THE PRODUCTION OF LOW-ENERGY NEUTRONS IN SOLAR FLARES AND THE IMPORTANCE OF THEIR DETECTION IN THE INNER HELIOSPHERE

Hua et al. (2002) approximated the center-of-mass neutron kinetic-energy distribution for the evaporation component with the expression

$$\begin{equation} d\sigma /dE_n \propto E_n^\gamma e^{-E_n/{T_{{\rm evap}}}}, \end{equation} \tag{ 1 }$$

where E_n is the neutron energy in the center of mass. This expression reproduces the general behavior of this component: a rise to a maximum followed by an exponential decay at higher energies. The parameter γ was fixed at a value of 5/11. Hua et al. (2002) assumed the parameter T_evap to be constant for all projectiles, targets, and projectile energies with a value of 2.5 MeV. However, nuclear-reaction theory associates T_evap with the density of states of the nucleus (Blatt and Weisskopf) so T_evap would be expected to depend on the projectile, target, and projectile energy.

Using TALYS, we calculated the energy-dependent evaporation neutron-production cross section for various projectiles, targets, and projectile energies. For α-particle reactions at energies >10 MeV nucleon⁻¹ and for proton reactions at energies >30 MeV, TALYS confirmed that the cross section does indeed fall exponentially with neutron energy as described by Equation (1). But we found that by varying T_evap as a function of projectile, target, and projectile energy, we could achieve better agreement of the cross sections calculated using Equation (1) with those calculated with TALYS. (We found that the parameter γ could be left unchanged with the value of 5/11.) In Figure 1(b) we show the energy-dependent evaporation neutron-production cross section for 60 MeV protons interacting with ²⁴Mg calculated with TALYS (solid curve), calculated using Equation (1) with T_evap = 2.5 MeV (dotted curve), and calculated using Equation (1) with the best-fitting value for T_evap for this reaction of 2.1 MeV (dashed curve).

In Table 1 we present values of T_evap that best fit the neutron-production cross sections calculated with TALYS for proton interactions with ¹⁶O, ²⁴Mg, and ⁵⁶Fe at several projectile energies. Similarly, in Table 2 we present values of T_evap for α-particle interactions. We see that for proton interactions with ¹⁶O and ²⁴Mg, the values are close to that assumed by Hua et al. (2002; 2.5 MeV). For ⁵⁶Fe the value is significantly less. For α-particle interactions, most values are quite different from that assumed by Hua et al. (2002). In all cases, T_evap increases with increasing projectile energy and decreases with increasing target mass. Because of the relatively weak dependence of T_evap on target mass, in the neutron-production code we will use the values calculated for ¹⁶O for all of the light-mass elements (C, N, and O) and the values calculated for ²⁴Mg for all of the intermediate-mass elements (Ne, Mg, and Si).

Hua et al. (2002) calculated energy- and angle-dependent neutron-production cross sections for the pre-equilibrium component using angle-dependent exponentials independent of projectile and target species as given by their Equations (2) through (9). Hua et al. (2002) compared such calculated cross sections with laboratory measurements and found very good agreement. We have now compared these pre-equilibrium cross sections with both laboratory measurements and those calculated with TALYS and confirmed this agreement. For example, the measured (Kalend et al. 1983) angle-integrated neutron cross section for 90 MeV protons interacting with ⁵⁸Ni is shown in Figure 2 (diamonds). Also shown is the pre-equilibrium component calculated using the Hua et al. (2002) procedure (along with the evaporation component calculated with TALYS). The combined total cross section fits the measured cross section very well. At high interaction energies, therefore, we will continue using the Hua et al. (2002) procedure for calculating the pre-equilibrium neutron-production cross section in the neutron-production code.

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Hua et al. (2002) combined the evaporation and pre-equilibrium components to produce the total neutron-production cross section by introducing an additional parameter, F_evap, the fraction of the total neutron production contributed by the evaporation component. (Note that the total neutron production is determined by the total production cross sections evaluated and presented by Hua et al. 2002 in their Figure 1.) Hua et al. (2002) derived F_evap for the various reactions from numerical results of Monte Carlo simulations (Alsmiller et al. 1967) of the cascade model, which is known to be accurate at high energies. For proton interactions, one set of values for the target nuclei was used at energies <40 MeV and another set for energies >40 MeV. For α-particle reactions, only one value was used for all interacting energies.

With TALYS, F_evap can be calculated as a function of projectile, target, and projectile energy. In Tables 1 and 2 we also show F_evap calculated with TALYS for proton and α-particle reactions, respectively. For proton reactions with lighter targets, the TALYS values are significantly higher than those used by Hua et al. (2002). For α-particle interactions with ¹⁶O and ²⁴Mg, the values are quite similar to the value used by Hua et al. (2002), and the values for α-particle interactions with ⁵⁶Fe are somewhat higher. Again, because of the relatively weak dependence of F_evap on target mass, in our neutron-production code we will use the values calculated for ¹⁶O for all of the light-mass elements (C, N, and O) and the values calculated for ²⁴Mg for all of the intermediate-mass elements (Ne, Mg, and Si).

In summary, we have shown that, at high ion-interaction energies, the original Hua et al. (2002) procedure used in the flare-loop code is generally adequate for calculating the energy- and angle-dependent neutron-production cross sections of both the evaporation and pre-equilibrium components. Using TALYS, we have improved its accuracy by introducing target and energy dependences for the procedure's parameters T_evap and F_evap. Because of its analytical simplicity and computational efficiency, we shall continue to use in the code the Hua et al. (2002) procedure (with the above improvements) for calculating neutron-production cross sections at high ion energies. The energy below which this procedure is no longer adequate and the new procedure used at these lower energies are discussed in the next section.

The flare-loop model (Hua et al. 1989, 2002) is represented by a set of parameters that describe the accelerated ions and the physical conditions of a magnetic loop in the solar atmosphere within which the ions are transported. These parameters are the energy spectrum and abundances of the accelerated ions, the magnetic convergence of the loop and the level of PAS within the loop, the composition and density–height distribution of the atmosphere, the loop length L, and the location of the flare on the disk relative to the directions of the neutron and gamma-ray detectors. The legs of the loop are perpendicular to the solar surface.

For the calculations presented in Section 3, we assume a coronal composition (Reames 1995) for both the accelerated ions and the ambient atmosphere (with a value of 0.1 for the ambient ⁴He/H ratio and a value of 0.2 for the accelerated α/proton ratio). The ambient ³He/H ratio is 3.7 × 10⁻⁵ (Mahaffy et al. 1998) and the accelerated ³He/⁴He ratio is 0.05. Studies of gamma-ray data from flares observed with the Solar Maximum Mission (SMM) and RHESSI (Mandzhavidze et al. 1997; Share & Murphy 1998; and G. H. Share 2011, private communication) generally indicate that the ambient medium and the accelerated particles have coronal abundances for elements heavier than ⁴He and the accelerated-ion α/proton ratio is at least as large as 0.2. Changing the ambient composition has little effect on neutron production. The atmosphere density–height distribution is the sunspot active region model given by Avrett (1981).

The composition of the accelerated ions in some flares may be similar to that of "impulsive" SEP events observed in interplanetary space where the abundances of elements heavier than ⁴He are enhanced relative to protons as compared with those of the corona (e.g., see Reames 1995). The accelerated ³He/⁴He ratio can also be enhanced, up to an extreme value of 1 or more. To investigate the effect on neutron and neutron-capture line production, we repeat the calculations assuming the Reames (1995) impulsive-event SEP composition ("³He-rich") for the flare-accelerated ions heavier than ⁴He relative to each other, but with an accelerated α/proton ratio of 0.2. This α/proton ratio is about a factor of six higher than that given by Reames (1995) for "average" impulsive events, and we have also increased the abundance of elements heavier than ⁴He relative to protons by this factor of six to maintain an α/O ratio of 50. The resulting accelerated Fe/proton ratio is larger than that of the corona by a factor of ∼60. We also assume an extreme accelerated ³He/⁴He ratio of 1. Where appropriate in the following, we will note the impact of assuming this impulsive-event composition.

Note that while we assumed an extreme value of 1 for the accelerated ³He/⁴He ratio for the impulsive-event composition, the flare-loop code currently does not consider neutron production by accelerated ³He with ambient heavy elements, only with ambient H and ⁴He. The neutron-production cross sections for ³He reactions with heavy elements can also have threshold energies below 1 MeV nucleon⁻¹ similar to the α-particle reactions. However, contribution to neutron production by these reactions would be significant only if the accelerated ³He/⁴He ratio were as large as the largest values observed in impulsive SEP events in space. The inclusion of these ³He reactions in the code involves significant effort evaluating numerous cross sections and is beyond the scope of this paper; they will be addressed in a subsequent paper. Where appropriate in the following, when the impact of the impulsive-event composition is considered, we will provide an estimate of the effect of these ³He reactions. The inverse reactions of accelerated heavy elements with ambient ³He are not significant because the ambient ³He/H abundance ratio is very low (∼10⁻⁵).

