Concepts for a deuterium-deuterium fusion reactor Roberto Onofrio1, 2 1Dipartimento di Fisica e Astronomia “Galileo Galilei”, Universit`adi Padova, Via Marzolo 8, Padova 35131, Italy 2Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA We revisit the assumption that reactors based on deuterium-deuterium (D-D) fusion processes have to be necessarily developed after the successful completion of experiments and demonstrations for deuterium-tritium (D-T) fusion reactors. Two possible mechanisms for enhancing the reactivity are discussed. Hard tails in the energy distribution of the nuclei, through the so-called κ-distribution, allow to boost the number of energetic nuclei available for fusion reactions. At higher temperatures − − than usually considered in D-T plasmas, vacuum polarization effects from real e+e and µ+µ pairs may provide further speed-up due to their contribution to screening of the Coulomb barrier. Furthermore, the energy collection system can benefit from the absence of the lithium blanket, both in simplicity and compactness. The usual thermal cycle can be bypassed with comparable efficiency levels using hadronic calorimetry and third-generation photovoltaic cells, possibly allowing to extend the use of fusion reactors to broader contexts, most notably maritime transport. I. INTRODUCTION high yield scintillating materials and hadronic calorime- try with third-generation photovoltaic cells. This could lead to energy conversion at an initially nearly compara- It is usually assumed that the first commercial fusion ble efficiency with respect to a thermal cycle, with un- reactors will be based on D-T mixtures, and within this known margins for improvement, and within a compact frame a well-defined path has been paved with the ongo- design. Neutron damage induced in the calorimeter and ing development of ITER, following the successful oper- the consequent possible decrease in the light yield indi- ation of JET and other tokamaks which have produced cate that applications of this energy conversion are lim- significant amount of fusion power. This path has two ited to about 1 MW of fusion power, more than enough recognized drawbacks: the shortage of natural tritium for maritime transport and for low power plants in low- sources [1, 2], and the irradiation damage caused by 14.06 density populated areas for instance. This paper should MeV neutrons, including the associated contamination of be considered as an overview of work in progress, and as the reactor. A lithium blanket to create tritium in situ such limited by the contingency of contributing to the is a nontrivial issue, as it must satisfy several compet- Festschrift, with three distinct research directions to be ing requirements, such as the need to breed tritium with pursued more quantitatively in the close future. easy extraction processes and generate heat, while sus- taining a large neutron flux. All these issues are avoided by using D-D reactors. Deuterium is easily available in II. ENHANCING THE REACTIVITY VIA water, the 2.45 MeV neutrons induce a irradiation dam- NON-BOLTZMANN ENERGY DISTRIBUTIONS age two orders of magnitude smaller than the one released in D-T fusion processes, and radioactive contamination The usual comparison between D-D and D-T fusion is mitigated and mainly contained to the tritium pro- rates relies on the hypothesis of Boltzmann energy dis- duced in one of the two channels of the fusion reaction. tributions. Considering the complex dynamics occurring However, the D-D cross-section in the interesting energy for a confined plasma, it is worth to scrutinize about the range is about two orders of magnitude smaller than the adequacy of this assumption. If the plasma is heated corresponding D-T cross-section, and therefore the re- by neutral beam injection, a steady state situation can quirements for igniting and self-sustaining the reaction occur in which the heating power is significantly higher are more demanding [3]. than the relaxation rate to equilibrium of the plasma it- In this contribution, we outline two proposals for en- self. Analogous situations already occur in low-density, arXiv:1910.12719v1 [physics.plasm-ph] 28 Oct 2019 hancing the reactivity of D-D fusion processes, and dis- low-temperature plasmas characteristic, for instance, of cuss the possibility of bypassing the thermal cycle for solar wind [4]. Under this circumstance the high-energy electricity production. More specifically, we discuss non- tail of the distribution may be enhanced – basically be- Boltzmann steady state configurations represented by cause of the large pile-up of energy which is hardly trans- power-law energy distributions, with an estimate of the ferred to lower energy particles – generating large devi- expected enhancement in the reactivity with