Correlated prompt fission data in transport simulations

EPJ manuscript No. (will be inserted by the editor) Correlated Prompt Fission Data in Transport Simulations P. Talou1 , R. Vogt2,3 , J. Randrup4 , M.E. Rising5 , S.A. Pozzi6 , J. Verbeke2 , M.T. Andrews5 , S.D. Clarke6 , P. Jaffke1 , M. Jandel7,9 , T. Kawano1 , M.J. Marcath6 , K. Meierbachtol8 , L. Nakae2 , G. Rusev7 , A. Sood5 , I. Stetcu1 , and C. Walker7 1 2 3 arXiv:1710.00107v3 [nucl-th] 2 Feb 2018 4 5 6 7 8 9 Nuclear Physics Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Nuclear & Chemical Sciences Division, Lawrence Livermore National Laboratory, Livermore, CA 94551, USA Physics Department, University of California at Davis, Davis, CA 95616, USA Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Monte Carlo Methods, Codes, and Applications Group, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, USA Nuclear and Radiochemistry Group, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Nuclear Engineering and Nonproliferation, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Department of Physics and Applied Physics, University of Massachusetts Lowell, Lowell, MA 01854, USA Draft as of February 5, 2018 Abstract. Detailed information on the fission process can be inferred from the observation, modeling and theoretical understanding of prompt fission neutron and γ-ray observables. Beyond simple average quantities, the study of distributions and correlations in prompt data, e.g., multiplicity-dependent neutron and γ-ray spectra, angular distributions of the emitted particles, n-n, n-γ, and γ-γ correlations, can place stringent constraints on fission models and parameters that would otherwise be free to be tuned separately to represent individual fission observables. The FREYA and CGMF codes have been developed to follow the sequential emissions of prompt neutrons and γ rays from the initial excited fission fragments produced right after scission. Both codes implement Monte Carlo techniques to sample initial fission fragment configurations in mass, charge and kinetic energy and sample probabilities of neutron and γ emission at each stage of the decay. This approach naturally leads to using simple but powerful statistical techniques to infer distributions and correlations among many observables and model parameters. The comparison of model calculations with experimental data provides a rich arena for testing various nuclear physics models such as those related to the nuclear structure and level densities of neutron-rich nuclei, the γ-ray strength functions of dipole and quadrupole transitions, the mechanism for dividing the excitation energy between the two nascent fragments near scission, and the mechanisms behind the production of angular momentum in the fragments, etc. Beyond the obvious interest from a fundamental physics point of view, such studies are also important for addressing data needs in various nuclear applications. The inclusion of the FREYA and CGMF codes into the MCNP6.2 and MCNPX-PoliMi transport codes, for instance, provides a new and powerful tool to simulate correlated fission events in neutron transport calculations important in nonproliferation, safeguards, nuclear energy, and defense programs. This review provides an overview of the topic, starting from theoretical considerations of the fission process, with a focus on correlated signatures. It then explores the status of experimental correlated fission data and current efforts to address some of the known shortcomings. Numerical simulations employing the FREYA and CGMF codes are compared to experimental data for a wide range of correlated fission quantities. The inclusion of those codes into the MCNP6.2 and MCNPXPoliMi transport codes is described and discussed in the context of relevant applications. The accuracy of the model predictions and their sensitivity to model assumptions and input parameters are discussed. Finally, a series of important experimental and theoretical questions that remain unanswered are presented, suggesting a renewed effort to address these shortcomings. PACS. 25.85.Ec, 24.10.Pa Nuclear fission; Monte Carlo transport simulations; MCNP; MCNPX-PoliMi; FREYA; CGMF 1 Introduction The nuclear fission process, known for over 75 years now, is at the core of many nuclear technologies and scientific studies in fields such as energy, defense, and astrophysics. Conceptually, it can be seen as a complex collective rearrangement of the nuclear many-body system. From our early qualitative description of this process in terms of the deformation of a charged liquid drop [1] to today’s quan- 2 P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations titative calculations based on macroscopic-microscopic [2, 3] or purely microscopic [4, 5, 6] descriptions, an enormous amount of data has been collected and a number of theoretical models proposed to account for the wide variety of fission signatures such as fission cross sections, fission fragment yields, fission half-lives, fission isomers, prompt and β-delayed neutron and γ-ray emission, ternary fission and fission fragment angular distributions. In addition, a large collection of integral data pertaining to the use of nuclear technologies and, in particular, to nuclear energy and defense applications, has been collected since the dawn of the atomic age. The existence of such a wide-ranging set of nuclear fission data may give the impression that everything in nuclear data is now well known. However, that is far from true. As new regions of the periodic table are explored, whether to understand how, when, and where the elements in the universe were initially formed, to understand exotic nuclear structure configurations away from the valley of stability, or to develop innovative nuclear technologies with distinct fuel and material compositions, more diverse and more accurate nuclear data as well as refined models are required to fill in gaps in data beyond the reach of even modern experimental techniques. In the specific case of nuclear fission, many fundamental questions remain. At the same time, modern applications require very high accuracy in quantities such as cross sections, angular distributions, and spectra. Studying correlated signatures of the fission process can help shed some light on both domains of interest. The various characteristics of a fission event are naturally correlated. However, such correlations are generally absent in the evaluated nuclear databases used by modern transport codes. Those correlations range from fission cross sections with fission fragment angular distributions, fission fragment yields with prompt fission neutrons and γ rays as well as correlations in the number, energy and angle of emission of neutrons and γ rays. In scenarios where average quantities dominate, such as the multiplication factor in a critical assembly, correlations are expected to play only a minor role. In other applications, like neutron multiplicity counting [7], however, the situation is quite different and great care must be taken to describe correlations and distributions adequately, such as the higher moments of the prompt neutron multiplicity distribution P (ν). In recent years, several parallel efforts to model the fission process on an event-by-event basis have led to the development of computer codes [8, 9, 10, 11] that can calculate many of these correlations. Integrating these codes into a transport simulation code like MCNP R 1 [12] repre- sents a major breakthrough for the accurate simulation of fission events in transport calculations. In this review, the various correlations that develop naturally in a fission event are described (Section 2) before discussing (Section 3.1) the CGMF and FREYA codes that simulate such events in detail. In Section 3.2, the MCNP6.2 code [12] is then briefly introduced and a more in-depth discussion of the fission models present in MCNP6.2 is given. The MCNPX-PoliMi code [13], developed at the University of Michigan, is an extension of MCNPX2.7. While not a standalone code, it has been at the forefront of the modeling of correlated fission data, primarily for detector development, safeguards and nonproliferation applications. The fission-specific developments made in MCNPX-PoliMi are also reviewed before discussing the integration of the fission event generators CGMF and FREYA into the new release of MCNP6.2. In Section 4, numerical results on correlated fission observables are compared to available experimental data. Those data span correlations between emitted particles, nn, n-γ and γ-γ; correlations between emitted particles and fission fragments; and time correlations in fission chains. The time correlations differ from the rest as they are not related to a single fission event but instead to a suite of fission events characteristic of multiplying objects. Because of the importance of those correlations in safeguards and nonproliferation applications, they are included here although they are not intrinsically within the scope of the event-by-event codes. Section 5 presents the status of the fission event generators discussed here, lists the fission reactions currently supported, and provides initial estimates of the sensitivity of the results to model input parameters and physics assumptions. A suite of new experimental and theoretical developments that are needed in order to improve the reliability and predictability of the fission event generators are also proposed. Finally, a broad summary is provided in Section 6. 2 The nuclear fission process This section introduces some of the basic concepts of fission physics that are most relevant to correlation studies. It begins with a brief description of fission theory and phenomenology. It then continues with a discussion of relevant fission observables with an emphasis on multiplicities, spectra and correlations. Finally, it concludes with an introduction to some of the experiments measuring correlations in fission, emphasizing those being carried out by some of the authors of this work. 1 MCNP R and Monte Carlo N-Particle R are registered trademarks owned by Los Alamos National Security, LLC, manager and operator of Los Alamos National Laboratory. Any third party use of such registered marks should be properly attributed to Los Alamos National Security, LLC, including the use of the designation as appropriate. For the purposes of visual clarity, the registered trademark symbol is assumed for all references to MCNP within the remainder of this paper. 2.1 Theoretical insights The fission of a heavy nucleus is generally described as a complex collective rearrangement of nuclear matter in which collective and single-particle effects play important roles. The traditional picture of the liquid-drop model, P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations proposed by Bohr and Wheeler in 1939 [1], provides a relatively simple basis for a qualitative understanding of many features of the fission process. However, only a more fundamental quantum description can explain certain wellknown fission observables, such as fission isomers, as well as provide more quantitative results. Fission occurs because the repulsive Coulomb force acting between the protons overwhelms the attractive nuclear force responsible for the nuclear binding. In its simplest representation, describing the minimum potential energy of the nucleus as a function of a single deformation parameter (e.g., quadrupole deformation) reveals a “fission barrier” that the nucleus has to overcome to undergo fission. In reality, the situation is much more complicated: the fission path is not restricted to a one-dimensional landscape. The fission barrier is often double-humped due to shell corrections leading, in particular, to the existence of fission isomers: the nucleus can emit an alpha particle or even heavier clusters during the fission process. In addition, nuclear dynamics along the fission path can strongly alter this much simpler static picture. In the simplest description of low-energy fission of actinides, the heavy nucleus breaks apart into two smaller fragments of unequal mass at the scission point. The preneutron emission heavy fission fragment yield is strongly peaked near AH ∼ 140. The corresponding peak in the light fragment yield is AL ∼ A0 −AH where A0 is the mass of the fissioning nucleus. There is a dip in the yield near symmetry, ∼ A0 /2. With increasing excitation energy, the asymmetric fragment mass yield distribution grows more and more symmetric, filling in the dip near symmetry. Fission fragments are characterized by a mass A, a charge Z, an excitation energy U , an angular momentum J, and a parity π. For each pair of complementary fragments produced in a fission event, the total excitation energy available to the fragments is a function of the Q value of the fission reaction and of the total energy carried away as kinetic energy. To a very good approximation, the two complementary fragments are emitted back-to-back and have opposite momentum vectors in the center-of-mass frame. However, except in the simplest case, spontaneous fission, the angular distribution of the fission fragments in the laboratory frame is not isotropic. Following Bohr’s interpretation [14] of early experimental fission fragment angular distributions observed in photofission reactions, the nucleus populates only a few well-defined fission transition states on top of the outer saddle barrier, defined by their quantum numbers (J, K, M ) where K is the projection of the angular momentum J on the fission axis, and M its projection on the beam axis. Most low-energy nuclear reaction codes such as CoH [15], used in modern nuclear data evaluations, rely on this representation of the fission barrier and transition states. Angular distributions of the fission fragments strongly depend on the specific sequence and density of these transition states. The primary fission fragments are significantly excited and quickly emit neutrons and γ rays to reach either their ground state or a long-lived isomeric state. Eventually, 3 these secondary fission fragments may further β decay, leading to fission products, which can themselves emit delayed neutrons and photons until they reach a stable configuration. This work is only concerned with prompt emissions. The definition of prompt is somewhat arbitrary since the time associated with the β-decay is unique to each secondary fission fragment species. In general, however, any particle emitted within a few hundred nanoseconds of fission, the time scale for late-time γ transitions, would be categorized as prompt. It is important to note that different experiments record prompt events differently because the detectors have different characteristics. Therefore, any comparison between experiment and theory has to take these into account. Most modern calculations, including the ones presented here, assume that all prompt neutrons are emitted from the fully-accelerated fragments. However, one cannot rule out the possibility that a small fraction of prompt neutrons is also emitted during the descent from the saddle to the scission point, or even dynamically right at scission, similar to the ternary fission process of α-particle emission. The search for such “scission” neutrons has been ongoing for a long time, but still with rather inconclusive results due to the difficulty of finding a unique signature. The average multiplicity of prompt neutrons, ν̄, depends on the average total excitation energy available in the fragments. Prompt γ rays are mostly emitted after the neutron evaporation cascade ceases. The average characteristics of prompt neutron emission are relatively well known, at least for a few selected and important spontaneous and neutron-induced fission reactions on key actinides. However, detailed correlated information is still lacking even for well-studied cases. In addition, few predictive capabilities are available aside from the average prompt fission neutron spectrum (PFNS), or more generally, the chi-matrix, relating the incident and outgoing neutron energies, hχ(E 0 , E)i. Even in this case, the accuracy of the predictions strongly depends on the availability of experimental data for neighboring fissile isotopes and/or energies. For a more comprehensive discussion of the physics and data on the PFNS of actinides, see the recent International Atomic Energy Agency Coordinated Research Project, “Prompt Fission Neutron Spectra of Actinides” [16]. Note that little to no discussion of correlated fission data can be found in this report, although some discussion of FREYA, CGMF and other Monte Carlo codes can be found in Ref. [16]. 2.2 Distributions and correlations in nuclear fission events Correlations arise naturally in the nuclear fission process. For instance, the angular distribution of the fission fragments has long been interpreted [14] as a signature of the presence of fission transition states at the top of the outer barrier. In turn, those transition states play a key role in the calculation of near-barrier fission cross sections. Specific fragmentations are also related to the particular fis- 4 P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations Average prompt neutron multiplicity (n/f) sion channels or modes that the nucleus goes through on its way to scission [17, 18]. Those particular fragmentations and the resulting structure and shape of the initial fission fragments strongly influence the number and energy of the prompt neutrons and γ rays that are subsequently emitted. This review is limited to the study of correlations among the prompt fission neutrons and γ rays, as well as their characteristics in specific fragmentations. No further discussion will be made of β-delayed emission. Even this limited scope already constitutes a vast and rich topic. 3.5 3 2.5 252 Cf 235 (sf) U (nth,f) 239 Pu (nth,f) 2 1.5 1 0.5 0 70 Average neutron kinetic energy (MeV) (a) 80 90 100 110 120 130 140 150 160 170 Fission fragment mass number 2.5 252 Cf 235 (sf) U (nth,f) 239 Pu (nth,f) 2 (b) 1.5 1 70 80 90 100 110 120 130 140 150 160 170 Fission fragment mass number Fig. 1. (Color online) (a) Average prompt fission neutron multiplicity data [19, 20, 21] as a function of the fission fragment mass, ν̄(A), for several actinides. (b) Data on the average prompt fission neutron kinetic energy in the center of mass frame as a function of fission fragment mass, hcm i(A) [22, 23, 24]. 