Monte Carlo Simulation Research on the Spontaneous Fission Yield of 240Pu

Home , Decay product, Fission product yield, Ternary fission

A dissertation submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in the Department of Nuclear & Radiological Engineering of the College of Engineering and Applied Sciences by

Tianyou Xie

University of Cincinnati Nov 2015

Committee Chair: Henry B. Spitz, Ph.D.

Abstract

This research describes results of mathematical simulations to predict the spontaneous fission yield of 240Pu thereby guiding selection of a suitable fission product for use as a radiochronometer for weapons grade plutonium.

The PUREX process is effective in separating plutonium and uranium in irradiated nuclear fuel from fission products, which may make it possible to use one or more of the spontaneous fission products from 240Pu as a radiochronometer. However, the fission product inventory from the spontaneous fission of 240Pu reported in the literature is quite variable.

The non-uniform induced fission probability at multiple neutron energies makes the induced fission yield too complex to be listed as a single value.

The neutron flux distribution from a source containing 240Pu is a combination of both spontaneous and induced fissions. Fission products generated in this source are due to a combination of spontaneous and induced fission processes, each with a unique neutron energy distribution and fission product yield. The physical conditions of the source (i.e., size, shape, etc.) will have a significant influence on the neutron flux distribution

2 and the induced fission probability associated with 240Pu. Thus, it is likely that there is a relationship between the combined (spontaneous and induced) fission yield and the physical conditions of the source. Exploring this relationship, a table of values will be generated to predict the combined fission product yield for any source geometry containing 240Pu. Minimizing variations in the source geometry should stabilize the combined fission product yield and provide a means to determine the spontaneous fission probability for 240Pu. This research has identified that the combined fission product yields from 97Zr and 138Xe exhibit less sensitivity to the physical source parameters than other fission products making them good candidates as radiochronometers for age dating weapons-grade plutonium.

Contents Abstract ...... 2 Chapter 1 Introduction ...... 7 1.1 Introduction ...... 7 1.1.1 Weapons-grade Plutonium ...... 8 1.1.2 Spontaneous Fission...... 9 1.1.3 Chronometer ...... 11 1.1.4 Monte Carlo Method & MCNPX ...... 13 1.1.5 Current Study ...... 15 1.1.6 Challenge of Chronometer Study...... 20 1.2 Objectives and Specific Aims ...... 22 Chapter 2 Neutron Energy Effect on Fission ...... 24 2.1 Background ...... 24 2.1.1 Fission Neutron Energy ...... 30 2.1.2 Simulations Using Fission Product Photopeaks ...... 31 2.1.3 Independent Fission Probability Study ...... 32 2.2 Method ...... 34 2.2.1 Single Neutron Energy Model ...... 34 2.2.2 Varied Energy Selection ...... 35 2.3 Result & Discussion ...... 36 2.4 Conclusions ...... 39 Chapter 3 Micro Position Model...... 41 3.1 Background ...... 41 3.2 Method ...... 44 3.2.1 Uniform Thickness Model ...... 47 3.2.2 Uniform Volume Model ...... 51 3.2.3 Neutron Energy Distribution...... 52 3.3 Result & Discussion ...... 53 3.3.1 Neutron Energy Distribution...... 53 3.3.2 Equal Thickness Model...... 55 3.3.3 Equal Volume Model ...... 58 3.3.4 Fission Yield Combination ...... 61

3.4 Conclusion ...... 62 Chapter 4 Macroscopic Parameter Model...... 64 4.1 Background ...... 64 4.1.1 Plutonium Fission Yield Curve ...... 64 4.2 Method ...... 65 4.2.1 Simulation Parameters Selection ...... 68 4.2.2 Macro Parameter Model ...... 69 4.3 Result & Discussion ...... 73 4.4 Conclusion ...... 83 Chapter 5 Conclusion ...... 85 5.1 Research Accomplishments ...... 85 5.2 Future Work ...... 86 References ...... 88 Appendix A ...... 90 Appendix B ...... 98 Appendix C ...... 102

Chapter 1

Introduction

1.1 Introduction

Plutonium is a radioactive actinide metal whose isotope, plutonium-239, is one of the three primary fissile isotopes1 (uranium-233 and uranium-235 are the other two), 2 Plutonium-241 is also highly fissile. A fissile isotope has a nucleus that may fission when struck by a thermal neutron.

Twenty radioactive isotopes of plutonium have been characterized. The longest-lived are plutonium-244, with a half-life of 80.8 million years, plutonium-242, with a half-life of 373,300 years, and plutonium-239, with a half-life of 24,110 years. All of the remaining radioactive isotopes have half-lives that are less than 7,000 years. This element also has eight metastable states, all with have half-lives less than one second.3

1.1.1 Weapons-grade Plutonium

The primary component of nuclear weapons is uranium or plutonium metal, enriched in a fissile isotope (233U, 235U or 239Pu). These isotopes are defined as Special Nuclear Material (SNM).4 The development of nuclear weapons has relied predominantly on the production of high-enriched uranium (HEU) or weapons-grade plutonium for fission primaries. However, spontaneous fission neutrons from 240Pu may cause a premature detonation of the device unless the weapon design accommodates this neutron source.4

The commercial nuclear power industry is a significant contemporary source of SNM. Plutonium is created by irradiation of 238U in a nuclear reactor, under tailored conditions of neutron bombardment and post irradiation decay, followed by radiochemical reprocessing. A major difference in used commercial nuclear fuel and weapons-grade plutonium is the concentration of 240Pu. Plutonium is considered as “weapons grade” if it contains at least

93% 239Pu and <7% 240Pu.5 Reactor grade plutonium contains more than 8%

240Pu. Plutonium is also used for civilian nuclear power production in mixed-oxide (MOX) fuel, which nominally contains 60% 239Pu and 25%

240Pu.6

Weapons grade plutonium is generated in nuclear fuel that has been irradiated for no more than 7 months. The economics of commercial nuclear power plants dictates that fuel be retained in a reactor core for about 4 years to achieve optimum performance and cost effectiveness. The typical 1-GWe nuclear reactor uses about 1 ton of fissile material per year which generates about 200 kg of plutonium. The annual global production of plutonium in commercial nuclear power reactors is about 70 t.7

1.1.2 Spontaneous Fission

In nuclear physics and chemistry, nuclear fission is either a nuclear reaction or a radioactive decay process in which the nucleus of an atom splits into smaller parts (lighter nuclei). The fission process also produces fast neutrons and photons (as gamma rays) with the release of a large amount of energy.

Fission is a form of nuclear transmutation because the resulting fragments are not the same element as the original atom. The two nuclei produced are most often of comparable but slightly different sizes, typically with a mass

9 ratio of products of about 3 to 2, for common fissile isotopes.8,9 Most fissions are binary fissions (producing two charged fragments), but occasionally (2 to 4 times per 1000 events), three positively charged fragments are produced, in a ternary fission. 10 The smallest of these fragments in ternary processes ranges in size from a proton to an argon nucleus.

Spontaneous fission, a rare decay mode occurring only in the heavy elements, gives the same results as induced nuclear fission. Like other forms of decay, spontaneous fission is an outcome of quantum tunneling. A neutron in a heavy element nucleus has a finite probability to tunnel through the fission barrier, classically, it could not surmount. In nuclear reactions, a heavy product nucleus can possess excitation energy comparable to, or more than, the height of the potential barrier, making fission a much more probable process.10

1.1.3 Chronometer

Radio-chronometers are useful tools in nuclear forensics that aid in ascertaining the elapsed time since a radioactive material was last purified.

Radionuclides linked to another by the processes of radioactive decay have relative concentrations that can be calculated by the simple laws describing the ingrowth of radioactive decay products from an initially pure parent. If a time exists at which only atoms of the parent are present, that time can be determined at some future time by measuring the concentrations of parent and decay products. The interval between the time that the sample was last purified and the time that it was subsequently analyzed is defined as the “age” of the material at the analysis time.15

Radiochronometers for plutonium are not limited to the isotopes produced by the process of α decay (viz., 240Pu/236U & 239Pu/235U). The 240Pu isotope decays almost completely by α emission; however, one decay in every

1.7×107 alpha decays occurs by spontaneous fission. In spontaneous fission, the atom subdivides into two roughly equal parts, with the fission yield being distributed over products covering a wide range of atomic numbers

11 and masses, similar to the yield distribution arising in the neutron-induced fission of 235U or 239Pu. Although the fraction of the atoms in a plutonium sample that undergo spontaneous fission decay does not appear significant, spontaneous fission neutrons create a limit on the concentration of 240Pu that can be present in weapons material. One gram of weapons-grade Pu containing 6% 240Pu by mass undergoes 1480 spontaneous fission (SF) decays every minute.4

The relative magnitudes of the spontaneous fission- and α- decay branches, coupled with the distribution of the fission-product yield over a large number of nuclides, favors the formation of 236U over any given fission product by at lease a factor of 109. However, as the decay rate of a radionuclide is inversely proportional to its half-life, the activities of many of the shorter-lived fission products are comparable to that of long-lived 236U.

The fission neutron yield, v, neutron energy spectra, and fission product yields associated with the spontaneous fission of 240Pu and the fissile 239Pu differ significantly and complicate selecting a fission product for radiochronometry. Significant efforts have been made to improve knowledge of induced fission product yields and their associated uncertainties for many of the chains.10

1.1.4 Monte Carlo Method & MCNPX

Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results; typically one runs simulations many times in order to obtain the distribution of an unknown probabilistic entity. The name comes from the resemblance of the technique for playing and recording results in a real gambling casino. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to obtain a closed-form expression, or unfeasible to apply a deterministic algorithm. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration and generation of draws from a probability distribution.11

In physics-related problems, Monte Carlo methods are quite useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures. Other examples include modeling phenomena with inputs exhibiting significant uncertainty, such as the calculation of risk in business and evaluation of multidimensional definite integrals with complicated boundary conditions.

In application to space and oil exploration problems, Monte Carlo–based predictions of failure, cost overruns and schedule overruns are routinely better than human intuition or alternative "soft" methods.12

Monte Carlo N-Particle Transport Code (MCNP) is a software package for simulating nuclear processes. It was initially developed in 1957 by Los

Alamos National Laboratory and has subsequently had 13 many major improvements. MCNP is distributed within the United States by the

Radiation Safety Information Computational Center in Oak Ridge, TN and internationally by the Nuclear Energy Agency in Paris, France. It is used primarily for the simulation of nuclear processes, such as fission, but has the capability to simulate particle interactions involving neutrons, photons, and electrons. Specific areas of application include radiation protection and dosimetry, radiation shielding, radiography, medical physics, nuclear criticality safety, detector design and analysis, nuclear oil well logging, accelerator target design, fission and fusion reactor design, decontamination and decommissioning. 14

MCNPX (Monte Carlo N-Particle eXtended) was also developed at Los

Alamos National Laboratory and is capable of simulating particle

14 interactions of 34 different types of particles (nucleons and ions) and 2000+ heavy ions at nearly all energies, including those simulated by MCNP.14

1.1.5 Current Study

The analysis of fission products from the spontaneous fission of 240Pu has been described by Moody et al. (2005) as a complementary method and a valuable tool for the chronometry of incomplete fuel reprocessing.15 These fission products may be used to determine processes involving metallurgical preparation of the plutonium. While the data cited by Moody et al. for weapons grade plutonium is modest in terms of activity (< 1 Bq of radioactivity) for each fission product isotope per gram, improvised nuclear devices (IND) or radiological dispersal devices (RDD) containing reactor grade and high burn plutonium would contain a significant quantity of fission products born from the spontaneous fission of 240Pu. Selected fission products from spontaneous fission of 1 gm of Pu containing 6% 240Pu are shown in Table 1.1.15

Table 1.1 Selected Spontaneous-fission products present in 1g of Pu containing 6% 240Pu after 1 year19

Nuclide Cumulative Yield (%) Half-life Atoms Activity (dpm) 85Kr 0.07 10.8 years 5.3e5 0.06 93Zr 3.0 1.5e6 years 2.3e7 2e-5 99Tc 6.9 2.1e5 years 5.4e7 3e-4 133I 8.2 20.8 hours 2.2e5 122 136Xe 7.5 Stable 5.8e7 137Cs 7.2 30.2 years 5.5e7 2.4 148Nd 1.6 Stable 1.2e7

Laidler and Brown (1962) investigated the mass distribution of spontaneous fission products from 240Pu radio-chemically by measuring the fission yields of 15 isotopes in the mass range 89 to 147.16 Their results are shown on

Table 1.2. They also described a series of changes in the yield-mass function for fission in various states of excitation of the same nucleus, since neutron-induced fission of 239Pu is essentially fission of 240Pu in excitation levels above the ground state.

