Photon and neutron production as in-situ diagnostics of proton-boron fusion

Short-pulse, ultra high-intensity lasers have opened new regimes for studying fusion plasmas and creating novel ultra-short ion beams and neutron sources. Diagnosing the plasma in these experiments is important for optimizing the fusion yield but difficult due to the picosecond time scales, 10s of micron-cubed volumes and high densities. We propose to use the yields of photons and neutrons produced by parallel reactions involving the same reactants to diagnose the plasma conditions and predict the yields of specific reactions of interest. In this work, we focus on verifying the yield of the high-interest aneutronic proton-boron fusion reaction $^{11}{B}(p,2\alpha){}^4{He}$, which is difficult to measure directly due to the short stopping range of the produced $\alpha$s in most materials. We identify promising photon-producing reactions for this purpose and compute the ratios of the photon yield to the $\alpha$ yield as a function of plasma parameters. In beam fusion experiments, the ${}^{11}{C}$ yield is an easily-measurable observable to verify the $\alpha$ yield. In light of our results, improving and extending measurements of the cross sections for these parallel reactions are important steps to gaining greater control over these laser-driven fusion plasmas.


I. INTRODUCTION
Short-pulse lasers offer new experimental approaches to creating and studying fusion plasmas.
In contrast to long-pulse lasers which have been a primary tool in inertial-confinement fusion (ICF), short-pulse lasers have pulse durations < 1 ps and use small focal spots to obtain peak intensities up to 10 23 W/cm 2 in a single pulse. Short-pulse lasers deliver their energy to the plasma in a time much shorter than the typical expansion timescale, and both electrons and ions achieve much higher momenta. These plasma conditions are far from the quasi-thermal equilibrium of ICF, where burn has recently been achieved [1], and the question is open whether or not the dynamics admit a pathway to net energy gain [2]. Short pulse lasers have successfully driven high-yield beam fusion experiments [3,4], which can in turn be translated into novel high-flux, ultra short-pulse ion [5][6][7] and neutron sources [8][9][10][11].
Short-pulse lasers can drive fusion in two ways: direct irradiation of a target containing the fusion reactants or laser-ion acceleration creating an ion beam that is dumped into a catcher/target.
To our knowledge, no experiment can claim to have optimized the fusion yield, and the efficiencies of these two methods for different candidate fusion reactions remains a topic of research. Naively, one expects direct irradiation to convert laser energy more efficiently into fusion yield, in part because fusion can occur both in the neighbourhood of the focus where all ion species are heated, and in the colder bulk of the target by ions accelerated out of the focal region. Anecdotally, recent experiments support this hypothesis [2].
Proper optimization will require greatly improved understanding and control of experimental outcomes compared to current capabilities. However the same laser properties, namely ultra shortpulse and typically small ∼ (10µm) 3 focal volume, make the plasma difficult to diagnose. Most interpretation is based on inference from the measured particle yields and spectra, sometimes supported by Monte Carlo or numerically-expensive kinetic laser-plasma simulations. Improving experimental diagnostics of laser-driven nuclear reactions has thus become a significant topic of discussion [12]. Our goal in this work is to identify new diagnostics providing information on the plasma conditions and nuclear reaction dynamics.
Out of the reactions studied with short-pulse lasers, we focus on the proton-boron-11 fusion reaction 11 B(p, 2α) 4 He, which is of particular interest because it releases 8 MeV into kinetic energy of the 3 α particles and no neutrons. The α particles themselves provide the most direct measure of the fusion yield, but because they deposit their kinetic energy into the surrounding medium very efficiently, only a small fraction of those produced escape the target. This problem is especially acute in direct irradiation experiments [5,[13][14][15][16][17][18], where the mean kinetic energy and density of the medium vary by orders of magnitude in different regions of the target, precluding a systematic analytic correction for α stopping.
