Can inertial ‘electrostatic confinement work beyond the ion-ion collisional time scale? W. M. Nevins Lawrence Livermore National Laboratory, Livermore, California 94550 (Received 20 March 1995; accepted 6 July 1995) Inertial electrostatic confinement (IEC) systems are predicated on a nonequilibrium ion distribution function. Coulomb collisions between ions cause this distribution to relax to a Maxwellian on the ion-ion collisional time scale. The power required to prevent this relaxation and maintain the IEC configuration for times beyond the ion-ion collisional time scale is shown to be greater than the fusion power produced. It is concluded that IEC systems show little promise as a basis for the development of commercial electric power plants. 0 I995 American institute of Physics.
I. INTRODUCTION and magnetic Iields.7 In the PolywellTM configuration,*-” a polyhedral magnetic cusp is used to confine energetic elec- Inertial electrostatic confinement (IEC) is a concept from trons. The space charge of these magnetically confined eiec- the early days of fusion research. Work on magnetic confine- trons then creates a potential well to confine the ions. ment fusion in the Soviet Union was begun by Sakharov and In estimating the importance of collisional effects on an others in response to a suggestion from Lavrent’ev that con- IEC fusion reactor, we will use the parameters in Table I. trolled fusion of deuterium could be achieved in an IEC These parameters generally follow those suggested by device.’ The concept was independently invented in the Bussard’ and KralLto We have adjusted the operating point United States by Famsworth.2 Inertial electrostatic confine- somewhat to take account of our more accurate calculation ment schemes require the formation of a spherical potential of the fusion reactivity (see Sec. III) and to ensure that the well. Low-energy ions are injected at the edge and allowed projected operating point is consistent with the model de- to fall into this potential well. If the ion injection energy is scribed in Sec. II. We assume a deuterium-tritium (DT) low, the ions have a low transverse energy, low angular mo- fueled IEC reactor because the power balance is most favor- mentum, and must pass near the center of the spherical po- able with this fuel, and we find power balance to be the tential well on each transit. The repeated focusing of the ions critical problem. at the center of the well results in peaking of the fuel density In this work a perfectly spherical potential is assumed, and greatly enhances the fusion rate relative to what would thus assuring that the ion focus can be maintained over many be achieved in a uniform plasma of the same volume and ion transit times (about 1 ,zs for the IEC reactor parameters stored energy. This strong ion focusing at the center of the of Table I). Ion-ion collisions act on a substantially longer potential well is the defining feature of IEC schemes. The time scale. The ion focusing that defines IEC systems is as- plasma configuration envisioned by proponents of IEC fu- sociated with a strong anisotropy in the ion distribution func- sion systems is illustrated in Fig. 1. tion. Ion-ion collisions tend to reduce this anisotropy on the Early spherical electrostatic traps2-4 required grids to ion-ion collisional time scale (about 1 s for the IEC reactor produce the confining potential. Calculations of grid cooling parameters of Table I). It is possible to maintain this non- requirement? indicated that this concept would require a equilibrium ion distribution function with sufficient recircu- grid radius greater than 10 m to achieve net energy output, lating power. The object of this paper is to compute the col- leading to an impractical reactor. It was suggested that the lisional relaxation rates and estimate the recirculating power concept could be improved by using a magnetic field to required to maintain an IEC reactor beyond the ion-ion col- shield the grid from the hot plasma; and in the Soviet Union lisional time scale. the concept evolved into an investigation of electrostatically Proponents of IEC systems often assume an ion distribu- plugged cusps (see Ref. 6 for an excellent review of this tion function that is nearly monoenergetic.2’9-” There is not tleld). In this evolution from a purely electrostatic confine- a necessary connection between maintaining the ion focus ment scheme into a .magnetoelectrostatic-confinement (which results from the dependence of the ion distribution scheme it appeared that a key advantage had been lost-the function on angular momentum) and the variation of the ion confining magnetic field lacks spherical symmetry so the distribution function with energy. However, some proponents strong ion focus is lost within a few ion transit times because (see especially Ref. 9) believe this to be a second key feature the angular momentum of the ions is not conserved in the of IEC systems because of the substantial increase in the absence of spherical symmetry. fusion rate coefficient for a monoenergetic distribution rela- Recently, there has been a resurgence of interest in elec- tive to that of a thermal ion distribution (but see Sec. III, trostatic confinement fusion. Two new concepts for forming where it is shown that this increase is not significant). the spherical potential well that do not involve internal grids Our approach in analyzing IEC systems is to develop a have been proposed-the PolywellTM and the Penning trap. simple model that contains the essential features described In a Penning trap a spherical effective potential well is by proponents of IEC systems (electrostatic confinement, formed in a rotating frame by a combination of electrostatic strong ion focusing, and a monoenergetic ion distribution
3804 Phys. Plasmas 2 (IO), October 1995 1070-664X/95/2( 10)/3804/l 6/$6.00 0 1995 American Institute of Physics
Downloaded 03 Jun 2009 to 128.174.163.154. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp frequency evaluated at the volume-averaged density. The ion Dense focus degrades progressively, so that it takes a time of order plasma vii1 to complete the relaxation to an isotropic Maxwellian. core with radius r. Clearly, some intervention is required if the nonthermal ion distribution is to be maintained beyond the ion collisional time scale. In Sec. VI we analyze two schemes proposed by proponents of IEC systems,g71’and conclude ‘that they will not be. effective in maintaining the nonthermal ion distribu- tion function. In Sec. VII we examine schemes for maintain- ing nonthermal ion distributions that rely on controlling the lifetime of ions in the electrostatic trap. We find that these schemes require the recirculating power be greater than the Tenuous bulk Cold fusion power. We conclude that IEC devices show little plasma with plasma promise as a means for generating electric power. However, radius a mantle they may be useful as a means of generating 14 MeV neu- trons for other applications. FIG. 1. The IEC plasma