Table 1.2 The observed yield of spontaneous fission products from a sample 91.7% 240Pu & 6.7% 239Pu.16

Nuclide No. of Observed fission determinations yield (%) 89Sr 2 0.80±0.32 91Sr 3 1.51±.017 97Zr 2 6.48±0.21 99Mo 3 6.82±0.07 105Rh 3 7.10±0.55 133I 3 8.20±0.12 135I 3 6.94±0.67 140Ba 4 5.99±0.22 141Ce 3 6.02±0.39

Bushuev et al. (2007) used photopeaks emitted by spontaneous-fission products observed in the photon energy spectrum from californium sources and plutonium samples to determine the yield of fission products.17 The yield of spontaneous-fission products was determined as the ratio of the number of counts in the gamma peaks of the fission products and the gamma peak at 642 keV of 240Pu. The results are shown in Table 1.3:

Table 1.3 The observed fission yield of spontaneous fission product of 41.46g 91.07% 240Pu 7.86% 239Pu sample with 1.07% mass 241Am.17

Fission product Energy of the γ rays measured Yield from 240Pu spontaneous fission % 92Sr 1383 2.92±0.14 94Rb 1309 1.23±0.04 94Sr 1428 3.94±0.06 96mY 1107 1.13±0.16 1751 133Sb 1096 3.26±0.18 1132 135I 1260 8.13±0.15 1796 136I 1321 3.65±0.40 138Xe 1768 7.60±0.29 138Cs 1436 6.85±0.18 140La 1596 5.70±0.18

The IAEA also has assembled a data base of spontaneous fission product yields collected from its member nations.18 Table 1.4 is a comparison of the results of the IAEA data base with the spontaneous fission yields listed on

Table 1.1 t Table 1.3.

Table 1.4 The cumulative neutron induced fission yield from IAEA data-base and other sources.

Fission Cumulative Neutron Induced fission yield (%) Spontaneous fission yield (%) Products TENDL- ROSFOND JENDL- ENDF- Laidler & Bushuev 12 2011 -2010 & 4.0 349 14 15 Moody Brown et al. JEFF-3.1 Sr 89 1.39 1.73 1.4 1.75 0.8 1.1 Sr 90 1.85 1.8 1.8 1.95 1.4 Sr 91 2.56 2.25 2.19 2.44 1.51 Sr 92 3.47 2.76 2.67 2.82 2.92 Zr 93 4.57 3.79 3.51 2.9 3 Rb 94 2.27 1.26 0.85 0.844 1.23 Sr 94 5.02 3.63 3.62 2.78 3.94 Zr 95 6.11 4.47 4.44 3.7 4.8 Y 96m 2.43 1.31 1.71 2.07 1.13 Zr 97 7.28 5.3 5.21 4.2 6.46 Tc 99 7.74 6.06 6 4.87 6.9 Mo 99 7.75 6.06 5.96 4.87 6.82 Ru 103 6.58 6.7 6.61 5.29 7.9 Rh 105 3.1 5.44 5.57 4.75 7.1 Ru 106 1.71 5.1 5.09 4.01 6.2 Pd 109 0.11 1.71 1.74 2.3 0.94 Ag 111 0.08 0.49 0.483 1.67 0.035 Cd 115 0.06 0.062 0.058 1.09 <0.03 Sb 125 0.065 0.087 0.089 1.61 0.05 I 131 0.406 3.57 3.34 4.46 2.34 2.3 I 133 2 6.97 6.7 5.39 8.19 8.2 Sb 133 0.876 1.41 1.96 0.611 3.26 Xe 133 2.005 6.98 6.71 5.55 8.2 I 135 5.17 6.81 6.72 3.76 6.39 8.13 Cs 135 5.3 7.56 7.22 5.34 7.8 I 136 3.9 1.58 2.34 1.5 3.65 Cs 137 7.74 6.53 6.55 4.51 7.2 Xe 138 7.8 5.28 5.7 2.97 7.6 Cs 138 8.23 6.22 6.31 5.05 6.85 La 140 7.71 5.5 5.7 3.85 5.7 Ba 140 7.71 5.49 5.5 3.79 5.99 Ce 141 7.38 4.72 5.49 3.59 6.02 5.7 Pr 143 6.64 4.52 4.49 3.1 4.78 Nd 147 3.21 2.15 2.12 1.79 1.22 2.1

1.1.6 Challenge of Chronometer Study

The present study identified four challenges that need resolution before determining whether a 240Pu fission product can be useful as a radiochronometer.

The first challenge is a lack of agreement on the spontaneous fission yield value for 240Pu. The yield for the same fission product can range from 2% to

8% depending upon the reporting source (e.g., 133I in Table 1.4). The confidence interval for fission yield of 89Sr shown on Table 1.2 is large compared to the mean as ±0.32% to 0.8%.

The second challenge is a lack of discussion about the spontaneous fission yield uncertainty in the literature to aid in resolving this issue.

The third challenge is that the 240Pu spontaneous fission yield has been determined by analyzing different types of samples, each with unique physical and chemical characteristics, using different methods such that any comparison of results is further obfuscated.

The fourth challenge is related to the selection of a fission product radiochronometer. Although hundreds of the fission products are produced by spontaneous and induced fission, the advantage and disadvantage of any one chronometer has yet to be determined.

1.2 Objectives and Specific Aims

The objective for this research involves evaluating variations in the reported spontaneous fission yield reported in the literature and describing how the induced fission yield and probability of 240Pu can be affected by the physical attributes of the samples (viz., size, shape, etc.) sample.

Aim #1: Developing various neutron energy models to characterize differences in spontaneous fission yields. Hypothesis: The fission product yields and induced fission probability are dependent on the energy of the neutron released during spontaneous fission of 240Pu.

Aim #2: Develop a micro sphere model of pure 240Pu to identify the relationship between the initial spontaneous fission and neutron- induced fission during the fission process. Hypothesis: The induced fission product yields from each spontaneous fission of 240Pu are a function of the neutron energy distribution, which is also a function of the location of the Pu240 atom in the sample.

Aim #3: Develop a macroscopic plutonium material model to identify how changes in the macroscopic parameters of a sample affect the neutron-induced fission product yields. Hypothesis: The induced fission product yields and probability are dependent on the physical attributes of a sample. The 240Pu fission product that is least affected by changes in sample attributes is likely the best choice for a radiochronometer.

Chapter 2

Neutron Energy Effect on Fission

2.1 Background

In a realistic experiment, a sample of weapons-grade plutonium always contains 240Pu and 239Pu isotopes. It is impossible to separate these isotopes using traditional methods because their mass is nearly identical. Likewise, during an experiment, one cannot differentiate fission products produced by the induced fission 239Pu or the spontaneous or induced fission of 240Pu .

Fission occurring in a sample containing plutonium is a combination of (a) pure spontaneous fission of 240Pu, (b) induced fission of 240Pu, or (c) induced fission of 239Pu. The spontaneous fission probability of 239Pu is very low compared to 240Pu and will be ignored in the following descriptions. In this thesis, the observed fission yield includes all three fission processes.

Spontaneous fission signifies only the “pure” spontaneous fission of 240Pu unaffected by neutrons from other sources.

The observed fission yield can be separated into three parts,

= , × , +

, × , +

, × , Equ.1 where

240 is the observed fission yield of Pu in an experiment,

240 , is the initial spontaneous fission yield of Pu*,

240 , is the fission yield of Pu that has been induced by

spontaneous fission neutrons from 240Pu or induced fission neutrons

from 239Pu,

239 , is the fission yield of Pu that has been induced by

the spontaneous or induced fission neutrons from 240Pu and induced

fission neutrons from 239Pu,

240 ,: The percentage of Pu spontaneous fission over

all fission,

240 ,: The percentage of Pu induced fission over all

fission

239 ,: The percentage of Pu induced fission over all fission

∗ , + , + , =1

The spontaneous fission yield is a function of the fission product probability due of a single 240Pu atom that is fixed in position for the given circumstances (benchmark).

Since induced and spontaneous fission cannot be distinguished in an experiment, equation 1 is modified to accommodate this constraint.

= , × , +

, × , Equ.1a

Where, , + , =1

Monte Carlo analysis has been used to simulate the fission processes occurring in a sample of weapons-grade plutonium with a focus on 240Pu fission. The value of , depends only on the physical property of the particle containing 240Pu.

The induced Pu fission product yield, ,, can be expressed as

, = ()() Equ. 2

Where

p(e):The independent induced fission probability at neutron energy e

D(e):The neutron energy distribution that induced the fission

The main sustaining source of neutrons that induce fission in the sample containing weapons-grade plutonium is from spontaneous fission of 240Pu.

Monte Carlo analysis can generate the induced fission probability per spontaneous fission, INF/SF,

/ = , ,

or the induced fission probability per spontaneous fission neutron, INF/SN.

, / = , ∗

Where

N Pu,induced is the number of Pu induced fissions in a given time period

240 , is the number of Pu spontaneous fissions in a given time period and

is the average number of emitted neutrons per spontaneous fission.

Given that the probability of an induced Pu fission or a spontaneous Pu240 fission can be written as

, , = ,+,

, , = ,+, then,

, , = ,+,

, , = ,+, ,

, , / = = , , 1+ / + , , and

, , = ,+,

, , = ,+, ,

, , = , + , , ,

1 = 1+ /

In this derivation, INF/SF can be used as important indicator to describe the contributions of induced fission to the observed fission yield. The, equation

1a can be simplified as

= × + , /

/ × Equ.1b , /

2.1.1 Fission Neutron Energy

Figure 2.1 Fission neutron spectra for 235U (red line) and 239Pu (blue line), approximated by Watt distributions.19

The median and average 239Pu fission neutron energy is 1.6 MeV, and 2.1

MeV, respectively, with 99.8% of all fission neutrons having energies below

10 MeV. Within a sample of plutonium material, the fission neutron energy spectrum will differ from Fig. 2.1 because of energy loss during transport.

2.1.2 Simulations Using Fission Product Photopeaks

The simulation is based upon photon emissions from spontaneous fission

products to determine the fission product yield. Three criteria were

adopted to select photons for the simulation from all the hundreds of

fission product photons emitted:

1. High γ energy during fission process.

2. High expected yield related to fission product.

3. High observed fission yield in current study

Table 2.2 Selected fission products that comply with the three criterions established in the simulation.

Fission Products Z γ peak energies Intensity (%) Current Study (kev) Yield (%) Sr 92 1383.93 90.00% 2~3 Sr 94 1427.7 94.20% 2~3 Zr 97 743.36 93.09% 4~7 Rh 105 318.9 19.10% 3~7 I 133 529.87 87.00% 5~8 I 135 1260.41 28.70% 3~8 Xe 138 1768 16.70% 5~7 Cs 138 1435.86 76.30% 5~8 Ba 140 537.26 24.39% 4~7

2.1.3 Independent Fission Probability Study

An important study on independent fission probability is the IAEA’s project

“Multi-platform EXFOR-CINDA-ENDF” (citation). Table 2.2, adapted from the IAEA study, lists the fission-product yields for incident neutron energies of 0.0253 MeV, 0.5 MeV and 14 MeV and demonstrates that the fission probability changes as the incident neutron energy changes. The majority of fission probabilities exhibit a positive relationship with incident neutron energy. There are a few exceptions to this relationship, such as 94Sr.,

Fission yields reported by the IAEA include only low (0.0253 MeV) , medium (0.5 MeV) and high (14 MeV) neutron energies. Result of this study provides fission probabilities over a continuous range of incident neutron energies from 0 to 4 MeV.

20 Table 2.2 The independent neutron induced fission yield of ENDF/B-VII.1 from IAEA data-base.

Independent Neutron induced fission- yields Fission Products 0.0253 MeV 0.5 MeV 14 MeV Sr 89 1.0570E-05 1.2190E-05 1.1291E-04 Sr 90 6.3019E-04 1.5752E-04 6.3019E-04 Sr 91 2.7242E-03 1.0462E-03 2.7242E-03 Sr 92 5.3954E-03 5.4842E-03 7.6426E-03 Zr 93 2.2400E-06 2.6500E-06 3.4283E-05 Rb 94 9.0655E-03 9.0713E-03 8.0443E-03 Sr 94 2.7140E-02 2.7296E-02 1.9941E-02 Zr 95 4.2656E-04 4.4153E-04 1.3913E-03 Y 96m 1.7181E-02 1.7836E-02 2.0743E-02 Zr 97 9.8490E-03 1.0140E-02 1.1741E-02 Tc 99 1.2300E-08 1.6900E-08 1.3797E-06 Mo 99 4.7060E-05 5.3530E-05 3.5315E-04 Ru 103 2.0200E-06 2.6800E-06 5.7138E-05 Rh 105 1.7600E-07 2.4300E-07 1.6596E-05 Ru 106 1.6152E-03 1.6900E-03 4.8687E-03 Pd 109 4.2800E-10 3.2300E-10 1.7806E-05 Ag 111 0.0000E+00 0.0000E+00 6.1087E-07 Cd 115 3.7100E-09 3.5800E-09 7.7084E-06 Sb 125 1.5800E-06 2.4300E-06 6.3273E-04 I 131 4.9640E-05 5.1990E-05 2.6031E-03 I 133 2.4118E-03 2.5856E-03 9.4849E-03 Sb 133 1.8231E-02 1.8621E-02 5.9595E-03 Xe 133 1.6240E-05 1.6590E-05 2.8817E-04 I 135 3.4474E-02 3.6606E-02 2.8756E-02 Cs 135 1.2815E-05 1.6570E-05 3.9086E-04 I 136 1.7957E-02 1.7010E-02 1.1956E-02 Cs 137 2.0813E-03 2.3841E-03 8.2417E-03 Xe 138 4.6140E-02 4.6155E-02 2.4952E-02 Cs 138 3.4160E-03 3.7510E-03 1.4168E-02 La 140 2.7480E-05 3.1880E-05 6.1988E-04 Ba 140 5.2978E-03 5.3862E-03 9.8560E-03 Ce 141 4.5100E-07 5.0200E-07 3.4903E-05 Pr 143 4.0400E-08 5.7200E-08 1.0148E-05 Nd 147 2.8500E-06 3.4600E-06 1.6628E-04

2.2 Method

2.2.1 Single Neutron Energy Model

The induced fission product probability is the probability that the observed fission product was induced by a neutron of a specific energy. This study of the independent neutron-induced 240Pu fission product probability begins by simulating a pure 240Pu material and a single energy neutron source. Of course, it is not possible to arrange these conditions experimentally. The geometry of the single neutron energy simulation model is illustrated in figure 2.2.

