International Conference on Reactor Physics, Nuclear Power: A Sustainale Resource Casino-Kursaal Conference Center, Interlaken, Switzerland, Septeer 14-19, 008 The ipact of 38 U resonance elastic scattering approxiations on theral reactor Doppler reactivity Deokjung Lee a,*, Kord Sith a, Joel Rhodes a a Studsvik Scandpower Inc., Idaho Falls, USA Astract The effects of accurate odeling of neutron scattering in 38 U resonances are analyzed for typical light water reactor (LWR) and next generation nuclear plant (NGNP) lattices. An exact scattering kernel is forulated and ipleented in a newly developed Monte Carlo code, MCSD (Monte Carlo Slowing Down), which solves a neutron slowing down in an infinite hoogeneous ediu and is used to generate resonance integral data used in the CASMO-5 lattice physics code. It is shown that the exact scattering kernel increases LWR Doppler coefficients y ~10% relative to the traditional assuption of asyptotic elastic downscatter for 38 U resonances. These resonance odeling iproveents are shown to decrease hot full power eigenvalues y ~00 pc for LWRs and ~450 pc for NGNPs. 1. Introduction NJOY (MacFarlane and Muir, 000) is coonly used to prepare cross section data for use in downstrea lattice physics codes (e.g., CASMO, CENTRUM, HELIOS, LANCER, PARAGON, etc.) and nuerous ultidiensional neutronics codes (e.g., MCNP, KENO, etc.). NJOY Doppler roadens cross sections for user-specified teperatures, and eploys an asyptotic scattering odel (i.e., assues that the nucleus is at rest in the laoratory syste) for neutron/nucleus elastic scattering interactions in the epitheral energy region. As a consequence of this assuption, all deterinistic neutronics codes eploying NJOY generated cross sections iplicitly ake the assuption that elastic scattering events in the epitheral range can e well approxiated y the asyptotic slowing down odel and upscattering events are ignored. Soe Monte Carlo codes approxiate the elastic scattering proaility density function (pdf) as (X-5 Monte Carlo Tea, 003): p( ( v ) v M ( V ) s r r μ, V ) =, (1) eff s ( vn ) vn where s (v) is the scattering cross section at 0K, eff s ( v n ) is the Doppler-roadened scattering cross section, v r is the relative velocity etween a neutron with a velocity v n and a target nucleus oving with a scalar velocity V, µ is the cosine of the angle etween the neutron and the target, and M(V) is the pdf for the Maxwellian distriution of target velocities. The pdf is often approxiated as: p( μ, V ) v M ( V ), () r where Eq. (1) has een siplified y assuing that the variation of s ( v r ) can e ignored (Carter and Cashwell, 1975). This assuption is quite good for * Deokjung Lee, deokjung.lee@studsvik.co Tel: +1 (08) 5-9854, Fax: +1 (08) 5-1187 1
light isotope scattering, ut is far fro true for heavy resonance scatterers such as 38 U. This assuption is rationalized y noting that heavy nuclides contriute little to neutron oderation ecause of the large ass difference etween neutrons and heavy nuclides. These assuptions have een previously questioned y nuerous researchers (Ouislouen and Sanchez, 1991; Bouland et al.; Kolesov and Ukraintsev, 006; Dagan and Broeders, 006). In fact, Ouislouen and Sanchez (1991) have shown that heavy nuclide scattering resonances can have a large effect on the average energy of scattered neutrons. They deonstrated that at certain energies near the resonance peaks the average scattered neutron energy actually exceeds the incident neutron energy. Studies have also shown that the asyptotic elastic scattering approxiation can lead to a significant underprediction of the Doppler feedack effect in LWRs (Bouland et al.; Kolesov and Ukraintsev, 006; Dagan and Broeders, 006). In this paper, the effects of various resonance scattering odels are investigated using an infiniteediu neutron slowing down code, MCSD, which is