2.2.1 Fragment-dependent characteristics The characteristics of prompt neutron and γ-ray emission depends strongly on the parent fission fragments. The measured average multiplicity of emitted neutrons is shown as a function of the pre-neutron emission fission fragment mass number A. It exhibits a well-known “sawtooth” shape for all actinides as shown in Fig. 1 (a). The average neutron multiplicity is foremost a reflection of how much excitation energy is present in the fission fragment from which the neutrons are evaporated. The gross structure observed in ν(A) reflects the interplay of the deformation energy and shell structure in the configurations of the fragments near scission. Very compact shapes, as predicted near AH ∼ 130, in the region of proton and neutron shell closures, would have little to no deformation and therefore very little extra energy for subsequent neutron emission. In addition, strong repulsive Coulomb forces will result in high kinetic energies. The complementary light fragments, near mass 122 in the case of 252 Cf(sf), will be strongly deformed further from shell closures. The difference between the light fragment shapes near scission and their ground-state configurations will result in higher excitation energies and thus higher neutron multiplicities at AL ∼ 122. The average kinetic energy of the neutrons emitted from each fragment depends on the nuclear structure and on the nuclear level density of the daughter fragment. Figure 1(b) shows the average prompt fission neutron kinetic energy in the center-of-mass of the fragments as a function of fragment mass. Note that since the neutron kinetic energy is measured in the lab frame, some modeling is required to obtain the result shown in Fig. 1(b). The average neutron kinetic energy, integrated over all fragments, is the first moment of the average prompt fission neutron spectrum. Thus, the hardness of the neutron spectrum depends significantly on the specific fragments that emitted the neutrons. For most nuclear applications, such details do not matter and only average quantities are relevant. In particular, the average prompt fission neutron spectrum and neutron multiplicity, ν are two observables that can be evaluated using simplified models [28] which do not require a detailed description of the sequential neutron evaporation process. However, observables such as the PFNS and neutron multiplicity distribution, P (ν), place important constraints on models that attempt to correctly describe neutron emission. Such constraints are particularly important when using those models in cases where experimental data are missing. The PFNS and P (ν), shown as a function of ν − ν to give the distributions a common center, are presented in Fig. 2 for the same isotopes as in Fig. 1. Prompt fission γ rays also depend on the parent fragment. The average prompt fission γ-ray spectrum (PFGS) is dominated by statistical γ rays with outgoing energies greater than 1 MeV. Significant structure appears mostly below 1 MeV [32, 29], reflecting specific γ transitions between low-lying excited states, as seen in Fig. 3. Experimental measurements of the average PFGS for 252 Cf(sf), 235 U(nth ,f), and 239 Pu(nth ,f) reactions are shown. The presence and intensity of each low-lying γ line depends mostly on the fission fragment yields resulting from these reactions. 1 0.1 0.01 252 Cf 235 (sf) U (nth,f) 239 Pu (nth,f) (a) 100 10−1 10 −2 252 Cf 235 (sf) U (nth,f) 239 Pu (nth,f) 10−3 0.001 0 2 4 6 8 1 10 Outgoing Neutron Energy (MeV) 0.4 2 3 4 5 Prompt γ−ray energy (MeV) 6 7 16 252 Cf 235 (sf) U (nth,f) 239 Pu (nth,f) (b) 14 γ−ray spectrum (γ/f/MeV) 0.3 P(ν) 5 101 (a) γ−ray spectrum (γ/f/MeV) Prompt Neutron Spectrum (n/f/MeV) P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations 0.2 0.1 (b) 12 10 252 Cf 235 (sf) U (nth,f) 239 Pu (nth,f) 8 6 4 2 0 0 −4 −3 −2 −1 0 1 2 3 4 0.2 Fig. 2. (Color online) (a) Prompt fission neutron spectrum for several actinides, taken from the ENDF/B-VII.1 library [25]. (b) Neutron multiplicity distributions for several actinides, both measured and evaluated. Each distribution is centered around the average neutron multiplicity. The data are from Santi and Miller [26] for 252 Cf(sf), and Holden and Zucker [27] for 235 U and 239 Pu (nth ,f). Figure 4 shows the average γ-ray multiplicity, N γ , Fig. 4(a), and energy per γ, hγ i, Fig. 4(b), as a function of the fission fragment mass. A similar sawtooth behavior can be seen for γ-ray multiplicity than for the prompt neutron multiplicity (compare Fig. 1(a) and Fig. 4(a)). A lower nuclear level density for fragments produced near the double shell closure at AH = 132 can explain the clear increase of hγ i(A) in this mass region, different from the dependence of the average neutron kinetic energy in Fig. 1(b) which shows no clear structure for this mass. 2.2.2 Multiplicity distributions The average prompt neutron multiplicity, ν̄, is a very important quantity for the accurate simulation of many nuclear applications. Evaluated nuclear data libraries rely almost exclusively on experimental data for this quantity. The so-called “standard” evaluations [34] rely entirely on experimental data and provide evaluated ν for selected 0.4 0.6 0.8 1 1.2 1.4 Prompt γ−ray energy (MeV) − (n/f) ν−ν Fig. 3. (Color online) Average prompt fission γ-ray spectra for 252 Cf(sf) and thermal neutron-induced fission of 239 Pu and 235 U, as measured by Billnert et al. [29], Gatera et al. [30] and Oberstedt et al. [31], respectively. isotopes with very high accuracy. For instance, the current evaluated uncertainty on ν̄ for 252 Cf(sf) is 0.13%! The comparison between a recent evaluation of P (ν) for 252 Cf(sf) by Santi and Miller [26], the earlier evaluation by Holden and Zucker [35] and data [36, 37] is shown in Fig. 5(a). The evaluated distributions for spontaneous fission of many Pu, Cm and Cf isotopes [26] are shown in Fig. 5(b) as a function of ν − ν to facilitate comparison. To a good approximation, those distributions can be represented by a Gaussian of width σν = 1.20, the red line in Fig. 5(b). Very little is known about the incident energy dependence of P (ν) for fast neutron-induced fission reactions. Only one such measurement by Soleilhac et al. [38] is known. Based on the observation that all measured neutron multiplicity distributions for spontaneous and thermal neutron-induced fission reactions are reasonably Gaussianlike, Terrell inferred a formula for P (ν) [39] ν X 1 P (n) = √ 2π n=0 Z (ν−ν+1/2+b)/σν −∞ exp(−t2 /2)dt, (1) P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations very well known for 252 Cf(sf). Unfortunately very little is known about the incident-energy dependence of the factorial multiplicity moments for fast neutron-induced fission reactions. 8 Pleasonton, 1972 7 (a) 235 U (nth,f) 6 0.5 5 4 0.4 3 2 (a) 0.3 1 252 0.2 0 Average γ−ray energy (MeV) Spencer, 1982 Holden and Zucker, 1985 Vorobyev, 2004 Santi and Miller, 2008 P(ν) Average γ−ray multiplicity (γ/fission) 6 Pleasonton, 1972 2 (b) Cf (sf) 0.1 235 U (nth,f) 0 0 1.5 1 2 3 4 5 6 7 8 9 ν (n/f) 0.4 1 exp. data σν=1.20 σν=1.08 0.3 (b) 80 90 100 110 120 130 140 Fission fragment mass (amu) 150 160 Fig. 4. (Color online) Average prompt fission γ-ray multiplicity (a) and average energy per emitted γ-ray (b) as a function of the fission fragment mass. The experimental data are from Pleasonton et al. [33]. P(ν) 0.5 0.2 0.1 0 −5 where t = (Einc − E)/(σν E0 ), E is the average excitation energy, and E0 is the change in ν with Einc . Thus Eq. (1) is often used to compute P (ν) as a function of Einc . The parameter b for each value of Einc is determined from the condition that X νP (ν; Einc ) = ν(Einc ) . (2) ν The value of b was found to be small in all cases. Terrell used [39] a Gaussian of width σν = 1.08 in his analysis (the blue line in Fig. 5(b)). The factorial moments of P (ν) are defined as νn = X ν ν! P (ν) . (ν − n)! (3) The first three moments are then given by ν1 = ν = hνi , ν2 = hν(ν − 1)i , ν3 = hν(ν − 1)(ν − 2)i . (4) (5) (6) These moments must be known very precisely for applications involving neutron multiplicity counting and they are −4 −3 −2 −1 0 1 − (n/f) ν−ν 2 3 4 5 Fig. 5. (Color online) Experimental [26, 36, 35, 37] and evaluated prompt neutron multiplicity distribution (a) in the case of 252 Cf(sf), and experimental neutron multiplicity distributions (b), expressed as a function of ν − ν for spontaneous fission of 236,238,240,242 Pu, 242,244,246,248 Cm, and 246,250,252,254 Cf (see Refs. [26, 40] and references therein). (Note that P (ν) supposes integer values of ν.) The prompt γ-ray multiplicity distribution, P (Nγ ), can also be used in non-destructive assay methods that rely on correlated γ-ray fission data. A negative binomial distribution was shown [41] to agree fairly well with experimental data. However, as can be seen in Fig. 6, even for 252 Cf(sf), recent measurements [42, 43] disagree significantly with past results [41]. Note that the “experimental’ data by Oberstedt [44] reported here corresponds to the result of a fit using a negative binomial distribution, and cannot be considered raw experimental data. The comparison of experimental data with model calculations of P (Nγ ) is complicated by the use of a specific γ-ray detector energy threshold below which no γ rays are measured, as well as by the time coincidence window between P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations the emitted γ rays and the fission trigger and have to be considered when comparing to model calculations. Both quantities have a significant impact on the reported experimental distributions since they are in part responsible for the differences between the Oberstedt and Czyzh data in Fig. 6. Other differences in unfolding techniques can likely account for the rest. 0.15 (a) 7 will be focused in the same direction, 0 degrees, while two neutrons emitted from complementary fragments will be emitted with an angular separation of close to 180 degrees. The initial conditions of the fragments dictate both neutron and γ-ray emission, thereby inducing natural correlations. Experimental data on 252 Cf(sf) from Nifenecker et al. [46] seem to indicate a positive correlation between the total γ-ray energy released and the number of neutrons emitted, as shown in Fig. 7. Nifenecker inferred the following relation between the neutron multiplicity and γray energy for a given fragment: E γ (A, KE) = (0.75ν(A, KE) + 2) MeV, 0.1 where A and KE represent the mass and kinetic energy of the fission fragments respectively. The line in Fig. 7 is the total γ-ray energy from a pair of complementary tot fragments, E γ = (0.75ν + 4) MeV. Recently, Wang et al. [47] measured correlations between the neutron and γ-ray multiplicities, as a function of the mass and total kinetic energy of the fragments, again in 252 Cf(sf). Figure 8 shows the strong and complex correlations observed for different fission fragment mass regions, indicating a potentially much more complicated situation than suggested by Eq. (7). Probability Oberstedt, 2015 Chyzh, 2014 Chyzh, 2012 Valentine, 2001 0.05 252 Cf (sf) 0 10−1 (b) 252 Cf (sf) −2 9 10−3 Oberstedt, 2015 Chyzh, 2014 Chyzh, 2012 Valentine, 2001 10−4 10 −5 0 5 10 15 Number of γ rays 20 25 Fig. 6. (Color online) Prompt fission γ-ray multiplicity distribution for 252 Cf (sf), as represented by a negative binomial distribution in Valentine [41], and as measured recently by Oberstedt et al. [42] and Chyzh et al. [43]. The distributions are shown on a linear scale (a) and a log scale (b) to highlight the mean and tail of the distributions respectively. The Oberstedt data were obtained with a 6 ns time coincidence window and 100 keV γ energy threshold [42]. The 2014 Chyzh data, obtained with the DANCE detector, employed a 10 ns time coincidence window and a 150 keV γ-energy threshold [43]. The 2012 Chyzh data [45] is also shown here. 2.2.3 Neutron-neutron and neutron-γ correlations Strong correlations are expected in prompt neutron and γray emission, due in part to the kinematic boost imparted to neutrons emitted from the same or complementary fragments. Two neutrons emitted from the same fragments Average total γ−ray energy (MeV) Probability 10 (7) 8 Nifenecker, 1972 −+4) MeV (0.75ν 7 252 6 Cf (sf) 5 2 3 4 5 Neutron multiplicity (n/f) 6 Fig. 7. (Color online) The average total prompt γ-ray energy is plotted as a function of the prompt neutron multiplicity measured by Nifenecker et al. [46]. Experimental points were digitized from Fig. 7 in Ref. [46]. For neutron-induced fission reactions, the (n, γf) process, first predicted theoretically in Refs. [48, 49], has been used to interpret variations of the average neutron multiplicity in the 235 U(n,f) and 239 Pu(n,f) reactions below ∼ 100 eV. This process leads to anti-correlations between ν̄ and N̄γ , since pre-fission γ rays increase N̄γ at the expense of the residual excitation energy available in the fragments for the emission of prompt neutrons. Limited data exist, as reported by Shcherbakov [50] (and references therein). An alternative explanation for the fluctuations 8 P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations 1.2 252 Cf(sf) Counts (arbitrary units) γ−ray multiplicity (γ/f) 5 4 3 2 Wang, 2016 85<A<123 124<A<131 132<A<167 Bowman, 1962 Skarsvag, 1963 1 0.8 0.6 252 0.2 0 1 0 0 0.5 1 1.5 2 2.5 3 Cf (sf) 0.4 3.5 Neutron multiplicity (n/f) Fig. 8. (Color online) The γ multiplicity is shown as a function of the neutron multiplicity for different fission fragment mass ranges, as measured by Wang et al. [47] in 252 Cf(sf). of ν̄ in the presence of resonances has been explored by Hambsch et al. [51, 52] as changes in the fission fragment yields in mass and kinetic energy which would influence the number of prompt neutrons emitted. An increase in kinetic energy would result in a smaller number of prompt neutrons but would not impact γ-ray emission, making this correlated measurement even more relevant. 20 40 60 80 100 120 140 160 180 θn−LF (degrees) Fig. 9. (Color online) Measured [53, 54] angular distribution of prompt fission neutrons with respect to the light fragment direction, in the case of 252 Cf(sf). The minimum neutron kinetic energy was 0.5 MeV. effect will also be small compared to the kinematic focusing. However, those contributions should not be neglected when trying to infer the contribution of scission neutrons by comparing the measured angular distributions to calculations. In the absence of a detector that can track the direction of the fission fragments, n-n angular correlations can also represent a signature of the fission process. Such cor2.2.4 Angular distributions relations will not exist in other neutron-induced reactions, Angular correlations between the fission fragments and such as (n, 2n), where the two neutrons would mostly be the emitted neutrons emerge naturally from the kinemat- emitted isotropically. In fission, because of the kinematic ics of the reaction. Assuming that neutrons are emitted focusing discussed above, the neutrons will follow the difrom fully accelerated fragments, the kinematic boost of rection of the fission axis. If the two neutrons are emitted the fragments from the center-of-mass to the laboratory from the same fragment, then their aperture will be very frame focuses the neutrons in the direction of the frag- small, ∼ 0 degrees. On the other hand, if the two neutrons ments. Therefore, it is expected, and observed, that neu- originate from each complementary fragment, then their trons are emitted preferentially near 0 and 180 degrees aperture will be close to 180 degrees. Here again, the disnear 0 and 180 relative to the direction of the light fragment. Figure 9 tribution of θn−n can be expected to peak 252 Cf(sf). illustrates this point in the case of 252 Cf(sf) where exper- degrees. This can be seen in Fig. 10 for imental data from Bowman [53] and Skarsvag [54] show A well-known feature of the fission process is the marked increased emissions at 0 and 180 degrees. In addition, the anisotropy of the fission fragment angular distribution in higher peak near 0 degrees indicatesthat more neutrons the laboratory frame. First observed in the photofission are emitted from the light fragment than from the heavy of thorium [59], this discovery was quickly interpreted by fragment in this particular reaction. A. Bohr [14] as the presence of discrete fission transition Another potential source of neutron emission anisotropy states on top of the fission barriers. Most recently, the is due to the rotation of the fragments. The average spin anisotropy coefficients have been measured for different of the initial fragments is often estimated to be ∼ 7h̄ − 9h̄, fission reactions and actinides at CERN [60], PNPI [61] and neutrons emitted from the fragments would tend to and LANL [62]. Prompt neutron polarization asymmealign in a plan perpendicular to the direction of the spin. tries were also measured recently at the HIγS facility in However, this effect is small compared to the kinematic the photofission of Th, U, Np, and Pu isotopes [63]. As focusing just discussed [55]. expected, the polarizations are strongly impacted by the Finally, neutrons emitted during the descent from sad- fission fragment angular distributions. Therefore, for fisdle to scission, at or near the neck rupture (scission), or sion reactions other than spontaneous fission or very lowduring the acceleration of the fragments could all con- energy fission reactions, the interpretation of any observed tribute to an increased anisotropy. It is generally believed n-n angular correlations should always be done by folding that more than 95% of the prompt neutrons are emit- the prompt neutron anisotropic emissions with the approted from the fully-accelerated fragments. Therefore this priate fission fragment angular distribution. P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations 9 Counts (arbitrary units) 2.5 252 Pringle, 1975 Gagarski, 2008 Pozzi, 2014 2 Cf (sf) 1.5 1 0 20 40 60 80 100 120 140 160 180 θn−n (degrees) Fig. 10. (Color online) Measured [56, 57, 58] neutron-neutron separation angle in the case of 252 Cf(sf). The neutron detection threshold for Gagarski [57] and Pozzi [58] data is 0.425 MeV, and 0.7 MeV for Pringle [56]. Fig. 11. (Color online) The low-energy Chi-Nu array consists of 22 6 Li glass detectors to measure the PFNS down to ∼ 10 keV. There is evidence [64] that prompt γ rays also exhibit anisotropic emission from rotating fission fragments. Such data can provide some information about the angular momentum vectors in the fragments and the multipolarity of γ emission. It can also help identify rotational and vibrational levels from stretched γ rays for specific fission fragments. 2.2.5 Time correlations Time correlations, in our case, are understood to be correlations between the arrival times of prompt neutrons in a fission chain, i.e., neutrons from different fission events, at a detector. Hence, this type of correlations is not intrinsic to a particular fission event but rather a property of multiplying fission objects. Because of its importance in nuclear assay applications, this topic is discussed at greater length in Sec. 4.8. 