Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
235 Computational Methods: General A Temperature Dependence Study of Alpha/Beta Cumulative Distribution Functions Based on S(α,β) Data Andrew T. Pavlou and Wei Ji Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180-3590, USA, [email protected], [email protected] INTRODUCTION When modeling a neutron transport process, the probabilities of neutron interactions with target materials must be accounted for properly. These interaction probabilities, or cross sections, are strong functions of the neutron energy and the target material temperature. We usually pre-calculate and store cross sections at many energies and temperatures to be used in neutron transport codes. For problems involving a coupling of neutronics and thermal hydraulics, pre-storing cross sections becomes memory intensive because of the fine temperature mesh needed to account for temperature feedback. For Very High Temperature Reactors (VHTR), many cross sections are needed on a fine temperature mesh to resolve the TRISO particle fuel and to account for high temperatures during accident scenarios like the Loss of Forced Cooling (LOFC) [1]. Methods have been developed in recent years to reduce cross section data storage in the resolvedresonance and epithermal energy ranges by generating the temperature-dependent cross sections on-the-fly [2,3]. In these on-the-fly methodologies, the energy distribution of the target nuclide is assumed to follow a MaxwellBoltzmann distribution, allowing for analytical expressions to be formed to drive the on-the-fly process. At thermal neutron energies, however, this assumption is not valid as chemical binding effects are dominant which complicates the calculation of the differential scattering cross section. A new strategy is needed to treat the temperature dependence of the double differential scattering data in the thermal energy region. This paper presents an on-thefly strategy to sample a neutron’s outgoing energy and scattering angle after a thermal scattering event for an arbitrary temperature. A method proposed by Ballinger [4] is introduced to construct probability density functions (PDFs) of momentum transfer (α) and energy transfer (β). The cumulative distribution functions (CDFs) of these PDFs are then constructed on a temperature mesh and an incoming energy mesh for common moderator nuclei. Regression models are then used to fit the temperature dependence of these CDFs. The best functional fits are chosen based on an error analysis procedure and the outgoing parameters are then sampled on-the-fly from the coefficients of the chosen functional fits. MOTIVATION In this work, we focus our attention on the incoherent inelastic differential scattering cross section. The inelastic portion of scattering involves a change in the neutron energy after the scatter while the incoherent portion of scattering ignores atomic plane interference effects between the neutron and target. This cross section is classically expressed as [4] ( E E ' , ' , T ) b 2kT E ' /2 e S ( , , T ), (1) E where E and E’ are, respectively, the incident and scattered neutron energy, Ω∙Ω’ is the scattering angle, σb is the bound atom cross section, kT is the ambient temperature and S(α,β,T) is the scattering law which contains the quantum translational, rotational, and vibrational motions of the neutron. The quantities α and β represent, respectively, momentum and energy transfer, ' 2 EE ' , A0 kT (2) ' , kT (3) where A0 is the target nucleus to neutron mass ratio and μ is the cosine of the scattering angle. Scattering law data are generated with the nuclear data processing code NJOY [5] and stored in ENDF thermal scattering files [6] for select moderator materials at specific temperatures and on an (α,β) mesh. The (α,β) mesh is chosen to allow interpolation between values within a specified fractional tolerance. The interpolation law used is moderatordependent. Modern Monte Carlo codes (MC21 and MCNP6) use a continuous-in-energy representation for the outgoing scattering parameters, requiring a very fine energy mesh. Even for a single temperature, the S(α,β,T) data can be large as shown in Table I for selected moderators at room temperature from the ENDF/B-VII.1 library. Because many temperature sets are often needed, the memory load needed can become unnecessarily large. The approach we take is based on CDFs of α and β from the S(α,β,T) data. In the next section, a method to Transactions of the American Nuclear Society, Vol. 110, Reno, Nevada, June 15–19, 2014 236 Computational Methods: General Table I. ENDF/B-VII.1 Continuous S(α,β,T) file sizes for selected moderator materials at room temperature Material File Size [MB] Graphite 24 Water 24.9 U in UO2 50 O2 in UO2 75 Zr in ZrH 56 H in ZrH 116 p( | E , T ) e /2 max max min min p( | , E , T ) construct these CDFs that can be used to sample the scattered neutron energy and angle of scatter at a single temperature is introduced. Then, these CDF forms at different temperatures and incoming energies are investigated for generally-used light material nuclei. The temperature dependence of