Fundamental limitations on plasma fusion systems not in thermodynamic equilibrium Todd H. Rider

Citation: Physics of Plasmas 4, 1039 (1997); doi: 10.1063/1.872556 View online: http://dx.doi.org/10.1063/1.872556 View Table of Contents: http://aip.scitation.org/toc/php/4/4 Published by the American Institute of Physics Fundamental limitations on plasma fusion systems not in thermodynamic equilibrium Todd H. Ridera) Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, Cambridge, Massachusetts 02139 ͑Received 5 June 1996; accepted 6 January 1997͒ Analytical Fokker–Planck calculations are used to accurately determine the minimum power that must be recycled in order to maintain a plasma out of thermodynamic equilibrium despite collisions. For virtually all possible types of fusion reactors in which the major particle species are significantly non-Maxwellian or are at radically different mean energies, this minimum recirculating power is substantially larger than the fusion power. Barring the discovery of methods for recycling the power at exceedingly high efficiencies, grossly nonequilibrium reactors will not be able to produce net power. © 1997 American Institute of Physics. ͓S1070-664X͑97͒01404-3͔

I. INTRODUCTION systems, fusion products, or other sources. This as- sumption sets a lower limit on the electron tempera- One of the most important challenges in modern physics ture and bremsstrahlung losses, in agreement with the is to identify the best approach to clean and efficient fusion stated goal of finding an optimistic bound on the per- power generation. Advanced aneutronic fuels such as formance of the fusion systems. 3He–3He, p-11B, and p-6Li would produce considerably less ͑iii͒ Likewise, it is optimistically assumed that the entire neutron radiation and radioactive by-products than more con- fusion reaction output power can be utilized. Conver- ventional fusion fuels like deuterium–tritium ͑D–T͒ and sion efficiency limitations are ignored, and power deuterium–deuterium ͑D–D͒, and furthermore they might permit high-efficiency direct electric conversion of the fusion losses are directly compared with the gross fusion energy instead of low-efficiency thermal conversion. Unfor- power Pfus . tunately, plasma systems which are essentially in thermody- ͑iv͒ In comparing collisional scattering effects, fusion, and namic equilibrium cannot break even against radiation losses bremsstrahlung radiation with each other, the density, with these aneutronic fuels;1 for this reason, it has been sug- spatial density profiles, and plasma volume do not matter, since all of these phenomena are two-body gested that plasma fusion systems which are substantially out 3 2 of thermodynamic equilibrium should be considered.2 As a effects and thus are proportional to ͐d x͓n͑x͔͒ ͑ne- further incentive for the study of nonequilibrium fusion plas- glecting the weak density dependence of the Coulomb mas, the somewhat more conventional fuel D–3He could be logarithm ln ⌳͒, in which n͑x͒ is the particle density made cleaner and more attractive if it were possible to sup- as a function of position. press undesirable D–D side reactions more than can be done ͑v͒ The regions of the plasma which have values of 3 2 in an equilibrium D–3He plasma.1 ͐d x͓n͑x͔͒͒ large enough to be of interest are ap- This paper will resolve the question of whether highly proximately isotropic. Otherwise they would be sub- 3 4 nonequilibrium plasma systems would be useful for fusion ject to counterstreaming, Weibel, and other insta- purposes, especially with regard to advanced-fuel fusion. bilities. Rather than limit the analysis to a particular type of nonequi- ͑vi͒ Although instabilities can prove to be a serious con- librium fusion reactor design, it would be wise to make this cern even in essentially isotropic nonthermal plasmas, study as generally applicable as possible. Accordingly, a they will be optimistically ignored here. minimum of assumptions will be made with regard to the ͑vii͒ Spatial variations of particle energies may be ne- plasma geometry, reactor confinement system, type of fuel, glected in regions of significant ͐d3x͓n͑x͔͒2. and other key parameters. Those assumptions which are ͑viii͒ The plasma is quasineutral and optically thin to made are as follows: bremsstrahlung. ͑ix͒ In the ion energy ranges of interest, the functional ͑i͒ Losses other than bremsstrahlung radiation and the dependence of the fusion reactivity ͗␴v͘ on the power required to keep the plasma out of thermody- fus mean ion energy ͗E ͘ is approximately independent namic equilibrium are ignored, so this analysis sets an i of the ion velocity distributions’ shapes if the distri- optimistic bound on the performance of nonequilib- butions are isotropic and the ion species have the rium fusion reactors. same mean energy, as shown explicitly in Ref. 5. ͑ii͒ Energy transfer from the fuel ions to the electrons is ͑x͒ The functional dependence of the bremsstrahlung ra- the only energy source available to the electrons; the diation power on the mean electron energy E is electrons cannot acquire energy from external heating ͗ e͘ approximately independent of the electron velocity distribution shape in the energy range of interest. a͒Present mailing address: 501 West A St., North Little Rock, Arkansas 72116. As demonstrated in Ref. 5, systems which violate the

