Progress in Nuclear Energy

Progress in Nuclear Energy 67 (213) 124e131 Contents lists available at SciVerse ScienceDirect Progress in Nuclear Energy journal homepage: www.elsevier.com/locate/pnucene Bondarenko method for obtaining group cross sections in a multi-region collision probability model C.L. Dembia 1, G.D. Recktenwald 1, M.R. Deinert * Department of Mechanical Engineering, The University of Texas at Austin, 1 University Station C22, Austin, TX 78715, United States article info abstract Article history: Received 27 July 212 Received in revised form 31 January 213 Accepted 1 February 213 Keywords: Collision probability method Equivalence theory Escape cross section Heterogeneous Resonance self-shielding Correct multigroup cross sections are essential to modeling the physics of nuclear reactors. In particular, the presence of resonances leads to the well-known self-shielding effect that complicates any procedure for obtaining group cross sections. The Bondarenko method for producing self-shielded group cross sections is widely used. For heterogeneous systems, the approach requires the use of an escape cross section that captures the probability that a neutron will leave a cell without interaction, and simple approximations are typically used for this purpose. Here we provide a concise derivation for how to determine group cross sections using the Bondarenko method with an extension to multi-region collision probability models. The accuracy of the method is demonstrated by comparing group cross sections derived this way with those produced by Monte Carlo simulations of thermal and fast spectrum reactors. Ó 213 Elsevier Ltd. All rights reserved. 1. Introduction Hundreds of thousands of energy grid points are required to fully resolve the energy dependence of neutron interaction cross sections. Even at the petascale, a brute force resolution of this dependence in Monte Carlo and discrete ordinate techniques is far from practical. As a result, reactor physics codes typically use a multigroup formulation where the neutron spectrum is represented by a few tens to hundreds of energy groups. Multigroup cross sections are then required that preserve the correct in-group reaction rates. However, producing these cross sections is not straightforward because the presence of in-group resonances affects the flux. Being able to produce the correct group-averaged cross sections then requires knowing the energy dependent flux within a group, which is often the quantity being sought. Furthermore, the attempt to model a heterogeneous system requires additional considerations. The traditional method for treating heterogeneity involves applying an equivalence relation to the background cross section of the Bondarenko method (Gopalakrishnan and Ganesan, 1998; Joo et al., 29; Kidman et al., 1972; Schneider et al., 26a; Stamm ler and Abbate, 1983). The desire to model next generation reactors with more complex * Corresponding author. E-mail address: mdeinert@mail.utexas.edu (M.R. Deinert). 1 The authors contributed equally to this work. geometries has led to the use of subgroup methods (Chiba, 23; Cullen, 1974; Herbert, 1997; Huang et al., 211). Self-shielding methods primarily differ in the accuracy with which they attempt to approximate the neutron flux within a group, and are well described in the reviews by Hwang (1982) and Herbert (27). The subgroup method is used when modeling next-generation reactors whose complex geometry precludes the use of the Bondarenko method. In this work we are concerned with modeling simple pin-cell geometries, in which case the Bondarenko method provides sufficient accuracy. The use of the Bondarenko method for heterogeneous systems requires the use of an effective escape cross section that describes the probability that a neutron may escape a resonance by leaving a region. There are a number of ways by which the escape cross section can be obtained. The methods vary in complexity, but most use the Wigner rational approximation and the mean chord length of a region in order to obtain an expression for a collision probability (either the first-flight escape probability or the fuel escape probability). Sometimes this approximation is adjusted by a Dancoff factor, Bell factor or by replacing the Wigner rational approximation with an N-term expression such as the one by Carlvik (Stamm ler and Abbate, 1983; MacFarlane and Muir, 1994; Herbert and Marleau, 1991; Yamamoto, 28). In the present contribution we provide a review and simplified derivation for the escape cross section using a collision probability model for the transport of neutrons from one reactor region to another. To demonstrate the accuracy of the approach we use the Bondarenko method to generate multigroup reaction and kernel 149-197/$ e see front matter Ó 213 Elsevier Ltd. All rights reserved. http://dx.doi.org/1.116/j.pnucene.213.2.1

