Neutrons in simulations of extensive air showers

. We study neutrons produced in simulations of extensive air showers. By using the Monte Carlo simulation package F luka , our examination is able to extend from the highest energy neutrons, produced in hadronic interactions, all the way down to thermal energies. The energy spectra, arrival times, and lateral distributions of neutrons at the ground are compared for di ff erent primary species, as are the longitudinal profiles of the neutron fluence. Direct comparisons are drawn with the analogous distributions for muons.


Introduction
Given their electrical neutrality and stability on the time scales in question, neutrons produced in air showers exhibit characteristics distinct from showers' electromagnetic and muonic components. A number of measurements of neutrons in air showers have been performed (for a summary, see [1][2][3]); however, a lack of reliable predictions as to the properties of the neutron cloud impedes its interpretation.
We use Fluka [4][5][6], a Monte Carlo simulation package for particle transport and interaction with matter, to investigate the properties of neutrons in extensive air showers. Air showers induced by proton, iron, and photon primaries ranging from sub-PeV to EeV in energy are covered in these studies. This work is motivated by the planned, ongoing, and existing deployments of scintillator detectors at large-area very high and ultra-high energy cosmic ray air shower arrays.
Fluka permits detailed simulation of neutron production and propagation in air showers and provides access to the characteristics of the neutron cloud down to even thermal energies, which are inaccessible with other air shower simulation programs. Fluka has been validated both with high-energy accelerator measurements (for a validation of neutron spectra, see [7]) and cosmic ray measurements at different atmospheric depths and under different conditions (for examples, see [8][9][10][11][12]). For the studies presented here, neutrons were transported down to 10 −5 eV. Earth's atmosphere was described by 100 layers with densities in accordance with the US Standard Atmosphere.

Energy spectrum and arrival times
The energy spectra of neutrons at sea level for vertical air showers induced by proton, iron, and photon primaries of * e-mail: david.schmidt@kit.edu 5.6 × 10 16 eV are shown in figure 1. These spectra are normalized such that the integral is equal to the total number of neutrons arriving at the ground.
As evidenced by the nearly identical neutron and antineutron spectra for energies above a few GeV, the highest energy neutrons are produced in baryon, anti-baryon pairs in hadronic interactions. These are the same hadronic interactions as those producing the pions and kaons that go on to decay to muons, and as with muons, the multiplicity of high-energy neutrons is much lower for photon-induced air showers as compared with hadronically-induced showers. The peak in the neutron spectrum between 50 and 150 MeV is the so-called quasi-elastic peak. Below 10 to 20 MeV, neutrons propagate through diffusion. The structures between 0.1 and 10 MeV are due to resonances largely dependent on the target material (in this case, air). The peak at the lowest energies exists for targets with sufficient hydrogen to fully thermalize the neutrons, which was the case for the soil simulated at sea-level.
The extension of the energy spectrum down to low, even thermal, energies stands in sharp contrast to that of muons, which suffer from decay and ionization losses during propagation. At the depth of sea-level, the neutron spectra for proton and iron primaries are strikingly similar, particularly for the highest energy neutrons. The source of this coincidence is further elaborated upon in section 3.
In figure 2, the energy spectrum of neutrons in air showers initiated by protons of 5.6 × 10 16 eV is broken down into the arrival times at the ground. The arrival time of a particle is defined here as the delay with respect to the time light would require to travel from the point where the primary first interacts to the impact point with the measurement plane. The broad range of neutron energies is reflected by a distribution of arrival times also covering many orders of magnitude. Whereas the bulk of muons arrive at the ground well within 1 µs, the majority of neutrons arrive after 1 µs and as late as milliseconds. This de-  layed arrival of the majority of neutrons provides a means of identifying them and differentiating them from muons. In figure 3, the energy spectra of neutrons and muons arriving at a depth of 675 g/cm 2 , which is similar to that of the IceTop surface detector array, are juxtaposed for ranges in energy and arrival time representative of those for which surface detector arrays are typically sensitive. Beyond 1 µs, the multiplicity of muons and neutrons are, to first order, comparable. The presence of an ice surface at the observation depth on the neutron energy spectrum appears to have only a very limited impact for the arrival times relevant for detection and only below ∼ 1 MeV.

Two coincidences
When observing the longitudinal profiles of neutrons, two coincidences may be observed that have implications for the scaling of neutron multiplicities with the primary energy and mass. Both coincidences arise due to the interplay between the energy in the hadronic cascade and the degree to which the resulting neutrons are attenuated prior to arrival at the depth of observation.
As a proxy for the longitudinal profile, the fluence for different particle species is calculated. This unitless quantity is defined as the product of the density of a given slice Comparing the red and black or gray spectra it is clear that the number of neutrons at these large distances and delays is comparable to the number of muons. These properties qualitatively agree with those reported by Linsley [2] and others.
For showers of the same energy at a deeper observation level, X det = 878 g/cm 2 with ground level similar to the Pierre Auger site, we show the spectra in Fig. 17. The observations are very similar to Fig. 16, with the main difference being a stronger contribution of the few MeV neutrons compared to the 100 MeV ones, and the smaller number of neutrons compared to muons. However, we expect that the last statement is only true at the given energy and that the situation is different at higher energies.

