy.An obvious candidate for the instability is two-stream instability because thereflected portion of the electron beam faces the incoming portion. According to thethreshold condition of the two-stream instability, the present beam with an estimatedbeam radius of 1. 5 mm has a minimum unstable wavelength of 60 cm, which correspondsto one-quarter of the trapping length. Therefore this instability presumably exists. Sincethe phase velocity of this instability is nearly equal to zero, we cannot expect such strongresonant wave-particle interactions as those appearing in Fig. X-29.The other possibility is an instability caused by coupling between (longitudinal)plasma waves and (transverse) backward cyclotron waves. The frequency is approxi-mately one-half the cyclotron frequency for the case of equal and opposite streamingvelocity beams. A relevant feature is that the phase velocity is very close to thestreaming velocity; thus resonant wave-particle interaction, if the unstable wave exists,can be expected.QPR No. 94INJECTION ENERGY 1000 VMAGNETIC FIELD 100 GBEAM CURRENT 120 pABASIC PRESSURE 4 x 10- 7Torr""- ::::iiiiii:i""'~"'"' ~""'""'~iiiiii iii," ""'~"~'"' """"' '::::~~ ii'~' ::::-r~~. :::1:::::::I:Iiii t: ::I:i:::::"':::::i" "'" 'u:::iiiliiiiil:ii" ~"'154(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)We observed the collector current signals. The oscillation on the signals has char-acteristically a frequency of ~50 MHz, which is almost independent of the magnetic field.The oscillation lasts a long time, and finally it ceases rather suddenly (for example, at30 sec in the 100-G field). The cessation time is approximately proportional to themagnetic field. Therefore the signal may be an indication of bunching of the charges,because of the two-stream instability, and the dependence of the cessation may beexplained by the focusing effect of the magnetic field to keep it dense enough to satisfythe instability condition.The oscillation of the wave electric field was observed by fixed electrostatic probes.We can distinguish two modes: one is the high-frequency mode, the other is the low-frequency mode. The low-frequency mode corresponds to the oscillation seen in thecollector signal. The high-frequency mode is observed to have a frequency of approxi-mately one-half the cyclotron frequency, as predicted for coupling between the plasmaand cyclotron waves. The detailed characteristics, such as spectrum and wavelength,are under investigation.The next pertinent question is whether the phase velocity of the unstable oscillationis larger or smaller than the stream velocity, with respect to the Landau damping. Thefrequency and wave-number formulas in the uncoupled-mode approximation by Maxumand Trivelpiece6 indicate that difference between the phase velocity and the streamvelocity is approximately one-thirtieth the stream velocity, or in terms of energy, theunstable wave lies at 1000 ± 60 V for the 1000-V stream (the plus sign is for the fastplasma wave, and the minus sign is for the slow plasma wave, coupled with the backwardcyclotron wave). These positions are approximately the positions where the peaks appearon the distribution function. Because of the large-amplitude oscillation, the trapping ofthe particles in the wave potential trough may be important in the wave-particle inter-action process. Further investigation of these points is under way.M. Murakami, L. M. LidskyReferences1. M. Murakami and L. M. Lidsky, Quarterly Progress Report No. 88, Research Lab-oratory of Electronics, M. I. T. , January 15, 1968, p. 220.2. M. Murakami and L. M. Lidsky, Quarterly Progress Report No. 89, Research Lab-oratory of Electronics, M. I. T. , April 15, 1968, p. 183.3. M. Murakami and L. M. Lidsky, Quarterly Progress Report No. 90, Research Lab-oratory of Electronics, M. I. T. , July 15, 1968, p. 145.4. T. H. Dupree, Phys. Fluids 9, 1773 (1966).5. T. G. Northrop, The Adiabatic Motion of Charged Particles (John Wiley and Sons,Inc. , New York, 1963).6. B. J. Maxum and A. W. Trivelpiece, J. Appl. Phys. 36, 481 (1965).QPR No. 94 155X. PLASMAS AND CONTROLLED NUCLEAR FUSIOND. Fusion-Related StudiesAcademic and Research StaffProf. D. J. RoseProf. R. J. BriggsGraduate StudentsG. L. Flint, Jr.Y. Y. LauA. E. Wright1. FEASIBILITY OF PULSED FUSION DEVICESRibe has proposed I a pulsed fusion device with a 10-cm vacuum wall radius R anda maximum B field of 200 kG. The vacuum wall is the magnetic-field-producing coiland withstands the stress thereby produced. The plasma has a temperature of 15 keVand a radius of half the wall radius. The burning time TT is three-eighths the fieldperiod T, and a cooling mechanism operates between pulses.In the present study we investigate the tritium-breeding, hoop-stress, wall-heating,thermal-stress, and cooling requirements in devices of this sort, using coils of copper-zirconium, niobium-zirconium, or TZM. The coils vary in size from 10 cm to 60 cm.Tritium BreedingWith an infinite blanket of 98% lithium and 2% niobium, the tritium blanket breedingis acceptable for TZM coils 7 cm or less in thickness. The TZM coils breed better thancomparable niobium-zirconium coils or copper-zirconium strengthened with TZMbacking. Copper-zirconium is inferior to TZM because copper nuclei slow down neu-trons that are then absorbed by the molybdenum nuclei before reacting with the lithium.For the 7. 