The Pennsylvania State University The Graduate School The College of Engineering THE EFFECT OF DELAYED NEUTRONS AND DETECTOR DEAD-TIME IN

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1 The Pennsylvania State University The Graduate School The College of Engineering THE EFFECT OF DELAYED NEUTRONS AND DETECTOR DEAD-TIME IN FEYNMAN DISTRIBUTION ANALYSIS A Dissertation in Nuclear Engineering by Mathieu Brener c 2016 Mathieu Brener Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2016

2 The dissertation of Mathieu Brener was reviewed and approved by the following: Gary L. Catchen Professor Emeritus of Nuclear Engineering Dissertation Advisor, Chair of Committee Arthur T. Motta Chair of Nuclear Engineering and Professor of Nuclear Engineering and Materials Science and Engineering Jack S. Brenizer, Jr. Professor of Mechanical and Nuclear Engineering Murali Haran Associate Professor and Undergraduate Chair, Department of Statistics Manoj Kumar Prasad Special Member Physicist at Lawrence Livermore National Laboratory Signatures are on file in the Graduate School.

3 Abstract I have explored the effect of delayed neutrons on neutron chains in a sub-critical nuclear system. Every fission has an approximately 0.7% chance of creating a delayed neutronthatwillgoontocauseanotherfission. Myhypothesiswasthatinsystems with a very high sub-critical multiplication, there will be enough delayed neutrons created that they will in effect become a stronger starting neutron source than the original source. I used a point Monte-Carlo style simulation of sub-critical neutron chains to explore the physics of neutron chains. The simulation was designed to keep the same assumptions behind the Hage-Cifarelli derivation and Prasad- Snyderman formulation, to preserve the statistical variations in the list-mode data found in a physical experiment, and to be as simple as possible while maintaining accuracy. I present this method and benchmark it against both measured data and a theoretical model. I also use the model to simulate a Feynman Beta curve and 1/M approach-to-criticality experiments. I modified the simulation to take into account delayed neutrons and found that delayed neutrons have no measurable effect on a sub-critical systems for subcritical multiplication (M) less then 200. I wrote an algorithm to take into account detector dead-time with multiple detectors and found that increasing the number of detectors is the only effective method of compensating for dead-time effects. I investigated the minimum number of events needed for the analysis methods to produce accurate results and found that, for low values of M, there are statistical variations of the calculated multiplicity that will produce inaccurate results. For high values of M, the statistical variation is low, but an insufficient number of events will produce M-values that are inaccurately low. iii

4 Table of Contents List of Figures List of Tables List of Symbols Acknowledgments viii x xi xii Chapter 1 Introduction 1 Chapter 2 Theory Subcritical Multiplication Multiplication Measurements The Feynman Histogram Moments of the Feynman Histogram Diven s Numbers The Meaning of Y m The Hansen-Dowdy Method of Feynman Analysis Efficiency Measurement Detector Die-Away Time and the Feynman-β Plot The Hage-Cifarelli Method of Feynman Analysis The Rossi-α Distribution and the Neutron Coincidence Counting Method Rossi-α Distribution iv

5 2.6.2 The Neutron Coincidence Counting Method of Multiplicity Analysis Similar Dissertation by Steven Nolen Prasad and Snyderman s Closed Form Solution of the Feynman Distribution The Relationship Between Number of Starter Events and Number of Detector Events The Ratio of Poisson Neutrons to Fission Neutrons (α) Chapter 3 Basic Simulation of Neutron Chains Physical Assumptions Method Number of Neutrons per Fission Source Selection Basic Chains Detector Efficiency Distribution in Time Caused by Detector Die-away Distribution in Time Caused by Source Decay Creating the Feynman Histogram Implementation of the Hansen- Dowdy Method Creating the Rossi-α Distribution Implementation of the NCC Method Parallel Processing Chains Combined Input Summary Chapter 4 Running Chains Results Benchmarking A Note About Simulation Parameters Extracting M from Chains /M Approach to Criticality Feynman-β Curve Time to Next Event Minimum Number of Events Needed for the Hansen-Dowdy Method Results v