C.L. Dembia et al. / Progress in Nuclear Energy 67 (213) 124e131 125 cross sections which we use with an in-house collision probability spectral solver to obtain a neutron spectrum. We compare the predictions of neutron spectrum and reaction rates for simulated fast and thermal spectrum reactors to a published benchmark (Rowlands et al., 1999) which is commonly used in the analysis of self-shielding methods (Herbert, 25) as well as to results produced using MCNPX 2.7.. 2. Group cross sections We define a group structure with G groups that span a range of energies from E G to E {ev}. The g-th energy group spans an energy range from E g to E g 1. Here we adopt the convention that the groups are ordered in descending energy so that the highest energy is E and the lowest energy is E G, and in general E g < E g 1. The width of an energy group is DE g ¼ E g 1 E g and is different for every group. The microscopic and macroscopic group cross sections for any interaction are respectively denoted as s g {barns} and S g {cm 1 }. It will be convenient to define a group flux f g {cm 2 sec 1 } given by the integral of f(e) {cm 2 sec 1 ev 1 } over the energies in group g: f g ¼ defðeþ (1) g where the integral over g indicates an integral from E g to E g 1. The group cross sections s g must be defined in such a way that they preserve the reaction rates R(E) ¼ Ns(E)f(E) {cm 3 sec 1 ev 1 } of the interactions that occur in the system, because it is the reaction rates that ultimately dictate the behavior of the reactor. Here N is the atom density {cm 3 } of the medium. The group cross section s g {barns} is given by (Lamarsh, 1972): desðeþfðeþ g s g ¼ (2) defðeþ g where s(e) is the energy dependent cross section {barns}. This averaging must be applied to all cross sections relevant to the problem (i.e. absorption, capture, etc). Similarly, the group-togroup scattering cross section s s;g)g is defined as: s s;g)g ¼ de de s s ðe)e ÞfðEÞ g g (3) f g The group cross section s s;g)g has units of barns, and the group-to-group scattering cross section s s (E)E ) has units of barns/ev. Equations (2) and (3) define the group cross sections, but they unfortunately depend on the flux f(e) between E g and E g 1. This presents a problem, as f(e) is the quantity we seek. If the flux f(e)is roughly constant through a group then the flux drops out of Eq. (2), and the flux is not needed to obtain group cross sections. This assumption is often valid, but cannot be used when the flux varies rapidly within a group as is the case in groups where the cross section exhibits resonances. To obtain group cross sections in these cases, the flux f(e) must be approximated and Eq. (2) must be solved by integration for the relevant interactions. However, it is not desirable to work in a spectral code with the point-wise data this integration requires. It is preferable to precompute this integration with a separate data processing code. The issue with this procedure is that the flux is necessarily problem-dependent, largely through the self-shielding effect described in the next section. In Sections 3 and 4 we introduce the background cross section method that allows this integration to be performed in a problem-independent manner. 2.1. Resonance self-shielding For a homogeneous system containing M nuclides, the macroscopic total cross section of the mixture is given by: S t ðeþ ¼ XM m N m s m t ðeþ (4) where N m is the number density of the m-th isotope in the mixture and S t is the total macroscopic cross section {cm 1 }. In this homogeneous medium, the neutron balance is given by the following: S t ðeþfðeþ ¼SðEÞ (5) where S t (E)f(E) {cm 3 sec 1 ev 1 } is the interaction rate at energy E and S(E) is the collision density and can be interpreted as the corresponding slowing down density if the neutron field is at equilibrium (Gopalakrishnan and Ganesan, 1998; Lamarsh, 1972). The idea with the Bondarenko method is to let S(E) be a smooth function that is functionally equivalent to the flux profile in the absence of resonance effects and will depend on the type of reactor. For a thermal spectrum S(E) would follow a MaxwelleBoltzmann distribution with a fission