Suprathermal-ion-driven fusion chain reactions in the pure deuterium system

It is argued that fusion chain reactions in the D-D system is feasible with supra-thermal deuterons in the MeV regime, with new generations of deuterons being generated either via neutron–deuteron or proton–deuteron collisions. The propagation of supra-thermal deuterons in an infinite, hot, dense deuterium target was studied using a Monte Carlo method that includes multiple nuclear reactions, electron and ion stopping, along with neutron and proton knock-ons. Over a wide range of densities we observed significant, albeit sub-critical chain reactions in the multi-keV temperature regime. At very high densities (over 1000 gcm−3) and temperatures (over 40 keV) we observed chain reactions that reached criticality. These results suggest that there is a case to re-assess the potential of inertial confinement fusion based on deuterium-heavy targets.


Introduction
Recently there has been a renewed interest in the possibility of chain reactions occurring in fusion systems via suprathermal ions [1][2][3][4][5][6].This is motivated by the potential for chain reactions to either enhance (complement) thermonuclear burn, or to form the basis for an advanced beam fusion concept that could lead to much higher beam fusion gains than have previously been thought possible.This is particularly important for the prospects for 'advanced' fusion fuels which include pure deuterium, D- 3 He mixtures, and p-11 B mixtures, as, at present, it does not appear feasible to achieve sufficient gain using these fuels in either inertial confinement fusion (ICF) schemes, or in magnetic confinement fusion (MCF) schemes [7,8].
Beam fusion [9,10], -where one attempts fusion by collision of an energetic fuel beam with either a static fuel target or another energetic fuel beam-is not normally considered Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. as a viable option for controlled fusion energy, as the energy losses due to Coulomb collisions result in a low gain at best.In the case of beam fusion the gain is often defined in terms of the ratio of the fusion energy yield to the beam energy in (G = E fus /E beam ).Although one obtains G ≪ 1 in the case of a 'cold' target, it was observed by Dawson [11] that G > 1 could in fact be achieved if the electron temperature of the target was sufficiently high.This stems from the reduction of the electron drag when v e,th > v i .Later studies by Santini [8] suggested that quite high gains might be possible for the DT system, however more sophisticated calculations by Sherlock [9] (which included the role of ion-ion stopping) showed that a peak gain in the range of 5-6 was more likely.Unfortunately, as Sherlock noted, a fusion system with G ∼ 6 is not a particularly feasible candidate for power production, but it was suggested that this could be exploited in other ways.
If it is at all possible to somehow extend the idea of beam fusion then it would seem likely (given the aforementioned studies) that this lies in driving chain reactions.Although the D-D reactions do produce products that can undergo further reactions (i.e.T and 3 He): and, the reactions involving these products cannot drive any further reactions, i.e.D + T −→ n + α, and 3 He + D −→ p + α.Thus the D-T,D-D, and D- 3 He based systems cannot support chain reactions without considering other processes.The same is true of the p- 11 B reaction which produces three alpha particles.There are, however, at least two candidates for processes that could enable chain reactions.Firstly there could be large angle elastic collisions between product ions and fuel ions in the target which result in a large momentum transfer to a fuel ion.This would then 'regenerate' the initial supra-thermal ion.This will be limited by the size of the relevant cross-section and by the range of the product ions.Secondly, in a sufficiently large fuel mass, the product neutrons can transfer momentum to fuel ions via elastic collisions which can also regenerate the supra-thermal fuel ions.It should be noted at this point that it is likely that neutrons would only be able to sustain a chain reaction (if at all possible) at target densities comparable to those obtain in ICF.The product of density and plasma size in MCF is too small to allow neutron energy to be re-captured, although product ions should remain confined and these could result in knock-ons.We note in passing, that, in principle, chain reactions could also be facilitated by auxillary nuclear reactions that can regenerate reactants.For the p-11 B fuel, the reaction α + 11 B → p + 14 C could be exploited [2].
The question of beam fusion is thus closely related to the question of whether or not suprathermal particles generated by fusion reactions could lead to an enhanced burn rate.The central idea is that the energetic fusion products (neutrons or ions) can impart multi-MeV energy to a fuel ion in the target, which then has a high probability of undergoing a rapid fusion reaction, or possibly even initiating a chain of fusion reactions.For the case of p-11 B this was recently analyzed by Belloni who concluded that the formation of chains of reactions was highly infrequent, and was unlikely to strongly affect the expected burn rate.One major factor that differentiates the deuterium system from the p- 11 B system is the presence of neutrons in the deuterium system, which will strongly affect the generation of new fuel ions under conditions where neutron scattering can occur.
It is possible that the beneficial effects of neutrons and suprathermal ions/products has already been observed in some simulations.For example, in Atzeni and Ciampi's [12,13] studies of the fast ignition of tritium-poor targets (most of the target being pure deuterium) it was noted that neutron transport and suprathermal product ions appeared to play an important role.However this was not fully quantified or investigated in the reports, and it is not clear whether some of the treatments of neutrons and suprathermals in their code was sufficient to fully resolve this issue.For these reasons, we regard this report as being complementary to the earlier work of Atzeni and Ciampi [12,13] and will hopefully provide impetus to the re-examination of deuterium heavy targets in ICF.
In this paper we have studied the prospect for chain reactions in a system consisting of suprathermal deuterons, and a pure deuterium target.This has principally been studied numerically using a Monte Carlo code for the case of an infinite, uniform target.We conclude that significant (k > 0.1) subcritical and even slightly super-critical (k ≳ 1) chain reactions can exist in at multi-keV temperatures (10-50 keV).Although significant subcritical chain reactions occur over a wide density range, chain reactions very close to criticality only occur at densities which are high even by the standards of ICF (>3000 gcm −3 ).The transfer of energy between deuteron generations via neutrons is essential, and since ICF with deuterium fuel is only really contemplated at areal densities where neutron stopping is significant ρr >5 gcm −2 , it would appear that the results presented in this manuscript could be highly relevant to ICF schemes using deuterium-heavy fuel .Since suprathermal chain reactions in D-D are more significant than has been previously thought, this work could therefore motivate a re-appraisal of the prospect and potential of deuterium-heavy fuel in ICF.
The paper is organized as follows : In section 2 the theoretical basis for chain reactions being possible is described.In section 3 the Monte Carlo model that is used for numerical simulations is described, and in section 4 the results of simulatlions performed using this model are presented, with the conclusions being stated in section 5.