6 4.3 Summary Chapter 5 Delayed Neutrons Calculating α Simulating Delayed Neutrons Using Elevated β Values Summary Chapter 6 Simulation of Dead-Time Paralyzable and Non-paralyzable Simulation of Count-Rate Only Non-paralyzable Paralyzable Results Simulation of Dead Time with Chains List Mode Multiple Detectors Dead-Time Runs and Results Dead-Time Correction Compared to Non-Dead Time Correction Hansen-Dowdy Method Compared to NCC Method NCC Method Percent of Detected Neutrons Lost Due to Dead-Time Summary Chapter 7 Conclusion Summary and Conclusion Basic Chains Delayed Neutrons Dead-time Advantages of Chains Practical Applications of Chains More Variation of Variables Advanced Dead-time Analysis Advanced Analysis of Minimum Number of Events Analysis of Neutron Chains at Critical Analysis of Thermal Neutrons vi

7 7.2.6 Final Thought Appendix A Complete Chains Algorithms 90 Bibliography 102 vii

8 List of Figures 1.1 The non-steady state nature of a multiplying system A plot of chain length vs k eff Source ν distributions for 240 Pu with delayed neutron contributions A diagram of a neutron well-counter A cut-away view of a neutron well-counter The formation of a Feynman histogram Feynman histogram vs Poisson distribution Feynman beta plot An example of a Rossi-α distribution Feynman histogram for various value of k eff Comparison of the LLNL experimental and therotical Feynman histogram, and Chains A plot of input values of M vs the output as given by the Hansen- Dowdy method /M plots with (α,n) and spontaneous fission sources Feynman-β plot from Chains A time to next event plot from Chains compared to one published by Prasad and Snyderman The dependence of the minimum number of events to get the true value of M on the input M-value The dependence of the number of neutron events recorded on the M-value and standard deviation for M= The dependence of the number of neutron events recorded on the M value and standard deviation for M= The dependence of the number of neutron events recorded on the M value and standard deviation for M=20 and efficiency of 25% for a 239 Pu system with 240 Pu starting neutrons The dependence of M on α for a 235 U system viii

9 5.2 The dependence of M on α for a 239 Pu system Source ν distributions for 240 Pu with delayed neutron contributions Plot of paralyzable and non-paralyzable dead-time from Knoll A count-rate only dead-time simulation without multiplication Examples of detector-to-pre-amplifier groupings Dead time correction compared to non-dead-time correction with M=3 for for paralyzable and non-paralyzable detectors Dead time correction compared to non-dead-time correction with M=15 for for paralyzable and non-paralyzable detectors The Hansen-Dowdy method compared to the NCC method, with dead-time The effect of dead-time on the NCC method The percent loss of detected events from detector dead-time for paralyzable and non-paralyzable systems The percent loss of detected events from detector dead-time for M=3 and M= ix

10 List of Tables 1.1 The initial hypothesis that the delayed neutrons increase with subcritical multiplication Example Y m values for specific sources Variables used with the Hansen-Dowdy formalism Variables used with the Hage-Cifarelli formalism P ν values for 240 Pu, 252 Cf, 235 U, and 239 Pu Comparison of the LLNL experimental Feynman histogram and theory, and Chains simulation The data table used to determine the minimum number of events for M=1.5 and efficiency of 1.5% for a 239 Pu system with 240 Pu starting neutrons The effect of delayed neutrons on a 235 U system with (α,n) starting neutrons Theeffectofdelayedneutronsona 239 Pusystemwith 240 Pustarting neutrons The effect of a non-physical β-value of 50% on delayed neutrons x

11 List of Symbols M T M L M HD Total subcritical multiplication. Leakage subcritical multiplication. The subcritical multiplication as calculated by the Hansen- Dowdy method. p The probability that a fission or source neutron will undergo a fission absorption. k eff Reactivity coefficient where M T = 1/(1 k eff ) ν or ν i ν s P ν Y or Y m Average number of neutrons emitted from induced fission. Averagenumberofneutronemittedfromspontaneousfission, usedmore generally in this dissertation for the average number of fissions emitted from the neutron source. The probability of getting ν neutrons per fission. τ Detector dead-time. The deviation of the variance to the mean for a Feynman histogram. β or λ The detector die-away time. Both symbols are used in the literature. ε Detector efficiency. α The ratio of Poisson neutron decays to spontaneous fission decays. ξ A uniformly distributed random variable between zero and one. Fs The rate of neutron source decays. T The time bin width used for Feynman analysis. xi