2.3 Measuring correlations in fission observables Experimental studies of the nuclear fission process have been rich and numerous since its discovery. Fission cross sections, average prompt fission neutron spectra, multiplicities and, to some extent, fission yields have been the focus of most efforts, typically driven by applications in nuclear energy and defense programs. However, experimental data on correlations between prompt emission and parent fission fragments are rather limited, and do not provide sufficient constraints on the input parameters of modern fission models and codes such as the ones presented here. In this section, our own efforts to measure these correlations are discussed. Other recent and future experiments that can nicely complement these efforts are also mentioned. Fig. 12. (Color online) The high-energy Chi-Nu array consists of 54 EJ-309 liquid scintillators designed to measure the PFNS up to ∼ 15 MeV with adequate statistics. The Chi-Nu arrays (Figs. 11 and 12) have been developed at Los Alamos National Laboratory (LANL) and are deployed at the Los Alamos Neutron Science Center (LANSCE) to measure the average PFNS of several actinides with great accuracy. In particular, 235 U(n,f) and 239 Pu(n,f) are studied over a broad range of incident neutron energies. Most past PFNS measurements have acquired data in the 700 keV to 7−8 MeV range of outgoing neutron energy. A significant number of neutrons are emitted below 700 keV but multiple scattering corrections, neutron background, and low-sensitivity of liquid neutron scintillators have prevented accurate measurements of the PFNS in this region. At the highest energies, statistics become poor and long acquisition times are necessary for an adequate measurement. In addition, most measurements have been performed for spontaneous and thermal neutron-induced fission reactions only. The Chi-Nu project aims to accurately measure the PFNS of neutroninduced fission of 235 U and 239 Pu for incident neutron energies from thermal up to ∼ 200 MeV and for outgoing neutron energies from 10 keV up to 15 MeV. 10 P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations Two Chi-Nu arrays have been built. The first one, Fig. 11, consists of 22 6 Li glass detectors designed to measure the low-energy part of the spectrum, down to ∼ 10 keV. The second one, Fig. 12, consists of 54 EJ-309 liquid scintillators that can be used to extend the spectral measurement up to ∼ 15 MeV with sufficient statistics. Thanks to their segmented nature, the Chi-Nu arrays can also be used to study n-n correlations as well as neutron energy and angular correlations. Using pulse-shape discrimination, they can also be used to study prompt fission γ rays. The analysis of a large amount of data already collected is now being performed specifically with correlations in mind. While the Chi-Nu arrays were not designed to extract correlated prompt data, the University of Michigan developed specific experiments [58] to measure those correlations. One of those experimental setups is shown in Fig. 13 and consists of 24 EJ-309 and 8 NaI(Tl) scintillators, arranged in two rings surrounding a centrally-located 252 Cf source. A somewhat different setup was used to measure correlations in the spontaneous fission of 240 Pu [65]. In that experiment, a ∼ 2 g plutonium sample was placed at the center of the detector assembly and neutron doubles were acquired within a 100 ns time window. To our knowledge, this was the first measurement of neutron-neutron correlations and neutron doubles for this reaction. Fig. 14. (Color online) The DANCE detector array is a 4π calorimeter made of 160 BaF2 crystals developed to measure capture cross sections with very small samples or/and very radioactive targets. surements of prompt fission neutrons and γ rays [68]. In its present configuration, NEUANCE consists of 21 23 mm × 23 mm × 100 mm stilbene crystals arranged cylindrically around the beam line with the target at the center inside the DANCE array. A picture of NEUANCE inside one of the hemispheres of DANCE is shown in Fig. 15. Fig. 13. (Color online) The array of EJ-309 and NaI scintillators used to measure neutron-neutron correlations in 252 Cf(sf) at the University of Michigan [65]. The DANCE setup (Detector for Advanced Neutron Capture Experiments), installed at the Lujan Center at LANSCE consists of 160 BaF2 crystals arranged in a 4π geometry, as shown in Fig. 14. Originally designed to measure capture cross sections on very small target samples and/or very radioactive materials, it is a very high efficiency calorimeter that can be used to study the prompt fission γ-ray multiplicity and energy spectrum [66, 67]. To enhance the capabilities of DANCE, a new detector array, NEUANCE, was developed to make correlated mea- Fig. 15. (Color online) The NEUANCE detector array [68] consists of 21 stilbene crystals arranged in a compact form to fit around the beam line at the center of the DANCE cavity. The NEUANCE stilbene detectors have excellent pulseshape discrimination (PSD) properties, allowing discrimination between neutron and γ-ray signals. A PSD plot for one of the stilbene detectors is shown in Fig. 16. A 252 Cf source with activity of 739.3 fission events per second was used in a recent measurement. The spectral intensities of γ rays and neutrons are shown in Fig. 17 by black and red P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations lines, respectively, for just one of the 21 stilbene crystals of the NEUANCE array. The thick lines represent the rates measured with the NEUANCE detectors for 252 Cf source and thin dotted lines are the background rates. 11 eter of 60 cm. Thirteen identical scintillators compose the top of the detection system. The detector was designed for fast multiplicity counting and assaying of fissile material. The fast scintillator decay time of a few ns allows faster count rates than 3 He well counters. The relatively tightly-packed system has an overall geometric efficiency of 50% (2π). Measurements have been carried out with spontaneous fission sources of 252 Cf and 240 Pu placed at the center of the detection system. Some results are discussed in Sec. 4. Fig. 16. (Color online) A pulse-shape discrimination (PSD) plot measured [68] with a 252 Cf source with one of the NEUANCE stilbene detectors. The upper band (outlined in red) is a result of detected neutrons while the bottom one (blue) corresponds to γ-ray events. 100 252 Counts / 10 keVee 10−1 Cf (sf) Fig. 18. (Color online) Photograph of the 77 liquid scintillator array used at LLNL to measure n-n correlations in 252 Cf(sf) and 240 Pu(sf) [69, 70]. γ rays γ−ray background neutrons neutron background 10−2 10−3 10−4 10−5 0 1 2 3 Energy (MeVee) 4 5 Fig. 17. (Color online) Gamma-ray (red lines) and neutron (black lines) spectral intensity observed [68] in one detector of the 21 stilbene detectors of the NEUANCE array. The thick lines represent the rates observed from the 252 Cf source and the thin lines are the background rates. The bump at 3.5 MeV in the ambient γ-ray background results from the saturation of signals from high energy γ-rays or neutrons (cosmic rays). See Ref. [68] for details. A detector setup to measure neutron-neutron correlations was also developed at Lawrence Livermore National Laboratory (LLNL), as shown in Fig. 18. It consists of 77 EJ-301 liquid scintillators, each read out by a single photomultiplier tube. Each tower of 8 scintillators is symmetrically arranged into octants with an array inner diam- As a step toward a more portable neutron-gamma detector setup, a small (six liquid scintillators) and flexible experimental setup has been built at LANL. Relative detector angles and distances from the fission source are adjustable. Data acquisition software provides list-mode data collection. The flexibility of this setup is important to validate transport simulations in a wide range of configurations to study n-n angle, multiplicity and energy correlations. Of particular interest is the measurement and characterization, via accurate simulations, of cross talk between adjacent detectors and scatter from surrounding objects. By definition, cross talk occurs when a particle recorded in one of the detectors scatters and triggers an adjacent detector. Simulations of the detector response for neutrons are in progress, while γ-ray capabilities will be added in the near future. Other experimental setups have been devised to measure various correlated fission data. At the JRC in Geel, Belgium, the SCINTIA array [52] has been developed to measure neutron energy and multiplicity in coincidence with the fission fragment mass. When complemented by γ-ray detectors, it can provide very useful information on n-γ energy and multiplicity correlations as a function of resonances in the n+235 U cross section, and help infer the respective, possibly complementary roles of the (n, γf) process and fission fragment Y (A,TKE) yield fluctuations. 12 P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations benchmarks for prompt fission neutron emission angular distributions, although only as part of other contributing reaction channels such as elastic and inelastic scattering. 3 Modeling prompt fission emission Fig. 19. (Color online) The prototype for a versatile, flexible, and portable neutron detector array [71] has been developed and is being tested at LANL to study neutron-neutron correlations in various geometrical configurations. The SOFIA (Study on Fission with Aladin) experimental program [72, 73], carried out at GSI, measures fission yields in inverse kinematics for a broad range of fissioning nuclei with very accurate information on the mass and charge of the fragments. The average neutron multiplicity can be inferred, albeit not for monoenergetic reactions. Measurements of the prompt neutron multiplicity distribution typically involves the use of a Gd-loaded scintillator tank [74] in order to capture and thermalize all prompt neutrons with an almost 100% efficiency. Many experimental data sets are available for spontaneous fission and, to a lesser extent, for thermal neutron-induced fission reactions. The measurement of P (ν) for Einc larger than thermal is rendered much more difficult due to the important background from neutron scattering in the surrounding material. Only one such measurement has been reported for 235,238 U(n,f) and 239 Pu(n,f) up to Einc = 10 MeV [38]. As shown in Fig. 5(b), the prompt neutron multiplicity distribution for actinides can typically be represented by a Gaussian distribution. The average multiplicity is 24 neutrons, with non-negligible contributions from up to 6-7 neutrons, see Fig. 5(a). In a few cases, the average neutron spectrum is known with great accuracy. However, very little is known about the dependence of the neutron spectrum on the neutron multiplicity. Is the average spectrum for 6 neutrons emitted the same as when only 2 neutrons are emitted? A similar question can be asked for the prompt γ rays, whose multiplicity distribution spans an even larger range, with Nγ up to ∼ 20 (see Fig. 6). Measuring the angular distribution of prompt neutrons and γ rays emitted in a particular fission reaction also brings useful information. However, it can be somewhat more difficult to interpret as it represents a convolution of the angular distribution of the fission fragments with the angular distribution of the emitted particles with respect to the direction of emission of the fragments. Finally, interesting quasi-differential measurements [75] of neutron scattering off fissile material provide useful This section introduces the two complete event fission models, CGMF and FREYA, that have been incorporated into the MCNP6 transport code. First the physics encapsulated in these two codes is described, along with some discussion of their similarities and differences. Next, a general introduction to radiation transport codes is presented, highlighting the concept of incorporating complete event models and how this can enhance the simulation of fission in such codes. The section ends with a brief demonstration that incorporation into MCNP6 does not affect the CGMF and FREYA results. 3.1 Complete event fission models Although various physics models and codes have been developed and used to describe different aspects of prompt fission neutron and γ-ray emission in specific limited studies, the models implemented in transport simulations and used to evaluate nuclear data libraries have been mostly limited to average multiplicity and spectra. For instance, the Los Alamos model [28] has been and is still used [76] for nuclear data evaluations of the χ(E 0 , E) matrix of the PFNS as a function of incident neutron energy for most evaluated libraries including the U.S. ENDF/B-VII.1 library [25]. This model provides an average spectrum with few adjustable parameters that can be tuned to match existing PFNS data. The accuracy of this approach is strongly limited by the availability of experimental PFNS data for neighboring nuclei and energies. This model makes use of several important physical assumptions, some of which have been lifted in modern extensions of the original model and averages over only a few mass yields. No additional detailed information, such as the average neutron multiplicity as a function of fragment mass ν(A) or the neutron multiplicity distribution P (ν), can be extracted. (See Ref. [16] for more details about evaluations with the Los Alamos model and its extensions.) In recent years, several computer codes have been developed to simulate the sequence of prompt neutron and γ-ray emission in detail. Event-by-event simulators have been implemented in fast numerical codes that can be integrated into transport simulations of fissioning systems. Here two such codes, CGMF and FREYA, are presented in some detail. Note that there are also several similar codes developed independently: FIFRELIN [10] developed at the CEA in France, GEF [11] developed at GSI in Germany and CENBG in France, FINE developed by Kornilov [77], and more recently EVITA [78], based on the TALYS deterministic code, developed by CEA in France. Some limited code comparisons can be found in Ref. [16]. A separate model code developed by Lestone at LANL [79] has been used P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations 13 successfully to simulate the neutron-neutron and neutronfragment correlations for 252 Cf(sf) and neutron-induced fission of 235 U and 239 Pu. Although this code uses more available experimental data as input, and therefore is more limited in scope, it represents a very viable, fast and complementary alternative to the efforts discussed in the present paper. There can be significant differences in the physics implemented in those different codes, hence one can expect differences in the calculated results, especially for more differential observables. Here only the broad outlines of the CGMF and FREYA codes are presented. For more detail, see the publicly available user manuals in Refs. [8] and [9] respectively and references therein. 3.1.1 The CGMF code Fig. 20. (Color online) Schematic representation [8] of the The CGMF code, developed at Los Alamos National Lab- decay of the fission fragments by successive evaporation of oratory, is a Monte Carlo implementation of the statisti- prompt neutrons and γ rays. cal Hauser-Feshbach nuclear reaction theory [80] applied to the de-excitation of the primary fission fragments. At every stage of the decay (see Fig. 20), the code samples Y (A), hTKEi(A), σTKE (A), or even Y (A,TKE) are availprobability distributions for the emission of neutrons and able for some limited isotopes and energies. In the case γ rays. Each fission fragment is described as a compound where no experimental data exist, which is particularly nucleus with an initial excitation energy Ei∗ , spin Ji and true for higher incident neutron energies, then the simparity πi . Neutrons are emitted, removing their kinetic en- ple five-Gaussian prescription used in FREYA, discussed ergy from the fragment intrinsic excitation energy, while in the following section, is also used in CGMF. The massdoing little to change the angular momentum J. On the dependent charge distributions Y (Z|A) are taken from other hand, γ-ray emissions, generally after all neutrons Wahl’s systematics [82]. are evaporated, tend to decrease J. Several nuclear modOnce a light fission fragment (ZL , AL ) is chosen ranels as well as nuclear structure information are needed in domly, its complementary heavy partner is obtained by order to perform these calculations, as discussed below. mass and charge conservation such that ZH = Z0 − ZL Typical results of a CGMF run can be collected as a (long) and AH = A0 − AL , where (Z0 , A0 ) are the charge and series of data strings that represent each fission event. mass of the fissioning parent nucleus. At this point, CGMF The initial characteristics of the fission fragment in mass, treats binary fission only, no ternary or more exotic fission charge, kinetic energy, excitation energy, spin, parity, and events are considered. The total excitation energy (TXE) their momentum vectors in the laboratory frame, as well available for these fragments is given by the Q value for as the kinematic information on all emitted neutrons and that particular split (QLH ) minus the total kinetic enphotons in the laboratory frame are recorded. The statisti- ergy carried away by these fragments (TKE), TXE = cal analysis of those recorded events provides the needed QLH − TKE, where output that can be compared to experimental data. All QLH = E0∗ + Mn (Z0 , A0 )c2 (8) types of distributions and correlations in multiplicity, en2 2 ergy and angular distribution can be inferred from such −Mn (ZL , AL )c − Mn (ZH , AH )c history files in a rather straightforward manner. For a particular fission reaction, such as 239 Pu(n,f) and Mn (Z, A) is the nuclear mass. The excitation of the with Einc = 2 MeV, CGMF requires the fission fragment fissioning nucleus depends on how the fission was initiated: yields in mass, charge and total kinetic energy, Y (Z, A,TKE) it vanishes for spontaneous fission, E0∗ (sf) = 0; it is given produced in this reaction as input. Those yields are sam- by E0∗ (γ, f) = Eγ for photofission and for (low-energy) pled using Monte Carlo techniques to obtain the initial neutron-induced fission it is equal to E0 (n, f) = Einc + Sn fission fragments from which the sequence of neutron and where Sn is the neutron separation energy. All nuclear γ-ray evaporations can start. Experimental information masses and binding energies are taken from the AME2012 on the fission fragment yields is rather scarce at this time Atomic Mass Evaluation [83], complemented by FRDMalthough important recent theoretical and experimental 2012 calculations [2] when no experimental data exist. developments, such as discussed in Sec. 2.3, show great The total excitation energy, TXE, available for neupromise. Depending on the reaction studied, different pre- tron and γ-ray emission is then shared among the two scriptions for the reconstruction of the full 3D distribu- complementary fragments. Several prescriptions exist for tion are used. When available, experimental information, sharing this energy. In its current version, CGMF uses a even partial, has been preferred [81] to less accurate phe- mass-dependent parameter RT (A) in order to best repronomenological models. For instance, experimental data on duce the experimental mass-dependent neutron multiplic- 14 P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations ity, ν(A). A second input parameter, α, is used to modify the initial spin distribution, given as   1 J(J + 1) ρ(J, π) = (2J + 1) exp − 2 , (9) 2 2B (Z, A, T ) where B 2 is defined in terms of the fragment temperature, B 2 (A, Z, T ) = α I0 (Z, A) , h̄2 (10) and I0 (Z, A) is the ground-state moment of inertia for the fragment (Z, A). The adjustable input parameter α can then be used to tune this initial spin distribution to reproduce the average prompt fission γ multiplicity. Once the initial conditions (Ui , Ji , πi ) in energy, spin and parity of each fragment are set, the Hauser-Feshbach statistical theory of de-excitation of a compound nucleus can be applied. Only neutrons and γ rays have a reasonable probability of being emitted for the fragments and energies considered. Charged particles are hindered by the Coulomb barrier. Also, the dynamical emission of particles or clusters, such as ternary α particles, is not considered. The probability for a fragment to emit a neutron of energy n is given by Pn (n )dn ∝ Tn (n )ρ(Z, A − 1, U − n − Sn )dn , (11) where ρ describes the nuclear level density in the residual nucleus (Z, A−1) at the residual excitation energy U −n − Sn , and Sn denotes the neutron separation energy. The neutron transmission coefficients Tn are obtained through optical model calculations. Because of the large number of fragments produced, only a global potential can be used. Until now, the global optical potential of Koning and Delaroche [84] has been used for most CGMF calculations. The transmission coefficients for photon emission are obtained from the γ-ray strength function fγ (γ ), assuming the Brink hypothesis, i.e., the equivalence between the (n, γ) and (γ, n) reaction channels, and using the Kopecky-Uhl formalism [85] Tγ (γ ) = 2π2l+1 fγ (γ ), γ At each stage of the decay of the fragments (see Fig. 20), the neutron and γ-ray emission probabilities are sampled and a particular transition picked, leading to a new configuration characterized by a new set of (Z, A, E ∗ , J, π). In addition, the kinematics of the neutrons, γ rays and fragments are followed exactly in the classical limit. Very small relativistic corrections are ignored. In the current version of the code, neutrons and γ rays are evaporated isotropically in the center-of-mass of the parent fragment. The (small) recoil of the fragments due to the emission of the particles is taken into account. The boost of the center-of-mass to laboratory frames is responsible for the strong focusing of the particles along the fission axis. Neutrons have a much higher probability of being emitted at high excitation energy, while γ rays compete mostly at lower energies. However, high spins can lead to larger decay width ratios, Γγ /Γn , allowing for the emission of γ rays for nuclear excitation energies higher than the neutron separation energy. An additional complication is the presence of long-lived isomers in the fission fragments. By default, CGMF calculates the prompt fission γ spectrum for a time coincidence window of 10 ns, which is typical of recent experimental setups used to measure this spectrum [31, 43]. However, the exact coincidence window for a specific experiment should be used to compare theory and experiment. In neutron-induced fission reactions with increasing incident neutron energy, neutrons can be emitted from the parent nucleus before it fissions, leading to multi-chance fission reactions, labeled as (n, n0 f), (n, 2nf), (n, 3nf), · · · where n0 indicates that the emitted neutron is not the same as the incident one. Above about 10 MeV, pre-equilibrium (PE) emissions can also occur, leading to a pre-fission neutron spectrum different than a compound nucleus evaporation spectrum. In this case, the emitted neutron is the incident neutron. The two-component exciton model [89] is used to calculate the probability of PE emission at a given incident neutron energy, as well as the PE neutron spectrum. Probabilities for first-, second-, third- and up to fourth-chance fission are calculated separately using the CoH nuclear reaction code [15]. (12) where l is the multipolarity of the electromagnetic transition. In CGMF, only E1, M1 and E2 transitions are considered. The strength function parameterizations of the RIPL-3 library [86] are used. In CGMF, as in most other Hauser-Feshbach codes, the fragment, i.e., the compound nucleus, is represented by a discrete level region at low excitation energies, completed by a continuum region at higher energies where discrete excitations cannot be resolved any longer. Known discrete levels are read in from the RIPL-3 library [86], which itself derives them from the ENSDF nuclear structure data library [87]. At higher excitation energies, the level density is calculated using the Gilbert-Cameron mixed model of a constant temperature followed by a Fermi gas region to represent the continuum. CGMF also implements Ignatyuk’s prescription [88] that dampens the shell corrections with increasing energy. 