these CDFs is determined by fitting to several regression models. The final functional form for the temperature dependence of alpha and beta at different values of CDFs will be constructed based on the best fitted regression model. On-the-fly sampling of the outgoing parameters can then be implemented using the obtained functional forms. max min S ( , , T )d e /2 S ( , , T )d d S ( , , T ) max min S ( , , T )d . , (7) (8) Assuming a given initial energy E at some temperature T, a value of beta can be sampled from Eq. (7). Then, once this is done, Eq. (8) is used to sample a value of alpha. From these sampled values, the scattered energy and cosine of the scattering angle can be directly calculated. Fig. 1 shows an example of the beta CDF generated for graphite at an incoming energy of 1 eV at five temperatures. METHODOLOGY The sampling procedure described here is based on the direct sampling method used by Ballinger [4]. The sampling is performed by first writing the double differential scattering cross section, Eq. (1), in terms of α and β, ( , , T ) b A0kT 4E e /2 S ( , , T ). (4) The sampling is then performed on a PDF form of this cross section, found by dividing by the total cross section integrated over all alpha and beta values, p( , | E , T ) ( , , T ) max max min min ( , , T )d d . (5) In Eq. (5), the minimum and maximum values of alpha and beta are readily found using Eqs. (2) and (3) for minimum and maximum energy transfer and momentum transfer. Eqs. (4) and (5) are combined to give p( , | E , T ) e /2 S ( , , T ) max max min min e /2 S ( , , T )d d . (6) This PDF is broken up into two PDFs; one conditional PDF in alpha and the other in beta, Fig. 1. Beta CDF for graphite at E = 1 eV for 300K, 650K, 1000K, 1500K and 2000K. The temperature dependence of the β CDF was examined at discrete CDF probability lines in the range [0,1] (15 probability lines per energy) and on a mesh of incoming neutron energies in the range [1E-5, 1] eV (71 total energies). In total, there were four variables considered in the analysis: Ein, β, T and Pβ, where Pβ is the β CDF probability. The α CDF temperature analysis consists of the four variables: β, α, T and Pα, where Pα is the α CDF probability. Although the α PDF/CDF is also dependent on the incoming neutron energy, this variable is ignored to reduce the data storage. Instead, the CDF is analyzed over the entire given α mesh instead of between αmin and αmax. Then, the α bounds are calculated as needed for the desired incoming neutron energy and the appropriate section of the α is sampled. Note that the β mesh for graphite consists of 96 values in the range [0, 80]. Functional expansions based on different regression models are examined to fit the β(T) and α(T) data at different values of alpha and beta CDFs. This Transactions of the American Nuclear Society, Vol. 110, Reno, Nevada, June 15–19, 2014 237 Computational Methods: General temperature dependence was performed on a mesh of 35 values in the range [300, 2000] K at 50 K increments for graphite. A code was developed to build the CDFs for beta and alpha at each temperature in the mesh from S(α,β,T) data obtained from NJOY. The β(T) data are then outputted at each beta and Pα in their respective meshes. Likewise, the α(T) data are outputted at each beta and Pα in their respective meshes. A visualization of this procedure is given in Fig. 2 for the beta data. In the figure, only five temperatures (300K, 650K, 1000K, 1500K and 2000K) are shown along with four probability lines (0.2, 0.4, 0.6 and 0.8) for one incoming energy value (1 eV). This is for visualization only. In actuality, 35 temperatures, 15 probability lines, 71 incoming energies and 96 beta values are used for the study. Fig. 2. Temperature dependence of the beta CDF for graphite for E = 1 eV. Along each CDF probability line, the temperature dependence is examined through fitting functions. The next section compares different temperature fitting functions to the alpha and beta CDFs for graphite. RESULTS To determine the best fit for the data, eight different regression models were considered, shown in Table II. The an’s are the fitting coefficients and N is the order of the functional expansion. To determine the best fit for β(T) and α(T), the root-mean-square error (RMSE) is calculated at every CDF probability, incoming energy, and beta value. The RMSE is given by NT RMSE ( i 1 i i ,est )2 N T ( N 1) , (9) Table II. Fits for β(T) and α(T) analysis Fit # Fit Fit # N 1 a T 2 a T n n 0 N n n 0 N n 3 a ( T )n 4 a ( T ) n n 0 N n0 n n Fit N n 5 a (ln T ) 6 a (ln T ) 7 a ( ln T ) n 8 a ( ln T ) n n n 0 N n n 0 N n 0 N n 0 n n n n where θi is the true value of beta or alpha at the current temperature, θi,est is the estimated value of beta or alpha at the current temperature from the fitting coefficients and NT is the number of temperature values used in the analysis. The true value of beta and alpha is found by running the LEAPR module of NJOY at the desired temperature to obtain the S(α,β,T) data. Then, Eqs. (7) and (8) are used to build the true PDFs. These RMSE values are averaged over the 15 CDF probability