P/373 USA

Study and Use of Rotating Plasma

By O. A. Anderson,* W. R. Baker,* A. Bratenahl,* H. P. Furth.f J. Ise Jr.,W. B. Kunkel* and J. M. Stone*

This paper is intended as a preliminary report onthe Homopolar device (axial magnetic field with radialelectric field), which is the rotating plasma configura-tion most thoroughly investigated thus far. A studyof this configuration for aerodynamic purposes wasreported in 1950 by H. С Early et al.,1 and a morecomplete report was issued in 1952.2 Some ion sourceshaving the same basic geometry were reported in 1955by L. P. Smith and J. Luce.3 The hydromagneticHomopolar machine was conceived by William R.Baker and Oscar A. Anderson in 1956, as a device forthe containment and heating of a thermonuclearplasma. More recently, there have been significantcontributions to this technique by Gow and Wilcox,4

and by Boyer, Hammel, Longmire, Nagle, Ribe, andRiesenfeld.5

THEORY

The analysis presented in this paper applies princi-pally to the "ideal" Homopolar, that is, the con-figuration free from electrode-sheath drops and otherdisturbing but remediable phenomena. The experi-ments described here have been carried out mostlyunder high-density (pinch-type) conditions whichfavor the creation of a totally rotating plasma. Asmodels of the Homopolar grow larger, it will becomepossible to use the present pinch-type technique ofplasma formation at much lower density, just as inthe case of the large toroidal stabilized pinch.6

The Rotational Equilibrium StateWhenever currents flow in plasmas across magnetic

field lines, as for example in most pinch-type con-figurations, the resultant Lorenz force must bebalanced by static or inertial plasma pressures.Where such pressure is lacking (e.g., in the low-/?stabilized pinch), the current flows along field lines orceases to flow. In the Homopolar, Fig. 1, a transientradial current is drawn while the plasma is acceleratedto its equilibrium rotation. In equilibrium, the drivingelectric field vanishes in the plasma rest frame, and the

* Radiation Laboratory, University of California, Berkeley,California.

t Radiation Laboratory, University of California, Livermore,California.

radial current ceases. The equilibrium velocity ofrotation is

(1)

The second term, a centrifugal effect, depends on e\mand, although small, results in different drift velocitiesfor electrons and ions. As a result, an azimuthalcurrent flows and the associated Lorenz force justbalances the centrifugal pressure of the plasma. If theplasma conductivity is finite, there is also an azi-muthal electric field resulting in a slow outward driftof plasma—a diffusive process impelled, so to speak,by centrifugal pressure.

A sudden application of electric field to the Homo-polar would produce trochoidal ion orbits as in Fig.2(B). In practice, the'applied voltage drops sharplyat the instant of ionization and then builds up againas the plasma rotation develops. The effective electricfield in the plasma in general rises slowly comparedwith an ion Larmor period. The ion orbits thereforeare likely to be of the form shown in Fig. 2(A).Although the direction of drift is the same for particlesof either sign, the mean radial displacements of ionsand electrons are in opposite directions. The effect issimilar to the displacement phenomenon in a dielectric.The associated dielectric constant, K, may be evalu-ated by ascribing the drift energy density of theplasma to the electric field energy density. Neglectingthe second term of Eq. (1), we have

KEZ

8тгwhence

(2)

(3)

where p is the mass density of the plasma.If the plasma in a rotational device may be regarded

as a dielectric, then the machine itself must resemblea capacitor. The Homopolar experiment is essentiallyequivalent to connecting an uncharged capacitor, theHomopolar, to a charged capacitor, the source con-denser bank, through an intermediate resistance andinductance (Fig. 3). In general there results a dampedoscillation with the current and voltage approximately90° out of phase, and equilibrium is reached with thecharge so distributed that the voltages on the twocapacitors are equal (Fig. 4). The current ceases at

155

156 SESSION A-9 P/373 O. A. ANDERSON et al.

equilibrium except for a trickle through the leakageresistance, JRL, representing viscous drag and heating.

The kinetic energy of the rotating plasma is theanalogue of stored energy in a capacitor \CV2.Another constant of the motion, the angular momen-tum Pe, is the analogue of the charge, Q — CV. Thetorque impulse due to the current through the Homo-polar must in the ideal case equal the angular momen-tum imparted to the plasma disc. It is easily shownthat, neglecting losses, we have

Pe =¡Hrdr¡Idt = ФС12тг, (4)

where Ф is total magnetic flux.The capacitor analogy may be carried to its logical

extreme with the addition of a ' 'crowbar" shortingswitch to the circuit of Fig. 3.