spectrum peak (MacFarlane, 2). Within a group Eq. (5) can then be used to express the behavior of the flux in terms of the macroscopic total cross section: fðeþ ¼ SðEÞ S t ðeþ Equation (6) shows that a resonance peak in S t (E) causes f(e) to dip correspondingly. Equation (6) can be used to approximate the behavior of the in-group flux (Bell and Glasstone, 197) and Eq. (2) becomes: s m g des m ðeþ SðEÞ ¼ g S t ðeþ de SðEÞ g S t ðeþ Here, s m g represents the microscopic group cross section for m-th nuclide in the mixture. Equation (7) canthenbeusedtogeneratethe appropriate group cross section for every individual reaction of interest. Note that it is the total cross section that always appears on the right hand side of Eq. (7) and a resonance in any of the interactions that contribute to the total cross section causes a depression in the flux. If the group size is small enough, S(E) is effectively constant and falls out of Eq. (7) (MacFarlane and Muir, 1994). 2.2. The background cross section A key contribution of Bondarenko (Bondarenko, 1964) was to separate the macroscopic total cross section into two terms: one that depends only on the point-wise cross section of nuclide m and a second term that encompasses all of the other isotopes in the mixture. Equation (4) is then written as: S t ¼ N m s m t þ X nsm N n s n t (8) The cross sections in Eq. (8) are continuous functions of energy, but we have omitted the energy argument for brevity. We factor out (6) (7)

126 C.L. Dembia et al. / Progress in Nuclear Energy 67 (213) 124e131 the number density of nuclide m, for which we desire the group cross section, from both terms: S t ¼ N m s m t þ s m (9) where, s m ¼ 1 N m X N n s n t (1) nsm where s m is commonly referred to as the background cross section. Combining Eqs. (7), (9) and (1) we obtain: s m g des m S ¼ g s m t þ s m S de g s m t þ s m (11) where s m t, sm, and S are functions of energy. It is clear from Eq. (11) that the group cross section for a specific isotope depends critically on the background cross section. 2.3. Heterogeneous systems and the equivalence relation Equations (1) and (11) apply when the medium through which the neutrons are traveling is homogeneous. However, nuclear reactors typically have distinct regions with different material compositions. In cases such as these a neutron at energy E could leak out of the region in which the flux is being computed, or it could undergo an interaction which removes it from the region in which flux is being computed. In either case, the flux at E would be reduced and Eq. (1) would need to be modified to capture the effect. An easy way to do this is by adding an additional term to s (E) in Eqs. (9)e(11) that takes escape from a region into consideration. We start by modifying Eq. (5): h S j t ðeþþsj eðeþi fðeþ ¼S j ðeþ (12) Here S j eðeþ {cm 1 } is the macroscopic escape cross section in the j-th reactor region. Note that S j (E), S t j (E) now apply specifically to the j-th region. Equation (12) constitutes a so-called equivalence relation because it allows us to treat the heterogeneous case identically to how we treated the homogeneous case by simply adding an effective cross section to the total cross section (Bell and Glasstone, 197). Equation (12) can be rearranged to give: S j ðeþ fðeþ ¼ S j t ðeþþsj eðeþ (13) We can expand the denominator, as we did in Eq. (8), to obtain: S j t þ Sj e ¼ N m s m t þ s m (14) where the background cross section for the m-th isotope in a region, j, is now given by: s m ¼ 1 N m! X N n s n t þ Sj e nsm (15) Here it is understood that s n t applies to the j th region. Equations (11) and (15) can be used to give the group cross section for any reaction, any nuclide, and in any region of a reactor provided that the correct escape cross section from that region can be formulated. Fig. 1. Self-shielding factor as a function of background cross section for uranium-238 at the 6.67 ev resonance. The raw data is produced by NJOY, and can be fit to the tanh function in Eq. (2) using the method in (Gopalakrishnan and Ganesan, 1998). Red dots identify the discrete values of the background cross section at which group cross sections are obtained from NJOY. These points form what is known as a dilution grid. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 2.4. The escape cross section in collision probability models Collision probability theory models the movement of neutrons between homogeneous reactor regions using transmission and escape probabilities (Lamarsh, 1972; Schneider et al., 26b). We define the collision probability P i)j ðeþ to be the probability that a neutron with energy E, born in region j (whether by scattering or fission), undergoes its next collision in region i. Accordingly, P i)i is the probability that a neutron born in region i collides next in region i. The escape cross section can be related directly to the probability that a neutron will leave a particular region of a reactor before colliding again: X j P i)j Se ¼ (16) S t þ S e isj Equation (16) can then be rearranged to provide the escape cross section in terms of the collision probabilities: 1 P P i)j 1 S j e ¼ S j isj B t@ P C P i)j A (17) isj Table 1 Dimensions and composition of the LWR from case 1 of the Rowlands benchmark. The fuel is enriched uranium dioxide, and is surrounded by a water moderator. A zirconium fuel rod is modeled by smearing the appropriate amount of zirconium across the water. Pin Diameter.8 cm Temperature 294. K Nuclide Density (#/b/cm 2 ) U-235.78 U-238.2264 O-16.46624 Annulus Pitch 1.2 cm Temperature 294. K Nuclide Density (#/b/cm 2 ) H-1.574461 O-16.286544 r-9.61594

C.L. Dembia et al. / Progress in Nuclear Energy 67 (213) 124e131 127 This formulation has been used to model the escape cross section in two region models in the past (Schneider et al., 26b, 27). The collision probabilities in Equation (17) depend on group cross sections. However, it is the group cross sections we are solving for. Thus, we solve for the group cross sections in an iterative fashion. First, we guess a value for s. Then, we compute group cross sections and collision probabilities. Using these collision probabilities, we compute the escape cross section and thus s. We terminate the iteration when our values for s converge. The iteration typically takes 3 steps or less to converge. 2.5. The self-shielding factor The infinite dilution (unshielded) cross section is defined as the limit of the group cross section as the background cross section goes to infinity It is common to define a self-shielding factor f as the ratio of the group cross section to the infinite dilution cross section: f ðs Þ¼ s gðs Þ s g ðs /NÞ (19) Accordingly, the quantity s g (s ) ¼ f(s )s g (s / N) is called the self-shielded cross section. Fig. 1 shows the self-shielding factor as a function of background cross section for the uranium-238 capture cross section at 6.67 ev. The figure shows that the self-shielding factor falls between zero and unity, and approaches unity as the background cross section increases. The function is commonly fit to a tanh curve: f ðs Þ¼Atanh½Bðlns þ CÞŠ þ D (2) where the constants A, B, C, and D are dimensionless fitting parameters. A method for obtaining these for a given set of selfshielding data is described in (Kidman et al., 1972). 3. Accuracy of the Bondarenko method Cross sections generated using Eq. (17) give excellent results compared to other computational approaches. To illustrate this, Eq. (17) was used to generate cross sections for an in-house multi-region collision probability code based on the fully benchmarked two-region VBUDS code (Schneider et al., 26b, 27). Infinite dilution and self-shielded cross sections were generated using NJOY (MacFarlane and Muir, 1994) for an infinite lattice light water reactor running fresh uranium dioxide fuel (Rowlands et al., 1999) Fig. 2. Self-shielded uranium-238 capture cross section in an LWR. The top graph provides the infinite dilution cross section and compares the self-shielded cross sections produced by our collision probability model with the equivalent group cross sections produced using MCNPX 2.7.. The middle graph presents the difference between our cross section and MCNPX s. The bottom graph shows the background cross section generated by our code as well as the escape portion of this background cross section. The relatively small background cross section leads to substantial selfshielding. R 2 ¼.99. Fig. 3. Self-shielded uranium-235 capture cross section in an LWR. The top graph provides the infinite dilution cross section and compares the self-shielded cross sections produced by our collision probability model with the equivalent group cross sections produced using MCNPX 2.7.. The middle graph presents the difference between our cross section and MCNPX s. The bottom graph shows the background cross section generated by our code as well as the escape portion of this background cross section. Since the background cross section is relatively large, little self-shielding occurs. R 2 ¼.99.

