Introduction
Nuclear fusion is the origin of energy in our sun and in the stars. The nuclear fusion in our sun (e.g., four protons combine into an alpha) is a weak interaction. Thus, the sun's hot plasma confinement is made possible due to the gravitational force of the sun's large mass. In a terrestrial laboratory, between the possible existing reactions, the first choice for controlled fusion energy is the deuterium–tritium (DT) strong interaction creating an alpha and a neutron with the release of 17.6 MeV per reaction. The DT has been chosen since the cross section and the rate of this process are the largest for the lowest practical temperatures of the order of 10 keV. The problem with the DT reaction is that it produces the undesired neutron that can activate radioactive materials.
The cleanest fusion reaction that avoids the neutron problem is the fusion of protons with 11B (pB11) that creates three alphas (Hora et al., Reference Hora, Eliezer, Kirchhoff, Nissim, Wang, Lalousis, Xu, Miley, Martinez Val, Mckenzie and Kirchhoff2017 and references therein). Using lasers, the first p–11B 1000 reactions, just above the level of sensitivity, were measured (Belyaev et al., Reference Belyaev, Matafonov, Vinogradov, Krainov, Lisitsa, Roussetski, Ignatyev and Andrianov2005). A combination of highly intense proton beams of energies above MeV produced by picosecond laser pulses intercepting a plasma created by a second irradiated laser beam produced more than one million pB11 reactions (Labaune et al., Reference Labaune, Baccou, Deprierraux, Goyon, Loisel, Yahia and Rafelski2013). At Prague PALS facility, the few hundred joules-nanosecond time duration iodine laser interacting with targets containing high boron concentration doped in silicon crystals produced one billion alpha particles (Picciotto et al., Reference Picciotto, Margarone, Velyhan, Bellini, Krasa, Szydlowski, Bertuccio, Shi, Margarone, Prokupek, Malinowska, Krouski, Ullschmied, Laska, Kucharik and Korn2014) and in an ELI meeting, more than 1011 alphas were reported per laser shot (Giuffrida, Reference Giuffrida2018).
The main problem in solving the energy problem with fusion on our planet for mankind is the difficulty to create it in a controllable and economical way with a positive energy balance. Two different distinctive schemes have been investigated in the past 60 years: (1) Magnetic confinement fusion (MCF) based on high-intensity magnetic fields (several teslas) confining low-density (1014 cm3) and high-temperature (~10 keV) plasmas for long or practically continuous times. (2) Inertial confinement fusion (ICF) based on rapid heating and compressing the fusion fuel to very large densities (Nuckolls et al., Reference Nuckolls, Wood, Thiessen and Zimmermann1972) and very high temperatures, larger than 5 keV for the DT fusion reaction. In order to ignite the fuel with less energy, it was suggested (Basov et al., Reference Basov, Guskov and Feoktistov1992; Tabak et al., Reference Tabak, Hammer, Glinsky, Kruer, Wilks, Woodworth, Campbell, Perry, Mason, Woodworth, Campbell, Perry and Mason1994) to separate the drivers that compress and ignite the target. First, the fuel is compressed, then a second driver ignites a small part of the fuel while the created alpha particles heat the rest of the target. This idea is called fast ignition (FI). The FI problem is that the laser pulse does not reach directly the compressed target; therefore, many schemes have been suggested (Guskov, Reference Guskov2013) including proton–boron fusion (Martinez-Val et al., Reference Martinez-Val, Eliezer, Piera and Velarde1996; Eliezer and Martinez Val, Reference Eliezer and Martinez Val1998).
