XPZLIB processing flow

The XPZLIB processing flow is shown in Fig. 1. The incident neutron data and thermal neutron scattering sublibraries are processed using the built-in modules in NJOY2016 [8], including MODER, RECONR, BROADR, UNRESR, THERMR, and GROUPR. The modules are used for format conversion, resonance reconstruction, Doppler broadening, unresolved cross-section computation, thermalization, and multi-group cross-section generation. The GROUPR output data are in GENDF format. The XPZR module was developed and integrated into NJOY2016 to process GENDF-format data into HDF5-format data in XPZLIB. In addition, radioactive decay and neutron-induced fission product yield data are processed using XPZR.

Fig. 1Full-screen View

XPZLIB generation flow

Cross-section and burnup data processing

The related fission data absent from the GENDF file, including the fission spectrum χ and neutron production cross-section νσ_f, are computed as follows: ${\chi }_{g_2}=\frac{{\displaystyle \sum _{g_1}{\sigma }_{\text{f},g_1\to g_2}}{\varphi }_{g_1}+{\chi }_{\text{d},g_2}{\displaystyle \sum _{g_1}{\nu }_{\text{d},g_1}{\sigma }_{\text{f},g_1}{\varphi }_{g_1}}}{{\displaystyle \sum _{g_2}{\displaystyle \sum _{g_1}{\sigma }_{\text{f},g_1\to g_2}{\varphi }_{g_1}}}+{\displaystyle \sum _{g_1}{\nu }_{\text{d},g_1}{\sigma }_{\text{f},g_1}{\varphi }_{g_1}}},$ (1) ${\nu }_{g_1}{\sigma }_{\text{f},g_1}={\displaystyle \sum _{g_2}{\sigma }_{\text{f},g_1\to g_2}}+{\nu }_{\text{d},g_1}{\sigma }_{\text{f},g_1},$ (2) where ${\chi }_{g_2}$: fission spectrum. ${\chi }_{\text{d},g_2}$: Delayed neutron fission spectra. ${\nu }_{g_1}$: Number of neutrons released per fission event. ${\nu }_{\text{d},g_1}$: Number of delayed neutrons released per fission event. ${\sigma }_{\text{f},g_1}$: Fission cross-section. ${\sigma }_{\text{f},g_1\to g_2}$: Prompt neutron fission matrix. ${\varphi }_{g_1}$: Weighting flux. $g_1$: Incident neutron energy group. $g_2$: Outgoing neutron energy group.

The delayed neutron fission spectrum and the number of delayed neutrons released per fission event can be expressed as ${\alpha }_i={\displaystyle \sum _{g_2}{\chi }_{i,g_2}^{\text{d},GENDF}},$ (3) ${\chi }_{i,g_2}^{\text{d}}=\frac{{\chi }_{i,g_2}^{\text{d},GENDF}}{{\alpha }_i},$ (4) ${\nu }_{i,g_1}^{\text{d}}{\sigma }_{\text{f},g_1}={\alpha }_i{\nu }_{g_1}^{\text{d}}{\sigma }_{\text{f},g_1},$ (5) where ${\chi }_{i,g_2}^{\text{d},GENDF}$: Delayed neutron fission spectrum of the i^th group in GENDF. ${\alpha }_i$: Fraction of the i^th group delayed neutron yield, note that ${\alpha }_i={\upsilon }_i^{\text{d}}/{\upsilon }^{\text{d}}$. ${\chi }_{i,g_2}^{\text{d}}$: Delayed neutron fission spectrum of the i^th group in XPZLIB. ${\nu }_{i,g_1}^{\text{d}}$: Number of i^th group delayed neutrons released per fission event. ${\nu }_{g_1}^{\text{d}}$: Number of delayed neutrons released per fission event.

In addition to fission-related data, scattering-related data are processed using XPZR. The original elastic scattering cross section in the GENDF file is obtained by assuming that the scattering kernel is in the static state. Thermal scattering models are required to consider the thermal motion of a scattering kernel [25, 26]. Generally, there are two thermal scattering models. One is the free gas model for free atoms, and the other is the S(α, β) model for bounded atoms, e.g., hydrogen nucleus in the water [27]. In the thermal energy region, the total and elastic scattering cross sections are modified by XPZR as follows: ${\sigma }_{\text{thermal\_total}}={\sigma }_{\text{total}}-{\sigma }_{\text{elastic}}+{\sigma }_{\text{thermal\_scattering}}$ (6) ${\sigma }_{\text{thermal\_elastic}}={\sigma }_{\text{thermal\_scattering}}$ (7) where ${\sigma }_{\text{thermal\_total}}$: Total cross section in the thermal energy region. ${\sigma }_{\text{thermal\_elastic}}$: Elastic scattering cross section in the thermal energy region. ${\sigma }_{\text{total}}$: Total cross section in the GENDF file. ${\sigma }_{\text{elastic}}$: Elastic scattering cross section in the GENDF file. ${\sigma }_{\text{thermal\_scattering}}$: Thermal scattering cross section.

