arXiv:1910.12719v1 [physics.plasm-ph] 28 Oct 2019 ics h osblt o obnn eetpors in progress recent combining we for Finally, possibility reactivity. the with the barrier, discuss of Coulomb the enhancement of consequent the lowering a a plasma, to vac- D-D lead from may pairs a uum electron-positron for real create achieved temperatures to currently possibility higher one to At the respect than with reactivities. reactivity the the Boltzmann-state of in estimate by enhancement an expected with represented distributions, configurations for energy power-law state cycle non- thermal steady discuss we the Boltzmann specifically, bypassing More dis- and of production. electricity processes, possibility fusion the D-D cuss of reactivity the hancing reaction the [3]. demanding re- self-sustaining more the and are igniting therefore the for and than quirements smaller cross-section, magnitude D-T of reaction. orders corresponding energy two fusion interesting about the the pro- is of in range tritium channels cross-section two D-D the the the to However, of contained one in mainly contamination duced and radioactive and mitigated processes, is released fusion one D-T the dam- than irradiation in smaller in magnitude a of available induce orders two easily neutrons age is MeV avoided 2.45 Deuterium are the issues water, reactors. these D-D sus- All using while flux. by heat, neutron large generate with compet- a tritium and taining several breed processes satisfy to extraction need must situ the easy it in as such tritium as requirements, create issue, ing to nontrivial blanket a lithium is A of reactor. contamination the associated 14.06 by the tritium caused including damage natural neutrons, two irradiation MeV the of has and 2], shortage path [1, This sources the produced power. drawbacks: have fusion recognized which of tokamaks other amount oper- and significant successful JET the ongo- of following the ation ITER, with this paved of within been development and has ing mixtures, path D-T well-defined a on frame based be will reactors nti otiuin eotietopooasfren- for proposals two outline we contribution, this In fusion commercial first the that assumed usually is It 1 iatmnod iiaeAtooi GlloGlli,Uni Galilei”, “Galileo Astronomia e Fisica di Dipartimento 2 eateto hsc n srnm,DrmuhClee 61 College, Dartmouth Astronomy, and Physics of Department r icse.Hr al nteeeg itiuino h n the of distribution energy the in tails possible Hard Two discussed. compl reactors. are successful fusion the (D-T) after deuterium-tritium developed for necessarily be to have hnuulycniee nDTpams aumpolarizatio vacuum plasmas, f D-T for in available nuclei considered energetic usually of number than the boost to allow h s ffso ecost rae otxs otnotably most contexts, broader to reactors fusion of b ph use can third-generation the cycle and thermal calorimetry hadronic usual using The levels t compactness. from contributio and benefit simplicity their can in system to collection due energy speed-up the Furthermore, further provide may pairs ervstteasmto htratr ae ndeuterium- on based reactors that assumption the revisit We .INTRODUCTION I. ocpsfradueimdueimfso reactor fusion deuterium-deuterium a for Concepts oet Onofrio Roberto ein n lntr eua.Tecaatrzto of characterization The HII nebulae. in planetary temperatures dis- and systematic electron regions solve determining to in application an crepancies for environ- [6] astrophysical and in ments, applications comprehen- a with for overview [5] sive instance for see decades, several since tosfo h otmn itiuin hs energy These devi- distribution. large named Boltzmann distributions, generating the – from trans- particles hardly ations energy be- is which lower basically energy to – of ferred pile-up enhanced large be the high-energy may of cause the distribution circumstance the this of Under tail of [4]. instance, low-density, wind for solar in characteristic, it- occur plasmas plasma already the low-temperature situations of higher Analogous equilibrium significantly to self. is rate can heated power relaxation situation heating the is state the than plasma steady which a the in injection, If occur the beam about neutral assumption. scrutinize by to this worth of is it adequacy dis- plasma, confined occurring energy a dynamics Boltzmann for complex of the hypothesis Considering the tributions. on relies rates use oeqatttvl ntecoefuture. close be the the to in to quantitatively directions more contributing research pursued distinct of as three contingency and with progress, the Festschrift, in by work limited of should overview such paper an low- This as in considered instance. plants enough be for power than areas low more populated for power, density and fusion transport of lim- maritime MW are for 1 conversion indi- about energy yield to this light ited of the