descried here. This code perfors a Monte Carlo siulation of neutron slowing down in the resolved resonance energy range and perits elastic scattering to e odeled with either the asyptotic or the exact scattering kernels. MCSD has een used to quantify the ipact of the asyptotic scattering approxiation. A ethod for capturing the physics of the exact scattering kernel and generating cross sections for downstrea deterinistic codes is presented. This ethod has een incorporated into the resonance treatent of CASMO-5 (Rhodes et al., 006), which is used to study the effects of the various scattering kernels.. Monte Carlo solver for neutron slowing down.1. Neutron slowing down prole The neutron slowing down equation for a hoogeneous ixture of three aterials in an infinite ediu can e forulated, as in NJOY (MacFarlane and Muir, 000), as: [ + ( E )] + + t E E ( s ) φ ( E ) = E / α, (3) K ( E E ) φ ( E ) de E ax E in sf s ( E ) K f ( E E ) φ ( E ) de where is the user-specified energy-independent ackground cross section, oderator section, and energy-independent scattering cross s is the adixed sf (E ) is the resonance aterial energy-dependent scattering cross section. The scattering kernel of the adixed oderator can e expressed y its asyptotic for 1 for α E E E K ( ) = (1 ) E E α E, (4) 0 otherwise where α = ( A 1) /( A + 1) and A is the ratio of the asses of the adixed oderator to a neutron. The scattering kernel of the resonance aterial is presented in the following sections... Asyptotic scattering kernel The scattering kernel of the resonance aterial can also e approxiated y the asyptotic kernel 1 for α f E E E K ( E E) = (1 α ) E. (5) f f 0 otherwise An ipleentation of this kernel in MCSD code is as follows. Consider a collision etween a neutron of speed v and a nucleus at rest in the laoratory syste (LS). The underlying assuptions for the asyptotic kernel are 1) the nucleus is initially at rest in the LS and ) the scattering is isotropic in the center-of-ass syste (CMS). Fro energy conservation and oentu conservation principles, the following relation can e derived easily: E A + Aμ + 1 =, (6) E A ( + 1) where E is the neutron energy efore the collision in the LS, E is the neutron energy after the collision in the LS, μ is the direction cosine of the scattered neutron, relative to v in the CMS, v is the neutron velocity efore the collision in the LS. Eq. (6) is used in the MCSD code for the asyptotic kernel with sapling of the angle, μ, ased on isotropic scattering..3. Exact scattering kernel The exact scattering kernel can e written as
K ( v v) = f 1 eff v ( v ) s v ( v ) p( v v) M( V) dμ dv, r s r (7) where s ( v r ) is the scattering cross section at 0K, eff s (v ) is the Doppler-roadened cross section, M(V) is the Maxwellian distriution, and p ( v v) is the proaility that the scattered neutron with initial velocity v will have final velocity v. The direct nuerical evaluation of p and the doule integration is coplicated, therefore a Monte Carlo ethod has een chosen to solve this neutron slowing down prole. Consider a collision etween a neutron of speed v and a nucleus of speed V in the LS. The relationships for the conservation of energy and oentu lead to an expression for the velocity of an elastically scattered neutron which can e written as (Bell and Glasstone, 1968) v V A A + vr + Vc v μ, (8) A + 1 A + 1 = c r where v + AV V c = and v r = V + v v Vμ, 1 + A v E are the neutron velocity and energy and and efore the collision in the LS, v and E are the neutron velocity and energy after the collision in the LS, μ is the direction cosine of the scattered neutron, relative to V c in the CMS, μ is the direction cosine of the nucleus efore collision, relative to v in the LS, V is the velocity of the target in the LS, V c is the velocity of the center of ass in the LS, and v r is the velocity of the neutron