key: cord-0145802-xfrhmzi2 authors: Belloni, Fabio title: On a fusion chain reaction via suprathermal ions in high-density H-$^{11}$B plasma date: 2020-12-28 journal: nan DOI: nan sha: 989540a86202538b33ab3b37bf42995f43f8e255 doc_id: 145802 cord_uid: xfrhmzi2 The $^{11}$B$(p,alpha)2alpha$ fusion reaction is particularly attractive for energy production purposes because of its aneutronic character and the absence of radioactive species among reactants and products. Its exploitation in the thermonuclear regime, however, appears to be prohibitive due to the low reactivity of the $^{11}$B fuel at temperatures up to 100 keV. A fusion chain sustained by elastic collisions between the alpha particles and fuel ions, this way scattered to suprathermal energies, has been proposed as a possible route to overcome this limitation. Based on a simple model, this work investigates the reproduction process in an infinite, non-degenerate $^{11}$B plasma, in a wide range of densities and temperatures which are of interest for laser-driven experiments ($10^{24} lesssim n_e lesssim 10^{28} {rm cm}^{-3}$, $T_e lesssim 100$ keV, $T_i sim$ 1 keV). In particular, cross section data for the $alpha$-$p$ scattering which include the nuclear interaction have been used. The multiplication factor, $k_infty$, increases markedly with electron temperature and less significantly with plasma density. However, even at the highest temperature and density considered, and despite a more than twofold increase by the inclusion of the nuclear scattering, $k_infty$ turns out to be of the order of $10^{-2}$ only. In general, values of $k_infty$ very close to 1 are needed in a confined scheme to enhance the suprathermal-to-thermonuclear energy yield by factors of up to $10^3$-$10^4$. The advanced fusion fuel (Dawson, 1981) based on a H-11 B mixture would exploit the reaction 11 B + p → 3α + 8.7 MeV which is particularly attractive as it involves only abundant and stable isotopes in the reactants, and no neutron in the reaction products. The reaction cross section shows a main, wide and Fisch, 2004) and, away from degeneracy, e.g. for ≳ 10 ⁄ when ≳ 150 keV and the boron-to-hydrogen ion concentration is lower than 50% (Levush and Cuperman, 1982) . In an infinite, homogeneous H 11 B plasma, an α particle emitted at a certain energy , is characterised by the multiplication (or reproduction) factor , , which is the average number of secondary α-particles generated via suprathermal processes during its slowing down. For the purpose of this work, the multiplication factor in terms of fusion events, , is well defined by the relation where is the α spectral distribution normalised to 1. Each fusion event gives rise to new generations of events along a geometric progression. The number of events adds up, on average, to 1 + + +. . . + +. .., where is the number of generations. Along these lines, it is not difficult to show that upon a thermonuclear rate per unit volume R, the time evolution of the cumulative number density of fusion events, ( ), is given by where is the average period between two consecutive generations, and the initial condition (0) = 0 has been assumed. Depending on whether i) < 1, ii) = 1, or iii) > 1, can respectively i) increase linearly with t, asymptotically to (1 − ) ⁄ , for → ∞, ii) increase quadratically with t, as it reduces to + 2 ⁄ , or iii) diverge exponentially, with the growth rate ⁄ . of plasma densities and temperatures which are of interest for current and future laser-based experiments. In particular, cross section data for the α-p scattering which include the nuclear interaction have been used. The aspiration is that results and conclusions can help inform the choice of parameters and the development of techniques in future experiments, with a view to maximising multiplication effects. We assume a two-temperature ( , ), H 11 B plasma with ions per unit volume (the subscript j stands for p or B); the density ratio ⁄ is denoted by . Indicating by , the energy of an ion just after the scattering by an α particle, we will consider as suprathermal those secondary ions for which , ≫ . In this limit, the number of ions of the species j scattered into the energy interval , , , + , through the path length of the α particle is given where is the differential