Theory
In this section we will consider the essential criteria necessary for a chain reaction to develop in the D-D beam-target system.Primarily we require that there is a very high probability of the primary D-D reactions occurring.If no drag due to Coulomb collisions acted on the deuterons then the mean free path due to D-D fusion reactions alone would be λ fus = 1/n t σ (with n t being the deuteron density in the target, and σ is the total crosssection for the D-D fusion reactions).Since there will be drag (which introduces a stopping length, λ s ), we therefore need to enter the regime where, λ fus /λ s < 1.There are two components to the drag : the electron induced drag and the ion induced drag.As previously noted, at sufficiently high temperatures the electron drag component can be strongly reduced, however this is not really viable for the ion component.We therefore consider the ion drag to represent the 'hard obstacle'.If we thus neglect the electron drag for now, the (half-energy) stopping length due to ion drag alone is [14,15], Substituting this into λ fus /λ s < 1 yields, We can re-express this in a more convenient form: From equation ( 5) one can see that this condition is more easily satisfied the smaller Z is, thus the D-D system is potentially well suited to being able to sustain chain reactions.The next issue is to examine the cross-sections, and the stopping number.In figure 1 the lab frame cross-sections for both D-D reactions and the D-T reaction are shown.
From figure 1 shows that 3-10 MeV, the total cross-section for the D-D reactions is about 0.2 barns.Thus even if we assume L ∼10, we can satisfy equation ( 5) if the deuteron energy is over 4.2 MeV.In figure 2 the stopping number is plotted versus density (log scale for density).
This shows that the stopping number could lie in the range of 10-20, in which case deuterons energies of up to 6 MeV could be required to satisfy equation ( 5).Overall we can thus conclude that it is feasible for multi-MeV deuterons to have sufficiently high reaction probabilities to sustain chain reactions in spite of ion-ion drag.This only holds provided that the electron-ion drag has been sufficiently reduced which implies that temperatures of at least several keV are reached [14,15].
The other main element of the chain reaction is the transfer of energy between 'generations' of deuterons.The main candidate for this are neutrons from either the neutron-bearing D-D reaction or D-T reactions.Unlike ions, the charge-free neutrons do not interact with the electrons and thus do not experience any stopping due to electrons.Neutrons will only lose their energy via elastic collisions with ions.What is therefore required is that the target areal density must be sufficiently large to ensure that this happens.In this study we assume that the target is infinite and thus neutron collisions are guaranteed.In reality, we would require ρr to exceed about 5 gcm −2 to ensure this occurs [7,12].Although this is large for ICF predicated on DT fuel, this is actually very comparable to the values of ρr anticipated for ICF predicated on deuterium heavy fuel [7,12,13].