relative to the target in the LS. Eq. (8) is used with the appropriate sapling of the angles μ and μ, and velocity of target, V. With the assuption that scattering is isotropic in the CMS, the direction cosine of outgoing neutron, μ, is sapled fro a unifor pdf in the range [-1, +1]..4. The MCSD code A siple infinite ediu neutron slowing down code, naed MCSD, was written to solve Eq. (3) with either the asyptotic or the exact scattering kernel..4.1 Source neutron sapling MCSD assues a 1/E distriution of source neutrons over a user-specified energy range and then solves the neutron slowing down prole until neutrons are either asored or scattered to an energy elow the resolved resonance region..4. Cross section data The MCSD code uses ENDF/B-VII.0 cross section data processed via NJOY to generate pointwise PENDF data of 38 U at 11 teperatures. MCSD linearly interpolates in the point-wise cross sections to evaluate cross sections at the requested neutron energies..4.3 Tallies Three ajor types of tallies are perfored in MCSD. First is siply an edit for verification purposes, in which the scattering kernel, K i (the proaility that an incident neutron is scattered into the energy in [E i-1, E i ]), is tallied y [ E, E ] ) T N Ki ( E i 1 i = i /, (9) where N is the total nuer of incident neutrons and T i = w j. [ E E ] E j i 1, i Second is the group-wise scalar flux, tallied as φ = d w, (10) g E j g j ln( 1 r) where d = and r is a rando nuer. t Third is group-wise reaction rate for interaction type alpha, tallied as Rα = α d w. (11) g E j g j Using these tallied quantities, group asorption cross section is calculated y Ra g a g =, (1) φ g and the asorption resonance integral is coputed as 3
Iag where = ag, (13) ag + is the user-defined ackground cross section which excludes the potential cross section of resonance isotope itself. The standard deviation of ean resonance integral, I, is calculated y I I =, (14) I M ( M 1) where M is the nuer of atches, 1 1 I = I and I = M M I and I is the resonance integral of atch..4.4 Analog exact scattering kernel One ipleentation of the exact scattering kernel in MCSD is the analog exact scattering kernel (AESK). In the AESK, the pdf for the velocity of the target nucleus, V, and the cosine angle of the collision etween a neutron and a nucleus, μ, can e written as (Ouislouen and Sanchez, 1991) p( μ, V ) = vr ( μ, V ) M ( V ), (15) where 3/ V kt M V V e ( ) = 4π (16) πkt is the Maxwellian distriution and T is the equiliriu teperature, k is the Boltzann constant, and is the ass of the target nucleus. This pdf has two independent variales, μ and V. The cross section,, is a function of these two variales such that they cannot e sapled independently. Therefore, one can choose to saple angle first, ased on the following pdf p ( μ ) = ~ ( μ ), (17) where, ~ ( μ ) = v (, V ) M ( V ) dv r μ. (18) 0 The pdf for the speed of the nucleus ecoes p V ) = vr ( μ ) ( μ, V ) M ( ), (19) ( 0 0 V where μ 0 is assued to e known, i.e., already sapled y Eq. (17)..4.5 Fast effective scattering kernel The AESK kernel in the previous section is coputationally inefficient ecause the right hand side of Eq. (18) has to e nuerically integrated in every collision sapling during the neutron slowing down process. In order to iprove the coputational efficiency of the MC siulation, an accelerated nonanalog version has een developed which will e called here as fast effective scattering kernel. In, the two independent variales are sapled separately using their own pdfs and the weight of the neutron is updated appropriately. The pdf for the angle etween the target nucleus and incident neutron is assued to e unifor (isotropic), and the pdf for the speed of nucleus is the Maxwell-Boltzann distriution M(V). The weight of neutron is updated y vr t ( vr ) + w = w v, (0) eff vr t ( v ) + v eff where t is the total cross