Since direct diagnostics of plasma conditions, such as probe laser pulses or atomic spectroscopy, remain an enormous technical challenge, we investigate other nuclear reactions for products whose yield or spectrum can be measured more reliably. Photons and neutrons are the best candidates, but not all reactions will yield enough photons or neutrons that can be identified as coming from a specific reaction. We introduce the yield ratio as a phenomenological tool to relate an easilymeasured yield to the yield of interest. Yield ratios have an established history in diagnosing the ICF plasma, where they can determine several of the important ρR parameters [19]. We have previously used the ratio between α and 11 C yields to determine the 11 B(p, 2α) 4 He reaction yield more accurately in beam-target experiments [20], and here we demonstrate its utility in direct irradiation experiments as well. The yield ratio eliminates normalization unknowns such as the local density of reactants, effective reaction volume and time, and takes as input a few model parameters, such as the mean ion kinetic energy, that can be determined from particle diagnostics. We conclude by identifying the two best candidate reactions for proxy measurements of the 11 B(p, 2α) 4 He yield in direct irradiation experiments and confirm 11 C as the best proxy in beam-target experiments.

II. ACCESSIBLE REACTIONS
The goal is to predict the outcomes and analyse the data from experiments on the aneutronic proton-boron fusion reaction 11 B(p, 2α)α. The 11 B(p, 2α)α cross section reaches ∼ 1 b around 650 keV center-of-mass (CM) energy, significantly higher than DD or DT fusion reactions because of the higher charge of boron. In fact, most other proton-boron reactions require even higher CM energy before the cross section approaches 100mb, and the high cross section of 11 B(p, 2α)α in the E < 1 MeV range is due to two identified resonances, related to above-threshold excited states of 12 C [21]. Recent work has resolved apparent normalization discrepancies in the measured cross section [22], resulting in re-evaluation of the process as a candidate for fusion energy.
With its cross section reaching ∼ 1 b already at 650 keV CM energy, lower than the thresholds of many other proton-initiated reactions on boron, 11  All data retried from sources in the EXFOR and JANIS databases, with the exception of 11 B(p, 2α) 4 He for which we use the concordant normalization of Ref. [22]. Q values marked with an * are computed from the mass difference of the initial and final states. Others are from the literature. a A recent measurement [23] finds ∼ 5× larger cross section than several previous measurements. To be conservative, we quote the older but more consistent results here. b There appear two distinct groups of cross sections measurements: two experiments find the cross section roughly 10× larger than the majority of others.
additional reactions that are naturally grouped as "primary" or "secondary". Primary reactions are initiated by the protons scattering on 11 B, 10 B or 14 N as present in typical boron or boronnitride solid targets. These are listed in Table I with peak cross section, the corresponding CM energy at which the peak cross section is found and the range of CM energy over which data is available. Secondary reactions are re-scattering of the α particles on the boron and nitrogen nuclei most prevalent in the environment. These are listed in Table II, similarly to the primary reactions. The databases contain additional processes, such as 14 N(p, n+p) 13 N , 14 N(p, n+ 3 He) 11 C, but the data is too sparse and the larger number of fragments generally makes the Q values for such reactions negative and large in magnitude. Consequently, their cross sections should have somewhat higher thresholds suppressing their contributions to the yields.
The corresponding cross sections are plotted in Figure 1. The global sets from EXFOR [24] frequently include inconsistent measurements, as for example the recently resolved normalization in the 11 B(p, 2α) 4 He cross section [22]. Cross section data is one significant source of uncertainty in our yield predictions. We have plotted the global data sets for each cross section without distinguishing their sources and in our calculations, we will use fits to these global data with a few exceptions described in the captions of Tables I and II   sections outside the range of available data. Instead, for numerical integrations we implement best fit curves that are forced rapidly to zero outside the range of experimental data. This choice almost certainly underestimates the yield for several processes. Notably the cross sections of 11 B(p, α) 8 Be, 10 B(p, γ) 11 C and all secondary reactions for which data is available seem likely to continue to increase with CM energy. However, the lack of information precludes quantifying the uncertainty in any attempted modeling of the cross section.