Figure 2.2 a) Pure 240Pu disk, thick 0.1mm, diameter 2cm. b) Uniform, mono-energetic neutron source positioned 0.1mm above 240Pu disk.

The combination of a very thin source emitting only mono-energetic neutrons minimizes the probability of generating secondary, neutron-

34 induced fission in the source such that photons associated with induced fission are produced by mono-energetic neutrons. The thin source also has a negligible effect on neutron direction or speed. This is similar to the conditions associated with a beam of mono-energetic neutrons.

2.2.2 Varied Energy Selection

The typical energy range of plutonium fission neutrons is 0 to 4 MeV. The simulation has been designed to accommodate a range of neutron energies for a realistic material by creating two energy bins, 0 to 1 MeV and 1 to 15

MeV. Neutron energy is incremented by 0.1 MeV in the lower energy bin where the change in fission cross-section with neutron energy is high, and by

1 MeV in the higher energy bin. The MCNPX code and output are provided in Appendix A.

2.3 Result & Discussion

The induced fission probabilities, shown on Fig 2.3, exhibit an exponential relationship at low neutron energy (viz., 0 to 1 MeV). Figure 2.3 illustrates that the fission product probabilities increase with increasing neutron energy.

0.006%

0.005%

0.004%

0.003%

0.002%

0.001% Independent Induced Fission Probability at at MeV Probability Fission Induced Independent 0.000% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Neutron Energy (MeV)

Sr-92 Sr-94 Zr-97 Rh-105 I-133 I-135 Xe-138 Cs-138 Ba-140

Figure 2.3 Induced fission probabilities due to neutrons with energy from 0 to 1 MeV based upon results of predicted fission product photon energy.

The predicted induced fission product probability for higher energy neutrons

(En > 1 MeV) is shown on Fig. 2.4 and has an appearance quite different

36 from Fig. 2.3. Although the general increasing trend with energy is still apparent, there are significant differences in the independent induced fission product probabilities between 3 MeV and 9 MeV. These differences are especially apparent for probability of Zr-97 and Rh-105, which exhibit unusual bumps at 4 MeV and 2 MeV, respectively. This observation implies that the induced fission yield probability resulting in this fission product is dominated at some specific neutron energies.

0.010% 0.009% 0.008% 0.007% 0.006% 0.005% 0.004% 0.003% 0.002% 0.001% 0.000% 0 2 4 6 8 10 12 14 16 Neutron Energy (MeV) Independent Induced Fission Probability at at MeV Probability Fission Induced Independent

Sr-92 Sr-94 Zr-97 Rh-105 I-133 I-135 Xe-138 Cs-138 Ba-140

Figure 2.4 Predicted independent induced fission probabilities from 0 to 15 MeV neutron energy based upon results of predicted fission product photon energy.

INF/SF has been shown to describe the related fission probability of induced fission to spontaneous fission. In this simulation, the induced fission

37 probability per spontaneous fission (INF/SF) has a similar outcome as the induced fission product probability. The induced fission probability has a positive relationship with the source neutron energy, which means the contribution of the induced fission to the total fission becomes stronger with increasing neutron energy.

It is seen from Fig. 2.5 that the INF/SF trend toward a plateau around 3 MeV and 8 MeV. This also suggests that the independent induced fission probability is dominated by some specific neutron energies.

7.00E-03

6.00E-03

5.00E-03

4.00E-03

3.00E-03 INF/SN at MeV MeV at INF/SN

2.00E-03

1.00E-03

0.00E+00 0 2 4 6 8 10 12 14 16 Neutron Energy (MeV)

Figure 2.5 Induced fission rate at 0 to 15 MeV neutron energy

2.4 Conclusions

Results of the single neutron energy model simulations performed in this research are similar to the independent induced fission probabilities reported by the IAEA and validate the simulation model. The IAEA independent fission probability varies at different source neutron energies. The main trend of the relationship reported by the IAEA for three neutron energies

(Figure 2.6) is that fission probability increases with the source neutron energy, except for in limited cases (i.e., I-135 and Sr-94).

4.00E-02

3.50E-02

3.00E-02

2.50E-02

2.00E-02

1.50E-02

1.00E-02

5.00E-03 IAEA Independent Fission Probability at MeV at Probability Fission Independent IAEA 0.00E+00 0.0253 0.5 14 Neutron Energy (MeV)

Sr 89 Sr 90 Sr 91 Sr 92 Zr 93 Sr 94 Zr 95 Y 96m Zr 97 Tc 99

Mo 99 Ru 103 Rh 105 Ru 106 Pd 109 Ag 111 Sb 125 I 131 I 133 Xe 133

I 135 Cs 135 Cs 137 Cs 138 La 140 Ba 140 Ce 141 Pr 143 Nd 147

Figure 2.6: IAEA fission yield prediction.

Results of the simulation described in this research offers more details about the induced fission probability Pu240. Both the independent induced fission probability distribution and the independent fission rate per neutron have two apparent maxima as the neutron energy increases from approximately 3

MeV to 8 MeV.

These conclusions demonstrate that the induced fission of 240Pu is very complex and is due to the non-uniform independent induced fission probability relationships with source neutron energy. The observed spontaneous fission yield is too complex to be stated as a single yield value and represents why there is such a large variation in the overall fission yield.

Chapter 3

Micro Position Model

3.1 Background

A neutron may interact with a nucleus with many reaction types. The probability of each type of reaction is very sensitive to the neutron energy and the target nucleus. Even the neighbor nuclides (as 239Pu and 240Pu), the reaction occurred may be dramatically different. Some of the prominent reactions are explained below:

Elastic scattering: The neutron and the nuclide share their kinetic energies when colliding. 21 Their total kinetic energy remains the same before and after the collision. If the nucleus is still before collision, it will start moving, and the neutron will slow down due to loss of kinetic energy. The residual nucleus remains in its ground state. This type of reaction is responsible for the majority of interactions of slow neutrons in nuclear reactors.

Inelastic scattering: The neutron and the nuclide share their kinetic energies when colliding. Their total kinetic energy after the collision is less than their total kinetic energy before the collision, while the difference is associating with the energy of excitation. If the nucleus is stationary before collision, the neutron must have kinetic energy more than excitation energy to produce a reaction. Inelastic scattering is threshold reaction where the threshold is the minimum neutron kinetic energy required to react possibly.

The excited nucleus subsequently de-excites γ decay.

Capture: The nucleus capture the neutron to form the next isotope (mass +1) that is in excited state when colliding. Then the excited nucleus subsequently de-excites γ decay.

(n,x) reaction: The nucleus capture the neutron to form the next isotope

(mass +1) that is in excited state when colliding. Then the excited nucleus subsequently de-excites decay by emitting particles such as a neutron, proton, deutron, α particle, etc. or a combination of such particles. If the emitted particle is an α, the interaction is designated as an (n,α) reaction. If a neutron and a proton are emitted, then it is called (n, p) reaction. If 2 neutrons are emitted, it is then (n,2n) reaction.

Fission: Nuclear fission describes a heavy nucleus splitting into two smaller nuclei (fission fragments). This reaction emits a large amount of energy with two or more neutrons, and gamma rays. When a neutron collides with a heavy nucleus, the neutron may be absorbed and the nucleus becomes excited to possibly fission. The resulting neutrons produced in fission are fast, with an average energy of 2 MeV. Error! Bookmark not defined.

The fission fragments themselves are formed in an excited state and de- excite generally by β, γ and neutron emissions. Neutrons emitted during fission are designated as prompt neutrons while neutrons emitted by fission fragments are designated as delayed neutrons. Similarly, prompt and delayed gammas are also emitted. About 80 % of the energy released in fission is carried away by the fission products (and the rest by the other particles), which in turn transfers the energy to the surroundings, making the energy recoverable. Some energy is carried away by particles known as neutrinos, which are chargeless and light, do not interact with any material, and hence their energy is not recoverable. The approximate distribution of fission energy through various emissions is given below: (Table 3.1) Error!

Bookmark not defined.

Table 3.1 Fission energy distribution through various emissions24

Kinetic energy of fission fragments 170 MeV Prompt radiations (γ and n) 12 MeV Delayed radiations (γ and n) 12 MeV Neutrinos (unrecoverable) 8 MeV Total 202 MeV

3.2 Method

Three of the 4 input parameters in equation 1a can be simulated, viz.,

, , , and ,. The remaining input,

,, is an unknown which is determined by the physical nature of 240Pu. Recall equation 1b:

= ,, + ,,

Based on the hypothesis that the induced fission yield of 240Pu is influenced by external physical parameters, the general function of the induced fission yields of Pu can be created:

, = , + × + × +⋯+ Equ. 3

Where,

,: The fission yield of Pu induced by a spontaneous or

induced fission neutron from Pu

: The baseline of induced fission yield neglecting external

physical parameters

: The influence of external physical parameter n.

: External physical parameter n’s value

: Error.

In the microscopic analysis, each location of a plutonium particle in a sample will have a unique set of physical conditions surrounding the particle that will affect the neutron energy, flux, etc. Thus, the observed induced fission yield can be expressed as the volume weighted average of local induced fission yield where

, = ∑(,() × ) Equ.4

Where,

: Fraction of sample volume impacting particle

: Total Volume of the Pu sample such that

= ∑

,(): Local induced fission yield for the given micro scenario

The local induced fission yield can be expressed by substituting equation 3 with micro scenario parameters:

,() = , + ∑ ∗ + Equ.5

Where,

, : The induced fission yield for the “Zero” scenario

where all the micro-parameter values are equal to 0

th : The influence coefficient of i micro-parameter

th : The i micro-parameter’s value given micro scenario

: Error.

Equations 4 & 5 describe the hypothesis that the induced fission yield of Pu is a function of both the neutron energy distribution (micro external parameters) and the location where a spontaneous fission occurs.

3.2.1 Uniform Thickness Model

The section describes simulations of spontaneous fissions occurring in a 1.5 cm diameter sphere of 240Pu that is divided into 20 equally thick concentric levels except for the center (level 1), which is a small ball of radius 0.55 cm.

(Fig. 3.1) The remaining 19 levels are each 0.05 cm thick concentric spheres. The outer surface of each level is labeled as surface 1 to surface 20.

The volume of level n is defined as the material between the surface n-1 and surface n.

Figure 3.1: Perspective View of the uniform depth Model

Simulations are performed in two parts, one involving neutron interactions and the second involving photon interactions.

Neutron simulations:

A total of 20 simulations of neutron interactions are performed, each with the spontaneous fission neutron source uniformly distributed in one of the 20 levels. The γ-ray spectra produced by induced fission products at each of the

20 surfaces are recorded. These simulations describe how the fission product gamma ray spectra will change with changes in the location of the source of spontaneous fission neutrons.

Gamma simulation:

Likewise, a total of 20 simulations are performed with a range of unique single-energy photon sources uniformly distributed in one of the 20 levels.

The selection of initial photon energies were described in section 2.1.3. The

γ-ray spectrum at each surface is recorded. The results of these simulations describe how gamma ray spectra will change with change origin of the photon source.

Given the location of the initial source of the spontaneous fission neutron and the resulting photon energy spectrum at each surface enables the photon emission probability at each level to be calculated. Adjusted for intensity,

48 the induced fission product yields can be used to reveal the initial spontaneous fission neutron source location. For example, consider a spontaneous fission neutron in level 1 and a 1384 keV photon from 92Sr somewhere in the sample. Simulation of neutron interactions described above gives the probability of observing a 1384 keV photon, Psurface(1384), due to an induced fission at each surface per spontaneous fission neutron release in the first level. The gamma simulation gives the probability of observing a 1384 keV photon, Plevel,surface(1384), at each surface per 1384 keV photon released in each level of the sample.

A set of functions can be created to determine the probability of a 1384 keV photon created by an induced fission in each level per spontaneous neutron released in the 1st level of the sample.

= , ×

……

= , ×

can be solved as probability of 1384 kev gamma partial released by the induced fission in each level per spontaneous neutron source released in the

1st level of the sphere.

Then ∑ × / is the induced fission yield of the fission product

92Sr. where,

: Average number of emitted neutrons per spontaneous fission

: Probability of observing the key energy photon per induced fission product.

In this example, the key energy photon is from the induced fission of 92Sr that occurred in level 1 of the sphere. This process is repeated for all the induced fission product yields listed on Table 2.2

The example of MCNPX code and output are attached in Appendix B

3.2.2 Uniform Volume Model

Figure 3.2 illustrates the same plutonium source as in the previous section but, except for the central volume, all levels now have equal volume The model is a small sphere of pure 240Pu divided into 20 equal-volume levels.

All levels have the same volume which is 1/20 of the total 240Pu volume.

The outer surface of each level is labeled surface 1 to surface 20.

Figure 3.2: Perspective View of the Uniform Volume Model

The same simulation scheme used for the equal-thickness model is applied to the uniform volume model.

The example of MCNPX code and output are attached in Appendix B

Table 3.2 Comparison of surface position for two models.