Accordingly, the scattering matrix is modified as follows: $\begin{array}{c}{\sigma }_{s,g_1\to g_2}={\sigma }_{\text{diffusion},g_1\to g_2}+{\sigma }_{n2n,g_1\to g_2}\\ +{\sigma }_{n3n,g_1\to g_2}+{\sigma }_{n4n,g_1\to g_2}\end{array}$ (8) where ${\sigma }_{\text{diffusion},g_1\to g_2}$: Diffusion matrix consisting of elastic and inelastic scattering matrices. This is equivalent to the thermal scattering matrix in the thermal energy region. ${\sigma }_{n2n,g_1\to g_2}$: (n, 2n) matrix. ${\sigma }_{n3n,g_1\to g_2}$: (n, 3n) matrix. ${\sigma }_{n4n,g_1\to g_2}$: (n, 4n) matrix.

In addition to cross-sectional data processing, XPZR can read radioactive decay and fission product yield data from the corresponding ENDF/B sublibraries and generate a compressed burnup chain. Nuclides with short half-lives and small fission yields, which are negligible for neutron transport calculations, can be grouped according to user-defined criteria. The typical criteria used in XPZR are a nuclide half-life of less than 30 days and a fission yield of less than 0.01%. XPZR can also accept a user-defined nuclide list and depletion channels, leading to different levels of detailed burn-up chains.

Automated processing system

A complete XPZLIB contains hundreds of nuclides, each with a corresponding NJOY input card and output file. To generate the input cards and manage the output files automatically, an automated processing system, PyNjoy2022, was developed based on the PyNjoy2012 system [15]. Fig. 2 shows the workflow of the PyNjoy2022 system.

Fig. 2Full-screen View

Workflow of the automated processing system PyNjoy2022

PyNjoy2022 contains an important Python script named PyNjoy.py, which provides the following functions: PyNjoy.pendf(): Processes the evaluated data in ENDF-6 format into pointwise cross-sectional data in PENDF format. PyNjoy.gendf(): Processes PENDF into group-wise cross-sectional data in GENDF format; PyNjoy.xpzlib(): Processes GENDF into XPZLIB data format via XPZR; PyNjoy.burnupxpz(): Generates depletion and fission yield data via XPZR.

The Input.py script contains a general XZPLIB description, for example, its weighting flux and energy group structure. It also contains the processing parameters of all the nuclides that must be included in XPZLIB. For each nuclide, Input.py calls the functions in PyNjoy.py to complete the data processing. First, PyNjoy.pendf() and PyNjoy.gendf() are used to generate PENDF and GENDF, respectively. Then, PyNjoy.xpzlib() is called to generate the cross-sectional data block of the current nuclide, which is automatically appended to the HDF5-format XPZLIB. After all nuclides have been processed, PyNjoy.burnupxpz() is called to generate the depletion data and add them to XPZLIB.

Released XPZLIBs

In general, a multi-group library with more detailed data is expected to have better accuracy and wider applicability, but at the expense of increased computational cost. Thus, it is important to use a sufficiently accurate XPZLIB for certain applications. As shown in Table 8, three typical XPZLIBs were processed with different numbers of energy groups, nuclides, and depletion reaction types to meet different user requirements. The three XPZLIBs were released via the Tsinghua cloud website (https://cloud.tsinghua.edu.cn/d/a7c675735ad8497f8a87).

Test cases

Table 9 lists the numerical cases based on HTGR and PWR fuel elements. For HTGR, two cases were taken from the standard fuel pebbles of HTR-10 [20] and HTR-PM [22]. The fuel pebble’s structure is shown in Fig. 8. It consists of a fuel region filled with dispersed TRISO particles [32] and a graphite shell. The TRISO particles consist of a spherical fuel kernel of UO₂ with multi-layer coatings, including a low-density pyrolytic carbon (PyC) buffer layer, inner high-density PyC layer, silicon carbide (SiC) layer, and outer high-density PyC layer. The detailed parameters of the two fuel pebbles and coated particles are listed in Table 10.