Phys. Plasmas 4 (4), April 1997 1070-664X/97/4(4)/1039/8/$10.00 © 1997 American Institute of Physics 1039 above assumptions would not offer advantages substantial The parallel velocity-space diffusion coefficient for a enough to be of particular value, even if such systems could particle with velocity vtest in the presence of isotropic, mo- be constructed. noenergetic field particles of the same species with speed v0 All quantities are in cgs units, with energies and tem- is7 peratures both measured in ergs, unless otherwise stated. For 3 2 ready comparison with equilibrium plasmas, the ‘‘tempera- ͱ␲ v0 v0 DʈϷ , ͑2͒ ture’’ T of a non-Maxwellian distribution with a mean par- 3ͱ6 ͩ vtestͪ ␶col ticle energy ͗E͘ is defined as Tϵ2͗E͘/3. In Ref. 1, it was shown that plasma systems which are where the usual definition of the collision time ␶col ͑Ref. 8͒ 2 intended to operate far from thermodynamic equilibrium and has been used with ͗E͘ϭ(3/2)TϷmv0/2: yet which have no specific provisions for maintaining such a ͱm͗E͘3/2 state will very rapidly relax to equilibrium. Methods of pas- ␶ ϵ . ͑3͒ col 4 sively maintaining nonequilibrium distributions have also 2ͱ3␲͑Ze͒ n ln ⌳ 6 been shown to be inadequate. Therefore the present discus- The time for a typical test particle to be collisionally sion will focus on systems which maintain nonequilibrium upscattered from the velocity v to the maximum allowed plasmas by active but otherwise arbitrary means. Such active 0 velocity v ϵv ϩv may be estimated as maintenance of nonequilibrium plasmas will entail certain fast 0 t minimum power requirements and limitations. 2 2 vt 3ͱ6 vt In Sec. II we will examine the limitations that affect ␶fastϷ Ϸ ␶col , ͑4͒ Dʈ ͩ v ͪ plasma systems which attempt to maintain substantially non- ͱ␲ 0 Maxwellian velocity distributions for the electrons or the where only the largest term has been retained. fuel ions. In Sec. III we will then present the fundamental By likewise keeping only the largest term of ⌬Efast and limitations that pertain to plasma systems in which two or 2 using ͗E͘Ϸmv0/2, one finds the energy upscattering to be more of the major particle species are at radically different mean energies. Using the results of Secs. II and III, in Sec. 1 2 2 vt ⌬Efastϭ m͑vfastϪv0͒Ϸ2 ͗E͘. ͑5͒ IV we will discuss the implications for controlled fusion. 2 v0 The final necessary assumption is that approximately half of the particles will be upscattered in energy and half II. NON-MAXWELLIAN VELOCITY DISTRIBUTIONS will be downscattered, so nfastϷn/2. By putting all of this information together, the recirculating power required to One way in which a plasma can deviate from thermody- hold the proper distribution shape despite like-particle colli- namic equilibrium is to have non-Maxwellian particle veloc- sions is found to be ity distributions. While no fusion system has perfectly Max- wellian distributions, the systems which will be considered ͱ␲ v0 n͗E͘ v0 n͗E͘ here are of interest because they deviate from the Maxwell- P Ϸ Ϸ0.24 . ͑6͒ recirc v ␶ v ␶ ian equilibrium in a much more marked fashion than is usual. 3ͱ6 t col t col A. Preliminary estimates The general form of this result will be confirmed by the more rigorous derivation. Before performing a rigorous derivation of the minimum The second case for which the minimum recirculating power requirements needed to maintain non-Maxwellian ve- power will be estimated concerns velocity distributions locity distributions, it is useful to make preliminary estimates which are nearly Maxwellian except that essentially all of the of these power requirements as a means of gaining physical very slow particles in the distribution are depleted. This situ- insight into the problem. Estimates will be made for two ation would be especially desirable for the electron distribu- different types of velocity distribution functions. For sim- tion in advanced-fuel fusion plasmas, so that far fewer than plicity only one particle species will be considered. the purely Maxwellian number of electrons would have First consider an isotropic beam-like velocity distribu- speeds slower than the ions. Because ion–electron energy tion in which the particles are centered around a mean speed transfer is mediated by those slow electrons,6 a large reduc- v0 with some ‘‘thermal’’ spread vtӶv0 on each side of the tion in the electron temperature and radiation losses would mean speed. Due to collisions, a certain number ͑actually a result, and the power balance for the advanced fuels would certain density͒ of the particles nfast will gain an amount of be considerably improved. energy ⌬Efast on a timescale of ␶fast . If the width of the Consider an electron distribution which looks superfi- distribution is to be kept from spreading beyond the allowed cially like a normal Maxwellian with a characteristic thermal vt , then one must extract a power density Precirc from the velocity vtf ϵ ͱ2T0f /me but has no particles at speeds below particles which have become too fast and give it to particles some velocity v0 , which is chosen such that it is comparable which have become too slow. This quantity Precirc is defined to ͑actually somewhat greater than͒ the ion thermal velocity as and obeys the relation v0Ӷvtf . Electron distributions which differ substantially from this while still keeping the slow nfast⌬Efast electrons depleted will deviate further from the Maxwellian Precircϭ . ͑1͒ ␶fast equilibrium state and hence be harder to maintain.