The novel scheme described here can be used for combinations such as (Eliezer and Mima, Reference Eliezer and Mima2009) helium3–deuterium (He3–D), deuterium–lithium6 (D–6Li), and proton–lithium6 (p–6Li), proton–lithium7 (p–7Li). In this paper, we suggest the clean (i.e., without neutrons) proton–boron11 fusion yielding 3α,
$${\rm p} + {}^{11}{\rm B}\to 3{}^4{\rm He} + 8.9\,{\rm MeV}$$This new approach to fusion is given schematically in Figure 1. Our reactor consists of a background plasma with densities of the order of a mg/cm3 of boron11 and hydrogen ions. A plasma channel or a solid target is irradiated by a high power laser that creates a shock wave containing proton particles with a flow energy (of the protons) in the domain of 300–1200 keV that enters the background plasma. Fusion boron alphas collide with protons of the plasma and accelerate them causing a chain reaction. The number of the alpha particles N α created in this process is given by
$$\eqalign{& N_{\rm \alpha} \sim3N_{{\rm \alpha} 0}\lpar {e^{{\rm \tau} /{\rm \tau}_{\rm A}}-1} \rpar \cr & {\rm \tau} _{\rm A}\equiv \displaystyle{1 \over {n_0\langle \sigma {v}\rangle}}} $$
“Fusion cross section (σ) times relative pB11 velocity (v)” is denoted by 
$\langle {\rm \sigma} v\rangle$ (in the range of 10−16 cm3/s to few times 10−15 cm3/s), N α is the number of the alpha particles created during the chain reaction for an initial number of alphas N α0. τ A is defined as the chain reaction time (also defined as avalanche time, Eliezer et al., Reference Eliezer, Hora, Korn, Nissim and Martinez-Val2016a; Hora et al., Reference Hora, Eliezer, Kirchhoff, Nissim, Wang, Lalousis, Xu, Miley, Martinez Val, Mckenzie and Kirchhoff2017). In our background, plasma n 0 is of the order of 1019 cm−3, 
$\langle {\rm \sigma}v\rangle$ ~10−15 cm3/s implying a time τ A~10−4 s. During an interaction time τ ~1 ms, we get an increase factor to the originally produced alphas by a factor of the order of 104. In order to keep the chain reaction going, our reactor contains a combination of an external electric field and a magnetic mirror confinement device for a pulse long enough to get a high energy gain.
Fig. 1. The fusion reactor schematic model. The two injected high power lasers, in the magnetic field's mirror, trigger the pB11 fusion in the vessel. The created alphas heat the target which flows into a cooler that recirculate the plasma fluid with a density of the order of 1 mg/cm3. The deposition of the energy is done continuously from the cooler, while the created alphas are confined by an external magnetic field.
In “The confinement and the chain reactions”, the confinement and the chain reactions are described. The conceptual design of the fusion reactor is given in “The New Reactor”. We end “Conclusions” with a short conclusion.
The shock wave initiating mechanism
In this system, two (or more) shock waves are created by very high irradiance lasers. The desired shock waves are semi-relativistic with a shock wave velocity of the order of 0.1 c, where c is the speed of light. The formalism of these shock waves was recently described in the literature (Eliezer et al., Reference Eliezer, Nissim, Raicher and Martinez-Val2014, Reference Eliezer, Pinhasi, Martinez-Val, Raicher and Henis2017). The laser-induced shock wave acts as an accelerator that accelerates fluid particles inside the shock wave domain (see Fig. 2) with proton and boron number densities n p and n B in a volume V. In our scheme, in this paper we consider two possibilities to create a laser induced shock wave: one for a solid target and the other for a gas target the shocked matter to velocities where the center of mass of p–B11 energy so that its “fusion cross section (σ) times relative pB11 velocity (v)” 
$\langle {\rm \sigma}v\rangle$~10−15 cm3/s gets large values. One case is the laser irradiates a solid that contains hydrogen and boron (Picciotto et al., Reference Picciotto, Margarone, Velyhan, Bellini, Krasa, Szydlowski, Bertuccio, Shi, Margarone, Prokupek, Malinowska, Krouski, Ullschmied, Laska, Kucharik and Korn2014; Giuffrida, Reference Giuffrida2018) in the entrance channel of Figure 1, or the other case is the laser creates a shock wave in the background gas.
Fig. 2. (a) The shock wave compression к = ρ/ρ0 as a function of the shock wave dimensionless pressure П = P/(ρ0c 2) is presented. The numerical values are obtained for Г = 5/3. The semi-relativistic domain is defined on the red line. (b) The dimensionless shock wave velocity u s/c and the particle velocity u p/c in the laboratory frame of reference are given versus the dimensionless laser irradiance ПL = I L/(ρ0c 3) in the domain 10−4 < ПL < 1.
The physics of shock waves is excellently summarized in Zeldovich and Raizer's book Physics of Shock Waves and High Temperature Hydrodynamic Phenomena (Zeldovich and Raizer, Reference Zeldovich and Raizer1966). The interaction of a high power laser with a planar target creates a one-dimensional (1D) shock wave (Fortov and Lomonosov, Reference Fortov and Lomonosov2010; Eliezer, Reference Eliezer2013). The theoretical basis for laser-induced shock waves analyzed and measured experimentally so far is based on plasma ablation. For laser intensities of 1012 W/cm2 < I L < 1016 W/cm2 and nanoseconds pulse duration, a hot plasma is created. This plasma exerts a high pressure on the surrounding material, leading to the formation of an intense shock wave moving into the interior of the target (Eliezer, Reference Eliezer2002). In this paper, we are interested in the semi-relativistic shock waves for solid or gas densities. Shock waves induced by lasers with irradiances in this regime are described by relativistic hydrodynamics (Landau and Lifshitz, Reference Landau and Lifshitz1987). Relativistic shock waves were first analyzed by Taub (Reference Taub1948) and in the context of laser–plasma interactions by Eliezer et al. (Reference Eliezer, Nissim, Raicher and Martinez-Val2014).
