RANDOMLY DISPERSED PARTICLE FUEL MODEL IN THE PSG MONTE CARLO NEUTRON TRANSPORT CODE

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1 Supercomputing in Nuclear Applications (M&C + SNA 2007) Monterey, California, April 15-19, 2007, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2007) RANDOMLY DISPERSED PARTICLE FUEL MODEL IN THE PSG MONTE CARLO NEUTRON TRANSPORT CODE Jaakko Leppänen VTT Technical Research Centre of Finland Lämpömiehenkuja 3 Espoo, P.O. Box 1000 FI VTT Finland Jaakko.Leppanen@vtt.fi ABSTRACT High-temperature gas-cooled reactor fuels are composed of thousands of microscopic fuel particles, randomly dispersed in a graphite matrix. The modelling of such geometry is complicated, especially using continuous-energy Monte Carlo codes, which are unable to apply any deterministic corrections in the calculation. This paper presents the geometry routine developed for modelling randomly dispersed particle fuels using the PSG Monte Carlo reactor physics code. The model is based on the delta-tracking method, and it takes into account the spatial self-shielding effects and the random dispersion of the fuel particles. The calculation routine is validated by comparing the results to reference MCNP4C calculations using uranium and plutonium based fuels. Key Words: high-temperature reactor, particle fuel, Monte Carlo, delta-tracking, PSG 1. INTRODUCTION High-temperature gas-cooled reactors (HTGRs) operate at temperatures exceeding 1000 K, which rules out the possibility to use metallic cladding and structural materials in the reactor core. The fuels are instead based on graphite, which also provides for neutron moderation. The present reactor concepts rely on two different designs. The so-called prismatic core is built from stacked hexagonal graphite blocks, loaded with small cylindrical fuel compacts. The alternative design is the pebble-bed core, which is basically a pile of spherical fuel pebbles. The similarity in the two designs is found at the microscopic level. Fissile material is packed into microscopic fuel kernels, coated by multiple material layers that provide for physical protection and keep the radioactive fission products from leaking into the surrounding medium. The coated fuel particles are dispersed in the graphite matrix in a completely random manner. There are thousands of particles in each macroscopic unit (fuel compact or pebble) and the total number of particles in the entire reactor core is counted in millions. It is clear that the accurate modelling of such geometry is not possible in reactor physics. In a way, the task is particularly difficult for continuous-energy Monte Carlo codes, which rely on very detailed modelling of geometry and interaction physics. Deterministic corrections have been developed for various deterministic transport codes, but the methods are not easily combined with Monte Carlo calculation. Since the simulation is carried out one neutron at a time, all approximations must be made directly in the geometry. The first obvious geometry approximation would be to homogenise the fuel particles in the graphite matrix. This approach fails because the neutron energy spectrum in a HTGR core is well thermalised and the mean free paths of thermal neutrons are comparable to the dimensions of the fuel kernels. If the heterogeneity of the fuel is not taken into account, spatial self-shielding effects are completely ignored, which results in large errors in the calculation.

2 Jaakko Leppänen The heterogeneity effects can be quite easily accounted for by modelling the medium as a regular three-dimensional array of fuel particles, embedded in a graphite matrix. This approximation, however, also has its problems. Fuel particles, arranged in a regular lattice, are effectively shielded by each other. Neutrons that are emitted or scattered in the appropriate direction may experience thick moderator regions, absent of any high-absorbing fuel kernels in their path. This, in some cases, may have a noticeable impact on neutron leakage and flux spectrum. The modelling of randomly-dispersed particle fuels using Monte Carlo neutron transport codes has been studied before (see e.g. Refs. [1 4]). This paper presents a simple particle fuel model, recently developed for the PSG Monte Carlo neutron transport code. The model is based on the idea introduced by Murata et al. [1] in The procedure is applied here with the delta-tracking method used by PSG. The methodology is introduced and the results compared to reference MCNP4C [5] calculations using both uranium and plutonium based fuels. The effects of heterogeneity and random dispersion are studied and the applicability and efficiency of the model discussed. 2. CODE OVERVIEW PSG [6] is a new Monte Carlo reactor physics code, developed at the VTT Technical Research Centre of Finland. The code is mainly intended for LWR fuel assembly calculations, but the geometry description allows the detailed modelling of almost any three-dimensional reactor configuration. PSG has the capability to generate all few-group constants needed in full-core nodal diffusion calculations, including diffusion coefficients using two fundamentally different methods. Group constant generation for deterministic reactor simulator codes is, in fact, one of the main motivations for PSG development. The code has already been successfully used for this task, although the applications are presently limited by the incapability to perform burnup calculation. PSG uses continuous-energy neutron interaction data, read from ACE-format data libraries. Reaction cross sections are reconstructed using an internal energy grid, which is the same for all isotopes. This results in a major speed-up in the calculation routines, as time-consuming iteration is reduced to minimum. The price of efficiency is that computer memory is wasted for storing redundant data points. All reaction channels relevant to fission applications are modelled according to classical collision kinematics and ENDF reaction laws. The procedures used for neutron transport are derived from the delta-tracking method, which was originally developed by Woodcock in the 1960 s [7]. Delta-tracking is essentially a rejection technique, that allows the random walk to be carried out without stopping the neutron at each material boundary. This eliminates the need to calculate surface distances, which simplifies the geometry routines and may lead to a considerable speed-up in complicated systems. One significant disadvantage of delta-tracking is that the track-length estimate of neutron flux is not available and integral reaction rates must be calculated using the collision estimator. This results in a major loss of efficiency in detector-type calculations, especially in small or optically thin regions of low collision rate. The efficiency of the collision estimator, however, is comparable to the track-length estimate if the integration is carried over a large region, and especially over the entire geometry. This is exactly the case in the generation of homogenised group constants in infinite fuel lattice calculations, which is the main intended application of PSG. The HOLE geometry package developed for the MONK Monte Carlo neutron transport code [4] includes a randomly dispersed geometry model based on the delta-tracking method. The details of the methodology are not known. 2/12

3 Randomly Dispersed Particle Fuel Model in PSG Other efficiency problems may emerge with delta-tracking if there are localised heavy absorbers in the geometry. Neutron path lengths are sampled using the so-called majorant cross section, which is the maximum of all material total cross sections in the system. Each sampled collision point is either accepted or rejected, with the probability given by the ratio of the actual material total cross section and the majorant. If the ratio is low, the efficiency of the rejection algorithm is poor and computing time is wasted for re-sampling the collision point. The loss of efficiency becomes pronounced when there are high-absorbing materials that occupy only small portions of the geometry. The problem is quite easily avoided in LWR lattice calculations, in which the heavy absorbers are located in well-defined material regions, such as control rods or burnable absorber pins. The solution in PSG is to calculate the optical distance to the nearest absorber region and sample neutron path lengths using a lower majorant cross section if the neutron is far from the absorber. It is obvious, however, that the same calculation routine cannot be directly applied if the high-absorbing regions are randomly dispersed throughout the geometry. This is the case in the particle fuel model introduced in the following. The efficiency problems of delta-tracking are discussed in Section PARTICLE FUEL MODEL IN PSG One of the characteristic features of delta-tracking is that surface crossings are not treated in any way. The geometry routine simply determines which material fills the cell at each tentative collision point. This feature forms the basis of the particle fuel model developed for PSG. The fuel particles are determined by randomly located spherical cells. Since it is not necessary to know the exact positions of the cells until a neutron actually collides inside one, the problem is reduced to calculating the probability that a particle is found at each sampled collision point. The methodology is introduced in the following. When a random point is selected inside a cell filled with N randomly dispersed fuel particles, the probability that the point is located inside a particle is given by the packing fraction: f = NV p V c = 4Nπr3 p 3V c, (1) where V c is the volume of the cell and the fuel particles are assumed to be spherical in shape with outer radius r p and volume V p. The packing fraction also yields the asymptotic probability that a neutron collides inside a fuel particle when the path length is long compared to particle dimensions and the collision point is located far from the cell boundaries. The probability given by Eq. (1) does not account for the fact that the existence of fuel particles in all previous collision points has already been fixed. The asymptotic probability hence breaks down if the sampled path length is short and the procedure requires an alternative approach. When the neutron starts from an arbitrary point in the graphite matrix and moves one step forward in the random walk, the distance to the nearest fuel particle in the direction of motion is exponentially distributed. In a way, the problem is very similar to the neutron transport process, in which the distribution of free neutron path length is also exponential. Murata et al. [1] have derived a relation between the mean particle distance and the particle radius and packing fraction: D = 4r p 3 ( ) 1 f 1. (2) The distribution functions can be written if it is assumed that this mean distance is equivalent with the 3/12

4 Jaakko Leppänen s r p θ Figure 1. Sampling the position of a randomly dispersed fuel particle. neutron mean free path in the transport process. The probability density function (PDF) is then written as: and the cumulative distribution function (CDF) as: dp(s) = 1 D e s/d ds (3) P(s) = 1 e s/d. (4) The inverse of the CDF can be used for sampling the optical distance to the nearest fuel particle: s = D log(1 ξ) = 4r ( ) p 1 3 f 1 log(1 ξ), (5) where ξ is a random variable, uniformly distributed between 0 and 1. It is relatively straightforward to show that the cosine of the angle in which the neutron enters the particle has a linear distribution function: where µ = cos θ. The CDF is written as: and the sampling can be carried out using: Figure 1 illustrates the geometry and the associated variables. dp(µ) = 2µdµ, (6) P(µ) = µ 2 (7) µ = ξ. (8) The above distributions are used for sampling the location of the nearest fuel particle in the direction of motion. New particles are sampled along the neutron flight path, starting from the previous collision point. The procedure is continued until the collision point lies inside or behind the sampled particle. If the neutron has passed through several particles between the previous and the new collision point, the path length is considered long enough for the asymptotic probability to hold, and the collision is sampled using the packing fraction. The particles are generated as temporary, super-imposed objects. This means that the previously generated particle remains in the geometry until a new one is encountered in the neutron flight path. The treatment allows neutrons to undergo multiple collisions inside the same particle, which accounts for the heterogeneity effects. The main calculation routine is described in Figure 2. Although the procedure is quite similar to sampling the next collision point in a neutron transport simulation, there is at least one crucial difference. Fuel particles, unlike the atomic nuclei in a medium, are clearly macroscopic objects. Since two particles may not overlap each other, the probability distribution 4/12

5 y y y Randomly Dispersed Particle Fuel Model in PSG c o l l i s i o n i n a c e l l fi l l e d w i t h r a n d o m l y d i s p e r s e d p a r t i c l e s c a l c u l a t e d i s t a n c e t o t h e e x i s t i n g p a r t i c l e n e u t r o n i s i n s i d e t h e p a r t i c l e e s i s t h e c o l l i s i o n p o i n t i n s i d e t h e e x i s t i n g p a r t i c l e? n o n e u t r o n i s i n t h e m a t r i x e s i s t h e d i s t a n c e t o t h e e x i s t i n g p a r t i c l e > 0? n o s a m p l e c o l l i s i o n u s i n g t h e a s y m p t o t i c p r o b a b i l i t y e s h a s t h e n e u t r o n p a s s e d t h r o u g h s e v e r a l p a r t i c l e s? n o g e n e r a t e a n e w p a r t i c l e i n n e u t r o n fl i g h t p a t h Figure 2. The main calculation routine in the particle fuel model developed for PSG. differs from the simple exponential distribution near existing fuel particles. A similar problem occurs close to cell boundaries. PSG always uses the exponential distribution for sampling the nearest particle position. If an overlapping of two particles occurs, the position is simply re-sampled. All calculations presented here were carried out in an infinite particle fuel geometry, and the problems related to cell boundaries are yet to be studied. It is basically possible to prevent the overlapping of particles with cell boundaries by defining a thin surface layer, composed only of the matrix material. If the thickness of the region is equal to the particle diameter, all particles encountered near the outer surface are super-imposed over the surface layer and fully enclosed inside the cell. 5/12

6 Jaakko Leppänen 4. CALCULATIONS Two types of fuel particles were included in the study. The geometry and material parameters were adopted from Ref. [8] and they are summarised in Table I. The fuel kernels are made of uranium and plutonium oxide and surrounded by a porous graphite layer. The purpose of this buffer zone is to contain the gaseous fission products escaping from the fissile material. The physical barrier coating the inner regions is made of silicon carbide, sandwiched between two pyrolytic carbon layers. The particles are dispersed in a graphite matrix and the particle density is characterised by the packing fraction. The plutonium oxide kernels are more absorbing compared to the uranium fuel, which leads to a stronger spatial self-shielding effect. The strong absorption is compensated for by smaller particle size and lower packing fraction. It is nevertheless anticipated that the various self-shielding and random dispersion effects are more pronounced in the PuO 2 fuel. Higher cross section also implies shorter path lengths in delta-tracking. This may lead to problems with the new geometry model, since the collision rate increases near the fuel particles, where the exponential probability distribution may not hold. The level of heterogeneity can be assessed by comparing the neutron mean free path in fuel to the diameter of the fuel kernel. The comparison in Figure 3 shows that there is a clear difference between the two fuel types, especially at low energy. All MCNP and PSG calculations presented in the following were carried out using the same ENDF/B-VI based cross section libraries, provided with the MCNP4C source code [9]. The capability to use the exact same data in both codes should eliminate all discrepancies originating from the evaluated nuclear data. This is essential for code validation, since the discrepancies in the fundamental data files are often a major source of uncertainty in the final results Reference Calculations PSG has mainly been used for LWR lattice calculations and prior experience with graphite-moderated systems is very limited. In order to validate the basic calculation routines, a series of reference calculations were carried out using MCNP4C and PSG. The reference cases covered both fuel types and geometry models consisting of infinite homogenised material regions, particles arranged in an infinite regular lattice and reflected cubical cells filled with 10, 50, 100 and 500 particles, dispersed in a completely random manner. The isotopic fractions were preserved in the homogenised models and particle dimensions and packing fractions in the heterogeneous geometries. The validation is restricted to infinite systems, mainly because it is not practical to model explicitly more than about 800 randomly dispersed fuel particles in the geometry. All calculations were carried out in the k-eigenvalue criticality source mode with 50 inactive and 500 active cycles of source neutrons. The results are summarised in Table II. The infinite multiplication factors calculated by the two codes are in a good agreement in all reference cases, which is an indication that PSG can be used for modelling graphite-moderated systems. It is clear that the spatial self-shielding effects in the fuel kernels have a major impact on neutronics. The difference between the homogeneous UO 2 fuel model and the corresponding heterogeneous regular lattice is about 8% in the multiplication factor. The effect is even greater in the plutonium fuel, in the order of 19%. There are systematic and statistically significant differences between the heterogeneous regular and randomly dispersed lattices as well. The impact on multiplication factor is about 0.2% in the uranium fuel and 0.3% in the plutonium fuel. It seems that the number of particles in the geometry has no impact on the results, which suggests that any of the models can be used for validating the new calculation routine. 6/12

7 Randomly Dispersed Particle Fuel Model in PSG Table I. Summary of the geometry and material parameters of the two fuel types included in the study. UO 2 fuel PuO 2 fuel Material densities (g/cm 3 ): Fuel kernel Porous buffer Inner pyrolytic carbon layer Silicon carbide layer Outer pyrolytic carbon layer Graphite matrix Particle radii (mm): Fuel kernel Porous buffer Inner pyrolytic carbon layer Silicon carbide layer Outer pyrolytic carbon layer Packing fraction (%): UO 2 kernel PuO 2 kernel 10 2 Ratio of neutron mfp to kernel diameter Neutron energy (MeV) Figure 3. The ratio of neutron mean free path in fuel to kernel diameter 7/12

8 Jaakko Leppänen Table II. Reference criticality calculations using MCNP4C and PSG. The statistical error estimates are absolute 1σ confidence intervals. Fuel Geometry model MCNP4C PSG k UO 2 Infinite homogenised medium ± ± Single particle in a regular lattice ± ± randomly dispersed particles ± ± randomly dispersed particles ± ± randomly dispersed particles ± ± randomly dispersed particles ± ± PuO 2 Infinite homogenised medium ± ± Single particle in a regular lattice ± ± randomly dispersed particles ± ± randomly dispersed particles ± ± randomly dispersed particles ± ± randomly dispersed particles ± ± PSG Particle Fuel Model The new particle fuel model was validated by comparing infinite multiplication factors, homogenised two-group cross sections and flux spectra inside the fuel kernels to reference MCNP4C and PSG results. The geometry models in the reference calculations comprised of 500 randomly dispersed fuel particles in a cubical cell surrounded by reflective boundary conditions. The results are in a good agreement with the reference calculations, although there are some statistically significant discrepancies as well. The infinite multiplication factors calculated using the new particle fuel model are ± for the uranium fuel and ± for the plutonium fuel. The values are reasonably close to the reference results in Table II. Homogenised two-group total, fission, capture and scattering cross sections are compared in Table III. It was decided to make the comparison to reference PSG results, in order to eliminate all additional discrepancies in the calculation. The differences are well below 1%, although in some cases they clearly exceed the random statistical variation. The largest discrepancies are found in thermal fission and capture cross sections and in the fast fission cross section in the UO 2 calculation. The comparison was repeated by varying the packing fraction between 1 and 30%. It seems that the differences grow along with the increasing particle density, which may be an indication that the exponential distribution fails when the collision density is high near the existing particles. The flux spectra inside the fuel kernels are compared to MCNP4C calculations in Figures 4 and 5. The statistical fluctuation is stronger in the PSG results, owing to the relatively poor efficiency of the collision flux estimator. The problem is emphasised in the PUO 2 fuel, where the particles are smaller and the packing fraction is lower. The figures show, however, that there are no systematic deviations in the distributions. A similar comparison was carried out for the flux spectra calculated in the graphite matrix, and the results were equally consistent. PSG has a tendency to over-estimate some of the group constants in the thermal energy group by about 0.5% compared to MCNP4C results. This discrepancy has been observed in LWR lattice calculations and similar differences were encountered in all the calculations in this study as well. The origin of this systematic over-prediction is not completely clear. 8/12

9 Randomly Dispersed Particle Fuel Model in PSG Table III. Two-group total, fission, capture and scattering cross sections calculated by PSG. Comparison between the new particle fuel model and a calculation in a reflected cubical cell filled with 500 randomly dispersed fuel particles. The error estimates are relative statistical errors (in %) and the energy group boundary is set to ev (1 = fast group, 2 = thermal group). Fuel Parameter PSG (dispersed) PSG (new model) (%) UO 2 Σ t (g = 1) 3.689E-01 (0.003) 3.689E-01 (0.004) (g = 2) 4.268E-01 (0.002) 4.268E-01 (0.002) Σ f (g = 1) 3.664E-04 (0.061) 3.681E-04 (0.085) (g = 2) 7.602E-03 (0.038) 7.578E-03 (0.052) Σ c (g = 1) 1.332E-03 (0.058) 1.329E-03 (0.083) (g = 2) 1.953E-03 (0.034) 1.947E-03 (0.046) Σ s (g = 1) 3.672E-01 (0.003) 3.672E-01 (0.004) (g = 2) 4.172E-01 (0.002) 4.173E-01 (0.002) PuO 2 Σ t (g = 1) 3.698E-01 (0.003) 3.698E-01 (0.004) (g = 2) 4.415E-01 (0.003) 4.414E-01 (0.005) Σ f (g = 1) 4.503E-04 (0.089) 4.502E-04 (0.115) (g = 2) 1.533E-02 (0.057) 1.529E-02 (0.087) Σ c (g = 1) 1.331E-03 (0.061) 1.327E-03 (0.081) (g = 2) 9.934E-03 (0.061) 9.907E-03 (0.092) Σ s (g = 1) 3.681E-01 (0.003) 3.680E-01 (0.004) (g = 2) 4.162E-01 (0.002) 4.162E-01 (0.003) Running Time and Efficiency It was discussed in Section 2 that the use of delta-tracking leads to certain efficiency problems when there are localised high-absorbing materials in the geometry. When the neutron is streaming through a medium in which the total cross section is low compared to the majorant, the probability of accepting a sampled collision point is low and computing time is wasted in the re-sampling loop. Randomly dispersed particle fuels are basically composed of small, high-absorbing fuel kernels, embedded in a graphite matrix where the total cross section is relatively low. The efficiency of the rejection algorithm basically depends on two factors: the difference between the total cross sections of fuel and graphite and the packing fraction, which determines the density of the fuel kernels. Figure 6 shows the efficiency of the rejection algorithm for both fuel types when the packing fraction is varied between 1 and 30%. Previous experience with LWR lattice calculations has shown that the calculation time turns to a steep increase when the efficiency falls below about 30%. This is clearly the case for the loosely-packed, high-absorbing plutonium fuel. PSG typically runs 8-15 times faster than MCNP4C in LWR lattice calculations. The factor in the single-particle reference calculations in Section 4.1 was 6.0 for the uranium fuel and 2.4 for the plutonium fuel. It should be noted, however, that no tallies were included in the MCNP calculations, which may affect the overall computing time quite significantly. A comparison between the new particle fuel model and the single-particle PSG calculation shows no significant difference in calculation time. 9/12

10 Jaakko Leppänen 2.5 PSG (new model) MCNP (dispersed) 2 Flux spectrum (A.U) Neutron energy (MeV) Figure 4. Flux spectra inside fuel kernels, comparison between MCNP4C and PSG (UO 2 fuel). 2.5 PSG (new model) MCNP (dispersed) 2 Flux spectrum (A.U) Neutron energy (MeV) Figure 5. Flux spectra inside fuel kernels, comparison between MCNP4C and (PuO 2 fuel). 10/12

11 Randomly Dispersed Particle Fuel Model in PSG UO 2 fuel PuO 2 fuel Efficiency Packing fraction Figure 6. Efficiency of the rejection algorithm in delta-tracking as function of packing fraction. 5. SUMMARY AND CONCLUSIONS High-temperature gas-cooled reactor fuels are composed of thousands of coated microscopic fuel particles containing the fissile material. The particles are embedded in a graphite matrix and dispersed in a completely random manner. This complicates the calculation, especially using continuous-energy Monte Carlo codes, which cannot model all particles explicitly, and yet are unable to apply any deterministic corrections to account for the spatial self-shielding and random dispersion effects. All approximations must instead be made directly in the geometry. A new randomly-dispersed particle fuel model was developed for the PSG Monte Carlo reactor physics code. The model takes advantage of the fact that it is not necessary to know the positions of individual fuel particles, until a collision occurs inside one of them. This feature is possible because of the delta-tracking method used by PSG. The probability to encounter a fuel particle at each sampled collision point depends on the particle packing fraction, and the distance to the nearest particle in the neutron flight path follows an exponential distribution.the method has a certain analogy with the neutron transport process, in which the free path length between two collision points is also exponentially distributed. The new geometry model was validated by comparison to reference MCNP4C and PSG calculations, in which 500 fuel particles were randomly arranged inside a reflected cubical cell. The results showed a relatively good agreement for both fuel types included in the study, although there were some statistically significant differences as well. These discrepancies may result from the fact that the exponential probability distribution fails when a collision occurs close to an existing fuel particle. The differences grow along with increasing particle density, and the problem requires some further studies before making any final conclusions on the applicability of the new model in calculations involving densely packed fuels. 11/12

12 Jaakko Leppänen The overall calculation time using the new model is comparable to a simple regular-lattice calculation with PSG. In both cases, the efficiency of the delta-tracking method becomes poor if the fuel kernels are high-absorbing and the particle packing fraction is low. Since this is often the case with HTGR fuels, it is considered necessary to develop methods to increase the efficiency in the calculation. PSG already deals with similar problems in LWR lattice calculations involving localised absorber rods, and it may be possible to extend the methodology to the new particle fuel model as well. The calculations in this study covered only infinite geometries. Since the random dispersion of fuel particles may also affect neutron leakage, it is necessary to validate the new model in finite geometries as well. The main reason why this was not done already in the present study, was the fact that it is quite difficult to produce reliable reference results for finite systems, since the number of explicitly modelled fuel particles is restricted by computational efficiency. Another interesting topic for future studies is the double-heterogeneity of pebble-bed reactors, in which the fuel pebbles containing the microscopic fuel particles are also randomly dispersed throughout the reactor core. REFERENCES [1] I. Murata, T. Mori and M. Nakagawa, Continuous Energy Monte Carlo Calculations of Randomly Distributed Spherical Fuels in High-Temperature Gas-Cooled Reactors Based on a Statistical Geometry Model, Nucl. Sci. Eng., 123, pp (1996). [2] F. B. Brown and W. R. Martin, Stochastic geometry capability in MCNP5 for the analysis of particle fuel, Ann. Nucl. Energy, 31, pp (2004). [3] T. Yamamoto, Extension of cross section homogenization methods for particle-dispersed media to layered particles, Ann. Nucl. Energy, 33, pp (2006). [4] N. Smith, M. Armishaw and A. Cooper, Current Status and Future Direction of the MONK Software Package, Proceedings of the 7th International Conference on Nuclear Criticality Safety (ICNC2003), Tokai-mura, Japan, October 20-24, [5] J. F. Briesmeister (editor), MCNP A General Monte Carlo n-particle transport code, LA M, Los Alamos National Laboratory (2000). [6] J. Leppänen, Current Status of the PSG Monte Carlo Neutron Transport Code, Proceedings of Advances in Nuclear Analysis and Simulation (PHYSOR-2006), Vancouver, BC, Canada, September (2006). [7] E. R. Woodcock et al., Techniques Used in the GEM Code for Monte Carlo Neutronics Calculations in Reactors and Other Systems of Complex Geometry, ANL-7050, Argonne National Laboratory (1965). [8] G. Hosking and T. D. Newton, Proposed Benchmark Specification for an HTR fuelled with Reactor Grade Plutonium (or Reactor Grade Pu/Th and U/Th), NEA/NSC/DOC(2003)22, OECD/NEA (2003). [9] J. S. Hendricks et al., ENDF/B-VI Data for MCNP, LA-1289, Los Alamos National Laboratory (1999). 12/12