Nuclear data generation & implementation for analog Monte Carlo simulation

. Nuclear data is in constant evolution as more experimental data is gathered, computational capabilities increase, and evaluators verify its validity by means of stochastic and deterministic simulations. The focus here is on the analog Monte Carlo simulation of nuclear reactions that produce more than two particles in the outgoing channel, which needs speciﬁc considerations to ensure the correlations between the particles and thus the conservation of energy and of translational and angular momenta. It is possible to adapt nuclear data and its exploitation to implement realistic reactions from the phenomenological point of view (as opposed to the historical need of variance reduction techniques), which increases computation time but allows the expansion of the transport codes capabilities. Simulation anomalies were found concerning the kinematical calculations of photon energies emitted from neutron-induced inelastic scattering (n,n’ γ ), as well as concerning the photon multiplicity of 155 Gd(n, γ ) due to the presence of a rotational band in 156 Gd. Recommendations are given for potential solutions for both anomalies.


Introduction
In the field of Monte Carlo simulation of nuclear reactions, in particular related to particle detector simulations, the reactions are simulated using the available nuclear data that describes the expected behavior of the reactions, and using a simulation algorithm capable of generating nuclear reaction events. This is done mainly by randomly sampling from the probability distributions that characterize the possible reactions channels, coupled to possibly reconsidering the simulation algorithm if the reaction requires further information that cannot be stored in a distribution.
In reactor physics applications, simulating reactions that produce more than two particles using Monte Carlo transport codes does not generally respect the correlations that result in the conservation of energy between the emitted particles from an individual reaction, as the main interest in reactor physics is the average behavior of macroscopic observables.
To be able to perform a so-called analog simulation, which consists in simulating all particles one by one (as opposed to non-analog which applies variance reduction techniques by adjusting particle weights), one must add extra features to the transport code to take into account the specific characteristics associated with nuclear reactions such as (n,n'γ) and (n,γ). Currently, not all nuclear evaluation files are updated to allow analog simulations, nor do all transport codes have event generation algorithms. As a result, individual multi-particle reactions are simulated with missing information and thus do not exactly reproduce the expected behaviors in the simulations. The

Neutron-induced inelastic scattering
Inelastic scattering is a particularly important reaction to take into account when modeling nuclear systems. The energy threshold of this reaction starts above a few keV for the heaviest nuclei and a few hundreds of keV for the lightest nuclei. These reactions are characterized by a transfer of the kinetic energy of the incident neutron into excitation energy of the target nucleus. The latter is then in an excited state. One can thus consider it is a two-body reaction on the outgoing channel: scattered neutron and excited nucleus. The knowledge of the kinematics of the scattered neutron in the center of mass (CM) reference frame is sufficient for the kinematical calculations for the recoil nucleus, the ensuing gamma cascade and the outgoing neutron in the laboratory (LAB) reference frame. In the first ENDF and JEFF libraries, the evaluations contained the angular distributions. However, they did not systematically include the decay data of the excited states by photon or electron emission. In the latest version JEFF-3.3, one can find 422 complete decay schemes out of 562 evaluation files.
The most complete description in terms of the ENDF-6 format is found in files of type MF=12, where each excited state declares the transition probabilities to the lower lying states. MF=12 can store up to 40 excited states, separated in sections MT=51 to MT=90, which are taken as the input data for creating the game of chance intended to reproduce a realistic gamma cascade that ends when the ground state is reached. This Monte Carlo implementation can be called analog or natural.
The non-analog description of this decay process is found in file MF=6, where a photon multiplicity (or weight) and transition probabilities are given for each excited state, so that only one photon is sampled from the declared transition probabilities. As only one photon is simulated with an adjusted weight, the computation time is reduced compared to analog simulation. Both analog and non-analog end results in terms of photon production spectrum should be the same on the average.
Because of data format limitations, simulations using MF=6 can perform kinematical corrections to the emitted photons, whereas with MF=12 photons are always simulated with the declared energy of the transition. Kinematical corrections become noticeable for the lighter nuclei, because they are more susceptible of gaining kinetic energy from the collision, which can boost the energy (measured in the LAB) of the emitted photons if they are emitted in the same orientation as the momentum of the recoil nucleus. This is called the Lorentz boost, which is analog to the Doppler effect.

