In the evaluated nuclear data files, cross sections are represented as a combination of tabulated cross sections and resonance parameters. In the thermal and fast energy regions, the cross sections are stored in the form of tabulated values with an interpolation law defining how to interpolate between the tabulated values. In the resolved resonance region (RRR), resonance parameters and corresponding formulae are given for reconstruction. In the resolved resonance region (URR), the resonance peaks are so close that the experimental resolution can hardly measure individual resonances. The resonance parameters cannot be explicitly given as the RRR, which are provided in the form of mean values and corresponding probability distributions among some given energy regions. In ENDF-6 format, the average resonance parameters for URR are given in Single-Level Breit-Wigner formula.
The main task of nuclear data processing codes is to translate the data in evaluations to a defined format used for transport, depletion or shielding calculation. At the present time, NECP-Atlas is capable of generating library for transport calculations. In the following part, the processing methods adopted in NECP-Atlas are briefly described.
1.Reconstruction and linearization
In RRR, the resonance parameters are given for reconstructing the cross section values on a fixed energy grid. To obtain the energy dependent cross sections, the resonance contribution is firstly calculated using resonance parameters and corresponding formula. Then the resonance contribution is added to the background cross sections.
One important procedure in nuclear data processing codes is to reconstruct the cross sections in RRR. NECP-Atlas can treat resonance parameters of the following formats: Single-Level Breit-Wigner, Multi-Level Breit-Wigner, Reich-Moore and R-Matrix Limited.
For URR, in the reconstruction process, the infinitely dilute cross sections are computed using the average resonance parameters and probability distributions. Generally, the energy points, where the average parameters are provided, are chosen for the reconstruction, and the cross sections at other energies are obtained by linear interpolation. However, for some isotopes, the original energy points in evaluations may not be sufficient to represent the shapes of cross sections. To overcome the potential drawbacks, NECP-Atlas adds some energy points among the original ones, and the cross sections at these energies are calculated using the unresolved resonance parameters which are obtained by linear interpolation. The interval of energy steps can be controlled by the users.
In principle, the cross sections at an energy point can be uniquely defined by the interpolation laws or resonance formulae, but it is a convention to convert the cross sections from their original nonlinear-tabulated form to linear interpolation form. The purpose of this treatment is to ensure the consistency of the sum cross section (such as total cross section) with the summation of its partial cross sections, and to prepare linearized data for the following processing procedures (such as Doppler broadening). The conventional interval-halving technique is adopted for linearization in NECP-Atlas. The output of energy grid points is set to be 9 significant digits to avoid that two adjacent energy points become so close that they are outputted as the same energy.
2 、Doppler broadening
In the evaluations, the nuclear data is stored as functions of relative kinetic energy between the incident neutron and target nucleus, which means that the neutron energy in evaluations is measured in the relative frame of reference. Before transport calculation, the neutron energy should be transformed from the relative to laboratory frame of reference, because the Boltzmann equation is expressed in the latter. At 0K, the target nuclei in a medium are stationary, so the neutron energies in the two frames of reference are identical. While in the real medium at a temperature, the target nuclei are moving about in a random manner with a distribution of velocity, therefore, neutrons with the same kinetic energy in laboratory frame of reference will have different relative kinetic energies when the target moves in different velocities. Therefore, we need define an effective cross section in laboratory frame of reference to preserve the average reaction rate in relative frame of reference. This is known as Doppler broadening. Besides, the velocity distribution of target nuclei is dependent on temperature, so that the effective cross section varies with the medium temperature.
The Doppler broadening equation is established by preserving the average reaction rate in the relative frame of reference and assuming that the speed distribution of target nuclei is a Maxwellian distribution and the angular distribution is isotropic for a given speed. In NECP-Atlas, the Kernel Broadening method is adopted to solved the Doppler broadening equation. This method is originally implemented in the SIGMA1 code , and now is used in most major cross section processing codes, such as NJOY and AMPX. In this method, the cross sections at 0K or any lower temperature are firstly represented as a series of linear-tabulated points. Except for this treatment of the cross sections, the Doppler broadening equation is solved exactly without any further approximations.
3、 Processing of nuclear data in URR
In URR, the evaluations provide mean values of resonance spacing, resonance partial width and their probability distributions. The energy dependent cross sections cannot be constructed as RRR. In NECP-Atlas, there are three different processing methods for the nuclear data in URR.