scattering cross section. The spectral distribution of these ions through the entire path length of the α particle is then where ⁄ is the stopping power of the α particle (taken as a positive quantity). For pure Coulomb scattering, is given by the well-known Rutherford cross section, , which reads, in terms of , , as where the z's are particle electric charges in units of the elementary charge e, the m's are particle masses, H is the Heaviside step function, and the endpoint energy , is given, from basic kinematics, by , = Denoting by the contribution to of the suprathermal population of the species j, it is straightforward to see that for an α particle of energy , , the spectrum of over , is linked to the spectrum of by where is the fusion probability of the ion throughout its thermalisation, and the factor of 3 is the number of α's per fusion event. , is calculated by numerical integration of Eq. (7) over , . In turn, is calculated by summing over the contribution of each ion species, i.e. Concerning the fusion probability in Eq. (7), it is easy to show (e.g. Giuffrida et al., 2020 ) that in the cold-ion approximation for the target species, and for ≪ 1 (which is our case, see Sect. 3), the following relation holds where ⁄ is the stopping power, is the fusion cross section, and is the CM energy of the p-11 B system, i.e. Calculations of the abovementioned quantities have been performed in conditions relevant to laser-driven fusion plasmas and are reviewed in Sect. 3. In computations, we have adopted the following input data or models and considerations. α spectrum, , By the sake of simplicity, we have used the crude two-group approximation according to which, on average, one α particle is emitted at energy , = 1 MeV and the other two at , = 4 MeV (Stave et al., 2011) . Consequently, , in Eq. (1) is given by where is the Dirac delta function. The analytic approximation of Nevins and Swain (2000) has been used below 3.5 MeV, and an interpolation of TENDL evaluated data (Koning et al., 2019) at higher energies (Fig. 1a ). For the α-p scattering, evaluated cross section data have been interpolated from SigmaCalc (Gurbich, 2016) . A comparison with is displayed in Fig. 2 , which shows that the nuclear contribution dramatically increases the cross section, within a factor of 3 for ≲ 2 MeV, up to a factor of 10 around 4 MeV and at high values of , , and by hundreds of times at progressively higher values of and , . Nevertheless, for > 4 MeV and , < 1 MeV, a wide area exists where ⁄ < 1, which is an effect of the interference between the Coulomb and nuclear scattering amplitudes (Perkins and Cullen, 1981) . For the α-11 B scattering, the Rutherford cross section has been used as the higher Coulomb barrier makes the nuclear contribution negligible at the energies concerned. The Spitzer-Sivukhin model (Spitzer, 1956; Sivukhin, 1966) has been used, in the form for a multicomponent (electrons and ion species), two-temperature H 11 B plasma detailed in Levush and Cuperman (1982) . Earlier, this model had also been used by Moreau (1976 Moreau ( , 1977 for slowdown calculations in high-density H 11 B plasma. With a view to the subsequent discussion, it is worth recalling the form of the electronic stopping power in the expressions of ⁄ and ⁄ , i.e. where the subscript q stands for α, p or B, and the functions Λ and are given by Sivukhin (1966) . Analogous formulas hold for the q-p and q-11 B components of the stopping power. With reference to Eqs. (4) and (9), it is useful to write the ion densities appearing therein in terms of the electron density, , and , i.e. Orders of magnitudes between 10 24 and 10 28 cm -3 have been considered for . As a term of reference, for amorphous boron in STP conditions, = 1.3 × 10 cm -3 , hence = 6.5 × 10 cm -3 . If one considers, by the sake of simplicity, only the electronic stopping power in the expressions of ⁄ and ⁄ [Eq. (12)], it is straightforward to note that the apparent linear dependence on in Eqs. (4) and (9) actually cancels out, leaving factors which depend on according to Eqs. (13). This implies that Eq. (7) and derived quantities depend on through the overall factor + ⁄ , which has a maximum at = 0.2 for = 1 and = 5. In reality, the dependence of the multiplication factors on is obviously much more complicate