1040 Phys. Plasmas, Vol. 4, No. 4, April 1997 Todd H. Rider ee The recirculating power which must be continually ex- If the distribution were allowed to relax for a time tE , the tracted from the tail of the electron distribution and given to number of slow electrons would approach this equilibrium the slow electrons to boost their energies and maintain the value but would still be less than it, so nslow will actually be ‘‘hole’’ in the center of the velocity distribution is somewhat less than the value on the right-hand side of Eq. ͑9͒. n ⌬E slow slow Slow electrons must be boosted up high enough in the Precircϭ ee , ͑7͒ tE velocity distribution that they will not immediately return to the depleted region. The exact amount of energy which they where nslow is the density of slow electrons that must con- must be given is not readily apparent in this simple model, tinually be acted upon, ⌬Eslow is the energy that must be ee but it should be comparable to the mean electron energy, given to each of them, and tE is the collision time for slow electrons of speed v interacting with Maxwellian ‘‘field’’ Eslowϳ͗E͘. 6 Putting all of this information together, one arrives at the electrons of temperature T0 f , conclusion that m2v3 2 ee e 3ͱ␲vtf 1 v0 tE ϭ 4 Ϸ ␶col . ͑8͒ v0 n͗E͘ 16␲e ne ln ⌳ 4v 4 ͩvtfͪ P ϳ . ͑10͒ recirc v ␶ ee tf col Here tE has been rewritten in terms of ␶col by using Eq. ͑3͒ 2 The numerical coefficient by which this expression should be with ͗E͘Ϸ(3/2)T0 f ϭ(3/4)mevtf. ee multiplied will be found from the rigorous derivation. Within a time period tE , the density of electrons which must be boosted in energy to prevent them from occupying the depleted region below vϭv0 will be comparable to the normal Maxwellian population of that region of velocity B. Rigorous derivation space, Consider a fairly general isotropic particle velocity dis-

3 tribution f (v) ͑for vу0͒ which peaks at some speed v0 and ne 4 4 v0 3 possesses characteristic widths vts and vtf on the slow and nslowϳ 3/2 3 ␲v0 ϭ ne . ͑9͒ ͩ ␲ v ͪͩ3 ͪ 3 ͩvtfͪ fast sides of the peak, respectively: tf ͱ␲

2 2 2 2 nK͕exp͓Ϫ͑vϪv0͒ /vts͔ϩexp͓Ϫ͑vϩv0͒ /vts͔͖ for vϽv0 , f ͑v͒ϵ 2 2 2 2 ͑11͒ ͭ nK͕exp͓Ϫ͑vϪv0͒ /vtf͔ϩexp͓Ϫ͑vϩv0͒ /vts͔͖ for vуv0 .

The normalization constant K is determined by the usual ϱ 4 f ץ 2 relation ͐f (v)4␲v2dvϭn. ϩ du f͑u͒u ϩ2͓f͑v͔͒ ϩ vץ v ͬ 3v͵ This distribution function, which is graphed in Fig. 1͑a͒, has many virtues. It can be set to a Maxwellian by the choice ϱ v u 2 v ϭ0, and even for other values of v it goes to the Max- ϫ du f͑u͒uϪ du f͑u͒u 1Ϫ 0 0 ͫ͵0 ͵0 ͩ vͪ wellian limit for large v. By varying the relative values of v0 , vts , and vtf , a wide variety of distribution shapes may be u ϫ 1ϩ , ͑12͒ studied. For example, for vtϵvtsϭvtf , Eq. ͑11͒ reduces to ͩ 2vͪͬͮ the beam-like distribution discussed in Sec. II A, as illus- 6 trated in Fig. 1͑b͒. Alternatively, for vtsӶv0Ӷvtf , Eq. ͑11͒ in which J͑v͒ is the collisional velocity-space particle flux: models the nearly Maxwellian electron distribution in which 16 2 Ze 4 ln f 1 1 v 4 ץ ⌳ ͒ ͑ the slow electrons are depleted, as was also discussed in Sec. ␲ J͑v͒ϭϪ 2 3 du f͑u͒u v 3 ͫ v ͵0ץ ͭ II A and is shown in Fig. 1͑c͒. Yet despite this high degree of m flexibility, the particular form of the distribution function in ϱ 1 v Eq. ͑11͒ allows one to obtain exact expressions for quantities 2 ϩ du f͑u͒u ϩf 2 duf͑u͒u vˆ, ͑13͒ such as the mean particle energy and the collision operator. ͵v ͬ v ͵0 ͮ For an isotropic but otherwise general distribution func- tion undergoing self-collisions, the collision operator may be where vˆ denotes the ‘‘radial’’ direction in velocity space. written as6,9 Note that the inclusion of the second term on each line v] ϭ0 and J(vϭ0)ϭ0, asץ/ f ץ] of Eq. ͑11͒ ensures that f vϭ0 ץ ϭϪ J is required for a self-consistent spherically symmetric distri- –t ͪ “vץ ͩ col bution. t)col and J, one mayץ/ f ץ) By using these expressions for f 1 v 2ץ 8␲2͑Ze͒4 ln ⌳ 2 4 determine the minimum recirculating power density Precirc ϭ 2 2 3 du f͑u͒u v ͫ v ͵0 needed to hold the non-Maxwellian distribution function ofץ m ͭ 3

Phys. Plasmas, Vol. 4, No. 4, April 1997 Todd H. Rider 1041 FIG. 2. A schematic diagram showing how to calculate the minimum recir- culating power required to maintain a given non-Maxwellian isotropic ve- locity distribution shape. This particular example shows the recirculating power needed to sustain a distribution qualitatively similar to that in Fig. 1͑b͒, but this general method may be extended to any isotropic but otherwise arbitrary velocity distribution, as described in Eq. ͑14͒.