applications in that and decrease cate calorimeter possible the consequent in un- compact induced the with a damage within cycle, Neutron and thermal design. improvement, a for to margins compara- respect known nearly with initially efficiency an could at ble This conversion energy cells. to photovoltaic lead calorime- third-generation hadronic with and try materials scintillating yield high estad aoa i azl ,Pdv 53,Italy 35131, Padova 8, Marzolo Via Padova, versit`a di O-OTMN NRYDISTRIBUTIONS ENERGY NON-BOLTZMANN h sa oprsnbtenDDadDTfusion D-T and D-D between comparison usual The I NACN H ECIIYVIA REACTIVITY THE ENHANCING II. tvlaccls osbyalwn oextend to allowing possibly cells, otovoltaic 7Wle aoaoy aoe,N 35,USA 03755, NH Hanover, Laboratory, Wilder 27 ,2 1, ce,truhteso-called the through uclei, to feprmnsaddemonstrations and experiments of etion ehnssfrehnigtereactivity the enhancing for mechanisms so ecin.A ihrtemperatures higher At reactions. usion easneo h ihu lne,both blanket, lithium barrier. the Coulomb of the absence he of screening to n aiietransport. maritime yasdwt oprbeefficiency comparable with bypassed e ffcsfo real from effects n etru DD uinprocesses fusion (D-D) deuterium κ dsrbtos aebe discussed been have -distributions, e + κ e − -distribution, and µ + µ − 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. At the same time, it is inter- + Eσ(E) dE κ+1 . (5) esting to notice that the range of temperatures required Z0 E for sustained D-D fusion reactions is not dissimilar from 1+ (κ−3/2)k T h B U i the one in which real electron-positron pairs can be gen- In Fig. 1 we show the dependence on temperature of the erated. These electron-positron pairs make the medium reactivities evaluated with averages over Boltzmann and in between the two nuclei endowed with an effective di- κ-distributions for both D-D and D-T fusion reactions. electric constant, and therefore in principle may reduce There is a significant advantage in using κ-distributions, the Coulomb barrier. which should be also beneficial for the D-T reactors at The production of particle-antiparticle pairs is a pro- relatively low temperatures, with reactivity gains of al- cess studied in finite temperature and density quantum most two orders of magnitude for temperatures in the few field theory, with applications to a variety of research con- KeV range. Fig. 1 also shows that the κ-distribution is texts ranging from astrophysics and cosmology to the ob- not consistently preferable to the Boltzmann distribution servation of quark-gluon plasma at proton colliders (see over the entire temperature range. The κ-distributions for instance [9–11]). The process is nonperturbative in are peaked at an energy lower than the corresponding character, and the particle-antiparticle production rate Boltzmann distribution, with a larger peak probability scales with the mass of the particle and the environmen- tal temperature as exp( 2mc2/k T ). Therefore, aim- (see for instance Fig. 3 in [6]). It is therefore under- − B standable that, for a given dependence of the fusion cross- ing for electrical polarizability effects, and considering the temperature achievable in thermonuclear fusion, we section on energy and with respect to the corresponding + − Boltzmann distribution, they can prevail at lower tem- initially limit the attention to electron-positron (e e ) peratures, then have an intermediate region of marginal pairs alone. Even in this case, if one considers plasma or no gain, and prevail again at higher temperatures. It temperatures of the order of 100 and 200 keV for in- is also worth to note that the D-D fusion reaction has a stance, the corresponding exponential suppression fac- tors are respectively 3.6 10−5 and 6.4 10−3. A de- Coulomb barrier of about 450 keV. This means that the × × high temperature behavior of the reactivity presented in tailed discussion of the effective potential between two Fig. 1a, purely based on tunneling phenomena, is a sort charges separated by a sea of electrons and positrons at of lower bound on its actual value. Ongoing work here finite temperature and density is available in [12]. For mainly consists in developing kinetic models capable, for temperatures low enough with respect to the production threshold of 2m c2 1 MeV and negligible chemical po- instance given the power level of neutral beam injection e ≈ heating, of finding if the plasma energy may be expressed tential, an effective Yukawa potential has been evaluated in terms of a κ-distribution, including predictions for the [12] value of κ and its dependence upon the experimental parameters. Characterization of non-Boltzmann energy e2 r distributions is of current experimental interest, as wit- V (r)= exp , (6) eff 4πǫ r  λ  nessed by recent work at the ASDEX upgrade tokamak − 0 − eff [8]. with the Yukawa range

