Plasma Phys. Conuol. Fusion 35 (1993) 929-940. Rinted in the UK

Alternate Fusion: Continuous Inertial Confinement

D. C . Barnes, R. A. Nebel, Leaf Turner, and T. N. Tiouririne

Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

Abstract. We argue that alternate fusion approaches should be pursued if 1) They do not require magnetic confinement superior to tokamaks; 2) Their physics basis may be succinctly stated and experimentally tested; 3) They offer near-term applications to important technical problems; and 4) Their cost to proof-of-principle is low enough to be consistent with budget realities. An approach satisfying all of these criteria is presented, based on continuous inertial confinement. E such an approach, the inertia of a nonequilibrium plasma produces concentrations of plasma density. Recent theoretical developments[l] indicate that fusion gain of order unity or greater may be produced in a system as small as a few mm radius! Confinement is that of a nonneutralized plasma. A pure electron plasma with a radial beam velocity distribution is absolutely confined by an applied Penning trap field. Spherical convergence of the confined electrons forms a deep virtual cathode near T = 0, in which thermonuclear ions are absolutely confined at useful densities. We examine the equilibrium, stability, and classical relaxation of such systems. A sketch of imiaediate and long-term experimental opportunities is given.

1. Introduction

We ask whether the excellent Confinementl21 observed in nonneutral (single species) plasmas confined in Penning type traps[3] might be applied to the controlled release of nuclear fusion energy. The challenge presented is to raise the reactivity (per unit volume) available from such a system to a practical value. Low reactivity is associated with the Brillouin density limit,[4] which constrains the plasma mass energy density (nMc*, n number density, M mass, c speed of light) relative to the magnetic field energy density (Bg/2pO, Bo magnetic induction, permeability of free space). In a recent letter,[5] we have shown how pervasive is this limitation. For a cold plasma in an a r b i t r e vacuum Bo, nMc2 may exceed B,2/2p0 by an arbitrarily large factor only in the presence of strong velocity shear (u /L , -up, U velocity, L, shear length, w, the plasma frequency).

Here, we describe a new approach[l] to .produce dense, station& nonneutral plasmas without significant velocity shear. Based on the observation that the Brillouin limit applies only in the volume averaged sense,[5] we consider enhancing the reactivity by inducing a k g e spatial variation of n over the confinement volume. This concept is related to earlier work on large amplitude compressional nonneutral plasma

0741-3335/93/080929+12$0750 01993 IOP Publishing Ltd 929

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oscillations,[6] which provide an enhancement by inducing large temporal variations of n. The present approach is also closely related to earlier self-collider ion concepts,['l] especially inertial-electrostatic cohement (LEC).[S]

We propose to inject low energy, low Po (canonical angular momentum) particles into a spherical Penning trap. These particles are accelerated radially inward to order 100 keV by the effective spherical parabolic vacuum well produced by the applied electromagnetic field. Convergence near T = 0 produces a dense, inertially coniined core. For fusion applications, a pure ion system might be used, but strong focusing occurs only for species with a common Q/M (Q charge). A D-D system could operate with an applied voltage of several 100 kV.[9] Altematively, a dense core may be formed in an electron machine. If neutrals are supplied to this core, they will be ionized, attain thermonuclear energies, and be electrostatically confined in the virtual cathode formed by the core electrons. This approach allows the confinement of a D-T or other mixed ion species plasma.

D C Barnes et a1

We now summarize this proposed approach.

2. Equilibrium and Stability

A well hown feature of nonneutral plasma confinem.ent is the equivalence of the Penning trap electromagnetic field to a parabolic well formed by an immobile, uniform background charge density. This effective background density is the Brillouin density associated with the uniform applied Bo = Bo?, and M . Thus, the motion of nonrelativistic partides in both Bo, and an applied electrostatic quadrupole Eo = -V(QE:/12M)(z2 - r2/2) (Q charge, z axial and T radial coordinates) may be described in a frame rotating with angular frequency Sl= -(Q/2M)Bo. In this frame,

where fi = B,Z/2p0McZ is the Brillouin density. A spherical system will then produce an effective potential parabolic well of height Qfia2/6e0 (U system radius, E,, permittivity of free space, mks units used except as noted). In the sequel, our nonrelativistic analysis is carried out in this rotating frame.