12 Acknowledgments I would like to give a special thanks to Professor Gary Catchen for advising me through this process. I wish to thank Dr. Manoj Prasad from Lawrence Livermore National Laboratory for serving as a special member of my dissertation committee, for providing technical advice, and for granting me access to a LLNL computer cluster for my simulations. I would also like to thank Professor Jack Brenizer for his support and advice during my time at Penn State. In addition. I would like to thank Professor Marek Flaska and Dr. Stephen Croft for providing technical advice, Professor Arthur Motta for supporting me as a TA, the staff at the Livermore Computing Help Desk for assisting me with remote access to the LLNL computer system. I would like to give a special thanks to Alissa Pendorf for the the amazing support she has given me. I would also like to thank my parrot, Pigwidgeon for being my constant companion over the years. xii

13 Chapter 1 Introduction At an undergraduate level, the students are taught that the time averaged number of neutrons in a subcritical reactor is n = Source (1+k eff +keff 2 +k3 eff...), which for k eff < 1 becomes n = Source 1/(1 k eff ), and we define the subcritical multiplication as M = 1/(1 k eff ). While this approach works fine for most applications, it misses important details. In a real subcritical system, especially when the source term is small, subcritical multiplication is not steady state, but instead is characterized by bursts of neutron chains, as shown in Figure 1.1. These chains can be as short as 1 neutron, but can be as long as hundreds of neutrons, even for relatively low M-values. This feature is shown in Figure 1.2. The probability distribution function(pdf) of these chains is called a Feynman distribution and it is non-poisson distributed. [1, 2]. Feynman distribution analysis is the method used to calculate the multiplication of the system and other parameters of the nuclear material and detector such as efficiency, dead-time, and die-away time. Feynman analysis is the only way to measure these parameters with a steady-state subcritical system; all other methods rely on changing reactivity. [3, 4, 5]. In the first part of this dissertation, I present a Monte Carlo simulation to calculate the PDF of these neutron chains. This simulation is zero dimensional in space, which is a valid assumption for small systems. The simulation also distributes the neutrons in time in order to properly simulate neutron die-away in the detector and overlapping fission chains. The simulation can also output list-mode data, which are when the time of every event in the detector is recorded.

14 2 Figure 1.1: A plot of the neutron reaction rate in time. The non-steady state nature of the system is clearly seen. The neutron chains in this plot was generated by Chains. The goal of this simulation was to accurately produce list-mode data of the recorded neutron events from a sub-critical system. In addition, the simulation was to keep the same assumptions behind the Hage-Cifarelli derivation and Prasad- Snyderman formulation. The only deviation from these assumptions was to add dead-time and delayed neutron effects. The simulation was also designed to preserve the statistical variations in the list-mode data found in a physical experiment, and to be as simple as possible while maintaining accuracy. I benchmark my simulation to data published by Lawrence Livermore National Laboratory (LLNL) and show that it accurately reproduces the data. I show that my simulation produces accurate results for subcritical multiplication values ranging from 1.5 to 100. I also explore the minimum number of detected-neutron events needed for analysis. In the second part of this paper, I expand the original simulation to take into account delayed neutrons. Currently, all published methods of Fenymnan analysis that I have read ignore the effects of delayed neutrons. This ansatz is reasonable

15 3 Figure 1.2: A plot of chain length vs k eff, from Nolen [6]. Please see Section 2.7 for a discussion of this plot. given that the β-fraction for uranium is approximately and thus delayed neutrons do not affect the measurement. However, I hypothesize that for large multiplication factors, the delayed neutrons produce two effects, the first is that they act as additional source neutrons and thus increase the rate of source neutrons in the system. This effect is illustrated in Table 1.1. The second effect is that, for spontaneous-fission-starting sources, such as 240 Pu, the delayed neutrons change the probability distribution function (PDF) of number of neutrons per starting source. The delayed neutrons become part of the probability distribution and effectively lower the ν s (average number of neutrons per spontaneous-fission) of the starting source. This effect in illustrated in Figure 1.3. While some of the conventional methods of neutron analysis can take into account sources that have (α, n) contamination with the spontaneous-fission, they all treat the effect as a constant, when it is in fact a function of multiplication factor. If the effect of delayed neutrons is not taken into account for very high-

