2.1. Fusion Chain via Intermediate Nuclear Reactions

The possibility of a fusion chain progressing via intermediate nuclear reactions in H-B targets has been presented in two works by Belyaev et al. [Reference Belyaev, Krainov, Zagreev and Matafonov11, Reference Belyaev, Krainov, Matafonov and Zagreev12]. The basic chain relies on the fact that a fusion-born α-particle can in turn react with ¹¹B through the reaction:

(2) $\begin{array}{}\alpha {+}^{11}\text{B}⟶\text{p}{+}^{14}\text{C}+0.8\text{MeV,}\end{array}$

which generates a high-energy proton. This proton can in turn fuse with ¹¹B, possibly giving rise to a chain reaction. However, there is the competing reaction:

(3) $\begin{array}{}\alpha {+}^{11}\text{B}⟶\text{n}{+}^{14}\text{N,}\end{array}$

which acts as a sink of α’s and generates neutrons. The neutron- and proton-generating reactions have a comparable cross section.

To compensate for the loss of α-particles in the fuel due to the neutron channel, and to reabsorb the neutron inside the fuel, one should exploit neutron capture on ¹⁰B, i.e.:

(4) $\begin{array}{}\text{n}{+}^{10}\text{B}⟶\alpha {+}^7\text{Li}.\end{array}$

This means that natural boron should be used in the fuel, or ¹¹B should be adequately supplemented with ¹⁰B.

Along all the reaction pathways in the chain, the authors find that the number of protons, α-particles, and neutrons in the fuel grows up as an avalanche over times of the order of 1 μs, approximately; cfr. Figure 1 in ref. [Reference Belyaev, Krainov, Matafonov and Zagreev12]. One has to remark, however, that this approach lies upon highly idealised assumptions, in particular,

1. (i) maximum values of the reaction cross sections have been used,

2. (ii) particles’ energy losses have been neglected, and

3. (iii) a too long confinement time is required, which is unrealistic for warm, solid-density fuel.

Shmatov [Reference Shmatov13] has finally shown that at least for temperatures up to 100 keV, only a tiny fraction of α-particles would be capable to react with ¹¹B because of their loss of energy in the fuel, thus preventing the development of the chain. On another note, if Belyaev et al.’s chain developed, fuel neutronicity would become considerably high, which would jeopardise the most appealing feature of H-¹¹B fusion.

It is also worth mentioning that experiments have been done by Labaune et al. [Reference Labaune, Baccou, Yahia, Neuville and Rafelski14] at LULI, France, to test the possibility of inducing a chain reaction in natural-boron or boron-nitride targets under irradiation by laser-accelerated protons (generated from a thin foil). Targets were solid or conditioned in a plasma state by laser irradiation. The authors intended to exploit several nuclear reaction pathways, as detailed in Figure 3. Even in the absence of a self-sustaining chain, they hoped that secondary reactions could substantially increase the energy yield compared with a pure p-¹¹B fusion scenario. While secondary reactions have successfully been induced and measured under such schemes, one has nevertheless to conclude that their rate is too low to induce any significant avalanche process or increase in the energy yield. For instance, in the case of a solid boron-nitride target, the overall number of the X(p,¹¹C) reactions, with X = ¹¹B or ¹⁴N (Figure 3(b)), was estimated at 10⁶ per shot by means of ¹¹C decay measurements. This figure appears to be at least 1000 times smaller than the number of ¹¹B(p, 3α) fusion reactions (>10⁹ per shot).

Figure 3:

Scheme of the main primary and secondary nuclear reactions produced by the interaction between a laser-accelerated proton beam and (a) a natural boron target and (b) a boron-nitride target. Reproduced from Labaune et al. [Reference Labaune, Baccou, Yahia, Neuville and Rafelski14], under the terms of the Creative Commons CC BY License.

2.2. Suprathermal Fusion Chain

The fact that three charged, massive, energetic particles are produced in the p-¹¹B reaction suggests that the fusion yield could effectively be enhanced by the elastic scattering of fuel ions to energies corresponding to the highest values of σ_f. This particularly applies to protons because of their higher charge-to-mass ratio compared with ¹¹B ions. While thermalising, some of the protons in these showers can undergo fusion, eventually setting a chain reaction up. At high matter density, moreover, α’s tend to lose energy mostly to plasma ions rather than to electrons. This happens when the electrons’ Fermi velocity (or their thermal velocity) becomes comparable to the α-particle velocity, while the ion thermal velocity remains substantially lower [Reference Gryzinski15–Reference Son and Fisch17].