3.1.2 The FREYA code The computational model FREYA, developed at Lawrence Berkeley and Lawrence Livermore National Laboratories, generates complete fission events, i.e. it provides the full kinematic information on the two product nuclei as well as all the emitted neutrons and photons. In its development, an emphasis had been put on speed, so large event samples can be generated fast. FREYA therefore relies on experimental data supplemented by simple physics-based modeling. In its standard version, to treat a given fission case, FREYA needs the fission fragment mass distribution Y (A) and the average total kinetic energy hTKEi(A) for the particular excitation energy considered. Y (A) is taken either directly from the measured yields or from a five-gaussian P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations 15 ∗ fit to the data, see Ref. [90] for details, which makes it pos- Esc = Q − TKE for the sampled value of TKE. The corresible to parameterize its energy dependence of the yields. sponding “scission temperature”, Tsc , which is scaled by ∗ 2 = (A0 /e0 )Tsc . In order to generate an event, FREYA first selects the the parameter cS , is obtained from Esc mass split based on the provided Y (A). The fragment FREYA explicitly conserves angular momentum. The charges are then sampled from the normal distributions overall rigid rotation of the dinuclear configuration prior suggested by experiment [90]. The linear and angular mo- to scission, caused by the absorption of the incoming neumenta of the two fragments and their internal excitations tron and the recoil(s) from any evaporated neutron(s), are subsequently sampled as described below. After their dictates certain mean angular momenta in the two fragformation, the fully accelerated fragments de-excite first ments. In addition, due to the statistical excitation of the by neutron evaporation and then by photon emission. In scission complex, the fragments also acquire fluctuations addition to spontaneous fission, FREYA treats neutron-inducedaround those mean values. FREYA includes fluctuations in fission up to Einc = 20 MeV. The possibility of pre-fission the wriggling and bending modes (consisting of rotations evaporations up to fourth-chance fission is considered as in the same or opposite sense around an axis perpendicuwell as pre-equilibrium neutron emission, as described in lar to the dinuclear axis) but ignores tilting and twisting Ref. [90]. (in which the fragments rotate around the dinuclear axis). FREYA contains a number of adjustable parameters that These dinuclear rotational modes are assumed to become control various physics aspects. They are listed here as a statistically excited during scission. They are therefore degroup but are also mentioned in the text where they were scribed by Boltzmann distributions, used. 2 (13) P± (s± ) dsx± dsy± ∼ e−s± /2I± TS dsx± dsy± , dTKE, an overall shift of TKE relative to the input TKE(A), used to adjust the average neutron multiplicity ν; y x e0 , the overall scale of the Fermi-gas level density param- where s± = (s± , s± , 0) is the spin of the normal modes with plus referring to the wriggling modes (having paraleters; x, the advantage in excitation energy given to the light lel rotations) and minus referring to the bending modes fragment. It is currently single valued and energy in- (having opposite rotations). The corresponding moments dependent but could be made mass dependent, like of inertia are denoted I± [55, 91]. The degree of fluctuthe RT (A) distribution in CGMF and FIFRELIN [10] for ation is governed by the ‘spin temperature’ TS = cS Tsc which can be adjusted by changing the parameter cS . The cases with sufficient data; cS , the ratio of the “spin temperature” to the “scission fluctuations vanish for cS = 0 and the fragments would then emerge with the angular momenta dictated by the temperature”. cT , the relative statistical fluctuation in the fragment ex- overall rigid rotation of the scission configuration (usually very small for induced fission - and entirely absent for citations. spontaneous fission). The default value, cS = 1, leads to So far, none of the FREYA parameters are assumed to de- S ∼ 6.2h̄ and S ∼ 7.6h̄ for 252 Cf(sf) and gives reasonL H pend on fragment mass. The dTKE is adjusted as a func- able agreement with the average energy of γ rays emitted tion of Einc to match ν(Einc ). As described shortly, the in fission (see Ref. [55] for details). prescription for the calculation of the level density paAfter accounting for the total rotational energy of the rameter is energy dependent even though e0 itself is not. two fragments, E , there is a total of E ∗ rot stat = E − Erot There is currently insufficient information available to as- remaining for statistical fragment excitation. Itsc is dissume any energy dependence of x, cS or cT . tributed between the two fragments as follows. First, a The emission of γ rays in FREYA is governed by cS . ∗ , is made accordpreliminary partition, Estat = ÉL∗ + ÉH There are two additional settings in FREYA that influence ing to the heat capacities of the two fragments which are the γ results: assumed to be proportional to the corresponding Fermi∗ gmin , the minimum γ-ray energy measurable by a given gas level density parameters, i.e. ÉL∗ /ÉH = aL /aH , where detector;  
tmax , the maximum half-life of a level during the photon Ai δWi ∗ −γUi a ( É ) = 1 + 1 − e , (14) i i decay process (which stops when it reaches a level have0 Ui ing a half-life exceeding tmax ). The quantities gmin and tmax are detector dependent with tmax corresponding to the time coincidence window for CGMF mentioned in the previous section. In the remainder of this section, the physics modeling in FREYA is described. For a given split of compound nucleus A0 into light and heavy fragments, AL and AH respectively, the fission Q value for FREYA is defined the same way as for CGMF, see the discussion around Eq. (8). For a given total fragment kinetic energy TKE, the energy available for rotational and statistical excitation of the two fragments is with Ui = Éi∗ − ∆i and γ = 0.05/MeV [92]. The pairing energy of the fragment, ∆i , and its shell correction, δWi , are tabulated based on the mass formula of Koura et al. [93]. The overall scale e0 is taken as a model parameter but it should be noted that if the shell corrections are negligible, δWi ≈ 0, or the available energy, Ui , is large, then ai ≈ Ai /e0 , i.e. ai is simply proportional to the fragment mass number Ai , and the energy-dependent renormalization is immaterial. The value determined in Ref. [90], e0 ∼ 10/ MeV, is used in the present studies. The level density treatment in FREYA is consistent with 16 P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations that of CGMF and close to the empirical results of BudtzJørgensen and Knitter in Ref. [94] If the two fragments are in mutual thermal equilibrium, TL = TH , the total excitation energy will, on average, be partitioned as above. But because the observed neutron multiplicities suggest that the light fragments tend to be disproportionately excited, the average excitations are modified in favor of the light fragment, ∗ ∗ ∗ E L = xÉL∗ , E H = Estat − E L , (15) where the adjustable parameter x is expected to be larger than unity. It was found that x ∼ 1.3 leads to reasonable agreement with ν(A) for 252 Cf(sf), while x = 1.2 is suitable for 235 U(n,f) [95]. After the mean fragment excitation energies have been assigned as described above, FREYA considers the effect of thermal fluctuations. In Weisskopf’s statistical model of the nucleus, which describes the excited nucleus as a degenerate Fermi gas, the mean excitation of a fragment ∗ is related to its temperature Ti by E i = ãi Ti2 [28, 96, 2 = 97] and the associated variance in the excitation is σE i ∗ 2 2 −∂ ln ρi (Ei )/∂Ei = 2E i Ti . Therefore, for each of the two fragments, an energy fluctuation δEi∗ is sampled from a ∗ normal distribution of variance 2cT E i Ti and the fragment excitations is adjusted accordingly, arriving at ∗ Ei∗ = E i + δEi∗ , i = L, H. (16) The factor cT multiplying the variance was introduced to explore the effect of the truncation of the normal distribution at the maximum available excitation. Its value affects the neutron multiplicity distribution P (ν). Previous work used the default value, cT = 1.0; cT ≥ 1.0 is expected. Energy conservation is accounted for by making a compensating opposite fluctuation in the total kinetic energy, ∗ TKE = TKE − δEL∗ − δEH . FREYA uses the RIPL-3 data library [86] for the discrete decays towards the end of the decay chain. The first stage is statistical radiation. These photons are emitted isotropically with an energy sampled from a black-body spectrum modulated by a giant dipole resonance, GDR, form factor, 2 ΓGDR Eγ2 dNγ E 2 e−Eγ /T . (18) ∼ 2 2 dEγ (Eγ2 − EGDR )2 + ΓGDR Eγ2 γ The position of the resonance is EGDR /MeV = 31.2/A1/3 + 20.6/A1/6 [100], while its width is ΓGDR = 5 MeV. It is assumed that each emission reduces the magnitude of the angular momentum by dS, the standard value being dS = 1 h̄. The RIPL-3 library [86] tabulates a large number of discrete electromagnetic transitions for nuclei throughout the nuclear chart, but complete information is available for only relatively few of them. However, by invoking certain assumptions, see Ref. [99], it is possible to construct, for each product species, a table of the possible decays from the lowest discrete levels, i.e. the level energies {ε` }, their half-lives {t` }, and the branching ratios of their various decays. Then whenever the decay process described above leads to an excitation below any of the tabulated levels, FREYA switches to a discrete cascade based on the RIPL-3 data. The discrete cascade is continued until the half-life t` exceeds the specified value of tmax . When comparing with experimental data, tmax should be adjusted to reflect the response time of the detection system. If the RIPL-3 tables do not include any transitions, in this case FREYA allows statistical excitation until near the yrast line and the remaining de-excitation occurs by emission of “collective” γ rays that each reduce the angular momentum by 2 h̄. When the γ-ray energy is below gmin , that γ ray is not registered in FREYA and does not count toward the total multiplicity. (17) The average TKE, TKE, has been adjusted by dTKE to reproduce the average neutron multiplicity, ν. The evaporated neutrons are assumed to be isotropic in the frame of the emitting nucleus, apart from a very slight flattening due to the nuclear rotation. Their energy is sampled from a black-body spectrum, dNn /dEn ∼ En exp(−En /Tmax ), where Tmax is the maximum possible temperature in the daughter nucleus, corresponding to emission of a very soft neutron [98]. FREYA generally assumes that neutron evaporation continues until the nuclear excitation energy is below the threshold Sn + Qmin , where Sn is the neutron separation energy and Qmin the energy above the neutron separation threshold where photon emission takes over from neutron emission. The value of Qmin is fixed at 0.01 MeV so that neutron evaporation continues as long as energetically possible, independent of angular momentum [99]. Neutron emission is treated relativistically in FREYA. After neutron evaporation has ceased, the excited product nucleus will emit photons sequentially. This emission is treated in several stages. The most recent version of 3.2 Transport codes 3.2.1 Overview The ultimate goal of advanced computational physics is to use physical data and numerical algorithms to simulate and predict the behavior of nature. More specifically, radiation transport codes intend to model the detailed behavior of radiated particles as they interact with various materials. In order for radiation transport codes to predict natural phenomena as accurately as possible, many details need to be realized: physical assumptions and numerical simplifications need to be minimized; nuclear data-like cross sections need to be well understood; and relevant experimental data are needed for proper validation in various application areas. While this list is not exhaustive, it does include the most basic components of an accurate and robust radiation transport code. To address the first point, the Monte Carlo method is widely considered to be the radiation transport method with the fewest physical and numerical approximations. P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations Because the approximations related to the spatial, angular and energy variables associated with the state of a particle are essentially negligible in continuous-energy Monte Carlo codes like MCNP, comparisons to a variety of experimentally measured quantities are feasible. Nuclear data and validation are addressed in this paper. Historically, MCNP has been used for many applications including, but not limited to, radiation shielding and protection, reactor physics and design, and nuclear criticality safety. A report on MCNP verification and validation is issued annually by the developers of MCNP [101], and provides details on the extensive verification and validation work necessary to ensure the trust in MCNP results in these applications. A similar goal exists for the new event-byevent nuclear fission models (FREYA and CGMF) introduced in MCNP for use in nonproliferation and safeguards applications. 3.2.2 Nuclear fission physics models in MCNP6 The latest release of MCNP6.2 not only includes the two new correlated fission multiplicity models, CGMF and FREYA, but it also includes fission multiplicity options dating back to many of the previous releases, including MCNP6.1.1 [102]. In MCNP, the bounded integer sampling scheme is employed by default to simulate secondary neutrons emitted from fission reactions. In this scheme, given the average number of neutrons emitted in fission, ν̄, when a fission event occurs the number of neutrons emitted is either the integer number n = bν̄c or n + 1. The probabilities for selecting n and n + 1 are chosen to preserve the expected value of ν̄. In the case of thermal neutron-induced fission of 235 U for instance, ν̄=2.42, and only ν values of 2 and 3 are sampled. Similarly, the production of photons from neutron interactions is done using the ratio of the photon production cross section, taking into account photon production interaction probabilities and photon yields, to the total interaction cross section. This ratio is the expected value of the number of photons produced per interaction at a given incident neutron energy. In general, if this ratio is small, then MCNP uses Russian roulette to determine if a single photon is produced. If this ratio is large, MCNP produces a few photons (less than ten) and gives a higher weight to each of these photons to preserve the overall expected photon production rate [103]. By enforcing the expected number and/or weight of the fission neutrons and photons produced, the expected values of quantities such as the flux, reaction rates, and criticality, keff , are also maintained. However, if the objective is to analyze the event-by-event nature of these reactions, such as simulating the behavior of neutron multiplicity counters, the detailed microscopic behavior of the particles (neutrons and photons) emitted during fission is needed. With the release of MCNP6, the capabilities in both MCNP5 [103] and MCNPX [104] were merged so that users have the ability to select various fission multiplicity treatments. Several sampling algorithms are available to sam- 17 ple from a Gaussian distribution for a given isotope based on data [7, 39, 105, 106]. While the MCNP input options do offer flexibility in simulating spontaneous and neutroninduced prompt fission neutron multiplicities, it is limited in its use in applications due to a variety of assumptions. First, the direction of travel of each neutron emitted in fission is independently sampled from an isotropic distribution in the laboratory frame. Next, the energy of neutrons emitted in each fission event is sampled independently from the same average PFNS. And finally, these specific features do not allow simulation of γ-ray multiplicities in spontaneous or neutron-induced fission. These limitations were somewhat lifted with the implementation of the LLNL Fission Library version 1.8 [40] in MCNPX and included in the recent releases of MCNP6. Before the Library was included, all photons produced from all neutron reaction channels in MCNP were sampled prior to the selection of the neutron reaction itself. While this does not bias the calculation of integral quantities such as flux and keff , it is still impossible to simulate fission event-by-event. While the LLNL Fission Library [40] addressed a few of the limitations of the standard MCNP multiplicity treatments, the issues with missing event-by-event energy, angular and particle correlations remained. With the explicit Monte Carlo modeling of the fission process done in both FREYA and CGMF, these last concerns are finally addressed. 3.2.3 The MCNPX-PoliMi code The MCNPX-PoliMi extension to MCNPX was developed to better simulate coincidence measurements and subsequent time analyses [107]. The PoliMi code includes built-in correlations for key spontaneously fissioning isotopes (252 Cf, 238 U, 238,240,242 Pu, 242,244 Cm) and for MCNPX-supported induced fission, event-by-event tracking, and conservation of energy and momentum on an event-by-event basis. MCNPX-PoliMi has the option to track and record event information collision-by-collision in specified detector regions. For each collision, key information is recorded: history number, particle number, particle type, collision type, target nucleus, collision cell, and collision time. This recorded collision information can be used to accurately model nonlinear detector responses event-by-event. Built-in spontaneous fission sources have prompt neutron multiplicity distributions and multiplicity-dependent neutron energy spectra. As the emitted neutron multiplicity increases, the neutron energy spectrum softens [13]. The MCNPX-PoliMi algorithm uses the average light and heavy fragment masses of each fissioning isotope to impart momentum from the fragments to the emitted neutrons. These built-in spontaneous fission sources also have prompt γ-ray multiplicity distributions. The 252 Cf photon energy is sampled from the spectrum evaluated by Valentine [108]. All other isotopes are sampled from a 235 U evaluation. Photons are emitted isotropically. Both neutrons and photons are generated from independent but full multiplicity and evaluated energy distributions for induced fission. 