Among these primary and secondary reactions, we identify promising candidates to be diagnostics. As mentioned above, the difficulty in verifying the 11 B(p, 2α) 4 He yield is greater in direct-irradiation type of experiments. For these we need a reaction occuring in parallel with a product that escapes the plasma unperturbed, such as neutrons and photons.   Tables I and II To strengthen identification of the originating reaction, a neutron or photon produced with definite energy is preferable. Exothermic reactions (Q > 0) are therefore better candidates, since we can expect the neutron or photon spectrum to peak at nonzero kinetic energy. Of the primary reactions Table I, only two satisfy these conditions, 11 B(p, γ) 12 C and 10 B(p, γ) 11 C, both emitting photons. The cross section data for both reactions is very limited. For the 10 B(p, γ) 11 C reaction especially, the trend in the available data suggests our calculations here may significantly underestimate the photon yield. Of the secondary reactions, three satisfy these conditions, 11 B(α, n) 14 N, 10 B(α, n) 13 N and 14 N(α, γ) 18 F.
As several of the candidate reactions have limited cross section data available and even more limited data on the spectrum of the outgoing neutron or photon, it would be a reasonable first step to verify yields with beam-target type of experiments. Since the target remains intact, we can measure a wider variety of reaction products, in particular unstable nuclides with half-lives much greater than the experiment duration. Several of the reactions selected by the previous approach also yield unstable nuclei, specifically 11 C, 13 N and 18 F with half lives on the order of 10 3 s.
Other unstable nuclides produced are listed in Table III. The significantly differing half lives make identification by reactivity straightforward with a Geiger counter placed near the target, though 14 C probably has too long a half life and 10 C too short a half life for reliable identification. 7 Be is undetectable by this means, but arises from a reaction without particular interest in this study. Now synthesizing and narrowing the list of promising reactions, two of the photon-and neutronproducing reactions stand out in utility. First, 11 B(p, γ) 12 C is promising to directly correlate with the reaction 11 B(p, 2α) 4 He, because it has the same initial state and produces a photon with energy significantly above most other products. The ∼ 100× larger cross section for 10 B(p, γ) 11 C make it a practical proxy in the near term, though it has a different initial state, which introduces additional uncertainty. Ultimately the yields of these two reactions will determine which is more useful in the experiment.
The second 10 B(α, n) 13 N could help verify the α yield. Since this reaction is the isospin partner of 10 B(α, p) 13 C, the cross sections are very similar in magnitude and energy dependence and its measurement would verify the role of proton-recycling from secondary reactions, proposed as an important mechanism in enhancing 11 B(p, 2α) 4 He yields in some experiments [25]. Moreover, 13 N is unstable and its yield can be measured independently in beam-target experiments. A small drawback to this reaction is that the cross section decreases sharply below CM energy of 3 MeV.
The majority of αs produced by 11 B(p, 2α) 4 He should have energy above this threshold, but αs also lose energy quickly in a cold medium and the yield is expected to be small.
Of the remaining two, (a) there is unfortunately no cross section data for 14 N(α, γ) 18 F, though it should produce a peak in the photon spectrum, and (b) the neutron produced in 11 B(α, n) 14 N is likely have very little kinetic energy, making detection difficult.
Of the radioactive nuclides, 11 C has already proven its utility in beam-target experiments providing a significantly higher-confidence estimate of the α yield than a direct measurement of the αs by CR-39 [20]. 18 F has been detected [20,25] but the absence of a cross section for its production limits the information gained. More surprisingly, 13 N has not been detected, probably due to a combination of shorter halflife and low yield making it difficult to distinguish from the 11 C signal.
Since better data exists for the 13 N production cross section and it is generated almost entirely by α scattering, we consider it the most important candidate for future experiments as a means to help verify the α yield. 14 O and 17 F are of little interest: in both production processes, Q < 0 so the neutron released is not a potential diagnostic in direct irradiation experiments.

III. YIELD EQUATIONS
Having winnowed the set of interesting reactions based on general criteria for good diagnostics of the fusion dynamics, we now evaluate the yields of the various products.
The challenge in deriving analytic expressions for the yield expression is two-fold. First, the ion momentum distribution function generally varies from shot to shot due to variations in the laser.
Therefore the ion distribution should be measured on each shot as well as possible and used in predictions. We address modeling related to incomplete measurement in Sec. IV.