If the crowbar switch is closed at a time when thereis voltage across the Homopolar, there must result asurge current through the Homopolar and crowbar(Fig. 5). There can be little doubt that the chargerepresented by the outpouring of the current gives a

373

Figure 1. Homopolar I

measure of the angular momentum stored in the rotat-ing plasma, since the crowbar current is opposite indirection to the charging current. The voltage andcurrent oscillations after crowbarring represent hydro-magnetic oscillations of the plasma similar to themotion of a torsion pendulum.

Plasma Containment by Centrifugal Force

In an ordinary mirror machine, only those ions arecontained that have a sufficiently large ratio ofazimuthal to axial velocity. Other ions are lost throughan "escape cone" in velocity space. Particles arecontinually diffusing into this cone by collisions. Athigher temperatures, diffusion—and hence loss rate—decreases while thermonuclear yield increases onaccount of the behavior of the appropriate crosssections. Much higher operating temperatures aretherefore required in a practical machine in order tocontrol the loss rate. The escape cone is a seriousinconvenience.

If one applies a radial electric field to a mirror

373.2

Figure 2. Particle trajectories in the Homopolar for (A) slowlyapplied field, (B) suddenly applied field

Equations of ¡on orbit: (A) d (log £r)/c/t <̂ co\; (B) d (log Er)dt >co¡, to ~ 12o>dî where ion Larmor frequency ш\ = eHz/2/r?iC and

angular drift velocity œd = Erc¡rHz

machine, there are at first components of E parallel toH, but since in general we have

nkT, (5)

the plasma will readjust itself so that E becomesorthogonal to H. Once this has happened, it turns outthat all first-order effects in E, tending either to con-tain or to expel particles, cancel out. The second-ordereffect in E permits containment of all particles up to acertain kinetic energy. In other words, for sufficientlylarge E the escape cone can be truncated arbitrarilyfar out in velocity space.

This proposition is seen to be plausible when onenotes that the azimuthal drift in the electric fieldproduces a centrifugal force tending to keep particlesfrom approaching the axis of drift. Since passagethrough the mirror involves approach to the machineaxis, there is thus an effective mirror repulsion whichis independent of particle sign and, like the centrifugalforce, increases with the square of E.

The idea of plasma containment in a mirror machineby means of the ' "gravitational" effect was first pro-posed by Baker and Anderson in 1956. A rigorousmathematical analysis was recently undertaken byNagle, Ribe, and Boyer leading to the elegant theoryof Longmire.5 An elementary derivation giving thesame results in the limit of weak mirror ratio, Rm, wasmade by H. P. Furth. It is found5 that the maximumaxial kinetic energy that can be contained is

(6)

IH

U

с„

SOURCE CROWBAR HOMOPOLAR

373 (OPTIONAL)

Figure 3. Equivalent circuit of the Homopolar experiment

STUDY AND USE OF ROTATING PLASMA 157

/5 20 25 30 35 40¿usees.

/5 20 25 30 35 40yusecs.

/0 /5 20 25 3O 35 40

200 JU J3

7X/OAR60N .-fOO/jt / J

.265 10 /5 20 25 30 35 40

yC/secs.

/0_ .- /5 20 25 30 35 40373 ^¿/S£>cs.

Figure 4. Current and voltage response in Homopolar I: upper traces, no magnetic field; lower tracesshow effect of decreasing pressure

in which W\\ and W± are the energies of thermalmotion parallel and perpendicular respectively to H.Note that the enhanced containment, second term ofEq. (6), depends on the magnitude of the rotationalkinetic energy of a particle, and also on Rm. Unfor-tunately, very large values of Rm appear to be of littleuse in achieving a high degree of enhanced contain-ment. It should be said, however, that the analysisleading to Eq. (6) is valid only in the limit where themirror field varies slowly over the particle trajectory(adiabatic limit).

It should also be noted that the containment effectis much smaller for electrons than for ions, given acommon velocity of rotation ve, but the consequencesneed not be serious. The loss of electrons through theescape cone sets up an axial electrostatic potentialequal to the mean energy of escaping electrons. Thispotential will be weak if the electron temperature iskept well below the ion temperature, as it should bein a practical machine. A very serious practical con-sideration is that, if the rotational kinetic energy ofthe ions is merely made equal to the mean ion thermal

energy, then the more energetic ions escape at aboutthe same rate as from an ordinary mirror machine. Itis necessary therefore to set the rotational energyequal to the energy of the most energetic ions onedesires to have participate significantly in the fusionprocess. For example, for Rm ~ 2, E — 4 kv/cm,H = 4 kilogauss, and ion density a few times 1013,one would have containment up to 10 kev. This issatisfactory for an experimental thermonuclearplasma device but not for an economic power-producing machine. For much higher energies, theover-all voltages required in large machines maypresent difficulties.