00-cm TZM coil there is no significant change in the T/n = 1. 35 breeding ratiofor variations in the amount of lithium coolant in the coil for cooling channels occupyingless than 7. 2% of the coil volume. Of the breeding, 90% comes from the L6(n, T) a reac-tion.Hoop StressThe hoop-stress considerations eliminate from consideration any coil made ofcopper-zirconium alone, since it cannot withstand the envisioned fields with a safetyfactor of two. Niobium-zirconium is also unsuitable. It has a low-yield stress at theThis work was supported by the U. S. Atomic Energy Commission (ContractAT(30-1)-3980).QPR No. 94 156(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)envisioned peak temperatures =1400*K. There is no serious problem with TZM coils.A copper-zirconium coil clad with TZM has been designed, but it would not permit tri-tium breeding.Wall HeatingThe wall heating is the most critical consideration. The heating in the coil comesfrom 5 sources: Bremsstrahlung, electrical resistance, neutron collisions, gamma-rayabsorption, and synchrotron radiation. The synchrotron radiation is only 0. 1% that ofBremsstrahlung; so it is neglected. Electrical and gamma heating are peaked towardthe inner wall, and the Bremsstrahlung radiation is deposited in a surface layer, 2 mmthick. Good engineering practice is not to permit operating temperatures to exceed halfthe melting temperature (on an absolute scale). This eliminates copper-zirconium coils(melting point 1353 0 K) from consideration. The wall heating for 200 kG, and fractionalburnup of 0. 09 for copper-zirconium is approximately 1400 0 K; for niobium-zirconium,2000 0 K; for TZM, 1700 0 K. The gamma heating is 26. 2 times the direct neutron heatingat the wall for the 7-cm TZM coil. Because of this, the field must be reduced toapproximately 180 kG. This does not permit reconsideration of copper-zirconium, sincethe initial coil temperature still prevents its use; nor of niobium-zirconium, since thegamma heating for it is approximately the same as for the TZM coil and the heating ismore severe in the niobium-zirconium coil than in the TZM coil.For the TZM coil, the maximum wall temperature isT = 1.11 X 10 19B4R ,' /2+ 7.73 X 10 1 9 B 4 R RTc T c T-9 2+ 4. 08 X 10 B + Tinitial ( K). (1)Here B is the field in gauss, Rc the coil radius in cm, and T the burning time in sec.Since TT c B, it is seen that for large machines (where B2R = constant from stressconsideration) the second term of Eq. 1 (the gamma and neutron contribution) dominatesthe first term (Bremsstrahlung heating) and the third term (electrical heating). This isobserved for the devices of interest (see Fig. X-30), where the wall heating is 1400*K.Of the temperature rise above the ambient value, 79% is due to gamma heating, 8% toelectrical heating, 10% to Bremsstrahlung heating, and 3% to direct neutron scattering.The fields and burning times necessary to achieve this wall heating are also given, asare the number of Bohm times TT/TB needed to achieve enough confinement. The netpower out, P out' is given byPout = 1{EPt+3(E-1)PE . (2)QPR No. 94 157(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)Here PT is theconverting tothermonuclearusable power.power, PE the electrical power, and E the efficiency ofThe factor 1/4 appears as the pulses are spaced1420140 1.0 - 1340 16 T Fig. X-30. The 7-cm TZM coil.600120 - 0.8 14I o _ _ _ _ _100 0. 4 - 12 -200 TT/rB80 0 B0.2 0.4 0.6R (METERS)3 pulse times apart to allow cooling. The factor 3/8 appears because the plasmaburns only during a portion of the pulse. The factor 3 accounts for the electricalloss in the feed slot, as well as in the coil. For radii less than 30 cm the net power outis negative. Thus it seems that a feasible device requires lower fields and larger radiithan those envisioned by Ribe.Thermal StressThe thermal stress attributable to the temperature gradient caused by the Brems-strahlung at the low fields required by wall heating are well below the yield stress of75K psi for TZM near 14000K. The thermal stress is also included in Fig. X-30.Cooling RequirementsThe cooling for the 7-cm TZM coil is easily handled by turbulent flow of liquid lith-ium in channels that are 1 mm in diameter and approximately 13 