Second, the ion momentum distribution is heterogeneous, with two or more distinguishable populations. Short-pulse high-intensity lasers deposit energy into a region 10-100 µm in radius from the focal spot, which we call the directly-irradiated (DI) volume. In this volume, electrons gain many MeV of energy and ions are likely to have more isotropic momentum distributions. Fast electrons pushed through and out of this volume, largely in the laser beam direction, can create large magnitude electrostatic fields that accelerate ions out of the DI volume. The precise ion acceleration mechanism and shape of the resulting ion spectrum depend on the thickness of the target. To maximize fusion yields, we assume the target is thick, that is greater than the stopping range of these high-energy ions, so that the probability of undergoing fusion is saturated. The fast ions are more likely to collide with at-rest ions much deeper in the bulk of the target, and the reaction kinematics are essentially those of beam fusion. We model the yield from each of these regions and ion populations separately, addressing reactions in the DI volume first and the beam fusion reactions second.
The dichotomy between the DI volume and the beam fusion region is artificial and the plasma will certainly contain some transitional regions. Given the dynamics described so far, these transitional regions are likely to contain electrons of intermediate kinetic energy 1 eV E e 1 MeV, both low energy ions and some beam ions and particle number densities similar to the initial state.
In this region, ion stopping is reduced compared to the cold limit and the fusion probability is similar to the beam fusion limit. Thus, the contribution to the yield can be thought of as a correction to the beam fusion yield since the length of such transitional regions is much less than the stopping range of the fast ions passing through.
The starting point to derive the yield is a classical expression for the total number of particles of type A produced in a 2-body collision, Here is the distribution function describing the probability of finding particle i = 1, 2 with momentum p + dp in the volume element x + dx at time t, σ is the cross section, and is the relative speed of the incident particles. We integrate over all initial particle momenta, all final states, and all space and time for the reaction to occur. For the reactions of interest here, sufficiently complete cross sections differential in solid angle are generally not available, and cross sections for the process exhibit resonances of nuclei with different quantum numbers, suggesting that the angular dependence will have a strong energy dependence, which we will not attempt to model here. We can focus on the total yield as the most relevant observable for both applications and practically available measurements from recent experiments.
We consider two models for our yield calculations, corresponding to the direct-irradiation and beam-target experiments.

A. Direct irradiation
The laser deposits a large amount of energy in the fusion target and reactions occur within the 10s of picosecond timescale that the ions are heated but before the target expands and its density drops. Although neither the electron nor the ion population can equilibrate in this short time, experimental ion spectra are frequently fit by a Maxwellian distribution, dN/dE ∝ e −βE . The parameter β is an inverse energy scale that characterizes the mean kinetic energy per particle of the distribution E /N = β −1 . We stress that local thermal equilibrium is certainly not achieved and we do not assume equilibrium distributions. The Maxwellian fit to experimental spectra is a phenomenological choice, enabling simple quantitative comparison between shots and facilities.
Maxwellian ion distributions is a strong simplifying approximation: kinetic simulations of shortpulse laser-target interaction have found the ion distributions can have significantly higher numbers of high-energy ions E i > β −1 than expected from a Maxwellian distribution. To account for the excess of high-energy E > β −1 ions, the single Maxwellian model can be improved by introducing a second Maxwellian distribution of smaller β. This second population often corresponds to the beam population that has significant directionality and is less likely to react with the larger β population, being accelerated out of the DI region by plasma fields on its boundary. Even so, the contribution is computed easily since the yield is linear in the distribution function and the yield in the double Maxwellian case can be derived by summing four yields corresponding to the four combinations of the two ion species' two β values. Therefore for simplicity and clarity here, we use the single Maxwellian.
Since the electrons have MeV-scale kinetic energy in the DI region, their stopping power is significantly reduced. While the high-intensity laser can drive large, short-lived, local increases in the electron density, the ion density varies from the initial value by a factor much less than 1, at least until the target expands significnatly on the 10s of picosecond timescale. Consequently, the proton and α stopping ranges (> 100 µm) are certainly larger than the radius of the DI region ( 100 µm), and we consider that the ion energy losses are negligible for the duration in the DI region.