16 4 Table 1.1: The initial hypothesis that the delayed neutrons increase with subcritical multiplication. Eventually the delayed neutrons dominate the source neutrons. The effective source neutron column is the sum of the starting neutrons and the delayed neutrons. Multiplication Factor Number of Starting Neutrons Neutrons From Fission Number of Fissions Delayed Neutrons , , Effective Source Neutrons multiplication systems, conventional methods produce erroneous results. One of the objectives of this dissertation is to determine the lower limit of sub-critical multiplication before delayed neutrons have an effect. In addition to studying the effect of delayed neutrons, I also simulate detector a) b) c) d) Figure 1.3: Source ν distributions for 240 Pu with delayed neutron-contributions. The figures show neutrons fractions of a) 0, b) 0.1, c) 0.5 and d) 1 with effective ν values of a) 2.154, b) 1.95, c) 1.55 and d) 1.36.

17 5 dead-time, both paralyzable and non-paralyzable, and explore the effect of using multiple detectors on the system dead-time. In this dissertation, I focus on two methods of neutron analysis, the Hansen- Dowdy method, which is based on Feynman histograms, and the Neutron Coincidence Counting Method (NCC), which is based on time-to-next events. All programming, data analysis and plotting for this dissertation was done in Mathematica.

18 Chapter 2 Theory 2.1 Subcritical Multiplication When a reactor is subcritical, it acts like a neutron amplifier, where the total neutron rate N is given by 1 : N = MS (2.1) where M is the subcritical multiplication factor and S is the source emission rate. The multiplication factor is related to the more commonly used k eff by: and k eff is defined as: M = (1+k eff +k 2 eff +k 3 eff) = k n eff (2.2) k eff = pν (2.3) where ν is the average number of neutrons emitted by fission for the multiplying nuclear material and p is the probability that a neutron causes a fission. The summation in Equation 2.2 can be carried out to obtain a simpler expression relating M to k eff : 1 Most of the equations in this chapter come from Neutron Specialist Handbook and Informational Text written by Los Alamos National Laboratory [7].

19 7 M T = Thus, the total multiplication is given by: 1 1 k eff (2.4) M T = 1 1 pν (2.5) However in real applications, one can only measure the neutrons that leak out of a nuclear system, thus the leakage multiplication is given as: M L = 1 p 1 pν (2.6) The leakage multiplication and total multiplication is related by the following expression 2 : where M T = M Lν 1 α ν 1 α (2.7) α = σ a σ f (2.8) and σ a is the absorption cross section and σ f is the fission cross section. This α is different from the α used in rest of this dissertation. 2.2 Multiplication Measurements The original theory behind neutron multiplicity measurements was developed by Richard Feynman, Frederic de Hoffmann, and Robert Serber during the Manhattan Project [2, 8, 1]. In a series of papers, they derived the underlying equations and mathematics relating to neutron fission chains and reactor noise analysis. The theory was greatly expanded in the 1980 s by H. Hansen, G. Dowdy, A. Robba, and H. Atwater at Los Alamos National Laboratory [3] and by W. Hage and D. 2 There is a contradiction with the equations. Combining Equation 2.5 and Equation 2.6 and solving for M t gives M T = M Lν 1. This contradiction can be explained by noting that this ν 1 result matches Equation 2.7 only if σ a = 0, which is of course not physically possible, but it can arise in some simplified theoretical models.

20 8 Figure 2.1: An example of the physical model behind this dissertation. The circle in the center is the nuclear material. The detector is polyethylene with 16 neutron detectors. Cifarelli of Italy [4] who independently derived equations that allow useful information to be extracted from the Feynman distribution. The theoretical framework was completed by N. Snyderman and M. Prasad at Lawrence Livermore National Laboratory in 2002(but not published until 2012)[5] when they derived the closedform solution to predicting the Feynman distribution. The bulk of the theory and method for this paper can be found in the Neutron Specialist Handbook [7] and Passive Nondestructive Assay of Nuclear Material (PANDA) [9]. Those two books provide a very good overview of the theory, methods, and applications of neutron multiplicity measurements. Another good reference is Active Neutron Multiplicity Counting by N. Ensslin [10]. Without loss of generality, the physical model that forms the basis for my dissertation is a ball of nuclear metal surrounded by a polyethylene well-counter with multiple neutron detectors. This model is shown in Figure 2.1. The nuclear materials modeled are uranium, plutonium, or neptunium. The theory and results in this paper can be equally applied to a more general system provided that it is a fast-neutron system and small enough that point kinetics is valid. The neutrons from an AmBe source are generated one-by-one when an alphaparticle emitted by the 241 Am nucleus undergoes an (α,n) reaction with the 9 Be nucleus. Americium only emits one alpha-particle per decay, and each (α, n) reaction also creates only one neutron, which fits the mathematical definition of a