We wish to exploit two properties associated with this effective parabolic vacuum well. First, rather than filling this well with a near thermal nonneutral plasma of density n < 5, we suppose that partides are injected at the top of the well, are accelerated toward 1' = 0, and form a focus there. The resulting spherical convergence can raise the effective reactivity well above that corresponding to 5. Assuming that particles are lost from the system before they thermalize, two cold radial beams will form from the injected particles and those reflected near T = 0. Because these two beams must have equal and opposite radial velocities and carry the same current across each radius, n = I/& (V beam velocity,' I a constant) over most of the volume. Further, U is slowly varying except near the inner and outer tuming points, so n F;: n=az/r2 (no the density near a) . Near T = 0, the beam distribution goes over to a nearly thermal plasma with temperature T, and density ns. This l/rZ variation

Alternate fusion: continuous inertial confinement 93 1

of n may be used to estimate the reactivity.

so that the reactivity is increased linearly by the radial convergence a/rc, compared to that of a uniform system with density n,.

distribution considered here, in operation consistent with high fusion gain. This OCCUIS

because particles can be lost only at the top of the well (near the material wails) where their energy is a very small fraction of their average energy. Thus, the energy confhement time, TE, is much greater than the particle confinement time T-, perhaps by four orders of magnitude. This allows for T,, < r,, the collision time, preventing thermalization, while 7~ > r ~ o u r ~ m , the fusion breakeven time.

We obtain Vlasov equilibria by specifying an equilibrium distribution function f , which determines n as a function of @ and r. Then (1) is integrated numerically starting from the regular solution near r = 0. Such a solution for a pure electron plasma is shown in Fig. 1.

Second, the presence of an potential well provides access to the nonmzmdi an

Figure 1. ws. radius; (b) Density us. radius.

Pure electron equilibrium solution: (a) Effective electrastatic potential, *

In this case, parameters at r = a were specified to represent a very cold, low energy electron beam injected over a small fraction of the total surface of the sphere. An equilibrium f (function of the total energy E = M v 2 / 2 + Qm, and the total angular momentum J = Mrwl) is chosen consistent with these boundary conditions. In the example shown, the distribution was chosen, as

if €1 < € < , IJI < J1 , f = { t else

and.'fo, €1, Ez, and Jl are chosen consistent with a directed energy of 50 eV, an injected radial temperature of T, = 5 eV, and an injected perpendicular temperature of Tl = of the surface of the sphere r = a. A VO of 120 kV was applied to a radius of a = 3 mm.

eV. The injected current was 125 mA over 2.5.

932 D C Barnes et a1

There is a large region (about the outer 80% of the radius) over which the injected electrons are accelerated inward. +side this region, the space charge associated with the central electrons slows the be- and forms a deep virtual cathode. The size of the final thermal plasma core is determined nearly directly by the injection TL. Outside of this core region, the density foUows a l/rZ variation, as previously noted, except for a thin boundary layer where the beam again stagnates.

If a source of ions is added to such a system, it is possible ‘to form a partially charge-neutralized plasma inside the region of the virtual cathode. We konsider a neutral gas background such an ion source. This background will supply ions distributed in’ space according to the electron density &d energy, and the neutral density. To obtain a self-consistent equilibrium, it is necessary to find the confined ion distribution function resulting from coilisional relaxation of this source, and couple this solution^ with the solution for Q. A complete treatment of this problem is not attempted here. Rather, we find an approximate solution assuming significant neutral pressure. In this case, there is a strong ion source and signiiicant charge-exchaage cooling from the neutrd background, and a low local ion temperature is expected in the portion of the plisma which is transpasent to neutrals. It appears that there may be better regimes of operation at lower neutral pressure, but their analysis is beyond our present scope. We have shown[l] that, for the parameter range of interest, the neutral background penetrates most of the plasma. However, within a =,all burnout radius, this background disappears. If rcr determined by electron convergence, is smaller than this burnout radius, a thermonuclear ion population may appear within the core where electron heating dominates neutr$ cooling.

