Contains reports on two research projects.United States Atomic Energy Commission (Contract AT(30-1)-1842

Lampis, G.

Tse, F. Y-F.

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PLASMA DYNAMICSProf. S. C. BrownProf. W. P. AllisM. L. AndrewsD. L. FlanneryE. V. GeorgeVIII. PLASMA PHYSICSAcademic and Research StaffProf. G. BekefiProf. J. C. IngrahamDr. G. LampisGraduate StudentsW. H. Glenn, Jr.P. W. JamesonR. L. KronquistD. T. Llewellyn-JonesJ. J. McCarthyW. J. MulliganG. L. RogoffD. W. SwainF. Y-F. TseA. APPROXIMATE SOLUTION OF THE COLLISIONLESS PLASMA-SHEATH EQUA-TION FOR BEAM-GENERATED PLASMA IN A PLANE GEOMETRYThe collisionless plasma-sheath equation for beam-generated plasma in a planegeometry has the form 1a [S+e(x)]- + + e-x [ /2 S() d,where(M/m)1/2 Va (2)pis a small parameter with typical values between 10-3 and 10-1, and nb/neo is theratio of the beam electron density to the plasma electron at the center. With the bound-ary cdin 1ary condition [21x=-- x=O0= 0, Eq. 1 can be integrated analytically with respect to x to give[s,(x)]- z + H(x) = 2 S [x-1/2 S'() d,40where H (x) = px + 1 - eIntegrating the right-hand side of Eq. 3 by parts and denoting S by Sa for a * 0 yieldsthe plasma-sheath equation in the forma2 [S (x)]-2 + H(x) =For a = 0, and since H(0) = 0, Eq. 4 is an Abel's integral equation with the solution*This work is supported by the United States Atomic Energy Commission (ContractAT (30-1)-1842).QPR No. 81[x-f]- 1 / 2 S a() dg.(VIII. PLASMA PHYSICS)S 1 d xS (x) = d (x-) - 1 / 2 H ( ) dg.By changing a - g - x, Eq. 5 givesS (x) =w(x-) 1 / H' () d .0-xFor H (x) = px + (1-e ), Eq. 6 reduces to the solution derived previously by Harrison2and Thompson (P=O):S (x) = 2 x1/2[+B(x)], (7whereB(x) exp[x 2 ( 2 -1)] das x - 0as x - ooWe found it convenient to use the function B(x) rather than the Dawson function thatHarrison and Thompson used. The function B(x) is related to Dawson function by therelationx1/2B(x) - e D(x /2) (9)For a * 0 we shall separate solution into two regions.Ra(x) - Sa(x) - So(x)For the inner solution we definefor 0 < x x -6,Cwhere xc = 0. 3444 with So(Xc) = 0, and 6 is a positive number such that R (x) << So(x) inthis region.Then the integro-differential equation satisfied by R (x) has the form2 x0Equation 10 also has the form of the Abel's equation with the solution(10)Ra(x) =- T (x-)-1/2 d{ [So()+Ra 2.(X-dSince R () << So() in this region, Eq. 11 can be writtena2 xR (x) ; a_a 2 Tr (12)(x-01/2 S t o I[So( )]3 d .QPR No. 81(11)02x(x-) - 1/ 2 Ra () d .(VIII. PLASMA PHYSICS)Equation 12 thus gives Ra(x) in terms of known functions. Note that Eq. 12 is good onlyfor x < x - 6 because at x = x S' is zero. For the outer solution we definec c oS (x) Sa(xc-6 ) + af (x) for x > x - 6, (13)a a a cwhere S (xc-6) is given by the inner solution. With Ho(x) = fx (x-.) - 1/2 S ( ) d asrequired by Eq. 4, and if we neglect R (x) for the inner region 0 < x - x - 6, Eq. 4becomes1 1-2 c - -1/24 [f ' (x)] - (x ) So-1/+ (x-) [S (x -6)+af (a)] dac-(x- )1/2 So() dx -1/2= (x- )/ [S (x c-)+af a()-So(g)] d . (14)x -6cAssume that af () << S (x c-6) - So () for small a. Then Eq. 14 gives1 ,i x  dgfa() x d(15)c (-1)/2 [Sa(c-6)-So ()] d 1Equation 15 gives fa(x) in terms of known functions. In this particular form, however,it is not very useful for computational purpose and more work is being done along thisline.It should be stated that the assumption af () << Sa(xc-5) - So () is rigorously justi-fiable but is omitted here for the sake of brevity.F. Y-F. TseReferences1. D. A. Dunn and S. A. Self, J. Appl Phys. 35, 113 (1964).2. E. R. Harrison and W. B. Thompson, Proc. Phys. Soc. (London) 72, 2145 (1959).QPR No. 81(VIII. PLASMA PHYSICS)B. LASER BREAKDOWNA laser breakdown experiment in gas is being performed. The plasma is producedby focussing in Helium with a 68-mm focal length lens, a laser beam of 40-MW averagepower, and 50-nsec width, produced by a switched ruby laser (see Fig. VIII-1). TheFig. VIII-1. Typical pulse from the ruby laser.Total energy: 2. 1 joules.VERTICAL : gv/divHORIZONTAL: 20 nsec/divHelium can be pressured up to 10 atm in a stainless-steel box with quartz windows towhich a light dump has been connected for quenching the part of the ruby beam notabsorbed by the plasma.1-7The plasma spark produced in the focus of the lens is cylindrically shaped with theaxis in the direction of the ruby beam. The plasma is being studied by crossing it withPIRANIGAUGEHe-Ne LASERI\ /IFig. VIII-2. Experimental arrangement.QPR No. 81(VIII. PLASMA PHYSICS)the light of a Helium-Neon laser, whose beam is at an angle of 90' with the ruby beam.See Fig. VIII-2.The gas laser light emerging from the spark is detected together with the emissionfrom the plasma by a photomultiplier tube (RCA 6199) in front of which a diaphragm sys-tem and a sharp interferential filter (7 A half-width centered at 6328 A) have been intro-duced. The 547 Textronix oscilloscope is triggered by an ITT FW114A photodiode lookingat the side of the ruby beam and collecting the Rayleigh scattered signal. Figure VIII-3shows the combination of the signals caused by the emission from the plasma and thelight from the gas laser, which has crossed the spark. On the left is shown