Given these conditions, the yield of a given process is straightforwardly derived from Eq. (1).
The reaction volume is a few times larger than the focal volume but is generally not known precisely.
Reactions will continue as long as the plasma remains relatively dense, up to several picoseconds, though this plasma "confinement time" as it is sometimes known is not well-determined either.
We therefore consider the yield per unit volume per unit time for a 2 → A + X reaction for nonrelativistic ions with Maxwellian distributions, where the parameters n i , m i , β i for i = 1, 2 are the number densities, masses and inverse mean kinetic energy of the two ion species. The remaining parameters are the reduced mass, m r , difference of inverse mean kinetic energy and combinations thereof, The result is even in β r as it must be since the choice of labels is arbitrary and the yield should always be positive. The integration variable corresponds to the magnitude of the relative velocity of the two ions. The integration will be performed numerically to use experimental data for the cross section σ A (E cm ), which is a function of the CM energy. Erf(z) is the usual error function with the normalization defined by The limit of equal mean kinetic energies is simplifies the result considerably to Considering our interest in particular reactions as in-situ diagnostics of the 11 B(p, 2α) 4 He reaction, we introduce ratios of yields to eliminate experimental unknowns. For reactions with the same initial state, e.g. p-11 B scattering, all the prefactors in Eq. (2) cancel. For example, to use the 11 B(p, γ) 12 C reaction as a diagnostic on 11 B(p, 2α) 4 He, we might consider the ratio in which all the mass-and β-dependent parameters are identical in the numerator and denominator.
Only the cross sections differ. For 2 → 2 reactions such as considered here, the spectra of produced neutrons and photons have been computed semianalytically showing that their widths and small shifts in the peak depend on the momentum distribution of scattering ions [26]. This allows the (approximate) β parameters of the ions to be retrieved by fitting spectra of the measured neutron or photon. Since the reactions have exactly the same initial state, potentially large scaling factors such as volume and time must be the same. Thus this yield ratio depends only on the mean kinetic energies of the two ion species. Considered as a function of these two energy scales, the ratio manifests the difference in the energy dependence of the cross sections, though less so than the beam-target experiments described below. In Eq. (6), the factor 3 has been included in the numerator to count the total number of αs produced for each 11 B(p, 2α) 4 He reaction. With this yield ratio, the number of 11 B(p, 2α) 4 He reactions is recovered by multiplying by the measured yield of photons identified as arising from this reaction.
Another reaction of interest for diagnostics is 10 B(p, γ) 11 C, which differs from 11 B(p, 2α) 4 He in the isotope of boron in the initial state. As a consequence, some prefactors remain in the yield ratio, The subscripts for boron parameters have been shortened to the isotope number for clarity. The measured constants in the prefactor, such as masses, are no trouble, but for this ratio to be useful we must argue that the ratio of densities remains nearly constant during the relevant period of plasma evolution. Since the charge is the same and the masses differ only by 10%, we suppose significant separation of isotopes from the initially uniform mixture can only develop slowly, on the same time scale (or longer) that the plasma expands and diffuses into free space. Note that dynamically, we expect the mean kinetic energy of the boron ions and protons to be similar, making the prefactor in parentheses close to 1. The remaining ratio of reduced kinetic energies is expected to be near unity for the same reason.
Another useful yield ratio could be 10 B(α, p) 13 C relative to 10 B(α, n) 13 N. The ratio would cancel dynamical unknowns such as the density of αs. The other proton-producing secondary reaction 11 B(α, p) 14 C could be added to the ratio to completely determine the secondary proton production, though the same remarks as above would apply to the prefactor. A measurement of the neutrons produced from 10 B(α, n) 13 N constrains the number of protons able to be recycled into the 11 B(p, 2α) 4 He reaction. In this case though the input is the α spectrum derived from all primary reactions, is a complicated function of energy and is expected to vary significantly as a function of mean ion kinetic energy. We consider its derivation beyond the scope of this paper.