Model
A numerical Monte Carlo [16][17][18][19] model (beams) was constructed to analyze the possibility that fusion chain reactions can occur in the D-D beam-target system.The model tracks a set of initial and secondary ions, with the number of initial ions being denoted by N init , and the maximum number of ions (initial plus secondary/product ions) being denoted by N max .The ions move through an infinite target that is completely homogeneous.
The nuclear reactions that are considered in the model are as follows [7,20]: p The model tracks deuterons, protons (from reactions (6) and ( 9)), tritons and helions.Since no reactions involve alpha particles as reactants, and we do not consider alpha-deuteron knock-ons, alpha particles are not tracked.
All ions (deuterons, protons, tritons, and helions) are subject to drag due to the target.Each ion is tracked from its initial energy to a final energy that is a specified fraction of the initial energy over a specified number of discrete steps.The distance moved by each ion over an energy interval is determined via, with the electronic and ionic drag terms being [7,14,15,21,22], and, where, Each reaction that the ion can undergo is assigned a 'tracker', R k , whose initial value is randomly generated using a uniform distribution over the interval 0 to 1. Across each energy interval, each tracker is reduced by n t σds, where σ is the cross-section at the energy mid-point of the interval.Tabulated cross-sections for most the relevant reactions were acquired from the ENDF/B-VIII.0library and linear interpolation using this tabulated data is employed in the code.This includes the reactions expressed by equations ( 6)- (10).Once a tracker falls below zero, a reaction is considered to have occurred.For reactions ( 6)- (9) this terminates the tracking of that ion.Where a reaction produces either a proton, deuteron, helion, or triton, this results in the 'spawning' of a new ion, however this only occurs if the current total number of ions is less than the hard limit in the code (N max ).For reactions ( 6)- (9), it is assumed that probability of product emission is isotropic in the zero momentum frame.
The generation of new deuterons, which can permit chain reactions to occur, is possible by the inclusion of two processes.The first is by elastic neutron collisions.Since, in this model, the target is considered to be infinite in extent, each neutron could make a large number of collisions and thus produce many deuterons.As each collision transfers about 4/9ths of the neutron energy on average, the number of collisions considered was limited to a specified number, M n .The second process is proton-deuteron elastic collisions.For this process, a cross-section was estimated from the experimentally reported cross-sections for large-angle (in the 150 • -170 • range) elastic scattering.It was decided to neglect alpha-deuteron scattering, as the ion drag scales as ∝ Z 2  1 Z 2 2 /v 2 , so the drag on the alphas will be 16 times larger (at the same energy).
The initial set of deuterons are all set to the same initial energy, denoted E D .The standard number of tracking steps from the initial to final energy being 100, and the standard final energy is set to 1% of the initial ion energy.Throughout this study we have employed N init = 10 5 , and N max = 1.2 × 10 8 .