section at 0K, t is the Doppler-roadened total cross section to the teperature T, and is the ackground cross section. The Doppler-roadened cross sections are defined y eff 1 α ( v ) = vr α ( vr ) M ( V) dv, (1) v where α is the type of reaction..5. Resonance scattering kernel tests The first tests for MCSD were to verify the ipleentation of the asyptotic and exact scattering kernels. For this purpose, neutrons were started at various energies and the resulting scattering kernels were tallied and copared with kernels pulished y Ouislouen and Sanchez (1991). For neutron energies just elow the peaks of the ajor 38 U asorption resonances (incident neutron energy 36.5 ev), the 4
exact kernels are quite different fro the traditionallyassued asyptotic kernels, as displayed in Fig. 1. Note that a larger fraction of scattered neutrons gain energy in a high teperature (1000K) aterial than in a low teperature (300K) aterial. Upscatter fractions coputed with MCSD are copared to those of Ouislouen and Sanchez (1991) in Tale 1, and it can e seen that there is very close agreeent, which one can interpret as a verification of the ipleentation of the exact scattering kernel in MCSD. Fig. 1. 38 U scattering kernels at 36.5 ev Tale 1 Upscatter percentage at 1000K Resonance (ev) 6.67 36.67 Neutron energy (ev) O&S 1) 6.5 8.03 83.40(0.04) ) 7.0 8.1 8.0 (0.007) 36.5 54.3 55.8 (0.056) 37.0 7.95 7.6 (0.011) 659.00 1.4 1.18 (0.034) 661.14 664.00 0.84 0.74 (0.00) 1) Ouislouen and Sanchez (1991) ) One Siga Standard Deviation.6. Variance reduction techniques Use of the exact scattering kernels rather than traditional asyptotic kernels akes the solution of the neutron slowing down prole uch ore difficult to statistically converge. This is expected since the Monte Carlo siulation of the exact kernels saples for the target nucleus velocity and angle relative to the incident neutron, and the neutron cross sections in the vicinity of a resonance are extreely sensitive to the relative velocity. Consequently, it is extreely iportant to take advantage of nuerous variance reduction techniques to iprove siulation statistics. The perforance of nuerous variance reduction techniques have een exained y solving the neutron slowing down prole over the neutron energy range of 53.6 ~ 7. ev for 38 U at 1000K and a ackground cross section of 60 arns and tallying resonance integrals fro 47.9 to 7.7 ev (36.67 ev resonance). The details of acceleration techniques will e descried in the following sections. Tale suarizes a calculation with all accelerations and without acceleration. Each siulation eployed 10 atches of 1 illion source neutrons. The accelerations reduce variance y a factor of 1 for the asyptotic kernel and a factor of 7 for kernel. The case with all accelerations takes a factor of two longer calculational tie. So the coputational efficiency is a factor of 5 iproved with accelerations. The individual effects of each acceleration technique have een investigated and they are detailed in the following sections. Tale Ipact of all acceleration techniques Acceleration Resonance Integral (1 siga) Asyptotic All 5.047 (0.0006) 5.84 (0.006) None 5.051 (0.0074) 5.83 (0.0191).6.1 Exact slowing down vs. 1/E One approxiation that is often ade when generating resonance integrals is that the source of neutrons is assued to e 1/E in the resonance range. If this assuption is valid, the slowing down prole could e solved without need for the rando coponent of source neutron energies that naturally result fro the siulation process. To test this assuption, two siulations were perfored: 1) source neutrons started fro aove 1000 ev and explicit slowing down (in hydrogen scatterer) past the 5
