# Analysis of the p-11B Fusion Scenario with Compensation of the Transfer of Kinetic Energy of Protons and Alpha Particles to the Gas Medium by the Electric Field

Authors: Mikhail L. Shmatov
Journal: Laser and Particle Beams
 (2022)

---

## Abstract

The energy balance of the p-11B fusion scenario with compensation of the transfer of kinetic energy of protons and alpha particles to the gas medium by the electric field is considered. It is shown that such scenario cannot provide the use of p-11B fusion reaction for power production due to the very low ratio of the energy release of the fusion reaction to the energy necessary for compensation. The upper boundary of this ratio is about 2×10−3.


---

## Body

## 1. Introduction

The influence of chain reactions on the rateR1 of p-11B fusion reaction(1)p+11B⟶3α+8.7MeV,is discussed since 1973 [1–12]. One of the chain reactions consists of the scattering of at least one of the three alpha particles, generated by reaction (1), on proton(s) with acceleration of the proton(s) to kinetic energies, corresponding to a relatively high cross-section σ1 for reaction (1) and the subsequent participation of the accelerated proton(s) in this reaction [1, 4, 6–12]. According to [1], at the temperature of 150–350 keV and the density of 1016–1026 cm−3, this chain reaction and other “nonthermal” effects result in an increase in R1 on 5–15%. The type of particles with such densities was not mentioned [1], but this detail is not essential because in plasma under consideration, the densities of all particles are comparable [8]. According to [2–4, 7, 9, 10, 12], at least if special measures are taken, the increase in R1 due to the chain reactions can be so high that it will provide the possibility of the use of reaction (1) for power production. The negative results of analysis of such assumptions from [2–4] are presented in [5, 6, 8, 11].In 2020, Eliezer and Martinez-Val [9] and Eliezer et al. [10] proposed p-11B fusion scenarios with the influence of electric and magnetic fields on protons and alpha particles in the gas medium. The main idea of the proposal is that during some time periods, time-dependent electric field should compensate approximately for the transfer of kinetic energy εp of a proton with εp≈εp∗, where εp∗ is εp corresponding to the largest value of σ1 for the collision of proton with the nucleus of 11B in the rest, to the medium and for transfer of the kinetic energy of the alpha particle to the medium [9, 10]. This compensation should increase the probability of participation of the protons in reaction (1) and that of “useful” acceleration of protons due to the scattering of alpha particles on them. The magnetic field should provide the realization of these scenarios in reactors with acceptable sizes [9, 10]. Below, it is shown that in the scenario proposed in [10], the ratio g of energy release of reaction (1) to the average value Ws of the energy spent for the initiation of one reaction (1) will be unacceptably low for power production.