In the following, we use the shock wave equations relevant to Figure 1. The relativistic shock wave Hugoniot equations in the laboratory frame of reference are given by
$$\eqalign{& (i){\rm} \displaystyle{{u_{\rm p}} \over c} = \sqrt {\displaystyle{{\lpar {P_1-P_0} \rpar \lpar {e_1-e_0} \rpar } \over {\lpar {e_0 + P_1} \rpar \lpar {e_1 + P_0} \rpar }}} {\rm} \cr & (ii){\rm} \displaystyle{{u_{\rm s}} \over c} = \sqrt {\displaystyle{{\lpar {P_1-P_0} \rpar \lpar {e_1 + P_0} \rpar } \over {\lpar {e_1-e_0} \rpar \lpar {e_0 + P_1} \rpar }}} \cr & (iii){\rm} \displaystyle{{{\lpar {e_1 + P_1} \rpar }^2} \over {{\rm \rho} _1^2}} -\displaystyle{{{\lpar {e_0 + P_0} \rpar }^2} \over {{\rm \rho} _0^2}} = \lpar {P_1-P_0} \rpar \left[ {\displaystyle{{\lpar {e_0 + P_0} \rpar } \over {{\rm \rho}_0^2}} + \displaystyle{{\lpar {e_1 + P_1} \rpar } \over {{\rm \rho}_1^2}}} \right]} $$P, e, and ρ are the pressure, energy density, and mass density accordingly, the subscripts 0 and 1 denote the domains before and after the shock arrival, u s is the shock wave velocity and u p is the particle flow velocity in the laboratory frame of reference, and c is the speed of light. We have assumed that in the laboratory the target is initially at rest. The equation of state (EOS) taken here in order to calculate the shock wave parameters is the ideal gas EOS
$$e_{\rm j} = {\rm \rho} _{\rm j}c^2 + \displaystyle{{P_{\rm j}} \over {{\rm \Gamma} -1}};\quad {\,j}={\rm 0,1}{\rm.} $$where Г is the specific heat ratio. We have to solve these Eqs (3) and (4) together with the following piston model equation (Esirkepov et al., Reference Esirkepov, Borghesi, Bulanov, Mourou and Tajima2004; Eliezer et al., Reference Eliezer, Nissim, Raicher and Martinez-Val2014),
$$P_1 = \displaystyle{{2I_{\rm L}} \over c}\left( {\displaystyle{{1-u_{\rm p}/c} \over {1 + u_{\rm p}/c}}} \right)$$It is convenient to use the following laser irradiance and pressure dimensionless variables in the solutions of the above equations
$$\Pi _{\rm L}\equiv \displaystyle{{I_{\rm L}} \over {{\rm \rho} _0c^3}}; \quad {\rm \Pi} = \displaystyle{{P_1} \over {{\rm \rho} _0c^2}}\equiv \displaystyle{P \over {{\rm \rho} _0c^2}}$$In the transition domain between relativistic and nonrelativistic shock waves, we get the following solutions for the shock wave parameters
$$\eqalign{& \displaystyle{{u_{\rm p}} \over c} = \sqrt {\displaystyle{{\Pi \lpar {2 + \Pi} \rpar } \over {\lpar {1 + \Pi} \rpar \lpar {\Gamma + 1 + \Pi} \rpar }}} {\rm} \approx 2\displaystyle{{{\lpar {\Pi_{\rm L}} \rpar }^{1/4}} \over {{\lpar {\Gamma + 1} \rpar }^{3/4}}} \cr & {\rm} \displaystyle{{u_{\rm s}} \over c} = \sqrt {\displaystyle{{\Pi (\Gamma + 1 + \Pi )} \over {\lpar {1 + \Pi} \rpar \lpar {2 + \Pi} \rpar }}} {\rm} \approx \sqrt 2 \displaystyle{{{\lpar {\Pi_{\rm L}} \rpar }^{1/4}} \over {{\lpar {\Gamma + 1} \rpar }^{1/4}}} \cr & \Pi = 2\Pi _{\rm L}\left( {\displaystyle{{1-u_{\rm p}/c} \over {1 + u_{\rm p}/c}}} \right)\approx 2\sqrt {\displaystyle{{\Pi _{\rm L}} \over {\Gamma + 1}}}} $$The compression к = ρ/ρ0 as a function of the dimensionless pressure П = P/(ρ0c 2) is given in Figure 2a for Г = 5/3. In order to see the transition between the relativistic and nonrelativistic approximation, one has to solve the relativistic equations in order to see the transition effects like the one shown in Figure 2a. The numerical solutions shown in Figure 2b give the dimensionless shock wave velocity u s/c and the particle velocity u p/c in the laboratory frame of reference versus the dimensionless laser irradiance ПL = I L/(ρ0c 3) in the domain 10−4 < ПL < 1. For the practical proposal, the inserted table shows numerical values in the area 10−4 < ПL < 10−2.