cm in length. Theinterfacial temperature difference is 139 0 C, and the temperature rise along the channelis 2500C. The channels are spaced so that the heat transfer coefficient is approximately7 X 104 Btu/hr ft 2 0 F in order to guarantee turbulent flow and reasonably long channels.This spacing is less than the thermal diffusion length in the coil, yet the channel volumenever exceeds 5% of the coil volume. The pressure drops needed for this cooling areapproximately 20-30 psi, and the pumping power needed to pump the coolant throughQPR No. 94 158(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)the channels does not exceed 8 kW/cm, which is negligible compared with the outputpower which is of the order of 105 W/cm.The present study indicates that the most feasible pulsed fusion devices will haveTZM coils, fields of 100 kG or less, and radii of 30 cm or more.G. L. Flint, Jr., D. J. RoseReferences1. F. L. Ribe, T. A. Oliphant, Jr., and W. E. Quinn, "Feasibility Study of a PulsedThermonuclear Reactor," Report LA-3294-MS, Los Alamos Scientific Laboratory,1965.QPR No. 94 159(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)2. END LOSS FROM A MAGNETIC MIRROR CAUSED BYFIELD FLUCTUATIONSConsider the magnetic mirror shown in Fig. X-31, which shows a finite-p plasmain the main part where the field inside the plasma is B 1 (uniform). At the mirrors, thefield is B > Bl . Suppose that B1 increases for some reason while Bo stays constant; forexample, a flute causes radial plasma loss which decreases P, but this has little effectin the region of the mirrors. Each trapped particle has its perpendicular velocity viincreased, hence has improved confinement with respect to the unperturbed loss cone.On the other hand, the loss cone is enlarged. Here is a question: Are particles sweptinto the loss cone, hence ejected from the region, or not? The question relates toobserved fluctuations in high-P mirrors, for example, the device ELMO at Oak RidgeNational Laboratory.Assume that a particle spends all but a negligible fraction of its time in the uniformfield region B 1 . For the particle, with velocities v ll parallel to the axis and v± in theperpendicular direction, we define the particle velocity direction p asptan p = vi/VII. (1)But the loss-cone angle 0m ismsin 8m = B 1 /B , (2)and 8 > 0 for trapped particles. Furthermore,p m2m vI2 - B1' (3)where i. is the (invariant) magnetic moment. From these relations, finddO p/dB 1 = sin p cos Op/2B 1 (4)dOm/dB 1 = tan Om/2Bl' (5)whencedO /dO = tan O sec z 0p/tan O . (6)m p m p pNow for the most vulnerable particle at the edge of the loss cone, m = 0 . Thus if B1mincreases, d and some particles are a ways thrown ut of the ystem axially.increases, dOm/dO > 1, and some particles are always thrown out of the system axially.mpQPR No. 94 160(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)----- B---D BoaFig. X-31. Magnetic mirror system.Conversely, decreasing B l does not lead to particle loss by this mechanism.More questions can be asked as a consequence. A first one is: Are enough particlesejected so that the total perpendicular plasma energy actually decreases (recall that theremaining particles are heated)? For a uniform filling of velocity angles out to the losscone, the answer is easy. From Eqs. 4 and 5, the loss cone encroaches into the distri-bution at a ratededO p 2m - dB =(tan m sin2 )/2B 1 , (7)dB dB 1 m mand the rate of change of pressure from all causes isdP P sin 60- -(8)dB B Z2cdB1 osoThe first term in the bracket represents adiabatic heating, and the second representsthe energy particles lost by the net dO/dB mechanism of Eq. 7. From Eq. 8, the totalenergy decreases if6 2sin 0 cos 0m > 1 m (9)2 32 cos Omwhich is true for 0m > 60. 80 (mirror ratio 1:31).Finally, can P 1 decrease enough that a mirror instability will develop? Here wemust haveBB 2I- + P +p (10)2o 1 fter 2o 0 1 beforeif so, the (presumed) equilibrium is upset by a lemon-pip (in Tennessee, watermelon-seed)QPR No. 94(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)effect. Since necessarily also2 2B B1 0or else no mirror would exist when the trap was empty, we see at once that there is noenergy source to drive the instability at low p.in the form+ <- 3x4(1-x)(2+x)At finite p, Eqs. 10 and 11 can be cast(unstable) (12)(13)x < 1 - ,where2x = sin 0mEquations 12 and 13 can be combined in the inequality(x+l)(x+2) < 3x3/2 (unstable),which cannot be satisfied for x < 1.effect.