sity required to hold that distribution constant despite like- particle collisions is

ϱ 1 f ץ 2 2 Precircϵ ͑dv4␲v ͒ mv ⍜͓J͑v͔͒, ͑14͒ tͪץͩͪ 2 ͩ 0͵ col

FIG. 1. Graphs of the isotropic velocity distribution of Eq. ͑11͒. The roles of where ⍜ is the unit step function. The physical meaning of the variables v0 , vts , and vtf are illustrated in ͑a͒. For vtϵvtsϭvtf as Eq. ͑14͒ is that the excess energy gained in collisions must shown in ͑b͒, Eq. ͑11͒ describes an isotropic beam-like distribution. For be removed from particles upscattered in velocity-space re- vtsӶv0Ӷvtf as illustrated in ͑c͒, Eq. ͑11͒ describes a nearly Maxwellian gions where J(v)Ͼ0. This energy can then be given to col- distribution in which the very slow particles have been depleted. lisionally down-scattered particles in regions where J(v)Ͻ0. For the distribution like that in Fig. 2, Eq. ͑14͒ reduces to

Eq. ͑11͒ constant despite self-collisions. Figure 2 shows a ϱ 1 f ץ 2 2 simple case in which the collisional velocity-space flux is Precircϭ ͑dv4␲v ͒ mv tͪץͩͪ v ͩ 2͵ positive above and negative below some dividing velocity d col vd . In other words, the dividing velocity is defined as the vd 1 f ץ 2 2 finite nonzero solution of the equation J(vd)ϭ0. If Nslow is ϭϪ ͑dv4␲v ͒ mv , ͑15͒ tͪץͩͪ 2ͩ 0͵ the number of particles which become too slow in a given col time period, the particles must be boosted up to the lowest in which vd may be found from the equation Nslow number of vacant states in the desired distribution v f ץ shape. Likewise, the Nfast particles which have become too d ͑dv4␲v2͒ ϭ0. ͑16͒ ͪ tץ ͩ fast must be decelerated to fill in the remaining vacant states ͵0 on the other side of the dividing velocity. If energy losses col from the distribution are neglected, the input power needed By Gauss’s divergence theorem and the relation t)colϭϪ“v–J, Eq. ͑16͒ may be seen to be simply aץ/ f ץ) to accelerate the slow particles may theoretically be entirely obtained by extracting from the fast particles the exact restatement of the earlier condition on vd , J(vd)ϭ0. amount of power needed to slow them down. This power is Equation ͑14͒ may be integrated numerically and ex- the minimum theoretical recirculating power. pressed in terms of the density n, mean particle energy ͗E͘, For a general, isotropic distribution ͑not restricted to the and collision time ␶col . distributions shown in Figs. 1 and 2͒, the appropriate math- For the important special case of Eq. ͑11͒ in which ematical definition for the minimum recirculating power den- vtsӶv0Ӷvtf , it is found that

1042 Phys. Plasmas, Vol. 4, No. 4, April 1997 Todd H. Rider TABLE I. Selected values of the function R0(v0/vtf). TABLE II. Selected values of the function R1(v0/vt).

v0/vtf R0(v0/vtf) v0/vt R1(v0/vt) 1/60 0.0637 0.01 5.81ϫ10Ϫ7 1/30 0.0644 0.1 5.63ϫ10Ϫ4 1/10 0.0687 0.5 0.0365 1/6 0.0749 1 0.0854 1/3 0.0957 3 0.148 1 0.183 10 0.221 30 0.253 100 0.265

v0 n͗E͘ P ϭR ͑v /v ͒ , ͑17͒ recirc 0 0 tf ͩv ͪ ␶ jected into other velocity-space regions of the particle distri- tf col butions by employing cyclotron resonance heating, particle where R0 is a slowly varying function whose values are beam injection, electromagnetic acceleration, or other meth- given in Table I. ods. Similarly, the recirculating power for the distribution of Due to the difficulties of precisely manipulating particles Eq. ͑11͒ in the beam-like case with vtϵvtsϭvtf is in narrowly defined regions of velocity space, realistic sys- tems for recirculating the power will probably have to re- v0 n͗E͘ P ϭR ͑v /v ͒ , ͑18͒ cycle considerably more than the minimum theoretical recir- recirc 1 0 t ͩ v ͪ ␶ t col culating power. Furthermore, realistic systems will involve in which R1 is a slowly varying function described in Table unlike particle collisions and instabilities that tend to in- II. crease the minimum required recirculating power, and all The recirculating power may be compared with the fu- specific foreseeable systems will also lose a significant sion power. It will be assumed that there are two fuel ion amount of the power in the process of recirculating it. Of species present, one of which is an isotope of hydrogen; x course, real fusion reactors will have many other power loss denotes the ratio of the density of the hydrogen isotope to the mechanisms as well. Aside from the actual losses on the density of the second ion species, and Zi2 represents the recirculating power, simply having to recycle an amount of charge state of the second ion species. The fusion power power comparable to or greater than the fusion power would density may then be written as1 make the reactor technologically cumbersome and relatively x W unattractive as a commercial power source. Because of all of P ϭ1.602ϫ10Ϫ19 n2͗␴v͘ E , these reasons, a nonequilibrium reactor design which is to be fus xϩZ 2 e fus fus,eV cm3 ͑ i2͒ considered promising should probably have a minimum re- ͑19͒ circulating power that is at least one order of magnitude where ne is the electron density, ͗␴v͘fus is the average fusion smaller than the gross fusion power. 3 reactivity in cm /s, and Efus,eV is the energy in eV released per reaction. If there is only one fuel ion species ͑which may C. Results 2 or may not be a hydrogen isotope͒, the factor x/(xϩZi2) in 2 1 In judging the performance of fusion systems, brems- Eq. ͑19͒ should be replaced by 1/2Zi . For the case in which particle species ‘‘a’’ is kept non- strahlung radiation will be considered. The bremsstrahlung Maxwellian with v ϵv ϭv , the recirculating power com- power loss density, including relativistic corrections, is given t ts tf 1 pared with the fusion power is in 2 P v m ⌺iZi ni Te recirc Ϫ6 0 e P ϭ1.69ϫ10Ϫ32n2 T 1ϩ0.7936 ϭ5.34ϫ10 R1͑v0 /vt͒ brem e ͱ e 2 P ͩ v ͪͱm ne ͫ mec fus t a ͭ

