2. The Upper Boundary of g

Eliezer et al. [Reference Eliezer, Schweitzer, Nissim and Martinez Val10] analyzed the situations when reaction (1) occurs in gaseous H₃ ¹¹B 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 [Reference Eliezer, Schweitzer, Nissim and Martinez Val10]. 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 [Reference Vasilevskiy13], we will estimate the lowest boundary $W_s^l$ of $\left(W_s\right)$ in gas medium consisting of atoms of ¹¹B with the density

(2) $\begin{array}{}{n}_{11\text{B}}\approx 2.5\times {10}^{18}{\text{cm}}^{-3},\end{array}$

and molecules of H₂ with the density

(3) $\begin{array}{}{n}_{\text{H}2}\approx 3.75\times {10}^{18}{\text{cm}}^{-3}.\end{array}$

At the conditions described in [Reference Eliezer, Schweitzer, Nissim and Martinez Val10], 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 $n_{11\text{B}}/n_{\text{H}2}$ corresponds to the ratio of the numbers of nuclei of ¹¹B and protons in the nonexisting free molecule of H₃ ¹¹B discussed in [Reference Eliezer, Schweitzer, Nissim and Martinez Val10]. The choice of $n_{11\text{B}}$ corresponds to an example presented on page 5 of Reference [Reference Eliezer, Schweitzer, Nissim and Martinez Val10] and is mainly important for an estimate of the typical proton path $l_{typ}=1/\left({\sigma }_1{n}_{11B}\right)$ , corresponding to one reaction (1). The estimate of $W_s^l$ presented below yields that this parameter is independent of $n_{11\text{B}}$ .

In the situation under consideration, the change $\text{d}{\epsilon }_p$ of ε_p on proton path dx is given approximately by

(4) $\begin{array}{}\text{d}{\epsilon }_p\approx \left(eE-k_{\text{H}2}^p\left({\epsilon }_p\right)n_{\text{H}2}-k_{\text{B}}^p\left({\epsilon }_p\right)n_{11\text{B}}\right)\text{d}x,\end{array}$

where e is the proton charge, E is the strength of the electric field, and $k_{\text{H}2}^p$ and $k_{\text{B}}^p$ are the parameters describing the transfer of ε_p to molecules of hydrogen and atoms of boron, respectively. The parameter $k_{\text{H}2}^p$ was calculated as

(5) $\begin{array}{}{k}_{\text{H}2}^p=2A_{\text{H}}{m}_u{S}_{\text{H}2}^p,\end{array}$

where A _H is the atomic mass of hydrogen, m_u is the atomic mass unit, and $S_{\text{H}2}^p$ is the stopping power of molecular hydrogen for proton. The parameter $k_{\text{B}}^p$ was calculated as

(6) $\begin{array}{}{k}_{\text{B}}^p\approx \frac{m_u}{2}\left(A_{\text{Be}}{S}_{\text{Be}}^p+A_{\text{C}}{S}_{\text{Cam}}^p\right),\end{array}$

where A _Be is the atomic mass of beryllium, $S_{\text{Be}}^p$ is its stopping power for proton, A _C is the atomic mass of carbon, and $S_{\text{Cam}}^p$ is the stopping power of amorphous carbon with the density of 2 g/cm³ for proton. The values of $S_{\text{H}2}^p$ , $S_{\text{Be}}^p$ , and $S_{\text{Cam}}^p$ from [Reference Berger, Coursey, Zuker and Chang14] were used.

The parameter $k_{\text{B}}^p$ was approximated by (6) due to the absence of data on the stopping power of boron for proton in [Reference Berger, Coursey, Zuker and Chang14]. 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) $\begin{array}{}P\left(Z\right)\approx \left(P\left(Z-\Delta Z\right)+P\left(Z+\Delta Z\right)\right)/2,\end{array}$

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 $P\left(Z=6,{\epsilon }_p=600\text{keV}\right)\approx 3797$ MeV cm² g⁻¹ and $P\left(Z=6,{\epsilon }_p=700\text{keV}\right)\approx 3440$ MeV cm² g⁻¹, calculated using $S_{\text{Cam}}^p$ from [Reference Berger, Coursey, Zuker and Chang14], with the same parameters, calculated using (7) and $\Delta Z=2$ . Substituting $S_{\text{Be}}^p$ and the stopping power of molecular oxygen for proton from [Reference Berger, Coursey, Zuker and Chang14] into (7), we obtain $P\left(Z=6,{\epsilon }_p=600\text{keV}\right)\approx 3773$ MeV cm² g⁻¹ and $P\left(Z=6,{\epsilon }_p=700\text{keV}\right)\approx 3424$ 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 $\le {\epsilon }_p\le 700$ keV (see below), the relative accuracy of (6) is of the order of 1% or even better.