Lorentz boost in 27 Al(n,n'γ)
The simulations implemented in the Monte Carlo code TRIPOLI-4 ® [1] consisted in one incident neutron of 5 MeV per batch in a 27 Al sphere, with the objective of assessing the impact of using either MF=6 or MF=12 as input data. As expected, when MF=12 is present in the evaluation file, the gamma cascade is simulated and more than one photon is detected per neutron history, whereas MF=6 simulations have only one photon. This can be verified by adding the simulated energies: in the case of MF=12, every calculated sum corresponds to an excited state of 27 Al.
However, when using MF=6 as input data, it was observed that the energy spread of the gamma energies due to the Lorentz boost did not correspond to the calculated values given by where E γc is the transition energy and θ c is the angle of emission of the photon in the center of mass of the (Al * ) system with momentum P Al * and total energy E Al * . When the calculations in Eq. 1 are done considering the (Al * + scattered neutron) center of mass system, then the values correspond to the simulation, which suggests that TRIPOLI-4 ® does not consider the right reference frame for its kinematical calculations.
The same analysis was performed using the Event Generator Mode (EGM) of the PHITS transport code [2] distributed by JAEA. EGM corresponds to what is called here analog simulation. The third state (MT=53) of 27 Al at 2.21 MeV was examined from the kinematical point of view, as it only decays to the ground state. Additionally, the 2.3 MeV transition from the eleventh state (MF=61) at 4.51 MeV to the third state was also examined. As shown on Figure 1, it was found that the spread in energy of transitions from level number 3→0 (in blue), as well as 11→3 followed by 3→0 (in red) correspond to the expected values calculated with Eq. 1 with a precision of 0.1 keV.   The number of evaluation files that contain MF=12 for the description of nuclear excited states increased from 239 in JEFF-3.2 to 422 in JEFF-3.3. Currently, there are still 68 evaluation files of the JEFF-3.3 library that contain MF=6 for the excited states, which means it is possible to implement a simulation analysis such as the one previously explained for the case of 27 Al. Additionally, it was found that 22 evaluation files of the JEFF-3.3 library contain an incomplete MF=12, in the sense that the number of declared excited states in MF=3 was higher than the number of declared excited states in MF=12. The missing sections of MF=12 were written in the ENDF-6 format by extracting the information from the RIPL-3 structure data, and sent to the JEFF community. Table 1 reports the added data.

Potential and effective contributions to the JEFF-3.3 library
Prior to the completion of the MF=12 that did not contain the same number of sections as in MF=3, the focus was on finding the evaluation files that only contain MF=6 (related to the excited states) in the JEFF-3.3 library and on creating MF=12 for them. As previously stated, there are 68 cases that can be worked on and it was possible to create 50 MF=12, as the information required is not available for all of them (structural and/or angular). Table 2 reports the created data. All the written data was tested by means of TRIPOLI-4 ® simulations that used either MF=6 or MF=12, where the photon production spectra resulted in the same shape for both MF=6 and MF=12 formats, and MF=12 simulations produced the expected gamma cascades.

Neutron capture reaction
The neutron capture reaction is a multi-stage process in which a number of photons are emitted from a nucleus that absorbs a neutron. Because the compound nucleus (post neutron capture) is in an excited state that reaches several MeV, its excited levels are considered in a quasicontinuum of states, and this effect starts to appear at lower energies for heavier nuclei. As photons are emitted from the quasi-continuum, the excited nucleus reaches lower energy states that are considered discrete if their separation is experimentally identifiable.
In terms of the ENDF-6 format, the energy and angle distributions that characterize the (n,γ) reaction can be found in the evaluation files in section MT=102 of file MF=6. The data is divided in two parts: when the declared energies are in decreasing order, each energy i represents a gamma transition from the well defined (or discrete) excited states, and have an associated probability p i . When the declared energies E j are in increasing order, the associated probability densities of emission P j are such that which is intended to reproduce the gamma emissions either from the quasi-continuum of states by sampling an energy in the interval [E j ;E j+1 ], or by emitting a gamma of energy i . Additionally, one can find in the Evaluated Nuclear Structure Data Files (ENSDF), in the E=TH dataset, all the gamma intensities adopted by the evaluators from experimental data, that are expected from (n,γ) events.