The neutral density profile may be shown[l] to be well approximated as a nearly constant density no for r > X and a negligible background for r 5 A, where X = a,vhn,r:/v, (or charge exchange cross section, vh wann ion velocity, average taken over warm ion distribution, n, core depsity within re, and v, characteristic thermal neutral velocity) is the characteristic radius for neutral burnout. For typical parameters, X - lo-%, so that large convergence is required to produce a thermonuclear ion core plasma in the case of significant neutral pressure.

With significant neutral pressure, there is a strong, distributed supply of cold ions. In steady state this source, must be balanced by collisional losses, since the nature of the equilibrium prevents the ion density from exceeding the electron density. These collisional losses will be s&ciently 1arge.only if the local ion temperature is low. In contrast, within the burnout radius, there is essentially no cold ion source, the electron density continues to rise with decreasing radius, and a deep virtual cathode, consistent with a thennonuclearion temperature appears. The ion energy distribution consistent with such a situation is nearly M w e l l i a n for deeply trapped ions Near the loss energy, the ion .distribution drops rapidly with energy, corresponding to a IOW “temperature”. In our assumed limit, the rate of this drop is dominated by an oversupply of ions new the loss energy, rather than upscatter of the thermonuclear core ions.

We model this situation by choosing the ion equilibrium density to correspond to strong cooling over the range of the virtual cathode where r > A, producing a typical edge plasma temperature of (say) 10 eV, and to a thermonuclear core for r 5 A. Using the approximate logarithmic variation of Qj with r, which results from a l / r 2 density variation, we iind that this profile may be produced by assuming that the ion density is some fraction of the electron density, which fraction varies from (say) 0.5 for 7 i_ X to nearly unity for X < T < ro (r,, radius of minimum of @). The Q! resulting from such a charge density profile will vary logarithmically with r with a slope which changes from

Alrernare fusion: continuous inertial Lonfinemenr 933

a sigdicant value for r 5 X to a small slope in the remainder of the virtual cathode. The slope dQ/dlogr is directly proportional to the local ion temperature (neglecting constants, logr - -logn/2 - qGJ2k~Ti).

Figure 2 shows a partially chargeneutralized equilibrium obtained in this manner.

-

'-I+ 8mo I

3- Io, IP I P 104 rm)

Partially chargeneutralized solution: (a) Effective electrostatic potential,

( C )

Figure 2. us. radius; (b) Density us. radius; (c) Perpendicular temperature vs. radius.

In this case, the ion density was chosen as 0.5 the electron density for r < m, and as 1 - lo4 of the electron density for 1 W 6 m< r < 7 . m= ro, the outer limit of the virtual cathode. Outside ro ions are unconfined, and the ion density is accordingly taken to vanish. A smooth transition between the three radial regions was provided by choosing appropriate narrow but smooth cutoff functions. All parameters were as for the pure electron solution previously discussed, except that the injected current was raised to 425 mA to produce a deep virtual cathode with partial charge neutralization from the ion species.

logr, we find a central ion temperature of 7.8 keV, roughly equilibrated uith the central electon temperature (Fig. 2(c)). Using the other consistent parameters, one calculates a D - T fusion power of about 50 mW, with an overall Q (energy gain), estimated to be a few To reach breakeven, we would be required either to provide colder electrons (leading to reduced rc) or to raise the applied voltage to order 300 kV. Such improvements could lead to a power density of 1 W/cm3 in a few watt system.

From the slope of Q va.