According to [Reference Nevins and Swain15], ${\epsilon }_p^{\ast }\approx 646.2$ keV and

(8) $\begin{array}{}{\sigma }_1\left({\epsilon }_p={\epsilon }_p^{\ast }\right)\approx 1.196\text{b}.\end{array}$

Let us denote the value of E corresponding to the condition $\text{d}{\epsilon }_p/\text{d}x=0$ , i.e., to the almost exact compensation of the transfer of kinetic energy of protons to the gas medium by the electric field, as E ₀. This value depends on ε_p ((4)). Equations (2)–(6) and (8)) yield that at ${\epsilon }_p={\epsilon }_p^{\ast }$ , $l_{typ}\approx 3.34\times {10}^5$ cm, $E_0\approx 24.9$ kV/cm, $e{E}_0{l}_{typ}\approx \left(k_{\text{H}2}^p\left(n_{\text{H}2}/n_{11B}\right)+k_{\text{B}}^p\right)/{\sigma }_1\approx 8.32\text{GeV}$ , and $\left(8.7\text{MeV}\right)/\left(e{E}_0{l}_{typ}\right)\approx 1.046\times {10}^{-3}$ .

At ${\epsilon }_p\approx {\epsilon }_p^{\ast }$ , $k_{\text{H}2}^p$ and $k_{\text{B}}^p$ decrease with increasing ε_p ((5) and (6) and [Reference Eliezer, Schweitzer, Nissim and Martinez Val10, Reference Berger, Coursey, Zuker and Chang14]). This results, in particular, in the impossibility to provide a stable motion of proton with such kinetic energy at constant E [Reference Eliezer, Schweitzer, Nissim and Martinez Val10]. The highest value of $1/\left(e{E}_0{l}_{typ}\right)$ corresponds to ${\epsilon }_p\approx 657.6$ keV, $E_0\approx 24.6$ kV/cm, $l_{typ}\approx 3.36\times {10}^5$ cm, $e{E}_0{l}_{typ}\approx 8.27$ GeV, and $\left(8.7\text{MeV}\right)/\left(e{E}_0{l}_{typ}\right)\approx 1.052\times {10}^{-3}$ . These values of $e{E}_0{l}_{typ}$ and $\left(8.7\text{MeV}\right)/\left(e{E}_0{l}_{typ}\right)$ can serve as $W_s^l$ and the upper boundary of g, respectively. It should be emphasized that the real value of $\left(W_s\right)$ can be much greater than $e{E}_0{l}_{typ}$ 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. [Reference Nagy and Végh16–Reference Raizer18]. 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 $W_s^l$ 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 [Reference Eliezer, Schweitzer, Nissim and Martinez Val10] for power production: in any case, $\left(W_s\right)$ will include $e{E}_0{l}_{typ}$ 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 [Reference Teller19, Reference Smirnov, Subbotin and Sharkov20]. According to [Reference Smirnov, Subbotin and Sharkov20], 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 η_d 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 [Reference Smirnov, Subbotin and Sharkov20]. This parameter should exceed ten even if η_d 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. [Reference Weaver, Zimmerman and Wood1] 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 [Reference Eliezer, Schweitzer, Nissim and Martinez Val10], 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 $\left(W_s\right)$ . This can be shown using equations, similar to (4)–(6), and the data from [Reference Eliezer, Schweitzer, Nissim and Martinez Val10, Reference Belloni11, Reference Berger, Coursey, Zuker and Chang14, Reference Landau and Lifshitz21] 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 $n_{\text{H}2}=\text{0}$ , the highest value of $1/\left(e{E}_0{l}_{typ}\right)$ corresponds to ${\epsilon }_p\approx 656.6$ keV, $e{E}_0{l}_{typ}\approx 4.30$ GeV, and $\left(8.7\text{MeV}\right)/\left(e{E}_0{l}_{typ}\right)\approx 2.024\times {10}^{-3}$ .