The ions are released at the top of the loop with an isotropic angular distribution. This isotropic release produces a downward isotropic distribution of ions entering the top of each leg of the loop. The redshifts of the de-excitation line centroids measured with SMM (Share et al. 2002) are consistent with a downward-isotropic interacting-ion distribution. (In the discussion of Section 4, we will also consider ions released as a downward beam directed along the axis of the loop.) We emphasize the distinction between the angular distribution of the ions when released at the top of the loop and the angular distribution of the ions when they interact after transport down the loop. As will be seen in the following sections, the latter angular distribution is what is important for the production of neutrons and their subsequent behavior.

For the accelerated-ion kinetic-energy distributions, all accelerated-ion species have the same spectrum (weighted by their relative abundances), and we will consider both mono-energetic ions and power-law distributions. The power law is differential in ion energy (per nucleon) with spectral index s (∝E^−s_ion) down to 1 MeV nucleon⁻¹ and then constant at lower ion energies (i.e., a "broken" power law). In the discussion (Section 4), we will also consider an unbroken power law below 1 MeV nucleon⁻¹.

Neutron measurements of the 1991 June 4 flare observed with OSSE (Murphy et al. 2007) at 1 AU showed that for that flare the ion spectrum above ∼100 MeV nucleon⁻¹ must steepen from the power law at lower energies. We will investigate the effect of such an ion spectrum on neutron production by performing the calculations using a power law with exponential roll-over; i.e., ∝E^−s_ion × exp (− E_ion/E₀) with roll-over energy E₀. E₀ → ∞ returns the spectrum to the original power law. Where appropriate in the following, we will note the impact of assuming this ion spectrum.

Below the transition region, the magnetic field strength is assumed to be proportional to a power δ of the pressure (Zweibel & Haber 1983). δ ≠ 0 corresponds to a converging magnetic field. For coronal and photospheric pressures of 0.2 and 10⁵ dyne cm⁻² (corresponding to atmospheric heights of approximately 2000 and 0 km, respectively) and associated magnetic-field strengths of 100 and 1600 G, respectively, δ ≃ 0.2. PAS is characterized by Λ, the mean free path required for an arbitrary initial angular distribution to relax to an isotropic distribution. In the model, the level of PAS is characterized by λ, the ratio of Λ to the loop half-length L_c (=L/2). No PAS corresponds to λ → ∞.

In the Monte Carlo simulation (see also Hua et al. 2002 and Hua & Lingenfelter 1987b for a complete discussion), the kinetic energy and direction relative to the energetic ion of each neutron is determined from the angle- and energy-dependent cross sections used by the code for each specific neutron-producing reaction. For those cross sections providing neutron information in the center-of-momentum frame, standard relativistic transformations to the laboratory frame are used (see, for example, Ramaty et al. 1979). Each neutron is followed, usually through many scatterings, until it either escapes from the solar atmosphere, decays, or slows down and is captured either radiatively on H to form deuterium with the emission of a 2.223 MeV gamma ray, or nonradiatively on ³He to form ³H with the emission of a proton. Only elastic scattering of the neutrons on hydrogen is considered since the contributions to neutron thermalization by scattering on He and heavier nuclei are negligible. The flux of neutrons escaping from the solar atmosphere is thus determined as a function of neutron energy and zenith angle.

Because effective neutron capture requires high density, it typically occurs deep in the photosphere after the neutrons have lost their initial energies and thermalized. Each 2.223 MeV gamma ray produced is followed until it either escapes from the solar atmosphere or is multiply Compton scattered to low energy (<20 keV). The flux of both the unscattered 2.223 MeV gamma-ray line and the Compton-scattered continuum escaping from the solar atmosphere is thus determined as a function of gamma-ray energy and zenith angle. Because most of the captures take place after the neutrons have thermalized at energies of ∼0.5 eV, the line energy is essentially unshifted and the line width is very narrow, with a Doppler broadened full width at half-maximum of only ∼0.1 keV.

For no magnetic convergence (δ = 0), the initial angular distribution of the accelerated ions is maintained as the ions propagate down the loop. For an isotropic angular distribution at the top of the loop, this results in a downward-isotropic angular distribution of ions when they reach the denser layers of the atmosphere and interact to produce neutrons. This is seen in Figure 4 where the interacting-ion angular distribution for δ = 0 as calculated by the code (black histogram) is shown versus μ = cos (θ), where θ is the angle between the normal to the solar surface and the direction of the ion. μ = 1 (θ = 0) is directed outward from the Sun. (Note that: the fluctuations are due to counting statistics associated with the Monte Carlo technique used by the code.)

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For no convergence, the height distribution of the neutron-producing interactions in the solar atmosphere simply reflects the ranges of the ions: the accumulated column density required for the accelerated ions to lose energy and interact as they move downward through the solar atmosphere. This range depends on the ion energy as is illustrated in Figure 5(a) which shows the fraction of interactions occurring deeper than a given height in the solar atmosphere for three accelerated-ion power-law spectral indexes: s = 2, 4, and 6. Harder spectra (smaller s) have more high-energy ions whose longer ranges shift the distributions to lower heights (higher densities).

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A converging magnetic field (δ ≠ 0) results in mirroring of the accelerated ions with the mirroring height depending on the initial pitch angle of the ion. Most ions interact near their mirroring heights (where the density is greatest), and the angular distribution of the interacting ions is therefore peaked in directions tangential to the solar surface (a "pancake" distribution). However, ions with initial pitch angles less than a certain value have mirror points so deep in the atmosphere that they are more likely to be lost on their first transit down the loop, either to nuclear reactions (including neutron production) or having their energies fall below reaction threshold energies. This angle defines the "loss cone." This can be seen in Figure 4, where the angular distribution for δ = 0.2 is shown (red histogram). The distribution peaks near μ = 0 (θ = 90°) and the excess near μ = −1 is due to interactions of ions with initial pitch angles within the loss cone.

Mirroring in a converging magnetic field also affects the height distribution of the neutron-producing interactions as ions are prevented from penetrating to the lower atmosphere. Neutron production occurs higher in the solar atmosphere, as can be seen by comparing panels (a) and (b) of Figure 5, where the height distributions for no convergence (δ = 0) and strong convergence (δ = 0.45) are shown.

PAS modifies the angular distribution of the ions as they propagate through the loop, replenishing the loss cone as ions are lost. This results in a more downward-directed angular distribution of ions when they interact to produce neutrons than without PAS. Hua et al. (2002) found that there is a limitation to this effect: When the rate of PAS exceeds the ion loss rate, the loss cone is "saturated." Saturation occurs for λ ≲ 20. The shift to more downward-directed interactions can be seen in Figure 4, where the angular distribution for saturated PAS (and δ = 0.2) is shown (green histogram).

PAS also affects the height distribution of the neutron-producing interactions, with production occurring deeper in the solar atmosphere as more ions are scattered downward into the loss cone rather than mirrored. This can be seen by comparing the height distributions in panels (b) and (c) of Figure 5 for no PAS (λ → ∞) and saturated PAS (λ = 20), respectively (and δ = 0.45).

We see that both magnetic convergence and PAS affect both the angular distribution of the accelerated ions when they interact to produce the neutrons and the heights in the atmosphere where those interactions occur. In addition, the ion kinetic-energy spectrum also affects the interaction height distribution. These interacting-ion angular and height distributions directly affect the spectra and yields of the escaping neutrons and the yields of the neutron-capture line. These spectra and yields are also affected by the composition of both the accelerated ions and the ambient medium (see the discussion of Section 4.1). While the loop length can affect the time dependence of both neutron production and the neutron-capture line, it has no effect on their time-integrated spectra and yields. There is minimal dependence of the interacting-ion angular distributions on the ion spectral index. Now, with the knowledge of the effects that s, δ, and λ have on the interacting ions, in the next two sections we discuss their effects on the resulting escaping-neutron spectra and yields and the yield of the neutron-capture line.

Except for very hard accelerated-ion spectra, the neutron-production height distributions in Figure 5(a)–(c) show that most neutrons are produced above 0 km, which corresponds to an overlying grammage of <10 g cm⁻². Hua & Lingenfelter (1987a) showed that at these heights those neutrons initially directed upward have a good chance of escaping from the solar atmosphere without scattering. Of the neutrons initially headed downward, a fraction may be scattered upward and escape with reduced energy, but most will lose their energy in the solar atmosphere with some being captured on H producing the neutron-capture line. Neutrons produced moving tangentially to the solar surface can be scattered as they propagate through the atmosphere, changing their direction and lowering their energy. Scattering is more significant for lower-energy neutrons because their scattering cross sections are larger.

Thus, neutron escape depends on the initial angular distribution of the neutrons, which, in turn, depends on the angular distribution of the interacting ions. Higher-energy neutrons will have an angular distribution similar to that of the interacting ions since such neutrons tend to be emitted in nearly the same directions as the initial ions. This is because in the center-of-momentum frame, higher-energy neutrons from pre-equilibrium reactions are emitted more in the forward direction, and this is enhanced by the transformation from the center-of-momentum frame to the laboratory frame, especially for the inverse reactions of accelerated heavy ions where the center of momentum has a large velocity. Lower-energy neutrons are emitted more isotropically, independent of the angular distribution of the interacting ions. This imposes an angular dependence on the escaping-neutron spectrum: the measured properties of the escaping neutrons will depend on the flare heliocentric observation angle θⁿ_obs, the angle between the normal to the solar surface at the flare and the direction to the neutron detector.