B. Beam-target
The beam-target experiment involves simpler kinematics. In the frame with the target material at rest, the center of mass energy is with the reduced mass given above by Eq. (3). The b subscript indicates a particle from the beam and the t subscript indicates a particle in the target. For pB scattering m r /m b 1.1. The momentum integral then only runs over the proton distribution. The target particle distribution function is nonzero only in the spatial region of the target material, and integrating over the beam axis and time convolves the projectile beam with the target distribution.
In standard beam-target experiments in order to maximize exposure, the target is placed on or adjacent to the anticipated axis of the ion beam. In the TPW experiment of particular interest, we were able to verify that the target material contained the cone of highest ion flux, which subtended an opening angle θ 0.3. We therefore assume that the transverse momentum of the beam is small relative to longitudinal momentum. These together imply we can reduce the beam momentum integral to the longitudinal momentum only and integrate the transverse position dependence into a 1-dimensional beam distribution function, where V t signifies the volume of the target. In this expression the longitudinal coordinate can also be considered as parameterizing the distance along the on-average straight-line trajectory; trajectories diverging from the beam axis would make a small geometric correction due to exiting through the side of the target rather than the opposite end. The constant density of the target has been taken outside the integral and the target length defined as L.
The beam distribution function f b has been reduced to its energy dependence dN b /dE.
The importance of stopping is seen by comparing the stopping range to the target dimensions.
The stopping range is defined where E 0 is the initial energy of the ion before interacting with the target and dE/dx from data is MeV α as 38 micron. However, target can be heated by both the ion beam and the even higher energy electrons that are accelerated out of the ion source by the laser driver. Using the fact that the stopping range is less than the target length even for the highest energy ions, the total energy deposited is just the total energy of the beam that enters the target. Even for the relatively high energy ions obtained from the Texas Petawatt, the total ion beam energy transferred to the target is at most ∼ 10% of the laser energy. For the upper limit on the Texas Petawatt, 10 J deposited into a hemisphere of radius equal to the 2mm stopping range, the specific heats of boron and boron nitride imply a temperature change ∆T 280 K 2.4-2.6 × 10 −2 eV). Without direct  [27,28]. Stopping ranges for lower energy ions are shorter, and ranges are generally less than the typical length ( cm) of the targets. The highest energy protons (E p 20 MeV) have a stopping ranges equal or greater than the target length, but their number and hence contribution is smaller by an order of magnitude or more. Neglecting this not-quite-stopped component therefore amounts to an error of ∼ 10% or less, smaller than the error propagated from the cross section and certainly smaller than the error due to the limited energy range of the cross section data. Therefore, to our working accuracy, the target can be considered "thick" in that almost all particles in the beam will be stopped. The conventional definition of the "thick-target yield" for a monoenergetic input is where dE/dx from data is conventionally positive and the expected minus sign is compensated by the flipping the limits on the integral. The density factor in converting tabulated dE/dx data into energy loss per unit length cancels with the density of target nuclei in the yield. Note that the integration can effectively be restricted to the energy range where the cross section is nonnegligible.
Since most of the cross sections have thresholds of order 1 MeV, Fig. 2 shows that finite temperature corrections to stopping matter only for T 1 keV. Raising the target temperature increases the projectile energy at which dE/dx achieves its maximum, but 1 keV is already much higher than can be realistically achieved in the beam-target experiment.
Thick target yields for all of the primary reactions in Fig. 3 show these properties. 11 B(p, 2α) 4 He and 11 B(p, α) 8 Be display the greatest sensitivity to the target temperature because the 11 B(p, 2α) 4 He cross section is largest and the cross section for 11 B(p, α) 8 Be is only available for CM energy 1M eV , where the dE/dx curves differ the most.
The total yield of the product nucleus A is obtained by integrating the thick target yield over the beam, weighted by the beam energy distribution dN/dE b , Yield ratios in the beam-fusion geometry, as in direct irradiation experiments, analytically eliminate dependence on geometric factors in the yield, such as target length and density. Less obviously, the overall normalization of the beam energy distribution also cancels in the ratio, since one could where N b is the total number of particles (that interact with the target) and f (E) is a normalized probability distribution for the ion energy. For example, the ratio of γs from 11 B(p, γ) 12 C to αs from 11 B(p, 2α) 4 He, is with the same stopping power dE/dx and normalized proton spectrum. Removing this dependence on the total number in the beam significantly reduces uncertainty in practice given the available on-shot beam measurements.