Distance to Center of 1.5 cm radius 240Pu sphere (cm)

Surface No. Equal thickness Model Equal volume Model

1 0.55 0.5526 2 0.6 0.6962 3 0.65 0.797 4 0.7 0.8772 5 0.75 0.9449 6 0.8 1.0041 7 0.85 1.0571 8 0.9 1.1052 9 0.95 1.1495 10 1 1.1906 11 1.05 1.229 12 1.1 1.2651 13 1.15 1.2994 14 1.2 1.3319 15 1.25 1.3628 16 1.3 1.3925 17 1.35 1.4209 18 1.4 1.4482 19 1.45 1.4746 20 1.5 1.5

3.2.3 Neutron Energy Distribution

In a realistic experiment, it is impossible to control the fission neutron source position. In this simulation, the fission neutron source is homogeneously distributed throughout the model. Simulations will show how changes in source geometry will affect the neutron energy flux at each surface.

3.3 Result & Discussion

3.3.1 Neutron Energy Distribution

0.5%

0.4%

0.3%

0.2%

0.1%

Neutron observation Probability at Surface at MeV Surface at at Probability observation Neutron 0.0% 0 2 4 6 8 10 12 14 Neutron Energy (MeV)

1 5 9 13 17 20

Figure 3.3 The fission neutron flux spectrum at surface 1, 5, 9, 13, 17 & 20 in equal thickness model when fission neutron source is homogeneously distributed within the 240Pu sphere.

The neutron flux spectrum for the equal thickness geometry is shown in the figure 3.3. There is very clear difference between the fluxes from surface 1 to 20, especially for neutrons from 0 to 4 MeV. The closer to the center the

53 lower will be the average and peak neutron energy and observation probability.

0.5%

0.4%

0.3%

0.2%

0.1%

Neutron observation Probability at Surface at MeV Surface at at Probability observation Neutron 0.0% 0 2 4 6 8 10 12 14 Neutron Energy (MeV)

1 5 9 13 17 20

Figure 3.4 The fission neutron flux spectrum at surface 1, 5, 9, 13, 17 & 20 in equal volume model for a homogeneously distributed plutonium source.

The neutron flux spectrum for the equal volume model is shown in the figure

3.4 and is nearly the same as the results for the equal thickness model shown on figure 3.3. The neutron flux at surface 17 is almost identical as the neutron flux at surface 20 likely because their distances to the center of the sphere are similar (1.4209 & 1.5 cm). These results demonstrate that the

54 neutron flux is very dependent upon the location within the plutonium model.

The clear difference between two models can be observed at surfaces 5 to 17.

The peak neutron energy decreases as one move closer to the center of the sphere.

3.3.2 Equal Thickness Model

To observe the relationship between the fission neutron source position and the fission yield, a moving average is used to smooth the trend of the data.

Although the fission yield of some fission products exhibit little change with the fission neutron source position (e.g., Xe-138), the yield of some other fission products do change significantly with position (e.g., Ba-140, Rh-105).

(Table 3.3 & Figure 3.5)

Table 3.3 Moving average of induced fission yield at each level* of multiple induced fission products in the equal thickness model MA** Induced Fission Yield (%) level Sr-92 Sr-94 Zr-97 Rh-105 I-133 I-135 Xe-138 Cs-138 Ba-140 2 1.773 1.746 6.778 11.227 7.710 5.448 3.050 3.831 6.385 3 1.769 1.750 6.795 11.395 7.727 5.465 3.033 3.829 6.377 4 1.765 1.754 6.770 11.264 7.675 5.468 3.015 3.834 6.391 5 1.767 1.759 6.766 11.182 7.663 5.509 3.008 3.834 6.412 6 1.760 1.764 6.774 11.075 7.648 5.496 3.025 3.838 6.389 7 1.761 1.766 6.804 11.011 7.657 5.512 3.011 3.836 6.369 8 1.765 1.766 6.807 11.016 7.639 5.485 3.007 3.831 6.361 9 1.764 1.769 6.789 10.943 7.644 5.460 2.984 3.818 6.356 10 1.765 1.769 6.773 10.949 7.641 5.436 3.000 3.820 6.329 11 1.761 1.774 6.776 10.952 7.629 5.423 2.989 3.832 6.262 12 1.775 1.768 6.793 10.957 7.637 5.429 2.993 3.854 6.216 13 1.772 1.768 6.803 10.886 7.642 5.452 2.990 3.849 6.143 14 1.767 1.760 6.799 10.819 7.656 5.475 2.985 3.836 6.090 15 1.757 1.756 6.775 10.848 7.648 5.491 2.995 3.828 6.065 16 1.749 1.756 6.770 10.841 7.639 5.479 3.004 3.827 6.087 17 1.756 1.757 6.764 10.857 7.637 5.456 3.017 3.837 6.115 18 1.758 1.756 6.740 10.768 7.634 5.463 3.029 3.841 6.138 19 1.761 1.751 6.732 10.848 7.642 5.481 3.030 3.852 6.158 *level: Level n is the material between surface n-1 and n. **MA: Moving Average at level n=(yieldn-1+yieldn+yieldn+1)/3

12.00%

10.00%

8.00%

6.00%

4.00% Induced Fission Yield (%) Yield Fission Induced

2.00%

0.00% 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 Distance to the center of 240Pu Ball

Rh-105 Xe-138 Ba-140

Figure 3.5 Moving average induced fission yield at multiple fission neutron position (equal thickness model)

As discussed in chapter 2, the ratio of induced to spontaneous fission

INF/SF is very important. Figure 3.6 shows that the closer to the center spontaneous fission occurs, the greater will be the INF/SF ratio, which implies a greater induced fission probability.

0.3

0.25

0.2

0.15 INF/SF

0.1

0.05

0 0.4 0.6 0.8 1 1.2 1.4 Spontaneous fission position

Figure 3.6 INF/SF when spontaneous fission at difference position (equal thickness model)

3.3.3 Equal Volume Model

The same outcome is observed with the equal volume model. The yield of some fission products do not change with the source of the fission neutron

(e.g., Xe-138). However, the fission product yields for others do change significantly with position (e.g., Ba-140, Rh-105). (Table 3.4 & Figure 3.7) the closer to the center spontaneous fission occurs, the greater will be the

INF/SF ratio, which implies a greater induced fission probability.

Table 3.4 Moving average of induced fission yield at each level* of multiple induced fission products in equal volume model MA** Induced Fission Yield (%) level Sr-92 Sr-94 Zr-97 Rh-105 I-133 I-135 Xe-138 Cs-138 Ba-140 2 1.778 1.747 6.756 11.358 7.660 5.438 3.021 3.811 6.393 3 1.777 1.768 6.757 11.342 7.661 5.402 3.049 3.827 6.373 4 1.779 1.771 6.742 11.191 7.670 5.421 3.028 3.829 6.350 5 1.770 1.770 6.748 11.142 7.655 5.428 3.041 3.829 6.378 6 1.757 1.749 6.755 11.041 7.686 5.429 3.023 3.820 6.337 7 1.749 1.738 6.775 10.977 7.670 5.408 3.037 3.819 6.261 8 1.770 1.735 6.781 10.858 7.705 5.415 3.000 3.828 6.221 9 1.771 1.757 6.797 10.893 7.708 5.390 3.017 3.828 6.232 10 1.793 1.774 6.775 10.817 7.719 5.399 2.978 3.846 6.293 11 1.795 1.783 6.732 10.905 7.672 5.393 2.997 3.856 6.320 12 1.811 1.774 6.713 10.884 7.658 5.412 3.007 3.849 6.351 13 1.800 1.778 6.745 10.992 7.627 5.427 3.075 3.830 6.265 14 1.789 1.776 6.776 11.037 7.645 5.479 3.069 3.817 6.199 15 1.794 1.766 6.780 10.961 7.630 5.440 3.075 3.818 6.054 16 1.802 1.754 6.795 10.859 7.637 5.419 3.050 3.804 5.996 17 1.786 1.757 6.807 10.578 7.639 5.361 3.078 3.809 5.942 18 1.769 1.776 6.783 10.613 7.647 5.398 3.063 3.804 5.946 19 1.757 1.787 6.772 10.750 7.657 5.426 3.061 3.822 5.996 *level: Level n is the material between surface n-1 and n. **MA: Moving Average at level n=(yieldn-1+yieldn+yieldn+1)/3

12.00%

10.00%

8.00%

6.00%

4.00%

2.00% Moving average Induced Fission Yield (%) Yield Fission Induced average Moving 0.00% 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 Fission Neutron Source position in 240Pu Ball (cm from center)

Rh-105 Xe-138 Ba-140

Figure 3.7 Moving average induced fission yield at multiple fission neutron position (equal volume model)

0.3

0.25

0.2

0.15 INF/SF

0.1

0.05

0 0.4 0.6 0.8 1 1.2 1.4 Spontaneous fission position

Figure 3.8 INF/SF when spontaneous fission at difference position (equal volume model)

3.3.4 Fission Yield Combination

Figure 3.9 compares how changes in the position of the spontaneous fission neutron source affects the induced fission yield with both models. The suffix

“sd” and “sv” with each fission product isotope in the figure legend indicates whether the data is from the equal thickness (sd) or equal volume (sv) model.

12%

10%

0% Moving average Induced Fission Yield (%) Yield Fission Induced average Moving 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Spontaneous Fission Neutron Source position in 240Pu Ball (cm from center)

Rh sd Xe sd Ba sd Rh sv Xe sv Ba sv

Figure 3.9 Moving average induced fission yield at multiple fission neutron position (combined)

Figure 3.9 demonstrates there are slight differences in the predicted induced fission product yields with “location” depending upon the model simulation, except for 138Xe.

There is also a small decrease in the end of the curve (i.e., 1.35 ~ 1.5 cm), for 105Rh and 140Ba that is likely due to the difference in material volume at the outer peripheral layers of the sample for the equal volume and equal thickness models. Also, the neutron flux decreases in the outer periphery of the material since there is no reflector to prevent neutrons from escaping.

None-the-less, the induced fission product yield for 138Xe remains constant which suggests that the independent fission probability offsets the neutron flux distribution effect.

3.4 Conclusion

The hypothesis that the neutron flux distribution anywhere within the sample is related to the position where the spontaneous fission occurs has been confirmed. Results of simulations based upon the microscopic model proposed in this thesis confirm the hypothesis that the induced fission product yield is related to the location where the spontaneous fission occurs.

The fission yield will change with combinations of the various independent fission neutron probabilities from Chapter 2. But for some fission products

(e.g., 138Xe ), these two effects offset each other resulting in a the fission yield that is stable.

These conclusions suggest that the induced fission yield of 240Pu is too complex to be described with a single value due to the relationship of the non-uniform independent fission probabilities with source neutron energy and the non-uniform neutron flux distribution. The various sizes and shapes of different samples will also affect the neutron flux distribution causing changes in the predicted spontaneous fission yield.

The main challenge with the microscopic model is that the volume of each level is still too large (around 5%) imposing a large variance on the induced fission neutron simulation and increased uncertainty on the location of the emitted photon.

Chapter 4

Macroscopic Parameter Model

4.1 Background

4.1.1 Plutonium Fission Yield Curve

Application of GAUSSIAN parameter theory with 239Pu results in a fission product yield distribution shown on Figure 4.2 consisting of 7 Gaussian peaks.22

Figure 4.2 The fission product yield curve for Pu-239

The fission products selected in the chapter 2 are identified by numbers on

Figure 4.2 and listed on Table 4.2. (There is no fission product was chosen on the Peak 6 & 7 because there is no fission product pass the criteria in the chapter 2)

Table 4.2 Peak location of selected fission products on Figure 4.2

92Sr 94Sr 97Zr 105Rh 133I 135I 138Xe 138Cs 140Ba

Peak 5 1 1 1 & 3 2 & 4 2 & 4 2 & 4 2 2

4.2 Method

In the macroscopic model analysis, reasonable macroscopic sample parameters included in the simulations are shape, volume, purity (% 240Pu) and shield. The expression for the induced fission product yield is

, = + + + + +

Equ.6 where:

: The influence coefficient of Shape

: The influence coefficient of Volume

: The influence coefficient of Purity

: The influence coefficient of Shield

: Dummy Variable of the shape

: Volume of the material

: Purity of 240Pu in the material

: Dummy Variable of different shield materials

*: The Base scenario fission product yield of a single

240Pu particle

*(At v=One particle volume, p=100%, no shield, the shape is ball)

And, the ratio of the induced to spontaneous fission can be expressed as:

/ = / + + + + +

Equ.7 where:

: The influence coefficient of Shape

: The influence coefficient of Volume

: The influence coefficient of Purity

: The influence coefficient of Shield

: Dummy Variable of the shape

: Volume of the material

: Purity of 240Pu in the material

: Dummy Variable of different shield materials

/*: The Baseline induced probability of a single

240Pu particle

*(At v=One particle volume, p=100%, no shield, the shape is ball)

The observed fission product yield from equation 1b is:

= , × +

/ , × Equ.1b

4.2.1 Simulation Parameters Selection

Four parameters have been selected for the macroscopic model simulations.

These parameters include volume, purity, shield and shape. Parameter values are listed on Table 4.3.