128 C.L. Dembia et al. / Progress in Nuclear Energy 67 (213) 124e131 as well as for an infinite lattice sodium cooled fast reactor fueled with uranium dioxide. The values of s were computed directly using Eqs. (15) and (17). We also present neutron spectrum, reaction rates, and criticality results for each of the two simulations. 3.1. Thermal cross section comparison The parameters of the thermal reactor are taken from the Rowlands benchmark (Rowlands et al., 1999), which is commonly used in the literature to evaluate resonance self-shielding methods (Gopalakrishnan and Ganesan, 1998; Herbert, 25). The benchmark provides 9 different cases; our light water reactor is modeled after case 1 and its geometry and composition are given in Table 1. The simulated reactor consists of an enriched uranium dioxide fuel pin in a square lattice surrounded by water, and both materials are at 294 K. We perform the transport calculation with 1 energy groups from 1 mev to 1 MeV. The Rowlands benchmark includes a zirconium fuel rod; we have treated this by smearing the same amount of zirconium evenly throughout the water. The cross sections are compared to their values at infinite dilution, as well as to the cross sections that MCNPX 2.7. provides for a simulation of this system, Figs. (2) and (3). The MCNPX cross sections are obtained from the reaction rates provided by a cell tally in conjunction with the appropriate tally multiplier. The MCNPX cross sections are appropriately assumed to be self-shielded. Results from our method are labeled CPM in the figures. Fig. 2 shows the capture cross section for uranium-238 in the energy range 1 eve1 kev. The top graph shows that there is substantial self-shielding of this cross section. This is expected, because uranium-238 is present in such a great concentration and many neutrons are absorbed in its resonances. The bottom graph shows the background cross section for uranium-238, as well as the escape portion of this background cross section. Over the entire energy range, the background cross section has a relatively small value, as we expect in the case where self-shielding is substantial. Though the error between MCNPX and our method is large in a few energy groups, the method is mostly able to compute the escape probability that yields the correct selfshielded cross sections. Fig. 3 shows the capture cross section for uranium-235, which is present in a much smaller concentration than is uranium-238. As a result, the background cross section is large and the infinite dilution cross section can be used as the group cross section. By comparing the bottom graph of Fig. 5 to the unshielded cross section in Fig. 4,it is evident that the background cross section for uranium-235 is dominated by the uranium-238 capture cross section. Fig. 4. Self-shielded uranium-238 capture cross section in a sodium-cooled fast reactor. The top graph provides the infinite dilution cross section and compares the self-shielded cross section produced by our collision probability model with the equivalent group cross section produced using MCNPX 2.7.. The middle graph presents the error between our cross section and MCNPX s. The bottom graph shows the background cross section generated by our code as well as the escape portion of this background cross section. The escape cross section is dominated by the sodium-23 resonance present in the coolant. The relatively small value for the background cross section gives rise to substantial self-shielding. R 2 ¼.99. Fig. 5. Self-shielded uranium-235 capture cross section in a sodium-cooled fast reactor. The top graph provides the infinite dilution cross section and compares the self-shielded cross section produced by our collision probability model with the equivalent group cross section produced using MCNPX 2.7.. The middle graph presents the error between our cross section and MCNPX s. The bottom graph shows the background cross section generated by our code as well as the escape portion of this background cross section. The escape cross section is dominated, as in Fig. (4), by the sodium-23 resonance present in the coolant. However, uranium-235 is present in a smaller concentration than uranium-238 and so its background cross section is not as large. R 2 ¼.99.

C.L. Dembia et al. / Progress in Nuclear Energy 67 (213) 124e131 129 Table 2 Dimensions and composition of a sodium fast reactor. The fuel is enriched uranium dioxide and is surrounded by a sodium moderator. A steel fuel rod is modeled by smearing chromium, iron, and nickel across the sodium. Pin Diameter 1.3727 cm Temperature 9. K Nuclide Density (#/b/cm 2 ) U-235.62999 U-238.18391 O-16.492181 Annulus Pitch 2. cm Temperature 6. K Nuclide Density (#/b/cm 2 ) Na-23.21618 Cr-52.1631 Fe-56.5392 Ni-58.717 3.2. Fast cross section comparison The fast reactor, whose geometry and composition is given in Table 2, consists of an enriched uranium fuel pin surrounded by a sodium coolant. A fuel rod is modeled by smearing a.5 cm thick steel rod (using only chromium, iron, and nickel) across the coolant. For this system, we use 42 energy groups from 4 ev to 1 MeV. The capture cross section for uranium-238 and uranium-235 are given respectively in Figs. (4) and (5). The shape of the escape cross section for both of these nuclides is dominated by the sodium cross Table 3 Comparison to Rowlands results of 3-group reaction rates in the LWR from case 1 of the Rowlands benchmark. The absorption rates for the uranium isotopes are provided. The collision probability results using group cross sections obtained with the Bondarenko methods are denoted by CPM. The numbers are normalized to 1, total absorptions in the system. The errors are slightly higher than in Table 4 as a result of how we have approximated the presence of the fuel rod. Nuclide Reaction MCNPX CPM Error (%) U-235 Fission 56,438 56,325.2 Capture 14,82 14,694.73 U-238 Fission 5126 534 4.17 Capture 22,848 22,87.1 section. Again, as expected, the background cross section for uranium-238 is much smaller than it is for uranium-235 because it is present in a higher concentration. As a result, the uranium-238 cross section is substantially self-shielded. 3.3. Thermal spectrum and reaction rates The flux obtained by our method for the thermal reactor, shown in Fig. (6), correctly captures all essential features of the MCNPX flux, including the three resonance dips in the epithermal region. The coefficient of determination R 2 between the two results, at.99, indicates a high level of accuracy of our method across the energy groups. Table 3 compares 3-group reaction rates from our method to those given in the Rowlands benchmark. The results are normalized to 1, absorptions in the system (in both the fuel and moderator). Table 4 compares our results to those obtained by an MCNPX simulation in which the zirconium fuel rod has been smeared evenly. Table 4 indicates a high level of accuracy of our method in computing reaction rates, as all errors are below 5%. However, it is clear from Table 3 that smearing the zirconium throughout the moderator introduces substantial error. 3.4. Fast spectrum and reaction rates The neutron spectrum in the fuel is compared in Fig. 7 to results obtained by MCNPX for the same system. The relative error between the two results is shown in the lower graph. The coefficient of determination R 2 between the two results is.99, which indicates a good correlation of the result across the energy groups. Table 4 Comparison to MCNPX of 3-group reaction rates in the LWR modified from case 1 of the Rowlands benchmark. The absorption rates for the uranium isotopes are provided. The collision probability results using group cross sections obtained with the Bondarenko methods are denoted by CPM. The numbers are normalized to 1, total absorptions in the system. All errors are below 5%, and are slightly smaller than in Table 3 because the fuel rod has been smeared in MCNPX just as it has been in our model. Fig. 6. Comparison to MCNPX of spectral flux in the fuel of the LWR from case 1 of the Rowlands benchmark. The top graph compares the fuel flux from our method ("CPM") to MCNPX results. The bottom graph provides the error between our method and MCNPX. R 2 ¼.99. Nuclide Reaction Group MCNPX CPM Error (%) U-235 Fission Fast 675 674.15 Res. 493 438 1.34 Thermal 49,424 49,555.27 Capture Fast 11 11 Res. 2371 2365.25 Thermal 8762 8795.38 U-238 Fission Fast 2765 2785.72 Res. 1 1 Capture Fast 2171 2161.46 Res. 15,218 15,33.56 Thermal 8395 842.3

13 C.L. Dembia et al. / Progress in Nuclear Energy 67 (213) 124e131 Table 6 Comparison to MCNPX of multiplication factor for both the fast and thermal reactor. The collision probability results using group cross sections obtained with the Bondarenko methods are denoted by CPM. Our model provides very accurate results in comparison to MCNPX simulations. There is greater error between our model and the results provided by the Rowlands benchmark because we have smeared the zirconium fuel rod in the benchmark across the moderator. Run Comparison CPM Error (mk) Fast 1.53634 1.539315 2.975 Rowlands 1.3911 1.4114 11.3 Rowlands in MCNPX 1.3975 1.4114 3.64 4. Conclusions We have provided a review and simplified derivation for the implementation of the Bondarenko method for obtaining group cross sections in multi-region collision probability models. We have used the results in an in-house collision probability model to show how the group cross sections obtained in this way compare to those generated from infinite lattice Monte Carlo simulations of thermal and fast spectrum reactors as well as with the Rowlands benchmark for a thermal spectrum system. The results confirm that the Bondarenko method, while simple, can yield excellent results. Fig. 7. Comparison to MCNPX of spectral flux in the fuel of the sodium fast reactor in the results. The top graph provides the infinite dilution cross section and compares the self-shielded cross section produced by our collision probability model with the equivalent group cross section produced using MCNPX 2.7.. Our method is able to capture the flux dips corresponding to resonances. The bottom graph provides the error between our method and MCNPX. R 2 ¼.99. Acknowledgments We would like to thank the United States Nuclear Regulatory Commission for grant NRC-38-8-946 which helped to support this work. Table 5 compares one-group reaction rates computed by our method to those obtained from MCNPX. The numbers are again normalized to a total of 1, absorptions in the entire system. The results indicate good agreement with MCNPX. 3.5. Criticality A summary of the multiplication factors for both the fast and thermal reactors is provided in Table 6. The thermal results are compared to both MCNPX and the results given by Rowlands benchmark. The error is less than 5 mk for comparisons with MCNPX, and the larger error with respect to the Rowlands results can be attributed to the fact that in our model we have smeared the zirconium fuel rod through the water. Table 5 Comparison to MCNPX of one-group reaction rates in a sodium fast reactor. The absorption rates for the uranium isotopes are provided. 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