Nuclear reaction model for deuterons below 150 MeV/n in FLUKA

. The lack of a dedicated model for nuclear reactions of deuterons at energies below 150 MeV / n has been a long-standing shortcoming of the general-purpose radiation-transport code FLUKA, preventing its use in applications either driven by deuteron beams or where reinteracting secondary deuterons play a signiﬁcant role. A dedicated modelling e ↵ ort has been undertaken to overcome this limitation. In this contribution, the essential ingredients of FLUKA’s novel model for low-energy deuteron nuclear inelastic interactions are brieﬂy described, including nucleon stripping to the continuum, nucleon transfer to discrete states, and elastic break-up. The resulting model has been extensively benchmarked against experimental data prior to its inclusion in FLUKA v4-2.0.


Introduction
FLUKA is a general-purpose code for the Monte Carlo simulation of radiation transport, developed and maintained at CERN and collaborating institutes since the 1960s [1-3]. The code tracks over 60 particle species, including leptons, hadrons, and ions, up to the PeV and down to the keV, with the particular case of neutrons, tracked down to thermal energies (fraction of meV). Hadronic and electromagnetic interactions are accounted for, leading to the natural coupling of hadronic and electromagnetic showers. FLUKA is further equipped with thoroughly tested built-in scoring and biasing capabilities. As a result, the code, as distributed by CERN, is nowadays used by a community of about 2000 active users worldwide, in fields ranging from accelerator, design, shielding, and operation, to medical applications, cosmic ray physics, radiation dosimetry, etc.
At high energies, deuteron reactions in FLUKA are treated via its external nucleus-nucleus event generators: rQMD between 150 MeV/n and 5 GeV/n [4,5] and DPMJET at higher energies [6,7]). For deuteron energies below 150 MeV/n, instead, only a proxy model was available upon user request prior to FLUKA v4-2.0, whereby the deuteron split into a neutron and a proton, with roughly equal energy sharing and mostly forward emission. This e↵ective model, while still subject to the correct reaction rate as per FLUKA's low-energy reaction cross section parametrization for ions in matter (based on [8]), did not capture the rich set of energy and angular spectral features exhibited by nuclear fragments emitted during low-energy deuteron reactions. Consequently, the use of FLUKA in applications driven by deuteron beam (or secondary deuteron) interactions was not recommended. The latter have however received increased attention recently, ⇤ e-mail: francesc.salvat.pujol@cern.ch not only as neutron sources [9], but also in the production of conventional and unconventional radioisotopes for medical applications [10,11], as well as in fusion studies [12], to name a few.
In order to extend the applicability of FLUKA also to these technical problems, a dedicated development effort has been undertaken, overcoming the aforementioned tentative interaction model. Particular attention has been devoted to capturing the essential reaction mechanisms for low-energy deuterons, discussed below in more detail: elastic break-up (the deuteron splits into a neutron and a proton without exciting the recoiling target nucleus), stripping to discrete levels (a nucleon-transfer reaction to a bound state of the compound nucleus, giving rise to a series of sharp peaks in the energy spectrum of the emitted nucleon), and stripping to the continuum (one nucleon is emitted via the Serber model [13] and the other interacts with the target nucleus via FLUKA's hadron-nucleus nuclear inelastic interaction model, PEANUT [14]). The contribution of these three reaction mechanisms to the energy spectrum of neutrons emitted from 7 Li under 40-MeV deuteron bombardment is discussed below. For deuterium and tritium targets, evaluated nuclear data have been employed [15]. Finally, a passage through the completefusion model in FLUKA's low-energy ion-ion nuclear inelastic interaction model facilitates the emission of heavy fragments from the target nucleus.
A systematic benchmarking e↵ort has been undertaken, assessing the performances of this novel model and revealing a very reasonable agreement with available experimental data at the di↵erential cross section level. The novel low-energy-deuteron nuclear reaction model discussed here was included in FLUKA v4-2.0.