range of interest, and ) source neutrons started fro a unifor 1/E source over the energy range of interest. Results of these two siulations are suarized in Tale 3. Tale 3 Ipact of source neutron sapling with rando sapling of neutron energies. Fro Tale 4 it is clear that forcing a unifor distriution of source neutrons reduces the statistical variation in resonance integral y a factor of approxiately two. Consequently, all susequent siulations reported here use the forced unifor source distriution to significantly iprove the statistics of reported results. Sources Resonance Integral (1 siga) Asyptotic Tale 4 Ipact of unifor source 1/E 5.047 (0.0006) 5.84 (0.006) Exact 5.038 (0.003) 5.84 (0.0080) Sources Resonance Integral (1 siga) Asyptotic These results deonstrate that the 1/E approxiation introduces only a very sall ias (~0.%) on the resonance integral relative to the difference etween the asyptotic and exact scattering odel (~ 4%) that is eing studied. It is also clear fro the relative statistics that starting neutrons in the 1/E spectra reduces the statistical variation of the resonance integral y a factor of roughly four. The 1/E assuption also reduces execution tie in half y eliinating the MC slowing down siulation over the energy region fro 1000 ev to the energy range of interest. Consequently, all susequent siulations reported here use the 1/E source assuption to significantly iprove statistics..6. Unifor 1/E sources When source neutrons are assued to e 1/E, the cuulative distriution function (cdf) for the source neutron energy is ln( E) ln( Ein ) P( E) =, () ln( E ) ln( E ) ax in where E and are the axiu and iniu ax Ein energy of siulated source neutrons. Consequently, source neutrons can e sapled as E ln( Ein ) + r ln( Eax / Ein ) = e, (3) where r is a rando nuer etween 0 and 1. However, since the source is assued to e 1/E and since one starts a fixed nuer of neutrons per atch, there is no need to randoly place the in energy. Instead, one can siply use the 1/E cdf to uniforly distriute the source neutrons. This avoids variations that result fro sapling source neutron energies. Tale 4 presents a coparison of two siulations, one with unifor 1/E source neutron energies and one Unifor 5.047 (0.0006) 5.84 (0.006) Rando 5.045 (0.0053) 5.81 (0.0059).6.3 Forced scattering interactions In the coputation of resonance integral, each collision with the ackground scatterer is assued to scatter the neutron past the resonance range. Consequently, any neutrons are reoved fro siulation y interaction with the ackground scatterer, and it is extreely inefficient to siply start another source neutron. Instead, it is advantageous to force scattering interactions with the resonant scatterer at each collision. A coparison of resonance integrals with analog and forced scattering is presented in Tale 5. Tale 5 Ipact of forced scattering Collisions Resonance Integral (1 siga) Asyptotic Forced 5.047 (0.0006) 5.84 (0.006) Rando 5.040 (0.0075) 5.7 (0.0058) These results deonstrate that at least a factor of two reduction of uncertainty is achieved y forcing scattering interactions. Since neutrons live longer when forced scattering is applied, execution ties increase y a factor of three, ut coputational efficiency is still iproved..6.4 Sapling of distance to collision In the hoogeneous infinite ediu studied here, the next collision is always in the sae ediu. Consequently, the distance to collision is only needed for track length estiates of fluxes and reaction rates. Since the distance to the next collision can e sapled with the cdf 6