The suprathermal fusion chain in an infinite, homogeneous H-¹¹B plasma can be effectively parameterised in terms of two multiplication (or reproduction) factors. An α-particle emitted at a certain energy $E_{\alpha ,0}$ in a primary fusion event is characterised by the multiplication factor $k_{\alpha }\left(E_{\alpha ,0}\right)$ , i.e., the average number of secondary α-particles generated via suprathermal processes during the slowing down of the primary particle. Likewise, the multiplication factor can be expressed in terms of fusion events. Then, one defines k _∞ as the average number of secondary fusion events per primary event. k _∞ can be estimated through the integration of k_α over the α-emission spectrum, $\phi \left(E_{\alpha ,0}\right)$ [Reference Belloni5]:

(5) $\begin{array}{}{k}_{\infty }={\int }^{\text{}}{k}_{\alpha }\left(E_{\alpha ,0}\right)\phi \left(E_{\alpha ,0}\right)\text{d}{E}_{\alpha ,0},\end{array}$

where ${\int }^{\text{}}\phi \left(E_{\alpha ,0}\right)\text{d}{E}_{\alpha ,0}=1$ . Strictly speaking, the concept of k _∞ is well grounded as long as the emitted α-particles have a comparable slowing-down time, and this quantity is in turn comparable to the (average) period between two consecutive generations of fusion events, τ_g. It is not difficult, then, to calculate the cumulative number of fusion events per unit volume at the time t, $n_f\left(t\right)$ , in regime of multiplication, upon the thermonuclear specific rate R. Depending on the value of k _∞, one can distinguish three cases for the time evolution of n_f (hence, of the energy yield):

1. (1) $k_{\infty }<1\Rightarrow n_f$ increases linearly with time, asymptotically to $Rt/\left(1-k_{\infty }\right)$ , for $t\gg {\tau }_g$ ;

2. (2) $k_{\infty }=\text{1}\Rightarrow n_f$ increases quadratically with time; in detail, $n_f\left(t\right)=R\left(t+t^2/2{\tau }_g\right)$ ; and

3. (3) $k_{\infty }>1\Rightarrow n_f$ diverges exponentially, with the growth rate $\text{ln}{k}_{\infty }/{\tau }_g$ .

It goes without saying that the capability to achieve a chain reaction with multiplicity k _∞ higher of or comparable to 1 would play a significant—if not indispensable—role in the possible exploitation of H-¹¹B fuel as an energy source.

The question is also how and how much a weak multiplication regime, namely when $k_{\infty }<1$ (and especially $k_{\infty }\ll 1$ ), can enhance the pure thermonuclear burn. Using the full expression of $n_f\left(t\right)$ , it is easy to calculate the ratio I of the suprathermal-to-thermonuclear energy yield in the confinement time τ_c [Reference Belloni5]; indeed, the total energy per unit volume is $n_f\left({\tau }_c\right)Q$ , the energy stemming from the sole thermonuclear burn is just $RQ{\tau }_c$ , and the suprathermal yield is given by the difference between the first two. I is shown in Figure (4) as a function of k _∞ for several orders of magnitude of the parameter ${\tau }_c/{\tau }_{\alpha }$ , where ${\tau }_g\approx {\tau }_{\alpha }$ is assumed and τ_α is the thermalisation time of the α particle at its most probable emission energy. Note that in typical ICF conditions, the quantity ${\tau }_c/{\tau }_{\alpha }$ can reach the order of 10³. One can distinguish the following noticeable limits for I. For ${\tau }_c/{\tau }_{\alpha }\gg 1$ and $k_{\infty }\ll 1$ , I scales as k _∞. On the contrary, when k _∞ approaches 1, I tends to $\left(1/2\right){\tau }_c/{\tau }_{\alpha }$ , which opens the possibility of very large increments in the energy output (consequently, high fusion gains). This means that k _∞ does not have to be necessarily larger than 1 to have a sizeable enhancement of the fusion yield.

Figure 4:

Suprathermal-to-thermonuclear energy ratio by the effect of a weak chain reaction. Republished from Belloni [Reference Belloni5]. © IOP Publishing Ltd. 2021.