Furthermore, the speed of sound in units of speed of light, c S/c, as a function of the dimensionless laser irradiance ПL = I L/(ρ0c 3) and the rarefaction velocity, c rw, is given by
$$\eqalign{& \displaystyle{{c_{\rm s}} \over c} = \sqrt {{\left( {\displaystyle{{\partial P} \over {\partial e}}} \right)}_{\rm S}} = \left( {\displaystyle{{\Gamma P} \over {e + P}}} \right)^{1/2} = \left[ {\displaystyle{{\Gamma \lpar {\Gamma -1} \rpar \Pi} \over {\Gamma \Pi + \lpar {\Gamma -1} \rpar {\rm \kappa}}}} \right]^{1/2} \cr & c_{{\rm rw}} = \displaystyle{{c_{\rm S} + u_{\rm p}} \over {1 + \left( {\displaystyle{{c_{\rm S}u_{\rm p}} \over {c^2}}} \right)}}} $$The time τ rw that the rarefaction wave reaches the shock front, for the case that the laser pulse duration is τ L, is
$$\tau _{{\rm rw}} = \displaystyle{{c_{{\rm rw}}\tau _{\rm L}} \over {c_{{\rm rw}}-u_{\rm s}}}$$As a numerical example, we take a laser dimensionless irradiance ПL yielding the desired velocities necessary for our scheme (see Fig. 2)
$$\eqalign{& \Pi _{\rm L} = {\rm} I_{\rm L}/({\rm \rho} _0{c}^3) = 8.3 \times 10^{-4}\Rightarrow \cr & \left\{ \matrix{{\rm \beta} \equiv u_{\rm p}/c = 0.035;\to u_{\rm p} = 1.05 \times 10^9\,({\rm cm/s}) = 4.77\left( {\displaystyle{{e^2} \over {hc}}} \right) \hfill \cr u_{\rm s}/c = 0.0465;\to u_{\rm s} = 1.40 \times 10^9\,({\rm cm/s}); \hfill \cr c_{\rm s}/c = 0.0226;\to c_{\rm s} = 0.69 \times 10^9\,({\rm cm/s}); \hfill \cr c_{{\rm rw}}/c = 0.0573;\to c_{{\rm rw}} = 1.72 \times 10^9\,({\rm cm/s}); \hfill \cr \Pi = P/({\rm \rho}_0c^2) = 1.61 \times 10^{-3}\to \hfill \cr P[{\rm bar}] = 1.45 \times 10^{12}\left( {\displaystyle{{{\rm \rho}_0} \over {1\,{\rm g/cm}^3}}} \right) \hfill} \right.} $$As an example for the shock wave created in the background plasma, relevant for our next section, we use
$${\rm \rho} _0 ({\rm g/cm}^3) = 10^{-3}\Rightarrow \left\{ \matrix{I_{\rm L}({\rm W/c}{\rm m}^2) = 2.24 \times 10^{18} \hfill \cr P[{\rm bar}] = 1.45 \times 10^9 = 1.45\,{\rm Gb} \hfill \cr \tau_{{\rm rw}} = 5.3\tau_{\rm L} \hfill} \right.$$For a given laser irradiance I L and energy W L, we estimate now the laser pulse duration τ L for our scheme in order to have a reasonable 1D shock wave. I L is a function of the flow velocity u p (fixed by ПL = 8.3 × 10−4) and the medium density ρ0, implying