because of the dependence on of the ion-ion components of the stopping powers. The optimum has to be calculated numerically and depends, moreover, on the specific set of parameters entering the equations. Its value and the corresponding maximum values of and , however, are in general not dramatically affected, as shown in Sect. 3. Minimising ⁄ and ⁄ in Eqs. (4) and (9), respectively, requires high values of both and in a classic (Maxwell-Boltzmann) plasma. As a precondition, one wants to deal with a fully ionized plasma in order to reduce the electronic component of the stopping power (Giuffrida et al., 2020) ; accordingly, should be much higher than the ionization energy of B 4+ , which is 0.34 keV for the isolated ion (Lide, 2000) and less in high-density matter (More, 1993 ). Even when is higher than the B 4+ ionization energy, electrons can still be Fermidegenerate at the high densities considered here. The Fermi energy , which scales as ⁄ , has been plotted in Fig. 3 for reference. In view of further reducing the electronic stopping power, we have verified that in our density domain it is convenient to work at low degeneracy, e.g. at ⁄ > 5, compared to the fully degenerate case (Son and Fisch, 2004; Giuffrida et al., 2020) . Accordingly, at a given , we have chosen to work with a classic plasma with > 5 , a condition which in our case ensures both full ionization and low degeneracy. On the other hand, at a given , the lower the more effective is the suprathermal energy transfer. Indeed, in the limit = 0, all the scattered ions are obviously suprathermal; moreover, from basic kinematics, the collisional energy transfer from the α particle to the ion occurs as long as the velocity of the former is higher than that of the latter. We have chosen to set = 1 keV, which is a good compromise among the needs to reduce the ion-ion component of the stopping power, increase the α-to-ion energy partition, and ensure a suprathermal spectrum as wide as possible ( ≪ , ≤ , ). As a matter of fact, the contribution to of protons with , < 10 keV is absolutely negligible; see Sect. 3. Incidentally, in a low-scheme the thermonuclear burn will be very modest and will just be used to seed the chain reaction, which is expected to The 11 B-ion contribution to in Eq. (8) , , it is immediate to recognise that the contribution to and , from protons with , ≲ 100 keV is negligible. This is obviously due to the extremely low fusion probability at those energies. One can also notice that the contribution to , of the low-energy α's (blue curve) is rather limited. A detailed analysis of the behaviour of , with and is shown in Fig. 6 . In panel a), curves have been generated for three different values of (10, 50 and 100 keV), keeping fixed at 10 26 cm -3 . As a term of comparison, a curve based only on the Rutherford α-p scattering has been calculated at = 50 keV. In panel b), has been varied by 1-decade steps from 10 24 to 10 28 cm -3 while keeping fixed at 100 keV. In all cases, the curves quickly drop below , ≃ 2 MeV. Above 4 MeV, their shape is approximately linear in the semi-log plot, meaning an exponential increase with , . A fit with a function of the form ∝ , on the curves of panel b) returns ≃ 0.45 MeV for 4 ≤ , ≤ 10 MeV, meaning that increases by a factor of about 2.5 each time , increases by 2 MeV. At a given , , increases with both and , as expected from stopping power considerations. The slope of the straight portion of the curves in Fig. 6 slightly increases with , whereas it is unaffected by variations of . In the latter case, the spacing between adjacent curves increases with regularity upon tenfold increments of . However, the logarithmic sensitivity of to is very limited; for instance, ⁄ < 0.12 at = 100 keV, whereas for = 10 cm -3 , ⁄ approximately ranges from 2.3 at = 10 keV to 1.8 at = 100 keV. We also remark the effect of the use of instead of in the α-p scattering. In Fig. 6a , the gap of the respective curves at = 50 keV progressively increases with , , resulting in a value of which e.g., at , = 10 MeV, is about 8 times higher when the nuclear interaction is taken into account. Despite this and the abovementioned favourable features, even at the highest values of , , and considered in this work, remains significantly lower than 