Observation of chain reactions
Taking a target density of n i = 1×10 33 m −3 , M n = 4, and E D = 6 MeV, we have a 'standard set' (A-I) that we will use as a reference point throughout what follows.The core results from this set are shown in table 1.We present the results through two important metrics.Firstly the mean gain, Ḡ, which is determined by the sum of Q-values (nuclear energy released or absorbed) over all particles divided by product of the number of initial deuterons and their initial energy, i.e. the initial energy input in suprathermal ions.Secondly, as described in section 3, the 'generation' of each deuteron is tracked.One can therefore determine the number of deuterons in each generation, and thus calculated an inter-generational multiplication factor, which we can write as, where N i is the number of deuterons in the ith generation.The initial set of deuterons is taken to the be zeroth generation.
Note that in some instances we write Ḡ as being greater than some value.These are cases where the simulation spawns so many new particles that it hits the maximum number allowed (i.e.N max ).In this case we have been unable to fully track the chain reaction, so we can only state the value reached with the maximum number of particles allowed for these simulations.
Note that these results are all given to two decimal places.
From table 1 one can see that we indeed observe chain reactions at all temperatures.The multiplication factor from the zeroth to first generation, k g,1 , is always relatively large, whereas the multiplication factors that relate the first to sixth generations are much smaller and generally quite consistent.This can be understood in terms of the differences in energy distributions between the zeroth generation and subsequent generations, and thus the values of k g,1 should be seen as a transient induced by the initial conditions.The values k g,2 to k g,6 (and beyond although these were not tracked) are much more interesting and are indicative of chain reactions that subsist for at least six generations.We can see that generally this leads to a subcritical chain (k < 1), however once the temperature hits 45 keV we observe that the chain becomes critical (k = 1).If the temperature is raised somewhat further then one can observe mildly super-critical chain reactions, and we have seen the average k value reaching 1.04 at 50 keV, and 1.08 and 55 keV, by slightly extending this set.In the following subsections we will now analyze the importance of different physical processes and chosen parameters in obtaining these results.

Role of neutrons
Tabulated results obtained from the beams calculations for M n =6,9,2,1 and without neutrons knock-ons can be found in appendix (see tables A1-A5).These sets, which we denote A-II-A-VI, use the same parameters as set A-I, only M n being varied.The effect on the chain reactions due to changing the  number of knock-ons per neutron (M n ), can be seen by looking at how Ḡ varies with M n at T = 40 keV, which is shown in figure 3. Figure 3 shows that the average gain obtained is strongly dependent on the number of neutron knock-ons per neutron, with strong chain reactions and high gains being achieved once we allow for 4 collisions per neutron, and as the number increases much beyond this the benefits become increasingly modest.It is therefore clear that the chain reactions that we observe are mainly enabled by neutron knock-ons transferring energy from reaction products to the next generation of energetic deuterons.It is important to note that even when neutron collisions are turned off that limited chain reactions (k ∼ 0.1) are still observed at 40 keV, which is due to the included of proton-deuteron knock-ons.

Role of proton-deuteron collisions
Next the role of the proton-deuteron knock-ons was examined by carrying out another set of simulations (set B) where the proton-deuterons collisions were turned off.These results can be found in tabulated form in table A6.For set B, n i = 1×10 33 m −3 , M n = 4, and E D = 6 MeV, in order to make the set comparable to set A-I.The values of Ḡ against T are compared in figure 4. Figure 4, shows that the average gain per initial deuteron is always higher when we include proton-deuteron knock-ons than when we do not.This difference becomes quite substantial from 30 keV onwards.In set A-I, above 40 keV, we obtain critical and super-critical chain reactions which fully consume the maximum number of particles in the simulation, leading to average gains of at least 240.These are not included on this plot.In constrast, in set B the chain reactions are still subcritical even at 60 keV, with slightly super-critical results only obtained at 70 keV.From this we can conclude that protondeuteron knock-ons, whilst not essential to the existence of strong chain reactions, has quite a strong influence of setting the lower temperature bound at which they occur.Although we have previously noted that MCF cannot benefit from neutron knock-ons, it is possible that MCF could benefit from the effect of proton and alpha knock-ons.