The earliest studies were not encouraging, however. In the early 70s, Weaver et al. [Reference Weaver, Zimmerman and Wood18] estimated the increase in the H-¹¹B reaction rate due to nonthermal effects to vary between 5% and 15% in the density range of 10¹⁶–10²⁶ cm⁻³ and in the temperature range of 150–350 keV. Subsequent calculations by Moreau [Reference Moreau9, Reference Moreau19] returned multiplication factors of the order of 10⁻² in a plasma with $100\le T_e\le 300\text{keV}$ , cold ions, and Coulomb logarithm $\ln \Lambda =5$ . In both cases, however, important details have not been given; moreover, only the Coulomb interaction has been taken into account in the α-ion scattering in the case of Moreau, or poorly known nuclear data have been used for this purpose in the case of Weaver et al.

The recent study of Putvinski et al. [Reference Putvinski, Ryutov and Yushmanov20] has substantially confirmed the findings of Weaver et al. [Reference Weaver, Zimmerman and Wood18]. The H-¹¹B reactivity has been calculated using a proton spectrum, which included kinetic effects at high energy: besides α-scattering, cooling on colder electrons ( $T_e<T_i$ ) and depletion of the spectrum tail by the fusion burn. The proton spectrum was self-consistently calculated by solving the steady-state Fokker–Planck equation upon a simple burn model. For reference parameters $T_i=300\text{keV}$ , $T_e=150\text{keV}$ , $n_B/n_p=0.15$ , and $E_{\alpha ,0}=4\text{MeV}$ , the resulting reactivity showed a 10% increase compared with its purely Maxwellian form. Note that this treatment is formally independent of absolute densities as long as a fixed value is used for $\ln \Lambda$ (details are not given, however). Also in this case, the nuclear interaction does not appear to have been taken into account in the α-ion scattering.

Recently, a supposed experimental manifestation of the suprathermal chain reaction has been the subject of some controversies [Reference Hora, Lalousis and Giuffrida21–Reference Belloni, Margarone, Picciotto, Schillaci and Giuffrida26], which have finally been resolved in favour of the impossibility to induce this effect in plasma conditions such as those achievable at the Prague Asterix Laser System (PALS), Czech Republic [Reference Picciotto, Margarone and Velyhan27, Reference Margarone, Velyhan and Krasa28]. A later study [Reference Belloni5] has confirmed that in high-density, nondegenerate H-¹¹B plasma, k _∞ turns out to be of the order of 10⁻² at most. The domain investigated is given by ${10}^{24}\le n_e\le {10}^{28}{\text{cm}}^{-3}$ , $T_i=1\text{keV}$ , $\text{max}\left(T_i,5E_F\left(n_e\right)\right)\le T_e\le 100\text{keV}$ , where E_F is the Fermi energy; $E_F\left(\text{keV}\right)=3.65\times {10}^{-18}{\left(n_e\left({\text{cm}}^{-3}\right)\right)}^{2/3}$ . This represents a low-T _i regime, where the thermonuclear burn is very modest and is just used to seed the chain reaction; the hope was that the suprathermal chain could drive the plasma burn towards ignition, by increasing T _i quickly.

If T _i is sufficiently low, one can assume that, at least for the first few generations, the suprathermal showers elicited by the α particles do not interact with each other and do not significantly affect the background (thermal) Maxwell–Boltzmann distribution of plasma ions. In a scenario of this kind, each primary α-particle or fusion event can be treated independently through a simplified model compared with more sophisticated kinetic-theory approaches, which are indispensable at high reaction rates; see, e.g., refs. [Reference Peres and Shvarts29–Reference Kumar, Ligou and Nicli31] for the case of DT fusion, and [Reference Putvinski, Ryutov and Yushmanov20] for H-¹¹B fusion. The simplified approach of ref. [Reference Belloni5], in particular, assesses whether the medium is multiplicative or not. Without entering details, the contribution to k_α of suprathermal H and ¹¹B ions (k_αp and k_αB, respectively) is calculated separately, i.e.:

(6) $\begin{array}{}{k}_{\alpha }\left(E_{\alpha ,0}\right)=k_{\alpha p}\left(E_{\alpha ,0}\right)+k_{\alpha B}\left(E_{\alpha ,0}\right).\end{array}$

Denoting by j the generic ion species and by E _j,0 its energy just after the scattering by an α particle, k_αj is related to the scattered ion spectrum, $\text{d}{N}_j/\text{d}{E}_{j,0}$ , and the fusion probability of j, P_j, by the relation:

(7) $\begin{array}{}\frac{\text{d}{k}_{\alpha j}}{\text{d}{E}_{j,0}}\left(E_{j,0};E_{\alpha ,0}\right)=3P_j\left(E_{j,0}\right)\frac{d{N}_j}{d{E}_{j,0}}\left(E_{j,0};E_{\alpha ,0}\right).\end{array}$

The spectrum $\text{d}{N}_j/\text{d}{E}_{j,0}$ in turn depends on the α-ion differential scattering cross section, σ_αj, and the α-particle stopping power, $\text{d}{E}_{\alpha }/\text{d}x$ , according to the relation:

(8) $\begin{array}{}\frac{\text{d}{N}_j}{\text{d}{E}_{j,0}}\left(E_{j,0};E_{\alpha ,0}\right)=n_j{\int }_{\left(3/2\right)T_i}^{E_{\alpha ,0}}{\sigma }_{\alpha j}\left(E_{\alpha },E_{j,0}\right){\left(\frac{d{E}_{\alpha }}{dx}\right)}^{-1}\text{d}{E}_{\alpha }.\end{array}$

P_j depends on σ_f and the stopping power of j, $\text{d}{E}_j/\text{d}x$ , in a similar fashion:

(9) $\begin{array}{}{P}_{p\left(B\right)}\left(E_{p\left(B\right),0}\right)=n_{B\left(p\right)}{\int }_0^{E_{p\left(B\right),0}}{\sigma }_f\left(E_{CM}\right){\left(\frac{\text{d}{E}_{p\left(B\right)}}{\text{d}x}\right)}^{-1}\text{d}{E}_{p\left(B\right)}.\end{array}$

From ref. [Reference Belloni5], there are several points to remark and discuss. First of all, the contribution of suprathermal ¹¹B ions to k_α and k _∞ is of the order of 1% only. Nevertheless, the effect of the nuclear interaction in the α-¹¹B scattering should be counterchecked, in the light of the elastic cross section measurements at $E_{\alpha }<5\text{MeV}$ performed by Spraker et al. [Reference Spraker, Ahmed and Blackston32]. In ref. [Reference Belloni5], ${\sigma }_{\alpha B}$ is calculated as the Rutherford cross section only.

In the case of the scattered proton, the complete elastic cross section, accounting also for the nuclear interaction, must definitely be used in calculations. In Figure 5(a), k_αp is plotted as a function of E_α,0 for different values of T _e and $n_e={10}^{26}{\text{cm}}^{-3}$ . As a term of comparison, a curve based only on the Rutherford α-p scattering cross section, σ_R, is shown for T _e = 50 keV. One can appreciate that for the most probable α-emission energy, 4 MeV, k_αp is more than twice that found for a pure Coulomb scattering. At $E_{\alpha ,0}$ = 10 MeV, the difference reaches a factor of 8.

Figure 5:

α-particle multiplication factor via suprathermal protons as a function of the initial α-energy, for different values of T _e at a fixed n _e (a), and of n _e at a fixed T _e (b). In (a), T _e values (in keV) are indicated next to the curves; σ_s is the complete α-p elastic scattering cross section, accounting also for the nuclear interaction. In (b), γ is the boron-to-proton ion concentration. Republished from Belloni [Reference Belloni5]. © IOP Publishing Ltd. 2021.

A parametric analysis shows that k_αp increases with both T _e and n _e , though it is much more sensitive to T _e . From Figure 5(b), one notes that k_αp drops quickly below $E_{\alpha ,0}\simeq 2\text{MeV}$ , while above 4 MeV, the shape of the curves is approximately linear in the semilog plot, meaning an exponential increase with $E_{\alpha ,0}$ (up to at least 10 MeV). Even at the highest values of n_e and T _e considered (10²⁸ cm⁻³ and 100 keV, respectively), k_αp (hence, k _∞) remains significantly lower than 1; for instance, $k_{\alpha p}=0.2$ for $E_{\alpha ,0}=10$ MeV, and $k_{\infty }\approx 0.01$ over the actual fusion spectrum. A fit of k_αp-vs-E _α,0 curves with an exponential function returns a common growth rate such that k_αp increases by a factor of about 2.5 each time E _α,0 increases by 2 MeV. At $n_e={10}^{28}c{m}^{-3}$ and T_e = 100 keV, one extrapolates $k_{\alpha p}=1$ for $E_{\alpha ,0}\approx 13.6\text{MeV}$ .