934 D C Barnes et a1

To establish additional experimental conditions, we estimate the neutral pressure required to maintain the central ion distribution. Extending previous methods[ll] to account for the strong radial density variation in an ad hoc manner, we estimate the central ion conhement time to be about 0.25 ms. Then using the electron impact ionization cross section, and the density profiles from the equilibrium solution, we 6nd that a pressure of 1.6. lo-' Torr is sufficient to supply the central ion losses. At higher pressures, a cold halo plasma will appear, as in our present solution, and the central temperature will be increased, for the same virtual cathode depth. Based on these considerations, we expect that a neutral pressure of IO-' - Torr is a likely operating condition to produce a core plasma of the type shown in Fig. 2.

Equilibria of this type will be realized if several additional conditions are met. The most obvious of these is stability. Given the cold beam distribution, it seems likely that the one-component plasma two-stream instability might be the most dangerous. As a first estimate, we may apply the idn i te slab stability condition.[l2] In this estimate, stability results.if k,Uo > w,~, where k, is the radial wave number, U0 the (single) beam velocity near r0, the radius where QQ is minimum, and wfl the plasma frequency there. From the l/rZ density,profile, no = E / 3 , for the density at r0. Now estimating k, x 2.5/a, U, G 6,a (GP corresponding to 8) gives the estimate Ic,U0/wfl = 4.3, or the plasma is stable by being 18 times underdense.

To quantify this stability picture, we consider the general stability problem for a cold, two-beam distribution of single species particles moving in a confining potential well. Consistent with our nonrelativistic treatment, we assume that themost dangerous instabilities will be electrostatic in nature. The equilibrium is described by a density and radial velocity associated with the outgoing (+ subscript) and with the incoming (- subscript) beams given by

and

The perturbation quantities are the electrostatic potential v+=-u-=u. , (3)

4 = 4(T)eXP(-iWt)Ylm(O, 4) (t time, p, O , $ spherical coordinates, x,,, spherical harmonic), the perturbed velocities

ii& = exp(-iwt)

and the perturbed density fi* = exp(-iwt)ij*n .

The stability of our underdense system may be determined by constructing a quad- ratic form.[l]

A = l T d T [- 1 1-1 d4 + - l(Z+l) 141z] U2n d r r 2 n

Alternate fusion: continuous inertial 'conjnemeni 935

where T = f77 dr/U is the range of the. radial coordinate r (V = ~ d r / d r ) between the inner r; and outer r, turning points, q d 4 = (d/drZ + d)$.

From (4), it follows that w2 is real. Thus, marginal stability corresponds to w = 0. In normalized variables, the marginal stability condition is

where the normalized quantities are i = r/T, U = T U / a , I: = r /a , 6 = n/ii., and 6; = Q2ETZ/MEa, and differentiation with respect to r is indicated by the primes.

Condition (5) is applied to the previous Vlasov equilibria We compute a stability margin fn by iinding a marginally stable cold beam problem which corresponds to a scaling of the warm equilibrium. We approximate the warm solution by a cold solution whose density agrees with the warm solution at the mid radius ro. The potential is taken as that from the warm solution. These two states agree very well except at the two turning points. Next, the stability margin of the cold solution is determined by scaling its density up until marginal stability occurs. The potential well is held constant for this scaling.

Specifically, a cold beam energy level €0 is defined near the top of the QQ well. Then, a cold beam equilibrium is defined by (2 ) and (3), plus the relations Eo = MU2/2+Q@ and I = fnnor:d2/M(€o - Q%), where f,, is the "safety margin" of stability, measuring how underdense is the warm equilibrium relative to cold marginal stability. Regularity requires that $', $", and $"' all vanish at i = 0 and i = 1. Consistent with these constraints, the trial function $' = [?(l - ?)I3 is chosen. Results for the equilibrium shown in Fig. 1 vary from fn = 85, for I = 0 to fn = 20 for I = 1000. These safety margins are very comparable to the previous margin of 18 estimated from the slab stability condition. The corresponding safety margins for the partially chargeneutralized system shown in Fig. 2 are fn = 16, for I = 0 and fn = 3.5 for I = 1000.