III. ENHANCING THE REACTIVITY VIA 2 1/4 2 ~ πmec mec VACUUM POLARIZATION EFFECTS λeff = 2 2 exp . (7) 2mec 2αemkBT  2kBT 

Another possible mechanism which can be used to en- where αem is the fine structure constant. hance reactivities consists in exploiting vacuum polariza- The above can be interpreted as a temperature- tion effects. The Boltzmann-averaged reactivity of the D- dependent Compton wavelength for the electron-positron T fusion process is maximum at about 70 keV with a rela- pairs, or alternatively as an effective, temperature- tively broad peak, therefore there is no gain in increasing dependent, photon mass. This leads to a space- the temperature of a D-T plasma above this value. As dependent weakening of the electrostatic repulsion be- a matter of fact, plasma temperatures of operating D-T tween two nuclei. In Fig. 2 we plot the dependence on tokamaks are in the 10-30 keV range at most. For D-D temperature in a range of interest for D-D fusion pro- plasmas it is instead advantageous to operate at higher cesses of various relevant length scales, namely the effec- temperatures, as the reactivity increases monotonically tive Yukawa range from Eq. (7), the Debye length of a until it reaches an even broader peak at temperatures of plasma with densities usually achieved in tokamaks, the about 1 MeV. For instance, the same reactivity of D-T distance of minimum approach between two nuclei with 2 fusion at the temperature of 10 keV is achieved for D-D a kinetic energy equal to kB T , rT = e /(4πǫ0kB T ) and, 4

0 10 considered no more than an estimate, providing an up-

-2 per bound to the reactivity enhancement, which can be 10 evaluated for instance using screening corrections as dis- -4 10 cussed in [13–15]. However, the more energetic nuclei

(m) λ Debye which are mainly responsible for fusion have a classical -6 r 10 p distance of approach much smaller than their average Debye r λ -8 T value rT , thus shielding is not effective in that part of , 10 λ

eff eff the trajectory. For the same reason, the screening ef-

λ -10

, 10 fect expected for a κ-distribution is even smaller. On the p

, r -12 other hand, the electron-positron plasma has a smaller T 10 r inertia than the ions, so it can adapt itself to the new -14 10 situation creating a time-dependent barrier with an in- termediate effectiveness. A dynamical response model is -16 10 1 2 required to accurately describe this effect. 10 10 T (keV) It seems that the spontaneous creation of electron- positron pairs at finite temperature is not enough to im- FIG. 2: Length scales relevant for screening the Coulomb prove screening of the Coulombian potential, and alter- repulsion between nuclei. The effective Compton wavelength native and more complex schemes should be considered. − for a finite temperature e+e plasma is evaluated according For instance, the possibility to enhance reactivity via con- 2 1/2 to Eq. (6). The Debye length λDebye = (ǫ0kB T/(ne )) is trolled microexplosions as a compression stage has been 3 evaluated for a typical ion density of n = 1020ions/m , with discussed in [16]. One could think to increase the screen- the consequent crossover between the screening length scales ing by injecting a large amount of electrons to upset the λDebye and λeff occuring in the 15-20 keV temperature range. densities of electrons and ions. The latter situation has also the advantage of improving sympathetic heating of the ions in case electron cyclotron resonance heating is for comparison, the (constant) distance at which strong implemented, due to the favorable heat capacity match- interactions give rise to fusion processes, of the order of ing. Unfortunately it is easy to show that the densities the proton radius rp. It is worth to notice that around required to have at least an electron-positron pair in be- T 17 keV the screening due to the electron-positron ≃ tween the distance of minimum approach of two nuclei plasma takes over the usual Debye screening, and be- are prohibitive. A less impossible idea is to intersect the comes dominant at higher temperatures. confined plasma with the focused beams of an electron- Unfortunately, in spite of the favorable scaling of the positron collider having enough energy to generate µ+µ− effective Yukawa range with temperature with respect to pairs. By replacing the electron mass with the muon mass the Debye length, the polarization effect is rather small. in Eq. (7), and provided the effective temperature of the This is already visible in Fig. 2 as the effective Compton µ+µ− gas is high enough, one can create a situation in wavelength, although decreasing by increasing tempera- which λeff rT . This is analogous to muon catalysis, but ture at variance with the Debye length, is still about four it is hard≃ to imagine that with few intersection points orders of magnitude larger than the average distance of between the core of the confined plasma and the e+e− minimum approach between the nuclei. This is confirmed collider there will be enough heating power to ignite fu- by a more quantitative analysis by introducing an aver- sion, unless a confinement geometry in which the whole age polarizability coefficient ǫr such that plasma and the e+e− beams coexist is designed.