This means that if we could boost the energy of α’s—let us say—above 10 MeV, we could substantially increase the multiplication factor. How might this be achieved? In principle, two ways can be identified at present. One way is to use high-energy protons to trigger the fusion reaction, which can be referred to as a kinematic boost; α particles with energies up to 20 MeV have recently been generated by Bonvalet et al. [Reference Bonvalet, Nicolaï and Raffestin33] in a laser-driven pitcher-catcher experiment. Another way is to accelerate the fusion-born α’s; for instance, in the same laser-induced electric field which accelerates the protons in direct-target-irradiation experiments. Evidence of this effect has recently been reported by Giuffrida et al. [Reference Giuffrida, Belloni and Margarone34]. With either of these means, it is not obvious, however, how E _α,0 could be kept so high for more than one generation. In the case of laser acceleration of the α particles, it is neither obvious how this effect, observed under irradiation of planar solid targets, could be reproduced on actual ICF targets.

It is also worth mentioning that the concept of a possible H-¹¹B fusion reactor has been proposed (but in a low-density plasma, in this case) [Reference Eliezer and Martinez-Val35,Reference Eliezer, Schweitzer, Nissim and Martinez-Val36], which is based on α’s acceleration by the application of an external electric field to counterbalance the stopping power and induce an avalanche of reactions.

To summarise, values of k _∞ very close to 1 are needed in an ICF scheme to enhance the suprathermal-to-thermonuclear energy yield by factors of up to 10³. Early computations by Weaver et al. [Reference Weaver, Zimmerman and Wood18] estimated the increase in the H-¹¹B reaction rate due to suprathermal effects to vary only between 5% and 15% in the density range of 10¹⁶–10²⁶ cm⁻³ ( $n_e\approx n_i$ ) and temperature range of 150–350 keV ( $T_e=T_i$ ). Subsequent calculations [Reference Moreau9, Reference Moreau19] returned multiplication factors of the order of 10⁻² in a plasma with $100\le T_e\le 300\text{keV}$ , T_i = 0, and $\ln \Lambda =5$ . Recently, Putvinski et al. [Reference Putvinski, Ryutov and Yushmanov20] have substantially confirmed the findings of Weaver et al., whereas in the high-density, low-T_i domain ${10}^{24}\lesssim n_e\lesssim {10}^{28}{\text{cm}}^{-3}$ , $T_e\lesssim 100\text{keV}$ , and $T_i\sim 1\text{keV}$ (non-degenerate plasma), it has been found $k_{\infty }\sim {10}^{-2}$ at most [Reference Belloni5]. This latest work has also shown that particularly for the α-p scattering, the complete elastic cross section, which includes the nuclear interaction, is needed in calculations. Furthermore, k_α has been found to increase exponentially with the α-particle energy, at least in the range of 4–10 MeV, with a growth rate that is independent of n_e. This exponential growth could in principle be exploited in cases where the energy of the fusion-born α’s is boosted, e.g., kinematically [Reference Bonvalet, Nicolaï and Raffestin33] or electrodynamically [Reference Giuffrida, Belloni and Margarone34]. Finally, no experimental evidence of suprathermal multiplication has been achieved so far.

2.3. Energy Multiplication Factor in a Beam-Driven Fusion Scheme

Fast ignition may provide an option to ignite non-DT fuels inasmuch as it significantly relaxes implosion symmetry requirements (compared with central hot-spot ignition) and allows for nonspherical target configurations or fuel seeding [Reference Atzeni and Meyer-ter-Vehn37]. Among the concepts for fast ignition, it has been proposed to use intense laser-accelerated proton beams [Reference Roth, Cowan and Key38]. In this approach, the energy deposited in the precompressed fuel by a proton beam generated outside the target bootstraps the fusion flame. Ideally, between 5 and 10% of the laser pulse energy is converted into kinetic energy of the beam as the result of the interaction of the pulse with a thin foil.

Here, we wish to emphasise that proton beam fast ignition is a particularly advantageous option for the case of H-¹¹B fuel. Indeed, while the protons transfer their energy to the plasma, additional heating is provided by in-flight fusion reactions. This is an exclusive effect of H-¹¹B fuel as it cannot obviously occur for other proposed fusion fuels under proton irradiation. On the quantitative ground, it is useful to make recourse to the so-called energy multiplication factor, a fundamental quantity in beam fusion. It is defined as the ratio of the fusion energy produced via in-flight reactions to the overall beam energy, i.e.:

(10) $\begin{array}{}F=\frac{P_p\left(E_0\right)Q}{E_0},\end{array}$

where F is the energy multiplication factor, E ₀ is the initial proton energy, Q is the fusion Q-value, and P_p is given by Equation (9) as long as it is sufficiently lower than 1. Denoting by E_b the beam energy, the overall energy deposited in the fuel, E_d, is then:

(11) $\begin{array}{}{E}_d=\left(1+F\right)E_b.\end{array}$

An estimate of F is important to set the value of E ₀ to be achieved in the laser acceleration of the protons.