(14)(15)Thus a mirror instability cannot develop from thisD. J. RoseQPR No. 94(11)162(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)3. EFFECT OF HIGH FUEL BURN-UP IN PROPOSEDDEUTERIUM-TRITIUM FUSION REACTORSA computer study of conditions expected inside a deuterium-tritium (D-T) fusionreactor for conditions of low fractional fuel burn-up (fb) is being extended to regionsof high burn-up. Results have been obtained for comparison with a fusion feasibilitystudy in which a different assumption about helium nuclei (a-particle) energy deposition 2was used.In summary, the model uses the following proposed energy balance. D + T ions areTable X-1. Plasma parameters for large fb exponential lifetimedistribution assumption.V. T. fb Q(keV) (keV)C l = 1. 100 64. 7 .0416 .202 3. 66C 2 =0.2 100 74. 0 .0656 . 155 5. 78100 84. 5 .0952 .125 8. 37S1. 100 93. 0 .1256 . 103 11.05V = 0 100 99. 1 .1544 .0855 13. 60e80 60.4 .0564 .132 6. 20J=1.80 66. 1 .0702 .121 7. 7280 77. 6 .1005 .105 11. 0580 93. 1 .1546 .0812 17. 0060 52. 7 .0609 .0961 8. 9560 58. 8 .0751 .0930 11. 0360 70. 8 .1055 .0861 15. 5060 87. 3 .1579 .0737 23. 240 45. 6 .0661 .0673 14. 5540 52. 0 .0805 .0695 17.7040 64. 2 .1100 .0701 24. 240 81. 2 .1605 .0655 35. 320 39. 5 .0721 .0462 31.720 45.9 .0863 .0509 38. 020 58.4 .1163 .0567 51.220 75. 5 .1651 .0565 72. 60 34. 3 .0786 .0317 -0 40.7 .0933 .03680 53. 0 .1227 .04490 70. 0 .1699 .0488QPR No. 94 163(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)injected with energy V i, for which input power is required. Inside the plasma, the ionshave energies in a presumed Maxwellian distribution at temperature T.. They are1confined with a mean lifetime T. and carry off power as they leave. Similar things canhappen to electrons, for which the symbols V e , T e , and Te apply. Also, electrons canlose energy via Bremsstrahlung, multiplied by an adjustable dimensionless constant C1 ,and modified synchrotron radiation, multiplied by a similar constant C 2 to account forreflectivity of the surrounding vacuum walls. Electrons and ions interchange energy byCoulomb interactions. Inside the plasma, D + T ions fuse and two-tenths of the fusionpower appears with the energetic a particles formed in the nuclear reaction. Thea density is small, but the a energy is high and hence the a pressure is substantial.The a cool in the plasma by dynamical friction on both electrons and ions; their energydistribution is not Maxwellian. By assumption, for this study, the lifetimes of the alphaparticles are exponentially distributed, with a time constant set equal to the ion lifetime.Table X-l shows some plasma conditions calculated for typical open-ended systems.In every case, electrons are injected with zero energy (Ve=0), the ion and electronconfinement times are the same, (J=Ti/T e=l), and an intermediate assumption is madeabout the escape of synchrotron radiation from the optically thick plasma (C 2 =0. 2, D=l,according to a previous report2). From the computer calculations, we find two quan-tities of interest for reactor feasibility, p and Q. is the probability of ion loss per900 effective Coulomb scatter, and Q is the ratio of fusion power to injection power.The results shown in Table X-1 are plotted in Fig. X-32, as contours of constant100 I80 V =100 6o 60, 80 Fig. X-32. Contours of constant V., -P, Q60 - 7 1< 40, on an fb-Ti scale.50 -20 1.0 bLi=0 / 1 C 2_0,240 - 20 / Vi= D = 1.0z 0 / Ve Oo 4, 0.05=50 J = :1.030 I0.03 0.05 0.1 0.2FRACTIONAL BURN-UP fbV i, , and Q on an fb-Ti scale. This graph shows that it will be more difficult tooperate in a region of high Q. For very optimistic confinement, 4 = 0. 1, the best Qwould be a bit more than 10, for an fb around 0.10. These results are more pessimistic2than ose in e fe ibility study, as can be seen by comparing contours of Q inthan those in the feasibility study, as can be seen by comparing contours of Q inQPR No. 94 164(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)Fig. X-33. The contours of Q are those from Fig. X-32, containing the assumption ofexponential distribution of a lifetimes. The contours of Q are for the same inputvalues, except for the assumption that the a completely thermalize, which were calcu-lated in the previous feasibility study. For a given fb' Q is lower than Q . Even thoughconfinement time is long at high burn-up, and the a deposit much of their energy, notall is deposited, some a escape early, and Q is deleteriously affected.100o 80- -7 0' =50 Fig. X-33.o 60, / Values of Q for exponential lifetime distri-o -, c=I 0 bution assumption compared with Q for0 10 / C2 =02 complete a-particle thermalization.40 - // = 10z Q=1 yQ50 Ve= Oo 020 Q=20 J I.O300.03 0.05 0.1 0.2FRACTIONAL BURN-UP fbThese two assumptions, fixed lifetime Ta for each a particle, or an exponentialdistribution with mean life T , in our view enclose what would be the true situation in afusion plasma. This begs the question of whether other effects not included in thisplasma model (such as non-Maxwellian electron velocity distribution) will alter theenergy balance still further.F. B. Marcus, D. J. RoseReferences1. F. B. Marcus and D. J. Rose, "Effect