Table 4.3 Parameter details

Volume (cm3) Purity (240Pu %) Shield Shape* 3.054 20% Air Ball 4.849 40% Glass Cylinder 1 7.238 60% Lead Glass Cylinder 2 10.306 80% Iron Disk 1 14.137 100% Lead Disk 2 18.817 Water 24.429 Depleted U 31.059 38.792 *: Cylinder 1 Radius=1 cm; Cylinder 2 Radius=1.5 cm; Disk 1 Height=0.9 cm; Disk 2 Height=0.5 cm.

4.2.2 Macro Parameter Model

Figure 3.4: Perspective View of the macroscopic parameter model. The figure illustrates a cubic sample, although several shapes have been simulated.

An example of the macroscopic model is shown in Figure 3.4. A material of given 240Pu content (purity) is uniformly distributed throughout a sample of given volume, size and shape. A spherical shield covers the sample, which is illustrated as a cube in this example. Simulations of neutron and photon interactions are performed separately.

Neutron simulations:

The spontaneous fission neutron source is uniformly distributed in the sample. The γ-ray spectra produced by induced fission products at the internal surface of the shield wall are recorded. This simulation will describe how the induced fission product gamma ray spectra occurring anywhere will vary with changes in values of the macroscopic model sample parameters.

Gamma simulations:

Likewise, selected single-energy photon sources are uniformly distributed within the sample of plutonium material. (The initial photon energies were selected in section 2.1.3.) The photon energy spectrum at the internal surface of the shield wall is recorded. The results of these simulations describe how the fission product gamma ray spectra will change with changes in values of the macroscopic model sample parameters.

Given the initial location of the spontaneous fission neutron and the resulting fission product photon energy spectrum at an internal surface of the shield wall enables the overall photon emission probability in the sample to be calculated. Adjusted for intensity, the induced fission product yields can be

70 used to reveal the macroscopic parameter values. For example, consider a spontaneous fission neutron in a Pu sample (Volume=3.054 cm3, Sphere,

100% 240 Pu, no shield) and a 1384 keV photon from 92Sr somewhere in the sample. Simulation of neutron interactions described above gives the probability of observing a 1384 keV photon, Psurface,fission(1384) at the internal surface of the shield wall, due to an induced fission in the sample. The gamma simulation gives the probability of observing a 1384 keV photon,

Psurface,gamma(1384), at internal surface of the shield wall per 1384 keV photon released in the Pu sample.

A simple function is created to determine the probability of a 1384 keV photon created by an induced fission in each level per spontaneous neutron released in the 1st level of the sample.

, = , × where N represents the probability of a 1384 kev fission product gamma created by an induced fission in the sample per spontaneous fission neutron released by 240Pu in the sample. Then × / is the induced fission yield of the fission product 92Sr where,

: Average number of emitted neutrons per spontaneous fission

: Probability of observing the key energy photon per induced fission product.

In this example, the key photon energy is emitted by the induced 92Sr fission product that was created in the sample with parameter values described above. This process is repeated for all the induced fission product yields listed on Table 2.2

The example of MCNPX code and output are attached in Appendix C

4.3 Result & Discussion

Simulations were performed to predict ratios of the induced to spontaneous fission probabilities for various values of the macroscopic model parameters.

Table 4.4a lists values of these ratios obtained by simulations.

Table 4.4a Selected simulated values of INF/SF obtained using the listed parameter values.

INF/SF Volume (cm3) Purity (240Pu) Shield Shape

16.620% 7.2382 60% Iron Cyl2

14.631% 4.849 80% Air Ball

22.561% 10.306 60% Glass Ball

32.798% 18.8166 40% Iron Ball

45.838% 31.0594 20% Lead Ball

13.679% 24.429 60% Air Dis2

11.849% 24.429 100% Water Dis2

22.442% 7.2382 100% Depleted U Cyl2

21.694% 14.1372 20% Lead Glass Dis1

20.571% 14.1372 100% Air Ball

Using the value of INF/SF in Table 4.4a as dependent variable and volume, purity, shield and shape as independent variables (shield and shape are dummy variables), a linear regression is performed to evaluate how changes in these macroscopic parameters affect the INF/SF ratio. The regression

73 coefficients are shown in the Table 4.4b. For example, using the volume coefficient, an increase in the sample volume of 1 cm3, the INF/SF ratio will increase 7.75E-03.

Table 4.4b INF/SF regression coefficients

INF/SF Intercept Volume Percent Percent2 Glass Lead G Iron

1.78E-01 7.75E-03 -9.19E-02 1.43E-02 1.56E-02 1.66E-02

Lead Water D. U. Cyl1 Cyl2 Dis1 Dis2

1.45E-02 8.65E-03 9.46E-02 -3.32E-02 -2.46E-02 -5.62E-02 -1.83E-01

Another linear regression was performed to evaluate how changes in the macroscopic parameter values affect the induced fission yield for a given fission product. The induced fission yield of a fission product was used as the dependent variable and the volume, purity, shield and shape as independent variables. The shield and shape are dummy variables. The regression coefficients are shown on the Table 4.5. A similar regression was performed that included a depleted uranium shield. Table 4.6 list results of the regression which includes a depleted U shield.

Table 4.5 Simulation Regression Ceofficients

92Sr 94Sr 97Zr 105Rh 133I 135I 138Xe 138Cs 140Ba 1.89E- 1.65E- 6.25E- 7.40E- 6.88E- 4.53E- 2.56E- 3.39E- 5.09E- Intercept 02 02 02 02 02 02 02 02 02 -4.20E- -2.61E- -1.58E- -1.65E- -1.98E- -1.57E- -8.40E- -8.03E- -7.94E- Volume 05 05 04 04 04 04 05 05 05 1.42E- 5.31E- 2.87E- -6.43E- 1.17E- 5.18E- 4.28E- 6.39E- Percent 03 04 02 04 02 03 03 03 -2.92E- -1.77E- -2.70E- Percent2 03 02 03 1.59E- 2.62E- 2.75E- 2.51E- 1.07E- 1.75E- -1.13E- Glass 04 04 03 05 03 04 03 1.20E- 1.66E- 6.46E- 3.30E- 5.38E- 6.06E- 8.80E- Lead G 04 04 03 04 04 05 04 3.21E- -1.67E- 1.23E- 3.06E- 8.76E- -3.30E- 2.56E- Iron 04 04 02 04 04 04 04 2.59E- -4.31E- 6.31E- 1.36E- 3.96E- -1.89E- 2.34E- Lead 04 05 03 03 04 04 03 -8.19E- 2.33E- -4.37E- 2.54E- 7.24E- -1.10E- -2.37E- Water 05 05 04 04 04 05 03 -1.38E- 1.33E- 6.97E- -6.45E- 7.33E- -1.65E- -1.72E- 4.75E- -6.79E- Cyl1 03 04 05 03 04 03 03 04 04 -8.42E- 3.32E- 5.83E- -2.17E- -1.10E- -2.71E- -2.57E- -6.77E- 1.38E- Cyl2 04 04 04 03 03 03 03 04 03 -2.87E- 1.65E- 3.43E- 8.22E- 9.44E- -1.37E- -5.57E- 3.31E- 5.20E- Dis1 04 04 03 03 04 03 04 04 03 7.44E- 6.83E- 7.60E- 1.38E- 1.03E- 2.75E- -3.32E- 1.16E- 5.49E- Dis2 04 04 03 02 02 03 04 03 03

Table 4.6 Simulation Regression that include a depleted uranium shield

92Sr 94Sr 97Zr 105Rh 133I 135I 138Xe 138Cs 140Ba 1.91E- 1.63E- 6.17E- 7.28E- 6.74E- 4.57E- 2.53E- 3.37E- 5.17E- Intercept 02 02 02 02 02 02 02 02 02 -2.07E- -6.82E- -9.89E- -8.89E- -5.50E- -3.46E- Volume 05 05 05 05 05 05 4.31E- 2.76E- 7.95E- 5.01E- 4.01E- Percent 04 02 03 03 03 -1.77E- -1.73E- 5.03E- Percent2 03 02 03 1.42E- 2.09E- -5.03E- 2.41E- -7.89E- 9.84E- 3.99E- 9.38E- -1.19E- Glass 04 04 04 03 05 04 04 05 03 1.03E- 1.12E- -2.38E- 6.12E- 2.26E- 4.48E- 3.62E- -2.07E- 8.21E- Lead G 04 04 04 03 04 04 04 05 04 3.04E- -2.21E- 5.00E- 1.19E- 2.02E- 7.86E- 3.84E- -4.11E- 1.98E- Iron 04 04 06 02 04 04 05 04 04 2.43E- -9.70E- 1.80E- 5.97E- 1.25E- 3.06E- -4.67E- -2.70E- 2.29E- Lead 04 05 04 03 03 04 04 04 03 -1.26E- -7.82E- -1.75E- -1.09E- 2.99E- 5.53E- 1.03E- -1.42E- -2.52E- Water 04 05 04 03 05 04 04 04 03 -3.94E- -3.91E- -1.52E- -1.11E- -1.55E- -1.12E- -5.46E- -9.60E- -1.34E- D. U. 03 03 02 02 02 02 03 03 02 -1.43E- 1.63E- 1.59E- -5.18E- 4.85E- -1.76E- -1.83E- 3.73E- -6.93E- Cyl1 03 04 05 03 04 03 03 04 04 -8.03E- 3.45E- 6.58E- -1.41E- -8.96E- -2.55E- -2.52E- -6.87E- 1.46E- Cyl2 04 04 04 03 04 03 03 04 03 -3.45E- 1.10E- 2.95E- 7.46E- 7.86E- -1.66E- -8.20E- 5.03E- 4.67E- Dis1 04 04 03 03 04 03 04 05 03 2.31E- 1.86E- 5.43E- 9.42E- 7.73E- 1.05E- -1.05E- 2.14E- 3.64E- Dis2 04 04 03 03 03 03 03 04 03

Although the addition of a depleted uranium shield has little, if any, affect on the baseline fission results, there is a change in fission product yields when a depleted uranium shield is included, likely because the uranium is fissionable. This suggests that the type of shield material is important in predicting fission product yield.

Figure 4.4 shows the baseline scenario 240Pu fission product yields that are symmetrically positioned on the fission product distribution curve. The shape of the simulated baseline 240Pu fission product yields on Figure 4.4 is similar to the shape of the conventional 239Pu fission product yield distribution curve shown on Figure 4.2.

8.0%

7.0%

6.0%

5.0%

4.0%

3.0%

Baseline Fission Yield (%) Yield Fission Baseline 2.0%

1.0%

0.0% 90 100 110 120 130 140 Fission Product Isotopic Mass Peak 1 Peak 2 Peak 3&4 Peak 5&6 symmetry points

Figure 4.4 Baseline scenario fission yields vs. Isotopic Mass of Fission products

Figure 4.5 shows how the fission product yield varies with the volume of the plutonium sample. The fission products located on the center of the peaks of the fission product yield curve (e.g., 97Zr or 138Xe) are more affected by changes in the volume of the plutonium sample that the fission products on the tails of the peaks (e.g., 94Sr or 138Cs). These results show when the sample volume increases, the kurtosis1 of the wide peaks becomes smaller

(i.e., flatter). The volume effect of fission decay products on both wide and narrow peaks (e.g., 105Rh or 135I) is stronger than those on the wide peaks

(e.g., 97Zr or 138Xe). This means the kurtosis of narrow peaks also decreases.

The same conclusion is observed for the peak tails. In the event that no fission decay products are on a peak (i.e., peak 7 on fig. 4.2), the kurtosis cannot be determined. In the summary, when the volume of plutonium material increases, the Gaussian peak describing fission yield becomes flatter.

1 In probability theory, kurtosis is a measure of the "tailedness" of the probability distribution of a real- valued random variable. A distribution with kurtosis less than 3 is said to be platykurtic. The smaller the kurtosis, a platykurtic distribution is more like the uniform distribution, which does not have positive- valued tails. Distributions with kurtosis greater than 3 are said to be leptokurtic. The larger the kurtosis, a leptokurtic distribution is more like Laplace distribution, which has tails that asymptotically approach zero more slowly than a Gaussian Distribution.

0.000% 90 100 110 120 130 140

) -0.005% 3

-0.010%

-0.015% volume Effect (%/cm Effect volume -0.020%

-0.025% Fission Product Isotopic Mass

Peak 1 Peak 2 Peak 3&4 Peak 5&6

Figure 4.5 Volume Effect on pure 240Pu Induced Fission Yields

The regression coefficient results suggest that all the fission daughters’ induced fission yields of 240Pu are sensitive to the Plutonium materials shape, and same shape’s effect on the fission yields of different fission daughters can be either positive or negative. (Figure 4.6)

1.20%

1.00%

0.80%

0.60%

0.40%

0.20% Shape Effect Shape 0.00% 92 94 97 105 133 135 138 140 -0.20%

-0.40%

-0.60% Fission Decay Product mass

Cyl1 Cyl2 Dis1 Dis2

Figure 4.6 Shape Effect on pure 240Pu Induced Fission Yields

The purity of the 240Pu (or the percentage of 239Pu in Plutonium material) has significant positive effect on almost all fission decay product yield, except

97Zr. The square of the purity has a significant negative effect on some fission product yields (105Rh & 133I), which means that the purity effect is not necessarily linear. However, by combining both purity and purity squared, the overall purity effect on the simulated fission product yield is positive.