$$\eqalign{& I_{\rm L}\,({\rm W/c}{\rm m}^2) = 2.2 \times 10^{18}\left( {\displaystyle{{{\rm \rho}_0} \over {{10}^{-3}\,{\rm g/c}{\rm m}^3}}} \right)\Rightarrow \cr & S\,({\rm c}{\rm m}^2) = {\rm \pi} R_{\rm L}^2 = 4.5 \times 10^{-7}\left( {\displaystyle{{{10}^{-3}\,{\rm g/c}{\rm m}^3} \over {{\rm \rho}_0}}} \right)\left( {\displaystyle{{W_{\rm L}} \over {1\,{\rm kJ}}}} \right)\left( {\displaystyle{{1\,{\rm ns}} \over {{\rm \tau}_{\rm L}}}} \right) \cr & R_{\rm L}\,({\rm \mu m}) = 0.12\left[ {\left( {\displaystyle{{1\,{\rm g/c}{\rm m}^3} \over {{\rm \rho}_0}}} \right)\left( {\displaystyle{{W_{\rm L}} \over {{\rm 1\,kJ}}}} \right)\left( {\displaystyle{{{\rm 1\,ns}} \over {{\rm \tau}_{\rm L}}}} \right)} \right]^{0.5} \cr & R_{\rm L}({\rm \mu m})\gg u_{\rm s}\tau _{\rm L}({\rm \mu m}) = 1.40 \times 10^4\,({\rm \mu m}/{\rm ns})\, {\rm \tau} _{\rm L}\lpar {{\rm ns}} \rpar } $$ To solve Eq. (12), we substitute the symbol 
$\gg$ by a factor of 5 equality, i.e., a laser diameter larger by a factor of 10 relative to the shock wave length during the pulse duration, u sτ L, we get
$${\rm \tau} _{\rm L}\,({\rm ns}) = 1.2 \times 10^{-3}\left[ {\left( {\displaystyle{{{10}^{-3}{\rm g/c}{\rm m}^3} \over {{\rm \rho}_0}}} \right)\left( {\displaystyle{{W_{\rm L}} \over {1\,{\rm kJ}}}} \right)} \right]^{1/3}$$Namely, for our scheme where I L = 2.2 × 1018 W/cm2, we need a laser pulse duration of 1.2 ps if the laser energy is 1 kJ.
The confinement and the chain reactions
In order to avoid proton and alphas losses to the wall of the vessel, we use a magnetic mirror confinement. For a longitudinal magnetic field inside the vessel, the transverse radius of the fuel container is at least 2R α, where R α is the alpha Larmor radius R α, is
$$R_{\rm \alpha} = \displaystyle{{{\rm \gamma} {\rm \beta} _\bot M_{\rm \alpha} c^2} \over {2{eB}}};\quad{\rm \beta}_\bot = \displaystyle{{{v}_\bot} \over c};\quad{\rm \beta} = \displaystyle{{v} \over c};\quad{\rm \gamma} = \displaystyle{1 \over {\sqrt {1-{\rm \beta} ^2}}} $$ M α is the alpha rest mass, e is the elementary charge, B is the applied longitudinal magnetic field, and 
$v_\bot$is the perpendicular velocity to the magnetic field. In our case, M α is about four times the proton mass and its kinetic energy is about 2.9 MeV, implying β = 3.87 × 10−2, γ−1 = 7.5 × 10−4 and for a magnetic field of 25 T, we get R α about 1 cm. The volume is controlled (Eliezer et al., Reference Eliezer, Tajima and Rosenbluth1987) by its maximum and minimum magnetic fields B max and B min accordingly where one of the important parameters is its mirror ratio R m defined by
$$R_{\rm m} = \displaystyle{{B_{\max}} \over {B_{\min}}} $$For R m = 1.5, one gets a (minimum) vessel volume V v given by
$$\eqalign{& V_{\rm v} = {\rm \pi} R^2L = 16{\rm \pi} R_{\rm \alpha} ^3 \cr & R = 2R_{\rm \alpha};\quad L = 4R_{\rm \alpha}} $$Therefore, our mirror confinement of alphas requires a minimum volume of about 50 cm3. We can increase the volume to sustain an appropriate fusion energy so that the temperature in the vessel is not more than a few electron volt.