1. Finally, the dependence on of , , , and the resulting is shown in Fig. 7 , in the range 0 ≤ ≤ 1 and for the most multiplication-effective (viz the highest) values of and explored in this work. As it is obvious, the curves vanish for → 0 (too few 11 B ions for the fusion reaction to occur) and decrease smoothly to 0 for > 1 (too few protons available for scattering). In between, a maximum occurs at values of which depend on and are however not far from 0.2, the value argued in Sect. 2 and used in the calculations above. More importantly, the differences in , , , and between the case = 0.2 and the respective optimal 's are limited to the order of 10%. We remark that in the conditions of Fig. 7 , the peak value for our estimate of is only of the order of 10 -2 . As anticipated, it is instructive remarking the physical reasons behind the very modest impact of suprathermal 11 B ions on the multiplication process. It is already evident from Eq. (10) for the p-11 B CM energy in the fixed-target reaction that the fusion probability of suprathermal 11 B ions is much smaller than that of protons, at the same particle energy. Indeed, is suppressed by a factor ⁄ , resulting in very low values of in Eq. (9). Moreover, at the same particle energy, ⁄ is larger than ⁄ , which still depresses the integrand in Eq. (9). On the opposite, at given , and ion energy, , ⁄ tends to be larger than , ⁄ , by a factor ⁄ = 2.3 when = is assumed in Eq. (4). Also, the 11 B suprathermal spectrum is slightly wider than the proton one, since , , ⁄ = ⁄ = 1.2. Nevertheless, the net result from Eq. (7) is that the 11 B contribution to in Eq. (8) is in any case much smaller than the proton one. Though the findings of Sect. 3 prevent the possibility of achieving the chain reaction in realistic conditions, it is of upmost importance to study how and how much a weak multiplication regime, i.e. when < 1 (and especially ≪ 1), can enhance the pure thermonuclear burn. In this respect, the ratio of the energy per unit volume produced during the confinement time , ℰ, to the energy stemming from the sole thermonuclear burn, ℰ , is just ( ) ⁄ , where ( ) is given by Eq. (2). We prefer to study this ratio in the form of the fractional increment ≡ (ℰ − ℰ ) ℰ ⁄ , which is equivalent to the suprathermal-to-thermonuclear energy yield, ℰ ℰ ⁄ , since the suprathermal energy component, ℰ , is obviously ℰ − ℰ . Explicitly, where is the spectrum-averaged thermalisation time of the α's, and we have used the fact that ≈ . Indeed, if one estimates as the average extinction time of the α-induced recoil shower, then ≈ + , where is the spectrum-averaged thermalisation time of the secondary protons; is obviously longer than the average thermalisation time of B ions, but much shorter than . One immediately notes that depends on the ratio ⁄ as a parameter. At the plasma densities considered here and for ~ 5 , generally turns out to be of the order of 1 ps or lower. With ~ 1 ns, ⁄ can then reach the order of 10 3 or 10 4 . Notice that for a selfsustaining chain reaction (i.e. ≥ 1), ⁄ represents the maximum possible number of αparticle generations within the time . Plots of as a function of are shown in Fig. 8 , for several orders of magnitude of ⁄ and < 1. In the limit ⁄ → ∞, Eq. (14) yields the asymptotic behaviour ~ (1 − ) ⁄ which, being independent of ⁄ , explains the saturation of the curves observed at high values of the parameter. For ≪ 1, one recognises the approximate scaling ~ in Fig. 8 . This means that in the plasma conditions investigated in this work, the burn enhancement due to the multiplication is of the order of 1% at most. In the limit → 1, Eq. (14) yields → (1 2 ⁄ ) ⁄ . This opens the possibility of very large increments in the energy output (and consequently, high fusion gains); however, at high ⁄ , raises steeply when → 1, so that has to lie very close to 1 to allow increments of the order of ⁄ being approached (e.g. ≈ 0.37 ⁄ when 1 − = ⁄ , for ⁄ ≫ 1). To summarise, our parametric analysis has shown that increases markedly with and less significantly with , with the optimum lying between 0.2 and 0.4. The achievable fusion energy is further enhanced by the ratio ⁄ . In the weak chain, however, the enhancement is quite limited moving from ⁄~ 10 to ⁄~ 100 while ≲ 0.5, and negligible beyond ⁄~ 100 while ≲ 0.9. We note, moreover, that there is a trade-off between the requirements for rising up on the one hand, and keeping ⁄ sufficiently large on the other hand, since also increases with (on the contrary, decreases with as ). For typical confinement times, however, values of ⁄ larger than at least 10 appear to be always ensured. We conclude this Section by making contact with previous representative findings for DT fuel. Peres and Shvarts (1975) have shown that a chain reaction via elastic recoils can proceed in a cold infinite DT plasma at densities above 8.4×10 27 ions/cm 3 . The optimum isotopic ratio is ⁄ = 0.72. In the analysis, they have also considered recoil-induced DD and TT fusion reactions and the scattering by their products. The main contributor to the chain reaction turns out to be the DT-born 14.1 MeV neutron, while the 3.5 MeV α particle contributes only a few percent. Indeed, if the neutron is disregarded, the medium is not critical even at 10 29 ions/cm 3 , the highest density considered by the authors. This observation is of interest for inertial confinement experiments, where the neutron can easily escape the compressed pellet. For a finite-temperature, infinite plasma, however, Afek et al. (1978) have found lower critical densities; for instance, 1.0×10 27 ions/cm -3 at = = 14 keV and ⁄ = 0.64. For the finite-temperature, finite-size case, Kumar et al. (1986) have estimated an upper bound of 0.5 for the suprathermal fusion probability associated to the DT neutron in a pellet compressed to a thickness ( ) of a few g/cm 2 , at a density of 6.0×10 25 ions/cm 3 (roughly 1000 times the solid density), = = 40 keV, and ⁄ = 1. We conclude that the suprathermal contribution to the fusion yield is substantially lower in H 11 B fuel compared to DT fuel, in similar plasma conditions and for cases of practical interest. We have investigated the possibility of a fusion chain reaction via α-recoiled ions in high-density, non-degenerate H 11 B plasma, under conditions which are of interest for laserdriven experiments (10 ≲ ≲ 10 cm -3 , ≲ 100 keV, ~ 1 keV). On the basis of a simple model, the multiplication factor for individual fusion events ( ) has been estimated in terms of the energy-dependent multiplication factor for individual α particles ( ), by averaging over the α emission spectrum. A spectral analysis of the suprathermal proton population and of the multiplication factors is also reported. We have found that the contribution of suprathermal 11 B ions to and is of the order of 1% only. In the case of the scattered proton, the complete elastic cross section, accounting also for the nuclear interaction, must be used in calculations. For instance, the value of for the most probable α emission energy (about 4 MeV) turns out to be more than twice that found for a pure Coulomb scattering. The spectral analysis shows that only protons with recoil energies higher than or comparable to 100 keV contribute to the multiplication factors. This important limitation is essentially due to the drop of the fusion probability at lower energies. The parametric analysis shows that increases with both and , though it is much more sensitive to . The optimum lies between 0.2 and 0.4. In general, quickly drops below , ≃ 2 MeV, while it increases nearly exponentially above 4 MeV, up to at least 10 MeV, the highest α energy considered in this work. Even for the highest values of and considered, (hence, ) remain significantly lower than 1; = 0.2 for , = 10 MeV, and ≈ 0.01. While ≲ 0.3, the fractional increment in the energy output relative to the thermonuclear burn scales linearly with , being practically insensitive to the parameter ⁄ and remaining, therefore, quite limited. On the contrary, for → 1, the burn enhancement approaches the order of magnitude of ⁄ , which can easily be made as large as 10 3 or 10 4 in experiments. Increasing above the order of 10 -2 , however, appears problematic in realistic laserdriven plasma conditions, meaning those presently achievable or likely to be achieved in the near future. 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The valuable support of D. Ostojić is also 20 gratefully acknowledged. All views expressed herein are entirely of the author and do not, in any way, engage his institution.