The yield ratio retains important information of the beam energy distribution. As seen in Fig. 1, different reactions have different thresholds, collision energies where the cross section approaches its maximum usually in the 0.1-1 b range. The yield ratio is greatly enhanced in case the beam energy distribution reaches the threshold of one reaction but not the other. Since laser-driven ion beams generally have a broad and decreasing energy distribution at low energy, the typical case is that a beam may contain ions of sufficient energy for a reaction with a low threshold but not for a reaction with a higher threshold. Thus for example the 11 B(p, 2α) 4 He has a peak cross section around 650 keV whereas the 11 B(p, n) 11 C has a peak around 7 MeV, so that a Maxwellian distribution with β −1 0.5-5 MeV would yield significant α particles but not 11 C. This effect is demonstrated in Fig. 4.

IV. MODELING AND RESULTS
Some modeling and assumptions have already been established in setting up expressions for the yield. We now discuss the details and quantitative inputs to the models.
As seen above in Tables I and II  However to avoid undue speculation and modeling, we assume the cross section vanishes quickly outside the available data range. Considering also the ion mean kinetic energy not likely exceeding ∼ 10 MeV, the numerical results for the yields are likely under estimates by a factor of a few, but not more than ten.

A. Direct irradiation
The yield ratio eliminates dependence on local, dynamic quantities, including effective reaction volume, confinement time and the densities. We need both the absolute kinetic energy scale and the relative kinetic energies of the two ion species. The absolute energy scale is determined by how efficientlly laser energy is transferred the plasma, which in turn generally depends on laser properties, such as total pulse energy, pulse length (if it is greater than ps-scale), and contrast.
For comparison between facilities we scan the absolute kinetic energy scale, using the proton mean kinetic energy as the reference. For intensities up to 10 22 W/cm 2 , we expect ion kinetic energies inside the target to be MeV-scale as the typical momentum obtained from a cycle of the laser field and so also from plasma-generated electrostatic fields.
The relative kinetic energy can be estimated from kinematics. For the same field strength and duration of interaction, the relative work done on ions of charge Z 1 , Z 2 and mass m 1 , m 2 is W 1 /W 2 = (Z 1 ) 2 m 2 /(Z 2 ) 2 m 1 . This suggests the typical energy of protons should be smaller than that of (fully-ionized) boron by factor of 2.5 (i.e. T p 0.4T B ). Good experimental measurements of ion kinetic energy distributions inside the laser-heated target a difficult to come by. Fortunately we find that the yield is mostly sensitive to the absolute kinetic energy scale, controlling how much of the ion distribution is above the threshold CM energy determined by the cross section. Once the threshold CM energy is achieved by a majority of the distribution, the yield becomes less sensitive to further increases in kinetic energy. At next order, the yield Eq. (2) is more sensitive to the kinetic energy of the heavier ion, due to the residual exponential dependence on ν. Note however that these yields are likely to increase somewhat for higher kinetic energy range if cross section data across a wider range of CM energy were available. These results are exhibited in Fig. 5.