In our scheme, we use the chain reaction as was explained recently in the literature (Eliezer et al., Reference Eliezer, Hora, Korn, Nissim and Martinez-Val2016a) and defined there as avalanche. This chain reaction is explained by the following elastic collisions: (1) the first collision is between the created alpha from the pB11 fusion (with an energy E α) and a proton in the vessel under consideration, (2) in the second step, this alpha collides with another proton in the vessel that (3) meets a boron in the vessel to fuse into three alphas. Therefore, after the alpha particle with an energy E α has its second collision with a proton and this proton collides with an 11B one gets in their center-of-mass system of reference an energy E cm(pB11)
$${E}_{{\rm cm}}({\rm pB}^{11}) = \left( {\displaystyle{{11} \over {12}}} \right)\left( {\displaystyle{{16} \over {25}}} \right)\left( {\displaystyle{9 \over {25}}} \right)E_{\rm \alpha} \simeq 0.21E_{\rm \alpha} $$ If the energy created in pB11 fusion is equally divided between the three alphas then E cm(pB11) ~600 keV. In general, recent experimental data (Feng, Reference Feng2020) give a large pB11 cross section for one alpha with energy about 6 MeV and the remaining 2.9 MeV is shared statistically by the other two alphas. The spectrum of alphas created in pB11 fusion does not change the concept of our fusion reactor. To evaluate the impact of this spectrum on the value of the number of the alpha particles created during the chain reaction N α (for an initial number of alphas N α0 created by the laser-induced shock wave), the value of 
$\langle {\rm \sigma}v\rangle$is required. This value depends on the cross section spectrum of the fusion reaction as explained in Eq. (24).
This published approach (Eliezer et al., Reference Eliezer, Hora, Korn, Nissim and Martinez-Val2016a) was criticized (Shmatov, Reference Shmatov2016; Belloni et al., Reference Belloni, Margarone, Picciotto, Schillaci and Giuffrida2018) and defended (Eliezer et al, Reference Eliezer, Hora, Korn, Nissim and Martinez-Val2016b). In this section, we show how to keep the chain reaction going for a time duration much larger than the laser pulse duration. This is achieved with an external magnetic field and an accelerating electric field (Bracci and Fiorentini, Reference Bracci and Fiorentini1982) acting as a cyclotron for protons and alphas. These fields prolong the avalanche process by overcoming the Bethe–Bloch energy loss (Bethe and Ashkin, Reference Bethe, Ashkin and Segre1953) of the protons and the alphas confined in the external magnetic field.
The Bethe–Bloch stopping power dT/dx is given by
$$\eqalign{& \displaystyle{{dT_{\rm A}} \over {dx}}({\rm erg}/{\rm cm}) = -\displaystyle{{4{\rm \pi} Z_{\rm A}^2 Z_{\rm B}e^4n_0} \over {m_{\rm e}c^2{\rm \beta} ^2}}\left[ {\ln \left( {\displaystyle{{2m_{\rm e}c^2{\rm \beta}^2{\rm \gamma}^2} \over I}} \right)-{\rm \beta}^2} \right] \cr & {\rm \beta} = \displaystyle{u \over c};{\rm \gamma}={\rm } \displaystyle{1 \over {\sqrt {1-{\rm \beta} ^2}}} = 1 + \displaystyle{{T_{\rm A}} \over {M_{\rm A}c^2}};{\rm} \cr & {\rm Non\ relativistic\ (NR):} T_{\rm A}{\rm}={\rm } \displaystyle{1 \over 2}M_{\rm A}c^2{\rm \beta} ^2{\rm}} $$
The projectile (e.g., proton in our case) with a mass M A and a charge Z Ae (e is the positive value of the electron charge) dissipates its energy into the medium (i.e., H3B) via interactions with the electrons of the medium. T A is the kinetic energy of projectile A, βc is the projectile velocity (β
$\gg$ 1/137), index B is the medium where its particles have a charge Z Be. The medium density is n 0 (atoms/cm3), m e is the electron mass, and I (~10 eV) is a phenomenological constant describing the binding of the electrons to the medium. We write the stopping power in the following practical units
$$\displaystyle{{dT_{\rm A}} \over {dx}}\lpar {{\rm eV}/{\rm cm}} \rpar = -1.65 \times 10^7Z_{\rm A}^2 Z_{\rm B}\left( {\displaystyle{{n_0} \over {{10}^{22}\,{\rm c}{\rm m}^{-3}}}} \right)\left( {\displaystyle{{0.04} \over {\rm \beta}}} \right)^2$$For our case, Z A = 1 (proton), Z B = 2 (H3B), β = 0.035 [see Eq. (10)] and for practical purposes, it is conceivable to take n 0 = 1019 (cm−3). The strength of the electric field is of the order of (dT A/dx)/(Z Ae), yielding an electric field of E = 43 kV/cm.