P ( d ) d = p ( x ) dx = 1 e 0 t d, (4) integral has the expected variation with nuer of histories per atch and nuer of atches. the distance to next collision is sapled fro ln( 1 r) d =. (5) t Rather than sapling for d, the expectation value 1/ t can e used directly for tallies. Results presented in Tale 6 deonstrate that a factor of at least six reduction in uncertainty is otained y using expectation values for the track length tallies of resonance integral. Tale 6 Ipact of collision distance sapling Distance Resonance Integral (1 siga) Asyptotic Expected 5.047 (0.0006) 5.84 (0.006) Rando 5.040 (0.0087) 5.76 (0.018).6.5 Neutron splitting and Russian roulette Traditional neutron splitting and Russian roulette are oth effective eans to reduce uncertainties. Tale 7 presents a coparison of resonance integrals when the neutron splitting and Russian roulette are not used. Tale 7 Ipact of neutron splitting and Russian roulette Resonance Integral (1 siga) Asyptotic Splitting and RR 5.047 (0.0006) 5.84 (0.006) No RR 5.046 (0.0005) 5.79 (0.007) No Splitting 5.045 (0.0007) 5.85 (0.0103) Splitting is extreely effective in iproving statistics and Russian roulette is effective in reducing execution ties (a factor of two), and oth are eployed in all susequent siulations..6.6 Siulation statistics A sensitivity study of the resonance integral statistics (with all accelerations eployed) to the nuer of neutron histories and nuer of atches is suarized in Tale 8. The uncertainty of resonance Tale 8 Sensitivity to neutron histories History (*1e+6) Batch Resonance Integral (1 siga) Asyptotic 1 10 5.047 (0.00065) 5.84 (0.0058) 1 40 5.046 (0.00048) 5.83 (0.0014) 1 160 5.046 (0.0005) 5.83 (0.0006) 1 640 5.046 (0.0001) 5.83 (0.00033) 4 10 5.046 (0.00030) 5.84 (0.00131) 16 10 5.046 (0.00019) 5.83 (0.00039).6.7 AESK vs. Spectra for oth an AESK and siulation are oserved to e nearly the sae, as expected ( spectru is displayed in Fig. ). The AESK siulation has a resonance integral uncertainty that is roughly a factor of three saller than the equivalent siulation. The execution tie of AESK siulation is 18,700 seconds, which is 470 ties longer than the siulation. Consequently, the overall perforance of is approxiates 50 ties ore efficient than AESK, and is used here for all susequent studies. 3. Neutron upscatter in scattering resonances NJOY and MCSD calculated spectra near the 38 U 36.67 ev resonance are depicted in Fig. for a 60 arn ackground scatterer. The NJOY spectru clearly atches the MCSD spectru coputed with the asyptotic kernel. The spectru shows noticeale flux reductions near the flux peak at 35.8 ev and a slight shift to higher energies (the region of higher 38 U capture cross section), which results in an increase in asorption resonance integrals. Note that the resonance integrals displayed in Tale 8 for the are ~ 4% larger than that of asyptotic kernel. 4. Resonance upscatter correction for CASMO-5 CASMO-5 uses Bondarenko self-shielding tales to interpolate resonance integral for given teperature, T, and ackground cross section,. Note that these resonance integrals are norally generated directly fro NJOY and do not account for the 7
resonance scattering effects discussed in this paper. The 38 U resonance upscatter effects are incorporated directly into CASMO-5 y using MCSD generated data rather than the NJOY data. CASMO-5 resonance integral tales have een generated for 18 ackground cross sections and 10 teperatures with 640M neutron histories (64M/atch * 10 atches) for each case. Fig.. Neutron spectru at 38 U 36.68 ev resonance 5. CASMO-5 LWR Doppler coefficients Heran, 006). Doppler coefficients were calculated fro eigenvalues at 600K (HZP) and 900K (HFP), and results are suarized in Fig. 3 and Tale 9. MCNP5 results in Fig. 3 are taken fro Mosteller (006) who also used ENDF/B-VII.0 data. It can e oserved fro these results that the upscatter correction akes fuel teperature coefficients (FTC) ore negative y 9-10%. In exaining the differences etween MCNP5 and CASMO-5, MCSD was also odified to use the sae elastic scattering assuptions of MCNP5. That is, Eq. () was used directly in the kernel to generate data for CASMO-5. Doppler coefficients coputed with this odel have een oserved to e even less negative than those without upscattering (e.g., -.18 pc/k at 4.5 w/o vs. -.1 pc/k of