Fig. 8Full-screen View

HTGR fuel pebble model

For the PWR, two cases were constructed using a typical PWR assembly pin and an MOX fuel pin [33]. Because the XPZ code is limited to treating one-dimensional (1D) geometries, the square-lattice model was converted to a volumetric-equivalent cylinder model, as illustrated in Fig. 9. The key parameters are specified in Table 11.

Fig. 9Full-screen View

(Color online) PWR pin cell model

Numerical results

The three released XPZLIBs were used in the XPZ code to simulate the HTGR and PWR cases. The XPZ code is used in lattice physics computations for 1D models, and is also used for decay heat estimation. XPZ uses the subgroup method with intermediate resonance approximation to treat the resonance effect [34-36]. The OpenMC Monte Carlo (MC) code [37], which uses a continuous energy cross-section library processed from the ENDF/B VIII.0 nuclear data, was used to provide the reference results. In the MC calculations, the neutron generation parameter was set to 2000, with 200 inactive generations and 10000 neutrons per generation.

Transport calculations were performed for all test cases. The resulting k_eff values and their differences from the OpenMC reference results are presented in Table 12. The k_eff standard deviation for the OpenMC calculations was < 20 pcm. It was found that the simplified XPZLIB provided marginally acceptable accuracy for all test cases. This is likely because the WIMS-69 energy group structure cannot account for complex resonance effects; therefore, its application is limited to typical PWRs [38, 39]. The standard and refined XPZLIBs produced identical results and exhibited good accuracy for all test cases. This is due to the use of the SHEM-361 energy group structure and the expanded resonance energy range.

In the burnup calculations of the HTGR cases, a power density of 74.074 W/gU and maximum burnup of 100 MWD/kgU were obtained for the HTR-10 pebble, with values of 97.146 W/gU and 110 MWD/kgU, respectively, for the HTR-PM pebble. In the burnup calculations of the PWR cases, a power density of 40.45 W/gU and a maximum burnup of 60 MWD/KgU was employed.

The k_eff dependence on burnup for the four cases is shown in Fig. 10, respectively. The k_eff standard deviation of the OpenMC reference calculations was < 40 pcm. For all test cases, the k_effresults obtained using the standard and refined XPZLIBs agreed well with the Monte Carlo reference result, with a maximum difference below 200 pcm. However, the simplified XPZLIB can provide acceptable accuracy only in the typical PWR fuel pin case, whereas it yields up to 1000 pcm errors in the HTGR cases. This is mainly due to the difference in the neutron spectra of the HTR and PWR cases. A simplified XPZLIB with a WIMS-69 energy structure was optimized based on the PWR neutron spectrum. The HTR cases require a more refined energy structure.

Fig. 10Full-screen View

k_eff with burnup for the (a) HTR-10 fuel pebble, (b) HTR-PM fuel pebble, (c) typical PWR fuel pin, and (d) PWR-MOX fuel pin

To compare the applicability of the XPZLIBs in nuclide transmutation simulations, important nuclides from the HTR-10 fuel pebble at the final burnup step were analyzed. Table 13 lists the atomic densities and relative errors of the standard and simplified XPZLIBs compared with those of the refined XPZLIB. The standard XPZLIB provided consistent results for most of the important nuclides compared to the refined library. Conversely, the simplified XPZLIB model exhibited considerable errors for some important nuclides. For instance, ¹³⁷Cs, an important isotope for burn-up measurements [40], presented a discrepancy of approximately 4%.

Decay heat calculations are crucial for nuclear reactor safety analysis. It is known that decay heat accounts for approximately 6% of the operational power at the moment of reactor shutdown. It decreases rapidly within several hours owing to the decay of short half-life nuclides; thereafter, it is dominated by the decay of middle and long half-life nuclides. For accurate decay heat simulations, it is necessary to consider short half-life nuclides; however, they are usually neglected in neutron transport calculations.

To test the applicability of the XPZLIBs for decay heat calculations, the HTR-10 fuel pebble was depleted to 100 MWD/kgU with a designed power density of 74.074 W/gU, followed by a cooling time of five days. The reference results were obtained using a superfine XPZLIB with an uncompressed burn-up chain containing 3821 nuclides and 12 depletion channels, which are comparable to the delicate libraries adopted by nuclide inventory calculation codes such as NUIT [41] and ORIGEN-S [42]. As shown in Fig. 11, the results obtained from the refined XPZLIB are in excellent agreement with the reference results, demonstrating its applicability in decay heat calculations. Both the standard and simplified XPZLIBs exhibited significant deviations from the reference results after a short shutdown period, as the two libraries lacked many short half-life nuclides.

Fig. 11Full-screen View

Decay heat results for the HTR-10 fuel pebble