A pulsed and oscillatory field is preferable because here it is possible to reach higher peak values than in the static field. In particular, one can use the oscillating electric field in conjunction with a magnetic field B at the cyclotron frequency ωc given by
$$\eqalign{& {\rm \omega} _{\rm c} = \displaystyle{{Z_{\rm A}{eB}} \over {M_{\rm A}c}}{\rm (Gaussian\ cgs\ units)};\quad {M}_{\rm A} = {AM}_{\rm p} \cr & {\rm \omega} _{\rm c}({\rm rad/s},{\rm proton}) = 9.58 \times 10^8\left[ {\displaystyle{B \over {10\,[{\rm Tesla}]}}} \right]} $$The breakdown field E ac for ac field is much higher than in the dc case and it is approximately given by
$$E_{{\rm ac}}\approx \displaystyle{{m_e{\rm \omega} _cc} \over {\rm e}}\approx 43({\rm kV}/{\rm cm})\left( {\displaystyle{B \over {25\,{\rm Tesla}}}} \right)$$The number density of the produced alpha particles n α in the nuclear fusion of pB11 is related to the proton number density,
$$n_{\rm p} = n_{{\rm p}0} + \delta n_{\rm p};\quad\delta n_{\rm p} = n_\alpha /3{\rm} $$The appropriate number densities (cm−3) of the boron11 and protons n B, n p are
$$\eqalign{& {\rm \varepsilon} = \displaystyle{{n_{\rm B}} \over {n_{\rm p}}};\quad{\rm} {\rm n}_0 = n_{\rm B} + n_{\rm p} = \lpar {{\rm \varepsilon} + 1} \rpar n_{\rm p} \cr & n_{\rm p} = 5.00 \times 10^{19}\left( {\displaystyle{\rho \over {{10}^{-3}\,{\rm g/c}{\rm m}^3}}} \right){\rm for\ \varepsilon} = 1/3 \cr & n_{_{\rm B}} = 51.66 \times 10^{19}\left( {\displaystyle{\rho \over {{10}^{-3}\,{\rm g/c}{\rm m}^3}}} \right){\rm for\ \varepsilon} = 1/3} $$We avoid the protons from decelerating with an external electric field [given by Eq. (21)] in the H3B medium so that the chain reaction yields the number density of the produced alpha particles n α,
$$\eqalign{& \displaystyle{{dn_{\rm \alpha}} \over {dt}} = n_{\rm B}\langle \sigma {v}\rangle \lpar {3n_{{\rm p}0} + n_{\rm \alpha}} \rpar \cr & n_{\rm \alpha} = 3n_{{\rm p}0}\left[ {\exp \left( {n_{\rm B}\langle \sigma {v}\rangle t} \right)-1} \right]\approx 3n_{{\rm p}0}\left[ {\exp \left( {\displaystyle{t \over {{\rm \tau}_{\rm A}}}} \right)-1} \right] \cr & {\rm \tau} _{\rm A} = \displaystyle{1 \over {n_{\rm B}\langle \sigma {v}\rangle}} \approx 4.8 \times 10^{-5}\left( {\displaystyle{{{10}^{-3}\,{\rm g/c}{\rm m}^3} \over {\rm \rho}}} \right){\rm for\ \varepsilon}={\rm 1/3}} $$The new reactor
In this paper, we suggest a clean proton–boron11 fusion reactor (Martinez Val and Eliezer, Reference Martinez Val and Eliezer2019). The concept of this reactor is distinctly different from ICF and MCF reactors. In this scheme, the particles are accelerated by a shock wave to velocities of the order of 109 cm/s so that the fusion cross section gets its maximum value.
We consider the fusion p + 11B → 34He +8.9 MeV. Our reactor consists of a background plasma with densities of the order of a mg/cm3 of boron11 and hydrogen ions. A plasma channel or a solid target is irradiated by a high power laser that creates a shock wave containing proton particles that enter the background plasma. Fusion boron alphas collide with protons of the plasma and accelerate them causing a chain reaction. The new created alphas are confined by external magnetic fields in the mirror vessel.
We use an existing fluid in our vessel filling the circuit shown in Figure 1. The 11B2H6 is the most popular and suitable compound with a density of 1.3 × 10−3 g/cm3. The fluid can easily be compressed and reach higher densities if required. Therefore, for our numerical examples, we take ρ = 0.001 g/cm3 and ε = 1/3.
For the case of creating a shock wave in the background plasma in Figure 1, we use two PW lasers with irradiances of 1018 W/cm2. The existence of these lasers is due to the chirped-pulse amplification technique developed more than 30 years ago (Strickland and Mourou, Reference Strickland and Mourou1985; Mourou et al., Reference Mourou, Barty and Perry1998). Today, the laser intensity has increased presently to a maximum value of 1022 W/cm2 at infrared wavelength (1.6 eV). Laser systems with an even higher power are worldwide under consideration and development, such as the ELI project in three countries (the Czech Republic, Hungary, and Rumania), XCELS in Russia, HIPER in the UK, and GEKKO EXA in Japan.