21 9 Figure 2.2: A cut-away view of an another variation of a neutron well counter. Figure by Alissa Pendorf. Poisson process. The distribution of neutrons that fall in a time-interval match the Poisson distribution. The Poisson distribution, which gives the probability of observing k (not to be confused with k eff ) number of counts with a mean of λ, is given by [11] The Feynman Histogram P(k) = e λλk, λ = 1,2,... (2.9) k! A 252 Cf source, on the other hand, emits 0-8 neutrons per spontaneous-fission decay, and its distribution is not described by a Poisson distribution. A histogram of the neutrons emitted by a multiplying system is called a Feynman histogram, andanexampleisshowninfigure2.4. Theneutronsemittedfromasingleneutron chain are correlated and thus a Feynman histogram is not Poisson. The Feynman histogram is calculated by forming a histogram of number of times n counts is recorded in a specified time bin interval and then normalized.

22 10 Figure 2.3: In the above example, there is one time where there are zero neutrons in a interval bin, four times where there is one neutron, three times were there are two neutrons, and two times where there are three neutrons. The Feynman histogram for the ten neutrons in this example would thus be {{0, 0.1}, {1, 0.4}, {2, 0.3,}, {3, 0.2}}. Figure by Alissa Pendorf. An example of the formation of a Feynman histogram shown in Figure Moments of the Feynman Histogram In a Feynman histogram, the variance is not equal to the mean. A commonly used metric for this measurement is the variance-to-mean rate ratio. Common interval bin times are between 20 µs and 200 µs. I call this histogram C n. The first, second, and third moments of the Feynman distribution are defined as follows: nc n n C = (2.10) Cn Figure 2.4: A Feynman histogram (solid bars) and the expected Poisson distribution (line) for a multiplying system on a log-normal plot. From page 48 of Neutron Specialist Handbook [7].

23 11 C 2 = C 3 = n 2 C n n (2.11) Cn n 3 C n n (2.12) Cn Reduced factorial moments are used with the Hage-Cifarelli theory described in Section 2.5 and are defined as follows: nc n n m 1! = 1! (2.13) C n n(n 1)C n n m 2! = 2! (2.14) C n n(n 1)(n 2)C n n m 3! = 3! (2.15) C n The moments are reduced factorial moments and are related by the following: m 2! = n(n 1)C n n=2 nc n n=1 m 1! = 1! = C (2.16) C n 2! = C n n=1 n=1 n 2 C n nc n n=2 2! C n n=1 = C2 C 2! (2.17) m 3! = n(n 1)(n 2)C n n=1 3! n=1 = n 3 C n 3n 2 C n +2nC n n=1 3! C n n=1 = C3 3C 2 +2C 3! (2.18) The moments for neutron-emission probabilities, which are parameters that de-

24 12 pend on the nuclear material in the system, are defined in a similar way [7]. Traditionally, these parameters representing the source neutrons are designated with an s-subscript and induced neutron values are designated with an i-subscript. These values are considered to be constants. This dissertation shows that the sourceneutron values are a function of the subcritical multiplication M of the system and that they should be considered variables for systems with high multiplication. Below are the definitions of the reduced factorial moments, which are commonly used in Feynman Analysis. ν = n np n (2.19) ν 2 = n n 2 P n (2.20) ν 3 = n n 3 P n (2.21) ν 1! = n np n (2.22) Diven s Numbers ν 2! = 1 n(n 1)P n (2.23) 2! n ν 3! = 1 n(n 1)(n 2)P n (2.24) 3! n The Hansen-Dowdy formalism, as shown in Section 2.4, uses a few special definitions related to ν. The D-number is defined as: D = ν(ν 1) ν = ν2 ν 1 (2.25)