18 P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations 3.2.4 Implementation of event-by-event models in MCNP6.2 γ−ray spectrum (γ/f/MeV) (a) 100 252 Cf (sf) 10−1 10−2 10−3 CGMF only CGMF−MCNP FREYA only FREYA−MCNP 10−4 10−5 10−6 −1 10 100 γ−ray energy (MeV) 101 10.0 (b) Average γ multiplicity (γ/f) In the MCNP6.2 release, both the FREYA and CGMF fission event generators are included. In the most recent prior version of MCNP, MCNP6.1.1 [12, 102], two low-energy neutronphoton multiplicity packages were released: the LLNL Fission Library [40] and the Cascading Gamma-ray Multiplicity (CGM) code from LANL [15]. Version 1.8 of the LLNL Fission Library [40] included neutron and photon multiplicity distributions but did not include any correlations between emitted particles by default. Likewise, the released version of the CGM code handles a variety of reactions, but does not include particle emission from fission reactions. The newest versions of these event generators, to be included in the MCNP6.2 release, are significantly improved over their predecessors by addressing some of these immediate deficiencies, as described below. The main MCNP6.2 source code remains separate from the event-by-event source codes. A clean interface was developed to call the necessary routines and pass the information to and from MCNP. In the worst case scenario, the interface caused an overhead of less than 1% on the total computation time. In any realistic transport calculation using these fission event generator models through the new interface, the added computational cost of the interface will be negligible due to the usual amount of computational time used in Monte Carlo codes tracking particles and looking up cross sections. As part of the routine MCNP code-integration strategy, several tests were performed to check that the integrated and standalone versions of the FREYA and CGMF codes provide equivalent results to an appropriate numerical accuracy. The following quantities were checked: average multiplicities, ν̄ and N̄γ , and energies, χ̄n and χ̄γ ; multiplicity distributions, P (ν) and P (Nγ ), and correlations in nγ multiplicities, P (ν, Nγ ), and n-n emission angles, Ω n · Ω n0 . Each simulation included approximately 106 fission events. While there are sometimes significant differences between the calculated results of CGMF and FREYA, in particular for γ-ray average energies and spectra, there is very good agreement between the results from the standalone codes and their MCNP-integrated counterparts, as shown in Fig. 21 for the average γ-ray spectrum of 252 Cf(sf) (a) and n-γ multiplicity correlations in the n(1.0273 MeV)+239 Pu fission reaction (b). The latter quantity represents an interesting correlation predicted by these models. Increasing the neutron multiplicity results in a decrease in the average photon multiplicity, as shown in Fig. 21(b). There is extremely good agreement between standalone and integrated codes. This shows that MCNP does generate the negative correlation between the neutron and photon multiplicities produced by both FREYA and CGMF. As a final verification test, the neutron-neutron angular correlations observed in these fission event generator models are shown in Fig. 22 for n(1.0273 MeV)+239 Pu. Note that these quantities are not readily available from MCNP6 in any standard output or tallies, but can only be computed by analyzing the list-mode data instead. 101 9.0 n(1.0273 MeV)+239Pu 8.0 7.0 CGMF only CGMF−MCNP FREYA only FREYA−MCNP 6.0 5.0 0 1 2 3 4 5 6 7 Neutron multiplicity (n/f) Fig. 21. (Color online) The γ-ray spectra of the 252 Cf spontaneous fission reaction (a) calculated using MCNP6, FREYA and CGMF. The average photon multiplicity, Nγ , as a function of the neutron multiplicity, ν, using MCNP6, FREYA and CGMF (b). 4 Simulation results Here results of simulations with CGMF and FREYA are shown for some fission observables and, where possible, compared to the data presented in Sec. 2.2. First the code results are compared for observables that depend on fragment mass and kinetic energy. Next, multiplicity-dependent spectral results are shown, followed by calculations of multiplicity distributions, for both neutron and γ emission. The results presented here from CGMF and FREYA use the inputs employed in the public versions of the codes unless otherwise noted. The remainder of the section is devoted to correlations. First, γ-n multiplicity correlations are discussed. The following two subsections are devoted to neutron-light fragment and neutron-neutron angular correlations. The last parts deal with time dependence of the results, first the dependence of γ emission on the time coincidence window of the detector, followed by a discussion of time-chain correlations in multiplicity counting. This last topic, while not pertaining only to a single event, is included here because of its importance to nuclear assay applications. Probability Density (arb. units) 4.5 CGMF only CGMF−MCNP FREYA only FREYA−MCNP 4.0 3.5 n(1.0273 MeV)+239Pu 3.0 2.5 2.0 1.5 −1.0 −0.5 0.0 0.5 1.0 Average Prompt Neutron Multiplicity (n/f) P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations 235 3 U (n,f) 1 0 80 90 100 110 120 130 140 Fission Fragment mass (amu) The average neutron multiplicity as a function of fragment mass, ν̄(A), calculated by CGMF and FREYA for several incident neutron energies in the 235 U(n,f) reaction are shown in Fig. 23. Near mass A = 132, characteristics of both neutron (N = 80) and proton (Z = 52) spherical shell closures, the average number of emitted neutrons reaches its minimum. There, the expected extra collective energy due to the deformation of the fragments near scission compared to their ground-state configuration is expected to be very small. On the contrary, the complementary fragment, near mass 104, is very elongated. The extra deformation energy will transform into an additional intrinsic excitation energy in the fragments after scission, eventually leading to the release of more prompt neutrons. The overall agreement between CGMF and FREYA is rather good. There is some discrepancy between the two results for A between 100 and 110, where FREYA emits fewer neutrons, and in the region between symmetry and A = 132 where FREYA emits slightly more neutrons. The evolution of this dependence as a function of excitation energy has been the focus of several studies in recent years [109, 110], although only limited and indirect experimental data exist. Those limited data sets, however, indicate that as the total excitation energy in the fragments increases, the average neutron multiplicity increases almost solely in the heavy fragments. At even higher energies, the situation becomes somewhat more complex since more and more neutrons are evaporated from the compound nucleus prior to fission, in the multi-chance fission process. The pre-fission neutrons cannot be attributed to either one of the fragments since they are not associated with the fragments. For example, at 10 MeV, ∼ 0.6 neutrons on average are emitted from the 236 U compound nucleus prior to fission in FREYA. Upcoming versions of the CGMF and FREYA codes will address this important question more thoroughly in the near future. 150 160 Fig. 23. (Color online) Average prompt neutron multiplicity as a function of fragment mass for the neutron-induced fission reaction on 235 U for thermal and 10 MeV neutrons, calculated by FREYA and CGMF. The average neutron kinetic energy in the center-ofmass neutron energy as a function of the fragment mass has been measured for several fission reactions, as shown in Fig. 1(b). The CGMF and FREYA results predicted for 235 U(n,f) are shown in Fig. 24. The dependence of the neutron kinetic energies on Einc is similar to that for ν(A). The A dependence shows a similar trend but the overall average energy is somewhat higher for FREYA. Average neutron kinetic energy (MeV) 4.1 Dependence on fission fragment mass and kinetic energy FREYA, Thermal FREYA, 10 MeV CGMF, thermal CGMF, 10 MeV 2 Neutron−neutron angle (cos(θ)) Fig. 22. (Color online) The neutron-neutron angular correlations for the n(1.0273 MeV)+239 Pu fission reaction calculated using MCNP6, FREYA and CGMF. 19 2.5 235 U (n,f) 2 FREYA, Thermal FREYA, 10 MeV CGMF, thermal CGMF, 10 MeV 1.5 1 0.5 80 90 100 110 120 130 140 Fission fragment mass number 150 160 Fig. 24. (Color online) The average center-of-mass neutron energy as a function of fragment mass for 235 U(n,f) with thermal and 10 MeV neutrons, calculated by FREYA and CGMF. Prompt γ-ray characteristics as a function of the fragment mass are also very interesting to study as they provide complementary information and constraints on the physics models of the fission event generators. In particular, the γ-n competition is governed by the distribution of angular momentum in the fragments. The average prompt γ-ray energy per photon, hγ i, as a function of fragment mass is shown in Fig. 25 for 235 U(nth ,f). The significant Average γ−ray energy (MeV) increase of hγ i for masses near 132 can be explained by the lower density of levels in these near-spherical fragments, thereby increasing the average energy of the γ transitions between excited levels. This result should be approximately independent of the fissioning nucleus since it depends on the characteristics of the fragments themselves. However, different fission fragment mass yields as observed in different fission reactions will lead to different hardness of the γ spectrum, from the convolution of Y (A) with γ (A) shown in Fig. 25. As noted earlier, the γ-ray multiplicity is very sensitive to the energy threshold, gmin in FREYA, and fission time coincidence window of the detector, tmax in FREYA. The calculations in Fig. 25 were made with the values for these quantities given in Ref. [33], gmin = 0.09 MeV and tmax = 5 ns. Given the sensitivity of the γ-ray multiplicity to these quantities, the average γ-ray energy shown in Fig 25 also has some sensitivity to gmin even though the total γ-ray energy is almost insensitive to the cutoff of most experiments. Pleasonton, 1972 CGMF FREYA 2 235 U (nth,f) 1 0.5 90 100 110 120 130 140 Fission fragment mass (amu) 150 7 235 U (n,f) 6 FREYA, Thermal FREYA, 10 MeV CGMF, thermal CGMF, 10 MeV 5 4 3 2 80 90 100 110 120 130 140 Fission fragment mass (amu) 150 160 Fig. 26. (Color online) The average prompt fission γ-ray multiplicity as a function of the fission fragment mass, for thermal and 10 MeV incident neutrons in the 235 U(n,f) reaction, as calculated with FREYA and CGMF. Measurements of the evolution of prompt neutron and γ-ray emission data as a function of the total kinetic energy of the fission fragments have been reported in a few fission reactions. The slope dν̄/dTKE is an indicator of how much energy is required to emit a neutron. Figure 27 shows ν̄ as a function of TKE for several 252 Cf(sf) measurements. They all exhibit a rather linear behavior in the range 180 MeV < TKE < 220 MeV. At energies TKE > 220 MeV, statistics are low since very little excitation energy is left for neutron emission. At TKE < 180 MeV, the measurements diverge. It has been suggested that the flatter low TKE behavior exhibited by some of the experiments is due to a nonlinear dependence of the average fission Q value with TKE. 1.5 80 Average γ−ray multiplicity (γ/fission) P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations 160 Fig. 25. (Color online) The average prompt fission γ-ray energy as a function of the fission fragment mass in the thermal neutron-induced fission reaction on 235 U. Experimental data are from Pleasonton et al. [33]. The situation with the average prompt fission γ-ray multiplicity as a function of fragment mass, is less clear however, as comparable experiments provide somewhat inconsistent results due to the different energy thresholds and time windows for different detectors. The FREYA calculations shown in Fig. 26 use gmin = 0.1 MeV and tmax = 10 ns. The results are essentially independent of Einc because, in FREYA, neutron emission continues until the neutron separation energy, Sn , is reached. Thus nearly the same residual excitation energy is left for γ emission in FREYA, regardless of Einc . A very similar conclusion is reached with CGMF, which treats the n-γ competition slightly differently. The two results are rather similar as a function of A except for A < 105 where the CGMF multiplicity is higher. 8 Average neutron multiplicity (n/f) 20 Bowman, 1963 Hambsch, 2011 Budtz−Jørgensen, 1988 Göök, 2014 CGMF CGMF, light fragments CGMF, heavy fragments FREYA FREYA, light fragments FREYA, heavy fragments 6 4 2 252 Cf (sf) 0 150 160 170 180 190 200 210 Total kinetic energy (MeV) 220 230 Fig. 27. (Color online) The average prompt fission neutron multiplicity as a function of the TKE of the fission fragments for 252 Cf(sf). The experimental data are from Bowman et al. [111], Hambschet al. [112], Budtz-Jørgensen and Knitter [94], and G`‘o`‘ok et al. [22]. P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations 252 16 6 Eγtot (MeV) 200 10 8 150 6 100 4 50 2 0 0 150 4 160 170 180 190 200 210 Total Kinetic Energy (MeV) 15 2 170 180 190 200 210 220 230 Total kinetic energy (MeV) Fig. 28. (Color online) Same as Fig. 27 but for selected fragment masses, compared to experimental data by Göök et al. [22]. Eγtot (MeV) 160 10 250 200 150 5 100 50 0 Figure 29 shows the predicted (a) and measured (b) Eγtot as a function of TKE. The experimental data were obtained with the DANCE array, in coincidence with two silicon detectors to measure the kinetic energy of the fragments [113]. While the very poor resolution of the fission fragment kinetic energies obtained with this preliminary setup prevents a fair comparison to the predicted results, a new experiment with improved energy resolution is planned. No detector response corrections have been applied to the experimental data in Fig. 29. The regularities in the calculation shown in the top panel of Fig. 29 can be interpreted as follows for a single decaying fission fragment. If TKE decreases, TXE increases and the excitation energy available in the fragment for particle emission increases. For fragment excitation energies lower than the neutron emission threshold, all the excitation energy is available for γ emission. If the excitation energy is above the threshold for neutron emission, the 220 300 (b) 0 150 250 12 Göök, A=110 Göök, A=122 Göök, A=130 Göök, A=142 CGMF FREYA Cf (sf) 300 (a) 14 Counts (arb. units) Average neutron multiplicity (n/f) 8 probability for emitting a neutron is larger than the probability of γ decay. In this case, the fragment A emits one neutron and the residual (A − 1) fragment is now produced with a much smaller residual excitation energy, resulting a rather low total energy available for γ emission. For still higher values of TXE, more energy is available to contribute to Eγtot until the TXE reaches the threshold for two-neutron emission. This pattern is repeated every time a new neutron threshold is reached, thereby explaining the somewhat regular pattern observed in Fig. 29. Counts (arb. units) The neutron multiplicity as a function of TKE has also been measured recently [22] as a function of fragment mass, as shown in Fig. 28. The pattern corresponds to the sawtooth shape of ν(A) with the largest ν̄(TKE) for A = 122, near the top of the sawtooth for 252 Cf(sf), as seen in Fig. 1 where as A = 110 and A = 142 are masses where the sawtooth is rising. Finally, A = 130 is near the doubly-closed shell, near the minimum of ν̄(A), giving the lowest ν̄(TKE). This behavior gives some insight into how much energy is needed to emit a neutron for a given fragment mass and deformation. The FREYA results for A = 110 and 142 are in good agreement with the data since, here, the agreement between FREYA and data on ν(A) is also very good. At A = 122 and 130, however, FREYA over- and underestimates ν(A) respectively with the single-valued parameter x governing the excitation energy sharing. The CGMF agreement is closer overall since it uses a mass dependent parameter, RT (A), to match ν(A). 21 0 0 50 100 150 Total Kinetic Energy (arb. units) Fig. 29. (Color online) CGMF-calculated (a) and DANCE experimental (b) data for the total prompt γ-ray energy as a function of TKE in 252 Cf(sf). Note the slightly different Eγtot limits on the y-axes and the arbitrary units for TKE on the DANCE data. The average total γ-ray multiplicity as a function of TKE has been measured by Wang et al. [47], and is shown in Fig. 30 in comparison with FREYA and CGMF results. The agreement between CGMF and the experimental data is remarkable, while FREYA tends to overpredict the γ multiplicity for most TKE values, as observed previously in Ref. [47]. The calculations use gmin = 0.05 MeV and tmax = 5 ns, as in Ref. [47]. P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations Average total γ−ray multiplicity (γ/fission) 22 Prompt Neutron Spectrum (n/f/MeV) 12 Wang, 2016 CGMF FREYA 9.65−0.0367*(TKE−140) 10 8 6 4 140 252 Cf (sf) 160 180 200 220 Total kinetic energy (MeV) CGMF, all ν CGMF, ν=1 CGMF, ν=3 CGMF, ν=5 FREYA, all ν FREYA, ν=1 FREYA, ν=3 FREYA, ν=5 0.1 0.01 252 Cf (sf) 0.001 240 0 2 4 6 8 10 Outgoing Neutron Energy (MeV) Fig. 30. (Color online) The average total γ-ray multiplicity as a function of the total kinetic energy of the fragments, as measured by Wang et al. [47]. Fig. 31. (Color online) Neutron multiplicity-dependent PFNS calculated for 252 Cf(sf). 10−1 The average prompt fission neutron spectrum (PFNS) and the average prompt fission γ-ray spectrum (PFGS) are not what would be called correlated data in the present context. Those quantities can generally, albeit not always, be found in the ENDF evaluated libraries, and are commonly used in transport simulations. What is not present, however, is the multiplicity-dependent PFNS and PFGS, as shown in Figs. 31 and 32, respectively. The CGMF-predicted results show a slight hardening of the neutron spectrum with increasing neutron multiplicity, and a much stronger softening of the γ spectrum with increasing γ multiplicity. It is important to note that CGMF predicts a much softer spectrum than the current standard evaluated result [34], which is most likely linked to an incorrect optical potential for neutron-rich, deformed nuclei. The FREYA results for the PFNS show very little dependence on the neutron multiplicity for this nucleus. The larger uncertainty for ν = 1 is because of the lower probability for single neutron emission from 252 Cf(sf) with its higher than average ν. The dotted lines in Fig. 32 correspond to the results of a “parameterized model” (PM) developed to interpret and fit the DANCE experimental results [67]. In this model, the total prompt fission γ-ray multiplicity is taken as a sum of two multiplicities, N1 + N2 = Nγ , one from each fragment, which are sampled from two independent distributions,