of Low Fuel Burn-up in Proposed Deuterium-Tritium Fusion Reactors," Quarterly Progress Report No. 93, Research Laboratoryof Electronics, M. I.T., April 15, 1969, pp. 122-130.2. D. J. Rose, "On the Feasibility of Power by Nuclear Fusion," ORNL-TM-2204,Oak Ridge National Laboratory, May 1968.QPR No. 94 165(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)4. SYNCHROTRON RADIATION FROM A FUSION PLASMAENCLOSED BY REFLECTING WALLSWe want to determine the synchrotron radiation intensity from a plasma enclosed byreflecting metal walls for proposed operating ranges of fusion reactors. First, thereflectivity of the surrounding walls is found. The electrical resistivity of possiblemetals for the wall is obtained from formulas and tables. The reflectivity at a givenfrequency is calculated for resistivities at different temperatures. With these we canfind an effective reflectivity F for an open-ended system. F is lowered primarilybecause of physical holes through which radiation can escape; variations in metal reflec-tivity caused by temperature and radiation wavelength will have little effect on theresults.In order to calculate the radiation from a plasma with a previously developed model,it is necessary to derive a synchrotron radiation coefficient C 2 , in the symbolism usedpreviously, which is the ratio of actual power radiated to the amount of energy thatwould be radiated if there were no reflectors present in the system. Since the area undera curve of spectral intensity vs frequency represents total radiated power, the ratio ofthe areas of power curves with reflectors to power curves without reflectors gives C 2.It is a very complex computational task to calculate the power radiated as a function offrequency at a given electron temperature T e. Partial results have been obtained forT = 25 keV and Te = 50 keV. The value of C 2 will be calculated for these two cases,and we shall show that it is approximately the same for both. We can therefore assumethat C 2 is roughly a constant and use it for intermediate values of T . A calculation iseperformed on the plasma model to find actual radiation level.Expressions 2 , 3 for electrical resistivity (ir) for Mo and Nb at different wall tempera-tures are used to calculate Table X-2. The surface reflectivity F , which is the fractionTable X-2. Resistivity and reflectivity coefficients.Electrical ReflectivityMaterial Temperature Resistivity -q F(oK) (- cm)Mo 800 18. 6 X 10 - 6 . 9848Mo 1100 26.7 X 10- . 9818-6Mo 1400 35. 1 X 106 . 9791Nb 800 36.7 X 10-6 9786-6Nb 1100 47.0X 10 . 9759Nb 1400 54. 6 X 106 . 9740QPR No. 94 166(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)of incident power reflected from a metal wall, is given in MKS units by 4WEorl= 1- 4 , co << 1 (1)For Table X-2, let w be the twentieth harmonic of electron synchrotron frequency obfor a magnetic field Bo of 5 T (experience shows that the power spectrum peaks at co20 wb). Note that a representative value is F = 0. 98, and that variations will be smallover different temperatures and frequencies. Next, an effective F may be found foran open-ended system. Assume a configuration of a closed cylindrical surface with ahole cut out at each end. Typically, the end holes with zero reflectivity take up 8% ofthe surface area; the rest would have F = 0. 98. We thereby obtain an effective F = 0. 9to be used in radiation calculations.The basic synchrotron radiation equation is5S cc SB[1 -exp(-aL)], (2)where S is the intensity of radiation at frequency w, SB is the spectral intensity ofblack-body radiation at temperature Te and frequency c. The quantity a is the energyabsorption coefficient for a ray propagating in the plasma, and L is a characteristiclength of the plasma. At the frequencies of interest, the Rayleigh-Jeans approximationfor SB is valid. Thus for fixed T e , for dimensionless frequency w/wb' we obtainS cc [1- exp(-aL)], (3)where b is the nonrelativistic synchrotron frequency at Bo . If the plasma is enclosedby reflectors with an effective reflectivity F, then the intensity of radiation SF is givenbySB[1 -exp(-aL)][1-F]S F cc (4)1 - F exp(-aL)Let S50' (aL) 5 0 , and SF50 denote S, (aL), and SF at Te = 50 keV, F = 0. 9, and simi-larly for S25, (aL)25, and S2 at T = 25 keV, F = 0. 9. Values for S50 and SF50 arein relative units, and values for (aL) 5 0 vs w/wb are available and are shown in4Table X-3 and in Figs. X-34 and X-35. These values are for 0 = 1 , wherew2L- PC (5)wbCwith plasma frequency w . This value of Y will be justified here. TrapezoidalPQPR No. 94 167Plasma synchrotron radiation with and without reflectors.QPR. No. 94b0 050 525Wb (aL) 50 S50 50 X (aL)Z5 SZ5 SrZ51 Large 1. 5 0. 1. 0 Large 1. 5 0. 22 Large 4. 0 0. 5 1. 0 Large 4. 0 0.43 Large 7. 0 0.8 1. 0 Large 7. 0 0. 74 Large 13. 0 1. 0 1. 0 Large 13. 