This observation suggests that the induced fission product yields of 239Pu and 240Pu are different because given all other macro parameters locked, the fission yields of fission products will change due to the volume ratio of the

240Pu over 239Pu. (Figure 4.7)

0% Purity effect Purity 92 94 97 105 133 135 138 140

-1%

-2%

-3% Fission Deca Product Mass

Percent Percent Square

Figure 4.7 Purity Effect on Pu Induced Fission Yields

The effect of shield material on fission product yield can be positive or negative depending on the material. However, the material was not observed to affect the yields for 97Zr and 138Xe. Light materials, such as glass and lead glass, have mainly a positive effect on the fission yields whereas heavy materials, such as iron and lead, may have a positive or negative effect depending upon the fission decay product. (Figure 4.8)

1.40%

1.20%

1.00%

0.80%

0.60%

0.40% Shield Effect Shield 0.20%

0.00% 92Sr 94Sr 97Zr 105Rh 133I 135I 138Xe 138Cs 140Ba -0.20%

-0.40% Fission Product Mass

Glass Lead G Iron Lead Water

Figure 4.8 Shield Effect on pure 240Pu Induced Fission Yields (without Depleted Uranium)

The regression shows a very strong negative affect on fission product yield when the shield contains depleted uranium. (Table 4.7) This is not unexpected since the depleted uranium is fissionable. Figure 4.9 shows results of the effect of shield material including depleted uranium.

1.50%

1.00%

0.50%

0.00% 92 94 97 105 133 135 138 140 -0.50% Shield Effect Shield

-1.00%

-1.50%

-2.00% Fission Daughters Z number

Glass Lead G Iron Lead Water D. U.

Figure 4.9 Shield Effect on pure 240Pu Induced Fission Yields (with Depleted Uranium)

The detail data set of MCNPX simulation is attached in Appendix C

4.4 Conclusion

The macroscopic parameter model confirms the conclusions of the microscopic model and demonstrates that the plutonium induced fission product yields can vary depending upon may physical characteristics of the sample, including how and where it is stored. These physical characteristics, as exemplified by parameter values used in the simulations, have different

83 effects on the induced fission product yields that can be either positive or negative and either linear or non-linear. The reason for the predicted changes in fission product yield is that different parameter values (i.e., storage conditions) create a change in the neutron flux distribution within the sample. The overall effect is very dependent upon which fission product is examined, an outcome that is important in selecting a fission product for use as a radiochronometer for 240Pu. The best radiochronometer would be one that exhibits the least sensitivity to changes in physical characteristics. This limited research observed that 97Zr or 138Xe may be good candidates as a radiochronometer. A more robust simulation that includes more than 4 parameter values is needed.

Chapter 5

Conclusion

5.1 Research Accomplishments

This research has investigated the reasons for variations in spontaneous fission product yields for 240Pu reported in the literature and hypothesized that four factors could be a source of this variation. These factors were included in a function (equation 1a) that could influence in the spontaneous

240Pu fission yield. As verification of the hypothesis, this research focused on the induced plutonium fission yield and induced probability per spontaneous fission to interpret the effects of neutron energy, neutron flux distribution and location of the initial spontaneous fission position in a sample. Simulations were performed that identified a relationship between selected parameter values and the inducted fission product yield. (Table 4.6)

Results of these simulations also provided ratios of the induced to spontaneous fission product yields (INF/SF). (Table 4.4) The fission products with the most stable induced fission product yields were found to be 97Zr or 138Xe and could be candidates as radiochronometers for 240Pu.

5.2 Future Work

The initial spontaneous fission yield is totally dependent on the physical characteristics of the fissile material. It is an unchanged input in the equation 1. As discussed in the chapter 1, there is no method to separate the initial spontaneous fission and induced fission. There is still no study on the pure initial spontaneous fission yield.

An approach to determine the initial spontaneous fission yield for 240Pu is described below.

1. Following the method in this paper, with much more input of varies status, the relationship index between induced fission yield and macro parameters can be figure out as table 4.6.

2. With the same process above, the INF/SF index can be offered as table 4.4.

In this case, to any fission daughter in any purified sample, indexes can offer the induced fission yield and INF/SF value.

3. With observed fission yield get from the realistic experiment and equation

1b, it is easy to calculate the initial spontaneous fission yield.

After this approach, there are three indexes created to fully describe the spontaneous fission process in Plutonium materials: Induced fission yield index, INF/SF index and Initial spontaneous fission index.

To create all three indexes, realistic experiment is necessary with assistant of the simulation process.

References

1. Stwertka, Albert (1998). "Plutonium". Guide to the Elements (Revised ed.). Oxford (UK): Oxford University Press.

2. EPA contributors (2008). "Fissile Material". Radiation Glossary. United States Environmental Protection Agency. Nov. 23, 2008.

3. NNDC contributors; Sonzogni (Database Manager), Alejandro A. (2008). "Chart of Nuclides". Upton (NY): National Nuclear Data Center, Brookhaven National Laboratory. Retrieved September 13, 2008

4. Samuel Glasstone and Leslie M. Redman, An Introduction to Nuclear Weapons (Atomic Energy Commission Division of Military Applications Report WASH-1038, June 1972), p. 12.

5. Gosling, F.G. (1999). The Manhattan Project: Making the Atomic Bomb. Oak Ridge (TN): United States Department of Energy.

6. WNA contributors (2006). "Mixed Oxide (MOX) Fuel". London (UK): World Nuclear Association. Retrieved December 14, 2008.

7. Wikipedia, https://en.wikipedia.org/wiki/Plutonium-239

8. M. G. Arora and M. Singh (1994). Nuclear Chemistry. Anmol Publications. p. 202.

9. Gopal B. Saha (1 Nov. 2010). Fundamentals of Nuclear Pharmacy. Springer. pp. 11

10. G. Friedlander, E. S. Macias, J. W. Kennedy, Nuclear and Radiochemistry, Wiley , (1981), ISBN 047186255X

11. Wikipedia, https://en.wikipedia.org/wiki/Monte_Carlo_method

12. Anderson, Herbert L. (1986). "Metropolis, Monte Carlo and the MANIAC". Los Alamos Science 14: 96–108

13. Cashwell, E.D.; Everett, C.J. (1959). A Practical Manual on the Monte Carlo Method for Random Walk Problems. London: Pergamon Press.

14. https://mcnpx.lanl.gov/opendocs/misc/FeaturesList.pdf

15. Kenton J. Moody, Ian D. Hutcheon, and Patrick M. Grant, Nuclear Forensic Analysis, CRC Press, Taylor & Francis Group, Boca Raton, FL (2005)

16. J.B. Laidler, F. Brown, Atomic Weapons Research Establishment, Aldermaston, Berksire (1962)

17. A. V. Bushuev, V. N. Zubarev, etc…, YIELD DETERMINATION OF 240Pu SPONTANEOUS FISSION PRODUCTS, Atomic Energy,(2007) 18. https://www-nds.iaea.org/exfor/endf.htm

19. MCNP - A general Monte Carlo N-Particle transport code, version 4B, ed. J.F. Briesmeister, LA-12625-M, LANL (1997).

20. https://www-nds.iaea.org/

21. http://www.nuceng.ca/igna/neutron_interactions.htm

22. Systematics of Fission-Product Yields, Arthur C. Wahl, April 25, 2002, Los Alamos National Lab

Appendix A

MCNPX input file sample for single neutron energy model:

C Pu240 fission product simulation C Cell Card 1 1 -19.8400 -1 2 -3 imp:n,p,#= 2 2 2 -0.001205 -4 #1 imp:n,p,#=1 22 0 4 imp:n,p,#= 0

C Surface Card 1 pz 0.01 2 pz 0 3 cz 2.0000 4 so 3.0000

C Data Card C Meterial Card M1 94240 -1.000 $ Pu -19.84 M2 6000 -0.000124 7000 -0.755268 8000 -0.231781 18000 -0.012827 $ Air C Control Card mode n p # C Scource Card sdef par=n pos=0 0 -0.01 erg=3 rad=d1 VEC=0 0 1 DIR=1 si1 h 0 2 sp1 -21 1 ACT DG=LINES DN=MODEL Fission=all f1:p 4 e1 0.001 1998i 2 nps 1e7

MCNPX output file sample for single neutron energy model:

1mcnpx version 27e ld=Thu Mar 03 08:00:00 MST 2011 11/12/14 20:23:05 *************************************************************************************** probid = 11/12/14 20:23:05 i=dem10.i o=dem10.o

************************************************************* * * * MCNPX * * * * Copyright 2007. Los Alamos National Security, LLC. * * All rights reserved. * * * * This material was produced under U.S. Government contract * * DE-AC52-06NA25396 for Los Alamos National Laboratory, * * which is operated by Los Alamos National Security, LLC * * for the U.S. Department of Energy. The Government is * * granted for itself and others acting on its behalf a * * paid-up, nonexclusive, irrevocable worldwide license in * * this material to reproduce, prepare derivative works, and * * works, and perform publicly and display publicly. * * Beginning five (5) years after June 1, 2006, subject to * * additional five-year worldwide renewals, the Government * * is granted for itself and others acting on its behalf * * a paid-up, nonexclusive, irrevocable worldwide license * * in this material to reproduce, prepare derivative works, * * distribute copies to the public, perform publicly and * * display publicly, and to permit others to do so. * * * * NEITHER THE UNITED STATES NOR THE UNITED STATES * * DEPARTMENT OF ENERGY, NOR LOS ALAMOS NATIONAL SECURITY, * * LLC, NOR ANY OF THEIR EMPLOYEES, MAKES ANY WARRANTY, * * EXPRESS OR IMPLIED, OR ASSUMES ANY LEGAL LIABILITY OR * * RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR * * USEFULNESS OF ANY INFORMATION, APPARATUS, PRODUCT, OR * * PROCESS DISCLOSED, OR REPRESENTS THAT ITS USE WOULD NOT * * INFRINGE PRIVATELY OWNED RIGHTS. * * * ************************************************************* 1- C Pu240 fission product simulation 2- C Cell Card 3- 1 1 -19.8400 -1 2 -3 imp:n,p,#= 2 4- 2 2 -0.001205 -4 #1 imp:n,p,#=1 5- 22 0 4 imp:n,p,#= 0 6- 7- C Surface Card 8- 1 pz 0.01 9- 2 pz 0 10- 3 cz 2.0000 11- 4 so 3.0000 12- 13- C Data Card 14- C Meterial Card 15- M1 94240 -1.000 $ Pu -19.84 16- M2 6000 -0.000124 7000 -0.755268 8000 -0.231781 18000 -0.012827 $ Air 17- C Control Card 18- mode n p # warning. photonuclear physics may be needed (phys:p). 19- C Scource Card 20- sdef par=n pos=0 0 -0.01 erg=10 rad=d1 VEC=0 0 1 DIR=1 21- si1 h 0 2 22- sp1 -21 1 23- ACT DG=LINES DN=MODEL Fission=all 24- f1:p 4 25- e1 0.001 1998i 2 26- nps 1e7

warning. total nu is now the default for fixed-source problems.

warning. 1 energy bins of tally 1 are below energy cutoff.

warning. use models for the following missing data tables: 7000. c 8000. c

1LAHET physics options: print table 41

lca ielas ipreq iexisa ichoic jcoul nexite npidk noact icem ilaq lca 2 1 1 23 1 1 0 1 0 0

lcb flenb(i),i=1,6 ctofe flim0 lcb 3.4900E+03 3.4900E+03 2.4900E+03 2.4900E+03 8.0000E+02 8.0000E+02 -1.0000E+00 -1.0000E+00

lea ipht icc nobalc nobale ifbrk ilvden ievap nofis lea 1 4 1 0 1 0 0 1

leb yzere bzere yzero bzero leb 1.5000E+00 8.0000E+00 1.5000E+00 1.0000E+01

warning. executing line+multigroup delayed-gamma mode 1cells print table 60

atom gram neutron photon heavy_ion photon wt cell mat density density volume mass pieces importance importance importance generation

1 1 1 4.97711E-02 1.98400E+01 1.25664E-01 2.49317E+00 1 2.0000E+00 2.0000E+00 2.0000E+00 -1.000E+00 2 2 2 4.98817E-05 1.20500E-03 1.12972E+02 1.36131E-01 1 1.0000E+00 1.0000E+00 1.0000E+00 -1.000E+00 3 22 0 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0 0.0000E+00 0.0000E+00 0.0000E+00 -1.000E+00

total 1.13097E+02 2.62930E+00

random number control 0.830206021468160E+14

minimum source weight = 1.0000E+00 maximum source weight = 1.0000E+00

5 warning messages so far. 1cross-section tables print table 100

table length

tables from file endf70a

particle-production data for ipt= 9 being expunged from 6000.70c particle-production data for ipt= 31 being expunged from 6000.70c particle-production data for ipt= 34 being expunged from 6000.70c no particle-production data for ipt= 35 from 6000.70c 6000.70c 58187 6-C - 0 at 293.6K from endf/b-vii.0 njoy99.248 mat 600 08/24/07

tables from file rmccsa

no particle-production data for ipt= 35 from 18000.35c 18000.35c 4724 endl85 ( 18) 11/01/85 temperature = 0.0000E+00 adjusted to 2.5300E-08

tables from file endf70j

no particle-production data for ipt= 35 from 94240.70c 94240.70c 436803 94-Pu-240 at 293.6K from endf/b-vii.0 njoy99.248 total nu mat9440 08/25/07 probability tables used from 5.7000E-03 to 4.0000E-02 mev.