the levelof the steady-state value of the power PE from the gas laser. When the breakdown hap-pens, the pulse presents a fast rise time z 30 nsec resulting from the strong emissionfrom the plasma. The emission then decays (see Fig. VIII-4) and meanwhile the He-Nelaser light is being absorbed. This causes the pulse amplitude to decrease back to theinitial value Pp and beyond. When the signal equals the initial steady-state value of theHe-Ne laser (before the breakdown) the following equality is verified:-2aRP * e + Pplasma(R, T) = P, (1)where PG is the power of the He-Ne laser which reaches the photomultiplier tube whenthere is no plasma in the geometry of the experiment; a is the absorption coefficient ofthe plasma for the laser frequency; R is the radius of the cylindrically shaped bubble of-aLplasma; Pf e is then the power from the gas laser attenuated by the plasma; P(R, T)is the power emitted by the plasma, reaching the photomultiplier tube in the band of thefilter and in the geometry of the experiment; and T is the temperature of the plasma,which is supposed to be homogeneous.Figure VIII-4 shows the shape of the emission from the plasma in the band of theinterferential filter. This picture was obtained by blinding the gas laser beam whilethe breakdown was being produced. By subtracting from the combination of the emissionand absorption signals (Fig. VIII-3) the signal of the emission alone (Fig. VIII-4), bothbeing detected in the same geometry, the optical thickness, aL, of the plasma is thusdetermined at every time of the plasma's life.This subtraction of data taken during different sparks, is allowed by the fairly goodreproducibility of the emission and absorption. By calculating theoretically, as a func-tion of the radius and of the temperature of the plasma, the value P plasma(T, R) of theemitted power impinging on the photomultiplier tube, and knowing R through measure-ments of the velocity of the radial expulsion of the bubble, the temperature T can be cal-culated at the instant t in which relation (1) is true, as a function of P . P is meas-ured by substituting for the photomultiplier a colorimetric cell (401 spectra physicspower meter), which calibrates the whole geometry. By using different values of theQPR No. 81ZERO LEVELCRITICAL GASLASER POWER PFig. VIII-3.Typical emission curvethe plasma.BREAKDOWN t* = 160 nsecVERTICAL : 20 mV/divHORIZONTAL : 100 nsec/divZERO LEVELFig. VIII-4. Typicalcurve.absorptionVERTICAL : 20 mV/divHORIZONTAL : 100 nsec/divGAS LASERSTEADY-STATELEVELZERO LEVELFig. VIII-5.Signal combination of the powersignals from the plasma andfrom the gas laser.VERTICAL : 10 mV/divHORIZONTALQPR No. 81fromBREAKDOWN100 nsec/div(VIII. PLASMA PHYSICS)power of the gas laser, the instant t can be shifted and a sampling of the temperatureof the plasma can be made. Thus far, samplings between 100 nsec and 1. 2 vsec havebeen obtained. By using very small-diameter diaphragms for limiting the light reachingthe photomultiplier tube, it is possible to eliminate completely the contribution from theplasma emission and allow only the detection of the gas laser beam. In this situation,clear pictures of the absorption of the He-Ne laser have been obtained (Fig. VIII-5),which show that the total absorption takes place in the early stages of the life of theplasma.Measurements of the radial expansion velocity are being made by offsetting by aknown distance the helium-neon beam in a plane normal to the ruby beam and measuringthe delay in the absorption effect.By moving in known amounts the position of the lens, focussing the ruby laser, andmeasuring the delay in the absorption on the helium beam, measurements of the expan-sion velocity along the cylindrical axis of the plasma are being made; previous meas-urements give values of 2. 5 X 105 cm/sec at 11. 5 atm pressure. Theoretical work isunder way on the calculation of Pplasma(R, T).Further experimental work will be related to the study of the dependence of T on thepressure of the gas and on improving the geometry of the detection.When the diameter of the plasma is large with respect to the acceptance of the dia-phragms, a simplication made by averaging on the solid angle through which the beamsfrom the plasma spark are received by the photomultiplier givesP e - aL + B(T) (1-e - aL) = Pf, (2)where is a constant depending on the geometry and on the interference filter charac-teristics, and B(T) is the Planck function.The factor (1-e - a L ) drops out and the quantity B(T) is proportional to the measuredP.B(T) =-PAt 110 nsec after breakdown, a temperature of 4 ev has been calculated in this way.G. LampisReferences1. S. L. Mandel'sh tam et al., Soviet Phys. - JETP 22, No. 1 (January 1966).2. Ya. Zel'dovich and Yu. Raizer, Soviet Phys. - JETP 20, No. 3 (March 1965).3. Yu. Raizer, Soviet Phys. - JETP 21, No. 5 (November 1965).4. G. Akor'yan and M. S. Rabinovich, Soviet Phys. - JETP 21, No. 1 (July 1965).5. D. Ryutov, Soviet Phys. - JETP 20, No. 6 (July 1965).6. R. G. Meyerand, Jr. and A. F. Haught, Phys. Rev. Letters 1, 9 (November 1, 1963).7. R. G. Meyerand, Jr. and A. F. Haught, Phys. Rev. Letters 13, 1 (July 6, 1964).QPR No. 81

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