Role of proton-deuteron disintegrations
The proton-deuteron disintegration reaction might be thought to have a significant effect on any potential suprathermal chain reaction, as it will act as sink or absorber of the energy in the To assess this we carried out a further set of simulations (set C) the results of set C are tabulated in table A7 which is identical to set A-I, except that the proton-deuteron distintegration reaction is disabled in set C. In figure 5 we compare Ḡ versus T in both sets.
Figure 5 shows that the effect of removing the disintegration reaction is very slight at low temperatures, with a modest effect at 30 keV.Suddenly there is very significant effect at 40 keV where, in set C, Ḡ shoots up to >422 while it is only about 36 in set A-I.What we therefore see is that the protondeuteron disintegrations shift the onset (in temperature) of critical and super-critical chain reactions down by about about 5 keV.Although it is therefore not greatly critical in terms of the key question of whether strong chain reactions occur or not, it is not completely negligible in terms of accurately determining their onset.

Role of initial deuteron energy
In all of the previous sets of results, the initial energy of the deuterons has been set to E D = 6 MeV.In set D, the temperature was fixed at 30 keV, and the initial deuteron energy was varied from 1 MeV to 8 MeV.A target density of n i = 1×10 33 m −3 , and M n = 4 was used, making the results comparable to set A-I and other sets.The tabulated results can be found in table A8.All of these results have been plotted in figures 6 and 7.
From both of these figures we can see two things.Firstly, the initial deuteron energy does have a strong effect on the average gain, which might be expected to some extent.However it is also clear that this is tied to k g,1 which is also  quite dependent on the initial deuteron energy.It also makes sense that the multiplication factor relating the zeroth and first generations will depend on the energy distribution of the initial generation.What is more interesting is what we observe for k g,i for i > 1, where is it clear from figure 7 that the multiplication factors for later generations are essentially independent of the energies of the zeroth generation.This reinforces the conclusion reached earlier (section 4.1) that the first generation represents a 'transient' before a chain reaction is established where the initial deuteron energies in each generation will likely have a distribution that differs substantially from the monoenergetic distribution used for the zeroth generation.Equally these results demonstrate that the choice of the energy of the zeroth deuteron generation does not greatly disrupt the formation of the 'natural' chain reaction from the  second generation onwards, although it will affect the absolute magnitudes of the populations.

Role of target density
In set E the effect of the target density was studied by varying the target density, n i , whilst choosing E D = 6 MeV, T = 45 keV, and M n = 4.The results of this set are tabulated in table A9.These results are plotted in figures 8 and 9.
In figure 8 we can see that up until the density pass 10 32 m −3 , the effect on the average gain is relatively modest, with it taking the density to change by a factor of 100 for the average gain to increase by a factor of three.The massive increase in the average gain at very high density is the result of the transition into criticality.In figure 9 we can see that the behaviour of the multiplication factors is much less dramatic, with the multiplication factors varying slowly across three orders of magnitude.We suggest that these findings are entirely sensible : if we refer back to equation ( 5) one can note that n i is not explicitly present, as it cancels out due to linear dependence on n i in both the reaction length and the stopping length.Instead the density is only present in the stopping number, L, where it resides within a logarithm function.This logarithmic dependence is very slow, which is exactly what we observe in these results.Although this suggests that critical chain reactions might only be viable at very high densities (> 1000 gcm −3 ) chain reactions that are mildly sub-critical (k > 0.1) still occur all the way down to about 3 gcm −3 , i.e. not too much above solid density.