is divided into three regions: a dense plasma core, a bulk plasma where the density falls approximately as l/r*, and a cold plasma mantle where the ions are reflected from the edge of the potential well and the mean ion kinetic energy is low. II. THE MODEL Two constants of the single-particle motion for an ion of species “s” in a spherically symmetric trap are the total en- function); and then to use this model as a basis for the cal- culation of collisional relaxation rates, estimates of the fu- ergy* sion power produced by the systems, and the auxiliary ppwer e=1/2m,u”+q,&); 0) required to maintain the nonequilibrium IEC configuration. A and the square of the particle’s angular momentum, successful IEC device must maintain a high convergence ra- tio, n/r”. We find .this to be a useful ordering parameter, and L2=(m,vXr)“. (2) use it freely to identify leading terms in the collisional relax- We consider weakly collisional systems, in which the ation rates and power balance. collision frequency (v) and fusion rate (FZ~~(crr~)~r) are small In Sec. II we present a model IEC ion distribution func- compared to the transit frequency (We), in the electrostatic tion and show that it reproduces the central features envi- well. At leading order in ulw, , the ion distribution function sioned by proponents of IEC systems. In Sec. III we compute is then a function of the single-particle constants of motion.- the averaged fusion rate coefficient for this distribution and We assume an ion distribution function of the form show that it is not substantially greater than that for a Max- wellian distribution with similar mean energy per particle. In j;=fs(k,L2). (3) Sec. IV we compute the collisional rate of increase in the The particle density at radius r is then given by angular momentum squared (L’) (which determines the rate of decay of the ion focus), and the collisional rate of increase n,(rj=r de dL2 J(ur,u:, in the energy spread of the ion distribution resulting from a( E,L2) f,(d)? collisions in the plasma bulk and core between ions with where the Jacobian between velocity space and (e,Lz). space large relative velocities. In Sec. V we compute the collisional is rates of change in (L2) and energy spread due to collisions in the plasma bulk between ions with small relative velocities. Gw:> T Assuming that the ion distribution is initially strongly fo- rr a(E,L2)’ = mk~r2u, (5) cused and nearly monoenergetic, this analysis indicates that the instantaneous rate of relaxation toward a Maxwellian is and (alru) Vii, where rO is the radius of the ion focus, a is the radius of the bulk plasma, and Vii is the ion-ion collision &.(E,L2)= JN. .= -i)
We consider ion distributions that are nearly monoener- TABLE I. Reference IEC reactor parameters. getic and strongly focused at the center of the sphere (i.e., distributions with low angular momentum). An ion ~&stribu- QUCUltity Symbol Value tion function with these properties is Potential well depth 4% SO.7keV Plasma radius n . Im (7) Core radius 1 cm where C, is a constant (to be evaluated below) and H(x) is a Volume-averaged density 2) 0.5X 10” me3 Peak ion density 3.3X10z3mm3 Heaviside function. ni0 In our calculations we consider a “square-well” poten- Fusion power Pfmion 590 Mw tial,
Phys. Plasmas, Vol. 2, No. IO, October 1995 W. M. Nevins 3805
Downloaded 03 Jun 2009 to 128.174.163.154. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp impurities), and that deuterium and tritium distribution func- tions have the same convergence ratio, so that
A substantial confining potential (f ho) is assumed in order n~(r)=nd(r)+n,(r)=2n,(r)=2n,(r). (15) to ensure that the dominant collisional effect is thermaliza- If the fusion rate coefficient, (ou)nr , were independent of tion of the ion distribution, rather than ion upscatter (in en- radius, then the fusion power would be given by ergy) followed by loss from the potential well. The ion num- ber density corresponding to our model distribution function d3r a:(r) is ~fw.ion=~ YDT(wU)DT n- 2%- n,(r)= -~r&~oYDT(cv)DT de,3,2, s r xC,G(E)H(Lg-L2), (8) where we have used the fact that, for the model distribution r
mdmt m=------(221 III. FUSION POWER GENERATION r md+rnt’ For a deuterium-tritium plasma, the total fusion power while the subscripts “d” and ‘9” label the ion species, deu- is given by terium and tritium. We have evaluated the ,u integral numerically using an
Pfusion=Y~~ d3r 4rh(r)(uu)DT(rl, (14) analytic fit to the center-of-mass fusion cross sections devel- I oped by Bosch and HaleI (which is accurate to within 2% where Yn,= 17.6 MeV is the fusion yield per event. We as- over the relevant energy range), We find significant variation sume an equal mixture of deuterium and tritium (with no in ( ~u)nT with both radius and potential well depth. The rate
3806 Phys. Plasmas, Vol. 2, No. 10, October 1995 W. M. Nevins
Downloaded 03 Jun 2009 to 128.174.163.154. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp (uu>E. has a peak value of 0.89X lo-‘l m3/s at a mean ion energy E= 113 keV (or T, =75 keV). The monoenergetic IEC
0.8 Phys. Plasmas,Vol. 2, No. 10, October 1995 W. M. Nevins 3807 Downloaded 03 Jun 2009 to 128.174.163.154. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp I&0=0, u) 0.0 1.0 2.0 3.0 4.0 5.0 U FIG. 3. Here I, ( ,uO.u) is displayed as a function of test particle speed (u) FIG. 4. Here I, ( /Q .u) is displayed as a function of test particle speed for for b=O.995 (corresponding to r= 1Or,,). The resonance at U= 1 results &=O, corresponding to radial locations in the plasma core, rs r,, . from self-collisions among comoving particles. between comoving particles leads to strong coupling be- to investigate this possibility we evaluate the collisional re- tween the transverse and longitudinal velocity dispersion of laxation rates of the ion distribution function. these bt%unS, as pointed out by Rosenberg and Krall.” We A. Collisional relaxation of ion anisotropy will return to this important effect in Sec. V. In this section we ignore the internal structure of these beams, focusing on Coulomb collisions will result in an increase in (L2> due the rate of increase in velocity dispersion due to collisions to transverse scattering (that is, scattering of the ion velocity, between ions in counter&earning beams. We remove the ef- so that it has a component in the plane perpendicular to G,). fect of collisions between comoving ions from our represen- In Appendix A we show that, in the limit L2-+0 (so that the tation of the collision integral by replacing the collision in- ion velocity is nearly radial over most of its orbit), the col- tegral ZL(po ,u> with I:(,u~ ,u), which has been cutoff at lisional rate of increase in the mean-square transverse veloc- ,~,=0.95, as described in the Appendix. In the bulk region ity for ions of species s and speed [where po= dm- 11, this modified collision inte- 24850 gral is well approximated by USE - (24) J fn, Lim[Zl(h,u)l= i A. is given by I*o-+1 The variation of ZI(,uo,u> with particle speed in