With no magnetic convergence (δ = 0), the resulting downward isotropic distribution of the interacting ions (see Figure 4) produces limb brightening of high-energy neutrons. In limb flares, more of the tangentially moving neutrons can escape toward the detector than in disk flares where they must first be scattered upward to escape toward the detector. For low-energy neutrons, which are more sensitive to scattering, the behavior is more complicated. Although they are more isotropic, what anisotropy there is results in limb brightening, as is the case for high-energy neutrons, but for such limb flares this competes with limb darkening due to the increased scattering. For steep ion spectra, neutron production is high enough in the atmosphere that scattering is less important, resulting in limb brightening of low-energy neutrons. For hard ion spectra, the increased scattering from deeper production results in limb darkening. For intermediate ion spectra, there can be a peak of the escaping low-energy neutron fluence at mid-heliocentric flare angles.

With magnetic convergence (δ ≠ 0) but no PAS, all escaping neutrons exhibit limb brightening because they are produced sufficiently high in the solar atmosphere (see panel (b) of Figure 5). The brightening is stronger for higher-energy neutrons since they are more tangentially directed similar to the interacting-ion directions (see Figure 4). When PAS is present, limb brightening is weaker because the neutrons are produced deeper in the atmosphere (Figure 5(c)) and preferentially in downward directions rather than tangentially (see Figure 4).

As we will show in Sections 3.2 and 3.3 (see Figures 7, 8, and 15–20), when compared with their production spectra, escaping-neutron spectra from limb flares are deficient in lower-energy neutrons because their scattering cross sections are larger. On the other hand, escaping-neutron spectra from disk flares are more abundant in lower-energy neutrons. These additional low-energy neutrons initially were higher-energy neutrons moving downward but escaped after several scatterings, changing their directions and reducing their energies.

Some of the neutrons produced moving downward can be captured on H to produce deuterium and the 2.223 MeV neutron-capture line. In addition, some of the tangentially moving neutrons can be scattered downward, and some of those can also be captured. Neutrons initially moving upward escape with essentially no chance of line production. So, similar to escaping neutrons, the probability of capture also depends on the angular distribution of the neutrons, which in turn depends on the interacting-ion angular distribution. For isotropic release of ions at the top of the loop and no PAS, changing δ from 0 (no magnetic convergence, downward-isotropic interacting-ion angular distribution) to δ = 0.45 (strong convergence, pancake distribution) can reduce the capture probability by up to a factor of two. PAS results in more downward-directed interactions and can increase the capture probability by up to a factor of two.

Because the capture line is produced deep in the atmosphere, attenuation of the line due to Compton scattering imposes a strong dependence of the observed line fluence on the flare heliocentric observation angle θ^2.223_obs (the angle between the flare normal and the direction to the gamma-ray detector). Although the line can be intrinsically the strongest line produced in a flare, its measured escaping fluence can be very weak if the flare is located near the solar limb.

The atmospheric height distributions of neutron capture for isotropic release of ions at the top of the loop with three spectral indexes (2, 4, and 6) and δ = 0 and 0.45 (all with no PAS) are shown in Figure 6. For all but the hardest accelerated-ion spectra, neutron production occurs high enough in the atmosphere (see Figure 5) that neutron capture will occur at essentially the same depth in the photosphere regardless of the interacting-ion angular distribution (i.e., independent of δ and λ). The resulting line attenuation therefore does not depend on δ and λ, although the line production (i.e., neutron capture on H) does, as discussed above.

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Neutrons from harder ion spectra are captured deeper because neutron production itself is deeper and because the neutrons have higher energies and therefore longer ranges. This results in more attenuation of the line and also a weak dependence on δ and λ. While strong PAS (or weak convergence) results in increased captures due to the more downward-directed angular distribution, this is offset by the increased attenuation due to the deeper captures of the hardest ion spectra.

We show in Figures 7 and 8 calculated neutron kinetic-energy spectra resulting from interactions of mono-energetic ions of six low projectile energies: 0.5, 0.75, 2, 5, 10, and 30 MeV nucleon⁻¹. These are escaping-neutron spectra at the Sun; i.e., just after transport through the solar atmosphere but before any modification due to neutron decay. Figure 7 is for a disk flare (θⁿ_obs = 0°) and Figure 8 is for a limb flare (θⁿ_obs = 85°). The neutron spectra are normalized to one accelerated proton of the given energy and are for isotropic release at the top of a loop with no magnetic convergence or PAS (δ = 0, λ → ∞), which results in a downward-isotropic interacting-ion angular distribution. The solid black curves are the total spectra, and spectra from the eight contributing reaction types (p + H, p + ⁴He, α + H, α + ⁴He, p + CNO, CNO + H, α + CNO, and CNO + ⁴He where CNO refers to all nuclear species heavier than ⁴He) are also shown. For the accelerated ³He abundance assumed, contributions from ³He reactions are not significant. The solid colored curves are for accelerated proton and α-particle reactions with ambient material and the dashed colored curves are the inverse reactions of accelerated heavy ions interacting with ambient H and ⁴He. We discuss these spectra below, demonstrating how the contributions from the various neutron-producing reactions change as the interaction energy increases and the flare heliocentric angle changes.

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Panels (a) and (b) of Figures 7 and 8 show disk- and limb-flare escaping-neutron energy spectra calculated for ion energies of 0.5 and 0.75 MeV nucleon⁻¹. Neutron production at these ion energies is primarily from interactions of accelerated α particles with the rare isotope ¹³C and the inverse reaction because all other reactions have effective threshold energies greater than ∼1 MeV nucleon⁻¹. The ¹³C reaction is actually exothermic (see Table 3, which shows threshold energies for neutron production by proton and α-particle reactions with various isotopes), but the accelerated ions must still overcome the Coulomb barrier. As a result, the effective ion-energy threshold for neutron production by this reaction is ∼500 keV nucleon⁻¹, lower than those for heavier isotopes with larger nuclear charge. Although the neutron-production cross section for this reaction is large due to the neutron excess of the nucleus, the isotopic fraction of ¹³C is only 1% and the neutron yield at these low ion energies is small compared with yields from higher-energy ions (see below).

Panels (a) and (b) of Figures 7 and 8 show that 0.5 and 0.75 MeV nucleon⁻¹ accelerated α particles can produce neutrons with energies of almost 5 and 6 MeV, respectively. This is due to the large energy available for these reactions: the total energy of the accelerated α particle (2 and 3 MeV) combined with the additional energy available from the exothermic reaction (the ¹³C reaction has a nuclear-reaction Q value of +2.2 MeV). Neutron production from the accelerated α-particle reactions at these low-energies is fairly isotropic and occurs relatively high in the solar atmosphere so there is little difference of the escaping spectra from the two flare locations. The slight excesses at low energies seen in the spectra from the disk flare compared to the limb flare are due to scattering of downward-moving higher-energy neutrons into the upward direction with loss of their original energy.

The larger total energy of the accelerated heavy ion (∼6 and 9 MeV for ¹³C) associated with the inverse reaction with ambient ⁴He can result in neutrons with energies up to ∼10 MeV. However, for the downward-isotropic interacting-ion angular distribution resulting from no magnetic convergence, neutrons with energies greater than ∼7 MeV are not seen from disk flares (Figure 7). These higher-energy neutrons tend to be produced in the same direction as the interacting ions (i.e., into the downward hemisphere), and to escape the Sun must first be scattered upward with reduction of their energies from their initial values. These higher-energy neutrons can be seen from limb flares (Figure 8) produced by those ions moving tangentially to the solar surface.

Note that when expressed as energy per nucleon, proton and α-particle reactions with heavy ambient nuclei have the same production cross sections as their corresponding inverse reactions. However, in the thick-target loop model assumed here, nuclear reactions compete with Coulomb energy losses of the accelerated ions which are proportional to Z²/A (where Z is the ion charge and A is the number of ion nucleons). Therefore, even if the accelerated and ambient populations have the same heavy element to H and ⁴He abundance ratios, the nuclear-reaction yields of the inverse reactions are reduced relative to those of the proton and α-particle reactions, as can be seen in Figures 7 and 8. Z²/A for ¹³C is the smallest of all elements heavier than ⁴He that are relevant for neutron production, so at these low ion energies where the α + ¹³C reaction and its inverse dominate (panels (a) and (b)), the relative contribution of the inverse reaction is the largest. As will be seen below, at higher ion energies neutron production is dominated by reactions involving heavier isotopes with larger Z²/A, and the relative contribution of the inverse reactions is less.