The Peta-Watt lasers create a semi-relativistic shock wave with flow velocities as given in “The Shock Wave initiating mechanism” and Figure 2b. The volume of this accelerated fluid is according to Eqs (10)–(12),
$$\eqalign{& V = S\lpar {{\rm \tau}_{\rm L} + \Delta t} \rpar u_{\rm s}\approx 5.3{\rm \tau}_{\rm L}u_{\rm sL} S \cr & V({\rm c}{\rm m}^3) = 2.5 \times 10^{-6}\left( {\displaystyle{{{10}^{-3}\,{\rm g/cm}^3} \over {{\rm \rho}_0}}} \right)\left( {\displaystyle{{W_{\rm L}} \over {1\,{\rm kJ}}}} \right)} $$
$$\eqalign{& N_{{\rm p}0} = n_{{\rm p}0}V = 1.25 \times 10^{14}\left( {\displaystyle{{W_{\rm L}} \over {1\,{\rm kJ}}}} \right), \cr & N_{\rm \alpha} = 3N_{{\rm p}0}\left[ {\exp \left( {\displaystyle{t \over {{\rm \tau}_{\rm A}}}} \right)-1} \right];\quad{\rm \tau} _{\rm A}\approx 4.8 \times 10^{-5}\left( {\displaystyle{{{10}^{-3}\,{\rm g/c}{\rm m}^3} \over {\rm \rho}}} \right) \cr & t\gg {\rm \tau} _{\rm A}:N_{\rm \alpha} \approx 1.65 \times 10^{14}\left( {\displaystyle{{W_{\rm L}} \over {1\,{\rm kJ}}}} \right)\exp \left( {\displaystyle{t \over {{\rm \tau}_{\rm A}}}} \right)} $$As a numerical example, we calculate for the case where the energy of the proton–boron11 nuclear fusion of 8.9 MeV per reaction is mainly divided between the three alphas so that we can define the gain G in this case by the ratio of W αN α to the laser energy W L, where W α = (8.9 MeV/3). For a reactor facility, a gain of 100 would be economically and technologically satisfactory. Using Eq. (26), we have
$$\eqalign{& G = \displaystyle{{N_{\rm \alpha} W_{\rm \alpha}} \over {W_{\rm L}}} = 7.8 \times 10^{-2}\exp \left( {\displaystyle{t \over {{\rm \tau}_{\rm A}}}} \right) \ge 100 \cr & \exp \left( {\displaystyle{t \over {{\rm \tau}_{\rm A}}}} \right) \ge 1282\to t[{\rm s}] \ge 7.15{\rm \tau} _{\rm A} = 3.0 \times 10^{-4}\left( {\displaystyle{{{10}^{-3}\,{\rm g/c}{\rm m}^3} \over {\rm \rho}}} \right)} $$For each laser pulse, we need an electric field with a duration of 0.3 ms in order to receive our chain reaction process. For a 100 MW power reactor, we need the power of 100 laser pulses. This can be accomplished with 100 lasers operating with a frequency of 1 Hz or for example with two lasers of 50 Hz.
Conclusions
Using a combination of laser–plasma interactions and magnetic confinement, a conceptual fusion reactor is suggested in this paper. Our reactor consists of a background plasma with densities of the order of a mg/cm3 and of boron11 and hydrogen ions. Since the temperature of this plasma is few eV, the well-known problem with a thermal plasma for pB11 is avoided, i.e., the radiation level is very low. The fusion process is started via a plasma channel or a solid target that is irradiated by a high power laser that creates a semi-relativistic shock wave. This accelerates a proton beam to reach a kinetic energy of 300–1200 keV where the pB11 fusion cross section is significantly large to produce three alphas
The fusion boron alphas collide with protons of the plasma and accelerate them causing a chain reaction as described in “The confinement and the chain reactions”. It is important to emphasize that in a thermal fusion reactor, the maximum possible theoretical gain is 8900/600~15 (for the case where the alphas have equal energy); however, in our case, due to the chain reaction process, the maximum gain is (8900/600)*(chain reaction factor) where the chain factor is given by 
$\rm exp[t/(n_0\langle{\rm \sigma} v\rangle)]$ [see Eqs (2) and (26)] and can be very large. In order to achieve this, a combination of an external electric field and a magnetic mirror confinement device keeps the chain reaction process going on for a pulse long enough to get a high energy gain.