V. LONGITUDINAL PROFILES AND SCALING
Following, the multitude of numerical coincidences observed in the energy spectra we want to further deepen our understanding of the exact origin and confirm the explana- tions given in the previous Sections. We use the fluence (cf. Eq. (1)) of particles scored in the 100 atmospheric layers of the simulation to analyse the longitudinal development of the shower in more detail.
In Fig. 18 we show the fluence profiles of muons for showers of different energies. By scaling inversely linear with energy we can highlight the expected scaling at any depth. With the slight exception of shower with an energy of E = 5.6⇥10 14 eV, the profiles are parallel for different energies and follow the expected scaling. It is also clearly visible that for photon showers -dashed lines rescaled with an additional factor 5 for visibility -the linear scaling is true, if losses due to attenuation are ignored. For practical purposes this is a good approximation given that the attenuation is of the order of 50% over four decades in energy.
For neutrons, the situation is different as already derived from the energy spectra in the previous Section. In Fig. 19  of atmosphere and the sum of the tracklengths of all particles of a given type therein normalized by the column density (i.e. thickness in g/cm 2 ) of the slice.

Energy scaling
The first coincidence is depicted in figure 4, where the longitudinal profiles of the neutron fluence are shown for proton and photon primaries over a range of energies. Each profile is scaled inversely proportional to the primary energy to highlight departure from a linear scaling with energy. Only neutrons with energies greater than 10 MeV were considered when constructing the profiles to narrow the focus on neutrons where detection is possible with typical air shower arrays. The maximum of the longitudinal profile of the neutron fluence scales with energy according to E β where β ≈ 0.9, analogous to muons. The increased depth of the maximum for primaries of higher energy results in a smaller degree of attenuation at a fixed observation depth, however. The result is that the number of neutrons reaching ground scales approximately linearly with energy for hadronic primaries at an observation depth of ∼ 850g/cm 2 , which approximately corresponds to the depth of the surface detector arrays of existing ultra-highenergy cosmic ray observatories. IceTop-like proton primary E = 5.6×10 16 eV X det = 675 g/cm 2 muons (total) muons, t ∈ (46.4, 1000) ns neutrons (total) with ground level neutrons, t ∈ (1, 21.5) µs with ground, t ∈ (1, 21.5) µs Figure 3. Energy spectra of muons and neutrons arriving at the ground for air showers induced by proton primaries of 5.6 × 10 16 eV. The energy spectra of all particles of each species are shown in addition to the spectra for arrival times representative of those for which surface detector arrays are typically sensitive.

Mass scaling
The second coincidence is depicted in figure 5, where the evolution of the neutron and muon fluences with atmospheric depth are juxtaposed for different species of primary particle. As before, only neutrons with energies greater than 10 MeV were considered in the calculation of the neutron fluence. The additional energy channeled into the hadronic component for heavier primaries manifests in a larger number of neutrons at the maximum. Due to their shorter removal length (100 g/cm 2 , as a rule of thumb), however, the number of neutrons is attenuated more quickly than that of muons. While a difference in the muon fluence of ∼ 25% remains between proton and iron primaries at an observation depth of ∼ 850g/cm 2 , very little difference in the neutron fluence is observed.

Lateral distribution
To examine the radial distribution of neutrons, we normalize the integral of the energy spectra above a given energy and normalize the results with the area of the given radial bin. We do the same for muons. The resulting lateral distributions are shown in figure 6 for proton, iron, and photon primaries of 5.6 × 10 18 eV at an observation depth of 878g/cm 2 . This primary energy and observation depth are characteristic of ultra-high-energy cosmic ray observatories. The distributions for neutrons are shown with a cut on the energy of 1 MeV and 1 GeV to highlight detectability and crudely differentiate between production mechanisms, respectively.
The lateral distribution of neutrons with energies above 1 MeV appears broader and has a slope less steep than that of muons. Similar to muons, the slope for proton primaries appears slightly steeper than that of iron primaries. An interesting coincidence is that for photon primaries, the total number of neutrons above 1 MeV is approximately equal to the number of muons, whereas for hadronic primaries, it is significantly less.

Conclusions
The neutrons produced in extensive air showers exhibit energy spectra and arrival time distributions distinct from those of the electromagnetic and muonic shower components. These distinct characteristics and the sheer abundance of neutrons make them of potential interest for airshower experiments.
Our predictions indicate that the considerable delay in the arrival times of the bulk of neutrons serves as a means of identifying their presence in measurements. We have also shown that an interplay between the energy in the hadronic component of air showers and attenuative effects results in a linear scaling with the primary energy of the number of potentially detectable neutrons arriving at observation depths typical of ultra-high-energy cosmic ray observatories. An additional coincidence resulting from the interplay between the same two effects also results in a striking similarity of the number of potentially detectable neutrons for different hadronic primaries. A lower abundance of neutrons in photon-induced air showers may provide additional information to differentiate them from hadronically-induced showers.
To verify the predicted characteristics of the neutron cloud with measurements, detailed detector simulations will be necessary.