Firstly, the effective self-shielded cross sections of different energy regions are calculated at various dilution conditions, which are used by deterministic transport methods. In NECP-Atlas, similar method with NJOY is used to calculate the effective cross sections. Like the reconstruction of infinitely dilute cross section mentioned above, NECP-Atlas adds some energy points among the original ones given in evaluations, and the cross sections at these energies are calculated using the unresolved resonance parameters obtained by linear interpolation.
Secondly, the probability tables used by Monte Carlo transport codes are calculated based on the unresolved resonance parameters. In NECP-Atlas, the ladder sampling method is used to generate the probability tables. In this method, for each spin sequence, a set of resonance energies in an energy range are sampled according to the resonance spacing distribution, and then the resonance cross section contributions at these energies are calculated using resonance parameters obtained by sampling the resonance partial widths. The above process is repeated for each spin sequence. A resonance ladder is obtained by accumulating the cross sections in all spin sequences, and then the probability table contributed from the resonance ladder is calculated. Enough ladders should be processed to guarantee the accuracy of the probability tables. The ψ-χ Doppler broadening method is adopted to get the probability tables at different temperatures.
Thirdly, the multi-band method is also implemented into NECP-Atlas to test different processing methods. In probability table methods, 20 cross section bands are always used, while in multi-band method, two band data could produce statistically identical keff results to those using 20 band data.
4、Generation of thermal scattering cross section
In the thermal energy region, the wavelengths of neutrons approach the size of molecules and the spacing of crystalline lattices. The target velocity will alter both the scattering cross sections and the secondary energy and angle distributions of scattered neutrons. The evaluations provide the thermal scattering law data of several bound molecules, such as hydrogen in water, graphite and beryllium. Different types of thermal scattering law can be processing by NECP-Atlas, e.g. coherent elastic scattering, incoherent elastic scattering and bound inelastic scattering. For the materials without thermal scattering law data, the free gas model is adopted to consider the effect of the target thermal agitation on the secondary energy and angle distributions.
5、 Treatment of resonance elastic scattering
In the epithermal energy regions, it’s customary to neglect the motion of the target nuclei, and the classical asymptotic transfer kernel is used to calculate the secondary energy and angle distributions of scattered neutrons. In this situation, a neuron always loses energy after colliding with the target. For the light nuclei, this assumption is valid, however, it is not the case for the heavy nuclei. After a neutron collides with the heavy nuclei in the epithermal energy region, it has a probability of gaining energy. Moreover, the heavy isotopes have resonance elastic scattering cross sections, which significantly increases the probability of gaining energy. Therefore, both the elastic scattering cross sections and transfer kernel should be Doppler broadened to incorporate the thermal agitation of target nuclei and the resonance elastic scattering. The Doppler broadening of elastic scattering cross sections is performed in the Doppler broadening module .
In NECP-Atlas, a new resonance elastic scattering kernel (RESK) processing module is developed to exactly Doppler broaden the transfer kernel to the desired temperature. In this module, the scattering kernels up to any Legendre order is exactly calculated for a series of incident neutron energies, and a precious interpolation method is developed to get the secondary energy and angle distributions at other incident energies.
6、 Generation of multi-group cross section
The multi-group cross sections used in deterministic transport calculation are generated by condensing the point-wise cross sections obtained by the methods described above. The calculation of weighting spectrum is a crucial part affecting the accuracy of multi-group cross sections.
There are two energy spectrum calculation methods to take into account the self-shielding effect in NECP-Atlas. The Bondarenko model is used to calculate the effective cross sections in the energy region where the resonance is narrow. For the RRR, a new point-wise neutron slowing-down spectrum calculation method is developed based on RESK, which can take into account the effect of neutron upscattering on the energy spectrum. The multi-group cross sections (such as fission and absorption cross sections) or resonance integral (RI) calculated using this spectrum are more accurate than those calculated using conventional asymptotic scattering kernel.
NECP-Atlas can also realize slowing-down spectrum calculation for 1D heterogeneous geometry, which is designed to generate the heterogeneous RI tables, because previous researches have indicated that the heterogeneous RI table provides more accurate results than the conventional homogeneous RI table.
7、 Transformation into different formats
The point-wise and multi-group cross sections obtained by the above processing procedures should be transformed into a specific format, before used in transport codes. At the present time, the point-wise cross sections and multi-group cross sections can be respectively transformed into ACE library and WIMS-D library by NECP-Atlas.