Nucleon stripping to the continuum
The reaction channel responsible for the bulk of the energy spectrum of emitted neutrons and protons from target nuclei under low-energy-deuteron impact is stripping to the continuum, a two-step process whereby one nucleon is emitted in the Coulomb and nuclear potential of the target nucleus and, in a second step, the conjugate nucleon penetrates the target nucleus and initiates a nuclear inelastic interaction.
The emission of the first nucleon (the neutron or the proton with equal likelihood) has been modelled relying on the Serber model [13], a semiclassical model which provides an e↵ective energy and angular distribution, sampled here in an uncorrelated way. Once a kinematically allowed four-momentum of the emitted (spectator) nucleon is sampled, the kinematics of the rest of the system can be closed, and the conjugate (participant) nucleon undergoes a nuclear inelastic interaction with the recoiling target nucleus via FLUKA's PEANUT model [14]. Depending on the sampled four-momentum of the spectator nucleon, there may not be enough available invariant mass to form the participant nucleon in addition to the recoiling target nucleus: if instead the direct formation of a compound nucleus is kinematically allowed, it is performed, and the recoiling excited compound nucleus passes through the evaporation and gamma de-excitation stages. If the kinematics for the conjugate system cannot be closed, the Serber sampling is rejected and attempted again until a kinematically allowed scenario is found.
The blue dashed curve in Figure 1 displays the contribution of the stripping-to-the-continuum mechanism to the energy spectrum of neutrons emitted at 30 from 7 Li under 40-MeV deuteron bombardment. The broad feature centered at around 15 MeV corresponds to neutrons emitted directly via the Serber model, while the low-energy peak corresponds to neutrons emitted from the nucleus as the conjugate nucleon interacts with it via FLUKA's hadronnucleus nuclear inelastic interaction model [14].

Nucleon transfer to discrete levels
The incoming deuteron may instead be stripped by the direct transfer of one of its nucleons to a bound state of the recoiling compound nucleus, left in either the ground state or one of a few low-lying excited levels. Should the compound nucleus be left in an excited level, the subsequent prompt relaxation to the ground state is performed via FLUKA's gamma de-excitation module. The angular distribution of the emitted nucleon can be obtained within the distorted-wave Born approximation [16], relying on the optical potential model (OPM) of An and Cai for deuterons on target nuclei [17], and on the Koning and Delaroche OPM for neutrons and protons on intermediate and heavy target nuclei [18], while using the OPM of Watson et al. for nucleons on light target nuclei [19].
A numerical database of angular distributions has been included to account for stripping to discrete levels for the explicit channels listed in Tab. 1, which covers most cases of practical relevance. For heavier target nuclei, the importance of nucleon transfer to discrete levels fades in comparison to stripping to the continuum, owing to their densely packed level structure. The brown dotted curve in Fig. 1 displays the contribution of nucleon transfer to discrete levels to the energy spectrum of neutrons from 7 Li(d,n) emitted at 30 , consisting in two peaks as a signature of the direct nucleon transfer process where the residual 8 Be is left in the ground state and first excited level, respectively. A possible 6 Li sample contamination may explain the experimental excess between 40 and 45 MeV.

Elastic break-up
The elastic break-up of the deuteron, whereby the deuteron splits into a neutron and a proton in the Coulomb and nuclear potential of the target nucleus (which remains in the ground state), has also been approached within the distorted-wave Born approximation, employing the aforementioned OPMs. While for stripping to discrete levels it has been necessary to fit OPM parameters [20] to available experimental data, for elastic break-up the calculation relies on standard OPM parametrizations without additional fits.
The cross section for this process is di↵erential with respect to the energy and polar angle of the emitted neutron, and the polar and azimuthal angles of the emitted proton [21][22][23]. An e↵ective database of di↵erential cross sections has been set up for sampling the energy and polar scattering angle of the emitted neutron. The polar (azimuthal) scattering angle of the proton is sampled in an inclusive way from an e↵ective angular distribution.
The dark green dot-dashed curve in Fig. 1 displays the contribution of deuteron elastic break-up to the energy spectrum of neutrons emitted in the reaction 7 Li(d,n) at 30 . It provides a central peak similar to that from stripping to the continuum which, although it has a di↵erent angular distribution, is essential to ensure a final good agreement with experimental data in the central neutron energy peak, mostly at forward angles.