tables from file mcplib04

6000.04p 3228 ENDF/B-VI Release 8 Photoatomic Data for 6-C mat 600 02/07/03 7000.04p 3270 ENDF/B-VI Release 8 Photoatomic Data for 7-N mat 700 02/07/03 8000.04p 3348 ENDF/B-VI Release 8 Photoatomic Data for 8-O mat 800 02/07/03 18000.04p 4772 ENDF/B-VI Release 8 Photoatomic Data for 18-AR mat1800 02/07/03 94000.04p 10527 ENDF/B-VI Release 8 Photoatomic Data for 94-PU mat9400 02/07/03

total 524859

warning. neutron energy cutoff is below some cross-section tables.

warning. 1 cross sections modified by free gas thermal treatment.

maximum photon energy set to 100.0 mev (maximum electron energy)

tables from file el03

6000.03e 2333 6/6/98 7000.03e 2333 6/6/98 8000.03e 2333 6/6/98

18000.03e 2341 6/6/98 94000.03e 2379 6/6/98

1particles and energy limits print table 101

particle maximum smallest largest always always cutoff particle table table use table use model particle type energy energy maximum maximum below above

1 n neutron 0.0000E+00 1.0000E+37 2.0000E+01 1.5000E+02 0.0000E+00 1.5000E+02 2 p photon 1.0000E-03 1.0000E+02 1.0000E+05 1.0000E+05 1.0000E+37 1.0000E+37 3 e electron 1.0000E-03 1.0000E+02 1.0000E+02 1.0000E+02 1.0000E+37 1.0000E+37 35 # heavy_ion 5.0000E+00 1.0000E+02 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

The following nuclides use physics models rather than data tables:

7000. c 8000. c

warning. material 1 has been set to a conductor.

decimal words of dynamically allocated storage

general 0 tallies 15656 bank 2146305 cross sections 524860

total 0 = 0 bytes

*********************************************************************************************************************** dump no. 1 on file runtpe nps = 0 coll = 0 ctm = 0.00 nrn = 0

8 warning messages so far. warning. unusual exponent for sampling source rad. nps = 1 nrn = 1 warning. tally not scored beyond last energy bin. nps = 25329 nrn = 184918 tal = 1 erg = 6.4538E+00

*********************************************************************************************************************** dump no. 2 on file runtpe nps = 1430183 coll = 8246 ctm = 15.02 nrn = 10505191

*********************************************************************************************************************** dump no. 3 on file runtpe nps = 2878306 coll = 16618 ctm = 30.05 nrn = 21144400

*********************************************************************************************************************** dump no. 4 on file runtpe nps = 4425547 coll = 25417 ctm = 45.06 nrn = 32494304

*********************************************************************************************************************** dump no. 5 on file runtpe nps = 5914288 coll = 33826 ctm = 60.06 nrn = 43431467

*********************************************************************************************************************** dump no. 6 on file runtpe nps = 7399440 coll = 42226 ctm = 75.06 nrn = 54332828

*********************************************************************************************************************** dump no. 7 on file runtpe nps = 8935038 coll = 50821 ctm = 90.07 nrn = 65599279

1problem summary

run terminated when 10000000 particle histories were done. + 11/12/14 22:02:39 C Pu240 fission product simulation probid = 11/12/14 20:23:05

neutron creation tracks weight energy neutron loss tracks weight energy (per source particle) (per source particle)

source 10000000 1.0000E+00 1.0000E+01 escape 9993978 1.0019E+00 9.9971E+00 nucl. interaction 499 5.0000E-05 1.5771E-04 energy cutoff 0 0. 0. particle decay 0 0. 0. time cutoff 0 0. 0. weight window 0 0. 0. weight window 0 0. 0. cell importance 5051871 2.5603E-01 2.5495E+00 cell importance 5096738 2.5607E-01 2.5499E+00 weight cutoff 0 0. 0. weight cutoff 0 0. 0. energy importance 0 0. 0. energy importance 0 0. 0. dxtran 0 0. 0. dxtran 0 0. 0. forced collisions 0 0. 0. forced collisions 0 0. 0. exp. transform 0 0. 0. exp. transform 0 0. 0. upscattering 0 0. 0. downscattering 0 0. 7.0177E-04 *photonuclear 0 0. 0. capture 0 8.4719E-07 5.5324E-06 (n,xn) 3946 1.9733E-04 1.8289E-04 loss to (n,xn) 1973 9.8666E-05 9.8663E-04 prompt fission 48295 2.4188E-03 5.1289E-03 loss to fission 11219 5.6187E-04 5.5820E-03 delayed fission 76 3.7987E-06 1.9064E-06 nucl. interaction 779 7.8000E-05 7.7913E-04 *prompt photofiss 0 0. 0. particle decay 0 0. 0. tabular boundary 1336 1.3440E-04 1.3050E-03 tabular boundary 1336 1.3440E-04 1.3050E-03 tabular sampling 0 0. 0. total 15106023 1.2588E+00 1.2556E+01 total 15106023 1.2588E+00 1.2556E+01

number of neutrons banked 5092831 average time of (shakes) cutoffs neutron tracks per source particle 1.5106E+00 escape 3.2477E+03 tco 1.0000E+34 neutron collisions per source particle 3.2218E-03 capture 2.7120E-02 eco 0.0000E+00 total neutron collisions 32218 capture or escape 3.2477E+03 wc1 -5.0000E-01 net multiplication 1.0020E+00 0.0000 any termination 5.2425E+03 wc2 -2.5000E-01

photon creation tracks weight energy photon loss tracks weight energy (per source particle) (per source particle)

source 0 0. 0. escape 44145 8.8461E-03 7.6800E-03 nucl. interaction 618 6.1800E-05 2.8381E-04 energy cutoff 0 0. 9.1658E-08 particle decay 63098 3.1604E-03 2.2296E-03 time cutoff 0 0. 0. weight window 0 0. 0. weight window 0 0. 0. cell importance 154 4.3985E-03 3.7298E-03 cell importance 43228 4.3719E-03 3.7266E-03 weight cutoff 0 0. 0. weight cutoff 0 0. 0. energy importance 0 0. 0. energy importance 0 0. 0. dxtran 0 0. 0. dxtran 0 0. 0. forced collisions 0 0. 0. forced collisions 0 0. 0. exp. transform 0 0. 0. exp. transform 0 0. 0. from neutrons 30015 6.2683E-03 5.4658E-03 compton scatter 0 0. 7.2775E-05 bremsstrahlung 4033 4.1378E-04 1.7340E-05 capture 20732 2.0624E-03 2.9626E-04 p-annihilation 116 1.3487E-05 6.8921E-06 pair production 58 6.7437E-06 1.9921E-05 *photonuclear 0 0. 0. *photonuclear abs 0 0. 0. electron x-rays 0 0. 0. *loss to photofis 0 0. 0. 1st fluorescence 8074 7.7753E-04 5.9355E-05 2nd fluorescence 2055 1.9326E-04 3.0230E-06 (gamma,xgamma) 0 0. 0. tabular sampling 0 0. 0. *prompt photofiss 0 0. 0. total 108163 1.5287E-02 1.1796E-02 total 108163 1.5287E-02 1.1796E-02

number of photons banked 100031 average time of (shakes) cutoffs photon tracks per source particle 1.0816E-02 escape 1.0335E+15 tco 1.0000E+34 photon collisions per source particle 2.4534E-03 capture 2.6166E+14 eco 1.0000E-03 total photon collisions 24534 capture or escape 8.8760E+14 wc1 -5.0000E-01 any termination 9.0404E+14 wc2 -2.5000E-01

heavy_ion creation tracks weight energy heavy_ion loss tracks weight energy (per source particle) (per source particle)

source 0 0. 0. escape 0 0. 0. nucl. interaction 0 0. 0. energy cutoff 0 0. 0. particle decay 0 0. 0. time cutoff 0 0. 0. weight window 0 0. 0. weight window 0 0. 0. cell importance 0 0. 0. cell importance 0 0. 0. weight cutoff 0 0. 0. weight cutoff 0 0. 0. energy importance 0 0. 0. energy importance 0 0. 0. dxtran 0 0. 0. dxtran 0 0. 0. forced collisions 0 0. 0. forced collisions 0 0. 0. exp. transform 0 0. 0. exp. transform 0 0. 0. tabular sampling 0 0. 0. multiple scatter 0 0. 0. bremsstrahlung 0 0. 0. photonuclear 0 0. 0. nucl. interaction 0 0. 0. elastic recoil 0 0. 0. elastic scatter 0 0. 0. particle decay 0 0. 0. capture 0 0. 0. (gamma,xgen_chg) 0 0. 0. tabular sampling 0 0. 0. total 0 0.0000E+00 0.0000E+00 total 0 0.0000E+00 0.0000E+00

number of particles banked 0 cutoffs particle tracks per source particle 0.0000E+00 tco 1.0000E+34 particle substeps per source particle 0.0000E+00 eco 5.0000E+00 total particle substeps 0 wc1 0.0000E+00 wc2 0.0000E+00

computer time so far in this run 99.57 minutes maximum number ever in bank 27 computer time in mcrun 99.19 minutes bank overflows to backup file 0 source particles per minute 1.0082E+05 dynamic storage 0 words, 0 bytes. random numbers generated 73404883 most random numbers used was 1369 in history 795915

range of sampled source weights = 1.0000E+00 to 1.0000E+00 1neutron activity in each cell print table 126

tracks population collisions collisions number flux average average cell entering * weight weighted weighted track weight track mfp (per history) energy energy (relative) (cm)

1 1 10153283 10192408 30082 1.5074E-03 9.5940E+00 9.8748E+00 1.0025E+00 3.4180E+00 2 2 15046129 12522947 2136 2.1450E-04 9.9260E+00 9.9780E+00 1.0025E+00 1.2748E+04

total 25199412 22715355 32218 1.7219E-03 1photon activity in each cell print table 126

tracks population collisions collisions number flux average average cell entering * weight weighted weighted track weight track mfp (per history) energy energy (relative) (cm)

1 1 308 99545 24523 2.4362E-03 9.1849E-01 9.1849E-01 2.0458E+00 4.5958E-01 2 2 43661 44227 11 2.3812E-06 8.6317E-01 8.6317E-01 2.0038E+00 1.1487E+04

total 43969 143772 24534 2.4386E-03 1heavy_ion activity in each cell print table 126

tracks population substeps substeps number flux average average cell entering * weight weighted weighted track weight track mfp (per history) energy energy (relative) (cm)

1 1 0 0 0 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 2 2 0 0 0 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

total 0 0 0 0.0000E+00 1summary of photons produced in neutron collisions

cell number of weight per energy per avg photon mev/gm per weight/neut energy/neut photons source neut source neut energy source neut collision collision

1 1 29994 6.26513E-03 5.46010E-03 8.71506E-01 2.19002E-03 4.15617E+00 3.62213E+00 2 2 21 3.17953E-06 5.67269E-06 1.78413E+00 4.16709E-05 1.48230E-02 2.64462E-02 3 22 0 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 total 30015 6.26831E-03 5.46577E-03 8.71969E-01

energy number of number cum number weight of weight cum weight interval photons frequency distribution photons frequency distribution

20.000 0 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 15.000 0 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 10.000 0 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 9.000 0 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 8.000 0 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 7.000 14 4.66433E-04 4.66433E-04 2.94252E-06 4.69429E-04 4.69429E-04 6.000 27 8.99550E-04 1.36598E-03 5.76711E-06 9.20043E-04 1.38947E-03 5.000 92 3.06513E-03 4.43112E-03 1.92591E-05 3.07245E-03 4.46192E-03 4.000 212 7.06314E-03 1.14943E-02 4.42227E-05 7.05496E-03 1.15169E-02 3.000 523 1.74246E-02 2.89189E-02 1.09493E-04 1.74677E-02 2.89846E-02 2.000 1777 5.92037E-02 8.81226E-02 3.72438E-04 5.94160E-02 8.84006E-02 1.000 5707 1.90138E-01 2.78261E-01 1.19064E-03 1.89946E-01 2.78346E-01 0.500 10272 3.42229E-01 6.20490E-01 2.14619E-03 3.42387E-01 6.20734E-01 0.100 9748 3.24771E-01 9.45261E-01 2.03708E-03 3.24981E-01 9.45715E-01 0.010 1519 5.06080E-02 9.95869E-01 3.14393E-04 5.01560E-02 9.95871E-01 0.000 124 4.13127E-03 1.00000E+00 2.58806E-05 4.12881E-03 1.00000E+00

total 30015 1.00000E+00 6.26831E-03 1.00000E+00 1tally 1 nps = 10000000 tally type 1 number of particles crossing a surface.

particle(s): photon

surface 4 energy 1.0000E-03 0.00000E+00 0.0000 2.0000E-03 0.00000E+00 0.0000 3.0000E-03 0.00000E+00 0.0000 4.0000E-03 5.20353E-07 0.8304 5.0000E-03 9.99655E-08 1.0000 6.0000E-03 0.00000E+00 0.0000 7.0000E-03 0.00000E+00 0.0000 …… 1.9940E+00 0.00000E+00 0.0000 1.9950E+00 4.20388E-07 1.0000 1.9960E+00 2.62214E-06 0.3945 1.9970E+00 1.81900E-06 0.3093 1.9980E+00 8.40699E-07 0.7071 1.9990E+00 4.20350E-07 1.0000 2.0000E+00 9.40665E-07 0.6408 total 8.12754E-03 0.0081 1analysis of the results in the tally fluctuation chart bin (tfc) for tally 1 with nps = 10000000 print table 160

normed average tally per history = 8.12754E-03 unnormed average tally per history = 8.12754E-03 estimated tally relative error = 0.0081 estimated variance of the variance = 0.0001 relative error from zero tallies = 0.0074 relative error from nonzero scores = 0.0033

number of nonzero history tallies = 18065 efficiency for the nonzero tallies = 0.0018 history number of largest tally = 6841795 largest unnormalized history tally = 1.84035E+01 (largest tally)/(average tally) = 2.26434E+03 (largest tally)/(avg nonzero tally)= 4.09053E+00

(confidence interval shift)/mean = 0.0000 shifted confidence interval center = 8.12786E-03

if the largest history score sampled so far were to occur on the next history, the tfc bin quantities would change as follows:

estimated quantities value at nps value at nps+1 value(nps+1)/value(nps)-1.

mean 8.12754E-03 8.12938E-03 0.000226 relative error 8.14482E-03 8.14612E-03 0.000160 variance of the variance 1.05268E-04 1.05701E-04 0.004111 shifted center 8.12786E-03 8.12786E-03 0.000000 figure of merit 1.51972E+02 1.51924E+02 -0.000319

the estimated slope of the 184 largest tallies starting at 1.02383E+01 appears to be decreasing at least exponentially. the large score tail of the empirical history score probability density function appears to have no unsampled regions.