The neutron spectra plotted in panels (a) and (b) of Figures 7 and 8 show structure at a few MeV due to the accelerated α-particle reactions. The structure results from the features present in the energy-dependent neutron-production cross sections at these energies as discussed in Section 2.3. This structure is smoothed out in the spectrum from the inverse reactions (dashed curve) due to the wider spread of neutron velocities resulting from the transformation from the center-of-momentum frame to the laboratory frame. Detection of this structure along with a lack of neutrons with energies greater than ∼10 MeV (or ∼7 MeV for a disk flare) would be a good indicator of ion acceleration only to energies less than ∼2 MeV nucleon⁻¹. The dotted curve in the figure is the total neutron spectrum calculated using the original Hua et al. (2002) procedure. This spectrum exhibits no structure and extends to higher neutron energies due to an inaccurate procedure used by Hua et al. (2002) at low interaction energies.

Panels (c) of Figures 7 and 8 show disk- and limb-flare escaping-neutron energy spectra calculated for an ion energy of 2 MeV nucleon⁻¹. Neutron production at this ion energy is primarily from interactions of accelerated α particles with the rare isotopes ²⁵Mg, ²⁶Mg, and ²²Ne and their inverse reactions (see Table 3). The isotopic fractions for these isotopes are large (∼21% for the sum of the ²⁵Mg and ²⁶Mg isotopes) as are their neutron-production cross sections, again due to the neutron excesses of their nuclei. Their abundance-weighted cross sections can be as large as 100 mbarn at 2 MeV nucleon⁻¹ (see Figure 1 of Hua et al. 2002), much larger than that of the α + ¹³C reaction. Z²/A for Mg is larger than for C so the relative contribution of the inverse reactions to total neutron production is less at this ion energy than at lower energies where neutron production is dominated by the ¹³C reaction. However, due to the larger total energies of the accelerated heavy ions, the inverse reactions dominate the production of neutrons with energies above ∼10 MeV. These high-energy neutrons can be seen from limb flares (Figure 8), but not from disk flares (Figure 7).

Because no proton reactions can contribute at this ion energy (see Table 3) and the inverse-reaction contribution to the total neutron production is less than 20%, the neutron yield depends almost directly on the α/proton ratio of the accelerated 2 MeV nucleon⁻¹ ions. This ratio is seen to be energy dependent in SEPs (e.g., Mewaldt et al. 2005) and can be as large as 0.25 at 0.5 MeV nucleon⁻¹. There is some structure in the spectrum at low neutron energies due to the accelerated α-particle reactions. Although at this ion energy the calculated spectrum is the sum of several reactions, the structures of their individual spectra are sufficiently similar that structure remains in the total neutron spectrum. The dotted curve in the figure is the total neutron spectrum calculated using the original Hua et al. (2002) procedure. Again, this spectrum exhibits no structure and extends to higher neutron energies due to an inaccurate procedure used by Hua et al. (2002) at low interaction energies.

Panels (d) of Figures 7 and 8 show disk- and limb-flare escaping-neutron spectra calculated for an ion energy of 5 MeV nucleon⁻¹. Here, the main contributing reactions are again those of α particles with heavy elements and their inverse reactions, but at this higher ion energy all neutron-producing isotopic species are contributing, not just the rare isotopes as at 2 MeV nucleon⁻¹ (see Table 3). Proton reactions and their inverse reactions are beginning to contribute but only with the rare, neutron-rich isotopes. The total spectrum again exhibits some structure due to the α-particle reactions. The difference between the neutron spectrum calculated using the new procedure and that calculated using the original Hua et al. (2002) procedure (dotted curve) is similar to that seen at 2 MeV nucleon⁻¹.

Panels (e) of Figures 7 and 8 show disk- and limb-flare escaping-neutron spectra calculated for an ion energy of 10 MeV nucleon⁻¹. While more of the reactions of accelerated protons with heavy isotopes and their inverse reactions are contributing, most of the neutron production is still from α-particle reactions with heavy elements, although no significant structure remains. The α + ⁴He reaction is beginning to contribute neutrons at a few MeV for the limb flare (Figure 8) but is significant for the disk flare only below ∼0.5 MeV (Figure 7). This is because in the center-of-momentum frame at projectile energies less than ∼15 MeV nucleon⁻¹ the α + ⁴He reaction is peaked in the forward direction (see Hua et al. 2002), resulting in an approximately downward-isotropic neutron angular distribution requiring scattering to produce upward-directed neutrons for a disk flare. Limb flares can produce escaping neutrons with energies in excess of 40 MeV at this ion energy. The neutron spectrum calculated with the new procedure and that calculated with the original Hua et al. (2002) procedure (dotted curve) are very similar.

Panels (f) of Figures 7 and 8 show disk- and limb-flare escaping-neutron spectra calculated for an ion energy of 30 MeV nucleon⁻¹. At this ion energy, all neutron-producing reactions are contributing except the p + H reaction (see Table 3). The α + ⁴He reaction is now the dominant contributor, and, because we did not change the procedure used for this reaction, there is essentially no difference between the spectrum calculated with the new procedure and that calculated with the original Hua et al. (2002) procedure (dotted curve). Limb flares can produce neutrons with energies in excess of 100 MeV at this ion energy.

The impulsive-event composition increases the level of the neutron spectrum from the inverse reactions shown in Figures 7 and 8 by about an order of magnitude, but the impact on the total neutron spectrum depends on the fraction such reactions contribute at each neutron energy. For the lowest ion energies (<2 MeV nucleon⁻¹) at which these reactions make a significant contribution, this increases the total neutron spectrum also by about an order of magnitude for limb flares and somewhat less for disk flares. For higher ion energies up to 10 MeV nucleon⁻¹, the increase to the total is less except at high neutron energies, where the inverse reactions can contribute such neutrons, especially for limb flares where the 10 MeV neutron yield is increased by about an order of magnitude for limb flares, less for disk flares. Above 20 MeV nucleon⁻¹ the inverse reactions are less important and do not add significantly to the total neutron spectrum. At these energies, the ³He + H reaction becomes important, contributing additional ∼10 MeV neutrons and increasing the total neutron yield at those energies by about a factor of three for limb flares and somewhat less for disk flares.

Using calculated escaping-neutron spectra such as those of Figures 7 and 8, we can compare which reactions contribute to the >30 MeV neutrons observed at 1 AU with those contributing to the 1–10 MeV neutrons observable in the inner heliosphere. Figure 9 shows the fraction that each of the eight reaction types contributes to the total production of >30 MeV neutrons as a function of the mono-energetic ion energy. Panel (a) is for a disk flare (θⁿ_obs = 0°) and panel (b) is for a limb flare (θⁿ_obs = 85°). At the lowest ion energies, the only reactions capable of producing 30 MeV neutrons are the inverse reactions of heavy elements with ambient ⁴He. Only these reactions have sufficient total energy to do so, but minimum ion energies of ∼5 MeV nucleon⁻¹ for limb flares and ∼15 MeV nucleon⁻¹ for disk flares are required. The higher energy required for disk flares is because the scattering needed to produce the upward-escaping neutrons significantly reduces their energies from their initial values so higher initial energies are required. At higher ion energies, the reactions of accelerated α particles with heavy elements become important for a narrow band (20–40 MeV nucleon⁻¹ for disk flares and 10–15 MeV nucleon⁻¹ for limb flares). Above 40 MeV nucleon⁻¹ for disk flares and 15 MeV nucleon⁻¹ for limb flares, the α + ⁴He reaction is dominant up to ∼400 MeV nucleon⁻¹ for disk flares and ∼40 MeV nucleon⁻¹ for limb flares. Above these energies, the p + ⁴He reaction dominates for disk flares, but the inverse α + H reaction is dominant for limb flares. Neutrons from the latter reaction are too downward directed to be significantly scattered into the upward direction from disk flares. Only at the highest ion energies (>2 GeV nucleon⁻¹ for limb flares and >700 MeV nucleon⁻¹ for limb flares) is the p + H reaction significant. We find essentially no dependence of the contributing-reaction fractions on either loop convergence or PAS.

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When the impulsive-event composition is assumed, for a disk flare the ³He + ⁴He reaction now accounts for about 50% of the >30 MeV neutron production for ion energies around 500 MeV nucleon⁻¹. For a limb flare, the inverse reactions with ambient H now account for about 20% of the production for ion energies around 20 MeV nucleon⁻¹ and the ³He + H reaction accounts for about 40% of the production for ion energies around 100 MeV nucleon⁻¹.

When only 1–10 MeV neutrons are considered, the reaction fractions are different, as shown in panels (a) and (b) of Figure 10 for disk and limb flares, respectively, for the coronal composition. Because these low-energy neutrons are generally produced more isotropically and higher in the solar atmosphere, there is less dependence on flare location than for >30 MeV neutrons. At the lowest ion energies, 1–10 MeV neutrons are produced predominantly by accelerated α-particle reactions with heavy elements for both flare locations, not their inverse reactions as for >30 MeV neutrons, because the extra total energy of accelerated heavy ions tends to produce neutrons with energies greater than 10 MeV. Above ∼10 MeV nucleon⁻¹ the α + ⁴He reaction dominates, and above ∼40 MeV nucleon⁻¹ the p + ⁴He reaction dominates. Above ∼500 MeV nucleon⁻¹, the p + H reaction contributes, and, for limb flares, proton reactions with heavy elements also contribute. The energies of ions capable of producing 1–10 MeV neutrons extend down to ∼0.5 MeV nucleon⁻¹. We find essentially no dependence of the contributing-reaction fractions on either loop convergence or PAS.