lines. Next, these beta RMSE values are averaged over the incoming energy mesh and the alpha RMSE values are averaged over the beta mesh. In doing this, a single averaged RMSE value is found for each functional expansion for each expansion order. Table III compares these averaged RMSE values for each fit for expansion orders 1,…,4 for the β(T) data. The same is shown for the α(T) data in Table IV. Fits 5-8 have been omitted from the table because their values are large compared to the other fits. Table III. Average RMSE for β(T) for graphite. N=1 N=2 N=3 N=4 Fit 1 0.1783 0.0793 0.0407 0.0228 Fit 2 0.0610 0.0202 0.0120 0.0092 Fit 3 0.1339 0.0489 0.0225 0.0133 Fit 4 0.0652 0.0174 0.0124 0.0091 Table IV. Average RMSE for α(T) for graphite. N=1 N=2 N=3 N=4 Fit 1 0.5473 0.2706 0.1631 0.1197 Fit 2 0.1758 0.1165 0.0671 0.0514 Fit 3 0.4498 0.1856 0.1220 0.0888 Fit 4 0.2373 0.1271 0.0798 0.0571 The best fit is the one with the lowest average RMSE. As the functional expansion order increases, the average RMSE decreases, but the storage of more coefficients is necessary. To determine which functional expansion order is best to use, the change in the average RMSE between adjacent expansion orders for a fit should be small. It was decided to choose Fit 4 with N=2 for the Transactions of the American Nuclear Society, Vol. 110, Reno, Nevada, June 15–19, 2014 238 Computational Methods: General β(T) data and Fit 2 with N=4 for the α(T) data. The total storage size of these coefficients is roughly 172 kB and can be used to sample secondary parameters for any temperature and any energy. This is a substantial improvement from the original 24 MB storage per temperature. To test the goodness of these coefficients, a simple Monte Carlo code was written to sample alpha and beta many times using only the coefficient files. Linear interpolation is performed to intermediate values in the energy and beta meshes when necessary. A plot of the energy (or beta) mesh versus the relative frequency of each sampled beta (or sampled alpha) reproduces the beta (or alpha) PDF. Integrating over this produces the CDF. This is then compared to the true CDF found from the true S(α,β,T) data at the incoming energy, beta value and temperature. Figs. 3 and 4 show the relative errors between the true values and the estimated values found from the fitting coefficients for the beta CDF and alpha CDF, respectively. From Figs. 3 and 4, the largest relative errors occur for high CDF probability values. This is because the standard deviation for the number of sampled values that fall inside the bins with CDF values greater than 0.999 is large, resulting in large errors. To remedy this, more samples could be run to obtain more values in these bins, lowering the standard deviation. However, for the majority of the CDF region, [0.001, 0.9], the largest relative error is 0.075 for beta and 0.736 for alpha. CONCLUSIONS Functional expansions were used to fit the temperature dependence of the alpha and beta CDFs constructed from S(α,β,T) data from graphite. For the β(T) data, second-order functional expansions in T -1/2 were used to sample beta. For the α(T) data, fourth-order functional expansions in 1/T were used to sample alpha. Both coefficient sets gave excellent results in the majority of the CDF range [0.001, 0.9]. The storage of coefficients is 172 kB and can be used to sample outgoing energy and angle for any incoming energy and any temperature onthe-fly. This is a large improvement from the current method which requires 24 MB of storage for each temperature. These coefficients are more compact than the current ACE data and can be combined with other onthe-fly methods in the future to complete the modeling of temperature effects for Monte Carlo codes. REFERENCES Fig. 3. Relative error in β from true and estimated CDF for graphite, T=875K, E=0.00518 eV. Fig. 4. Relative error in α from true and estimated CDF for graphite, T=875K, β=1.865. 1. R.B. VILIM, E.E. FELDMAN, W.D. POINTER, T.Y.C. WEI, “Initial VHTR Accident Scenario Classification: Models and Data”, Argonne National Laboratory, Nuclear Engineering Division Status Report (2004) 2. G. YESILYURT, W.R. MARTIN, F.B. BROWN, “Onthe-Fly Doppler Broadening for Monte Carlo Codes”, Nucl. Sci. Eng., Vol. 171, pp. 239 – 257 (2012) 3. E.E. SUNNY, W.R. MARTIN, “On-the-Fly Generation of Differential Resonance Scattering Probability Distribution Functions for Monte Carlo Codes”, Proc. of the International Conference on Mathematics and Computational Methods Applied to Nuclear Science & Engineering (M&C 2013), Sun Valley, Idaho (2013) 4. C.T. BALLINGER, “The Direct S(α,β) Method for Thermal Neutron Scattering”, Proc. Int. Conf. on Math. and Comp., Reac. Phys., and Env. Anal., Portland, Oregon, April 30 – May 4, 1995, American Nuclear Society (1995) 5. R.E. MACFARLANE, et al., “The NJOY Nuclear Data Processing System, Version 2013”, Los Alamos National Laboratory Report LA-UR-12-27079 (2012) 6. CSEWG, “ENDF-6 Formats Manual”, Brookhaven National Laboratory Report BNL-90365-2009 Rev. 2 (2011) Transactions of the American Nuclear Society, Vol. 110, Reno, Nevada, June 15–19, 2014