The relative insensitivity of 10 B(p, γ) 11 C to ion mean kinetic energy is probably an artifact of the limited data range available for the cross section, which causes the thick target yield to plateau rapidly above ∼ 8 MeV. The greater sensitivity of 11 B(p, γ) 12 C to the mean kinetic energy could allow the yield ratio to be used in a more conventional manner: by measuring both yields in the ratio, one determines to high accuracy the mean kinetic energy of ions in the target. This method is in fact how yield ratios are commonly used in heavy-ion collisions [29]. In particular, if photons from both processes 11 B(p, γ) 12 C and 10 B(p, γ) 11 C can be detected, the ratio of these photon yields alone could probe the mean kinetic energy of ions in the target. We expect the sensitivity of the photon ratio can only be established with more cross section data.  We first compare the total α yield to the 11 C yield in boron and boron-nitride targets. As shown in Fig. 4, for the cross sections and expected temperature of the target, beam energy losses are well-approximated by the cold limit, which we use here. In boron targets, 11 B(p, 2α) 4 He is the dominant source of α particles, while in boron-nitride targets the 14 N(p, α) 11 C process can contribute a similar number. Therefore, for BN targets we show both the total α to 11 C ratio and the 11 B(p, 2α) 4 He yield relative to the (total) 11 C yield, so that the measured 11 C number can be both compared to the measured α yield (e.g. in CR-39) and used to estimate the number of Photons are also produced by the 11 B(p, γ) 12 C and 10 B(p, γ) 11 C processes in the beamtarget geometry. Detection of the photons in the beam-target experiment is a natural proof-ofprinciple/validation step before using the photon measurement to diagnose the direct-irradiation experiments. As shown in Fig. 7, the photon yields are 5 to 6 orders of magnitude smaller than the alpha yield, and display different dependence on the beam β parameter in beam-target experiments compared to the quasi-thermal plasma of direct-irradiation. The yield of 10 B(p, γ) 11 C peaks around 3 MeV due to the narrow range of energies for which cross section data is available; this peak may disappear with more complete cross section data.

V. CONCLUSION
We have thus arrived at a list of reactions and their products with diagnostic potential.
1. 11 C is produced by three reactions in Table I with cross sections of several hundred millibarns in 5-20 MeV center-of-mass energy range. The third channel, 10 B(p, γ) 11 C, generally contributes less than 10 −4 to the total yield. Although it requires a higher proton energy, in experiments on an ultra high-intensity laser, 1 11 C is produced for every 10-100 α particles from the 11 B(p, 2α) 4 He reaction, see Fig. 6. Modulo some uncertainty in modeling the proton beam, the 11 C yield thus offers a strong, easily-measured signal to corroborate or substitute more direct measurements of the α yield [20].
2. 11 B(p, γ) 12 C has the same initial state as the 11 B(p, 2α) 4 He, so that in both direct-irradiation and beam-target experiments, macroscopic factors such as density, volume and time, as well as physical constant prefactors cancel in the yield ratio. The drawback to this process is its low cross section and higher threshold: only 1 photon per million 11 B(p, 2α) 4 He reactions is expected, according to Fig. 5. That suggests ∼ 10 3 − 10 4 such photons were produced in recent experiments such as Refs. [5,7,20]. Pending a direct measurement though the photon should be easily distinguishable at an energy 10 MeV.
While these prefactors are expected to be order 1, due to the similar dynamics of boron-10 versus boron-11 in a laser-heated target, they introduce additional uncertainty, which also grows with the duration of the fusion. The yield ratio 4. 10 B(α, n) 13 N has not been evaluated here, but is an excellent candidate for corroborating the α yield if the neutron number can be measured. This reaction could also determine the importance of p recycling by virtue of its probably near-unity yield ratio to the isospin partner reaction 10 B(α, p) 13 C. A rough estimate of the thick target yield for a 4 MeV α suggests that ∼ 10 −5 of the αs produced may be converted to neutrons by this process. The neutron is slow enough to be easily identified by time-of-flight spectrometers.
One other reaction could be a good proxy for α yield if its cross section were independently measured in conventional nuclear scattering experiments: 14 N(α, γ) 18 F. This reaction has the benefit of producing an unstable nuclide so that its yield can be first checked in beam-target experiments. We have also noted that the ratio between the two photon-producing processes, 11 B(p, γ) 12 C and 10 B(p, γ) 11 C, could provide a measurement sensitive to the mean kinetic energy of the ions in the plasma. However its accuracy is currently severely limited by the little cross section data available for 10 B(p, γ) 11 C.
Considering the experimental interest and potential applications of the 11 B(p, 2α) 4 He reaction and other laser-driven fusion reactions, we strongly recommend increasing engagement with the accelerator-nuclear physics community to improve cross section measurements and add photon and neutron diagnostics. Particle yields and yield ratios can become a powerful tool to determine laser-driven fusion plasma conditions, in the same way they have been thoroughly developed for probing nuclear-matter plasmas.