Uncertainty analysis
Tests have been carried out to assess the extent to which errors affect the results that have been presented in the previous sections.An example of this are tabulated in tables.For a given set of parameters the simulations were repeated five times and the standard error as a fraction of the mean values were examined.It was found that the standard error on both the average gain and the multiplication factors did not exceed 2% of the mean value and was often less than 1%.

Conclusions
In this paper we have examined the possibility that fusion chain reactions can occur in the pure deuterium system, and to what extent any chain reactions approach criticality.The ideal case of an essentially infinite target that is uniform and at at constant temperature and density has been examined by means of a Monte Carlo model.Sub-critical chain reactions have been observed over a wide range of densities and temperatures in the multi-keV range.Throughout that parameter space, the occurence of sub-critical chain reactions leads to gains (defined as the total energy yield from nuclear reactions induced by one initial deuteron divided by the initial energy of that deuteron) in excess of one.At sufficiently high temperatures and densities chain reactions that reached and slightly exceeded criticality (k > 1) were observed.
We believe that the results presented in this work could have significant implications for pursuing ICF with targets where deuterium is the majority component (a.k.a.'tritium poor targets').Although some aspects of the results presented here could be seen as being promising for an advanced beam-fusion concept, the regions of parameter space where this is most true are the same high temperature and high density regions that are only contemplated in ICF.We therefore take the view that it is better to view these results in terms of how the supra-thermal ions generated by thermal fusion will lead to enhanced burn in deuterium-heavy targets.Since it can be expected that every tritium ion generated that D-D reaction channel will react relatively rapidly, this means that we can expect (at sufficiently high ρr) that DT neutrons will be able to produce supra-thermal deuterons via knock-ons.This implies that, provided ρr is sufficiently large, on average there will be one supra-thermal chain reactions for every two thermonuclear D-D reactions.Since this leads to substantial gain in many cases this implies that the chain reactions will increase the energy released by a substantial multiplicative factor.Thus there is a case that chain reactions could produce significant enhanced burn in deuterium-heavy targets.This effect appear not to have been considered in important previous studies.Potentially, by affecting the fusion heating rate, it could affect the ignition conditions (e.g.ignition energy).A more important question concerns the extent to which a burn rate is enhanced by supra-thermal chain reactions would alter the burn-up fraction, as this could have an important bearing on the gain that could be achieved.Even at 50 keV, the fusion time of a 1-2 MeV deuteron is ten times smaller than the thermal fusion time, which suggests there could be substantial improvements to the burn-up fraction.This could be particularly important for re-assessing the future of D-D fusion, as previous studies have shown that the region of parameter space for exploiting D-D fusion might be quite limited, even using a DT 'seed' region to facilitate ignition [23].

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Figure 1 .
Figure 1.Plot of the cross-section of the D-D and D-T reactions in terms of incident energy.

Figure 4 .
Figure 4. Plot of Ḡ against T for sets B and A-I as indicated.This figure provides a comparison of turning the proton-deuteron knock-ons on and off.Note that value from set A-I at 45 keV not plotted as this exceeds 240.

Figure 5 .
Figure 5. Plot of Ḡ against T for sets C and A-I as indicated.This figure provides a comparison of turning the proton-deuteron disintegrations on and off.

Figure 6 .
Figure 6.Plot of Ḡ against E D for set D.

Figure 7 .
Figure 7. Plot of k g,1 -k g,4 against E D for set D.

Figure 8 .
Figure 8. Plot of Ḡ against n i for set E. Note that result for 1 × 10 33 m −3 is not plotted as value exceed 245.

Figure 9 .
Figure 9. lot of k g,1 -k g,4 against n i for set E.

Table A9 .
Results from set E where the target density is varied (E D = 6 MeV, T = 45 keV, and Mn = 4).