P (Ni ) = Ci (2Ni + 1) exp −Ni (Ni + 1)/c2i , (19) where i = 1, 2. The terms Ci = C1,2 are constants to ensure proper normalization of the probability distribution. Following approximate relations based on the statistical model of the compound nucleus, the probability to emit one γ ray with energy εγ is given by (20) γ−ray spectrum (1/f/MeV) 4.2 Multiplicity-dependent spectra χ1 (εγ ) = D1 ε2γ exp (−(a1 + b1 Nγ )εγ ) , 1 235 U (nth,f) 10−2 CGMF FREYA PM 10−3 Nγ=5 10−4 2 Nγ=10 Nγ=15 3 4 5 Outgoing γ−ray energy (MeV) 6 Fig. 32. (Color online) Multiplicity-dependent prompt fission γ-ray spectra calculated with CGMF, FREYA, and the parameterized model (PM) for 235 U(nth ,f), for Nγ = 5, 10, and 15. All the calculations are normalized to unity. for multiplicity N1 and χ2 (εγ ) = D2 ε3γ exp (−(a2 + b2 Nγ )εγ ) , (21) for multiplicity N2 . In Eqs. (20) and (21), Nγ is the total γ-ray multiplicity, while D1,2 are normalization constants that ensure that Z ∞ χ1,2 (εγ ) dεγ = 1 . (22) 0 Therefore, for a given multiplicity Nγ , the multiplicitydependent prompt-fission γ-ray spectrum (also unit normalized) is given by: χNγ (εγ ) = 1 Nγ PNγ N1 =0 {N1 P (N1 )χ1 (εγ )+ N2 P (Nγ − N1 )χ2 (εγ )} . (23) The six model parameters {a1,2 , b1,2 , c1,2 } were fit to the DANCE data for 235 U(nth ,f) [114], 239 Pu(nth ,f) [66], P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations and 252 Cf(sf) [67]. This model, while not used in CGMF and FREYA is useful to speed up calculations and correctly predict the tails of the multiplicity-dependent PFGS. However, predictions based on this simplified statistical model would fail to predict the low-energy part of the spectrum where non-statistical transitions between discrete excited states in fission fragments cause strong fluctuations in the average total γ-ray spectrum as observed repeatedly for various fissioning systems [31]. DANCE PM CGMF FREYA 100 10−3 10−6 10−9 4.3 Multiplicity distributions P (ν) and P (Nγ ) Neutron multiplicity and neutron coincidence counting methods are employed to assess the mass and multiplication of fissile materials in safeguards and international treaty verification. In these methods, the arrival times of neutrons in detectors are recorded and analyzed. Bursts of neutrons are indicative of the presence of fissile material and are used to statistically infer the fissile material properties. The first, second and third factorial moments of the prompt neutron multiplicity distribution P (ν), Eqs. (4)(6) are important input data to these techniques. Higher moments (fourth and fifth) are even being considered [116] as a way to extract more useful information about fissile materials. The neutron multiplicity distributions of most common spontaneously fissioning nuclei, see Fig. 5(b), and important thermal-neutron-induced fission reactions are relatively well known. These distributions have been revisited recently by Santi and Miller for many spontaneous fission cases [26]. Recall, however, that much less is known about the energy dependence of these distributions. The Zucker and Holden evaluation [117] in Fig. 34 is based on the single data set of Soleilhac et al. [38] and Terrell’s model [39], see Eq. (1). The CGMF calculations for neutroninduced fission reactions on 239 Pu up to 20 MeV incident neutron energy are shown in Fig. 34. CGMF model input parameters RT (A) and α were adjusted to match the experimental ν and N γ values below 5 MeV only. The agreement with Terrell’s model at higher energies and for the higher moments of the distribution is quite good. In this comparison, the FREYA parameters cS and e0 were fixed by 252 Cf(sf) data while cT was adjusted to the thermal point to improve agreement with the shape of P (ν) but was left energy independent. The dTKE, adjusted to agree with ν̄ for all energies, is the only energy dependent parameter in FREYA. 10−12 10−15 10 −18 Mγdet=1 (top), 3, .., 13, 15 (bottom) 1 2 3 4 5 6 γ−ray energy (MeV) Fig. 33. (Color online) The prompt fission γ-ray spectra for γ detector multiplicities, 1 ≤ Mγdet ≤ 15, in steps of Mγdet = 2, observed [67] in the DANCE detector array in the spontaneous fission of 252 Cf, are compared with the results of the PM (red lines), CGMF (cyan lines) and FREYA (blue lines). These spectra correspond to the raw data as measured by DANCE and have not been processed to unfold the complicated detector response. Instead, Monte Carlo output from the PM, CGMF and FREYA have been processed through a GEANT4 simulation [115] of DANCE response. All curves are unit normalized, but scaled det down by a factor of 10Mγ −1 . Factorial Moments of P(ν) Prompt fission γ−ray spectra (γ/f/MeV) The PM results are compared with CGMF and FREYA calculations for 235 U(nth ,f) in Fig. 32. There is very good agreement between the PM and CGMF above γ > 2 MeV. The FREYA calculations exhibit the same trend but with a somewhat harder slope at higher γ-ray energy. A followup comparison between the model calculations propagated through the GEANT4 model of DANCE and the experimental results for 252 Cf(sf) is shown in Fig. 33. The γray spectra shown correspond to a γ-ray detector multiplicity, Mγdet , indicative of how many DANCE detectors are fired in coincidence. The relation between Nγ and Mγdet is not trivial and has to be simulated through GEANT or MCNP simulations. The dependence of the DANCE γray spectra on Mγdet is very well reproduced by the calculations. The same trends observed in Fig. 32 for 235 U(nth ,f) are seen here. 23 FREYA CGMF Holden−Zucker 100 ν3 ν2 10 ν1 239 Pu(n,f) 1 0 5 10 15 Incident Neutron Energy (MeV) 20 Fig. 34. (Color online) The first three factorial moments ν1 (red), ν2 (blue) and ν3 (green) as defined in Eqs. (4)-(6), of the prompt neutron multiplicity distribution P (ν) for the 239 Pu(n, f) reaction, as a function of incident neutron energy. 24 P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations The prompt γ-ray multiplicity distribution P (Nγ ) measured [42] for 252 Cf(sf) is compared to CGMF and FREYA calculations in Fig. 35. This distribution and in particular the average γ-ray multiplicity N γ is very sensitive to the threshold energy and time coincidence window since fission considered. However, the overall shape of the CGMFcalculated distribution is very nicely represented by a negative binomial distribution NB(r, p) with r=14.286 and p=0.633, in agreement with the distribution inferred by Oberstedt et al. [42], albeit with a γ energy threshold of 80 keV instead of the 100 keV reported by the authors. The FREYA results, although in rather good agreement with the average multiplicity, are narrower than the data. This is still under investigation. 0.15 (a) Oberstedt, 2015 CGMF (80 keV) CGMF (100 keV) FREYA NegBin(14.286,0.633) Probability 0.1 252 It is worth explaining the calculated trends. The calculated ν(TKE) decreases with increasing TKE in all mass regions, see Fig. 28. However, for FREYA, there is somewhat of a plateau for TKE < 175 MeV in the low mass region. Also, the dependence of the photon multiplicity with TKE is more complex and changes with mass region: it slightly increases with TKE until TKE ∼ 185 MeV and then remains relatively independent of TKE in the low mass region; it is independent of TKE for masses near symmetry; and it decreases with TKE in the high mass region, leading to the behavior shown in Fig. 36 when ν(TKE) and Nγ (TKE) are plotted against each other. The decrease in γ-ray multiplicity for ν > 2.5 is due to the lower Nγ at low TKE. When averaged over all masses, the Nγ (TKE) for FREYA is as observed in Fig. 30. In general, for a positive for Nγ relative to ν, both must decrease with increasing TKE. Cf (sf) 0.05 0 (b) Probability 10−2 252 Cf (sf) 10−3 10 Oberstedt, 2015 CGMF (80 keV) CGMF (100 keV) FREYA NegBin(14.286,0.633) −4 0 5 10 15 Number of γ rays 20 experiments were conducted with a 252 Cf fission source and fragment detectors surrounded by a large spherical gadolinium-loaded liquid scintillator tank. Prompt neutron and γ-ray detections were discriminated primarily through timing cuts. Recently, Wang et al. [47] expanded upon Nifenecker’s work by measuring the γ-ray multiplicity as a function of the neutron multiplicity for the light (85 < A < 123), symmetric (124 < A < 131), and heavy (132 < A < 167) fission fragments, shown in Fig. 8. For these experiments, two surface barrier detectors were used to estimate the fragment mass. A high purity germanium detector was used to count γ rays and a LS301 liquid scintillator was used to count neutrons. Figure 36 shows a slightly increasing, a strongly increasing, and a non-monotonic trend in γ-ray multiplicity as a function of neutron multiplicity, for those three mass regions respectively. Each data point is a 3 MeV-wide bin in TKE. While the predictions from FREYA also show some fragment mass dependence, the experimentally observed trends are not well reproduced. (Note that the FREYA calculations differ from those in Ref. [47], performed before the RIPL-3 lines were included.) The CGMF results exhibit the same trends as the FREYA calculations. 25 Fig. 35. (Color online) The prompt γ-ray multiplicity distribution P (Nγ ) for 252 Cf(sf) measured by Oberstedt et al. [42] is compared to CGMF and FREYA results on linear (a) and logarithmic (b) scales. The negative binomial function with parameters r=14.286 and p=0.633 represents a best fit to the experimental data. 4.4 γ-n multiplicity correlations Nifenecker et al. [46] found that the average total γ-ray energy from 252 Cf(sf) was linearly proportional to the average neutron multiplicity, see Eq. (7) and Fig. 7. The A recent experiment performed at the University of Michigan [65] used a scintillator detector array (see Fig. 13) to measure correlations between prompt neutrons and γ rays from fission. The measured time cross-correlations are in relatively good agreement with the results simulated by PoliMi, FREYA, and CGMF, as shown in Fig. 37 for the cross-correlation time distributions for n-n, n-γ, γ-n, and γ-γ coincidences. Note that the γ-γand n-n correlations are expected to be symmetric around ∆t = 0 while n-γand γ-n correlations are reflected around ∆t = 0 since the time axis represents the difference between the time of detection in detector 1 and detector 2. Therefore an nγevent (a photon detected in 1 and a neutron in 2) results in ∆t < 0 while a γ-n event results in ∆t > 0. The three simulated cases utilize PoliMi for particle transport but vary the fission models to include CGMF and FREYA as well as the PoliMi model. Rather large discrepancies appear in the γ-γ correlation points (cyan), most likely due to Counts per fission (counts/f) γ−ray multiplicity (γ/f) P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations 132<A<167 5 4 3 γ−ray multiplicity (γ/f) 4 3 10−7 10−8 10−9 −40 −20 0 20 40 60 and FREYA results. A small increase in calculated neutron energy is observed in the calculations with the number of coincident detections for both neutron and γ coincidences, primarily between one and two emitted neutrons and γrays. More data are required to resolve any trend in the neutron energy. Wang, 2016 FREYA CGMF 1 γ−ray multiplicity (γ/f) 10−6 Fig. 37. (Color online) 252 Cf(sf) cross-correlated time distributions for neutron and γ coincidences for measured (points) and simulated (lines) data from Ref. [65]. The three simulated cases utilize PoliMi for particle transport and vary the fission models: CGMF, FREYA and the built-in PoliMi model are employed. 124<A<131 2 nn nγ γn γγ Experiment PoliMi CGMF−PoliMi FREYA−PoliMi ∆t (ns) 1 5 −5 10−10 −60 Wang, 2016 FREYA CGMF 2 10 25 5 γ coinc. 1 2 3 4 4 3 85<A<123 Wang, 2016 FREYA CGMF 2 1 0 0.5 1 1.5 2 2.5 3 3.5 Neutron multiplicity (n/f) Fig. 36. (Color online) Prompt γ-ray multiplicity as a function of the neutron multiplicity ν, recently measured by Wang et al. [47] for 252 Cf(sf), and compared to FREYA and CGMF calculations. The mass selection is performed on the mass A of the initial pre-neutron emission fission fragments. background contamination which could be removed with the use of a fission chamber. Neutron energies were estimated by time-of-flight with a γ-ray trigger. Tables 1 and 2 show the mean neutron energy as a function of coincident particle detections for the measurements as well as the simulated PoliMi, CGMF, PoliMi-1 CGMF-PoliMi FREYA-PoliMi 2.344(2) 2.411(9) 2.45(4) 2.5(3) 2.320(6) 2.37(3) 2.4(1) - 2.397(6) 2.49(3) 2.4(2) - Meas. 2.475(2) 2.48(1) 2.49(6) 2.4(4) Table 1. (Color online) The average detected neutron energy (in MeV) by time-of-flight over the sensitive range of the detectors, 1.1 − 6.6 MeV, as a function of the number of γray coincidences. Omitted entries had too few statistics. The parentheses record the uncertainty in the last significant figure. The two-particle coincidence events were binned by the number of neutrons and γ rays detected within the 80 ns time-coincidence window. The measurements are shown in Fig. 38(a) and (b) respectively as the ratio of calculated to experimental results, C/E. The simulations overpredict the observed counts for all neutron coincidences except zero. Despite a basic background subtraction, the number of coincidences is underpredicted at zero because background photon coincidences contribute disproportionately. However, the observed discrepancies do not neces- n coinc. 1 2 3 4 P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations PoliMi-1 CGMF-PoliMi FREYA-PoliMi 2.347(2) 2.359(9) 2.34(6) 2.3(5) 2.322(6) 2.34(3) 2.4(2) - 2.400(6) 2.44(3) 2.5(2) - Meas. 2.475(2) 2.52(1) 2.51(9) 2.4(9) Table 2. Average detected neutron energy (in MeV) by timeof-flight over the sensitive range of the detectors, 1.1−6.6 MeV, as a function of the number of neutron coincidences. Omitted entries had too few statistics. The parentheses record the uncertainty in the last significant figure. Ratio C/E of detected neutron number 26 1.6 1.4 PoliMi CGMF−PoliMi FREYA−PoliMi 1.2 1 0.8 252 Cf (sf) 0.6 0 4.5 Neutron-fragment angular correlations 2 3 4 5 1.3 1.2 PoliMi CGMF−PoliMi FREYA−PoliMi (b) 1.1 1 0.9 252 0.8 Cf (sf) 0.7 0 Angular correlations between the fission fragments and the emitted neutrons emerge naturally from the kinematics of the reaction. Assuming that neutrons are emitted from fully accelerated fragments, the kinematic boost of the fragments from the center-of-mass frame to the laboratory frame induces a significant focusing of the neutrons in the direction of the fragments. Therefore, it is expected, and observed, that neutrons are emitted preferentially near zero and 180 degrees from the direction of the light fragment. Figure 39 illustrates this feature in the case of 252 Cf(sf). The ratio of neutrons emitted in the direction of the heavy fragment relative to the light fragment direction corresponds rather well to the average number of neutrons emitted from the light relative to the heavy fragment. Both the CGMF and FREYA calculations compare rather well with the data. A more detailed view of this process can be obtained by isolating the contribution from each mass split. This was recently achieved experimentally by Göök et al. [118] as shown in Fig. 40 for 235 U. The data were taken with incident neutron energies 0.3 eV ≤ Einc ≤ 60 keV, with a mean energy of 1.6 keV. The CGMF and FREYA results for thermal-neutron induced fission are also shown. The agreement is very satisfactory between the experimental and calculated results, except in a few cases: CGMF overestimates the experimental data near 180 degrees for the 106/130 mass split while the FREYA calculation overestimates the data for θn−LF > 140 degrees for AL = 96 and 106. 1 Neutron coincidences Ratio C/E of detected mean photon number sarily reflect a problem with the calculated n-γ correlations and may instead be due to inaccuracies in the calculated PFNS. The C/E for γ multiplicity relative to neutron coincidences agrees well, C/E ∼ 1, for zero counts. In this case PoliMi alone overpredicts while CGMF and FREYA underpredict. (Note that PoliMi employs a Watt distribution for the PFNS except for 252 Cf(sf) where the Mannhart evaluation is used directly.) Here, C/E is within 10% of unity. (a) 1 2 3 4 5 Neutron coincidences Fig. 38. (Color online) (a) Ratio of calculated to measured (C/E) neutron number as a function of neutron coincidences. (b) Ratio of calculated to measured (C/E) average number of γ rays as a function of neutron coincidences. 4.6 n-n and n-γ correlations In the absence of a fission fragment detector, correlations between the prompt neutrons and γ rays are invaluable to distinguish a fission reaction from other neutron-induced reactions such as inelastic scattering. Because of the boost imparted to the prompt neutrons emitted from the excited fission fragments, n-n angular correlations are expected to be peaked near 0 and 180 degrees. On the other hand, neutrons emitted in an (n, 2n) reaction, for example, would be emitted isotropically in the laboratory frame, except at higher incident neutron energies where the pre-equilibrium component would tend to focus the neutrons slightly more in the direction of the incident beam. Observing significant enhancements near 0 and 180 degrees is therefore a clear signature of a fission event. Figure 41 shows neutron-neutron angular distributions in the spontaneous fission of 252 Cf, as measured by several experimental groups [56, 57, 58], compared to calculations using CGMF and FREYA. The Pozzi [58] and Pringle [56] data agree very well, except at the lowest angles where crosstalk corrections should be taken into account. The effect P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations Bowman, 1962 Skarsvag, 1963 CGMF FREYA Counts (arbitrary units) 1 0.8 0.6 252 Cf (sf) 0.4 0.2 0 0 20 40 60 80 100 120 140 160 180 θn−LF (degrees) Fig. 39. (Color online) The neutron-light fragment angular distribution for 252 Cf(sf). The experimental data are from Bowman [53] and Skarsvåg [54]. Only neutrons with kinetic energies above 0.5 MeV were analyzed. 2 dν/dΩ (n/str) (a) FREYA CGMF Göök, 2016 2.25 1.75 (b) FREYA CGMF Göök, 2016 AL=91 AL=96 1.5 1.25 1 0.75 0.5 0.25 0 2.25 dν/dΩ (n/str) (c) FREYA CGMF Göök, 2016 2 1.75 (d) FREYA CGMF Göök, 2016 AL=101 determined by the excitation energy sharing, x for FREYA and RT (A) for CGMF. Also of interest is the evolution of these n-n correlations as the neutron energy threshold increases, as shown in Fig. 42 with the Gagarski [57] data. The results obtained using MCNPX with FREYA are overlaid in Fig. 42. For each energy-matched pair of data with simulation, the integral are matched in the range 40 < θn−n < 140 degrees. The lower angular range is excluded to avoid the region at low θn−n where no cross-talk correction has been applied. The correlation rises with threshold energy at 0 and 180 degrees. One may expect a steeper rise since the higher energy neutrons are more likely to be emitted in the direction of the fragment before the boost to the laboratory frame. Note that in Fig. 42 the FREYA calculations are run through a full detector simulation while the calculations in Fig. 41 were not. 2.5 Counts (arbitrary units) 1.2 27 252 Pringle, 1975 Gagarski, 2008 Pozzi, 2014 FREYA CGMF 2 Cf (sf) 1.5 1 AL=106 0 1.5 1.25 1 0.75 20 40 60 80 100 120 140 160 180 θn−n (degrees) 0.5 0.25 0 0 20 40 60 80 100 120 140 160 180 Θn−LF (degrees) 20 40 60 80 100 120 140 160 180 Θn−LF (degrees) Fig. 40. (Color online) The angular distributions of prompt fission neutrons in the thermal-neutron-induced fission reaction on 235 U, for different fragment mass splits. The experimental data are by Göök et al. [118]. Only neutrons with kinetic energies above 0.5 MeV were analyzed. of cross talk is evident in the Pozzi point at θn−n = 22 degrees which has not been corrected for this effect. The Gagarski [57] data lie higher than the other two data sets at the largest angles. The experiments use somewhat different cutoffs: Pringle [56] had a minimum neutron energy cutoff of 0.7 MeV, compared to the Gagarski measurement with a cutoff of 0.425 MeV [57]. The Pozzi [58] data used the same cutoff as the Gagarski result shown here. The FREYA calculations agree well for θn−n < 90 degrees while CGMF somewhat underestimates here. At angles close to back-to-back neutron emission, the calculations both overestimate the data. The dependence of the calculated correlation on θn−n for a fixed neutron energy threshold is Fig. 41. (Color online) Neutron-neutron angular distribution for 252 Cf(sf). Experimental data are from Pringle [56], Gagarski et al. [57] and Pozzi et al. [58]. No detector response was folded onto the calculations. The experimental neutron detection thresholds are 0.7 MeV for Pringle [56] and 0.425 MeV for Gagarski [57] and Pozzi [58] data. Neutron-neutron angular correlation measurements of Pu(sf) were performed recently [65] at the European Commission Joint Research Center, in Ispra, Italy. Figure 43 shows the experimental points (red) in comparison with MCNPX-PoliMi and FREYA simulations, with cross-talk events removed. The cross-talk contribution was estimated at each detector angle pair using MCNPX-PoliMi simulations and removed from the measured doubles [65]. The MCNPX-PoliMi results (purple) overpredict the measured values below 100 degrees and agree above, whereas the FREYA results overpredict the data above 80 degrees but agrees below. Note that the FREYA calculation in Fig. 43 is using a value of the x parameter discussed in Sec. 3.1.2 that has not been adjusted to ν(A) data since none are available. The default value of x = 1.2 from Ref. [9] was used here, giving a stronger correlation for θn−n > 90 degrees. 