the ; (Ad)=247 @‘UJl(&,U,4, plasma core (6 ro) is shown in Fig. 4. At a given radius the collisional rate of increase in L2 is where simply related to the collisional rate of increase in the trans- verse velocity dispersion, (26) dL= =m,2r2 f {AL&. dr (30) following Book,16 we have defined collisions sls,- 4~q,2q~rn,~LnA,,~ The rate of increase in L2 varies over the ion orbit. However, vo = (27) at the ion densities and energies projected for an IEC reactor Pn,‘u; the change in L2 due to colhsions during a single orbit is and small. Hence, we average the collisional change in L2 over the ion orbit to eliminate the rapid time scale associated with (28) ion orbital motion, and obtain the bounce-averaged rate of change in L2: is the cosine of the angle between v and i, at which the model IEC distribution function goes to zero for the given radius. (: L2),,,,iI,si,=~”F ~lc,,isioib ”C31s- ) The variation of the collision integral ZL(po,u) with The radial dependence of the transverse cohision inte- particle speed is shown in Fig. 3 for a typical location in the gral, Il[~(r),u), is shown in Fig. 5. Except for a small plasma bulk, r= 1Or,. At each location in this region region about the plasma core (6 3ro), 1: is well approxi- (ro 3808 Phys. Plasmas, Vol. 2, No. 10, October 1995 W. M. Nevins Downloaded 03 Jun 2009 to 128.174.163.154. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp grades. The actual rate of degradation can be substantially 1.o 1-1 higher. For example, asymmetries in the confining potential ~L~oWro), u) 0.8 - may occur due to the inherent lack of symmetry in the mag- netic fields needed to confine the electrons that generate the potential well,18 asymmetries associated internal or external electrodes, asymmetries associated with the apparatus that injects the ions into the trap, or due to waves and instabilities.lg Even very small asymmetries in the confining potential can substantially increase the rate at which the ion I I distribution function relaxes toward isotropy because they 0.0 2.0 4.0 6.0 8.0 10.0 scatter longitudinal velocity into transverse velocity at a ra- r/r0 dius r-a (so for a fixed hvl we generate the maximum change in L2); and because a “collision” occurs between each ion and the confining potential once each bounce pe- FIG. 5. Radial variation in the transverse collision integral, I&~~(r),u] is displayed versus r/./r0 for a test particle with speed u=l. The structure at riod. Hence, we may estimate the rate of change in L2 due to r/rO=I is associated with the cutoff in I{ at /+=0.95 applied for rlr$l. asymmetries in the confining potential as w 1 -m .--.- $ (L2)aSplXl~trieS (37) (~v~“lnj) is nearly independent of radius. Hence, we may i.40 176 ’ approximate the bounce-average collisional rate increase in where “m” is the mode number and 84 is the magnitude of L” for ions of species s as the asymmetry in the confining potential. For the 50 keV , deuterons in the reference IEC reactor described in Table I, v;‘= l/r,= 1.1 MHz, while ( vt’d)vol-2.23 Hz. Hence, the bounce collisions~m~u,2C n /,,~vs,~ frequency is larger than the collision frequency by a factor of 5' s 7.6X 10”: Clearly, even very small asymmetries in the con- fining potential can lead to relaxation of the anisotropy in the X (32) ion distribution function at a faster rate than Coulomb colli- sions. ~c&c%>w (33) B. Collisional relaxation of the ion energy distribution where It is still important to examine the collisional relaxation of the monoenergetic ion energy distribution function be- c,- (34) cause the rate of thermalization in energy has important im- -? bs+hJlus ’ plications regarding the effect of collisions between comov- the sum on s ’ includes field ions of the opposite species ing ions on the evolution of the ion anisotropy (see Sec. V). streaming both parallel and anti-parallel to the test ions of In the Appendix it is shown that the collisional rate of species s; and L~=~m,u,a)2 is the value of (L’) for an increase in the longitudinal velocity dispersion of ions of isotropic, monoenergetic ion distribution-that is, an ion dis- species “s” is given by tribution that yields a constant ion density throughout the trap. For a DT plasma we find cd=6.5 and c,=5.4. We were motivated to introduced the volume-average f &“‘s=~ 4?‘u:$ ~,,h,,u,r)- scattering rate, The variation of the longitudinal collision integral with speed 4--dfluLn &d (n,,)~, , in the plasma core ( r Phys. Plasmas,Vol. 2, No. 10, October 1995 W. M. Nevins 3809 Downloaded 03 Jun 2009 to 128.174.163.154. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp The bounce-averaged collisional rate of increase in the 1.0 ion energy is 0.8 - I,,(cl*=OAJ) (C)orbit= 2 I,” g $ $ (Au~>~ 0.6 - s s -F !fc J!2- I” d’ n,P(r)r,,[po(r),u,r, n,r Us’ 0 a (40) where FIG. 6. Here I,,(/.Q,,u) is displayed as a function of test particle speed for ,q,=O, corresponding to radial locations in the plasma core, rare, masses (when this condition is satisfied the center-of-mass We have computed the orbit integral numerically for a DT frame is nearly identical to the lab frame for collisions be- plasma with equal deuterium and tritium fractions, finding tween counterstreaming ions in the plasma bulk). The more dp1.73 and d,=1.91. general result of Eq. (39)-that collisions in the bulk plasma At constant ion convergence ratio (a/t-,), the ion energy do not cause appreciable energy diffusion for any choice of distribution would relax to a Maxwellian in a time of order, the relative ion energies-follows from the assumption that the scattering angle is small (which is always the case for the (43) dominant contribution to the Coulomb collision operator), together with the fact that, in the bulk plasma region, the For our reference IEC reactor of Table I this works out to velocities of the colliding particles are nearly colinear. For e-5.4 ms. small-angle collisions the momentum transfer between the colliding particles can be obtained by treating the interaction as a perturbation, and integrating along the unperturbed (i.e., parallel, straight-line) orbits. When the impact parameter is V. COLLISIONS BETWEEN COMOVING IONS finite (as required for small-angle collisions), it is easily seen When mapped to radial locations in the bulk plasma that the momentum transfer (the time integral of the force on ( ro< r FIG. 7. Radial variation in the longitudinal collision integral, I,;[~~(r),u], is displayed for a test particle with speed u= 1. The structure at r/ro=l is (4% associated with the cutoff in 1,; at p‘. =0.95 applied for r/r+ I. 3810 Phys. Plasmas, Vol. 2, No. IO, October 1995 W. M. Nevins Downloaded 03 Jun 2009 to 128.174.163.154. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp is the potential gradient scale length at ~=a. The effect of rates computed in Sec. IV. These rates of increase in the these edge collisions is to transfer energy between the longi- beam longitudinal temperature and (L’) for ions of species s tudinal and transverse degrees of freedom. That is, to couple are the energy spread of the ion beam to the quality of the ion dTf) focus. (50) Edge collisions are omitted from our square-well model dt core coUisions because r+ vanishes for a square well. However, collisions and between comoving ions in the plasma bulk have the same effect-i.e., these collisions couple the longitudinal and d(L2) transverse degrees of freedom, the collisional rate is large = c,Li( z@),, . (51) dt bulk collisions because the relative velocity of the &moving ions is small, and they act over most of the ion orbit, rather than just at the A. Nearly monoenergetic ions, (7js)/qs@o)c$(rfjla2) plasma surface. Collisions between comoving ions change the beam velocity dispersion at a rate We follow Kogan”’ in computing the collisional relax- ation of the beam velocity dispersion between the longitudi- d(Au”) WPO nal and transverse degrees of freedom. In the spirit of the -J- 2 q&0( @ho,. (46) model IEC ion distribution function of Sec. II, we begin by dt comoving ions m@v > considering a beam with a finite ion convergence radius (so Since rJa is expected to be less than unity, we conclude that that (L2)>O) and a nearly monoenergetic ion distribution bulk collisions between comoving ions dominate the edge function, such that Ty)( a) Z=T(is) or, equivalently, Tf’lq, +. collisions emphasized in Ref. 11. These collisions are in- <0.5r$a2. Then Ty)>T(IS) everywhere in the well, and the cluded in our square-well model, and will be examined in local rates of change in Ty) and Tf) due to collisions among detail in the remainder of this section. comoving ions are20 Following Rosenberg and Krall, we resolve the singular- ity associated with the delta function in the ion distribution dT;“) (52) function by modeling the internal structure of the beam-like dt comoving ions ion distribution function in the plasma bulk by assuming that it is a drifting bi-Maxwellian. The longitudinal and trans- and verse temperatures of the beam are chosen to reproduce the dTy) 1 longitudinal and transverse velocity dispersion discussed in a-5 (53) dt Sec. IV. The ion distribution function has a longitudinal ve- comoving ions locity dispersion, It follows that the local rate of change in (L2) due to colli- sions among comoving ions is Tp (Au:)= m,, (47) 3 4L2) +-a--- dt 4 (54) and a transverse velocity dispersion, comoving ipns These rates need to be averaged over that portion of the (L2) 2Ty(r) ion trajectory with rSr, , where W:>(r)= jgq~= m,. (48) s p s------Tr0 20r Note that the transverse velocity dispersion of the beam, c Jiq I. O (Au:), and the transverse temperature, T(,s), are simply re- lated to (L*) and the ion focal radius, r. [see Eq. (49) below]. is the radius at which the cutoff in the longitudinal collisional Hence, we only introduce one new parameter, Tf), to de- integrals introduced in Sec. IV to remove the effect of colli- scribe the internal structure of the counterstreaming ion sions among comoving ions becomes effective (recall that beams. we have taken ~,=0.95). The orbit-averaged rates are The transverse temperature, Ty’, is a strong function of radius. This strong radial variation in Ty’ results from the fact that different groups of ion orbits intersect at each radial location. Hence, we find it convenient to compute the con- (56) tribution of collisions among comoving ions to the transverse and velocity dispersion of the ion “beam” by relating it to d(L2)ldt. We may then compute the local value of T’,“‘(r) and the ion focal radius from the relations (57) TIqr)=T’,S)(a)$ I_q ($ Lg lq Both core collisions and collisions among comoving I r- 2 ’ Or 2m,r . ions lead to an increase in the longitudinal velocity disper- In the absence of collisions between comoving ions, the sion. However, even for a relatively poorly focused ion dis- velocity dispersion would increase at the (bounce-averaged) tribution, rola < G/S c ,-0.21, the decrease in (L2) due to Phys. Plasmas, Vol. 2, No. 10, October 1995 W. M. Nevins 3811 Downloaded 03 Jun 2009 to 128.174.163.154. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp collisions among comoving ions will dominate the increase due to collisions in the bulk plasma and (L*) will decrease. As a result, the ion distribution will rapidly evolve until T’,s)=~,s’ at r=a. B. Moderately thermalized ion distributions, -@#‘)vo,, 631 sva2)~ +%dPo~ k&3 where e-2.718... is the base of the Naperian logarithms. We are led to consider the effects of collisions among comoving ions in the limit that the longitudinal velocity dis- persion is large compared to the transverse velocity disper- sion, T{,*‘> Ty’. In this limit the local rates of change in q,‘) C. Strongly thermaiized ion distributions, and tic’ due to collisions among comoving ions are*’ 7;fs’ks4Jo~ kfw> dTf’ Finally, we consider the regime q: ‘/qs +. 3 i( t-i/r:) =5x lo-*. In this limit the longitudinal beam temperature is dt comoving ions greater than the local value of the transverse beam tempera- ture everywhere in the plasma bulk, and the orbit-averaged (58) rates of change in the beam velocity dispersion are given by dTf ) and - dt comovingi,.,*-$l? ‘n(Y) %lcomo,., io”; ILL ($j I’* 8TLS’ r2 XLn - -7 qs40v;fs. (59) q&0 r. ( 1 and It follows that the local rate of increase in (L*> due to colli- sions among comoving ions is d@*) dt comoving ions Jet 4%or * (651 (60) D. Time evolution of 7’f), (L’}, and ~-,,/a We perform the orbit average by dividing the ion orbit We are now ready to examine the time evolution of lon- into a portion at small radius, r< rx , where Ty)( r)> Tf’, gitudinal velocity dispersion as measured by T:,S”/q,$o, and and a portion at large r, r> rx , where q)> T’,s)(r), using the transverse velocity dispersion as measured by (L*) or the expressions for the local rate of change in the longitudi- rola. Summing the term describing the rate of increase in nal and transverse temperatures appropriate for each region. Ti’) due to core collisions [as given by Eq. (50)] with the The radius rx , where Ty’( rx) = fl,“‘, is given by appropriate term describing the rate of change in Tft’ due to both collisions between comoving ions [from Eq. (56), (62), (61) or (64), as appropriate], we obtain an expression for the total rate of change in the longitudinal beam temperature, Averaging these rates over the ion orbit, we obtain ex- pressions for the rates of change in the longitudinal and (66) transverse beam temperatures valid for longitudinal beam fgltota,=;G(g. ~)(g2s40 M7,1. temperatures in the range The function G,(Tf,S’/q,q50,rola) is displayed in Fig. 8 for deuterium ions in a DT plasma. We see that G, is weakly varying with both ~,s’/qs~o and rola. A further decrease in rola beyond 10V4 results in only a very small downward shift relative to the rola = 10Y4 curve of Fig. 8; whiIe at smaller values of Z’~“/~,C$~ the function G, goes to G~=d~+$Ln[~~~, [z for any rola. We find that dTt,“ldt is positive for at1 values (62) of Tf)lq,#o and rola. Hence, collisions will result in a and monotonic increase of Ti”’ with time. 3812 Phys. Plasmas, Vol. 2, No. 10, October 1995 W. M. Nevins Downloaded 03 Jun 2009 to 128.174.163.154. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp fi a Tip 1 H s-c,- 8 2 i 4d0<2(r0/a)’ i * However, for larger values of @)/qs& the orbit-averaged effect of collisions between comoving ions weakens, so that both (L2) and Ta”’ increase monotonically with time when ~,d’lqdf$O>0.017rolu. The transition from collisional focusing to collisional de- focusing can be understood by examining the leading terms in the expression for d(L2)ldt in the moderately thermalized regime, FroI I/’ a 2GTf)/qs&G $(r$rz), 4L2) L:(v&bl dt (69) FXG.8. Thevariation of the longitudinalbeam heating rate, including both core collisionsand collisionswith comovingions for deuteronsin a DT .where we have used Eq. (61) to express ( rxlro)2 as plasmais displayedfor Q/U= lo-’ (solidline), rola= 10e3(long dashes), $Tf)Iq,&. It follows that the transition from focusing to andro/a = 10V4(short dashes). defocusing for deuterium in a DT plasma occurs when Thd) d--m r0 - ---PO.017 3. (70) The behavior of the transverse velocity dispersion as i-i&? 16Cd a measured by (L’) is more interesting. Summing the term After initial transients, in which the ion focal radius may describing the rate of increase in (L2) due to core collisions decrease in size while Tf)lq,&, increases, the system will [as given by Eq. (51)], with the appropriate term .describing reach a state in which ~)/qs~,,Zrola. The longitudinal the rate of change in (L2) due to both collisions between temperature and ion convergence then satisfy the equations comoving ions [from Eqs. (57), (63), or (65), as appropriate] we obtain an expression for the total rate of change in the (71) G”), where we have taken Gd=1.3 (valid for and Tf’/q,r$, (68) -r&as lo-“; see Fig. 8); and The function Hs(T~)Iqs~o ,rola) is displayed in Fig. 9. ;(+4.3($-1(V&l; (73 When Tf)/g,& is small (less than 0.0 17 rola for deu- terium in a DT plasma) collisions between comoving ions where we have taken Hd=6.4 (valid for T~d)lqd@-rola; dominate the bulk plasma collisions so-that the net effect is see Fig. 9), and eliminated (L’) in favor of rolg using Eqs. rapid decrease in (L’) (i.e., a rapid decrease in the size of the (49). ion focus, ro). For the smallest values of .Tf)/q,ycjoH, as- These equations have the solution ymptotes to I;;4,2.9dm (73) . . and H&IP)/q&r roW (74) s -z ~*?;?- .,,,,.: , , ,,,,,,: -, :1 5.0 We conclude that -collisions between counterstreaming , ions will initially lead to rapid thermalization of the distribu- 0.0 ;’ i-” tion of ion radial velocities [at a rate of order Wrd( @>~~ll- Once TII@) has increased to the extent that Tf’lq,&$O.O 17rola this process will be accompanied by the spreading of the ion focus. Finally, when Tf’/q,r& Zr,,/a, the increase in T~)/q,& and rola will proceed in concert at a rate of order (alr,,)(v;S(s)vol. As the ion focus spreads the rate of increase in the focal radius decreases so that it takes a time of order FIG. 9, The variationin the totaltransverse beam heating rate as measured by the rateof increasein (L’), includingboth bulk collisionsand collisions betweencomoving ions, is displayedfor deuteriumions in a DT plasma with rola=lO-* (solid line), rola=lO-a (long dashes),and rola=10-4 for the ion distribution function to relax to is.otropy and for (shortdashes) as a functionof Thd)lqd+o. the IEC configuration to be destroyed. Phys. Plasmas, Vol. 2, No. IO, October 1995 W. M. Nevins 3813 Downloaded 03 Jun 2009 to 128.174.163.154. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp Our conclusions regarding the time evolution of the IEC distribution function is very different from that reached in Ref. I I, where steady-state, beam-like solutions to the ki- netic equation were found. Two key reasons for our com- pletely different results are the following. (1) The artificial constraint imposed in Eq. (11) of Ref. 11, which prevents collisions between counterstreaming ion beams from producing any net increase in the velocity dis- persion (i.e., heating) of the ion beams J (2) The neglect of the dominant term in the evolution of the beam temperature-the increase in the longitudinal ve- locity dispersion of the beam due to collisions in the plasma core. @JoWV) When this problem is treated correctly we see that there are no beam-like steady-state solutions to the kinetic equa- FIG. 10. An orbit-averaged fusion rate coefficient (solid curve) and orbit- tion. averaged energy diffusion rate (dashed curve) versus potential well depth, 6, for deuterons in a DT plasma. The ion density, convergence ratio, etc. are taken from Table I. However, the relative magnitude of fusion and col- VI. SCHEMES FOR MAINTAINING A STRONGLY lisional rates are insensitive to these parameters. FOCUSED ION DISTRIBUTION THAT DO NOT WORK We have shown that ion-ion collisions will cause the well depths considered (5 kVGq$G300 kV). We conclude IEC distribution to relax toward an isotropic Maxwellian. At that fusion reactions rates are not sufficient to materially ef- the high energies and relatively low volume-averaged densi- fect the form of the ion energy distribution function. ties proposed for IEC devices, the ion collisional time scale The time required for complete relaxation of the ion an- is rather long--(#d)vo, is about 2.2 Hz (as compared to an isotropy is longer (by -alrc) than the instantaneous time ion bounce frequency of 1.1 MHz)-for the IEC reactor pa- scale for spreading of the ion energy distribution. Hence, one rameters of Table I. Hence, it may be possible to prevent this might hope that the loss of fuel ions through fusion reactions collisional relaxation through some process that acts only could maintain the ion anisotropy. We may estimate the re- weakly on the ion distribution function. We consider two sulting ion core radius by replacing “t” in Eq. (73) with the such schemes that have been proposed by proponents of IEC inverse of the orbit-averaged fusion reaction rate. After a bit fusion reactors in this section. of manipulation, this estimate takes the form A. Fusion reaction rates ( &orbit $=7.3 (77) Bussard’ makes the rather surprising claim that the fu- M”4DTLlrbitP1* sion reactions in an IEC device will remove fuel ions at a We conclude that the removal of fuel ions by fusion reactions rate sufficient to maintain a nearly monoenergetic ion distri- occurs at a rate that is insufficient to maintain an ion focus. bution function. In making this claim Bussard recognizes The situation regarding maintenance of the ion anisotropy is that the fusion reaction rate must be greater than the colli- essentially the same as that regarding maintenance of a non- sional energy-scattering rate if the loss of fuel ions by fusion Maxwellian ion energy distribution because the orbit- reactions is to substantially alter the ion distribution function. averaged fusion rate decreases as the ion focus spreads, In Sec. V we showed that the orbit-averaged collisional rate thereby making fusion reactions less