0 1.35 Large 19. 0 1.5 0. 725 1. 3 13.8 1.86 14. 7 30. 0 2. 5 0. 710 1. 2 21.3 2.97 5. 6 39. 0 3. 5 0. 513 0. 71 20. 0 3. 68 2. 88 49. 0 5. 0 0. 297 0. 33 14.4 4. 19 1. 37 50. 0 6. 5 0. 157 0. 12 7. 8 3.910 0. 750 41.5 8. 0 0. 118 0. 064 4. 9 3. 311 0.463 34. 0 8. 7 0. 080 0. 052 2. 7 1.912 0. 288 28. 5 9.0 0. 067 0. 017 1.9 1.713 0. 177 22. 0 9. 0 0. 067 0. 001 1.5 1.414 Small 18. 0 9. 0 0. 059 Small 1. 1 1. 115 Small 15. O0 8. 5 0. 058 Small 0. 9 0. 916 Small 11. 5 8.0 0. Small 0. 0.17 Small 8. O0 6. o 0. Small 0. 0.18 Small 5. 0 5. 0 0. Small 0. 0.19 Small 3. 0 3. 0 0. Small 0. 0.20 Small 2. 0 2. 0 0. Small 0. 0.Table X-3.16810-110-20 - 0 110- - %12 141618 2010-6 I I II I 1 1 I0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Frequency w/wbFig. X-34. Dimensionless absorption coefficient aL/Y for cyclotron radiation as afunction of dimensionless frequency w/wb, for a plasma with electron tem-perature Te = 50 keV. (After J. L. Hirshfield, D. E. Baldwin, and S. C. Brown.)Black body No reflection(Rayleigh-Jeans) - 90 reflection- - - 90% reflection12213.4ba - 100 RJ (A= 10keT- -= 50 key. 0.50= k 25 keyS= 1030 I o o--Fig. X-35.Relative cyclotron radiation loss from aplasma at Te = 50 keV for = w2L/wbC =3 410 and 10 with and without 90% reflectingwalls. Short arrows indicate the frequencyw* at which aL = 1.Fig. X-36.Relative cyclotron radiation loss froma plasma slab at T e = 50 keV, 25 keV,10 keV forA = o L/bC= 10 . In thiscase A is .Y. (After B. A. Trubnikovand V. S. Kudryavtsev. 7 )QPR No. 94 169(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)integrations of S 5 0 and SF50 yield96. 5C (T = 50 keV)- 399. 0. 24. (6)2 e 399. 3Now let us find C 2 (Te = 25 keV). Define the ratio X = S 2 5 /S 5 0 . Values for S 2 5 andX are available 7 and are shown in Table X-3 and in Fig. X-36. From Eq. 3,(aL) 2 5 = n - X1 -exp[-(aL)50] (7)Values for (aL) 2 5 in Table X-3 are found from Eq. 7. From Eqs. 2 and 4,S2511-F]S F25 (8)1 - F exp[-(aL)25Values for SF25 in Table X-3 are found from Eq. 8. Trapezoidal integrations of S 2 5and SF 2 5 yield29.1C (T = 25 keV) - 115.1 = 0. 25. (9)2 e 115. 1For Y= 104 and F = 0. 9, C2 has nearly the same value at T e = 25 keV and T e= 50 keV.Therefore, it seems reasonable to adopt C 2 = 0. 25 for intermediate values of T eTable X-3 may be used for calculations for other F.With a previously developed model of a fusion plasma,1 we find a completely self-consistent solution, with C2 = 0. 25. Relevant quantities are the magnetic field at thevacuum wall (Bm), the magnetic field at the plasma surface (Bo), the radius of wallfacing the plasma in a cylindrical geometry (rw), the neutron power deposition per unitwall area (S ), a dimensional parameter characteristic of the system (D), the ion tem-perature (Ti), ion and electron injection voltages (V i and V ), fractional fuel burn-up (fb ) , total synchrotron radiation per fusion event (Uc), with each fusion event giving3500 keV to the plasma. Results have been calculated 8 for T = 40 keV and are shownein Table X-4 for comparison with results from the previous feasibility study. Othersymbols are described there.The thing to note here is the ratio C 2 /D, which is a multiplicative factor in the syn-chrotron radiation term of the model. This ratio is considerably larger in the self-consistent case. The radiation is much greater than was hitherto estimated, as isshown by the term Uc; fb is much higher, which will also justify a model assumptionof complete alpha-particle fusion energy deposition in the plasma. This raises T.and makes energy available for the increased radiation. Note that the answer isQPR No. 94 170(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)Table X-4. Comparison of previous resultsfor an open-ended system.with self-consistent resultsT = 40 keVeC = 1.J = 1.V. = 40 keVV = 0 keVeB =5TB = 10 TmF =3mS107 2S =10 W/m2Input C 2Input DAssumed 2Computer calculated T.prDUaiUczefbUxUcFeasibilityStudy1.10462. 8 keV.2841. 23X10 20/m 31. 24x 104.591954 keV1542 keV. 07280. 3 keV259 keVSelf-consistent.25.7410479. 3 keV.331.31X 10 20/m 31. 24X 104.741954 keV1542 keV.13479. O0 keV1345 keVself-consistent, with calculated D = 0. 74, and 2 is close to 104. Thus we have shownthat C2 may be considered a constant over a range of Te, and that the synchrotron radia-tion is considerably larger than had been assumed in previous calculations.Solutions to the plasma energy-balance equations that are internally self-consistentto this extent are hard to come by, and we have calculated just one. There are manyothers, and our judgment is that more attractive solutions lie at lower fractional burn-up(0. 08?) than was found here (0. 