respect to ations from the Boltzmann distribution. These energy Boltzmann-state reactivities. At higher temperatures distributions, named κ-distributions, have been discussed than the one currently achieved for a D-D plasma, the since several decades, see for instance [5] for a comprehen- possibility to create real electron-positron pairs from vac- sive overview with applications in astrophysical environ- uum may lead to a lowering of the Coulomb barrier, with ments, and [6] for an application to solve systematic dis- a consequent enhancement of the reactivity. Finally, we crepancies in determining electron temperatures in HII discuss the possibility for combining recent progress in regions and planetary nebulae. The characterization of 2 -15 -15 10 10 D-D D-T -16 -16 10 10 /s) /s) κ=2 3 3 -17 -17 10 10 κ=4 v> (cm v> (cm σ σ < < -18 -18 Boltzmann 10 κ=2 10 κ=4 Boltzmann -19 -19 10 1 2 10 1 2 10 10 10 10 T (keV) T (keV) FIG. 1: Reactivities for D-D (left) and D-T (right) fusion processes averaged over Boltzmann energy distributions (black) and over two κ-distributed energies, κ = 2 (blue) and κ = 4 (red), in the 2-100 keV temperature range. Reactivities and temperatures in the two plots are expressed with the same scales allowing for an easier comparison. The D-D reactivity is obtained summing over both reaction channels, D(d,p)T and D(d,n)3He. these κ-distributions requires the introduction of two pa- scribed in terms of the S-factor capturing the nuclear rameters, the kinetic temperature, an effective tempera- physics of the fusion process and softly dependent on the ture such that the energy per unit of particle U can still energy (modulo possible resonance phenomena) and a be written as U = 3kBTU /2 as in the Boltzmann case, factor which incorporates the tunneling process through and the κ-parameter (with values in the 3/2 <κ< + the Coulombian barrier, typically determined through a range). The energy probability density is expressed as [6]∞ Wentzel-Kramers-Brillouin (WKB) approximation C(κ) E1/2 S(E) BG P (E)= , (1) σ(E)= exp . (2) (k T )3/2 κ+1 E −√E B U 1+ E (κ−3/2)kB TU h i where BG is the Gamow constant. Here, the S-factor is fitted with a Pad´epolynomial, following the notation 1/2 3/2 where C(κ) = 2Γ(κ + 1)/[π (κ 3/2) Γ(κ 1/2)]. introduced in [7], that is The Boltzmann distribution is− recovered in− the κ + case, with the kinetic temperature of the κ distribu-→ ∞ A1+ E(A2 + E(A3 + E(A4 + EA5)))) tion tending to the temperature Tcore of the Boltzmann S(E)= . (3) distribution interpolating its “core” distribution, i.e. the 1+ E(B1 + E(B2 + E(B3 + EB4))) region of energies with the most probable population, The numerical values of Ai and Bj are determined with a T = [1 3/(2κ)]T . The concrete value of κ is usu- core − U best fit and tabulated in Table IV in [7]. The correspond- ally determined from a best fit of the observed energy ing values of the cross-sections are reliable within few % distribution and, as far as we are aware of, there is not in the 3-400 keV energy range, enough for our discussion. yet a kinetic model to predict its value. For now, we will Fusion reactivities are then evaluated by averaging the assume values of κ as typically inferred by space plasma product of the fusion cross-section and the relative ve- physics, to have a common-sense, perhaps questionable, locities v between the two nuclei over the corresponding benchmark. energy distribution. For Boltzmann-distributed energies, Since most of the fusion reactions occur in the high- this implies energy tail of the energy distribution, and the κ- distributions are characterized by a hard, power-law tails, 1/2 +∞ a first exercise may consist in evaluating the gain in reac- 8 1 −βE σv = 3/2 dEEe σ(E), tivity for various fusion reaction by using κ-distributions h i πmr (kB Tcore) Z0 en lieu of Boltzmann ones. To be fair in the comparison, (4) we will compare κ-distributions with effective tempera- where mr is the reduced mass of the system made of ture TU to the Boltzmann distributions with the corre- two colliding nuclei and β the inverse temperature β = −1 sponding core temperature Tcore as defined above. We (kBT ) . The reactivity depends on temperature, which have used parameterized cross-sections for D-T and the allows for a comparison to the parametrized fusion reac- two channels of the D-D fusion process from [7]. As tivities tabulated in Table VII of [7]. In the case of the customary for fusion processes, the cross-section is de- κ-distribution, we have 3 fusion only at the temperature of about 300 keV. Pur- suing this high-temperature scenario obviously opens up 1/2 8 C(κ) challenging technological issues due to the heat irradi- σv = 3/2 h i πmr (kB TU) × ated on the first wall, and to the large energy losses due ∞ to electron bremmstrahlung.