effective as a mecha- of increase in the beam velocity dispersion is nism for removing fuel ions before they are scattered further in angle. (V%?rbit= 2* ~~ltota~~dr(~j(v~dt 4 ,& ‘“)“~,, (75) J 6. Maintenance of ion anisotropy with a “cold” while the orbit-averaged fusion rate for the deuterons is plasma mantle given by The basic idea inspiring the work of Rosenberg and Krall was that it might be possible to control the ion distri- (76) bution function in an TEC device by manipulating the ion distribution function in the neighborhood of the ion injection Note that the orbit-averaged fusion rate scales with the core point. In Sec. V we demonstrated that the calculation per- convergence ratio and ion density as (alrO)(n,)vO,---that is, formed in Ref. I I is in error, and that collisions between ions in exactly the same manner as the rate of increase in the with low reIative velocities cannot prevent thermalization of longitudinal velocity dispersion. We computed the orbit- the ion distribution function. However, perhaps this only averaged fusion rate following the methods described in Sec. demonstrates that the wrong problem was addressed both in III. This rate is plotted, together with the orbit-averaged en- Sec. V and in Ref. 11, In this section we consider the related ergy diffusion rate versus the potential well depth in Fig. 10. problem in which the ion distribution function at the injec- We see that the orbit-averaged energy diffusion rate substan- tion point is treated as a boundary condition. We are then tially exceeds the orbit-averaged fusion rate at all potential able to force the transverse temperature to go to zero at the 3814 Phys. Plasmas, Vol. 2, No. 10, October 1995 W. M. Nevins Downloaded 03 Jun 2009 to 128.174.163.154. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp lip of the potential well, which we now take at (E- L2/2m,a2) =O [rather than at (E- L2/2m,a2) = +qs&], so that this boundary conditionwill have greater -fo influence on the distribution of trapped ions. _----f2 ______f4 It is convenient to adopt as velocity-space coordinates u, -_---- the particle speed normalized to the thermal velocity, and ,u, ill b the cosine of the angle that the particle makes with the nor- mal when it strikes the surface of the spherical square-well potential. Note that both of these velocity-space coordinates can be expressed as functions of E and L”, FIG. 11. The iirst four even Legendre harmonics of the steady-state ion U= 2(e+ 4s450) distribution function when a zero transverse temperature boundary condition J Ts ’ is applied at u=3. so that u and ,U are themselves constants of the single- particle motion. We assume that the steady-state ion distribu- The bulk of the phase-space density is contained in fn( u), tion function is nearly an isotropic Maxwellian (and verify providing the a posteriori justification for the use of the test this assumption a posteriori), so that we may use the usual particle collision operator. test-particle collision operator. In the high-velocity limit The effective radius of the ion focus may be computed (~k2) the steady-state kinetic equation may then be written as a function of the ion speed from the root-mean-square as value of the distance of closest approach, (80) r,&u)=a Jfi=a JF. (84) We see that there is essentially no ion focusing at speeds less We look for solutions in the form than twice the ion thermal velocity. We conclude that it is not possible to maintain a strongly anisotropic ion distribution f(uJd=~ fLL(U)PL(Pu)v (81) function in an IFC device solely by controlling the form of the ion distribution function at the ion injection point. where the PL&) are Legendre polynomials of index L, and the fL( u) satisfy VII. SCHEMES FOR MAINTAINING A STRONGLY FOCUSED ION DISTRIBUTION THAT WORK, BUT REQUIRE TOO MUCH POWER (821 We have shown that two mechanisms proposed by pro- We solve this equation on the domain 0~ u < u,, ponents of IEC systems to maintain a strongly nonthermal =dw, taking as our boundary condition a plasma ion distribution function will not be effective. In the absence with finite phase-space density and zero velocity spread at of intervention the ion distribution function will thermalize the ion injection point, in an ion-ion collision time. In this section we propose an approach for maintaining a strong ion focus in an IEC sys- tem based on the assumption that some means can be found f(u (83) to control the ion lifetime in the electrostatic trap, i.e., we assume that ions can be intentionally “pumped” out of the Expanding this boundary condition in a Legendre series, we electrostatic trap before they have time to fully thermalize. obtain f,Ju,,)=fs(L+ i) for L even, and JFL(u)=0 for L Such schemes are plausible since the ion lifetime is in fact odd. limited in electrostatic traps that use grids (due to the finite The solutions of the kinetic equation with this boundary grid transparency), and in Penning traps (due the leakage of condition are shown in Fig. 11 for the first four even Leg- ions through the poles of the trap). Similar schemes were endre harmonics. We see that our highly anisotropic bound- proposed (and studied extensively), in connection with main- ary condition has an appreciable effect on the distribution taining an anisotropic ion-distribution in the thermal barrier function only at the highest speeds, uS2. The L=O Legendre cell of tandem mirrors.z3 harmonic, fc(u), is the sum of a Maxwellian plus a small A. Power required to “pump” the ion distribution constant term required to match the boundary condition at u = u,, . The density and temperature of this Maxwellian When an ion is pumped from the trap it is replaced by an must be determined from consideration of energy and par- ion with zero angular momentum and zero energy (i.e., with ticle balance in analogy to the Pastukhov solution of the an ion whose kinetic energy within the potential well is problem of electron confinement in magnetic mirrors.2”22 4&c). Assuming that the lifetime of a typical ion is rpumpr Phys. Plasmas, Vol. 2, No. 10, October 1995 W. M. Nevins 3815 Downloaded 03 Jun 2009 to 128.174.163.154. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp we can model the effects of ion pumping by adding appro- 2-PNi ~pmnp priate sink terms to Eqs. (66) and (68), yielding P .wmr . - ‘-pump Tito,=;Gs (;)q.~o2M ’?,,,-(8.5) j; GdH~)“2q~oNil~j3’~(~~‘“)M1. (91) which comes to about 23 CW for the IEC reactor of Table I and (which produces only 590 MW of fusion power). W2> The power balance in this operating mode can be char- - =HsLf(@),,- 0. 