3. Classical Collisions

This stable, nonthermal state is maintained by providing a throughput of particles. In a manner similar to an IEC system, electrons are supplied near the top of the well, and removed there before they thermalize. Recalling that the well exists in a rapidly rotating frame for low Po particles, one finds that particles should be produced near r = 0 and injected through a s m a l l portion of the "polar" region (near the z-axis). Almost all injected orbits are ergodic, because of the central potential hill (virtual cathode for our electron system). In the absence of collisions then, electrons wil l be confined for a number of transits, forming an essentially spherically symmetric system, and be lost when they again encounter the injection area. The mean confinement time will be simply the radial (round trip) transit time divided by the fraction of the spherical surface represented by the injection area. For our example, electron confkement time is r, = 2 . lo-' s. B y comparison, the beam-beam collision time, based on conditions at ro is

Even though collisions are very weak, we wish to maintain extremely cold beam conditions. Temperatures achieved will be determined by a competition between

= 0.56 s. Thus, collisions have little effect.

936

thermalization and replacement of these partially thermalized particles by injected beam particles. Under the conditions assumed (two cold electron beams passing through a warm ion background) there me two important such processes, perpendicular beam heating and isotropization (equipartition). The former results from collisions between the two electron beams and between the beam electrons and the essentially stationary ions. The latter results from intrabeam collisions.

We neglect the dependence of interbeam collisions on beam velocity in favor of the much stronger l / r z density dependence. (Equivalently, the confining potential is approximated as a square well.) We fix the Coulomb logarithm at 20. Further, we approximate the isotropization frequency as that for a nearly isotropic plasma whose temperature is the maximum of T, and Tl. Each of these approximations can be argued to introduce less than a factor of two error into OUT calculations.

D C Barnes et a1

With these assumptions, our model equations are

where VT = 8 . 10-’Zn/[max(T,,TA)]3’2, v l = 8 . (K, relative beam velocity with relative energy WO), v; = 1.3 . 108n/V,”, and Ib is an energy conserving term from isotropization, which we consider subsequently. The find terms of (6) and (7) represent the thermal energy gained (lost) by particle injection (loss) at the boundary with T,,, and TA.,, the injection temperatures. We have chosen to express T and WO here in electron volts. All other units remain mbs. Note that n is the total electron density, while the electron-electron collision frequencies are proportional to the single beam density n/2. Note also that WO is four times the single beam energy.

From (6) and (7) we End that, for small v , T, and rZTL are slowly varying functions which are constant on the radial bounce time scale. We then evaluate the bounce average of the collisional terms. Denoting by s = r2TL and taking the bounce average, steady solutions of (6) and (7) satisfy

and

where SO again refers to the injection conditions, and ( ), denotes the bounce average. The final term of (9) is evaluated by noting that equipartition does not affect

the bounce averaged thermal energy. Since this energy is T = (T,+2s(l /rz) , ) /3 , conservation of energy under equipartition requires

The time averages may be replaced by the radial average. Assuming r,/a < 1,

Alternate fusion: continuous 'inertial confinement 937

Using (10) and (11) and the previous expressions for v l and vi, and including collisions with ions only in the ion region r < r g , (9) may be rewritten as

Finally, we evaluate the right hand side of (8) using the asymptotic behavior for large and small radius(l1, to obtain a single nonlinear equation for the normalized equipartition radius 3

where s1 = ~ o + r , a T , ~ / 2 + 3 . 8 ~ 1 O - ' ~ n ~ r ~ ~ ~ ( l + 4 r o / a ) / W ~ ~ ~ , is the perpendicular adiabat which would be achieved without equipartition, and z is d&ed by ./(.a)' = T,.

In case there is suificient beam heating (e.g. r, large enough) z becomes large, reflecting a significant TL. In this case, the second term on the left of (14). (representing equipartition) is negligible, there is small radial heating (T, - Z o ) and s - SI. A regime of greater interest for accessing strong radial convergence occurs when there is weaker heating (e.g. r, small enough). In this case, the co&cient of the equipartition term becomes large, compared to that of the right-hand side (SI - T ~ , so ratio scales as T,?'~). In this case, there is a cold root (small z) near the zero of the factor containing the logarithm, given approximately by x - er,/a. Physically, if the radial beam temperature is low enough, equipartition will be rapid (high intrabeam collision rate), and perpendicular heating will appear as a rise of T, above the injection condition.

