On fusion chain reactions in 11 B targets for laser driven aneutronic fusion

The work presented in this letter suggests that it is possible to enhance the yield in laser driven aneutronic fusion devices by fusion chain reactions. This mechanism will be described using the example of aneutronic fusion between an incoming high-energy proton beam and a 11 B target. Such fusion reactions create alphas that can again fuse with a 11 B particle in a dense solid state target. An improved target design will be shown that enhances the recycling of fast alpha particles that are created from fusion reactions. It will also be argued that such alpha recycling may have already been observed in experiments, although it was attributed to another, more complex physical mechanism.


Introduction
Different schemes for obtaining fusion reactions have been proposed but in recent years, concepts for aneutronic fusion reactors emerged [1][2][3][4].One particular novelty, laser driven fusion devices with solid targets, consisting of boron, which can react with incoming proton beams, have been studied [5,6].In Refs.[6] and [7] an unexpected high yield of around 10 9 alpha particles was observed and explained by the occurrence of so-called avalanche proton boron reactions.The basis for these avalanche reactions was believed to be inelastic collisions between alpha particles and protons.It was argued that the alphas, which were produced during the fusion processes, transferred kinetic energy to protons through multiple inelastic collisions.These protons were accelerated up to about 660 keV, which falls into the energy range for the maximum fusion cross section between proton and boron.However, we argue that due to the isotropy of particle ejection after a fusion event and directional nature of an incoming proton beam as it was used in the aforementioned papers such a process is highly unlikely.There is also another paper by M. Shmatov that also argues against the proposed mechanism of avalanche reactions [9].Thus, in this work a much simpler mechanism is proposed that can explain the extraordinary yields of alpha particles observed by the cited authors.It will be argued that in a densely packed solid state fusion target, it is much more likely that some of the produced 4 He particles will directly fuse with some of the neighbouring 11 B atoms in the lattice.Such a fusion chain reaction, on the other hand, is able to produce again protons with a kinetic energy (around 730 keV) that matches the energy at the maximum reaction cross section for protonboron fusion reactions (about 620 keV) very well.These thoughts will be outlined in the next section of this paper.

II. Physical basics
As  It can be seen from Fig. 1 that half of the fusion products that are created at or near the surface of the target, will be ejected from the target backwards into the direction of the incoming proton beam.This loss will be accounted for in the subsequent calculations.First, we will have a look at the number of fusion events F that is normalised to one m³: Where n p and n B are the proton and boron number density, σ is the (energy dependent) fusion cross section and D is the thickness of the boron target (100 nm for the PALS experiment).It has to be noted that D as well as the densities are taken to be constant in this case.In general an expanding gas or plasma plume may form at the point of proton impact that causes D to increase and the density n B to go down locally, which is neglected in this work.It has also to be emphasized that the number of fusion reactions that are obtained by using Eq. ( 1) are normalised to /m³.Hence, the actual number of fusion reaction that occurs in a thin layer will be considerably less.The same normalisation to /m³ is applied to the proton and boron densities as well.In order to explain the aforementioned enhanced alpha yields, it is sufficient to compare only the ratios of incoming protons to created 4 He particles and secondary protons, respectively.For simplicity we will only look at the maximum value of the fusion cross section and assume that the incoming proton beam has gamma ray when fusing with a 11 B atom.Each of those reactions is depicted in more detail in the following Fig. 2, which also illustrates that the kinetic energy of the proton produced matches the kinetic energy at the peak cross section for proton boron fusion and can, thus, be 'recycled' for further fusion reactions.

Figure 1 :
Figure 1: Schematic spread of alpha particles created through aneutronic proton boron fusion events.

1 Figure 2 :
Figure 2: Fusion chain reaction through the creation of a 731.9 keV proton, which is fed into a second fusion cycle in the

28 92 Figure 3 :
Figure 3: optimised target design for a maximum alpha93 145 that a laser-to-proton conversion efficiency for 146 laser proton acceleration of about 15 % was 147 experimentally achieved [11].Thus, the energy 148 gain in such a case is already very close to break 149 even, even when the laser to proton energy 150 conversion efficiency is taken into account.It is 151 reasonable to assume that the advances in laser 152 technology and laser particle acceleration will 153 only further improve those results in the near 154 future.

Table 1 :
Fusion reactions used in this paper, their maximum cross section and kinetic energy at the maximum cross section, taken from the EXFOR data base[12].