134). Such solutions would naturally have lower Te'hence lower synchrotron radiation loss. Thus the discontentful situation here will besomewhat ameliorated; but it seems that the synchrotron radiation losses in mirrors islikely to exceed those casually implied heretofore.F. B. Marcus, D. J. RoseReferences1. D. J. Rose, "On the Feasibility of Power by Nuclear Fusion," ORNL-TM-2204,Oak Ridge National Laboratory Report, May 1968.2. For Mo, see International Critical Tables.QPR No. 94 171(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)3. For Nb, unpublished information from David J. Rose.4. R. B. Adler, L. J. Chu, and R. M. Fano, Electromagnetic Energy Transmissionand Radiation (John Wiley and Sons, Inc. , New York, 1960), p. 431.5. D. J. Rose and M. Clark, Jr., Plasmas and Controlled Fusion (The M. I. T. Press,Cambridge, Mass., 1961), Chap. 11.6. Ibid. , pp. 248, 249.7. B. A. Trubnikov and V. S. Kudryavtsev, "Plasma Radiation in a Magnetic Field,"in Proc. Second International Conference on Peaceful Uses of Atomic Energy, Geneva,1958, Vol. 31, p. 93.8. F. B. Marcus, "Effects of Fuel Burnup and Synchrotron Radiation in Fusion Reac-tors," S,B. Thesis, M. I. T., 1969.QPR No. 94 172(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)5. STABILITY OF A RELATIVISTIC ELECTRON LAYERWe have continued our study of the stability of a relativistic gyrating electron beamas previously formulated. , 2 In the last report, 2 we presented a dispersion relation forthe beam in vacuum. Now, we shall examine this dispersion relation in some detail,and in the last part of this report we shall consider the case in which background plasmais included. Nyquist stability methods have been used to study these dispersion relations.Stability of the Beam in VacuumThe dispersion relation for a neutralized relativistic electron beam in vacuum,located at radius R with thickness T, can be written 21D1 (, ) + - (b +b_) = 0, (1)where(1 1 2 o + 6 2 T - 6r0 r + 6rDD, = , ) K + K 2 log + K (2)In these equations,-u W (R) .TK= 2 2 2(4)b= , - (3)b z (5)+ jcow 0E 0/o =r2b = jwE (6)1,2and the remaining symbols are the same as those used previously.The map of the lower half of the cut r plane on the D 1 plane is illustrated inFig. X-37. The Nyquist criteria indicate that the beam is stable if the contour l , in1the D -plane (Fig. X-37) does not enclose the point - - (b +b ). Therefore, for all1 3 +lossless, "inductive" structures, b + + b_ < 0, the beam is stable; while for lossy andsome capacitive structures, b+ + b_ > 0, instability is predicted.QPR No. 94 173(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)Im7 I.BRANCHCUTSRe ?7Fig. X-37. Map of the lower half of r~ plane on D 1 plane.It is of some interest to compute the growth rates for capacitive structures, underthe assumption that the beam is "tenuous." For such cases, K is small, and thus as afirst approximation, Eq. 1 becomes- (b +b_) + K_ 1r 0.n + 81 -.The omitted terms involve factors of K 2 and are of higher orderparameter. Equation 7, when solved for r, yieldsin -, the expansion= ++br=- +b+, t- b _2+b ) T (b +b_)Note that the first term under the radical is assumed small. Thus, stability results if> K4RT(b +b_)+2(b+ +b_)QPR No. 94(9)Im D1Re D1174(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)2W Tpwhere - is known as the "loading factor" and is essentially equal to the ratio of0wRthe self-magnetic field to the applied magnetic field. Equation 9 indicates that, even ifthe structure is capacitive, the beam is stable when the wave number f is large enough.As f increases, the assumption that K is small becomes better (K~1/f), and the growthrate, because of the omitted terms, becomes diminishingly small. As an example, con-2 2sider an electron beam parametrized by wp/W = 0. 2 and R/T = 20, located in vacuuminside a (capacitive) structure with dimensions a/R = 2/3 and b/R = 1.4. For this con-figuration, Eq. 9 predicts stability for f > 7. We have also shown that when >i 7 thegrowth rate attributable to the omitted terms (~K 2 ) in Eq. 2 is very small.As a closing remark, we note that the present analysis is consistent with that ofBriggs and Neil 3 as we let T -+ 0. In so doing, only the last term under the radical inEq. 8 is important. Furthermore, the series expansion should not be pushed any fartherbecause we have omitted terms of higher order in rn when we formulated the constitutiverelation for J."Stability of the Beam in a Cold PlasmaWe shall now consider the effects of a cold plasma on a relativistic E-layer of inter-mediate thickness. The inclusion of background plasma will modify the previous formu-lation in the following way: An additional contribution to the total current arising froma background plasma is now introduced into the constitutive relation. Needless to say,the values of the wave admittances b ± at the beam edges are also modified because ofthe presence of plasmas; otherwise, the analysis is identical