asyptotic kernel), and they are in closer agreeent with the MCNP5 FTCs. By iplication, the MCNP5 scattering odel produces FTCs even ore discrepant, relative to the exact scattering kernel, than the siple asyptotic odel. The ipact of accurate odeling of resonance scattering grows rapidly with fuel teperature. For instance, in the 4.5 w/o enriched PWR lattice, the odel produces eigenvalues that are lower than the asyptotic odel y 110 and 11 pc at 600K and 900K, respectively. At 300K, the difference is only 3 pc. This is extreely fortunate, as any MCNP5 cold criticals calculations have een used in the validation of ENDF/B-VII.0 data (Olozinsky and Heran, 006). 5.1. Doppler Pin-cell Benchark The UO pin-cell Doppler enchark of Mosteller (Mosteller, 006, 007) has een analyzed with CASMO-5 and ENDF/B-VII.0 (Olozinsky and Tale 9 Fuel teperature coefficients Enrichent (w/o) Asyptotic Kernel Kernel HZP HFP FTC 1) HZP HFP FTC FTC Diff (%) 0.711 0.66510 0.65891-4.71 0.66451 0.65777-5.13-9.0 1.6 0.96076 0.9506-3.17 0.95989 0.95039-3.47-9.5.4 1.0991 1.08950 -.70 1.0981 1.08761 -.96-9.6 3.1 1.1773 1.16713 -.47 1.1767 1.1651 -.71-9.7 3.9 1.4008 1.955 -.30 1.3900 1.747 -.53-9.7 4.5 1.7556 1.6484 -.1 1.7446 1.673 -.43-9.7 5.0 1.9990 1.8908 -.15 1.9878 1.8696 -.36-9.7 1) FTC = (1/k hzp 1/k hfp ) *1E+5 / 300 (pc/k) 8
FTC (pc/k) -1.5 -.5-3.5-4.5-5.5 MCNP5 CASMO w/ Asyptotic Kernel CASMO w/ Exact Kernel 0 1 3 4 5 6 U Enrichent (w/o) Fig. 3. Fuel teperature coefficients of UO pin 5.. Resonance Scattering in NGNP Fuel To evaluate the ipact of 38 U resonance scattering on NGNP fuel, a siple infinite ediu of TRISO particles was exained. Typical design paraeters of high teperature reactor (HTR) pele fuel are shown in Tales 10 and 11, where the fuel kernel coating layers have een siplified to a single layer of SiC. Tale 11 presents a siple transforation of the pele sphere containing 100 sphere fuel kernels to a cylinder pin-cell prole that can e coputed y CASMO-5. The radius of cylinder fuel has een chosen to preserve the ean chord length of spherical fuel kernel. The thickness of the cylinder clad was coputed to preserve the volue ratio of coating layer to fuel kernel. Pin pitch has een adjusted to preserve the fuel to oderator (coating layer + graphite atrix) volue. Tale 1 presents results for oth LWR and HTR fuels coputed with exact scattering kernel and the asyptotic scattering kernel. At typical HFP conditions, the exact scattering kernel effects in a PWR and an HTR are ~00 pc and ~450 pc, respectively. The effect of exact scattering kernel is insensitive to uraniu enrichent in oth reactors, ut very sensitive to fuel teperature. Tale 10 HTR aterial specifications Nuer of Kernels/Pele 1000 Kernel Fuel UO UO Density [g/cc] 10.36 Coating Material Coating Density [g/cc] 3.4 Moderator Material Graphite Density [g/cc] 1.6 Tale 11 HTR diensions: sphere vs. cylinder odel Spherical Model SiC @ 1000K Graphite@1000K Cylindrical Model Radius [c] Kernel 0.05 Fuel Pin 0.01667 Thickness [c] Coating 0.0 Clad 0.0358 Radius [c] Pele.5 Pin Cell 0.6960 Fuel/Mod Ratio Pele 0.0115 Pin 0.0115 Tale 1 Reactivity effect of exact scattering kernel Fuel Tep (K) Enrichent (w/o) PWR HTR Asyptotic Diff. (pc) Asyptotic Diff. (pc) 300 900 1350 4 1.5994 1.5963-31 1.1814 1.181 8 8 1.40550 1.40519-30 1.33661 1.33671 10 1 1.46695 1.46668-8 1.38310 1.3831 11 4 1.3609 1.3401-09 1.13643 1.13360-83 8 1.38006 1.37794-1 1.501 1.4914-86 1 1.441 1.4391-01 1.30066 1.9795-71 4 1.98 1.1947-35 1.09730 1.0987-443 8 1.36643 1.3684-358 1.118 1.073-450 1 1.4775 1.4434-340 1.6186 1.5758-48 9