240 Counts (arbitrary units) 28 P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations 12 11 10 9 8 7 6 5 4 3 2 1 0 En > 425 keV En > 550 keV En > 800 keV En > 1200 keV En > 1600 keV open: FREYA full: data 252 Cf (sf) 0 20 40 60 80 100 120 140 160 180 θn−n (degrees) Fig. 42. (Color online) Same as Fig. 41 but for different neutron energy detection thresholds. The data are from Gagarski et al. [57]. tions were calculated by comparing the number of neutronneutron coincidences obtained from simulating time-correlated neutrons from spontaneous fission to those obtained from simulating single neutron emission from the same spontaneous fission neutron spectrum. More details on the experimental setup, cross-talk correction, and analysis can be found in Ref. [70]. An MCNPX model of the detector setup was developed and FREYA simulations were folded in to obtain the results shown with open symbols. In this case, the angular correlations were used to adjust the value of x used, to x = 1.3, since these correlations are sensitive to the excitation energy sharing [70]. The number of fissions simulated with FREYA was equivalent to the 23 hours of data taking in the experiment. The agreement between experiment and simulations is very good for most energies and angles. A default MCNPX calculation without FREYA would have resulted in flat distributions, except for a peak at 0 degrees due to neutron cross talk before subtraction of cross-talk. Count rate (doubles/fission/pair/Sr) 4.7 Late-time emission of prompt γ rays 0.028 measured MCNPX−PoliMi FREYA−PoliMi 0.026 0.024 0.022 0.02 0.018 240 0.016 Pu (sf) 0.014 20 40 60 80 100 120 140 160 180 θn−n (degrees) Fig. 43. (Color online) The neutron-neutron angular distribution for 240 Pu(sf) was measured recently [65] (70 keVee distribution in Fig. 10 of Ref. [65]) and compared to MCNPXPoliMibuilt-in fission model and FREYA simulations with x = 1.2. The neutron detection threshold for all results is 0.65 MeV, equivalent to 70 keVee in light output. Results on n-n angular correlations in spontaneous fission were also obtained at LLNL with the detector setup shown in Fig. 18, with a 240 Pu source located at the center. The measurements were carried out using a 4.5 mg sample of 240 Pu (98% pure) of intensity 4,590 neutrons/s. The contribution of fission neutrons originating from other plutonium isotopes present in the sample are negligible. The data were obtained over a 23-hour period. Only fission neutrons detected within a 40 ns time window were considered correlated. Figure 44 shows the measured cross-talk corrected correlations for several neutron kinetic energy cutoffs with full symbols. The angle-dependent cross-talk corrections were estimated by Monte-Carlo simulation. The correc- Prompt γ-ray emissions can be delayed due to the presence of long-lived isomers in the fission products [119]. Depending on the specific half-lives of those isomers, the observed prompt fission γ-ray spectrum and multiplicity can change significantly with the time coincidence window. This is particularly true if one singles out a specific fragment, through, for example, γ-ray tagging. Figure 45 shows the relative cumulative γ-ray multiplicity as a function of time since fission, normalized to 1.0 at 5 µs, before any β-delayed contributions. A typical time coincidence window used for identifying prompt fission neutrons, in coincidence with a fission event, is a few nanoseconds. Up to 8% of the prompt γ rays are emitted after 5 ns since fission. The determination of the detected γ-ray multiplicity can therefore be biased by as much as 8% for 239 Pu and 235 U thermal neutron-induced fission reactions. In addition, if one uses prompt fission γ rays to estimate a neutron detector efficiency, as discussed in Refs. [120, 121], the effect of these late prompt γ rays is to artificially bias the efficiency curve for the most energetic neutrons. Some differences in the calculations may be expected since CGMF and FREYA treat gaps in the RIPL-3 tables in different ways. Although MCNP6.2 does not include this time dependent γ-ray information, plans are to include it in a future version of the code. 4.8 Time chain correlations Methods based on time-correlated signals have been developed over many decades to characterize fissile materials [122, 123, 124]. Starting in the 1940s, Neutron Multiplicity Counting (NMC) techniques have enabled quantitative evaluation of masses and multiplications of fissile materials. In NMC, sequences of thermal neutron captures are recorded in 3 He tubes. The 3 He(n, p) reaction P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations produces two ions that generate charges within the gas. These charges are collected by a voltage-biased wire running through the tube. To determine the dominant constituents of the measured objects, the sequences were split into time windows and the numbers of neutrons arriving in each window were recorded to build statistical count distributions. Some materials such as 252 Cf simply emit several neutrons effectively simultaneously without multiplication (independent fission), whereas others like uranium and plutonium multiply the number of neutrons by subsequent, time-correlated fissions, to form bursts of neutrons. This multiplication translates into unmistakable count distribution signatures. To determine the type of materials measured, one can reconstruct measured count distributions with theoretical ones generated by a fission chain model. When the neutron background is negligible, the theoretical count distributions can be completely characterized by a few parameters: the mass of the object; the multiplication M ; the α-ratio (the ratio of the rate of (α, n) source neutrons to the rate of spontaneous fission neutrons); and the neutron detection efficiency . For such reconstruction to be successful, the precise knowledge of the multiplicity distributions of the isotopes is important. Indeed, to determine parameters of the object to be characterized by neutron multiplicity counting measurements, it is necessary to solve equations involving factorial moments of the multiplicity distribution, given in Eqs. (4)-(6). Any error in this distribution will thus lead to errors in the parameters of the reconstructed object. The neutron capture cross section in 3 He is only large enough to record fission neutrons after they have been thermalized in a moderating material. Therefore the time windows must be at least tens to hundreds of microseconds long to collect sufficient counting statistics to pick up counts from the same spontaneous fission or fission chain. In the case of a strong neutron source such as plutonium, many fission chains will thus be generated within individual time windows and therefore overlap within a window. While the neutron time correlations of interest are generated by individual fission chains, the signal received by the 3 He tubes is a convolution of multiple fission chains. To disentangle the contributions from separate fission chains, counting with 3 He requires high statistics and thus long measurement times. Scintillators, on the other hand, can detect fission neutrons directly without the need for a moderator. Scintillators detect neutrons through inelastic scattering, primarily on hydrogen, emitting a recoil proton and producing prompt scintillation light. Consequently, counting happens on a nanosecond time scale. To detect a correlation signal, microsecond time windows are not required, as with 3 He, but only of order ∼ 100 ns. These shorter time windows enormously reduce the number of overlapping chains within a window, so that windows encompass neutrons from far fewer fission chains. In terms of fissile material detection and authentication, scintillators prove to be more efficient than 3 He tubes. Indeed, smaller time gates allow a larger number of sam- 29 ples to be studied in a given measurement time, leading to reduced uncertainties. Figure 46 shows the uncertainties on the reconstruction of the mass and multiplication of a simulated Beryllium-Reflected Plutonium (BeRP) ball, often used in criticality-safety measurements [125]. When one reconstructs these quantities, the uncertainties on mass and multiplication are much smaller with the scintillators (a) than with the 3 He tubes (b), even for six times longer 3 He measurements. (Note the difference in the axes for the two types of detectors.) It has also been shown that these new fast counting signatures can help distinguish fast multiplication from thermal multiplication in a system where neutrons restart chains after they have been thermalized by a moderator [126]. Discrimination between neutrons and γ rays opens a new door to coincidence counting applications. Whereas one has historically focused on neutron multiplicity counting, correlations between neutrons and γ rays can now be measured. Feynman’s original point model theory, which uses the factorial moments of P (ν) given in Eqs. (4)-(6) to determine the composition of an unknown object, was originally developed for neutron detectors based on 3 He. The neutron capture cross section of this isotope is such that these detectors are blind to fast neutrons and most sensitive to neutrons that have undergone thermalization, a process which requires fission neutrons to down-scatter for tens of microseconds. To model the fission neutron detection, the original point model theory assumed that neutrons were detected on a time scale much longer than the fission chain evolution time scale: all neutrons in a chain were assumed to be emitted at time zero and then slowly diffused to the detectors over a time scale of tens of microseconds. While this is a good approximation for 3 He detectors, scintillators can detect γ rays and fast neutrons within nanoseconds of their production, long before the fission chain has ended. Therefore, Feynman’s original assumption of instantaneous fission chain evolution and slow neutron diffusion to the detectors no longer holds on a nanosecond time scale. Feynman’s theory has recently been extended to the detection of γ rays [127, 128, 129] and to fast counting [130]. In terms of data visualization, the shorter time scale over which scintillators detect neutrons enables the study of fission chains on the time scale over which they evolve. By plotting time intervals between fast neutron detections as a function of time, one can easily observe fission chains when fissile isotopes are present. Indeed, when no fissile materials are present, two bands are observed in Fig. 47: one around 10 ns, which is mainly due to individual cosmic-ray-induced fast neutrons registering multiple counts in adjacent detectors (neutron cross-talk) and a second around 0.1 s, which represents the average time interval between cosmic-ray secondaries interacting with the lead pile. The region between the two time bands is empty. When fissile materials are present, the gap between the two bands in Fig. 47 fills in, as shown in Fig. 48. The band around 10 ns is now a mix of fast neutrons coming from in- 30 P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations dividual fission reactions, fast neutrons coming from fast fission chains, and neutron cross-talk. The second band around 0.1 s represents the average time interval between spontaneous fissions as well as cosmic-ray secondaries interacting with the lead pile. The empty region has now filled in. The vertical streaks? filling the gap between the two bands point to the presence of neutron bursts due to fission chains that evolve over time scales of multiple microseconds. Such fission chains can only be due to neutrons thermalizing and restarting new fission chains after a thermal neutron induced fission of 235 U. Figure 49 focuses on a single fission chain observed in Fig. 48 at time 277 s. It shows the accumulation of fast neutrons as a function of time. The step of 14 neutrons emitted over less than 20 ns is indicative of a fast-neutron fission chain. In contrast to 3 He tubes, which detect only thermal neutrons, liquid scintillators detect neutrons above a higher energy threshold with minimum kinetic energies of 500 keV to 1 MeV. There is a threshold because the recoil protons do not produce a sufficiently unique scintillation light pulse to confidently distinguish them from the light pulse produced by γ-ray Compton interactions. Reducing this threshold is an area of very prolific research, and the discrimination between neutrons and γ rays has improved over the years through material research and advanced signal processing. A further advantage of scintillators is their ability to determine the neutron energies. The total light collected from the fast proton recoil in the scintillator is statistically proportional to the incident neutron energy. This enables a statistical energy spectrum to be determined which, for example, can be used to distinguish plutonium metal from plutonium oxide [131]. 5 Correlated fission data: status and challenges This section presents the current status of CGMF and FREYA and discusses some of the future challenges that have to be met. The first part deals with a brief discussion of the sensitivity of the results to input data and parameters. It then goes on to assess the current relevance of these codes for important applications. The next part highlights some of the modeling issues these codes need to address and describes some ways that current theory improvements may help address these issues. Since CGMF and FREYA depend upon data for model constraints, some of the basic experimental needs for modeling are touched upon. Finally, the current capabilities of the codes are briefly summarized. 5.1 Accuracy and sensitivity of fission event generators While modern codes such as the ones presented in this paper can predict many prompt fission data of interest, it is equally important to judge how accurate those predictions are and how sensitive they are to model assumptions and input parameters. Here an initial and somewhat limited attempt to answer this question is reported. The sensitivity of the calculated observables to a particular model parameter can be readily determined by simply varying that parameter. However, because the model parameters are not necessarily mutually independent, it is important to consider their correlations. Furthermore, by sampling the parameter values from probability distributions obtained by analyzing the experimental errors on the input data and performing a statistical analysis of the observables generated by the codes it is possible to identify the most critical measurements needed for improving the predictions. An important ingredient of these calculations is the pre-neutron emission fission fragment yields in mass, charge and kinetic energy. In particular, the kinetic energy carried away by the fission fragments determines, to a large degree, the intrinsic excitation energy left in the fragments and thus the number of prompt neutrons that they will evaporate. Any uncertainty in hTKEi will therefore produce an uncertainty in ν̄. Some preliminary studies of the sensitivity of the calculations to the input yields have been made in the case of 252 Cf(sf), for which a covariance matrix associated with the yields Y (A, Z, TKE) was estimated based on experimental data. Yields sampled from this covariance matrix were then used as input to FREYA and results on prompt neutrons and γ rays analyzed. Preliminary results were reported in [132] confirming the strong anti-correlation between hTKEi and ν̄. In the case of 252 Cf(sf), ν̄ is a “standard” [34] and is known with a reported uncertainty of ∼ 0.13%. This very small uncertainty places stringent constraints on the values of hTKEi, much stronger than the reported experimental uncertainty of 1.5 MeV [133]. Although ν̄ is very sensitive to hTKEi, the average prompt fission neutron spectrum, n-n angular correlations, and all γ-ray data are only weakly impacted by any change in hTKEi. Such sensitivity studies are very useful to assess what can be considered as robust predictions by the event generators, regardless of the precise knowledge of the input parameters, as opposed to quantities that show great sensitivity to those same parameters. Accurate simulations of critical assemblies are very sensitive to any change in the underlying nuclear data. Typical examples include the GODIVA and JEZEBEL critical assemblies, made almost entirely of 235 U and 239 Pu respectively, which are considered as the most accurate criticality-safety benchmarks and play a somewhat disproportionate role in validating evaluated nuclear data libraries. Their simulations are very sensitive to the PFNS and ν̄. At this moment, calculations of those two quantities by FREYA and CGMF cannot reproduce the quality of results present in evaluated libraries such as ENDF/BVII.1. Therefore the use of those two codes for very accurate criticality-safety applications should be avoided. A complete uncertainty quantification study would require more than a simple estimate based on parameter sampling. Assumptions and limits of the phenomenological physics models used in the description of prompt neu- P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations tron and γ-ray emission carry some systematic and correlated biases that are more difficult to assess. Another important example of target accuracy for fission event generators can be found in the assay of nuclear materials. To assay a spontaneous fission source like 252 Cf surrounded by unknown materials, a system of two equations needs to be solved for the source strength Fsf and the neutron detection efficiency : R1 = νFsf R2F = ν2 /ν (24) (25) where R1 is the measured neutron count rate in the detectors. (Note that νFsf = R1 / is the source rate on the y-axis of Fig. 46.) R2F is the measured number of correlated neutron pairs relative to the number of counts, sometimes referred to as the Feynman correlated moment. The spontaneous fission multiplicity distribution produced by FREYA is very close to the distribution from the Santi evaluation [26]. However, when solving Eqs. (24) and (25) for  and Fsf , the small differences between the two results lead to differences of 0.13% in the neutron detection efficiency  and 0.8% in the 252 Cf source strength. 