Plasma HeatingIf a thermonuclear kinetic energy density can be

imparted to a rotating plasma, one may hope in ageneral way that some of this will be converted intodisordered motion. Since the Homopolar, unlike theordinary mirror machine, has no simple ' 'escape cone"in velocity space, there is no objection in principle toparticle collisions. Various diffusive processes of

158 SESSION A-9 P/373 O. A. ANDERSON et al.

thermalization are therefore conceivable. The situa-tion is similar to that in the stabilized pinch. In bothcases a long-lived equilibrium state can readily becreated, and one looks next for processes that willconvert stored nonthermal energy density into thermalenergy density at a suitable rate.

The plasma of the Homopolar, like the plasma ofthe stabilized pinch, is heated in the process of outwarddiffusion. This is just the ordinary Joule heating pro-cess. Since the velocity distribution in the rotatingplasma tends to go as l/r. there is also viscous dragbetween plasma layers. The associated heating effectis comparable to Joule heating. Probably there canalso be some turbulent mixing of plasma layers,resulting in additional thermalization.

While the charging current of the Homopolar isdrawn, there results a pinch-type compression in theaxial direction. It can be shown that this effect isalways sufficient to produce a moderate pinch awayfrom the wall, but insufficient to compress the plasmaappreciably, provided the energy density of the plasmarotation is comparable to that of the axial magneticfield.

Aside from these built-in heating mechanisms, thereare also various artificial means one can employ. Forexample, superimposing a rapidly oscillating com-ponent on the radial electric field can set up a standinghydromagnetic half-wave in the axial direction. Thewave propagation velocity is the Alfvén speed,

vs = cK-i, (7)

and amounts to only 3xlO7 cm/sec for a typicaldielectric constant of К = 10е. The viscous dragbrought into play by the associated velocity gradientsalong field lines is large and should result in a rapidconversion of directed kinetic energy into heatenergy. Experience with deuterium pinch devices hasshown that щго2 X 2 x 1016 cm-1 for optimumperformance, where /0 is the tube radius and n0

~i—i—г

ZSKr/S Кб

"I 1—Г

xr

(A)

1 ' Г"

/30//АЯ6ОА/

J 1 1 1 1 1 I I I i I i Iо io го 30 -to

373 TfMEji/seC

the initial particle density. It is an interesting coinci-dence that an Alfvén half-wave length A/2 at thedeuteron cyclotron frequency is just such as to giveиД2/4 = 2 x 10" cm-1.

As an alternative to impressing an oscillatoryelectric field, one can simply perturb the cylindricalsymmetry of the Homopolar. Once a certain level ofthermal and rotational energy has been established itcan also be enhanced by radial compression as in amirror machine.

EXPERIMENTAL RESULTS ON HOMOPOLAR I

Homopolar I is illustrated in Fig. 1. The dischargespace is bounded by a central electrode 3 in. in dia-meter, an outer electrode 9f in. in diameter, and apair of Vicor glass plates f in. apart that He againstthe pole faces of an electromagnet. The glass serves toseparate the plasma discharge current from the returncurrent that flows over the copper-plated pole faces.The discharge current is supplied by a condenserbank, usually 48 \i&. This source is connected to theouter electrode through 48 coaxial cables like the oneshown in the figure. Deuterium, helium, neon, argon,and iodine vapor have been employed in these hydro-magnetic experiments.

At the time of breakdown, the plasma is pinchedaway from the glass. The radial pinch current, typi-cally of the order of 50 ka, produces a torque as itflows across the axial magnetic field, and the torqueaccelerates the plasma into the rotational state. Thepinch phase is accompanied by a rapid fall of theapplied voltage, which subsequently rises again duringthe acceleration phase. The high voltage then decaysslowly during the rotation state because of losses andinternal heating. The current shows a rise during thepinch and early acceleration phase, but then fallsagain to a very low value as the equilibrium rotationstate is reached.

Under certain circumstances the voltage and currenthave an oscillatory character but in most of the experi-ments the discharge is approximately criticallydamped. The phase of the voltage and current arealways such as to demonstrate the capacitorlikebehavior of the Homopolar. Accordingly, the Homo-polar is represented in Fig. 3 as the capacitor CH, andthe loss mechanism (which includes internal heating)by the shunt resistance Rj,.