rT − 3 ǫ 1 = drr2 exp( r/λ ). (8) r 3 3 eff IV. ENERGY CONVERSION BY HADRONIC rT rp Zrp − − CALORIMETERS AND HIGH-EFFICIENCY such that the effective potential in Eq. (6) is obtained PHOTOVOLTAIC CELLS via the substitution ǫ0 ǫrǫ0. The weakening of the electrostatic repulsion due→ to quantum vacuum screen- Since a D-D reactor does not need a lithium blan- ing expressed by Eq. (8) is considered in the classically ket, this allows for revising the conversion of the neutron forbidden volume used for the WKB evaluation of the energy into electrical energy. In particular, the tradi- tunneling rate, i.e. between the nuclei distance rp (of the tional thermal cycles have an overall efficiency limited to order of a proton radius) and the distance of minimum about 30 %. Recent developments in scintillating materi- approach rT . After integration and series expansion of als, decades-long experience with hadronic calorimeters, the exponential term, one gets an effective relative per- and progress in photovoltaic conversion may allow for mittivity by which the Coulomb electrostatic repulsion is an alternative scheme bypassing the thermal cycle while weakened as ǫ 1+ r /(4λ ) which is negligible for achieving comparable efficiency. The idea is to mimic r ≈ T eff rT /λeff << 1. the solar energy collection. In the latter case, the Sun The gain in reactivity evaluated in this way should be is emitting with a peak around the visible region of the 5 electromagnetic spectrum, and the few eV of the photons dle seems in sight. The production and control of κ- in this regions are adequate to create electron-hole pairs distributed energy for the plasma requires a kinetic ap- in semiconductors. In our case, since fusion in reactors proach related, for instance, to the dynamics of neutral must occur at higher temperatures to reach reasonable beam injection heating. Exploiting quantum vacuum ef- amounts of power, a wavelength shifter of the produced fects from existing e+e− pairs at temperatures one order photons is required. Therefore, neutrons emitted from of magnitude higher than usually achieved in tokamaks, half of the D-D fusion reactions are progressively ab- or from intersecting e+e− beams with the plasma to pro- sorbed in a calorimeter producing ionization and scintil- duce µ+µ− pairs for efficient Coulomb shielding, will re- lation. A material scintillating with a light yield of 30% quire a radical revision of the design of current fusion has been recently discovered [17] and third generation experiments. Calorimetry and photovoltaic panels need photovoltaic cells now under development are expected to to be integrated into the unique goal of energy conver- reach 70% efficiency [18]. Therefore, combining the two sion to bypass the limitations in efficiency intrinsic to the technologies, efficiencies of the order of 20 % seem within thermal cycle, opening up a vast spectrum of applications reach. The power of the reactor in this approach seems for fusion energy through compact, low-power plants. limited by the radiation damage induced by neutrons in This project could be developed in parallel to the exist- the calorimeter, with detectors at the Large Hadron Col- ing ITER-DEMO plan on D-T fusion, with no subtrac- lider at CERN representing a state of the art design in tion of human and financial resources to these existing allowed neutron flux (estimated to 2 1017 neutrons/cm2 and already planned facilities. By now, the margins of − for an integrated luminosity of 3000× fb 1 [19]), and the safety for embracing a sustainable, continuous, control- light yield has been found to decrease with increasing lable and clean source of energy before irreversible cli- neutron fluence [20]. mate changes are extremely limited. As such, the ITER- Such a scheme can be feasible, in light of the maximum DEMO project and similar ones under development need affordable neutron flux, for compact low and medium to be developed at the maximum speed and extent. power fusion reactors, with applications to decentralized Apart from obviously requesting more resources to the electricity production in regions requiring low power den- society at large, we believe that extra-resources on par- sities (for instance rural regions, and as a complement to allel designs as the one sketched here could be available intermittent, renewable sources like wind and solar en- once a consistent part of the scientific community will ergy), and especially in the sector of maritime transport, acknowledge where the most urgent, impactful priorities with the prospect of a virtually unlimited range of the are. In this regard, it is reasonable to expect that the vessels. Container vessels, due to their emissions and currently disproportionate emphasis on quantum infor- sheer number, significantly contribute to air and water mation technologies – mainly driven by military and fi- pollution [21]. Notice that, due to the compact design of nance secrecy issues – will fade away, especially once re- a fusion reactor, the usual request for low-cost per unit searchers will recognize the large ratio between promised of surface of a photovoltaic cell is not a priority as in ex- and delivered results [22]. A balanced transfer of hu- tensive solar power plants. This could lead to the search man and financial resources from quantum information for higher efficiency–higher price photovoltaic cells, still technology to research in controlled plasma fusion will be convenient especially if many (now unaccounted for) en- highly beneficial for our unique and irreplaceable Planet vironmental costs are considered. Earth.

V. CONCLUSIONS Acknowledgments

We have discussed somewhat futuristic approaches to It is a special honor to contribute to the Festschrift for nuclear fusion both in enhancing reactivity of D-D fu- colleague and friend Lev Petrovich Pitaevskii, whose out- sion processes, and in delivering electricity without the standing contributions to some of the themes discussed use of a thermal cycle. It is possible that various as- in this note, plasma physics, quantum field theory, and pects of this proposal may become more realistic with statistical physics, will continue to be inspirational for dedicated efforts, as no fundamental technological hur- many generations to come.

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