Besides, the NECP-Atlas is developed as a platform to research more accurate nuclear data processing methods. It has been connected with a new 3D whole-core high-fidelity transport code NECP-X. NECP-Atlas can provide all the cross sections needed by NECP-X.
8、 Depletion system compression
In NECP-Atlas, a compression method based on quantitative significance analysis on the Basic Unit Compression Operation (BUCO) was proposed. This method identifies basic unit compression operation in the fine depletion system which is based on the evaluated nuclear data library. For each BUCO, a quantitative evaluation indicator is defined with accuracy consideration of both neutronic and target nuclides aspects. Then these quantitative indicators which obtained based on the depletion calculation of some representative problems are used to compress the depletion system under prescribed acceptance criteria. Depletion system of different sizes can be obtained by changing the acceptance criteria. In the meanwhile, a method that makes combined usage of target nuclide definition and decay heat precursors was proposed to make the compressed depletion systems capable of delivering accurate decay heat calculation results.
Numerical results and analysis demonstrated that the obtained compressed depletion systems for PWR application could lower both the computation time and the storage requirements by over 90% and approximately 80% respectively for depletion calculation, and the saving effect was about 80% and 50% for neutronic calculation. Compared to depletion systems obtained by conventional WLUP method, calculation results of an exemplar problem showed that the accuracy degradation of kinf at deep burnup was minimized from more than 500 pcm to less than 100 pcm, while the accuracy degradation of nuclide number densities decreases from over 10% to less than 1%.
9、 Generation of broad-group sheliding library
The output files in GENDF (groupwise ENDF) format containing multi-group cross sections can be processed to generate broad-group library for calculation of shielding problems. There are three steps in the process.
Firstly, output files in GENDF format are processed to generate fine-group mater library. This library takes in account thermal scattering correction. Some data are recount, such as the total cross section. And some data are generated which are not included in the output files in GENDF format, such as Bondarenko factors and fission spectrum.
Secondly, a 1D model is needed to calculate using SN transport code, such as DORT. When reading the fine-group master library to obtain problem-dependent cross sections, resonance self-shielding is processed by the Bondarenko method and Dancoff factors are used to account for heterogeneous effects. Fluxes calculated in some zones of this model are adopted as weight functions to collapse fine-group library.
Thirdly, the broad-group library are obtained by collapsing the fine-group library. This library is able to provide cross section directly for SN transport code to calculate shielding problems.
10、 Tsl data calculation
In the thermal energy range, neutron energies are comparable to the atomic or molecular characteristic rotational, vibrational or translational energies of the scattering system. Thermal neutron scattering process is affected by the chemical bonding and thermal motion in the scattering system. Thermal neutrons can exchange energy with the scattering system by deexciting or exciting energy states of the system. In addition, the de Broglie wavelength of thermal neutrons approaches the interatomic distances in solids and liquids, and the interference effects arising due to the interaction between thermal neutrons and the scattering system must be taken into account. Consequently, thermal neutron scattering is affected by the atomic binding details of the scattering system and its structural and dynamical properties. In the Evaluated Nuclear Data File (ENDF), thermal neutron scattering in an atomic system is described using the thermal scattering law (TSL), which is the fundamental data to calculate thermal scattering cross sections. The main work is to prepare the TSL data for moderator materials which are not provided in ENDF.
Thermal scattering cross section are usually divided into three different parts:
• Coherent elastic. Important for crystalline solids like graphite or beryllium. In NECP-Atlas, a technique based on the anisotropic displacement parameters (ADPs) method is implemented to consider the anisotropy of interatomic forces and the correlation of the forces from different directions, removing the cubic approximation and atom site approximation used in the conventional method.
• Incoherent elastic. Important for hydrogenous solids like solid methane, polyethylene, and zirconium hydride. The Debye-Waller integral and effective temperature are calculated from the frequency spectrum.
• Inelastic. Important for all materials. Phonon expansion method is used process the continuous component of the frequency spectrum, the free gas model or the effective width model are used for the diffusion component in liquid material, and discrete harmonic oscillator model are used to describe the vibrational/rotational motion in polyatomic molecules. Finally, the intermolecular neutron interference in the inelastic TSL is taken into account by using the Sköld approximation.