Typical oscillatory behavior is shown in Fig. 4.The upper pair of traces, with magnetic field switchedoff, show voltage and current in phase. In this caseCH in Fig. 3 should be replaced by a short circuit.The period of oscillation is Г8 = 2TJ-[(L8+.LH)C8]*.The lower traces, with 13 kilogauss magnetic field showthe ~90° phase shift between voltage and current.As the gas pressure is reduced, the period, which isgiven by

(8)

(В)Figure S. Superposition of a number of crowbar signals

is also reduced, in quantitative agreement with thereduction of the dielectric constant K, Eq. (3), ofCH. It will further be noticed that the equilibrium

STUDY AND USE OF ROTATING PLASMA 159

/JO- 02

SO

О - 0

50

COl/LOMBS

D

í .1 I \ F 1 i 1

* ^ 66// AfiGOM7.5M/5 KG-SOURCE CAPAC/TY 48//f!HOMOPOLAR САЯ 5.7/zf

, 1 , 1 . 1 .

0 2 4- 6 3 to t2 14 16

373 ТШ£ JJSeC

Figure 6. Analysis of data of Fig. 5(A)

voltage, which is proportional to CS/(CS+CH), in-creases with decreasing gas pressure, again in com-plete agreement with Eq. (3).

In order to demonstrate that the Homopolar is, infact, a device that stores energy as â condenser does,к is useful to apply the low-inductance short circuitor "crowbar" already mentioned. Figs. 5(A) and 5(B)show, in an overlay of a number of traces, the generalbehavior of the crowbar signal as a function of timeafter charging. The period of the oscillation is

SP/Л/¿.OOP

/Ousec/c/T? - •

/7.4 X&

373 7/OtFigure 7. Spin loop signal with deuterium; also shown is visible

light outputField 17.4 kilogauss, pressure 400 ¡г Hg; voltage 7 kv

It is quite clear that the Homopolar presentsinteresting possibilities as a practical capacitor. Ratesof change of current of ~5 x 1011 amp/sec, up to peakcurrents of 300 ka, are readily obtainable in Homo-polar I. The low inductance, < 0.001 (jh, and highdielectric constant, > 106, result in a very compactsystem of electrical energy storage. Further considera-tions of the plasma Homopolar as a hydromagneticcapacitor will be published elsewhere.7

Besides showing the comparatively long lifetime ofthe rotational state, the crowbar has provided a usefuldiagnostic technique to determine the dielectric con-stant under variations of pressure and magnetic field.Since the vacuum capacity of the Homopolar is0.75 ¡¿[if, К is given by

С = 0.75 x Ю-12 К = V- = QjV, (9)

where V is voltage at time of crowbar and the integralis taken over the first quarter cycle of the crowbarcurrent. It was possible to vary К between the limitsof 106 to 108 and, over this variation, the ratio of theobserved dielectric constant to that calculated fromEq. (3) falls in the limits

1 < KobsIKth < 2. (10)

Both the nature of the observed limits on the varia-tion of KobsfKth and the behavior of Сн with timeprovide diagnostic information. The lower limit of thevariation is taken to mean that essentially all theinitial gas has become a rotating plasma. The upperlimit can be understood in a qualitative way by com-parison with an idealized case. It has already beenmentioned that when the centrifugal plasma pressurebecomes comparable with the magnetic field energy

Figure 8. Doppler shift of 3888 Â He and some impurity lines.The spectra are for normal and reversed magnetic fields. The

comparison spectrum is He and Hg

density, the plasma leans heavily on the magneticfield lines, bending them outward, increasing themirror ratio. In the idealized case the conductivity istaken to be infinite so that in the absence of diffusionacross magnetic field lines, plasma and field distorttogether. It is easy to show that the capacity is modi-fied as follows:

2 Щг21п)

where H = фН0 is the distended magnetic field,r\y r% are the inner and outer radii of the Homopolar,

160 SESSION A-9 P/373 O. A. ANDERSON et al.

and CHO is the unmodified capacity. If all the magneticfield is crowded against the outer electrode, thenСн = 2.45Сно- This upper limit is compatible withthe experimental values of Коьа/К^- If the finiteplasma conductivity is taken into account, then thecalculation becomes much more complicated. Oneobvious conclusion is that when plasma is actually lostby resistive diffusion out of the system, then Сн willbe diminished. A reduction in Сн after long times hasbeen observed under conditions of large E/H, althoughunder most conditions Сн remains essentially con-stant during the lifetime of the rotational state.Above 10 kv, Homopolar I appears to break downacross the glass insulators.