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When the impulsive-event composition is assumed, the contribution to 1–10 MeV neutron production from α-particle reactions with ambient heavy elements is reduced at ion energies less than ∼10 MeV nucleon⁻¹ for a disk flare as the contribution from the inverse reactions of accelerated heavy elements with ambient ⁴He increase. These reactions now contribute more than 50% of the production for ion energies below 2 MeV nucleon⁻¹ for disk flares and 80% of the production for ion energies below 10 MeV nucleon⁻¹ for limb flares. Also for limb flares, the contribution of the α + ⁴He reaction at ion energies around 20 MeV nucleon⁻¹ is reduced as contribution from the ³He + H and ³He + ⁴He reactions increase, now accounting for ∼70% of the production at these energies.

After escaping the solar atmosphere, neutron spectra such as those of Figures 7 and 8 begin to show the effect of attenuation due to neutron decay, increasing with increasing distance from the Sun. As an example, Figure 11 shows time-integrated neutron spectra at three distances D = 10 solar radii (R_s), 0.5 AU, and 1 AU (215 R_s) from mono-energetic 10 MeV nucleon⁻¹ ion interactions for a disk flare. The spectra are normalized to one accelerated proton. The dramatic decrease of the low-energy neutron fluence with increasing distance to the detector is clearly seen, due to both the D² factor and the smaller fraction of low-energy neutrons surviving to reach the detector.

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Figure 12 shows calculated fluences of 1–10 MeV neutrons for the coronal composition as a function of the ion kinetic energy for a neutron detector located at three distances from the Sun D_n = 10 R_s, 0.5 AU, and 1 AU for flares occurring at three heliocentric angles θⁿ_obs = 0°, 60°, and 85°. The fluences are normalized to one accelerated proton of the given ion energy and calculated for isotropic release of ions at the top of a loop with no convergence or PAS (δ = 0, λ → ∞), which results in a downward-isotropic interacting-ion angular distribution. The fluence increases rapidly up to ion energies of ∼50 MeV nucleon⁻¹ because the reaction cross sections rise rapidly from their various threshold energies. The flattening above 50 MeV nucleon⁻¹ is because at those energies most of the cross sections have reached their maxima and flatten with energy (see Figure 1 of Hua et al. 2002). For low-energy ions, 1–10 MeV neutrons are produced sufficiently isotropically and at sufficiently shallow depths to show little dependence on the flare heliocentric angle. Neutrons of energy 1–10 MeV from higher-energy ions are produced deep enough to show significant attenuation for limb flares. To calculate an actual 1–10 MeV neutron fluence at a neutron detector, the value from the curve at the detector distance should be multiplied by the number of mono-energetic accelerated protons of the given energy. Conversely, a measured fluence can be converted into the required number of accelerated protons by dividing it by the value on the curve.

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When the impulsive-event composition is assumed, the 1–10 MeV neutron fluence from limb flares is increased by about a factor of five for ion energies less than 10 MeV nucleon⁻¹. Above an ion energy of 10 MeV nucleon⁻¹, the fluence is increased by about a factor of two, falling to less than a 50% increase by an ion energy of 50 MeV nucleon⁻¹. For disk flares, the increase is about a factor of two below an ion energy of 2 MeV nucleon⁻¹ and less than 50% for higher ion energies. We note that neutron production from the reactions of accelerated ³He with ambient heavy elements, which are not currently included in the code, would be significant only for ion energies less than ∼1 MeV nucleon⁻¹, and then only if the accelerated ³He/⁴He ratio were as large as 1. If it were, the 1–10 MeV neutron and neutron-capture line yields from mono-energetic ions shown in Figures 12 and 14 would increase at most by a factor of ∼3.

In Figure 12 we also show the calculated fluence of >30 MeV neutrons for the coronal composition (dashed curves) at a neutron detector located at D_n = 1 AU for the three flare locations. We clearly see limb brightening for these higher-energy neutrons at all ion energies. The >30 MeV neutron fluence at 1 AU overwhelms the 1–10 MeV fluence at 1 AU for all ion energies greater than a few tens of MeV nucleon⁻¹.

When the impulsive-event composition is assumed, the >30 MeV neutron fluence is increased by less than a factor of two for ion energies greater than ∼40 MeV nucleon⁻¹. Below that energy, the fluence is increased by a factor of ∼15 for a disk flare and a factor of ∼30 for a limb flare as the inverse reactions of heavy elements with ambient ⁴He dominate production.

Figure 13 shows the fraction that each of the eight reaction types contributes to the production of the 2.223 MeV neutron-capture line as a function of the mono-energetic ion energy. Panel (a) is for a disk flare (θⁿ_obs = 0°) and panel (b) is for a limb flare (θⁿ_obs = 85°). Comparing these distributions with those for 1–10 MeV neutron production shown in Figure 10, we see that they are very similar, with only minor differences. This similarity is because the energies of the captured neutrons producing the bulk of the capture-line photons that actually escape the solar atmosphere without scattering are near 10 MeV. Higher-energy neutrons tend to be captured too deep and the capture photons suffer too much attenuation. Lower-energy neutrons tend to decay in the solar atmosphere without being captured. For disk flares, at higher ion energies the contribution of the α + ⁴He reaction is larger than for 1–10 MeV neutron production because the more downward-directed angular distribution of neutrons from these reactions results in more captures and less escaping neutrons. When the impulsive-event composition is assumed, the change to the reaction fractions contributing to neutron-capture line production is the same as that for 1–10 MeV neutrons discussed above.

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Figure 14 shows calculated fluences of the 2.223 MeV neutron-capture line, ϕ_2.223, as a function of ion kinetic energy for a gamma-ray detector located at 1 AU for flares occurring at three heliocentric angles θ^2.223_obs = 0°, 60°, and 85°. The fluences are normalized to one accelerated proton of the given ion energy and calculated for isotropic release of ions at the top of a loop with no convergence or PAS (δ = 0, λ → ∞), which results in a downward-isotropic interacting-ion angular distribution. Similar to the neutron fluence, there is significantly more line attenuation for limb flares from higher-energy ions because the neutrons from these higher-energy ion interactions are captured deeper in the photosphere. To calculate an actual capture-line fluence at the detector, the value from the curve should be multiplied by the number of mono-energetic accelerated protons of the given energy. Line fluences at gamma-ray detectors located at other distances to the Sun can be calculated by adjusting the value from the curves by the appropriate D² factor.

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When the impulsive-event composition is assumed, the neutron-capture line fluence is increased by about a factor of five for ion energies less than 10 MeV nucleon⁻¹ at all flare locations. Above an ion energy of 10 MeV nucleon⁻¹, the fluence is increased by about a factor of two, falling to less than a 50% increase by an ion energy of 100 MeV nucleon⁻¹.

Shown in Figures 15–17 are neutron kinetic-energy spectra from a disk flare (θⁿ_obs = 0°) for accelerated-ion spectra with power-law indexes s of 2, 4, and 6, respectively. These indexes span the range derived for flares from previous analyses of flare data (e.g., Ramaty et al. (1996) found that ∼95% of the gamma-ray line-producing flares observed with the SMM gamma-ray detector had power-law spectral indexes between 3 and 6). These are escaping-neutron spectra at the Sun (i.e., just after transport through the solar atmosphere but before any modification due to neutron decay). Shown in Figures 18–20 are the corresponding neutron spectra calculated for a limb flare (θⁿ_obs = 85°). All of the neutron spectra are normalized to one accelerated proton with energy greater than 30 MeV (i.e., N_p(> 30 MeV) = 1) and calculated for isotropic release of ions at the top of a loop with no convergence or PAS (δ = 0, λ → ∞), which results in a downward-isotropic interacting-ion angular distribution. The solid black curves are the total spectra, and the colored curves are spectra from the eight contributing reaction types. The solid colored curves are for accelerated proton and α-particle reactions with ambient material and the dashed colored curves are the inverse reactions of heavy accelerated ions interacting with ambient H and ⁴He. We discuss these spectra below, demonstrating how the contributions from the various neutron-producing reactions change as the accelerated-ion spectrum and the flare heliocentric angle change.

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For neutron spectra from the hardest ion spectrum (s = 2, Figures 15 and 18), neutron production is dominated by the p + H, p + ⁴He, and α + H reactions at all neutron energies and for both flare locations. Since we did not change the code procedures for these reactions, the neutron spectra are very similar to those calculated using the original Hua et al. (2002) procedures (the dotted curves in the figures). At high neutron energies, the fluence is larger for a limb flare; i.e., there is limb brightening of high-energy neutrons for the downward-isotropic interacting-ion angular distribution resulting from assuming no magnetic convergence. At low neutron energies, the fluence is larger for the disk flare; the excess at low energies is due to scattering of downward-moving higher-energy neutrons into the upward direction with loss of their original energy. Thus, for this angular distribution, there is limb darkening of low-energy neutrons.

