4. Results and Discussion

Comparing Figures 2a and 2b, we can see that for the boron solid, the proton beam can only penetrate to the surface, whereas for the boron plasma, it can penetrate to a longer distance. This difference can be explained as follows. The boron solid has a large resistivity, and the boron plasma, with abundant free electrons, has a much lower resistivity. Ohm’s law (4) reveals the fact that the large difference in resistivity will lead to a significant difference in the electric field generation. As shown in Figure. 2c and 2d, the maximum value of the electric field in the boron solid is more than 100 times stronger than that in the boron plasma. Such a strong electric field in the boron solid will greatly prevent the beam from penetrating deeper into the target.

Imitating the experimental measurement method, we have recorded the energy spectra of α particles escaping from the left simulation boundary in the range of 0 to 6.5 MeV, which are plotted in Figure 3. Comparing the cases of the boron plasma without electromagnetic fields (5N-noEB) and the boron solid without electromagnetic fields (N-noEB), we find that when electromagnetic fields are not calculated in the simulations, there are about 40% more α particles produced by the p¹¹B fusion reactions in the laser-ablated boron solid (boron plasma). This difference is attributed to degeneracy effects, which do not play a role in the solid boron target but become non-negligible after laser ablation, as mentioned above. A theoretical explanation can be given here. For degenerate electrons, their velocity distribution is governed by the Fermi ̶Dirac (FD) statistics as follows:

(5) $\begin{array}{}{f}_{FD}\left(v_e\right)=\frac{2m_e^3}{{\left(2\pi \hslash \right)}^3{n}_e}\frac{1}{\exp \left(\beta \left(E_e-\mu \right)\right)+1},\end{array}$

where m^e is the electron mass, $\beta =1/k_B{T}_e$ , E_e is the electron energy, and μ is the chemical potential. The dielectric function of degenerate electrons can be expressed as [Reference Norreys, Batani and Baton39].

(6) $\begin{array}{}\epsilon \left(k,\omega \right)=1+\frac{1}{4\pi k_F{a}_0^2{z}^3}\left(g\left(u+z\right)-g\left(u-z\right)\right).\end{array}$

Here, a ₀ is the Bohr radius, $k_F=m_e{v}_F\hslash ={\left(3{\pi }^2{n}_e\right)}^{1/3}$ , $u=\omega /k{v}_F$ , $z=k/2k_F$ , and

(7) $\begin{array}{}g\left(x\right)={\int }_0^{\infty }\frac{ydy}{\exp \left(D{y}^2-\beta \mu \right)+1}\ln \left(\frac{x+y}{x-y}\right),\end{array}$

where $D=E_F\beta$ is the degeneracy parameter. Finally, the stopping power of degenerate electrons can be obtained by the widely used dielectric formalism [Reference Néstor40–Reference Casas, Andreev, Schnürer, Barriga-Carrasco, Morales and González-Gallego43].

(8) $\begin{array}{}sp=-\frac{dE}{dz}=\frac{2{\left(Ze\right)}^2}{\pi v^2}{\int }_0^{\infty }\frac{dk}{k}{\int }_0^{kv}d\omega \omega \mathrm{Im}\left(\frac{-1}{\epsilon \left(k,\omega \right)}\right).\end{array}$

For the convenience of analysis, it is instructive to take advantage of the stopping power per unit density (SPPUD) to evaluate the influence of degeneracy effects

(9) $\begin{array}{}s{p}^{\prime }=-\frac{sp}{n_e}.\end{array}$

Figure 4 shows the numerical results of (9) for different electron densities. It can be seen that if the electron density is increased from $2.52\times {10}^{23}$ cm ⁻³ (density of the boron solid) to $1.26\times {10}^{24}$ cm ⁻³ (density of the boron plasma), SPPUD of the electrons is decreased. In our cases, the yield of α particles produced by the p ¹¹ B fusion can be expressed as [Reference Giuffrida, Belloni and Margarone18].

(10) $\begin{array}{}{N}_{\alpha }=\frac{3N_p{n}_e}{Z_i}{\int }_0^{E_0}\sigma \left(E\right){\left(\frac{dE}{dz}\right)}^{-1}dE=\frac{3N_p}{Z_i}{\int }_0^{E_0}\frac{\sigma \left(E\right)}{s{p}^{\prime }}dE,\end{array}$

where N_p is the number of protons, Z_i is the charge number of the boron ion, andσ(E) is the cross section of the p ¹¹ B fusion. (10) reveals the relation between the yield of α particles and the SPPUD of the electrons and implies that the proton beam propagating in the high-density boron plasma will have more chances to collide with boron nuclei, generate the p ¹¹ B fusion and produce α particles, which is consistent with our simulation results about the gap between the cases of the boron plasma without electromagnetic fields (5N-noEB) and the boron solid without electromagnetic fields (N-noEB) in Figure 3. Both the theory and the simulations indicate that degeneracy effects have an influence on the p ¹¹ B fusion. Nonetheless, quantitatively speaking, they are not the primary factor that causes the significant difference in the yield of α particles in the experiments of Labaune et al. since, as shown in Figure 3, they can only increase the yield by about 40%.