A calculation of F vs E ₀ for beam-driven H-¹¹B fusion has been carried out by Moreau [Reference Moreau9]. This author considered protons injected into a ¹¹B plasma with warm electrons and cold ions. The results are shown in Figure 6 for several values of T_e. In all cases, there is a maximum at E ₀ around 1 MeV; at high values of T_e, other two maxima appear just below 3 MeV and between 4 and 5 MeV, respectively. It is hard, however, to achieve a multiplication factor better than 30%. Anyway, Moreau’s calculation should be redone with a more accurate fusion cross section (which was barely known at that time) and stopping power model (Sivukhin’s model [Reference Sivukhin39] was used).

Figure 6:

Energy multiplication factor vs proton injection energy for various electron temperatures. Protons are injected into a ¹¹B plasma with cold ions and warm electrons. The results are independent of plasma density when the value of $ln\Lambda$ is fixed. Reproduced from Moreau [Reference Moreau9] with the permission of the publisher. © IAEA 1977.

Note that for a Maxwellian plasma, F is formally independent of density when the value of $\text{ln}\Lambda$ is kept fixed. This comes from an implicit cancellation of the density in the product between n_B and ${\left(\text{d}{E}_p/\text{d}x\right)}^{-1}$ in Equation (9), with a residual density dependence holding through the expression of $\text{ln}\Lambda$ in $\text{d}{E}_p/\text{d}x$ [Reference Belloni5]. This residual dependence is very weak, however. It is also worth noticing that in a fully degenerate plasma, the electronic component of $\text{d}{E}_p/\text{d}x$ would become independent of n_e and proportional to the proton velocity, under certain conditions. (In general, the stopping power of an ion in a fully degenerate plasma scales roughly linearly with n_e. However, when the velocity of the ion is much smaller than the Fermi velocity and the parameter $r_s=\left(m{e}^2/{\hslash }^2\right){\left(3/4\pi n_e\right)}^{1/3}$ —where m is the electron mass—is much smaller than 1, the stopping power becomes independent of n_e and proportional to the ion velocity. The condition $r_s\ll 1$ holds for $n_e\gg {10}^{24}{\text{cm}}^{-3}$ ) [Reference Son and Fisch17, Reference Fermi and Teller40, Reference Lindhard41]. As long as the ion-ion component of $\text{d}{E}_p/\text{d}x$ can be neglected, P_p would then become truly proportional to n_B. This effect could boost P_p towards 1 even at low (possibly sub-MeV) values of E ₀, given the linear velocity dependence of the electronic stopping power. As a consequence, F could rise up significantly, well above unity.

Another aspect to emphasise is that the energy multiplication factor can be further increased if suprathermal chain reaction effects take place. It is easy to show that in this case, E_d is augmented by a term $S_{l}F{E}_b$ compared with Equation (11), where the factor S_l depends on the number of generations, l, and is essentially a partial summation of the geometric series with common ratio k _∞, according to the relation:

(12) $\begin{array}{}{S}_l\left(k_{\infty }\right)+1=\sum _{i=0}^l{k}_{\infty }^i=\frac{1-k_{\infty }^{l+1}}{1-k_{\infty }}.\end{array}$

Explicitly,

(13) $\begin{array}{}{E}_d=\left(1+F+S_{l}F\right)E_b,\end{array}$

where the energy multiplication factor can be redefined as follows:

(14) $\begin{array}{}{F}_{supr}\stackrel{\scriptscriptstyle\mathrm{def}}{=}\left(1+S_l\right)F.\end{array}$

Note that S_l converges to $k_{\infty }/\left(1-k_{\infty }\right)$ when $k_{\infty }<1$ and l is sufficiently large, while $S_l⟶l$ when $k_{\infty }⟶1$ . Now, if one was capable to keep the chain going for just two generations with k _∞ sufficiently close to 1, assuming F = 0.3, Equation (13) would return $E_d\approx 2E_b$ , which is quite a significant amplification. It is particularly relevant to this case what has been mentioned in the previous section, namely, that k _∞ could be made close to 1 by exploiting the kinematic boost of the proton beam.