Evaluation of the 235 U prompt ﬁssion neutron spectrum including a detailed analysis of experimental data and improved model information

. We present an evaluation of the 235 U prompt ﬁssion neutron spectrum (PFNS) induced by thermal to 20-MeV neutrons. Experimental data and associated covariances were analyzed in detail. The incident energy dependence of the PFNS was modeled with an extended Los Alamos model combined with the Hauser-Feshbach and the exciton models. These models describe prompt ﬁssion, pre-ﬁssion compound nucleus and pre-equilibrium neutron emissions. The evaluated PFNS agree well with the experimental data included in this evaluation, preliminary data of the LANL and LLNL Chi-Nu measurement and recent evaluations by Capote et al. and Rising et al. However, they are softer than the ENDF/B-VII.1 (VII.1) and JENDL-4.0 PFNS for incident neutron energies up to 2 MeV. Simulated effective multiplication factors k eff of the Godiva and Flattop-25 critical assemblies are further from the measured k eff if the current data are used within VII.1 compared to using only VII.1 data. However, if this work is used with ENDF/B-VIII.0 β 2 data, simulated values of k eff agree well with the measured ones.


Introduction
The prompt fission neutron spectrum (PFNS) of 235 U is a quantity of interest for reactor physics and global security. Thus, it has been a subject of in-depth study within an IAEA coordinated research project on "PFNS of actinides" [1]. The 235 U PFNS at an incident neutron energy E inc = thermal was also selected as a reference PFNS within the standards project and a preliminary evaluation by Capote et al. [2]-mainly based on experimental datais available. This evaluation and a recent one by Rising et al. [3] for E inc ≤ 5 MeV support both a softer PFNS than the ENDF/B-VII.1 [4] (VII.1) and JENDL-4.0 [5] PFNS. The current β2 version of ENDF/B-VIII.0 [6] (VIII.0β2) contains Capote et al. data for E inc = thermal, Rising et al. data for 0.5-5 MeV and VII.1 data above.
Here, we present an evaluation of the 235 U PFNS induced by thermal to 20-MeV neutrons. This evaluation is based on modeling all physics processes relevant for this incident energy range, and a detailed analysis of experimental data and their uncertainties, which are briefly summarized in Sect. 2. Evaluated results are shown in Sect. 3 compared to data of VII.1, VIII.0β2, JENDL-4.0 and Rising et al. [3] at E inc = thermal. The evaluated data were used to simulate the effective multiplication factors of the Godiva and Flattop-25 critical assemblies presented in Table 1.

Input for evaluation
The vector of evaluated data ψ and the associated covariance matrix Cov ψ are obtained by a generalized least a e-mail: dneudecker@lanl.gov squares algorithm, which combines experimental data N, model values χ and their covariances Cov N and Cov χ . The design matrix S and its transpose S t are calculated by linear interpolation [7] and account for the outgoing neutron energy E and E inc of experimental data and model values.
The experimental covariance matrix Cov N was estimated by partitioning the uncertainties of each experiment (with standard deviations k ) according to their k sources, estimating associated correlations Cor k i, j for uncertainties between data points i and j and combining them to Cov N i j = k k i Cor k i, j k j . The covariance matrix between the same and different experiments shown in Fig. 2 was estimated using the newly developed uncertainty quantification program for experimental data "ARIADNE". This program is being developed to facilitate estimating detailed uncertainties and covariances of experimental data. To date, it includes the uncertainty quantification algorithms of Section III.M of [1] for estimating PFNS uncertainties. It will be extended in the future to estimate uncertainties of other observables.
Additional uncertainties were included in Cov N based on MCNP studies [17,18] of the experiments highlighted in Fig. 1. These MCNP studies uncovered possible biases  Model information: The "CoH" code [19] was used to calculate model PFNS χ for first and multiple-chance fission processes and the associated covariance matrix Cov χ by means of the Los Alamos model [20] combined with the Hauser-Feshbach [21] and exciton models [22]. The total spectrum χ in the laboratory frame, considering up to 4 th -chance fission processes, is given by The spectra φ j (E) are the pre-fission neutron spectra of j = {1, 2, 3} neutron(s) emitted before fission in compound and pre-equilibrium processes as described in Ref. [7]. The variables P f i correspond to the i = {1 st , 2 nd , 3 rd , 4 th }-chance fission probabilities. They are calculated by fitting fission barrier heights and curvatures to P f i obtained from VII.1 235 U total and i th -chance fission cross sections. The spectra χ i are weighted with P f i and ν i , the average number of neutrons emitted. The spectra χ i (E) are those of neutrons emitted promptly from the i th -chance fission fragments. The first-chance fission spectrum χ 1 (E) is described by an extended LAM [23] as a sum of effective light and heavy fission fragment neutron emission spectra, χ L1 and χ H 1 , weighted with the average light and heavy neutron multiplicities, ν L1 and ν H 1 , with the temperature distribution I (ε) of [24].
The variable E f denotes the kinetic energy per nucleon of the fission fragments and σ c,x the inverse compound nucleus formation cross section. The maximum temperature of the fission fragments is given by T m,x = √ E * / a x , with the average level density parameter a x = A x /11 dependent on the mass number A x . The average total excitation energy E * = E r + E inc + B n − TKE is calculated from the average energy release E r , E inc , the neutron binding energy B n and the total kinetic energy TKE averaged over the fission fragment distribution. A value of 170.5 MeV is taken for TKE at E inc = thermal and a linear decrease is assumed for higher E inc . The slope of this linear function was obtained by fitting to TKE data of [25]. The energy release is parametrized following Ref. [26]. The parameter b in Eq. (3) defines the strength of the anisotropy of the neutron evaporation in the centerof-mass frame with center-of-mass energy ε and assumes a value of 0.1. For multiple-chance fission spectra χ i (E) with i > 1, the average excitation energy of the pre-fission nucleus is corrected for the energy removed by the prefission neutrons. The covariances Cov χ (E i .E j ) associated with model values χ are calculated by sampling parameter values p l , calculating χ ( p l ) and computing from those the covariance estimator as described in Ref. [7].