For neutron spectra from the s = 4 ion spectrum (Figures 16 and 19), neutron production at high neutron energies is again dominated by reactions whose procedures have not been changed, and the spectra are very similar to those calculated using the original Hua et al. (2002) procedures (the dotted curves in the figures) at both flare locations. For the limb flare (Figure 19) at low neutron energies (less than about 5 MeV), α-particle reactions with heavy targets contribute significantly. The low-energy structure in the neutron-production cross sections for these heavy-element reactions calculated using the new procedure can be seen in the calculated neutron spectra and produces a small change in the total spectrum for the limb flare (compare with the dotted curve). For the disk flare, the contribution of this reaction to low-energy neutron production is less since the neutrons are produced more in the downward direction and their escape is less likely.

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For neutron spectra from the steepest ion spectrum (s = 6, Figures 17 and 20), neutron production at high neutron energies is again dominated by the α + ⁴He and α + H reactions, and the spectra are very similar to those calculated using the original Hua et al. (2002) procedure (dotted curves). At neutron energies below about 10 MeV, neutron production is dominated (by more than a order of magnitude) by α-particle reactions with heavy targets for both flare locations. The revised procedures for calculating the cross sections for these reactions result in structure in the total neutron spectra at these energies; the spectra look similar to those from mono-energetic ions at 5 MeV nucleon⁻¹ shown in Figures 7 and 8. Neutron detectors with good instrument resolution at low neutron energies may be able to resolve this structure.

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When the impulsive-event composition is assumed, for steep ion spectra the inverse reactions of heavy elements with ⁴He produce more low-energy (<30 MeV) neutrons, increasing the yield of <5 MeV neutrons by a factor of 2–3 for limb flares but less than a factor of two for disk flares. For hard ion spectra, the inverse heavy reactions do not add significantly to the neutron yield. For these ion spectra, the ³He + H reaction increases the >100 MeV yield by about 50% for limb flares and the ³He + ⁴He reaction contributes >100 MeV neutrons, increasing the 1 GeV neutron yield by almost an order of magnitude.

When the ion spectrum is assumed to be a power law with exponential roll-over, the effect on the neutron spectrum as E₀ decreases from large values is a reduction at all neutron energies. For steep ion spectra, the reduction is most significant at neutron energies >10 MeV, steepening the spectrum. For hard ion spectra the reduction is significant at all neutron energies because for these spectra, many low-energy neutrons are produced by interactions of high-energy ions (see the discussion of contributing ion energy ranges below). The reduction is more significant at high neutron energies, again resulting in a steepening of the neutron spectrum.

Using calculated escaping-neutron spectra for the coronal composition such as those of Figures 15–20, we can compare which reactions contribute to the >30 MeV neutrons observed at 1 AU with those contributing to the 1–10 MeV neutrons observable in the inner heliosphere. Figure 21 shows the fractions that the various reactions contribute to the production of >30 MeV neutrons as a function of ion spectral index s. Panel (a) is for a disk flare (θⁿ_obs = 0°) and panel (b) is for a limb flare (θⁿ_obs = 85°). For disk flares at most spectral indexes, only the α + ⁴He reaction produces these higher-energy neutrons sufficiently isotropically that they can easily escape into the upward direction for the downward-isotropic interacting-ion angular distribution resulting from no magnetic convergence. For the hardest spectra, the p + ⁴He also contributes. For the limb flare, the α + H and the α + ⁴He reactions dominate >30 MeV neutron production at all spectral indexes. We find essentially no dependence of the contributing-reaction fractions on either loop convergence or PAS.

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Because >30 MeV neutron production is dominated by the α + ⁴He and α + H reactions, the increased heavy-element abundances of the impulsive-event composition do not significantly change the reaction fractions contributing to these neutrons. However, >30 MeV neutron production from accelerated ³He reactions is now significant. For a disk flare, the ³He + ⁴He reaction is isotropic enough to produce these higher-energy neutrons in the upward direction, sharing neutron production with the α +⁴He reaction almost equally for ion indexes steeper than about 2. For a limb flare, the ³He + H reaction now shares equally with the α + ⁴He and α + H reactions at all ion spectral indexes.

When the ion spectrum is assumed to be a power law with exponential roll-over, the relative contribution to >30 MeV neutron production by the α + ⁴He reaction for hard ion spectra increases as E₀ decreases from large values. For disk flares, for E₀ < 100 MeV, >30 MeV neutron production is dominated by this reaction for all ion spectral indexes. For limb flares, this reaction is comparable to the α + H reaction for hard ion spectra when E₀ is less than 50 MeV nucleon⁻¹.

Panels (a) and (b) of Figure 22 show reaction contribution fractions for the production of 1–10 MeV neutrons from a disk and a limb flare for the coronal composition, respectively. Again, there is little difference between the two flare locations because these low-energy neutrons tend to be produced more isotropically and higher in the solar atmosphere. The fractions of the inverse reactions of accelerated heavy ions interacting with ambient H and ⁴He and the accelerated α-particle reaction with ambient H are increased somewhat for the limb flare because the large velocity of the center of momentum of these reactions results in a neutron angular distribution similar to the downward-isotropic interacting-ion distribution and subsequent limb brightening for these reactions. The reaction fractions for 1–10 MeV neutrons are very different from those for >30 MeV neutrons. For spectra steeper than s ∼ 5, α-particle reactions with ambient heavy elements now dominate with some contribution from the corresponding inverse reactions for the limb flare. The α + ⁴He reaction contributes significantly only for indexes near s = 4–5. For spectra harder than s ∼ 4, the p + ⁴He reaction dominates for both flare locations. We find essentially no dependence of the contributing-reaction fractions on either loop convergence or PAS.

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For the impulsive-event composition, the inverse-reaction contribution to 1–10 MeV neutron production increases for ion spectral indexes steeper than 4. For limb flares, these inverse reactions dominate line production because the neutrons can more easily escape, accounting for >80% of the production for s > 6. For disk flares the inverse-reaction contribution becomes only approximately equally to that of the α-particle reactions with ambient heavy elements. For hard spectra there is some additional contribution from the ³He + H reaction.

When the ion spectrum is assumed to be a power law with exponential roll-over, the impact on the reaction contribution fractions is similar for both flare locations. For steep ion spectra there is essentially no change in the reaction fractions as E₀ is decreased from large values. For hard ion spectra, as E₀ decreases, the contribution of the α + ⁴He reaction (threshold energy of ∼10 MeV nucleon⁻¹) increases at the expense of the p + ⁴He reaction (threshold energy of ∼25 MeV nucleon⁻¹). For E₀ < 30 MeV nucleon⁻¹ the α + ⁴He reaction dominates.

After escaping the solar atmosphere, neutron spectra such as those of Figures 15–20 begin to suffer attenuation due to neutron decay, increasing with increasing distance from the Sun. As an example, Figure 23 shows the resulting neutron spectra at three distances D = 10 R_s, 0.5 AU, and 1 AU from accelerated ions with spectral index s = 4 for a disk flare. The dramatic decrease of the low-energy neutron fluence with increasing distance to the detector is clearly seen, due to both the D² factor and the smaller fraction of low-energy neutrons surviving to reach the detector.

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Figure 24 shows the calculated fluence of 1–10 MeV neutrons (solid curves) as a function of the ion spectral index s for a neutron detector located at three distances from the Sun D_n = 10 R_s, 0.5 AU, and 1 AU for flares occurring at three heliocentric angles θⁿ_obs = 0°, 60°, and 85°. The fluences are normalized to N_p(> 30 MeV) = 1 and calculated for isotropic release of ions at the top of a loop with no convergence or PAS (δ = 0, λ → ∞), which results in a downward-isotropic interacting-ion angular distribution. At all detector distances, we see very minor limb brightening from ion spectra steeper than s ∼ 5 for this angular distribution of interacting ions; the neutrons are produced at sufficiently shallow depths in the atmosphere such that attenuation at the limb is not significant. Neutrons of energy 1–10 MeV from harder ion spectra are produced deep enough to show significant limb darkening. The dramatic increase of the 1–10 MeV neutron fluence with decreasing distance to the detector is again clearly seen. To calculate an actual neutron fluence at the detector, the value from the curve should be multiplied by the number of accelerated protons with energy greater than 30 MeV.

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For the impulsive-event composition, the 1–10 MeV neutron yield is increased for all ion spectral indexes. For a disk flare, the yield is increased from about 30% for an ion spectral index s = 2 to a factor of ∼2 for s > 6. For limb flares, the impact of the additional inverse reactions for steep ion spectra is greater because the neutrons can more easily escape; the 1–10 MeV neutron yield increases from about 30% for an ion spectral index s = 2 to a factor of 5–6 for s > 6. We note that neutron production from the reactions of accelerated ³He with ambient heavy elements, which are not currently included in the code, would be significant only for ion energies less than ∼1 MeV nucleon⁻¹, and then only if the accelerated ³He/⁴He ratio were as large as 1. If it were, the 1–10 MeV neutron and neutron-capture line yields for power-law spectra shown in Figures 24 and 26 would increase at most by a factor of ∼3.

