Free Gas Model Research Articles

Investigation of the impact of thermal neutron scattering cross sections and angular distributions on criticality calculations

The present study focuses on the influence of neutron scattering cross section and angular distribution when employing the Thermal Scattering Law (TSL) on criticality calculation benchmarks, in comparison with the Free Gas Model (FGM). The benchmarks sensitive to beryllium oxide (HMT-027) and graphite (HMT-026) as well as some simplified Pressurized Water Reactor (PWR) benchmarks are interpreted using the OpenMC code to assess the influence of TSL. The results indicate that the influence of TSL mainly attributes to both the total scattering cross section and the secondary angular distribution characterized by the average cosine of the scattering angle. The HMT-027 and HMT-026 series benchmarks respectively show the more significant influence of one of these two effects. For the simplified PWR benchmark, employing the TSL of H in H2O weakens the neutron moderation effect, which results in a reduction of around 100 pcm in the calculated keff, while the combined influence of cross section and secondary angular distribution is negligible for the TSL of UO2 fuel.

Single-shot laser-driven neutron resonance spectroscopy for temperature profiling

The temperature measurement of material inside of an object is one of the key technologies for control of dynamical processes. For this purpose, various techniques such as laser-based thermography and phase-contrast imaging thermography have been studied. However, it is, in principle, impossible to measure the temperature of an element inside of an object using these techniques. One of the possible solutions is measurements of Doppler brooding effect in neutron resonance absorption (NRA). Here we present a method to measure the temperature of an element or an isotope inside of an object using NRA with a single neutron pulse of approximately 100 ns width provided from a high-power laser. We demonstrate temperature measurements of a tantalum (Ta) metallic foil heated from the room temperature up to 617 K. Although the neutron energy resolution is fluctuated from shot to shot, we obtain the temperature dependence of resonance Doppler broadening using a reference of a silver (Ag) foil kept to the room temperature. A free gas model well reproduces the results. This method enables element(isotope)-sensitive thermometry to detect the instantaneous temperature rise in dynamical processes.

Radial Oscillations of Strange Quark Stars Admixed with Dark Matter

We investigate the equilibrium structure and radial oscillations of strange quark stars admixed with fermionic dark matter. For strange quark matter, we employ a stiff equation of state from a color-superconductivity improved bag model. For dark matter, we adopt the cold free Fermi gas model. We rederive and numerically solve the radial oscillation equations of two-fluid stars based on general relativity, in which the dark matter and strange quark matter couple through gravity and oscillate with the same frequency. Our results show that the stellar maximum mass and radius are reduced by inclusion of dark matter. As to the fundamental mode of the radial oscillations, the frequency f0 is also reduced comparing to pure strange stars, and f02 reaches the zero point at the maximum stellar mass with dM/dϵq,c=0. Therefore, the stability criteria f02>0 and dM/dϵq,c>0 are consistent in our dark matter-mixed strange quark stars with a fixed fraction of dark matter. We also find a discontinuity of f0 as functions of the stellar mass, in contrast to the continuous function in pure strange stars. And it is also accompanied with discontinuity of the oscillation amplitudes as well as a discontinuous in-phase-to-out-phase transition between oscillations of dark matter and strange quark matter.

Discrepancy across various OpenMC versions due to thermal neutron scattering law

The performance of neutron transport calculations is heavily reliant on the fidelity of nuclear data. The Free Gas Model (FGM) is no longer applicable at low neutron energy range due to nucleus binding effects, necessitating the implementation of the Thermal Scattering Law (TSL). The present study systematically reports the unexpected discrepancies in keff calculations with TSL across different versions of OpenMC, an open-source Monte Carlo code developed by MIT. Based on four representative benchmark series (HMT-026, HMT-027, BEAVRS, and TRIGA fuel rod), the analysis reveals that updates in OpenMC v0.13.0 significantly contribute to these discrepancies, sometimes up to ~1000 pcm. After further review, a modification in the source code for coherent elastic scattering (suggestion #1949) may affect the sampling of the cosine of the coherent scattering angle. Additionally, recompilation of OpenMC v13.2 using the older segment code aligns reactivity calculations closely with version 0.12.2 and thus confirms the above conclusion. Nevertheless, both the developers and we are still uncertain whether this update is correct until now. A systematic examination of the source code is necessary in the near future. Another direct conclusion from the present study is the importance of including diverse benchmarks for code verification and validation.