Simplified (N,G) Algorithm (SNGA)
In order to exploit all the existing nuclear data, and avoid analytical models such as the use of nuclear level densities and photon strength functions, we have implemented an algorithm named SNGA for Simplified (N,G) Algorithm, that simulates gamma emission from the quasi-continuum followed by the gamma cascade of discrete states, based on the Monte Carlo method. The SNGA is indeed more elaborate than what one typically finds in the reactor transport codes, but less elaborate than the DICEBOX code [3] distributed by the IAEA, which is the reference for our analysis.
The SNGA, which is based on the Monte Carlo method to generate (n,γ) analog events (i.e. all particles are simulated one by one), is made with the expectation that this method will reproduce the physical behavior of the neutron capture reaction. As it can be observed on Figure  2, the SNGA produces a continuous spectrum as well as discrete lines of emission. It relies on the structure data found in the RIPL-3 library, to produce realistic cascades from well defined excited states, working alongside the probability distribution defined in the MF=6, MT=102 in the JEFF-3.3 library for photons emitted from the quasicontinuum of states. Firstly, the DICEBOX analysis suggests that 1% of the neutron capture events bring the nucleus to its ground state, which corresponds to the rightmost peak on Figure 2 and is identified with the reaction Q-value. Then, according to the ENSDF 95 Mo(n,γ) E=TH dataset, 14% of events bring the nucleus to a known energy level, after which the decay scheme is followed. Finally, because in 85% of first gamma emission events the nucleus remains in its quasi-continuum of states, it was assumed that the conditional probability of remaining in the quasi-continuum is 85%, in which case the photons are sampled from the MF=6 MT=102 distribution with the additional constraint that the sampled photon carries less energy than the current available energy, decreased at each gamma emission. This condition can be put aside if the number of emitted gammas is the sampled multiplicity minus one, in which case the last photon carries all the available energy.
Further analysis of the reaction process simulated by the SNGA reveals that the individual behavior of the emitted photons is not exactly as in the DICEBOX simulation. This is due to the fact that the SNGA relies solely on the available data. As such, it requires constraints to ensure the conservation of total energy, for example the instruction for the last photon to carry all the remaining available energy if all gamma emissions of the capture event were sampled from the MF=6 MT=102 distribution. The con-sequence of such limitations are visible in the emission densities found in the outcome of the simplified simulation, shown on Figure 3, where the emission densities of the last photon are represented.  It was found from EGM PHITS simulations of 155 Gd(n,γ) that the resulting gamma multiplicity distribution has two maxima, suggesting that 10-photon cascade events are the second most frequent. The reason for this is that an excited state of 156 Gd * at 3.995 MeV with spinparity J π = 16 + is often reached which provokes a 9photon cascade. This state should not be reached as it lies on a rotational band gap, which consists of several excited states with angular momenta J>10. The transition is highly improbable as the intial state reached has J=2, and photons are unlikely to carry more than 3 units of angular momentum.
The SNGA was adapted to read the structure data stored in the PHITS source code, which contains all the relevant information found in ENSDF, including the spin of the excited states. Cascades simulations were implemented imposing |∆J| ≤ 3 for all transitions (called Spin considerations on Figure 4), where ∆J is the difference in angular momentum of the inital and final excited state. The control case consisted in simulating the decay scheme of ENSDF without considering the ∆J of transitions (called No spin considerations on Figure 4).
Although the two maxima do not appear in our simulations, the average number of photons is still too high, as can be seen from Table 3. The reasons for the anomaly found using PHITS are that the 3.995 MeV state is disproportionately reached and also because of how the data in the source of PHITS is constructed. Whenever ENSDF does not declare branching ratios, these are declared in PHITS as decays towards the closest state, which creates long cascades in the case of 156 Gd * . The full library of nuclides available to PHITS simulations was checked in case the rotational band situation is repeated, and it was found that 198 Hg also yields a double-hump gamma multiplicity distribution and that 199 Hg has a rotational band as well.
Source γ multiplicity at 10 −5 eV JEFF-3.  Table 3. Comparison of the average number of emitted photons in the 155 Gd(n,γ) reaction

Conclusions
The analog simulation of the gamma cascade of the neutron-induced inelastic scattering can be implemented when the evaluation file uses the data format of MF=12 for gamma production from the excited states. The data format defines the photon energies in the laboratory system, so the Lorentz boost is not calculated by TRIPOLI-4 ® . The analog simulation of gamma cascades from neutron capture is less flexible in terms of nuclear data compared to the inelastic reaction because the absorbing nucleus changes its nature, and because the high excitation energy reached introduces the need of modelling the quasi-continuum of states. The limitations imposed by the ENDF-6 data format are one motivation of the Nuclear Energy Agency for creating the Generalised Nuclear Database Structure. This more modern nuclear data format can bring solutions to the issues mentioned in this work.