Figure 3: The energy spectra of α particles escaping from the left simulation boundary in the range of 0 MeV to 6.5 MeV: (1) the blue line (N-noEB), the boron solid without electromagnetic fields; (2) the red line (5N-noEB), the boron plasma without electromagnetic fields; (3) the blue triangle line (N-EB), the boron solid with electromagnetic fields; (4) the red square line (5N-EB), the boron plasma with electromagnetic fields. The yellow patch corresponds to where it cannot be measured in the experiments.

Figure 4: SPPUD as a function of the proton energy. For the blue solid line, the electron density is $n_e=2.52\times {10}^{23}$ cm ⁻³, corresponding to the density of the boron solid, and for the red dotted line, the electron density is $n_e=1.26\times {10}^{24}$ cm ⁻³, corresponding to the density of boron plasma.

It can be seen in Figure 3 that there is a large gap between the cases of the boron solid without electromagnetic fields (N-noEB) and the boron solid with electromagnetic fields (N-EB), which indicates that in terms of the boron solid, collective electromagnetic effects have a huge influence on the number of fusion reactions and the yield of α particles. As mentioned above and shown in Figure 2c, when the proton beam is injected into the boron solid, a strong stopping electric field will be generated. On the one hand, it can greatly increase the energy loss of the proton beam and prevent the beam from penetrating. Recently, Ren et al. presented a piece of experimental evidence on the significantly enhanced energy loss of a laser-accelerated proton beam in the dense ionized matter [Reference Clauser and Arista44], which is similar to the case we are describing. On the other hand, (10) shows that if the energy loss of the beam increases, the number of fusion reactions and the yield of α particles will decrease accordingly. For the boron plasma, the gap between the cases of the boron plasma without electromagnetic fields (5N-noEB) and the boron plasma with electromagnetic fields (5N-EB) is not that large because, compared with the boron solid, the boron plasma has a much lower resistivity and, according to Ohm’s law Eq. (4), the generated electric field will also be smaller, as displayed in Figure 2c and 2d. Therefore, collective electromagnetic effects in the boron plasma are not as significant as in the boron solid. Collective electromagnetic effects described in this paper are a kind of nonlinear effects caused by a large number of injected ions. Previously commonly used single-particle theories and simulation models cannot be used here. Collective electromagnetic effects depend on many factors, such as the current density of the proton beam, the resistivity of the boron target, and the flow velocity of plasma electrons.

For the cases of the boron solid without electromagnetic fields (N-EB) and the boron plasma with electromagnetic fields (5N-EB), both degeneracy effects and collective electromagnetic effects are taken into account. The gap in the yields of α particles between these two cases is about a tenth of a second, which is in good agreement with the results at dt = 1.2 ns in the experiments of Labaune et al. As we have discussed above separately, the gap here originates from two aspects: degeneracy effects and collective electromagnetic effects. They exert influences on the number of fusion reactions by changing the energy loss of the proton beam. To be specific, the more energy the proton beam losses during its transport in boron targets, the smaller the number of fusion reactions between protons and boron atomic nuclei will be. Readers may notice that the specific numbers of recorded α particles in our simulations are greater than those in the experiments. Actually, it is caused by the difference in the total number of injected protons between our simulations and their experiments. As shown in Eq. (10), the yield of α particles produced by the p ¹¹ B fusion is proportional to the number of protons. If the total numbers of protons in our simulations is greater than that in the experiments, then there will be an equal multiple difference in the yields of α particles. In this work, we are concerned with the difference in the α-particle yields produced in different states of boron targets rather than the specific numbers. From this perspective, our simulations are indeed in good agreement with the experiments.

Eventually, it should be mentioned that while α particles produced by the p ¹¹ B fusion are propagating in boron targets, they are simultaneously heated and being stopped by the background particles [Reference Ren, Deng and Qi45–Reference Malekynia, Ghoranneviss, Hora and Miley47], which, as a matter of fact, will alter the initial energy spectrum of α particles. This indicates that degeneracy effects and collective electromagnetic effects influence not only the yield of α particles but also their energy spectrum or velocity distributions. Degeneracy effects can be considered to be isotropic if local fluctuations of the boron density and temperature are ignored, but it is not the case with collective electromagnetic effects. For the α particles moving forward (the opposite direction of the electric field), their energy loss will be increased, whereas for the α particles moving backward (the same direction as the electric field), they will be accelerated by the electric field and gain energy. Whether one tries to use the p ¹¹ B fusion to obtain a net energy output to solve the energy crisis or view the p¹¹ B fusion as α-particle source, the influences of degeneracy effects and collective electromagnetic effects on the energy evolution of α particles could be a topic worthy of in-depth study in future work.