Calculation of thermal neutron scattering data of MgF2 and its effect on beam shaping assembly for BNCT

MgF2 as a moderator material has been extensively used in the beam shaping assembly (BSA) that plays an important role in the boron neutron capture therapy (BNCT). Regarded as important for applications, the thermal neutron scattering data of MgF2 were calculated, based on the phonon expansion model. The structural properties of MgF2 were researched by the VASP code based on the ab-initio methods. The PHONOPY code was employed to calculate the phonon density of states. Furthermore, the NJOY code was used to calculate the thermal neutron scattering data of MgF2. The calculated inelastic cross sections plus absorption cross sections are in agreement with the available experimental data. The neutron transport in the BSA has been simulated by using a hybrid Monte-Carlo-Deterministic code NECP-MCX. The results indicated that compared with the calculation of the free gas model, the thermal neutron flux and epithermal neutron flux at the BSA exit port calculated by using the thermal neutron scattering data of MgF2 were reduced by 27.7% and 8.2%, respectively.

Using effective temperature as a measure of the thermal scattering law uncertainties to UOX fuel calculations from room temperature to 80°C

The effective temperature Teff is an important physical quantity in neutronic calculations. It can be introduced in a Free Gas Model to approximate crystal lattice effects in the Doppler broadening of the neutron cross sections. In the last decade, a few research works proposed analytical or Monte-Carlo perturbation schemes for estimating uncertainties in neutronic calculations due to thermal scattering laws. However, the relationship between the reported results with Teff was not discussed. The present work aims to show how the effective temperature can measure the impact of the thermal scattering law uncertainties on neutronic calculations. The discussions are illustrated with Monte-Carlo calculations performed with the TRIPOLI-4® code on the MISTRAL-1 benchmark carried out in the EOLE facility of CEA Cadarache (France) from room temperature to 354 K (80°C). The uncertainty analysis is focused on the impact of the thermal scattering laws of H2O and UO2 on the neutron multiplication factor keff for UOX fuel moderated by water. When using the H2O and UO2 candidate files for the JEFF-4 library, the variation range of Teff leads to a keff uncertainty of 2.3 pcm/K, on average. In the temperature range investigated in this work, Teff uncertainties of ±20 K for H2O and ±10 K for UO2 give uncertainties on the multiplication factor that remains close to ±50 pcm. Such a low uncertainty confirms the improved accuracy achieved on the modelisation of the latest thermal scattering laws of interest for light water reactors. In the future evaluated nuclear data libraries, uncertainty budget analysis associated with the low neutron energy scattering process will be a marginal contribution compared to the capture process.

Temperature perturbation method using on-the-fly treatment of the cross-sections in the resolved resonance region

A negative fuel temperature reactivity coefficient is an important reactor physical parameter that indicates the inherent reactor operating safety. However, existing Monte Carlo perturbation methods still cannot accurately and efficiently predict the derivative of the k-eigenvalue with respect to temperature. The main difficulty lies in calculating the derivatives of the microscopic cross-section with respect to the temperature. This paper presents a Temperature Perturbation Method (TPM) that incorporates on-the-fly cross-section treatment as well as the free gas model into the differential operator method to calculate derivatives of the k-eigenvalue with respect to fuel temperature. TPM was then used to predict the k-eigenvalue at various temperatures for the Mosteller pin model. The differences between TPM and the direct difference method are less than 3% within 3 times of standard deviation. The results indicate that TPM can accurately and efficiently estimate the derivatives of the k-eigenvalue with respect to the fuel temperature.

IMPROVEMENT OF SCALE-XSProc MULTIGROUP CROSS SECTION PROCESSING BASED ON THE CENTRM POINTWISE SLOWING DOWN CALCULATION

The SCALE-XSProc multigroup (MG) cross section processing procedure based on the CENTRM pointwise slowing down calculation is the primary procedure to process problem-dependent self-shielded MG cross sections and scattering matrices for neutron transport calculations. This procedure supports various cell-based geometries including slab, 1-D cylindrical, 1-D spherical and 2-D rectangular configurations and doubly heterogeneous particulate fuels. Recently, this procedure has been significantly improved to be applied to any advanced reactor analysis covering thermal and fast reactor systems, and to be comparable to continuous energy (CE) Monte Carlo calculations. Some reactivity bias and reaction rate differences have been observed compared with CE Monte Carlo calculations, and several areas for improvement have been identified in the SCALE-XSProc MG cross section processing: (1) resonance self-shielding calculations within the unresolved resonance range, (2) 10 eV thermal cut-off energy for the free gas model, (3) on-the-fly adjustments to the thermal scattering matrix, (4) normalization of the pointwise neutron flux, and (5) fine MG energy structure. This procedure ensures very accurate MG cross section processing for high-fidelity deterministic reactor physics analysis for various advanced reactor systems.