## 2. The Upper Boundary ofg

Eliezer et al. [10] analyzed the situations when reaction (1) occurs in gaseous H311B or other hydride of 11B with a density of 1019 cm−3 or of the order of 1019 cm−3 and temperature of about 1 eV or few eV. Ionization of this gas is supposed negligible [10]. Since a free molecule of H3B does not exist and at the temperature above 700°C all hydrides of boron dissociate into boron and hydrogen [13], we will estimate the lowest boundary Wsl of Ws in gas medium consisting of atoms of 11B with the density(2)n11B≈2.5×1018cm−3,and molecules of H2 with the density(3)nH2≈3.75×1018cm−3.At the conditions described in [10], the medium containing atoms of boron and molecules of hydrogen will also contain atoms of hydrogen and ions, but this is not essential for the analysis of the acceptability of attainable values of g for power production. The ratio n11B/nH2 corresponds to the ratio of the numbers of nuclei of 11B and protons in the nonexisting free molecule of H311B discussed in [10]. The choice of n11B corresponds to an example presented on page 5 of Reference [10] and is mainly important for an estimate of the typical proton path ltyp=1/σ1n11B, corresponding to one reaction (1). The estimate of Wsl presented below yields that this parameter is independent of n11B.In the situation under consideration, the changedεp of εp on proton path dx is given approximately by(4)dεp≈eE−kH2pεpnH2−kBpεpn11Bdx,where e is the proton charge, E is the strength of the electric field, and kH2p and kBp are the parameters describing the transfer of εp to molecules of hydrogen and atoms of boron, respectively. The parameter kH2p was calculated as(5)kH2p=2AHmuSH2p,where AH is the atomic mass of hydrogen, mu is the atomic mass unit, and SH2p is the stopping power of molecular hydrogen for proton. The parameter kBp was calculated as(6)kBp≈mu2ABeSBep+ACSCamp,where ABe is the atomic mass of beryllium, SBep is its stopping power for proton, AC is the atomic mass of carbon, and SCamp is the stopping power of amorphous carbon with the density of 2 g/cm3 for proton. The values of SH2p, SBep, and SCamp from [14] were used.The parameterkBp was approximated by (6) due to the absence of data on the stopping power of boron for proton in [14]. This equation corresponds to the assumption that the product P of the stopping power of the medium, consisting of atoms or molecules of one chemical element with atomic number Z, on the atomic mass of this element depends on Z approximately linearly and, therefore,(7)PZ≈PZ−ΔZ+PZ+ΔZ/2,where ΔZ is a small natural number, for example, unity or two. In order to demonstrate that at least in some situations, the accuracy of (7) is rather high, let us compare PZ=6,εp=600keV≈3797 MeV cm2 g−1 and PZ=6,εp=700keV≈3440 MeV cm2 g−1, calculated using SCamp from [14], with the same parameters, calculated using (7) and ΔZ=2. Substituting SBep and the stopping power of molecular oxygen for proton from [14] into (7), we obtain PZ=6,εp=600keV≈3773 MeV cm2 g−1 and PZ=6,εp=700keV≈3424 MeV cm2 g−1. Thus, in these cases, the relative accuracy of (7) is better than 1%. This allows us to assume that at 600 keV ≤εp≤700 keV (see below), the relative accuracy of (6) is of the order of 1% or even better.According to [15], εp∗≈646.2 keV and(8)σ1εp=εp∗≈1.196b.Let us denote the value ofE corresponding to the condition dεp/dx=0, i.e., to the almost exact compensation of the transfer of kinetic energy of protons to the gas medium by the electric field, as E0. This value depends on εp ((4)). Equations (2)–(6) and (8)) yield that at εp=εp∗, ltyp≈3.34×105 cm, E0≈24.9 kV/cm, eE0ltyp≈kH2pnH2/n11B+kBp/σ1≈8.32GeV, and 8.7MeV/eE0ltyp≈1.046×10−3.Atεp≈εp∗, kH2p and kBp decrease with increasing εp ((5) and (6) and [10, 14]). This results, in particular, in the impossibility to provide a stable motion of proton with such kinetic energy at constant E [10]. The highest value of 1/eE0ltyp corresponds to εp≈657.6 keV, E0≈24.6 kV/cm, ltyp≈3.36×105 cm, eE0ltyp≈8.27 GeV, and 8.7MeV/eE0ltyp≈1.052×10−3. These values of eE0ltyp and 8.7MeV/eE0ltyp can serve as Wsl and the upper boundary of g, respectively. It should be emphasized that the real value of Ws can be much greater than eE0ltyp due to acceleration of secondary charged particles, i.e., molecular ions of hydrogen, protons, ions of 11B, and electrons created by the fast protons considered above and alpha particles, etc. [16–18]. At sufficiently high temperature, the acceleration of electrons and ions arising due to thermal ionization can also be important. The problem of the possibility of electric breakdown in the gas medium under consideration can probably be solved only experimentally. The presented estimate of Wsl corresponds to the assumption that the magnetic field prevents the acceleration of electrons and relatively slow molecular ions of hydrogen, protons, and ions of 11B by the electric field. However, the accuracy of this estimate is sufficient for the reliable qualitative conclusion about the unacceptability of the scenario proposed in [10] for power production: in any case, Ws will include eE0ltyp and, therefore, g will be too low. The reason is that the efficiency of the use of any fusion reaction for power production will be determined, in particular, by the cost of electricity [19, 20]. According to [20], for the inertial fusion energy power plant with conversion of fusion energy into thermal energy and subsequent conversion of 30–35% of the latter into electricity, the cost of electricity will be acceptable when the product of the target gain on the driver efficiency ηd exceeds ten. The target gain is the ratio of fusion energy release of one microexplosion to the energy delivered to the target for ignition of the microexplosion [20]. This parameter should exceed ten even if ηd is close to unity and is an analog of the parameter g. Thus, g of the order of 10−3 and less is not sufficient for power production involving conversion of fusion energy into thermal energy. Note that Weaver et al. [1] discussed briefly the potential feasibility of power production in the regime of subignition operation corresponding to g<1. In any case, g of the order of 10−3 and less seems to be too low even for this regime.Note also that in the scenario proposed in [10], the acceleration of alpha particles, if it is not suppressed by the magnetic field, will not provide the effective acceleration of protons and, therefore, will serve mainly as a process increasing Ws. This can be shown using equations, similar to (4)–(6), and the data from [10, 11, 14, 21] for the analysis of the motion of alpha particles and the transfer of their kinetic energy to protons. The compensation of deceleration of protons in the gas medium consisting mainly of atoms of 11B will also not provide sufficiently high values of g: at nH2=0, the highest value of 1/eE0ltyp corresponds to εp≈656.6 keV, eE0ltyp≈4.30 GeV, and 8.7MeV/eE0ltyp≈2.024×10−3.


## 3. Conclusion

The scenario proposed in [10] cannot be used for effective power production due to the very low attainable g, the upper boundary of which is about 10−3. A decrease in nH2/n11B2 down to zero can result only in an approximately two-fold increase in the upper boundary of g. The real value of g can be much less than its upper boundary.



---

*Source: 7473118-2022-06-28.xml*