We find this cold solution in the specific c a e of partial charge neutralization displayed previously in Fig. 2. In this case, U = 3 . 10-3m, ro = 7 . lO-*m, no = 2.8 . T,, = 5eV, and $0 = 9 . 10-''eV - mz. We find that the cold root exists at x = 4.6 . ~10-4. The corresponding temperatures are T, = 18eV and Tlo = 3.8. 10-6eV. Because of the numerous approximations involved in (6-7), we interpret this cold solution as indicating the possibility that equipartition may be useful for maintaining a cold beam condition, rather than as a prediction of actual performance. Numerous additional effects which have been neglected must be included in a future, more consistent treatment. One major limitation'of the pres& calculation is the neglect of collisional scattering near the beam turning points, particularly the inner point, where a dense, hot core plasma exists.

Examining (13), we find that there are two distinct 'repimes.

4. Experimental Realization

The required injection parameters may be realized by a small electron source whose extracted beam is accelerated, expanded spherically, then decelerated before entering the injection area. The source radius r, should be small enough that the electromagnetic contribution to Pe be s m a l l compared to MUor,, the "thermal" Po associated with the final focus. For our example, rs < 1.6. IO-' m is required. The source should also be small enough that its emittance is extremely low. If a beam of radius rI is expanded without emittance growth to the injection radius rI = 19. m, Tl = IO-* eV implies r, = 3 . m for a typical source temperature of T, = 0.4 eV. The emittance limit thus provides the most restrictive condition.

938 D C Barnes et a1

The required 425 mA may be supplied from a tungsten needle cathode operated in thermally assisted field emission mode.[l2] Such sources have a typical size of a few 1000 angstroms, so the emittance limit seems to be the beam transport optics. Given the domnant effect of equipartition, an injection temperature an order of magnitude larger appears acceptable. Thus, careful design and alignment of the source optics will be necessary, but appears physically possible.

EIectrons ejected through the injection aperature will be focused near the needle cathode. They should be collected there at low energy. The collector will need to be biased slightly positive relative to the cathode, by a few times the T, of the beam electrons. Thus, it seems that the power required for operating the electron source is of order 425 mA-30 V= 13 W, representing quite a modest paver density, even for such a small system. Based on a neutral pressure of less than Torr, ion losses are comparable nearly equal tothese electron losses. Thus, an overall fusion gain of about

results from a few dozen mW of released fusion power. The background neutral pressure is quite transparent to the energetic beam electrons. There is a beneficial effect from collisions between any thermalized electrons (which represent a parasitic loss by filling up the vacuum well with uninteresting particles) and the background neutrais. Such collisions will remove these thermalized electrons from the trap by the well hown cross-field transport mechanism of neutral dominated Penning traps.[l3] At a neutral pressure of Torr, this loss time is less than 2p s. Under the same condition, the dominant source of thermal electrons is ionization of the background gas. This leads to an estimate of average thermal electron to beam electron density ratio of about 2 %.

A skiking technology issue for a system of the type considered here is the very large applied electric field. This field is much larger than surface fiashover limits over any known dielectric surfam. If vacuum insulated gaps are used, the required field map be produced while avoiding vacuum spark breakdown. A voltage of greater than 100 kV may be supported across a 1 mm gap between well conditioned stainless steel electrodes.[l4] It does not appear that the additional surface loading produced by particle and radiation losses from the plasma will significantly affect this holdoff capability. These loads are hges t where the electrical stress is smallest (near the cylindrical axis) and represent several orders of magnitude less power density than that pnesent at surface microfeatures under conditions leading to breakdown.[l5] Larger applied voltages would present the cumbersome requirement for multiple gaps, since required gap spacing increases much more rapidly than total voltage. Such designs appear possible, but wil l require careful design and development.