25 13 where 3 ν(ν 1) = n ν(ν 1)P n (2.26) The Diven s parameter is defined as: Diven s parameter = ν(ν 1) ν 2 (2.27) It is interesting to note that for a Poisson distribution, the Diven s parameter is equal to 1. These values are considered to be constants from the nuclear material (D 1 ) and starting neutron source (D 0 ). Like the ν values, D 0 is also a function of M. The original theory behind neutron chains was developed at Los Alamos Scientific Laboratory between 1943 and 1946 by R.P. Feynman, F. de Hoffman, R. Serber and others [2, 8, 1]. The Feynman distribution analysis was named from this work. 2.3 The Meaning of Y m A key value in Feynman analysis is Y m, which is an indicator of how far the histogram deviates from a Poisson distribution. In Poisson statistics, the variance is equal to the mean, however in neutron multiplicity analysis the variance becomes greater than the mean as the multiplication increases. Y m is calculated from the Feynman histogram using Equation Y m = C2 C C 1 (2.28) In order for a Poisson distribution to represent a source, the source must emit 3 The Neutron Specialist Handbook has a typographical error. If Equation 2.26 were instead ν(ν 1) = n(n 1)P n, then n(n 1)P n = λ 2, it for a Poisson distribution would make n n ν(ν 1) ν 2 = λ2 = 1. If the version shown in the Neutron Specialist Handbook is used, then λ2 ν(ν 1) n(n 1)P n = (λ 1)λ. It would make = (λ 1)λ, which is not equal to 1, and n ν 2 thus this expression violates the statement that the Diven s parameter is equal to 1 for a Poisson distribution. λ 2

26 14 Table 2.1: Example Y m values for specific sources. From Neutron Specialist Handbook, page 58 [7]. The Beryllium Reflected Plutonium (BeRP) ball is a 6 kg ball of α-phase 239 Pu that was originally used in criticality experiments. It is not currently reflected by plutonium. The negative value for the AmLi source is not elaborated on in the reference. Description Y m BeRP ball in a shipping container BeRP ball bare Cf source AmLi source PuO 2 sample a single particle at a time, and the probability of decaying must be represented by dn = λn. The second requirement is easily satisfied from the fundamental dt physics of any spontaneous radioactive decay, the first is violated when a source can emit more then one particle at at time, such as spontaneous-fission or neutrons coming from a multiplying nuclear system. I tacitly assume that the time scale for neutron emission following fission is very fast compared to the neutron-slowingdown time scale. A Poisson distribution from a purely random source, such as an AmBe source, has a Y m value of 0, while Feynman distribution with multiplicity from the 252 Cf has a Y m value greater then 0. A table of Y m values for some sources is shown as Table 2.1. More details can be found on page 58 of the Neutron Specialist Handbook [7]. 2.4 The Hansen-Dowdy Method of Feynman Analysis In 1983 Robba, Dowdy, and Atwater published Neutron Multiplication Measurements Using Moments of the Neutron Counting Distribution [3] in which they derived the foundation to the neutron analysis method known as the Hansen-Dowdy Formalism. In their paper, they present a method for using the Feynman distribution to measure the subcritical multiplicity of a nuclear system. The specific application used in the paper is the Thor Core in a well-counter. Thor was a tho-

27 15 Table 2.2: Variables used with the Hansen-Dowdy formalism. From Neutron Specialist Handbook, page 59 [7]. Symbol C C 2 T 0 β τ D 0 D 1 D 0 ε t C NSS Ψ R R = S 0 ν 0 F 0 S 0 F 0 Definition First moment of Feynman histogram Second moment of Feynman histogram Window width of Feynman histogram Inverse neutron lifetime Detector dead-time D-number for starter neutrons D-number for multiplying material AverageD 0 numberforstarterneutrons,taking into account random starter neutrons Detector total efficiency Total count time Total counts Total neutron source strength Transmission ratio Randoms ratio Ratio of the number of random starter neutrons to the number of correlated starter neutrons Uncorrelated neutron rate ( (α,n) rate) Induced fission rate rium reflected plutonium critical assembly that was built and run at Los Alamos National Laboratory; and, after it was disassembled, the plutonium core was used in sub-critical neutron measurements. The Hansen-Dowdy formulas used to calculate M T, the total multiplication, are given below using the definitions in Table 2.2. M T = D 0 D 1 + (1 D 0 ) D 2 + 4Y C (2.29) 1 AD 1 Y C = Y m +2τ C T 0 1 4τh (2.30)