6-W acterized by the fusion gain, dt total rww Equations (85) and (86) have the steady-state solution (92) f-0 4 -= (87) a J 3 fu#%ol~p”mp For the IEC reactor of Table I, we find Qn7~0.026. In fact, QnT will certainly be below this limit, as this estimate does and not take account of the power required to maintain the po- tential well and support energy losses in the electron channel. The estimate of the upper limit on the fusion gain de- (88) pends on the potential well depth, (PO,only through the com- bination where Eq. (49) was used to eliminate (L2) in favor of rola. In these steady-state solutions the two parameters de- scribing the ion distribution function (rola and Tf’lq,q50) Numerical evaluation shows this to be a very slowly increas- are replaced by a single parameter, [7pumpin Eqs. (87) and ing function of $. for 4oZ=50 kV. The value increases by (SS)]. Alternatively, we may choose rola as the single pa- 10% as +a is increased from 50 kV to 75 kV, and by an rameter characterizing the steady-state ion distribution func- additional 10% as I$~ is increased to 300 kV. Assuming that tion. Equation (87) then determines the required pumping there must be some penalty to increasing (PO,we take QIo-75 time, kV, yielding a 10% improvement in QnT (to QoTSO.028). The fusion gain also increases with decreasing ion conver- 3 1 (rolaJ2 (89) gence ratio, alro. Since a high ion convergence ratio is a Tpuw= 4 H, ( &yk, 1 defining feature of IEC systems, we will require alro~lO, The fusion gain is then limited to QoTC0.091.25 while Eq. (88) determines the longitudinal velocity disper- sion at the chosen convergence ratio. Equation (88) has been solved numerically (including the weak dependence of G, and H, on rola and 7$s’Iq,+o illustrated in Figs. 8 and 9) to B. Effect of a ‘“hard core” on pumping power determine G,( rola) and H,( rola) for these steady-state so- requirements lutions. The residual dependence on the convergence ratio is It has been suggested26that the ion pumping power re- weak-for deuterons in a DT plasma, Gd increases mono- quirements can be mitigated by introducing a “hard core” in tonically from 1.56 to 2.40 and H, decreases monotonically the electrostatic potential profile. We note that it is not en- from 6.3 to 6.21 as rola decreases from 10-l to 10v4. For tirely clear how such a hard core is to be established since the IEC reactor parameters of Table I (t-,/a = 10b2), we ob- the ion charge producing this hard core must be provided by tain Cd* 1.69, Hd=6.12, T6d’lq,&o--2.8X 10e3, and 7rurnp ions that have sufficient energy to traverse the core (so these -5.5 #L&s. ions, at least, will undergo energy diffusion due to core col- The energy required to remove an ion from the trap is lisions). Nevertheless, we devote this section to estimating estimated by the energy spread in the ion distribution, the ion pumping requirements that would pertain if such a hard core potential can be produced. w It was concluded in Sec. IV B that a leading cause of ion epump~msU,(AU:)“2=qs~o 2 - (90) d-- 4s40’ energy diffusion in IEC devices is collisions in the vicinity of the plasma core. This effect can be removed by setting the Lower vaIues of epumpare thermodynamically allowed,24 but coefficient d, equal to zero. Note that this does not entirely is very difficult to see how lower pumping energies could be eliminate ion energy diffusion-there remains the contribu- achieved in IEC systems as they have been described in the tion to ion energy diffusion from bulk collisions between literature to date. Assuming a potential well depth of comoving ions, as described in Sec. V. The analysis of the 4bo-50.7 keV to maximize the DT fusion rate coefficient, it pumping power requirements presented in the previous sec- will require about 3.8 keV to remove each ion. tion still applies. However, the residual dependence of G, The total power required to maintain the ion distribution and H, must be recomputed with dd set equal to zero. We function is then find that Cd now varies from 0.361 to 0.889 and Hd varies 3816 Phys. Plasmas, Vol. 2, No. IO, October 1995 W. M. Nevins Downloaded 03 Jun 2009 to 128.174.163.154. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp from 5.15 to 5.78 as rola decreases Tom 10-l to 10-4. The This analysis is based on a particular model ion distribu- optimized fusion gain (which still occurs at rala=O.l and tion function. While the reactor operating point has -been &=75 kV) is increased to Qn+O.21. optimized over the parameters of this model, it is possible These very disappointing limits on the fusion gain fol- that a more attractive power balance could be obtained by low from the fact that the averaged fusion rate is small com- further optimization of the form of the ion distribution func- pared to the averaged collisional rate [see Eq. (92) and Fig. tion. A serious effort to perform such an optimization would lo], so that the power required to maintain the nonthermal require the development of a bounce-averaged Fokker IEC ion distribution is substantially larger than the fusion Planck code in (e, L2) space. However, it seems most un- power generated. The IEC reactor power balance is opti- likely that such optimization will increase Q by the factor of mized by allowing the ion distribution to thermalize as much -100 (or -SO if hard core potentials can be maintained) as is consistent with the operating regime in question (i.e., required to achieve an acceptable recirculating power frac- choosing the lowest ion convergence ratio, alro, consistent tion for an economic power plant. Hence, we conclude that with an IEC configuration). The power required to maintain inertial electrostatic confinement shows little promise as a an ion distribution function that retains the defining charac- basis for the development of commercial electrical power teristic of an IEC system (alrcZl0) is 11 times greater than plants. the fusion power that this system might produce. If a “hard The analysis does not place a lower limit on the unit size core” potential can be formed the reactor performance can be of an lEC reactor. Such a lower limit on the unit size will improved by a further factor of 2.3 to Qb+O.21. Reactor (presumably) follow from an analysis of electron energy con- studies indicate that an economic DT fusion power reactor finement and the energy cost of maintaining the spherical requires a much more favorable energy balance, QDT>lO. potential well. This leaves open the possibility that IEC- Hence, there appears to be no prospect that an economic based reactors may prove useful as a means of generating a electrical power generating reactor can be developed based modest flux of 14 MeV neutrons for applications other than on an inertial electrostatic confinement scheme. power generation, such as such as assaying, neutron imaging, materials studies, and isotope production. In such applica- tions a small unit size (PfusionSl kW) and, hence, smaller VIII. CONCLUSIONS unit cost might compensate for modest values of Qnr . We have presented a model for the ion distribution func- tion in an inertial electrostatic confinement system. This ACKNOWLEDGMENTS model is shown to reproduce the essential features of IEC This work was motivated by presentations of R. W. Bus- systems-electrostatic confinement, strong centra1 peaking sard and N. A. Krall on the promise of IEC-based fusion of the ions, and a nearly monoenergetic energy distribution. reactors. It has benefited