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 1997! 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. @S1070-664X~97!01404-3# 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 ~iii! 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–T! and sion efficiency limitations are ignored, and power deuterium–deuterium ~D–D!, 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- ~iv! 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 x@n~x!# ~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 L!, in which n~x! 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 ~v! The regions of the plasma which have values of 3 2 in an equilibrium D–3He plasma.1 *d x@n~x!#! 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- ~vi! 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 ~vii! Spatial variations of particle energies may be ne- plasma geometry, reactor confinement system, type of fuel, glected in regions of significant *d3x@n~x!#2. and other key parameters. Those assumptions which are ~viii! The plasma is quasineutral and optically thin to made are as follows: bremsstrahlung. ~ix! In the ion energy ranges of interest, the functional ~i! Losses other than bremsstrahlung radiation and the dependence of the fusion reactivity ^sv& 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. ~ii! Energy transfer from the fuel ions to the electrons is ~x! 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 ^ e& approximately independent of the electron velocity distribution shape in the energy range of interest. a!Present 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- Ap v0 v0 Di' , ~2! ture’’ T of a non-Maxwellian distribution with a mean par- 3A6 S vtestD tcol ticle energy ^E& is defined as T[2^E&/3. In Ref. 1, it was shown that plasma systems which are where the usual definition of the collision time tcol ~Ref. 8! 2 intended to operate far from thermodynamic equilibrium and has been used with ^E&5(3/2)T'mv0/2: yet which have no specific provisions for maintaining such a Am^E&3/2 state will very rapidly relax to equilibrium. Methods of pas- t [ . ~3! col 4 sively maintaining nonequilibrium distributions have also 2A3p~Ze! n ln L 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 [v 1v may be estimated as maintenance of nonequilibrium plasmas will entail certain fast 0 t minimum power requirements and limitations. 2 2 vt 3A6 vt In Sec. II we will examine the limitations that affect tfast' ' tcol , ~4! Di S v D plasma systems which attempt to maintain substantially non- Ap 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 DEfast and limitations that pertain to plasma systems in which two or 2 using ^E&'mv0/2, 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 DEfast5 m~vfast2v0!'2 ^E&. ~5! 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 nfast'n/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 Ap v0 n^E& v0 n^E& here are of interest because they deviate from the Maxwell- P ' '0.24 . ~6! recirc v t v t ian equilibrium in a much more marked fashion than is usual. 3A6 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 vt!v0 on each side of the tion in the electron temperature and radiation losses would mean speed.