with the one we havedescribed in some detail previously.The model consists of a metallic cylinder of inner radius a and outer radius b,between which the E-layer, located between rl and r 2 , is immersed in a blanket of cold(collisionless) plasma. We are still concerned with the negative-mass mode whose onlynonvanishing field quantities are Hz, Er, and E0 , and whose frequencies are near kwoThe total perturbation current now isJ = J(beam) + J(plasma)ow _ Wplo EA d o p2 pl odr 2 + 2 2 cr-jWE rL c+ O dr + 2 2 ( - E r-j wE0) 0 rQPR No. 94 175(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)Here,2Cp2 = plasma frequency of the E-layer22 = plasma frequency of the background plasmapleB0= -- ,-=oo .C m OOWith this expression substituted in Maxwell's equations, and after eliminating E andrH , we arrive at the following differential equation:d )(KI+K2) rE 0 _ d r d(rE)dr 2 1 dr 22 Kr 2 2 Kir2C Cd 2(rE ) 2 r(K 1 +K 2 ) rE@SK1 2 2l2r_ Klr2o2 d+ o dr (rE ) + KIE = 0, (10)W - f0 dr100where2p lKL = 1 2 2C2piKand K 2 = op2/7 oThe remaining task is to solve Eq. 10 subject to appropriate boundary conditionsat r1 and r 2 . Note that Eq. 10 has a simple mathematical structure (though the algebrais complicated when we series-expand its solution), in the sense that it has only onesingularity, namely at a point where ( - two = 0, in the region of interest since theQPR No. 94 176(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)point r' = fc/0c-K 1 does not lie between r l and r 2 for the parameters of interest here(K > 1.4 for w p 30 . Therefore, the series solution expanded about the singu-larity converges rapidly enough at the beam edge, in contrast to the vacuum case.We have obtained a series solution for = rE6 about the singularity 0 = w - = 0.2 0For convenience, we set = c/r and 2 a constant across the beam. With the serieso plsolution substituted in the boundary conditions at rl and r 2 , we obtain the following dis-persion relation.I F bl (0+60) cD() (0-8)P a + +IPI Fa + b(z(2-65) + c2(2+62) (02+6)=0, (11)whereK 2y QKZ,M1,2 KKK jr 2, 1 K 2 b + K K 22 r I r 22 -c (In the analysis up to this point we have imposed no restriction on the plasma densitydistribution outside the beam. We have considered a simplified case in which the plasmablanket extends uniformly from the inner to outer walls. Furthermore, if the plasmadensity is high enough (>101 0/cm3 for modes 2< o = 9), the fields outside the beam decayrapidly, and the edge impedance is approximately that of an infinite plasma. Thisassumption simplifies the calculation of b considerably. Under this assumption, weobtained the Nyquist plot of D for various particular values of T/R, (, J and pl Theresult is as follows. For a given value of T/R, and I, there is a critical plasma den-sity (which increases with f at fixed and T/R) above which the Nyquist contour of DQPR No. 94 177(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)does not encircle the origin (see Fig. X-38). In other words, the negative mass insta-bility can be suppressed by a high-density plasma background. A numerical examplewith T/R = 1/20, = 4% is illustrated in Fig. X-38.y=9, ~=4 %, =_R 20iIi-AUNSTABLEi-H-STABLEN 'C-C2 x 1010 4 x 1010Fig. X-38. Critical plasma density as a function of k.Analytically, we can showditions for stability accordingthat the following set of inequalities yields sufficient con-to the general dispersion relation, 11a + bl - a - b Z < 0c 2 - 2b260 > 0c1 + 2b6 > 0al 2 <01 2802(12)and the numerical data agreed with this result. It is of some interest to note, accordingto our data, that in the stable regime, b+ +b_ <0, and in the unstable regime, b++b_ >0,so that the stabilization at higher plasma densities is basically an inductive mediumeffect. Also, the curve in Fig. X-38 has a remarkable resemblance to the result ofBriggs, which was obtained from a model with zero thickness.Y. Y. Lau, R. J. BriggsQPR No. 943 x 1010 5 x 1010 6 x 1010178(X. PLASMAS AND CONTROLLED NUCLEAR FUSION)References1. Y. Y. Lau and R. J. Briggs, Quarterly Progress Report No. 92, Research Labo-ratory of Electronics, M. I. T., January 15, 1969, pp. 266-271.2. Y. Y. Lau and R. J. Briggs, Quarterly Progress Report No. 93, Research Labo-ratory of Electronics, M. I. T., April 15, 1969, pp. 126-130.3. R. J. Briggs and V. K. Neil, "Negative-Mass Instability in a Cylindrical Layer ofRelativistic Electrons," J. Nucl. Energy, Part C, Vol. 9, pp. 207-227, 1967.4. R. J. Briggs, "Effects of Cold Plasma on the Negative-Mass Instability," UCID-15001,University of California, Lawrence Radiation Laboratory, Livermore, California,July 15, 1966.QPR No. 94 179

Landau Damping and Debye Shielding

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Landau Damping and Debye Shielding

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