Results
Evaluated results: The evaluated PFNS at E inc = thermal in Fig. 3 agrees well with experimental data of [8][9][10][11], the Rising et al. and the VIII.0β2 evaluation (Capote et al.) within its 1-σ uncertainty bounds. It is distinctly softer than the VII.1 and JENDL-4.0 evaluations. This trend to a softer PFNS can also be observed for E inc = 1.5 MeV. The data of Lestone et al. [14] and Knitter et al. [15] agree similarly well with all three evaluations. However, the preliminary Chi-Nu data [27]-which are not included in any of these evaluations-support a PFNS softer than VII.1. The higher evaluated PFNS at low E is partially caused by extension made to the LAM as shown in Ref. [23], but also by the experimental database chosen. The latter point is supported by the agreement of this evaluation with VIII.0β2 at E inc = thermal, which is mainly based on experimental data. Both evaluations include data of the Starostov et al. measurement series [8,9] which were not considered for the VII.1 evaluation.
At   [8-11, 14, 15], preliminary Chi-Nu data [27] and evaluations of [2][3][4][5][6] for E inc = thermal, 1.5, 6 and 14 MeV. Experimental data are scaled with respect to this work.  hundred keV at E inc = 6 MeV arises from the fact that only pre-fission and prompt fission neutrons are counted [7]. The tendency of this evaluation to softer PFNS compared to VII.1 and JENDL-4.0 for E inc ≤ 2 MeV is also visible in the evaluated mean energies in Fig. 4. The evaluated mean energies are nearly the same as those calculated from the VIII.0β2 evaluation for E inc ≤ 5 MeV. The mean energies of this work agree qualitatively with the JENDL-4.0 evaluation for E inc = 5-11 MeV. Above E inc = 11 MeV, the evaluated mean energies differ widely. No experimental data are currently available to guide evaluations for E inc > 11 MeV, but the Chi-Nu project might provide insight in the future.
Benchmarking  (9) multiplication factor k eff of Godiva is higher by 29 pcm, and further from unity, than if only VII.1 data are used. This slightly worse k eff was expected as VII.1 data are adjusted to yield a simulated value close to the measured Godiva benchmark. In addition, this evaluation differs visibly from the VII.1 PFNS in Fig. 3 leading to differences in the simulated k eff . An effect of a similar size (31 pcm) can be observed for k eff of Flattop-25 in Table 1.
However, if this evaluation is used with VIII.0β2 data, the simulated k eff of Godiva is very close to unity and differs from the value simulated with only VIII.0β2 data within the Monte Carlo (MC) sampling uncertainties. The Flattop k eff simulated with VIII.0β2 data are improved by more than 200 pcm compared to values simulated with VII.1. Again, k eff factors differ within their MC sampling uncertainties if simulated with this evaluation combined with VIII.0β2 compared to using only VIII.0β2. These similar values of k eff concur with the fact that the VIII.0β2 PFNS and the current evaluation agree for E inc ≤ 5 MeV within the evaluated uncertainties of this work (see Figs. 3-4).

Summary and outlook
An evaluation of the 235 U PFNS induced by neutrons of thermal to 20 MeV was presented. The covariances of experimental data [8][9][10][11][12][13][14][15][16] were estimated in detail using information provided in Refs. [17,18]. The Los Alamos model [20] combined with the Hauser-Feshbach [21] and the exciton models [22] were used to model prompt fission, pre-fission compound nucleus and pre-equilibrium neutron emissions. The evaluated results agree reasonably well with PFNS and mean energies of recent evaluations by Capote et al. [2] at E inc = thermal and Rising et al. [3] for E inc ≤ 5 MeV, which are included in the VIII.0β2 library. The PFNS of these three evaluations are softer than those of VII.1 and JENDL-4.0 for E inc ≤ 2 MeV (Figs. 3-4). Preliminary Chi-Nu data [27] support this tendency of a softer PFNS. Calculated effective multiplication factors k eff of Godiva and Flattop-25 using this evaluation combined with VIII.0β2 are close to the measured benchmarks ( Table 1). The k eff of Flattop-25 is better than the one simulated using VII.1. Further benchmark studies-such as calculating k eff of thermal solution critical assemblies-need to be undertaken.