Thermal neutron scattering properties of Bismuth crystal filter

ABSTRACT Accurate thermal neutron scattering cross-section data of the Bismuth (Bi) filter is required for high-fidelity calculations of the thermal neutron beam’s performance after the filtration in reactor column. In this paper, the phonon density of states of Bi was calculated using the ab initio method, based on which the thermal neutron scattering cross-section data was generated by the introduction of Bi coherent scattering model. By comparing the numerical results with the experimental data, the applicability of the model is verified. The model is then applied to simulate the neutron beam-shaping effect in the filter with an incident neutron spectrum in a research reactor column. Compared with the free gas model, the thermal neutron fraction was enhanced remarkably with an accurate phonon spectrum, and the phenomenon of Bi neutron filtration can be captured.

Extended Validity of the Energy Dependent Scattering Kernel within the Boltzmann Transport Equation

The scattering term within the Boltzmann equation was for decades approximated either through the Legendre polynomial for deterministic solvers or by /free gas model approach for Monte Carlo solvers. This, to some extent, inaccurate approach led to the assumption in several cases that the scattering term can be further “tuned” to simplify complex mathematical solver of the transport equation, mainly by reducing considerably the computation time, assuming no consequences on the physical results. The introduction of the resonant dependent scattering kernel to MONTE CARLO code and in particular the experimental validation within the resolved resonance range for Uranium and Thorium, in RPI (Renselear Polytechnic Institute), pointed out that the scattering term cannot be taken as a second order negligible term, but rather should be accurately regarded for any solution of the transport equation. Corollary to the considerable high impact of that “physical” scattering kernel, this study extends its importance beyond the epithermal resolved resonance range and aims to proof that the scattering kernel can and should be accurately dealt also at higher energies at least up to about several tens of keV. Moreover, in the debate between the purely quantum mechanics treatment known as the Optical Model -OM and the Doppler broadening based classical approach, the latter seems to be correct for this extended energy range. Strictly speaking, Newton’s Laws for the scattering kernel evaluation remain intact, yet in accordance to their quantum mechanics based integrated scattering cross section values as was shown for example by the well-known fundamental Breit Wigner formula.

On-the-fly temperature-dependent cross section treatment under extremes in RMC code

Neutron transport relevant to inertial confinement fusion always involves extremes, in which the physical quantities are extremely high, widely distributed and changes rapidly with space and time. In order to solve the memory and efficiency problems in nuclear data storage and processing, the on-the-fly temperature-dependent cross section treatment technique was investigated and developed under extremes in RMC code. Different strategies were adopted for different energy regions, i.e., the free gas model in the thermal region, the TMS method with 0K basis cross section temperature in the resolved resonance region, and the infinite dilution cross section in the unresolved resonance region, whereas the high energy region above the unresolved resonance region was not treated currently. The test results of Godiva sphere and plutonium sphere show that the on-the-fly technique has high accuracy, but the efficiency of the proposed technique still needs to be improved for some cases, and the optimization work with the elevated basis cross section temperatures is ongoing.

Investigation on the impact of thermal-kernel effect of lithium hydride on the reactivity of nuclear propulsion particle bed reactor

Due to the thermalization of lithium hydride (LiH) in nuclear propulsion particle bed reactor (PBR), thermal-kernel effect of LiH and its impact on the reactivity of PBR was studied. The phonon properties of LiH have been analyzed by the first-principles simulations using the VASP and PHONON codes. Applying the generated phonon density of states, the ACE-format thermal-neutron cross-section library of LiH has been generated by NJOY code. Neutronic modeling and simulation of PBR were done in this paper. The impact of LiH thermalization on the reactivity feedback of PBR was studied and analyzed. From the numerical results, it can be observed that the thermal effect of LiH was positive to the reactivity of PBR. The approximation of the free-gas model introduces significant error over 2000 pcm (10−5). In order to find the thermal mechanism of LiH, detailed research and analysis towards the thermal expansion and thermal dissociation have been also done. It was found that the thermal-kernel effect finally made the most contribution. In order to improve the safety of PBR reactor, the mixed moderator method was designed to reduce or resolve the positive moderator temperature effect.