2 4 2 2 ͑xϩZi2͒ Zana ln ⌳ Te 3 Te W ϫ 2 , ͑20͒ ϩ1.874 2 ϩ 2 3 , ͑21͒ x ne ͗␴v͘ E ͱ͗E ͘ ͩ mec ͪ ͬ & mec cm fus fus,eV a,eV ͮ 2 To apply Eq. ͑20͒ when vtsӶv0Ӷvtf , one should make the in which the electron temperature Te and rest energy mec substitutions vt vtf and R1 R0 in that equation. are in eV. Although these→ calculations→ of the minimum recirculat- The D–T, D–3He, and D–D fuels can theoretically pro- ing power apply to any possible means of recycling the duce net power when they are burned in a plasma which is power and are not restricted to a particular method, it may be essentially in thermodynamic equilibrium. Such systems will helpful to give specific examples of systems for recycling the not be considered here, since their optimum performance is power. In principle, power may be selectively extracted from discussed in detail in Refs. 1 and 5. Unfortunately, although particular velocity-space regions of the particle distributions the minimum bremsstrahlung power loss from such systems via high-voltage charged particle direct electric converters, is in principle tolerably small in comparison with the fusion electromagnetic radiation from the particles, or other means. power, for D–3He and D–D it is not as small as one might This power may then ͑in principle͒ be processed and rein- wish. Furthermore, for 3He–3He, p-11B, and p-6Li plasmas