from stimulating discussions with Using this model distribution function we are able to test key B. Afeyan, D. C. Barnes, R. H. Cohen, T. J. Dolan, R. Durst, claims made by proponents of IEC systems. We find the R. Fonck, T. K. Fowler, L. L. Lodestro, G: H. Miley, J. following: Perkins, T. Rider, and D. Ryutov. Particular thanks are due to ill After averaging over collision angle and volume, the S. W. Haney for providing the numerical solution to the ion peak in the effective fusion rate coefficient for DT kinetic equation described in Sec. VI. (( uv)$=9.OX 1o-22 m3/s at $~a%50keV for the IEC This work was performed under the auspices of the U.S. distribution versus 8.9X1O-22 m3/s at T=75 keV for a Department of Energy by the Lawrence Livermore National thermal distribution) or D3He (( av)$u,*2.8~ LO-22 Laboratory under Contract No. W-7405ENG-48. m3/s at &o- 140 keV vs 2.5X 10vz2 m3/s at T=250 keV) reactions are not significantly higher than the peak in the APPENDIX: COLLISIONAL RATE COEFFICIENTS FOR corresponding thermal rate coefficient. A MONOENERGETIC IEC DISTRIBUTION (2) Ion/ion collisions will cause the ion distribution function FUNCTION to relax toward a Maxwellian at an instantaneous rate In this appendix we compute the rate of increase in the that is enhanced relative to the ion-ion collision fr& transverse and longitudinal velocity dispersion, (Au:) and quency (evaluated at the volume-averaged density) by (Au;), for a test particle of species s and velocity v=u&, one power of the convergence ratio, a/r0 . colliding with field particles of species s’. The distribution In the absence of a competing process, the ion distribu- (31 function of the field particles is taken to be the monoener- tion will fully relax to an isotropic Maxwellian on the getic IEC model distribution function defined in Sec. II. ion-ion collisional time scale (evaluated at the volume- Since the model IEC velocity distribution function varies as averaged density). a function of radius, we expect these rates of increase in the The means of preventing this relaxation of the ion dis- (4) longitudinal and transverse velocity dispersion to vary with tribution function so far proposed by proponents of IEC radius. schemes are not effective. Rosenbluth, McDonald, and Judd27 give a compact ex- The energy cost of maintaining an anisotropic ion distri- (5) pression for the rate of increase in the velocity dispersion due bution function through control of the ion lifetime is to Coulomb collisions. Following these authors we use the greater than the fusion power that would be produced by symbols (Ati:) and (Au;) in this appendix only to denote the the IEC device. rate of increase in the velocity dispersion rather than the Phys. Plasmas, Vol. 2, No. 10, October 1995 W. M. Nevins 3817. Downloaded 03 Jun 2009 to 128.174.163.154. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp velocity dispersion itself. In the main text these symbols are (U,,-p ’)i$- Jm(il cos ++i, sin 4) used to denote the velocity dispersion. These rates are given +zt= by It follows that the integral over an azimuthal angle acts only (Av Av)s=T (Av Av)ssp, on the diads, yielding where (Av Av)~~~,the rate of increase in the velocity disper- 1 --/id2 ( I - i$i,) sion in species s due to collisions with particles of species s’, (z+- l-2/4’U,Yr) is given by (Av Av)~~, = Because there is only one preferred direction in velocity x Iv-V’I. space (&J, we are able to write (Av Av), in the form Defining characteristic speeds v, for species s, and using the (hv Av)~~~=~(Au~)~~~(~-~~~~)+(A~~)~,~~~~~, vector identity where the rate of increase in the transverse velocity disper- & ,v-vq= !g, sion is given by we can write the rate of increase in the velocity dispersion as (Av~),,~=~Y~J~ 2 1’ d,uL) “:‘rr’,~’ -I (Av Av)~~,= v;“‘u,~ 2 1 d3v’ j=Jv’) $, s 1 X where w=(v-v'), G=w/IwI, 1 is the identity tensor, and, iij u,2,+ 1 --2p?Ms’ following Book,16 we define 1 l-/P $,$,- 4dq& Ln A,,1 - a (u,2,+ 1 -2#u’us,)3’2 ’ ug = ii msus2 3 and the rate of increase in the longitudinal velocity disper- The monoenergetic IEC distribution evaluated at radius sion is given by r may be written in spherical coordinates as f$,(v,)- nsq H(lP’l-Po) S(u)-us,), (AU&~,= @‘u,2 $ /;,dp’ “:;‘,T’ 2msr 2(1-Po) where the principle axis of the spherical coordinates is taken l-/P parallel to &, ,u’ is the cosine of the angle between v’ and (u,2,+ 1-2#t4,)3’2 &, IZ,’ (r) is the local value of the field ion density as given by Es. (9), Two integrals remain to be evaluated, 24s’40 us,= - dCt’ J m,r 1 -I-&2p’u and 1 Jl +l4*-2~& =....-- TGt-0, u 2(1-Po) po= i ;iqp, r>r(). and The integral over the field ion speed is easily performed, /J l-$2 yielding ~2(/-w)= 1 2(~-/-%~ f (1+U2-2p54)31~d~’ (Av A+,,, = @‘v,2 2 1; ,d& I,‘” 2 H;;$c’ l-/Z (1 +u2-2/U)-“2 0 =- u 2(1-luo) 2p (1 +U2-24’12 -7 2(1 -fiol 2 (l+U2-2p)3’* -52 2(1-iuo) * The only dependence on the azimuthal angle, 4, comes Combining these results, we obtain the rate of increase through the unit vector in the transverse velocity dispersion, 3818 Phys. Plasmas, Vol. 2, No. 10, October 1995 W. M. Nevins Downloaded 03 Jun 2009 to 128.174.163.154. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp l- 1 p~~Icr;~Po7~)l=2 Ius,f 11 and the rate of increase in the longitudinal velocity disper- and sion, where ‘A. Sakharov, Memoirs (Random House, New York, 1992). p; 139. IL(POP) ‘R. L. Hirsch, J. Appl. Phys. 384522 (1967). 30. A. Lavrent’ev, Uk. Fiz. Zb. 8, 440 (1963). 40. A. Lavrent’ev, Investigations of an Electmmagnetic Trap. Magnitnye Lovushki Vypusk (Naukova Dumka, Kiev, 1968), p. 77 [for an English translation, see Atomic Energy Commission Report No. AEC-TR-7002 (Rev)]. - ~2(l,~~-~2(ruo,u!+~2(-~o,u)-z2(-l,u) 50. A. Lavrent’ev, Ann. NY Acad. Sci. 251, 152 (1975). 4(l-Poj 1 6T. J. Dolan, Plasma Phys. Controlled Fusion 36, 1539 (1994). 7D. C. Barnes, R. A. Nebel, and L. Turner, Phys. Fluids B 5, 3651 (1993). and ‘R. W. Bussard, “Method and apparatus for controlling charged particles,” U.S. Patent No. 4,826,646 (2 May 1989). 9R. W. Bussard, Fusion Technol. 19, 273 (1991). ‘ON. A. KraIl, Fusion Technol. 22, 42 (1992). “M. Rosenberg and N. A. Krall, Phys. Fluids B 4, 1788 (1992). For &=1 (corresponding to radial locations in the bulk ‘*H. S. Bosch and G. M. Hale, Nucl. Fusion 32, 611 (1992). plasma, where rSro), the lEC distribution function corre- 13G. H. Miley, J. Nadler, T. Hochberg, Y. Gu, 0. Bamouin, and J. Lovberg, sponds to two counterstreaming beams. We can remove the Fusion Technol. 19, 840 (1991). 14J F Santarius, IS. H. Simmons, and G. A. Emmett, Bull. Am. Phys. SOC. contribution of collisions between comoving particles from ;9, ‘1740 (1994). the collision integrals, IL and Z,,, by replacing the upper limit “G . H . Miley and H. Towner, Conference on Nuclear Cross Sections and of 1 in the integrals I, and I, with /..+ Phys. Plasmas, Vol. 2, No.. 10, October 1995 W. M. Nevins 3819 Downloaded 03 Jun 2009 to 128.174.163.154. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp