MULTIGROUP CROSS SECTION COLLAPSING OPTIMIZATION OF A HE-3 DETECTOR ASSEMBLY MODEL USING...

International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011)

Rio de Janeiro, RJ, Brazil, May 8-12, 2011, on CD-ROM, Latin American Section (LAS) / American Nuclear Society (ANS) ISBN 978-85-63688-00-2

MULTIGROUP CROSS SECTION COLLAPSING OPTIMIZATION

OF A HE-3 DETECTOR ASSEMBLY MODEL USING

DETERMINISTIC TRANSPORT TECHNIQUES

Mi Huang and Ce Yi

Department of Nuclear and Radiological Engineering

University of Florida

202 Nuclear Sciences Bldg., Gainesville, FL 32611

[email protected]; [email protected]

Kevin L. Manalo and Glenn E. Sjoden

Nuclear and Radiological Engineering and Medical Physics Program

Woodruff School of Mechanical Engineering, Georgia Institute of Technology

770 State St, Atlanta, GA 30332

[email protected]; [email protected]

ABSTRACT

Multigroup optimization is performed on a neutron detector assembly to examine the validity of

transport response in forward and adjoint modes. For SN transport simulations, we discuss the

multigroup collapse of an 80 group library to 40, 30, and 16 groups, constructed from using the

3-D parallel PENTRAN and macroscopic cross section collapsing with YGROUP contributon

weighting. The difference in using P1 and P3 Legendre order in scattering cross sections is

investigated; also, associated forward and adjoint transport responses are calculated. We conclude

that for the block analyzed, a 30 group cross section optimizes both computation time and

accuracy relative to the 80 group transport calculations.

Key Words: SN transport, detector response, multigroup, collapse, adjoint

1. INTRODUCTION

Transport simulation offers tremendous opportunities for highly detailed information for neutron

spectroscopy applications. Concurrently, computer architectures continue to rapidly evolve to

multicore CPUs and GPUs, which aids all engineering software simulation. However, there still

remains a need to remain practical if one wishes to reduce computation time from several hours

to minutes. Multigroup macroscopic cross section collapsing is one obvious treatment. In

particular, we examine the validation of a multigroup collapse sequence for a fixed source

detector problem using the PENTRAN 3-D SN Transport code and the YGROUP Multigroup

cross section collapsing code.

We are interested in the construction of neutron detector assembly models for SNM detection, for

which counts are optimized to specific energy ranges. The evaluation and selection of moderator

materials was previously performed to optimize four distinct energy bands [1]. We choose to

examine 1 of 4 blocks (corresponding to each energy band) from the prior study, and present a

Mi Huang, et al.

2011 International Conference on Mathematics and Computational Methods Applied to

Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011

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cross section view of the basic detector model in Figure 1 [1] . The dimensions along a

prescribed x-axis are given as 16 cm concrete, 1 cm Hafnium, 1.5 cm High-Density Polyethylene

(HDPE), 1 cm He-3, 7.5 cm HDPE. The specific energy range targeted by this block design is

approximately neutrons from 1.0 MeV to 3.68 MeV.

We began our study with an 80 group library (upper energy values listed in Appendix A) and

worked to collapse the library further. An 80 group structure may be computationally

burdensome, particularly with upscatter, as wall clock times for transport (in parallel, with 80

CPUs) require on order of 7 to 8 wall clock hours (S30 Legendre-Chebyshev quadrature [960

directions per mesh cell] with a P5 Legendre Scattering order) to appropriately converge the

adjoint block models. In order to improve computational efficiency yet preserve group accuracy,

an optimal strategy would entail reduction of 80 energy groups to an even broader group

structure, perhaps somewhere between 10 to 20 energy groups, without (hopefully) any loss of

fidelity of the calculation. Originally, we performed adjoint transport calculations using the

detector as an adjoint source. In order to complement these runs, we perform standard forward

transport calculations, with the application of a half-isotropic (directed towards detector),

uniform surface source located just outside the front detector assembly face. For an individual

detector assembly, with paired calculations of forward and adjoint transport, we can apply the

results and input them to the YGROUP code, which optimizes the energy group collapse using

several options.

2. CALCULATION OF FORWARD AND ADJOINT TRANSPORT RESPONSE

In this section we briefly discuss the evaluation of forward and transport response (in the He-3

detector), since this is computed in this work, and is used to evaluate model consistency. Similar

analyses have been performed [2], but we now consider calculation of response in terms of the

multigroup collapse; we proceed to outline the basic theory and calculation steps.

Figure 1. Detector Assembly Block

Multigroup Cross Section Optimization of a Detector Assembly Model



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2.1. Multigroup Cross Section Collapsing

The YGROUP code performs the cross section collapsing [3], and was developed as part of a

code (funded through computational NNSA research with Sandia National Laboratory) to speed

up deterministic particle transport simulations by reducing the number of discrete energy groups

while maintaining computational transport accuracy [4]. The YGROUP code helps to achieve

this through application of the contributon approach originally developed by Alpan and

Haghighat to automate collapsing/group selection [5]. This contributon collapsing option aims

to divide the broad group bin structure uniformly in the sense of relative contributons from

each fine group. Contributons are defined by

3 *, , , ,V

C E d r d r E r E

(1)

Where and * are the forward flux and adjoint functions, respectively. Equation (1) specifies

the energy-dependent contributon for a particular calculation objective (as defined by the adjoint

function). For example, if the detector response is the objective, the detector response cross

section can be used as the adjoint source. The calculated adjoint function is a measurement of a

particles importance to the detector.

The group-dependent discretized contributon (only considering zeroth order moments) is given

by:

*g gg ii

C V (2)

where iV

is the discretized fine mesh volume with the same material in the model, g and *g

are the average forward scalar flux and inverse adjoint function. Therefore, Cg is a measurement

of a Group g particles importance to the objective. By using Cg as the weighting function during

cross section data collapsing, the broad group cross section could conserve the detector response

better than using forward flux as weighting function.

YGROUP operates in the following manner. First, forward and adjoint deterministic transport

calculations are performed on a smaller problem model, or on one section of a large problem

model representative of problem physics using a fine group structure. Then, the calculated

forward flux and adjoint function moments are used by YGROUP to collapse the fine group

cross section library and generate a problem-dependent broad group cross section library.

Finally, the broad group library is used for new transport calculations on the full scale/ refined

problem model, and results are compared to evaluate the effectiveness of the group collapse.

YGROUP provides several weighting options to both determine group structure, and also

collapse the cross sections. Users can also specify fine groups in specific energy ranges of

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interest to be reserved after collapsing. YGROUP also can be used to evaluate the Feynman-Y

asymptote characterizing neutron multiplicity, although this feature is not used here.

2.2. Transport Response

Since the user must select a material as the basis for flux weighting, in all detector blocks, we

considered two materials/regions in ths study: He-3 + HDPE (evenly weighted 50%, 50%) in the

detector assembly, as well as for He-3 alone (100%). There are other considerations one could

make, such as using all materials defining the assembly, although we are primarily concerned

with the neutron response of the He-3 tube. The detector assembly, with forward transport (with

an isotropic uniform energy source, as mentioned before) and adjoint transport models (aliased to

the absorption responses in He-3) were validated with regard to convergence by computing

detector response based on both the forward flux and independently using the adjoint response,

calculated using the expression for detector response R :

*det,g g g gR q (3)

From equation (3) , it is clear that detector response can be obtained by complete integration of

the source distribution with the adjoint functionfor any arbitrary source distribution.

Therefore, R can be computed directly from the results of either of several forward transport

computations for each neutron source distributions, or one single adjoint transport computation

with coupling to each source density distribution. The true calculated ratio between these two

methods of computing responses should be unity, but deviations from unity result typically as a

result of numerical truncation error; in theory, forward response and adjoint response

independently computed should be equal to each other. In the final results, we will identify

common trends by examining response ratios.

2.2.1. Response Calculation in Discretized Form

First, we discuss the calculation of transport detector response in discretized form. We can

simply sum over the groups, from 1 to G, using index g. The units are shown in square brackets.

3det, det, det 21

1 #

sec

G

g g

g

V cmcm cm

(4)

We assume that the detector is usually comprised of a volume and not a surface. The units show

clearly that we expect a response rate.

For adjoint transport, the equivalent response is given by




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33

*

,

1'

, #sec

1)( cm

cmVq srcgsrc

g

g

gsrc

(5)

By unit analysis, the forward source term provides a group dependent rate (and can be per unit

volume, the importance function is unitless, and the volume will cancel the preceding term. So

again, we arrive with adjoint response having the same units.

But if we define a surface source (in our case a J surface source), then the respective adjoint

equation changes to the form of

22,

*

1

, #sec

1))(( cm

cmAJq srcgsrc

g

g

gsrc

(6)

We use a convention where we normalize the response by dividing by the integral source term.

The normalization step is important for cross-comparison to Monte Carlo and other transport

codes.

2.3. A Brief Description of Codes Used

PENTRAN is a parallel discrete ordinates SN solver in Cartesian Coordinates [6]. There are

many recent advances with the code, with both high performance computing and adaptive

differencing recently investigated [7]. In a parallel computation environment, files are also

created based on the decomposition strategy used. Post-processing tools are quite useful, such as

the PENDATA code, which extracts parallel data from PENTRAN. For the analysis presented in

this paper, we have also incorporated appropriate scripting to auto-calculate transport response

from PENTRAN using the PENGRAB code.

As previously discussed, material fluxes associated with full multigroup transport are used to

serve as weighting functions for the calculations performed in YGROUP. To be clear, both the

selection of groups collapsed and subsequent reweighting of collapsed cross sections are handled

by YGROUP. As an example, if a user requests 35 groups, YGROUP may enforce collapsing to

38 groups. Users can also supply a fixed list of fine groups to collapse into a broad group. The

options provided by YGROUP should be considered carefully especially when collapsing to less

than 10 groups. For this study, we use the contributon weighting option available with the

collapsing code.

2.3. Model Parameters for Transport Codes

For the SN transport model, the forward model applies a half-isotropic source in the directed

toward the detector assembly, on the detector assembly face. Vacuum boundary conditions are

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used in the transport calculations. All models were run using S30P1 and S30P3, using a Legendre-

Chebyshev (Pn-Tn) quadrature. All or some models employed both angular and group

decomposition, with a minimum of group decomposition performed. Instead of a standard

multigroup iteration, we applied the Hiromoto-Weinke multigroup iteration method as applied by

the PENTRAN code, which performs a single iteration in each group while stepping toward

convergence. Also, we applied the default DZ-DTW-EDI adaptive spatial differencing strategy.

The inner flux tolerance (infinity norm) was set to 1.0E-03.

3. RESULTS

A series of 16 (2 x 4 x 2) transport calculations were performed on the detector assembly block

to investigate model performance. They are broken down into the combinations of : forward v.

adjoint, 16 vs. 30 vs. 40 vs. 80 groups, P1 vs. P3 Legendre order in scattering cross section.

The first two figures give a detailed view of an 80-group flux in the detector in Figure 2, and also

of the adjoint current importance (exiting thru the detector face) in Figure 3.

Figure 4 is a more general examination of the spatial distribution of the adjoint importance

function for the mean-weighted and max-weighted energy groups of the P1-80 group (red block

is the detector assembly, outside of the block is air). We can see that the peaks in spatial adjoint

importance occur in the polyethylene, outside of the He-3.

A combination of all the case results for the adjoint are given by Figure 6. Similarly,

combinations of case results are given for forward transport in Figure 7. Immediately, we can

see calculation fidelity is not well maintained with the 16 group structure in red in the epithermal

range, and is more apparent in the adjoint transport.




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Figure 2. Detector Flux (in He-3 region) in

Forward Transport for 30 Groups and 80

Groups.

Figure 3. Partial Adjoint Leakage (at the Detector

Assemblys Front Surface) for 30 Groups and 80

Groups.

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The average energy as calculated by Figure 4 is calculated (along a prescribed x-axis, 1D) is given

in equation (7), where gE is the average energy for group g, is the polar angle, and ,adj

g x

is the group adjoint function.

1 0

1 0

,

,

Gadj

g g

g

Gadj

g

g

E d x

E

d x

(7)

Defining the partial adjoint leakage function as

0

,adj adjg gJ x x d

(8)

Figure 4. Adjoint Importance along x-axis at

Average Forward Group 11 (of 80 groups) and

Max Forward Group 13 (of 80 groups).




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we can simplify equation (7) to

1

1

Gadj

g g

g

Gadj

g

g

E J x

E

J x

(9)

Figure 5. Adjoint Transport - Relative Jx-1/2 Adjoint Leakage Function for All Groups on Log-

Log scale.

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Another interesting examination to be made is comparing P3 vs. P1 Legendre order.

Ratios by energy are given on a Log-Log scale in Figure 7 and 8 for adjoint and forward

transport, respectively. From Figure 7, the 80 group case oscillates more due to a better

representation of cross section detail. An effective smoothing occurs as a function of increasing

group collapse. It is also apparent that the 16 group case is losing information in the epithermal

range, decreasing accuracy. Figure 8 showcases the ratio of P3 to P1 as a function of log energy;

we see that scatter effects are noted in the fast energy range (as expected), likely due to

improvements in hydrogen scatter cross section in polyethylene surrounding the He-3.

Figure 6. Forward Transport - Relative Detector

Flux for All Groups on Log-Log scale.




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Figure 7. Adjoint Transport P3/P1 Partial

Adjoint Leakage Ratio for All Groups on

Log-Log scale.

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Finally, we present the tabulated results of the response, presenting forward response, adjoint

response, and their ratio (adjoint/forward) in Table I for P1 and Table II for P3. If we ratio the

results, we can affirm that P3 is higher in the forward and adjoint cases than compared to P1.

As was previously discussed in the basic theory section, we note that all forward/adjoint

response ratios are consistently close to 1 (worst percent difference was 9.7 % in the P3-16 group

case). There is a monotonic trend decrease with increasing groups, which may also be indicative

of Legendre term expansion impact with more scattering cross section values.

Figure 8. Forward Transport - P3/P1 Detector

Flux Ratio for All Groups on Log-Log scale.




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Table I. P1 Transport Results: Forward Response, Adjoint Response, and Response Ratio

P1; # Groups Forward Adjoint RR

16 2.784 10-12

2.889 10-12

1.037

30 1.260 10-12

1.255 10-12

0.9961

40 1.325 10-12

1.276 10-12

0.9630

80 1.164 10-12

2.500 10-2

0.9317

Table II. P3 Transport Results: Forward Response, Adjoint Response, and Response Ratio

P3; # Groups Forward Adjoint RR

16 3.143 10-12

3.448 10-12

1.097

30 1.416 10-12

1.496 10-12

1.056

40 1.490 10-12

1.517 10-12

1.018

80 1.304 10-12

1.274 10-2

0.9773

Table III. P3 to P1 Ratio: Ratios of Forward Response, Adjoint Response, and Response

Ratio

# Groups Ratio of Forward

Ratio of Adjoint

Ratio of RR

16 1.129 1.194 1.057

30 1.124

1.192

1.061

40 1.125

1.189

1.057

80 1.120

1.175 1.049

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3. CONCLUSIONS

From examination of a variety of transport models, computations using the 30 group library

collapsed using the contributon approach in YGROUP yielded the minimum group structure

while preserving problem accuracy. The P1-30 group calculation, and associated multigroup

cross sections from YGROUP contributon weighting process, yielded a reduced transport

computation time (< 10 minutes), and that response is calculated to a reasonable accuracy with

response ratio calculated to 0.996. Computation times of all cases are listed in Appendix A.

Future work will continue with similar analysis of the three other detector blocks targeting the

distinct neutron energy bands of 31.8-369 keV, 0.369-1.0 MeV, and 3.68-17.3 MeV [1].

ACKNOWLEDGMENTS

This research was made possible with support by NNSA.

REFERENCES

1. Ghita, G., G. Sjoden, and J. Baciak, "On Neutron Spectroscopy Using Gas Proportional

Detectors Optimized by Transport Theory". Nuclear Technology, (2009). 168(3): p. 620-

628.