Phys. Plasmas, Vol. 4, No. 4, April 1997 Todd H. Rider 1043 TABLE III. Comparison of recirculating power and bremsstrahlung radia- TABLE IV. Comparison of recirculating power and bremsstrahlung radia- tion power with gross fusion power for nearly Maxwellian electron distri- tion power with gross fusion power for isotropic, beam-like electron distri- Ϫ16 3 butions with the slow electrons depleted. ͑The ions are Maxwellian.͒ butions. ͑The ions are Maxwellian. For DT, ͗␴v͘fusϭ8.54ϫ10 cm /s and Efusϭ17.6 MeV.͒ Fuel ͗Ei͘ ͗Ee͘ ͗␴v͘fus Efus Ϫ16 3 ¯ mixture ͑keV͒ ͑keV͒ ͑10 cm /s͒ ͑MeV͒ Pbrem/Pfus Precirc/Pfus Fuel ͗Ei͘ ͗Ee͘ Precirc/Pfus Precirc/Pfus mixture ͑keV͒ ͑keV͒ Pbrem/Pfus ͑v0/vtϭ1͒ ͑v0/vtϭ10͒ D–3He ͑1:1͒ 150 39 1.67 18.3 0.093 5.2 D–D 750 170 1.90 3.7 0.18 2.6 D–T͑1:1͒ 75 63 0.007 7.3 190 3He–3He 1500 160 1.25 12.9 0.50 5.6 D–3He ͑1:1͒ 150 108 0.19 61 1600 p-11B ͑5:1͒ 450 35 2.39 8.7 0.50 52 D–D 750 315 0.35 35 900 p-6Li͑3:1͒ 1200 22 1.60 4.0 0.50 330 3He–3He 1500 160 0.50 85 2200 p-11B ͑5:1͒ 450 35 0.50 350 9100 p-6Li ͑3:1͒ 1200 22 0.50 870 23 000 which are essentially in thermodynamic equilibrium, the bremsstrahlung radiation losses are prohibitively large.1 Since ion–electron energy transfer is mediated by the order to prevent additional neutron production and radioac- comparatively small number of electrons which are moving tivity from reactions of daughter nuclei. Leaving the fusion more slowly than the ions,6 one obvious method of lowering products in the plasma would substantially alter the perfor- the electron temperature and hence the radiation losses in an mance of only two of the fuels. The effective E and P advanced fuel reactor would be to actively deplete those fus fus for D–D would increase by a factor of 5.85 due to burnup of slow electrons. An appropriate non-Maxwellian electron dis- bred T and 3He, but then large numbers of unpleasant 14 tribution may be described by Eq. ͑11͒ with v Ӷv Ӷv , ts 0 tf MeV D–T neutrons would be produced. Also, this perfor- where v Ϸ 2v ϵ 2ͱ2T /m ͑in which the i1 species is the 0 ti1 i i1 mance increase would not be large enough to make most of lighter of the two fuel ion species͒ and v ϳv tf te the D–D systems considered in this paper truly feasible. Al- ϵ 2T /m . ͱ e e lowing the 3He bred by p-6Li to burn up with exogenous D Table III summarizes the minimum recirculating power would effectively improve the performance of p-6Li by a requirements needed to maintain such an electron distribu- factor of 5.5, but it would increase the neutron production tion shape and lower the bremsstrahlung radiation losses while still not rendering the p-6Li systems considered in this P for several different fusion fuels. For this and subse- brem paper feasible.͒ quent tables, the ion energies and fuel mixtures have been Similarly, Tables IV and V reveal the difficulty of main- chosen to approximately minimize the ratio of bremsstrah- taining nearly monoenergetic velocity distributions lung losses to fusion power, and the Coulomb logarithm has ͑v ϵv ϭv and v уv ͒ for electrons and ions, respec- been set at 15, an optimistic value for a magnetic fusion t ts tf 0 t tively. Such beam-like distributions have been proposed for reactor. ͑The lower Coulomb log of inertial confinement fu- use in a number of different nonequilibrium fusion ap- sion does not alter the results enough to change this paper’s proaches, such as inertial-electrostatic confinement,11 conclusions about the viability of various fusion ap- migma,12 and related ideas.13–15 The recirculating power lev- proaches.͒ els for beam-like electrons are clearly prohibitive for all of For D–3He and D–D, the electron energies in Table III the cases in Table IV, regardless of the degree of sharpness have been chosen to reduce the bremsstrahlung losses to half of the distribution peaks. Of all of the beam-like ion cases of what they would be in the equilibrium state,1 and for the considered in Table V, only D–T plasmas might be able to other fuels the electron energies have been chosen to limit operate with an acceptable recirculating power level ͑see the bremsstrahlung to half of the fusion power. Fusion reac- Ref. 15 for an example͒, and even then only when the total tivities are drawn from Ref. 10. D–T is not included in the ion population does not deviate too greatly from thermody- table, since its radiation losses can in theory be made quite namic equilibrium. ͓For D–T and D–3He in Table V, Eq. small even with perfectly Maxwellian electrons. As shown in ͑20͒ has been used to estimate the effects of collisions be- the table, the recirculating power levels are substantially tween unlike ions as well as those between like ions by tak- larger than the fusion power. If the mean electron energy is ing into account the differences in mass and charge between lowered below the values in the table, the recirculating the species.͔ power will increase; if the electron energy is raised, the bremsstrahlung losses will increase. More precise tailoring of the electron distribution shape can lower the recirculating 5 power levels somewhat, but the improvement is far from TABLE V. Comparison of recirculating power with gross fusion power for being large enough to be truly useful. Therefore, all of the isotropic, beam-like ion distributions. ͑The electrons are Maxwellian.͒ systems in Table III fail to meet the criterion for a promising Fuel ͗E ͘ P /P P /P nonequilibrium reactor concept as defined above. ı recirc fus recirc fus mixture ͑keV͒ ͑v0/vtϭ2͒ ͑v0/vtϭ10͒ ͑As discussed in the Introduction, the ultimate goal of D–T ͑1:1͒ 75 0.3 3 this investigation is to examine the cleanest possible fusion 3 approaches. Therefore, in these calculations it has been as- D– He ͑1:1͒ 150 2 20 D–D 750 1.1 9.6 sumed that the fusion products are somehow removed from 3He–3He 1500 4.3 38 the plasma before they can undergo any further reactions, in