Benchmarking
Experimental energy spectra for nuclear fragments (n,p,t,d, 3 He,↵) emitted from a series of target isotopes ( 2 H to 238 U) under bombardment with deuterons at various energies (from Coulomb barrier up to 100 MeV) have been systematically extracted from EXFOR [24], leading to over 600 individual test cases, against which the model outlined above has been benchmarked. Overall, very reasonable agreement has been obtained for the (d,n) channel, as displayed by the energy spectra of neutrons in Fig.  2, emitted at the indicated angles from 9 Be, 27 Al, nat Cu, and 93 Nb under bombardment with 102-MeV deuterons. Note that the comparison is in absolute units of mb MeV 1 sr 1 . The dominant central feature in these spectra at fairly forward emission is well reproduced by the Serber model (complemented by the weaker though relevant contribution of elastic break-up), in conjunction with FLUKA's hadron-nucleus interaction model [14], which accounts in its evaporation stage for the emission of low-energy neutrons, giving rise to the low-energy rise. The discrete peaks, clearly visible in the case of 9 Be(d,n), are well reproduced by our distorted-wave model for nucleon transfer to discrete levels. A fixed Gaussian time-of-flight resolution function has been applied following the details quoted in the original experimental references in order to obtain the actual broadening of the latter features. Figure 3 displays the energy spectrum of protons emitted at the indicated angles from 12 C, 27 Al, 58 Ni, and 238 U, under bombardment with 99.6-MeV deuterons. The combination of a model for nucleon stripping to the continuum, elastic break-up, and nucleon transfer to discrete levels gives a very reasonable agreement also for the (d,p) channel. Note that, as opposed to the (d,n) spectra above, no smearing has been applied on the simulated spectrum for 12 C(d,p), leading to the sharper peaks shown in the figure. The second panel in Fig. 3 exhibits the acceptable agreement with experimental data far from forward emission, for protons emitted at 60 , while the lower two panels showcase the good agreement obtained also for heavier targets.
The lowest panel of Fig. 4 displays the energy spectrum of 238 U(d,p), with 99.6-MeV deuterons, for protons Red dots correspond to experimental data extracted from EXFOR [24], while blue lines correspond to 50,000 events sampled with FLUKA employing the model described in this paper (statistical error bars have ben omitted for clarity). A constant neutron timeof-flight resolution has been applied, following the original data references. emitted backwards at an angle of 120 , yielding an excellent agreement with experimental data also for backward emission. Finally, the remaining panels in the figure display the reasonable agreement obtained for less intense channels involving the emission of heavier fragments at various angles: 27 Al(d, 3 He) and 58 Ni(d,↵) for a deuteron kinetic energy of E d = 80 MeV, and 208 Pb(d,t) for E d = 70 MeV. The discrepancy in the top panel is attributed to a direct proton pick-up process, presently not accounted for.

Conclusions
A model for nuclear inelastic interactions of deuterons with energies below 150 MeV/n has been developed and included in FLUKA, overcoming a longstanding lack of this general-purpose code for the Monte Carlo simulation of radiation transport. To account for stripping to the continuum, our approach relies on the Serber model, coupled to FLUKA's hadron-nucleus interaction model [14]. Nucleon transfer to discrete levels and elastic break-up are modelled on the basis of the distorted-wave Born approximation. For deuteron reactions on deuterium and tritium, evaluated nuclear data are employed. A dedicated benchmarking e↵ort led to the validation of the model against experimental data at the double-di↵erential cross-section level. This new module has been included in FLUKA v4-2.0, allowing the users to apply the code for the first time to problems driven by low-energy deuteron beams (and/or reinteracting deuterons).   Al(d, 3 He) E d = 80 MeV ✓ n = 60 ± 2