The 1–10 MeV neutron yield is reduced when the ion spectrum is assumed to be a power law with exponential roll-over. For steep ion spectra (s > 6), the yield is reduced by less than 5% at both flare locations for E₀ = 100 MeV nucleon⁻¹. But for hard ion spectra (s = 2), the yield for E₀ = 100 MeV nucleon⁻¹ is only 20% of that without roll-over for a disk flare and 40% for a limb flare.

In Figure 24 we also show the calculated fluence of >30 MeV neutrons for the coronal composition (dashed curves) at a neutron detector located at D_n = 1 AU for the three flare locations. We clearly see limb brightening for these higher-energy neutrons at all ion spectral indexes, although it is relatively less for hard spectra because of the increased scattering losses due to deeper production that even these higher-energy neutrons suffer. For limb flares, the >30 MeV neutron fluence at 1 AU overwhelms the 1–10 MeV fluence at 1 AU for all ion spectral indexes harder than s = 6, an index steeper than those measured for typical flares. But for steeper ion spectra, the neutron spectra are so steep that the 1–10 MeV neutrons dominate regardless of losses due to neutron decay. For disk flares, however, the reduction of upward-escaping high-energy neutrons is so great that only for ion spectra harder than s ∼ 4.5 does the >30 MeV neutron fluence actually dominate. For a typical flare index of s = 4 and θⁿ_obs = 60°, at 1 AU the 1–10 MeV neutron fluence is ∼7% of the >30 MeV neutron fluence.

The increase of the >30 MeV neutron yield due to the impulsive-event composition is not dramatic. For the limb flare, the increase is about 50% for an ion spectral index of 2, increasing monotonically to a little more than a factor of two for s = 8. The increase for a disk flare is similar, maximizing at about a factor of two for an ion index of about 3.

Because they are made by higher-energy ions, the reduction of the >30 MeV neutron yield when the ion spectrum is assumed to be a power law with exponential roll-over is larger than that of the 1–10 MeV neutrons, especially for hard ion spectra. For E₀ = 100 MeV nucleon⁻¹ and an ion spectral index of 2, the >30 MeV neutron yield is reduced by about a factor of 30 for a disk flare and 15 for a limb flare. For a steep index of 6 and E₀ = 100 MeV nucleon⁻¹, the yield is reduced by about a factor of two for both flare locations.

Figure 25 shows the fraction that each of the eight reaction types contributes to the production of the 2.223 MeV neutron-capture line for the coronal composition as a function of the ion spectral index s. Panel (a) is for a disk flare (θⁿ_obs = 0°) and panel (b) is for a limb flare (θⁿ_obs = 85°). The fractions for the two flare locations are very similar, with the α + H reaction, which is significant only for harder spectra, being somewhat larger for the disk flare.

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The neutron-capture line reaction fractions shown in Figure 25 are similar to those for 1–10 MeV neutron production shown in Figure 22. This is because the energies of the captured neutrons producing the bulk of the capture-line photons that actually escape the solar atmosphere without scattering are near 10 MeV. As noted above, higher-energy neutrons tend to be captured too deep and the capture photons suffer too much attenuation. Lower-energy neutrons tend to decay in the solar atmosphere without being captured. The most significant differences between the capture-line and 1–10 MeV neutron-production fractions are for disk flares. For harder spectra, the contribution of the α + H reaction is larger for neutron-capture line production, and for steeper spectra, the contribution of the inverse reactions of accelerated heavy elements on ambient ⁴He is larger. The angular distributions of neutrons from these reactions are more downward directed so their contribution to escaping neutrons is less.

The impact on the neutron-capture line reaction fractions of assuming the impulsive-event composition is similar to that for the 1–10 MeV neutrons but with less dependence on flare location. For both flare locations, the impulsive-event composition causes the reaction fractions of the α + heavy ambient elements and their inverse reactions to switch roles, with the latter reactions now dominating capture-line production for ion power-law indexes steeper than ∼5. For harder ion spectra, the increased abundance of ³He causes the ³He + H reaction to contribute up to ∼25% for ion spectral indexes from 2 to 4.

When the ion spectrum is assumed to be a power law with exponential roll-over, the impact on the reaction contribution fractions is similar for both flare locations. For steep ion spectra, there is essentially no change in the reaction fractions as E₀ is decreased from large values. For hard ion spectra, as E₀ decreases, the contribution of the α + ⁴He reaction increases at the expense of the p + ⁴He reaction. For E₀ < 30 MeV nucleon⁻¹, the α + ⁴He reaction dominates.

Figure 26 shows the calculated fluence of the 2.223 MeV neutron-capture line for coronal abundances, ϕ_2.223, as a function of the ion spectral index s for a gamma-ray detector located at 1 AU for flares occurring at three heliocentric angles (θ^2.223_obs = 0°, 60°, and 85°). The fluences are again normalized to N_p(> 30 MeV) = 1 and calculated for isotropic release of ions at the top of a loop with no convergence or PAS (δ = 0, λ → ∞), which results in a downward-isotropic interacting-ion angular distribution. Unlike the 1–10 MeV neutron fluence, the capture line is limb darkened at all ion spectral indexes, and there is significantly more limb darkening from harder spectra because neutron captures from higher-energy ion interactions occur deeper in the photosphere. To calculate an actual capture line fluence at the detector, the value from the curve should be multiplied by the number of accelerated protons with energy greater than 30 MeV. Line fluences at other distances to the gamma-ray detector can be calculated by adjusting the values from the curves by the appropriate D² factor. Note that the curves of Figure 26 differ from those given in Figure 13 of Murphy et al. (2007) for δ = 0 and no PAS due to the different accelerated-ion compositions assumed.

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For both flare locations, assuming the impulsive-event composition causes the capture-line yield to increase monotonically as the ion spectrum steepens; by about 30% for s = 2 to a maximum of a factor of 5–6 at s = 8.

When the ion spectrum is assumed to be a power law with exponential roll-over, the neutron-capture line fluence is reduced for all ion spectral indexes, with the most effect for hard spectra. For a disk flare and E₀ = 100 MeV nucleon⁻¹, the reduction is a factor of six for an ion spectral index s = 2 but only by 10% for an index of 6. For a limb flare and E₀ = 100 MeV nucleon⁻¹, the reduction is a factor of five for s = 2, but only by 5% for s = 6.

The neutron-capture line fluence curves of Figure 26 exhibit a minimum around spectral indexes of ∼4.5, with greater yields from both steeper and harder spectra. This specific shape is due to the (arbitrary) choice of normalizing the accelerated-proton spectrum to have unit integral above 30 MeV. Other choices of normalization would modify the shape. Note that the 1–10 MeV neutron fluence curves of Figure 24 would be more like those of the capture line if the fluence of neutrons of all energies were considered (see Figure 13 of Murphy et al. 2007). Restricting consideration to 1–10 MeV neutrons reduces their relative yields from hard spectra as neutrons tend to be produced with energies above 10 MeV and low-energy neutrons are more easily scattered, especially for limb flares.

Compton scattering of 2.223 MeV line photons in the solar atmosphere reduces their energy and the residual photons form a continuum of emission at energies less than 2.223 MeV. For an accelerated-ion spectral index s = 4, Figure 27 shows the spectrum of this emission (including the line itself) from a disk flare (θ^2.223_obs = 0°) and a limb flare (θ^2.223_obs = 85°). The spectra have been normalized to have the same yield in the line. At photon energies near the line (greater than ∼1 MeV), the limb flare produces more scattered continuum relative to the line than the disk flare since the path length traveled by the photons is greater. This is also true at lower photon energies but the Compton-scattering cross section at low energies is so strong that many of these residual photons cannot escape from the limb flare, and the resulting relative continuum escaping the solar atmosphere is in fact less than for the disk flare. This general behavior of the scattered-continuum spectrum holds for all ion spectral indexes.

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The ratio of the yield in the scattered continuum to that of the line is a strong function of both the location of the flare on the disk and the accelerated-ion spectral index. The detailed behavior of the ratio depends on the definition of the energy range used to calculate the scattered-continuum yield. Figure 28 shows this ratio as a function of the accelerated-ion spectral index s for three flare heliocentric angles (θ^2.223_obs = 0°, 60°, and 85°) and for two definitions of the energy range (>200 keV and >1 MeV). For both definitions, the ratio increases for harder spectra (smaller s) because the deeper captures associated with harder spectra always produce more scattered continuum relative to the line. When the continuum definition is >1 MeV, the ratio increases as the flare location nears the limb because there is more escaping scattered continuum in this energy range relative to the line for limb flares (see Figure 27). But when the definition is >200 keV, the limb flare produces less escaping continuum relative to the line than the disk flare because of the strong attenuation of the <1 MeV photons (see Figure 27).