1044 Phys. Plasmas, Vol. 4, No. 4, April 1997 Todd H. Rider TABLE VI. Comparison of recirculating power and bremsstrahlung radia- fusion reaction rate or suppress undesirable side reactions. tion power with gross fusion power for the active refrigeration of electrons. Unfortunately, the temperatures of two ion species equili- ͑The ions and electrons are Maxwellian.͒ brate on the order of ͱmi /me faster than the temperatures of 16 ͗Ei͘ ͗Ee͘ ions and electrons interacting with each other, so attempts Fuel ͑keV͒ ͑keV͒ Pbrem/Pfus Precirc/Pfus at energy decoupling between two ion species meet with the D–3He ͑1:1͒ 150 39 0.093 1.9 same fate as ion–electron energy decoupling, as shown ex- D–D 750 170 0.18 0.9 plicitly in Refs. 1 and 5. All currently available techniques 3He–3He 1500 160 0.50 6.2 for potentially decreasing the energy transfer rate between p-11B ͑5:1͒ 450 35 0.50 33 the ion species and lowering the recirculating power levels 6 p- Li ͑3:1͒ 1200 22 0.50 320 are insufficient for the present task.5 Therefore, fusion sys- tems which attempt to decouple the relative energies of two fuel ion species do not appear to be feasible. It has not actually been necessary to assume that the plasma is in steady state, either for these cases of interspecies III. DIFFERENT PARTICLE SPECIES AT RADICALLY energy differences or for the earlier situations with non- DIFFERENT MEAN ENERGIES Maxwellian distributions. For virtually all of the cases con- sidered, it has been shown that the power flow in the plas- As has been mentioned, it would be desirable to keep the ma’s phase space corresponding to particle species adjusting mean electron energy much lower than the mean ion energy their relative energies and velocity distribution shapes is con- in an advanced fuel reactor, in order to minimize the brems- siderably larger than the fusion power. As a result, even strahlung and synchrotron losses. If the ion temperature is pulsed systems in which the plasma is actively reordered held constant and Coulomb friction with the ions is the only back to the desired nonequilibrium state at the end of each energy source available to the electrons, the electron tem- pulse would not be useful; the power involved in reordering perature will equilibrate to a somewhat lower value than the the plasma for the next pulse would exceed the fusion power ion temperature, since the electrons lose energy by derived from the pulse. radiation.1 A hypothetical system for keeping the electron temperature lower than this equilibrium value would have to continually extract a minimum recirculating power from the IV. IMPLICATIONS FOR ADVANCED-FUEL FUSION electrons and return it to the ions in order to keep the ions Because of the neutron production and radioactive in- and electrons ‘‘decoupled’’ in energy. In this case the mini- ventory associated with D–T and D–D fusion, it has been mum recirculating power is P ϵP ϪP , where the recirc ie brem observed that fusion reactors could be made much more de- ion–electron energy transfer rate P is given in Ref. 1, ie sirable if cleaner, more advanced fusion fuels could be used.2 2 If they could be successfully employed, the advanced 0.3T Z n m 3 3 11 6 Ϫ28 e i i p aneutronic fuels ͑ He– He, p- B, and p- Li͒ would be very Pieϭ7.61ϫ10 ne 1ϩ 2 ͚ 3/2 ln ⌳ ͩ mec ͪ i miTe attractive reactor fuels due to the very low neutron produc- tion and radioactive inventories associated with them. Unfor- Z2n 2/3 i i me Ti W tunately, there appears to be no way to produce net power ϫexp Ϫ 3.5͚ ͑TiϪTe͒ 3 , ͑22͒ ͫ ͩ i ne mi Te ͪ ͬ cm with any of these fuels. If they are burned in a plasma which is essentially in thermodynamic equilibrium, the electron 1 in which m p is the proton mass, the temperatures and elec- temperature and hence the radiation losses will be too large. tron rest energy are in eV, and the Coulomb logarithm is As revealed in Table VI, actively cooling the electrons while ln ⌳Ϸ24Ϫln(ͱne/Te). maintaining the ion temperature by somehow recirculating Table VI gives the recirculating power levels required to power from the electrons back to the ions would require one lower the electron temperature in various fuel mixtures to recycle much more power than the fusion power, regard- enough that the bremsstrahlung radiation losses will be sub- less of the specific mechanism for actually returning the stantially reduced from their usual equilibrium values. For power. An alternate method of lowering the electron tem- each of the fuels listed in the table, the amount of power perature would be to actively deplete the very slow electrons which must be recycled is clearly much too large in compari- that mediate ion–electron energy transfer, but this technique son with the fusion power. Methods of passively6 or actively would still require prohibitively large amounts of recirculat- ͑see the previous section͒ depleting the slow electrons to ing power, as shown in Table III. As has been discussed, the reduce the ion–electron energy transfer rate and hence the power recycling requirements also rule out boosting the fu- required recirculating power are insufficient to improve the sion reaction rate over the Maxwellian-averaged value by outlook for ion–electron energy decoupling. Likewise, all keeping one fuel ion species at a substantially different mean other presently available techniques are unable to reduce the energy than the other. recirculating power to manageable levels.5 Thus fusion sys- D–3He is a fusion fuel which can break even against tems that attempt to actively cool the electrons can be ruled radiation losses in an equilibrium plasma, but it is plagued by out. D–D side reactions which produce neutrons and tritium and A related idea would be to maintain two fuel ion species thus keep the fuel from being as clean as would be desired. at significantly different temperatures in order to boost the In an equilibrium plasma, operating much more 3He-rich