======

results of 10 statistical checks for the estimated answer for the tally fluctuation chart (tfc) bin of tally 1

tfc bin --mean------relative error------variance of the variance------figure of merit-- -pdf- behavior behavior value decrease decrease rate value decrease decrease rate value behavior slope

desired random <0.10 yes 1/sqrt(nps) <0.10 yes 1/nps constant random >3.00 observed random 0.01 yes yes 0.00 yes yes constant random 10.00 passed? yes yes yes yes yes yes yes yes yes yes

======

this tally meets the statistical criteria used to form confidence intervals: check the tally fluctuation chart to verify. the results in other bins associated with this tally may not meet these statistical criteria.

estimated asymmetric confidence interval(1,2,3 sigma): 8.0617E-03 to 8.1941E-03; 7.9955E-03 to 8.2603E-03; 7.9293E-03 to 8.3265E-03 estimated symmetric confidence interval(1,2,3 sigma): 8.0613E-03 to 8.1937E-03; 7.9951E-03 to 8.2599E-03; 7.9290E-03 to 8.3261E-03

fom = (histories/minute)*(f(x) signal-to-noise ratio)**2 = (1.008E+05)*( 3.883E-02)**2 = (1.008E+05)*(1.507E-03) = 1.520E+02 1 some tally scores were not made for various reasons:

beyond last bin not in

tally user segment mult angle energy time 1 0 0 0 0 3086 0 1status of the statistical checks used to form confidence intervals for the mean for each tally bin

tally result of statistical checks for the tfc bin (the first check not passed is listed) and error magnitude check for all bins

1 passed the 10 statistical checks for the tally fluctuation chart bin result missed all bin error check: 2001 tally bins had 74 bins with zeros and 1893 bins with relative errors exceeding 0.10

the 10 statistical checks are only for the tally fluctuation chart bin and do not apply to other tally bins.

the tally bins with zeros may or may not be correct: compare the source, cutoffs, multipliers, et cetera with the tally bins.

warning. 1 of the 1 tallies had bins with relative errors greater than recommended. 1tally fluctuation charts

tally 1 nps mean error vov slope fom 512000 7.9609E-03 0.0362 0.0020 3.3 128 1024000 8.0587E-03 0.0255 0.0010 3.1 139 1536000 8.0454E-03 0.0208 0.0007 10.0 142 2048000 8.0807E-03 0.0180 0.0005 10.0 144 2560000 8.1080E-03 0.0161 0.0004 10.0 144 3072000 8.1280E-03 0.0147 0.0003 1.4 144 3584000 8.0768E-03 0.0137 0.0003 1.7 146 4096000 8.1417E-03 0.0127 0.0003 10.0 147 4608000 8.1147E-03 0.0120 0.0002 10.0 149 5120000 8.1294E-03 0.0114 0.0002 10.0 148 5632000 8.1206E-03 0.0108 0.0002 1.5 148 6144000 8.1255E-03 0.0104 0.0002 10.0 149 6656000 8.1616E-03 0.0100 0.0002 10.0 149 7168000 8.1557E-03 0.0096 0.0001 10.0 149 7680000 8.1739E-03 0.0093 0.0001 10.0 149 8192000 8.1698E-03 0.0090 0.0001 10.0 150 8704000 8.1557E-03 0.0087 0.0001 10.0 150 9216000 8.1434E-03 0.0085 0.0001 10.0 150 9728000 8.1238E-03 0.0083 0.0001 10.0 151 10000000 8.1275E-03 0.0081 0.0001 10.0 152

*********************************************************************************************************************** dump no. 8 on file runtpe nps = 10000000 coll = 56752 ctm = 99.19 nrn = 73404883

11 warning messages so far.

run terminated when 10000000 particle histories were done.

computer time = 99.57 minutes

mcnpx version 27e Thu Mar 03 08:00:00 MST 2011 11/12/14 22:02:39 probid = 11/12/14 20:23:05

Appendix B

MCNPX input file for microscope models Neutron Simulation (Output file is similar to the output in Appendix A)

C Pu240 fission product simulation C Cell Card 11 1 -19.8400 -11 imp:n,p,#= 1 12 1 -19.8400 11 -12 imp:n,p,#= 1 13 1 -19.8400 12 -13 imp:n,p,#= 1 14 1 -19.8400 13 -14 imp:n,p,#= 1 15 1 -19.8400 14 -15 imp:n,p,#= 1 16 1 -19.8400 15 -16 imp:n,p,#= 1 17 1 -19.8400 16 -17 imp:n,p,#= 1 18 1 -19.8400 17 -18 imp:n,p,#= 1 19 1 -19.8400 18 -19 imp:n,p,#= 1 20 1 -19.8400 19 -20 imp:n,p,#= 1 21 1 -19.8400 20 -21 imp:n,p,#= 1 22 1 -19.8400 21 -22 imp:n,p,#= 1 23 1 -19.8400 22 -23 imp:n,p,#= 1 24 1 -19.8400 23 -24 imp:n,p,#= 1 25 1 -19.8400 24 -25 imp:n,p,#= 1 26 1 -19.8400 25 -26 imp:n,p,#= 1 27 1 -19.8400 26 -27 imp:n,p,#= 1 28 1 -19.8400 27 -28 imp:n,p,#= 1 29 1 -19.8400 28 -29 imp:n,p,#= 1 30 1 -19.8400 29 -30 imp:n,p,#= 1 31 2 -0.001205 30 -31 imp:n,p,#= 2 32 0 31 imp:n,p,#= 0

C Surface Card 11 so 0.5500 12 so 0.6000 13 so 0.6500 14 so 0.7000 15 so 0.7500 16 so 0.8000 17 so 0.8500 18 so 0.9000 19 so 0.9500 20 so 1.0000 21 so 1.0500 22 so 1.1000

23 so 1.1500 24 so 1.2000 25 so 1.2500 26 so 1.3000 27 so 1.3500 28 so 1.4000 29 so 1.4500 30 so 1.5000 31 so 50

C Data Card C Meterial Card M1 94240 -1.000 $ Pu -19.84 M2 6000 -0.000124 7000 -0.755268 8000 -0.231781 18000 -0.012827 $ Air 0.001205 C Control Card mode n p # C Scource Card sdef par=sf pos= 0 0 0 rad=d1 si1 0 0.5500 sp1 -21 2 ACT DG=LINES DN=MODEL Fission=all *f4:n 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 f14:n 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 f11:p 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 e11 0.001 1998i 2 nps 5e6

MCNPX input file for microscope models Gamma Simulation (Output file is similar to the output in Appendix A)

C Pu240 fission product simulation C Cell Card 1 1 -19.8400 -1 imp:n,p,#= 1 2 1 -19.8400 1 -2 imp:n,p,#= 1 3 1 -19.8400 2 -3 imp:n,p,#= 1 4 1 -19.8400 3 -4 imp:n,p,#= 1 5 1 -19.8400 4 -5 imp:n,p,#= 1 6 1 -19.8400 5 -6 imp:n,p,#= 1 7 1 -19.8400 6 -7 imp:n,p,#= 1 8 1 -19.8400 7 -8 imp:n,p,#= 1 9 1 -19.8400 8 -9 imp:n,p,#= 1 10 1 -19.8400 9 -10 imp:n,p,#= 1 11 1 -19.8400 10 -11 imp:n,p,#= 1 12 1 -19.8400 11 -12 imp:n,p,#= 1 13 1 -19.8400 12 -13 imp:n,p,#= 1 14 1 -19.8400 13 -14 imp:n,p,#= 1 15 1 -19.8400 14 -15 imp:n,p,#= 1 16 1 -19.8400 15 -16 imp:n,p,#= 1 17 1 -19.8400 16 -17 imp:n,p,#= 1 18 1 -19.8400 17 -18 imp:n,p,#= 1 19 1 -19.8400 18 -19 imp:n,p,#= 1 20 1 -19.8400 19 -20 imp:n,p,#= 1 21 2 -0.001205 20 -21 imp:n,p,#= 2 22 0 21 imp:n,p,#= 0

C Surface Card 1 so 0.5500 2 so 0.6000 3 so 0.6500 4 so 0.7000 5 so 0.7500 6 so 0.8000 7 so 0.8500 8 so 0.9000 9 so 0.9500 10 so 1.0000 11 so 1.0500 12 so 1.1000 13 so 1.1500 14 so 1.2000 15 so 1.2500 16 so 1.3000 17 so 1.3500 18 so 1.4000

100

19 so 1.4500 20 so 1.5000 21 so 50

C Data Card C Meterial Card M1 94240 -1.000 $ Pu -19.84 M2 6000 -0.000124 7000 -0.755268 8000 -0.231781 18000 -0.012827 $ Air 0.001205 C Control Card mode p C Scource Card sdef par=p erg=d2 pos=0 0 0 rad=d1 cel=1 si1 0 1.5 sp1 -21 2 si2 l 1.38393 1.4277 .74336 .3189 .52987 1.26041 1.768 1.43586 .53726 .0001 sp2 d .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 ACT DG=None f11:p 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 e11 0.001 1998i 2 nps 1e7

101

Appendix C

MCNPX input file for macro-scope models Neutron Simulation (Output file is similar to the output in Appendix A)

C Pu240 fission product simulation C Cell Card 1 1 -19.8400 -1 imp:n,p,#= 1 2 7 -18.90 1 -2 imp:n,p,#= 2 3 2 -0.001205 2 -3 imp:n,p,#= 4 4 0 3 imp:n,p,#= 0

C Surface Card 1 so 2.1 2 so 2.6 3 so 50

C Data Card C Meterial Card M1 94240 -1.000 $ Pu -19.84 M2 6000 -0.000124 7000 -0.755268 8000 -0.231781 18000 -0.012827 $ Air 0.001205 M3 5011 -0.040066 8016 -0.539559 11023 -0.028191 13027 -0.011644 14000 -0.377220 19000 -0.003321 $Borosilicate 2.230 M4 8016 -0.156453 14000 -0.080866 22000 -0.008092 33075 -0.002651 82000 -0.751938 $Lead Glass 6.220 M5 82000 -1.000000 $Lead 11.35 M6 26000 -1.000000 $Iron 7.874 M7 92235 -0.002500 92238 -0.997500 $Depleted Uranium 18.90 C Control Card mode n p # C Scource Card sdef par=sf pos= 0 0 0 rad=d1 si1 0 2.1 sp1 -21 2 ACT DG=LINES DN=MODEL Fission=all *f1:n 1 2 f11:p 1 e11 0.001 1998i 2 nps 7e5

102

MCNPX input file for macro-scope models Gamma Simulation (Output file is similar to the output in Appendix A)

C Pu240 fission product simulation C Cell Card 1 1 -19.8400 -1 imp:n,p,#= 1 2 7 -18.90 1 -2 imp:n,p,#= 2 3 2 -0.001205 2 -3 imp:n,p,#= 4 4 0 3 imp:n,p,#= 0

C Surface Card 1 so 2.1 2 so 2.6 3 so 50

C Data Card C Meterial Card M1 94240 -1.000 $ Pu -19.84 M2 6000 -0.000124 7000 -0.755268 8000 -0.231781 18000 -0.012827 $ Air 0.001205 M3 5011 -0.040066 8016 -0.539559 11023 -0.028191 13027 -0.011644 14000 -0.377220 19000 -0.003321 $Borosilicate 2.230 M4 8016 -0.156453 14000 -0.080866 22000 -0.008092 33075 -0.002651 82000 -0.751938 $Lead Glass 6.220 M5 82000 -1.000000 $Lead 11.35 M6 26000 -1.000000 $Iron 7.874 M7 92235 -0.002500 92238 -0.997500 $Depleted Uranium 18.90 C Control Card mode p C Scource Card nps 1e6 ACT DG=None sdef par=p erg=d2 pos=0 0 0 rad=d1 cel=1 si1 0 3.952 sp1 -21 2 si2 l 1.38393 1.4277 .74336 .3189 .52987 1.26041 1.768 1.43586 .53726 .0001 sp2 d .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 f11:p 1 e11 0.001 1998i 2

103