Doppler broadening of neutron-induced resonances using ab initio phonon spectrum

Neutron resonances observed in neutron cross section data can only be compared with their theoretical analogues after a correct broadening of the resonance widths. This broadening is usually carried out by two different theoretical models, namely the Free Gas Model and the Crystal Lattice Model, which, however, are only applicable under certain assumptions. Here, we use neutron transmission experiments on UO2 samples at $T=23.7$ K and $T=293.7$ K, to investigate the limitations of these models when an ab initio phonon spectrum is introduced in the calculations. Comparisons of the experimental and theoretical transmissions highlight the underestimation of the energy transferred at low temperature and its impact on the accurate determination of the radiation widths $\Gamma_{\gamma_{\lambda}}$ of the 238U resonances $\lambda$ . The observed deficiency of the model represents an experimental evidence that the Debye-Waller factor is not correctly calculated at low temperature near the Neel temperature ( $ T_{N}=30.8$ K).

Comparison of the thermal neutron scattering treatment in MCNP6 and GEANT4 codes

To ensure the reliability of simulation tools, verification and comparison should be made regularly. This paper describes the work performed in order to compare the neutron transport treatment in MCNP6.1 and GEANT4-10.3 in the thermal energy range. This work focuses on the thermal neutron scattering processes for several potential materials which would be involved in the neutron source designs of Compact Accelerator-based Neutrons Sources (CANS), such as beryllium metal, beryllium oxide, polyethylene, graphite, para-hydrogen, light water, heavy water, aluminium and iron. Both thermal scattering law and free gas model, coming from the evaluated data library ENDF/B-VII, were considered. It was observed that the GEANT4.10.03-patch2 version was not able to account properly the coherent elastic process occurring in crystal lattice. This bug is treated in this work and it should be included in the next release of the code. Cross section sampling and integral tests have been performed for both simulation codes showing a fair agreement between the two codes for most of the materials except for iron and aluminium.

The Influence of Resonance Scattering to the Doppler Reactivity Coefficient

The paper presents the results of an evaluation of the effect of scattering on resonances in calculating the Doppler reactivity coefficient using the data preparation algorithms implemented in the GRUCON processing program. A comparison of the free-gas model with the resonance scattering model and published results of calculations performed using other methods of data preparation is presented. On the benchmark of Mosteller for light water grids with various fuel compositions it was shown that taking into account resonances in the differential cross sections of elastic scattering of uranium-238 leads to a shift of the Doppler reactivity coefficient by ~ 10% towards negative values, thereby increasing the negative feedback with the temperature of the fuel.

On-the-Fly Generation of Differential Resonance Scattering Probability Distribution Functions for Monte Carlo Codes

Current Monte Carlo codes use one of three models: (1) the asymptotic scattering model, (2) the free gas scattering model, or (3) the S(α,β) model, depending on the neutron energy and the specific Monte Carlo code. This paper addresses the consequences of using the free gas scattering model, which assumes that the neutron interacts with atoms in thermal motion in a monatomic gas in thermal equilibrium at material temperature T. Most importantly, the free gas model assumes the scattering cross section is constant over the neutron energy range, which is usually a good approximation for light nuclei, but not for heavy nuclei, where the scattering cross section may have several resonances in the epithermal region. Several researchers in the field have shown that the exact resonance scattering model is temperature dependent, and neglecting the resonances in the lower epithermal range can underpredict resonance absorption due to the upscattering phenomenon mentioned above, leading to an overprediction of keff by several hundred pcm. Existing methods to address this issue involve changing the neutron weights or implementing an extra rejection scheme in the free gas sampling scheme, and these all involve performing the collision analysis in the center-of-mass (CM) frame, followed by a conversion back to the laboratory frame to continue the random walk of the neutron.The goal of this paper was to develop a sampling methodology that (1) accounted for the energy-dependent scattering cross sections in the collision analysis and (2) was performed in the laboratory frame, avoiding the conversion to the CM frame. The energy dependence of the scattering cross section was modeled with even-ordered polynomials (second and fourth order) to approximate the scattering cross section in Blackshaw’s equations for the moments of the differential scattering probability distribution functions. These moments were used to sample the outgoing neutron speed and angle in the laboratory frame on the fly during the random walk of the neutron. Results for criticality studies on fuel pin and fuel assembly calculations using methods developed in this paper showed very close comparison to results using the reference Doppler-broadened rejection correction scheme.