Phys. Plasmas, Vol. 4, No. 4, April 1997 Todd H. Rider 1045 than a 1:1 fuel mixture in order to suppress D–D reactions V. CONCLUSIONS leads to radiation losses which are too large and a fusion In this paper we have derived fundamental power limi- power which is too small. Lowering the electron temperature tations that apply to virtually any possible type of fusion and bremsstrahlung losses by active cooling of the electrons reactor in which the electrons or fuel ions possess a signifi- ͑Table VI͒ or by active depletion of the very slow electrons cantly non-Maxwellian velocity distribution or in which two ͑Table III͒ in order to permit the use of more 3He-rich fuel major particle species are at radically different mean ener- mixtures would require intolerably large amounts of recircu- gies. Analytical Fokker–Planck calculations have been used lating power. Likewise, using highly nonequilibrium ion to accurately determine the minimum recirculating power populations in order to improve the ratio of the D–3He and that must be extracted from undesirable regions of the plas- D–D reactivities would also involve the recirculation of too ma’s phase space and reinjected into the proper regions of much power in comparison with the fusion power, as has the phase space in order to counteract the effects of colli- been discussed. sional scattering events and keep the plasma out of thermo- Therefore, the large amounts of power which must be dynamic equilibrium. In virtually all cases, this minimum recycled in order to sustain nonequilibrium fusion plasmas recirculating power is substantially larger than the fusion prevent such systems from being useful for burning the rela- power, so barring the discovery of methods for recirculating tively clean advanced aneutronic fuels or for reducing the the power at exceedingly high efficiencies, reactors employ- radioactivity of D–3He fusion. ing substantially nonequilibrium plasmas will not be able to Furthermore, the results of this paper indicate that fusion produce net power. approaches such as inertial-electrostatic confinement,11 migma,12 and other ideas13,14 which attempt to employ highly nonequilibrium plasmas will probably not even be ACKNOWLEDGMENTS able to produce net power with D–T, and they certainly will P. J. Catto and T. H. Dupree were extraordinarily helpful not be able to produce net power with any of the other fusion in the course of this research. Productive discussions with L. fuels. ͑Most of these proposed approaches do not even have M. Lidsky, P. L. Hagelstein, L. D. Smullin, W. M. Nevins, mechanisms for recirculating power to stay out of equilib- and D. C. Barnes are also gratefully acknowledged. rium, and so they would quickly relax to equilibrium.1 Even The author was supported by graduate fellowships from if they had such mechanisms, those mechanisms would be the Office of Naval Research and from the Department of limited by the constraints found in this paper.͒ This funda- Electrical Engineering and Computer Science at the Massa- mental and broadly applicable limitation is in addition to chusetts Institute of Technology. This work was performed certain design-specific flaws which have already been noted in partial fulfillment of the degree requirements for a Ph.D. with some of these approaches.1,17 in Electrical Engineering and Computer Science from MIT. Some observations should also be made regarding the connections between this paper’s results and the method pro- 18 posed by Snyder et al. for channeling fusion product en- 1T. H. Rider, Phys. Plasmas 2, 1853 ͑1995͒. ergy to fuel ions. Most of the performance improvement re- 2L. M. Lidsky, J. Fusion Energy 2, 269 ͑1982͒. 3 ported in Ref. 18 comes from reducing the heating of E. A. Frieman, M. L. Goldberger, K. M. Watson, S. Weinberg, and M. N. Rosenbluth, Phys. Fluids 5, 196 ͑1962͒. electrons by mechanisms other than Coulomb friction with 4H. P. Furth, Phys. Fluids 6,48͑1963͒. the fuel ions. In comparison, this paper has assumed from the 5T. H. Rider, Ph.D. thesis, Massachusetts Institute of Technology, 1995. 6T. H. Rider and P. J. Catto, Phys. Plasmas 2, 1873 ͑1995͒. outset that there is no heating of electrons by mechanisms 7 other than friction with the fuel ions, and the paper has pro- D. V. Sivukhin, in Reviews of Plasma Physics, edited by M. A. Leontovich ͑Consultants Bureau, New York, 1966͒, Vol. 4, pp. 93–241. ceeded to examine broad categories of approaches for im- 8D. L. Book, NRL Plasma Formulary ͑Naval Research Laboratory, Wash- proving reactor performance still further. Thus this paper has ington, DC, revised 1987͒. 9 focused on potential approaches beyond those which are pro- W. M. MacDonald, M. N. Rosenbluth, and W. Chuck, Phys. Rev. 107, 350 ͑1957͒. posed in Ref. 18; the results of this paper, as discouraging as 10See National Technical Information Service Document No. ORNL TM- they may seem, are actually inherently more optimistic than 6914 ͑J. R. McNally, Jr., K. E. Rothe, and R. D. Sharp, Fusion Reactivity the results reported by Snyder et al. Graphs and Tables for Charged Particle Reactions, Oak Ridge National The numerical results in Ref. 18 also show that in the Laboratory Report ORNL TM-6914, 1979͒. Copies are available from the National Technical Information Service, Springfield, VA 22161. absence of an active particle cooling system, two fuel ion 11R. W. Bussard and N. A. Krall, Fusion Tech. 26, 1326 ͑1994͒. species cannot be kept at substantially different mean ener- 12B. C. Maglich, Nucl. Instrum. Methods A 271,13͑1988͒. gies, as shown analytically in Ref. 1. Furthermore, the nu- 13D. C. Barnes, R. A. Nebel, and L. Turner, Phys. Fluids B 5, 3651 ͑1993͒. 14 merical results of Snyder et al. demonstrate that changing N. Rostoker, F. Wessel, H. Rahman, B. C. Maglich, B. Spivey, and A. Fisher, Phys. Rev. Lett. 70, 1818 ͑1993͒. from Maxwellian to non-Maxwellian ion distributions would 15J. M. Dawson, H. P. Furth, and F. H. Tenney, Phys. Rev. Lett. 26, 1156 alter the fusion reactivity by at most a few percent, provided ͑1971͒. that the ions are at the same mean energy; this change is in 16L. Spitzer, Jr., Physics of Fully Ionized Gases, 2nd revised ed. ͑Wiley– agreement with the results in Ref. 5 and is much too small to Interscience, New York, 1962͒. 17W. M. Nevins, Phys. Plasmas 2, 3804 ͑1995͒. alter this paper’s conclusions about the viability of various 18P. B. Snyder, M. C. Herrmann, and N. J. Fisch, J. Fusion Energy 13, 281 fusion approaches. ͑1994͒.

1046 Phys. Plasmas, Vol. 4, No. 4, April 1997 Todd H. Rider