Figure 6 shows an analysis of the data of Fig. 5(A).Curve A gives the charge Q of Eq. (4) at the variouscrowbar times. Curve D is the true charging currentobtained from A by differentiating the curve. It isseen to be less than the observed input current,Curve B, and starts somewhat later. Evidently excesscurrent flows through the device without increasingangular momentum of the plasma, and thus representscurrent flowing in Rj, of Fig. 3. This current is asso-ciated with viscosity of the initially cold gas and withviscous drag on the walls. From Eq. (1) one obtainsve ~ 3 x 106 cm/sec, with a maximum of 6 x 10е

cm/sec. The corresponding argon energies are 180 evand 660 ev. Сн remained constant at 5.7 ¡d for thetime of observation; in this case i£obs/-Kth = 1.0.

A distinction should be made between long-livedrotational effects and actual containment. The con-tainment of the plasma in the rotating state dependson the notion of trapping by centrifugal force in atrough of magnetic field, but actual containmentmust depend on whether the plasma is in fact leaningon the field, or whether it is leaning on the walls.Evidence that the plasma is indeed leaning on thefield is shown in a typical magnetic probe or spin loopsignal, the third trace of Fig. 7. This signal wasobtained by a probe coil located between the glassplate and the pole piece.

The coil is oriented to detect the enhancement ofthe radial component HT produced by the rotatingplasma. Thé second term in Eq. (1) for the velocityve results in an azimuthal current density

je = /mflrH ~ £2/Я3. (12)This current is the source of Hr, the centrifugallyproduced distortion of the magnetic field. As proofthat the component Hr is produced in the mannerstated, it is observed that Hr changes sign with rever-sal of В but not with reversal of E. It is gratifying tosee that Hr has the same general lifetime as the

rotating state. One concludes that the plasma isleaning on the field Unes for the duration of therotating state.

For final proof that the plasma is in a rotating state,the Doppler shift of spectral lines may be observed.Provision was made for spectrographic observationsalong a line of sight passing within \ in. of the innersurface of the outer electrode. By photographing theHomopolar spectrum with the magnetic field first inone direction, then in the other, one can observe shiftsin the lines due to the Doppler effect. The shifts aresymmetrical about reference lines with no field.Figure 8 shows the shift of 3888 Â He and someimpurity Unes under the conditions 8 kv, 17 kilogauss,600 ¡л. Hg of He gas for normal and reversed directionof the magnetic field. The lower comparison spectrumis for He and Hg. The shift corresponds to a velocityof ~3xlO 6 cm/sec. It is expected that highervelocities would be found near the central electrode.

ACKNOWLEDGEMENTS

We thank Chester M. Van Atta for stimulatingencouragement and Stirling A. Colgate and HerbertF. York for helpful discussions. We are indebted toAllen N. Kaufman, Lloyd Smith, Kenneth Watson,and Ted Northrop for theoretical support; to JohnWier for the designs and Elliot B. Hewitt for earlyexperimental work, and to technicians under ArthurHartwig and Louis Biagi.

REFERENCES

1. H. C. Early and W. D. Dow, Supersonic Wind at LowPressure Produced by Arc in Magnetic Field, Phys. Rev.,79, 186 (1950).

2. H. С Early, H. L. Smith and D. C. Lu, Electric WindPhenomena, Summary Report No. 1, Engineering ResearchInstitute, University of Michigan (November 1952).

3. L. P. Smith and J. Luce, Conferences on ThermonuclearReactions at Princeton, N.J. (June 1954) and UCRLLivermore, California (February 1955).

4. J. D. Gow and J. M. Wilcox, Conference on ThermonuclearReactions, Washington, D.C. (February 1958).

5. K. Boyer, J. E. Hammel, C. L. Longmire, D. E. Nagle,F. L. Ribe and W. B. Riesenfeld, Experimental andTheoretical Discussion of Ixion, a Thermonuclear Device,P/2383, Vol. 31, these Proceedings.

6. P. С Thonemann, R. Carruthers, D. W. Fry, R. S. Pease,G. N. Harding, S. A. Ramsden, E. P. Butt, E. McWhirter,D. J. Lees, A. Dellis, S. Ward and A. Gibson, The Produc-tion of High Temperatures and Nuclear Reactions in a GasDischarge, AERE, Harwell, Nature, 181, 217 (1958).

7. O. A. Anderson, W. R. Baker, A. Bratenahl, H. P. Furth,W. B. Kunkel and J. M. Stone, The Hydromagnetic Capa-citor, to be submitted to the Journal of Applied Physics.