Fi- 3 shows a conceptual sketch of a small experimental arrangement implementing these concepts.

5. Summary and Conclusions

We have described a new approach for producing dense, thermonuclear plasmas with striking con6nement properties in a Penning trap. A m c e of cold electrons at the boundary, coupled with a corresponding sink of nearly cold electrons there, leads to a radial beam velocity distribution for the con6ned electrons. Strong spherical convergence forms, a deep virtual cathode near the spherical origin. Ions may be coniined in this cathode and will achieve thermonuclear energies.

Equilibria have been found with properties suitable for fusion applications. Because the mean density is constrained to be below the Brillouin density, electrostatic two-

Alternate fusion: continuous inertial confinement 939

m 3. Conceptual arrangement for experimental test of proposed concept: (a) Owall experimental arrangement; (b) "Ront end" of experimental arrangement. Fhdiu ot hitiDg sphere is 3 nun.

stream stabiity is assured. The stability limit is found from analysis of a quadratic formulation of the dispersion relation to be several times (order 10) the equilibrium density. When classical collisions are considered, it is found that equipartition may be effective in limiting the perpendicular temperature to very low values, consistent with the desired IO' spherical convergence. A cursory power balance analysis shows that a few m W of fusion power might be produced in a pea-sized trap with an energy gain of greater than lO-3 . We conclude that such systems would provide useful laboratory fusion sounes in a very manageable system.

Additional features of the proposed approach have not been considered in detail here, but deserve mention. Synchrotron radiation is low because of the spherical symmetry of the confined charge distribution. We have not considered relativistic effects, which would introduce an order 10% non spherical electron mass variation. Such effects d d clearly need compensation, to achieve the sharp focus desired. The spatial distribution of the electron mass may be easily computed. Electromagnetic effects have not been considered here. We estimate their importance to be of order lO-3. Thus, it seems unlikely that electromagnetic effects wiU qualitatively modify the condusions of the present analysis.

We cl- with a brief comment on the relation of the proposed concept to the generation of fusion power. There appear two possible approaches. First, a massively modular system could be considered. Even if millions of modules were to share a common magnet, electrostatic, vacuum, fuel. blanket, and control system, however, it seem uniikely that the cost per module could be made low enough to be of economic interest, although one cannot rule out that innovative engineering could make such an appmach feasible. A second approach would be to replace the low-order multipole

940

fieIds (magnetic dipole, electrostatic quadrupole) of the spherical Penning trap with a higharder multipole system. Such a system could provide an effective spherical potential to coniine a surface electron plasma a few mm thick in a macroscopic system (a k 30 un). To reach a useful edge plasma densty, one still requires an applied voltage of order 100 kV over a few mm, but the total voltage might be as lav as 100 kV. Spherical convergence of injected electrons would still be required to reach a desired power density of 1 W j d . The economics of such scaling appear to be more favorable than those associated with the massively modular approach. This occm because the volume scales with the cube of the number of multipole elements, rather than h l y with the number of parallel modules. Additionally, the localization of the effective vacuum potential to the surface of the machine allows more e0icient volume utilization. Whatever the feasibility of such long-range developments, we conclude that the physics developed here is SuEciently promising and interesting that a timely experimental investigation of the concepts presented is justified.

Acknowledgments

We are grateful to Profs. Dan Dubin, Tom O'Neil, and Fred Driscoll of the University of California, San Diego who provided numerous comments related to the work presented here. One of us (D. C. B.) gratefully acknowledges useful discussions with Prof. Eerb Berk of the Institute for Fusion Studies. This work was supported by the US. Department of Energy and the Los Alamos National Laboratory.

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D C Barnes et a1

[l] Bgmes D C, TurnerL and Nebel R A 1993 Production and Application of Deme Penning Trap Plasmas (Los Alamos: Los Alamos National Laboratory Fleport LA-UR 93-31) submitted to Phys. Flqids B

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