5.2 Challenges for theory and modeling Fission event generators such as FREYA and CGMF, coupled with a transport code such as MCNP, provide a very powerful simulation tool for fission reactions, producing a wealth of correlated data on prompt fission neutrons and γ rays in multiplicity, energy and angle for every single fragmentation in mass, charge, kinetic energy, excitation energy and spin of the initial fragments. Given this daunting task, it is remarkable that such codes provide as good results as they do for a large quantity of data. However, fundamental theoretical questions about the fission process and the de-excitation of the fission fragments remain. Here a few are mentioned that should be explored in the future to improve the quality and predictive power of these unique tools. Accurate pre-neutron emission fission fragment distributions are a crucial input for those codes to produce reliable results. Depending on the type of output data one is interested in, the accuracy with which those yields need to be known varies. The total kinetic energy distribution Y (TKE) is important to accurately predict the prompt neutron multiplicity distribution P (ν). Several promising theoretical efforts [134, 135] are underway to generate fragment yields in mass, charge and kinetic energy and provide, for the first time, a rather predictive capability for this quantity. Using the results of such model predictions in fission event generators is still at the early stage but results are very promising. For a particular pair of fission fragments, for which the ground-state masses are relatively well known, and for a given TKE, the energy balance of the reaction gives the total excitation energy that will eventually be released through prompt neutron and γ-ray emission. How energy is shared between the two fragments at scission has been 31 the object of various theoretical studies [109, 110], with reasonable success. The importance of the extra-deformation energy induced by the collective deformation of the fragments near scission compared to their ground-state shapes, and the role of the level density in the fragments remain to be quantified more precisely. There is rather conclusive evidence that the average angular momentum of the fission fragments following scission is much higher (∼ 8h̄) than the value observed in low-energy compound nuclear reactions (∼ 4h̄). Various mechanisms for producing such high angular momentum have been discussed [136, 137, 138]. A quantifiable and predictive theory of the angular momentum distribution for a given fragmentation in mass, charge and kinetic energy remains to be developed. Again, microscopic calculations may be best equipped to address this point. The Hartree-Fock+BCS theoretical framework using a Skyrme nucleon-nucleon interaction was used [136] in the discussion of angular momentum production in the quantum pumping process. All those considerations, which point to the initial conditions of the fission fragments immediately following scission can only be addressed theoretically through an appropriate treatment of the dynamics of the fission process from the saddle to the scission point. Semi-classical macroscopic-microscopic approaches to this problem, which rely on a description of the fissioning nucleus via a set of shape parameters and which treat the dynamics with random walk or Langevin-type equations, have enjoyed some recent successes in describing the initial fragment yields [135, 139, 140]. Microscopic approaches such as the Time-Dependent Hartree-Fock [141], the Time-Dependent Generalized Coordinate Method [5] or even the TimeDependent Superfluid Local Density Approximation [6] are promising methods for describing fission dynamics in a more fundamental, microscopic approach. Fission dynamics right around the scission point is important for at least one other reason. So-called “scission neutrons”, emitted right at the time of the rupture of the neck have been postulated for many years, and have been used at various times to explain discrepancies between observed and calculated PFNS and angular distributions of neutrons. However, such interpretations are highly modeldependent and lead to a large spread of predictions for this extra source of neutrons. That is not to say that such a neutron source cannot exist. However, its existence should be addressed through a more consistent approach, leaving less room for fitting. Neither scission neutrons nor neutrons emitted during fragment acceleration are included in CGMF and FREYA. A high value for the initial angular momentum of the fission fragments could lead to anisotropic neutron emission in their center-of-mass frame. Such an assumption led Terrell [96] to derive an analytical formula that has been commonly used to infer a softer PFNS, in better agreement with selected experimental data. The anisotropy parameter used in this formula is adjusted to reproduce the PFNS below a few hundred keV, regardless of the neutron angular distributions, which are often unknown. Model 32 P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations predictions of the angular momentum distribution should be able to constrain this parameter more effectively. FREYA and CGMF treat the de-excitation of the fission fragments in the framework of the statistical WeisskopfEwing and Hauser-Feshbach nuclear reaction theories respectively. They rely on extensive databases and systematics of nuclear reaction model parameters, such as those of the RIPL-3 “Reference Input Parameter Library” [86]. Many of these models are phenomenological in nature and their parameters have been tuned to reproduce experimental data available in subsets of the nuclear chart, most extensively near the valley of stability. Some of the difficulties noticed in reproducing the PFNS in well-known fission reactions, e.g., 252 Cf(sf) with the present calculations, are certainly due in part to the inaccuracy of such models. Improvements in a global deformed optical potential applicable across a large suite of deformed, neutronrich nuclei, are necessary. In those models, the γ decay probabilities are estimated using the strength function formalism. Significant efforts have been devoted to this topic in recent years, leading in particular to the discovery of the importance of the “scissors mode” to the (n, γ) cross section calculations [142]. Systematics for the description of this mode throughout the nuclear chart are being developed. It will be interesting to study the impact of these systematics on the predictions of fission event generators. The nuclear level densities used to represent the continuum of states above the known energy region of resolved excitations are an important input to statistical nuclear reactions. They are commonly described using a constant temperature formula at low energy with a Fermi gas formula at higher energies. For many nuclei away from the valley of stability the parameters entering in these formulae are obtained from systematics not necessarily accurate in these regions. In the present context, the level densities can also be used to share the excitation energy between the fragments. 5.3 Experimental needs and status A tremendous amount of work has been accomplished recently in measurements of the fission fragment yields for various fission reactions and at a number of excitation energies. The SOFIA experimental program [72, 73] at GSI, Darmstadt, has been measuring fission product yields with unprecedented accuracy through Coulomb excitation in reverse kinematics. While such data are not monoenergetic, they still provide very valuable benchmarks for theoretical developments. The SPIDER project at LANSCE is a 2E-2v experiment measuring fission fragment yields for incident neutron energies from thermal up to 200 MeV. So far, n+235 U and n+238 U data have been released [143] with a two-arm spectrometer configuration. The incident neutron energy dependence of fission product yields was also measured at TUNL [144] for several important actinides using monoenergetic neutrons through (d, p) and (d, t) reactions. Other efforts such as the VERDI [145] and EXILL [146] experiments are also trying to provide new information on fission fragment yields and their dependence on excitation energy. Measurements of hTKEi as a function of incident neutron energy in the fast region and above were performed recently at LANSCE, providing invaluable data for 235,238 U(n,f) [143 147] and 239 Pu(n,f) [148] for Einc up to ∼ 200 MeV. These data have been used to constrain CGMF calculations below the second-chance fission threshold in order to accurately reproduce the energy dependence of the average neutron multiplicity. The initial rise of hTKEi at low incident energies observed in 235,238 U remains somewhat of a mystery and constitutes a test of current theoretical approaches to the fission dynamics. Other measurements of similar systems would be useful. Note that high-energy resolution measurements of hT KEi in the resonance regions have been performed [51, 112] to study the correlation between TKE and ν fluctuations. Experiments aimed at measuring correlations between prompt neutrons and γ rays as a function of fragment characteristics are obviously ideal to benchmark the type of studies presented here. The average neutron multiplicity as a function of the fragment mass, ν̄(A), has been measured for only a handful of spontaneous and low-energy fission reactions. Only two experimental results have been reported for higher-energy neutrons: 237 Np(n,f) and 235 U(n,f) [149, 150], although additional information can be somewhat inferred from proton-induced fission reactions [151]. The lack of good experimental data on ν̄(A, E ∗ ) as a function of increasing excitation energy, as well as on the average γ-ray multiplicity as a function of fragment mass and excitation energy, N̄γ (A, E ∗ ), is preventing the emergence of a clear theoretical model of the energy sorting and angular momentum production mechanisms at scission. Recent results [22] on the PFNS and the neutronlight fragment angular distributions θn,LF as a function of the fragment mass are very useful for testing assumptions made about the energy sharing mechanisms while at the same time providing constraints on the magnitude of any anisotropy parameter, often introduced rather arbitrarily in PFNS evaluations. Such measurements should be extended to neutron-induced reactions up to at least Einc = 20 MeV. Measurements of the average neutron multiplicity ν̄ as a function of TKE have been reported for various lowenergy and spontaneous fission reactions. All fission event generators such as FREYA and CGMF have been able to reproduce the observed trend for 252 Cf(sf) but have failed to reproduce the ones reported for 235 U(nth ,f) and 239 Pu(nth ,f). Recently, Hambsch argued [52] that all previously reported experimental trends have been biased due to poor mass and energy resolution and that more recent and more accurate results would tend to agree better with theoretical calculations. 5.4 Current capabilities At this time, the FREYA and CGMF fission event generators now integrated into MCNP6.2 can compute correlated P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations fission data for a limited set of fission reactions and isotopes. Those are listed in Table 3. Only spontaneous fission and neutron-induced fission have been considered so far. Photofission reactions play a special role. A simple hack of both code input files, with yields and TKE based on these quantities in neutron-induced fission, can be used to calculate photofission reactions, but only in an approximate way, which could lead to incorrect results if used improperly. Isotope n+233 U n+235,238 U 238 U 239,241 n+ Pu 238 Pu 240,242 Pu 244 Cm 252 Cf Einc (MeV) 0−20 0−20 sf 0−20 sf sf sf sf FREYA 3 3 3 3 3 3 3 3 CGMF 3 3 3 3 3 Table 3. List of isotopes and fission reactions that the current versions of FREYA and CGMF can handle. This list will eventually be extended to all isotopes and fission reactions, in particular photofission, in the near future. It is very important to note that, at this point, the use of those fission event generators should be limited to the simulations of correlated data only. As discussed earlier, the average prompt fission neutron spectra predicted by those codes are not necessarily in agreement with the evaluated PFNS present in libraries such as ENDF/B-VII.1. Two main reasons contribute to this situation. First, evaluated PFNS are often obtained by combining model calculations and experimental data [16]. In some cases, only experimental PFNS data are used, as in the case of the 252 Cf(sf) standard. Next, the FREYA and CGMF model parameters have not (yet) been tuned to obtain a better agreement with the evaluated PFNS. One should note that the evaluated PFNS are not necessarily correct. Because they are part of a coherent ensemble of evaluated data, which has been validated against various integral benchmarks, it is, however, difficult to modify them without negatively impacting the benchmark results e.g. by modifying other quantites such as the neutron multiplicity distribution. At this stage, the correlated fission option in MCNP6.2 cannot be used for criticality safety calculations. In the future, however, the hope is to be able to reconcile the event generator calculations with benchmark simulations. 6 Conclusions and future work Fission event generators such as FREYA and CGMF that can follow the de-excitation of fission fragments through neutron and γ-ray emissions in detail provide a unique and powerful view into the post-scission physics of a nuclear 33 fission reaction. They can be used to push for a more fundamental understanding of the fission process as well as for developing new powerful applications that implicitly or explicitly make use of the natural correlations between particles emitted from the fragments. Fundamental questions about nuclear fission remain unanswered or only partially answered to this day, such as: What are the configurations of the nascent fragments near the scission point? What is the possible contribution of scission or pre-scission neutrons on the average total prompt fission neutron spectrum? Are the prompt neutrons emitted from fully accelerated fragments only? Are prompt γ rays emitted in competition with prompt neutrons? Fission event generators are based on the statistical nuclear theory of compound nuclei, which should apply quite well in the case of excited fission fragments. Significant differences exist between FREYA and CGMF in the way they compute the probabilities of neutron and γ-ray emissions at a given stage in the de-excitation cascade. Those differences, instead of being a drawback, can help reveal interesting and overlooked features of the fission process. However, both codes rely on a suite of phenomenological model parameters whose values remain somewhat uncertain. In most cases, those parameters have been tuned to nuclear reactions close to the valley of stability, as opposed to the neutron-rich region where fission fragments are produced. Work to better understand and quantify the uncertainties associated with the model parameters, as well as provide reasonable bounds on the applicability of the physics models used in those cases, is in progress. From a more applied point of view, fission event generators can already be efficiently used to extract much more information from a set of neutron and γ-ray data than by simply relying on average quantities. For instance, neutron chain reactions cannot be simulated properly if only an average neutron multiplicity is used in the transport calculations. Experimental data and systematics on P (ν) have been used for a long time but, in most cases, assuming that the neutron energy spectrum does not depend on how many neutrons are emitted. The more realistic descriptions used in these Monte Carlo codes provide a way to go beyond such approximations. Also, most P (ν) systematics concern spontaneous fission or thermal neutroninduced fission reactions only. Both FREYA and CGMF can now provide results up to 20 MeV incident neutron energies. Angular correlations between emitted neutrons are also accessible since the successive emissions are obtained from any fission event. Such data can be invaluable for unambiguously detecting a fission signature. Prompt γ rays are simulated either using an appropriate statistical γ-ray strength function model or from specific transitions between known discrete nuclear levels. Those discrete lines show up mostly in the low-energy part of the γ-ray spectrum, and the calculations reproduce some of the most recent measurements on a few fission reactions quite well. As some of those discrete states are isomers whose halflives range from a few nanoseconds to a few microseconds, the prompt γ-ray spectrum changes in time, and can be used to tag specific fission fragments. 34 P. Talou et al.: Correlated Prompt Fission Data in Transport Simulations As has been seen throughout this work, FREYA and CGMF can reproduce many fission data reasonably well, although the validity and accuracy of the predictions depends significantly on the specific quantity of interest. It also depends to a large extent on many accurate experimental data that are already known for a particular fission reaction. For instance, a wealth of experimental information has been accumulated on the spontaneous fission of 252 Cf and further predictions of prompt neutron and γray observables from FREYA and CGMF are therefore more trustworthy for this nucleus than in making similar predictions for lesser known actinides, such as Cm. The incident neutron energy dependence of the results is also more questionable as multi-chance fission and pre-equilibrium components become important. Some predictions such as neutron-neutron angular correlations are also fairly robust against variations in the model input parameters while others are very sensitive to particular inputs, e.g., P (ν) varies strongly with the distribution in total kinetic energy of the pre-neutron emission fission fragments. The past few years have seen a resurgence in the accurate measurement and modeling of fission fragment yields in mass, charge and kinetic energy, Y (Z, A, TKE). An accurate representation of those yields is critical to providing reliable predictions for many prompt fission neutron and γ-ray observables. Several experimental efforts are already underway to address some of the most important limitations of past experiments, such as poor mass resolution and low statistics and thus offer accurate measurements of the fission fragment yields of many isotopes as a function of excitation energy for the first time. Nuclear theories have also made great strides toward developing rather predictive tools, with several approaches in healthy competition. Evaluated nuclear data files related to fission yields should see large improvements in the next few years, thereby significantly improving our understanding of prompt fission data as well. Finally, experiments aimed at capturing correlated fission data are also being devised, like DANCE+NEUANCE at LANL. While it is possibly foolish or even misguided to search for an all-encompassing experimental fission setup that would measure everything, it is also quite interesting and important to study correlated data as they place stringent constraints on theoretical models of fission, which in turn help devise more predictive tools. In the near future, the plan is to significantly extend the number of fission reactions, energies and isotopes that FREYA and CGMF can handle. Uncertainty quantification and the optimization of model input parameters to reproduce most of the experimental data is also underway. It is an important step towards ensuring good agreement between simulations and very accurate criticality benchmarks while at the same time predicting correlations beyond anything than can be reasonably stored in a tabulated file. The power of incorporating such fission event generators in MCNP will become more and more evident as time goes by and the simulations get faster and more reliable. Acknowledgements We are grateful to A. J. Sierk, M. B. Chadwick and J. P. Lestone for discussions. This work was supported by the Office of Defense Nuclear Nonproliferation Research & Development (DNN R&D), National Nuclear Security Administration, US Department of Energy. The work of P. T., M. E. R., M. T. A., P. J., M. J., T. K., K. M., G. R., A. S., I. S. and C. 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