Effect of Tensile Stress on Hall Coefficients of Nickel-Base Superalloys

This paper reports on Hall Effect measurements on nickel-base superalloys and their stress dependency. The work is motivated by the desire to develop a nondestructive method of characterizing the near-surface protective residual stress in metals. Our approach is based on the assumption that the Hall coefficient deviates under the stress. It is anticipated that stress measurements based on the Hall Effect are less contaminated by cold work and other effects than conductivity-based measurements such as eddy current. The paper focuses on the two superalloys, Inconel 718 and Inconel 600. The challenge is that Hall coefficients are small in metals, and the stress-induced changes are even smaller. To measure the small effect, the lock-in technique was used, with AC injected current and AC magnetic field. It was found that the Hall coefficients indeed vary proportionally to the stress. The proportionality coefficients are significantly larger than what are estimated from the volumetric effect in a free carrier gas model. The temperature and injected current dependences of the Hall coefficients were also measured, while no dependence on the magnetic flux density was observed.

Symmetry energy of nucleonic matter with tensor correlations

The nuclear symmetry energy (E_sym(\roh)) is a vital ingredient of our understanding of many processes, from heavy-ion collisions to neutron stars structure. While the total nuclear symmetry energy at nuclear saturation density (\rho_0) is relatively well determined, its value at supranuclear densities is not. The latter can be better constrained by separately examining its kinetic and potential terms and their density dependencies. The kinetic term of the symmetry energy, E_sym^kin(\rho_0), equals the difference in the per-nucleon kinetic energy between pure neutron matter (PNM) and symmetric nuclear matter (SNM), often calculated using a simple Fermi gas model. However, experiments show that tensor force induced short-range correlations (SRC) between proton-neutron pairs shift nucleons to high-momentum in SNM, but have almost no effect in PNM. We present an approximate analytical expression for E_sym^kin(\rho) of correlated nucleonic matter. In our model, E_sym^kin(\rho_0) = -10 MeV, which differs significantly from +12.5 MeV for the widely-used free Fermi gas model. This result is consistent with our analysis of recent data on the free proton-to-neutron ratios measured in intermediate energy nucleus-nucleus collisions as well as with microscopic many-body calculations, and previous phenomenological extractions. We then use our calculated E_sym^kin(\rho) in combination with the known total symmetry energy and its density dependence at saturation density to constrain the value and density dependence of the potential part and to extrapolate the total symmetry energy to supranuclear densities.

Monte Carlo Fuel Temperature Coefficient Estimation by an Adjoint-Weighted Correlated Sampling Method

It is very time-consuming to obtain a high-precision Monte Carlo (MC) estimate of the fuel temperature reactivity coefficient (FTC) through direct subtraction of two reactivity values from MC calculations at two different fuel temperatures. As an alternative to the direct subtraction MC estimate of the FTC, this paper presents a new method based on the adjoint-weighted correlated sampling technique. The new method translates the change in fuel temperature as the corresponding changes in both the microscopic cross sections and the transfer probabilities in scattering kernels described by the free gas model. The effectiveness of the new method is examined through continuous-energy MC neutronics calculations for pressurized water reactor pin cell and CANDU pressurized heavy water reactor lattice problems. The isotope-wise and reaction-type–wise contributions to the FTCs in the two problems are examined for two free gas models: the constant-cross-section and the resonance-cross-section models. It is demonstrated that the new MC method can predict the reactivity change due to fuel temperature variation as accurately as the conventional, more time-consuming direct subtraction MC method.

Revisiting the hot matter in the center of gamma-ray bursts and supernovae

Aims. Hot matter with nucleons can be produced in the inner region of the neutrino-dominated accretion flow in gamma-ray bursts or during the proto-neutron star birth in successful supernovae. The composition and equation of state of the matter depend on the dynamic β equilibrium under various neutrino opacities. The strong interaction between nucleons may also play an important role. We plan to extend the previous studies by incorporating these two aspects in our model. Methods. The modification of the β-equilibrium condition from neutrino optically thin to thick was modeled by an equilibrium factor χ ranging between the neutrino-freely-escaping case and the neutrino-trapped case. We employed the microscopic Brueckner-Hartree-Fock approach extended to the finite temperature regime to study the interacting nucleons. Results. We show the composition and chemical potentials of the hot nuclear matter for different densities and temperatures at each stage of β equilibrium. We also compare our realistic equation of states with those of the free-gas model. We find that it is important to properly describe the neutrino opacity and the strong interaction between nucleons, and they should be taken into account in model calculations.