2. Sjoden, G.E., "Deterministic adjoint transport applications for He-3 neutron detector

design". Annals of Nuclear Energy, (2002). 29(9): p. 1055-1071.

3. Yi, C. and G. Sjoden, "YGROUP User Manual: Multigroup Cross Section Data

Collapsing Code Using Contributon Weighting Scheme". (2010), University of Florida.

4. Yi, C., et al. "Computationally Optimized Multigroup Cross Section Data Collapsing

Using the YGROUP Code". in PHYSOR 2010. (2010). Pittsburgh, PA: American Nuclear

Society.

5. Alpan, A. and A. Haghighat, "Development of the CPXSD methodology for generation of

fine-group libraries for shielding applications". (2005), La Grange Park, IL: American

Nuclear Society. 14.

6. Sjoden, G. and A. Haghighat, "PENTRAN - Parallel Environment Neutral-particle

TRANsport, Code Users Guide/Manual, Version 9.4X.5". (2008), HSW Technologies

LLC.

7. Sjoden, G., et al. "Automatically Tuned Adaptive Differencing Algorithm For 3-D Sn

Implemented in PENTRAN". in International Conference on Mathematics,

Computational Methods & Reactor Physics (M & C 2009). (2009). Saratoga Springs,

New York: American Nuclear Society.

International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011)

Rio de Janeiro, RJ, Brazil, May 8-12, 2011, on CD-ROM, Latin American Section (LAS) / American Nuclear Society (ANS) ISBN 978-85-63688-00-2

APPENDIX A

The wall-clock times for PENTRAN calculations are provided in Tables A.I and A.II for forward

and adjoint transport, respectively. Calculations were performed on nodes of dual quad-core

E5405 Xeon processors, on up to 5 nodes. The 16 group calculations were performed on 2 nodes,

and the rest were performed on 5 nodes. There is a minimum of 4 GB available per node.

Tables A.III A.V give the YGROUP collapse results with associated broad group/contributon

weighting. Table A.VI is the 80 group energy structure used.

Table A.I. Forward Transport Wall Clock Times in Seconds

# Groups P1 P3 CPUs

16 172

(2:52 min) 622

(10.3 min) 16

30 323

(5.3 min)

1249 (21 min)

30

40 1252

(20 min)

1664 (28 min)

40

80 2268

(38 min) 6755 sec (112 min)

40

Table A.II. Adjoint Transport Wall Clock Times in Seconds

# Groups P1 P3 CPUs

16 376

(6.25 min) 1411

(23.5 min) 16

30 734

(12.25 min) 2647

(44 min) 30

40 1567

(26 min) 5573

(93 min) 40

80 6367

(106 min) 14751

(246 min) 40

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Table A.III. 40 Group Collapse Results

Broad group

#

Fine group range

Broad group wt.

Broad group

#

Fine group range

Broad group wt.

1 1 to 1 8.49395E-01 21 40 to 41 1.94042E-03

2 2 to 3 6.34773E-05 22 42 to 43 4.75340E-03

3 4 to 5 6.74168E-05 23 44 to 45 2.18759E-03

4 6 to 7 8.92143E-05 24 46 to 47 2.89751E-03

5 8 to 9 7.55904E-05 25 48 to 49 2.62035E-03

6 10 to 11 8.42971E-05 26 50 to 51 1.79396E-03

7 12 to 13 1.10408E-04 27 52 to 53 9.96407E-04

8 14 to 15 2.70830E-05 28 54 to 55 3.09489E-03

9 16 to 17 1.73748E-05 29 56 to 57 3.49418E-03

10 18 to 19 3.81938E-05 30 58 to 59 3.43235E-03

11 20 to 21 3.71920E-05 31 60 to 61 5.98998E-03

12 22 to 23 4.35763E-05 32 62 to 63 4.79148E-03

13 24 to 25 4.68327E-05 33 64 to 65 1.04344E-02

14 26 to 27 6.11489E-05 34 66 to 67 1.08146E-02

15 28 to 29 5.52832E-05 35 68 to 69 2.31466E-02

16 30 to 31 5.12889E-05 36 70 to 71 2.29577E-02

17 32 to 33 7.76942E-05 37 72 to 73 1.43598E-02

18 34 to 35 1.21201E-04 38 74 to 74 2.68466E-02

19 36 to 37 3.31489E-04 39 75 to 76 7.17634E-04

20 38 to 39 1.01197E-03 40 77 to 80 9.24418E-04




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Table A.IV. 30 Group Collapse Results

Broad group

#

Fine group range

Broad group wt.

Broad group

#

Fine group range

Broad group wt.

1 1 to 1 8.49395E-01 16 58 to 61 9.42234E-03

2 2 to 5 1.30894E-04 17 62 to 65 1.52259E-02

3 6 to 9 1.64805E-04 18 66 to 67 1.08146E-02

4 10 to 13 1.94705E-04 19 68 to 68 8.50400E-03

5 14 to 17 4.44578E-05 20 69 to 69 1.46426E-02

6 18 to 21 7.53859E-05 21 70 to 70 1.43243E-02

7 22 to 25 9.04090E-05 22 71 to 71 8.63349E-03

8 26 to 29 1.16432E-04 23 72 to 72 9.68713E-03

9 30 to 33 1.28983E-04 24 73 to 73 4.67268E-03

10 34 to 37 4.52691E-04 25 74 to 74 2.68466E-02

11 38 to 41 2.95239E-03 26 75 to 75 4.33843E-04

12 42 to 45 6.94100E-03 27 76 to 77 6.05606E-04

13 46 to 49 5.51786E-03 28 78 to 78 2.85618E-04

14 50 to 53 2.79037E-03 29 79 to 79 2.97477E-04

15 54 to 57 6.58907E-03 30 80 to 80 1.95068E-05

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Table A.V. 16 Group Collapse Results

Broad group

#

Fine group range Broad group

weight

1 1 to 1 8.49395E-01

2 2 to 10 3.41808E-04

3 11 to 19 2.31248E-04

4 20 to 28 2.09249E-04

5 29 to 37 6.16458E-04

6 38 to 46 1.12705E-02

7 47 to 55 1.00260E-02

8 56 to 64 2.19866E-02

9 65 to 67 1.69703E-02

10 68 to 68 8.50400E-03

11 69 to 69 1.46426E-02

12 70 to 70 1.43243E-02

13 71 to 71 8.63349E-03

14 72 to 73 1.43598E-02

15 74 to 74 2.68466E-02

16 75 to 80 1.64205E-03




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Table A.VI. 80 Group Upper Energies (MeV)

Group Upper MeV Group Upper MeV Group Upper MeV

1 20 31 0.03 61 4e-07

2 17.333 32 0.025 62 3.5e-07

3 15.683 33 0.017 63 3.25e-07

4 12.84 34 0.013 64 2.75e-07

5 10 35 0.0095 65 2.25e-07

6 8.1873 36 0.006 66 1.75e-07

7 6.434 37 0.00374 67 1.5e-07

8 4.8 38 0.00155 68 1.25e-07

9 4.304 39 0.00055 69 1e-07

10 3 40 0.00021 70 7e-08

11 2.479 41 0.000108 71 5e-08

12 2.354 42 3.7e-05 72 4e-08

13 1.85 43 1e-05 73 3e-08

14 1.5 44 5e-06 74 2.53e-08

15 1.4 45 4e-06 75 1.5e-09

16 1.356 46 3.05e-06 76 1.2e-09

17 1.317 47 2.38e-06 77 1e-09

18 1.25 48 1.86e-06 78 7.5e-10

19 1.2 49 1.45e-06 79 5e-10

20 1.01 50 1.3e-06 80 1e-10

21 0.82 51 1.12e-06

22 0.75 52 1.08e-06

23 0.6 53 1.04e-06

24 0.49952 54 1e-06

25 0.33 55 8.5e-07

26 0.27 56 8e-07

27 0.2 57 7e-07

28 0.1 58 6.25e-07

29 0.073 59 5.5e-07

30 0.045 60 5e-07

MULTIGROUP CROSS SECTION COLLAPSING OPTIMIZATION OF A HE-3 DETECTOR ASSEMBLY MODEL USING...

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