The 38 U exact scattering kernel has its iggest ipact in the groups containing the 36.67 ev and 0.87 ev resonances, with 5% and % asorption increases, respectively. Since the 38 U asorption reaction in the HTR fuel is approxiately 50% larger than in the PWR fuel, reactivity effects are proportionally larger. 6. Conclusions The asyptotic elastic scattering odels used in the epitheral energy range in NJOY and Monte Carlo codes lead to ~10% under prediction of Doppler coefficients of LWR lattices. Reactivity effects fro ipleenting the exact scattering kernel are ~00 pc for PWRs and ~450 pc for NGNPs at HFP. Results presented here (coputed via a totally independent coputational approach) confir the oservations of nuerous other researchers (Ouislouen and Sanchez, 1991; Bouland et al.; Kolesov and Ukraintsev, 006; Dagan and Broeders, 006). Consequently, until such tie as the NJOY resonance elastic scattering odel is iproved, NJOY data used in downstrea deterinistic codes will introduce large systeatic errors in theral reactor eigenvalues and Doppler coefficients. The siple odel presented here for ipleentation of resonance elastic scattering effects in CASMO-5 provides one exaple for circuventing current NJOY liitations. However, direct iproveent in NJOY odeling of resonance elastic scattering is a preferale long- ter path. Likewise, Monte Carlo codes that ake the asyptotic scattering approxiation for heavy nuclide resonance scattering, like MCNP5, MCNP4C, MCNPX, O5R, SAM-CE, VIM, RCP, MVP, and TART, all currently suffer siilar shortcoings (Brown, 008). Until such tie as these approxiations are iproved, caution should e used when considering Monte Carlo calculations as reference solutions for low-enriched uraniu theralspectru reactors. It is straightforward to ipleent the ethods presented here for sapling of target velocity and target nucleus/incident neutron scattering angle into any Monte Carlo code. However, otaining statistically eaningful results will e far ore difficult with sapling of the exact scattering kernels, and the required neutron histories ay ake ost theral reactor applications ipractical. It should e noted that any of the acceleration ethods discussed here are not directly applicale to general Monte Carlo applications, and additional work in the Monte Carlo counity will e required to efficiently address the resonance scattering issues detailed here. References Bell, G. I., Glasstone, S., 1968. Nuclear Reactor Theory. Roert E. Krieder Pulishing Co. Bouland, O., Kolesov, V., Rowlands, J.L., The Effect of Approxiations in the Energy Distriutions of Scattered Neutrons on Theral Reactor Doppler Effects. JEF/DOC-486. Brown, F., 008. Personal Counication. Carter, L.L., Cashwell, E.D., 1975. Particle-Transport Siulation with the Monte Carlo Method. Los Alaos Scientific Laoratory. Dagan, R., Broeders, C.H.M., 006. On the Effect of Resonance Dependent Scattering-kernel on Fuel Cycle and Inventory. PHYSOR-006, Vancouver, Canada. Kolesov, V.V., Ukraintsev, V.F., 006. Teperature Effects and Resonance Elastic Cross Section Influence on Secondary Energy Distriutions of Scattered Neutrons in the Resolved Resonance Region. PHYSOR-006, Vancouver, Canada. MacFarlane, R.E., Muir, D.W., 000. NJOY99.0 Code Syste for Producing Pointwise and Multigroup Neutron and Photon Cross Sections fro ENDF/B Data. PSR-480/NJOY99.00, Los Alaos National Laoratory, Los Alaos. Mosteller, R.D., 006. Coputational Benchark for the Doppler Reactivity Defect. LA-UR-06-968, Los Alaos National Laoratory. Mosteller, R.D., 007. ENDF/B-V, ENDF/B-VI, and ENDF/B-VII.0 Results for the Doppler-defect Benchark. M&C+SNA 007, Monterey, CA. Olozinsky, P., Heran, M., 006. Special Issue on Evaluated Nuclear Data File ENDF/B-VII.0. Nuclear Data Sheets, Volue 107, Nuer 1. Ouislouen, M., Sanchez, R., 1991. A Model for Neutron Scattering off Heavy Isotopes that Accounts for Theral Agitation Effects. Nucl. Sci. Eng., 107, pp. 189-00. Rhodes, J., Sith, K., Lee, D., 006. CASMO-5 Developent and Applications. PHYSOR- 006, Vancouver, Canada. X-5 Monte Carlo Tea, 003. MCNP A General Monte Carlo N-Particle Transport Code, Version 5. LA-CP03-045. 10