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An important question to ask is what accelerated-ion energy ranges are responsible for neutron and neutron-capture line production? For 1–10 MeV neutron production, Figure 29 shows the fractions that ions with energies in four energy windows (0–1, 1–10, 10–30, and >30 MeV nucleon⁻¹) contribute as a function of ion spectral index from a disk flare (θⁿ_obs = 0°) for a neutron detector located at 10 R_s (solid curves) and 1 AU (dashed curves). For ion spectra with indexes steeper than ∼5.5, 1–10 MeV neutrons are produced almost exclusively by 1–10 MeV nucleon⁻¹ ions. For ion spectra with indexes harder than ∼4.5, 1–10 MeV neutrons are produced almost exclusively by ions with energies >30 MeV nucleon⁻¹. For a narrow range of indexes near s = 5, 1–10 MeV neutrons are produced almost equally by ions from the three >1 MeV nucleon⁻¹ energy windows. While ions with energies less than 1 MeV nucleon⁻¹ are capable of producing neutrons with energies greater than 1 MeV, they never contribute significantly because the neutron-production cross sections of higher-energy ions are so much larger that these ions always dominate even 1–10 MeV neutron production. We find essentially no dependence of the contributing-ion-energy fractions on either loop convergence or PAS.

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Figure 29 shows that there is a weak dependence of the ion-energy window fractions on detector distance from the Sun. For neutrons of a given energy there is no dependence on distance since decay only attenuates the neutron fluence and cannot change the distribution of ion energies contributing to their production. But when a range of neutron energies is considered, more neutrons with energies near the upper end of the range survive transport to the detector than those near the lower end, and the population shifts to higher neutron energies with increasing detector distance (see Figure 23). Since higher-energy neutrons generally are produced by higher-energy ions, the range of relevant ion energies for the range of neutron energies correspondingly also shifts to higher energies with increasing detector distance.

There is also a weak dependence of the ion-energy window fractions on the heliocentric angle of the flare. As the flare location changes from center to limb, fewer 1–10 MeV neutrons are produced by ions with energies >30 MeV nucleon⁻¹ and more by 10–30 MeV nucleon⁻¹ ions. This is because the interactions of the higher-energy ions occur deeper in the atmosphere and the neutrons produced at the limb are more effectively attenuated by scattering as they transport tangentially out of the solar atmosphere. This effect is evident only for the harder ion spectra and essentially results in a decrease of the >30 MeV nucleon⁻¹ ion window fractions for s < 5 by about 10%–20% and a corresponding increase of the 10–30 MeV nucleon⁻¹ window fractions. The contributions from ions <10 MeV nucleon⁻¹ are essentially unchanged since their interactions occur higher in the solar atmosphere and are less affected by scattering as they escape.

Note that the ion-energy window fractions of Figure 29 directly reflect the contributing-reaction fractions of Figure 22. Steep spectra produce 1–10 MeV neutrons primarily via accelerated α-particle reactions with ambient heavy nuclei, which have low production threshold energies (a few MeV nucleon⁻¹ or lower; see Table 3), while hard spectra produce neutrons primarily via the p + ⁴He reaction and its inverse, which have a high threshold energy (∼26 MeV nucleon⁻¹). Ion spectra with intermediate indexes also produce neutrons via the α + ⁴He reaction, which has an intermediate threshold energy (∼10 MeV nucleon⁻¹).

The impulsive-event composition does not change the ion energies responsible for 1–10 MeV neutron production. For steep ion spectra, while the inverse reactions now dominate, they have the same production cross sections when expressed as energy per nucleon as the corresponding accelerated proton and α-particle reactions. For hard spectra, the production cross section for the ³He + H reaction has an energy dependence similar to that of the p + ⁴He.

Assuming the ion spectrum is a power law with exponential roll-over has little impact on the ion-energy window fractions contributing to 1–10 MeV neutron production until E₀ is less than about 20 MeV nucleon⁻¹ and then only for spectral indexes harder than s ∼ 5. For such ion spectra, the 10–30 MeV nucleon⁻¹ ion contribution becomes comparable to or exceeds that of the >30 MeV nucleon⁻¹ ions.

Figure 30 shows the fractions that ions with energies in the four energy windows contribute to >30 MeV neutron production for the coronal composition as a function of ion spectral index for a neutron detector located at 1 AU from a disk flare (θⁿ_obs = 0°, solid curves) and a limb flare (θⁿ_obs = 85°, dashed curves). For disk flares, these higher-energy neutrons can only be made by >30 MeV nucleon⁻¹ ions. This is because for the downward-isotropic interacting-ion angular distribution resulting from no magnetic convergence, the downward-directed high-energy neutrons must be scattered (with subsequent loss of energy) to escape from a disk flare, and their initial energies must therefore be significantly higher requiring higher-energy ion reactions. Even for limb flares, >30 MeV neutron production is dominated by >30 MeV nucleon⁻¹ ion reactions, with 10–30 MeV nucleon⁻¹ ions only contributing (<20%) for the steepest ion spectra, although such ions can only produce neutrons with energies not much greater than 30 MeV. We note that in the figure the 0–1 and 1–10 MeV nucleon⁻¹ ion fractions are essentially zero and cannot be seen.

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The impulsive-event composition has little impact on the ion energies contributing to >30 MeV neutron production. The inverse reactions do not contribute significantly to these neutrons and the energy threshold and dependence of the ³He + ⁴He reaction cross section is almost identical to those of the α + ⁴He reaction (see Hua et al. 2002).

Assuming the ion spectrum is a power law with exponential roll-over has little effect on the ion energies contributing to >30 MeV neutron production for disk flares. These neutrons must have high initial energies to escape after scattering upward and so can only be made by high-energy reactions. As E₀ is reduced, the yield of >30 MeV neutrons is significantly reduced as discussed above. For limb flares, there is little effect on the contributing ion energies until E₀ is reduced to less than about 20 MeV nucleon⁻¹. For such roll-over energies, 10–30 MeV nucleon⁻¹ ions produce about half of the neutrons for ion spectral indexes steeper than s = 6.

Figure 31 shows the fractions that ions with energies in the four energy windows contribute to neutron-capture line production for the coronal composition as a function of ion spectral index from a disk flare (θ^2.223_obs = 0°). While ions with energies less than 1 MeV nucleon⁻¹ are capable of producing the line, only for the steepest spectra are these ion energies relevant. For ion spectra with indexes steeper than ∼5.5, the line is produced almost exclusively by 1–10 MeV nucleon⁻¹ ions. For ion spectra with indexes harder than ∼5, the line is produced almost exclusively by ions with energies >30 MeV nucleon⁻¹. For a narrow range of indexes near s = 5.5, the line is produced almost equally by ions from the three >1 MeV nucleon⁻¹ energy windows. Similar to the ion-energy fractions for 1–10 MeV neutron production, for a limb flare there are fewer capture-line photons produced by ions with energies >30 MeV nucleon⁻¹ and more by 10–30 MeV nucleon⁻¹ ions because the higher-energy interactions produce neutrons that are captured too deep for the photons to escape unscattered from a flare near the limb.

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Note that the distributions of ion energy-window fractions for the production of the capture line shown in Figure 31 are similar to those for 1–10 MeV neutron production shown in Figure 29. This is because the energies of the captured neutrons producing the bulk of the capture-line photons that actually escape the solar atmosphere without scattering are near 10 MeV.

Similar to the 1–10 MeV neutrons, the impulsive-event composition has little impact on the ion energies contributing to the line. For steep ion spectra, while the inverse reactions now dominate, they have the same production cross sections when expressed as energy per nucleon as the corresponding accelerated proton and α-particle reactions. For hard spectra, the production cross section for the ³He + H reaction has an energy dependence similar to that of the α + H (see Hua et al. 2002).

Again similar to the 1–10 MeV neutrons, assuming the ion spectrum is a power law with exponential roll-over has little impact on the ion-energy window fractions contributing to neutron-capture line production until E₀ is less than about 20 MeV nucleon⁻¹ and then only for spectral indexes harder than s ∼ 5. For such ion spectra, the 10–30 MeV nucleon⁻¹ ion contribution becomes comparable to or exceeds that of the >30 MeV nucleon⁻¹ ions.

An alternative view of the ion-energy contributions to neutron-capture line production is shown in Figure 32 where the effective ion-energy range for line production from a disk flare is shown as a function of the accelerated-ion spectral index. This effective-energy range is defined as in Murphy et al. (2007): We identify the ion energy where the line yield is maximum for a given power-law spectral index, and then define the effective ion-energy range to be that within which the yield has fallen to 50% on each side of the maximum. For steep ion spectra, the neutrons are produced primarily by accelerated α-particle reactions with ambient heavy elements and their inverse reactions whose lower threshold energies result in lower effective ion energies. For hard spectra, the accelerated proton reaction with ambient ⁴He and the inverse reaction dominate whose higher threshold energy shifts the effective range to higher energies. For intermediate indexes, both reactions contribute resulting in a broad effective-energy range.

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