28 16 h = C T 0 + εβd 1(M T 1) 2 (2.31) A = T T 0 εg(βt)e βt (2.32) g(βt) = 1 1 e βt βt (2.33) T = T 0 τ (2.34) D 0 = ν 0(ν 0 1) ν 0 = ν2 0 ν 0 1 (2.35) D 1 = ν 1(ν 1 1) ν 1 = ν2 1 ν 1 1 (2.36) D 0 = D 0 1 R (2.37) ε = C 1 ( )Ψ (2.38) T 0 NSS The randoms ratio, which is the ratio of(α, n) starting neutrons to spontaneous fission starting neutrons, is defined as: R = S 0 ν 0 F 0 (2.39) The transmission ratio Ψ is used to compensate for moderating material around a source that would reduce the apparent neutron source strength. This value is not in the original Hansen-Dowdy papers. Also note that M T appears in Equation 2.31, thus it is necessary to iterate Equation 2.29 in order to correctly calculate M T.

29 Efficiency Measurement Using a 252 Cf source in the detector in the absence of any other nuclear material, one can also calculate the detector efficiency using Equation This measurement is based on the fact that the ν value for 252 Cf is known (3.757) [7] and the multiplication factor is 1. ) Ym Cf = ǫd Cf T τ 0 (1 1 e βτ e βτ (2.40) T 0 βτ Detector Die-Away Time and the Feynman-β Plot Because a neutron must be moderated before it is detected in a thermal system such as shown in Figure 2.1 and Figure 2.2, there is a time delay between when the neutron is emitted and when it is detected. This delay is described by a decaying exponential distribution of times. In this type of analysis the neutron lifetime (or detector die-away time) is defined as the average time it takes for a neutron to be detected in the system. This value can be measured by plotting Y m as a function of interval size, and example of this plot, called a Feynman-β plot, is shown in Figure 2.5. The efficiency, nuclear constants and the e βτ term are collapsed into a single-magnitude constant. Solving Equation 2.41 for β gives a measured value for detector die-away time. This experiment is commonly done by measuring Y m for a variety of time interval sizes and then curve fitting those data to Equation Y m = K T 0 τ T 0 (1 1 e β(t 0 τ) β(t 0 τ) ) (2.41) 2.5 The Hage-Cifarelli Method of Feynman Analysis In 1985 W. Hage and D.M. Cifarelli published a paper, On the Factorial Moments of the Neutron Multiplicity Distribution of Fission Cascades, in which they derived some of the underlying probability theory relating to neutron chains [12]. They expanded on the theory in 1986 when they published Models for a Three-Parameter

30 18 Figure 2.5: AFeynman-β plotandthebestfittoequation2.41. FromtheNeutron Specialist Handbook page 63 [7]. Analysis of Neutron Signal Correlation Measurements for Fissile Material Assay, which outlines a method to calculate the leakage multiplication from the Feynman histogram [4]. The basis of the Hage-Cifarelli method is to solve Equation 2.42, a cubic equation, for M, the leakage multiplication factor. The definitions of the terms are intricate and are given below, using the definitions found in Table 2.3. α 0 +α 1 M L +α 2 M 2 L +α 3 M 3 L = 0 (2.42) The variables in Equation 2.42 are defined as the following. α 2 = α 1 = R 2 ε 2 α 0 = R 3 ε 3 (2.43) [ νs3! 2 ν ] I2! ν S2! ν I1! 1 [ ] R 1 νi2! ν S3! ν I3! + 2R 2ν I2! ε(ν I1! 1) ν S2! ε 2 (ν I1! 1) (2.44) (2.45) α 3 = 2R 2ν I2! ε 2 (ν I1! 1) α 2 (2.46)

31 19 Table 2.3: Variables used with the Hage-Cifarelli formalism. From Neutron Specialist Handbook, page 68 [7]. Symbol R 1 R 2 R 3 M L τ F s S α ε ν S1!,ν S2!,ν S3! ν I1!,ν I2!,ν I3! T M λ m 1!,m 2!,m 3! Definition Singles counting rate Doubles counting rate Triples counting rate Leakage multiplication Time interval (Feynman window width) Fissions per second Random neutron rate from (α,n) reactions Detector efficiency First, second and third reduced factorial moments for spontaneous-fission First, second and third reduced factorial moments for induced fission Total measurement time Detector die-away time, also inverse neutron lifetime First, second and third reduced factorial moments of the Feynman histogram where F s = 1 [ R 2 T M ε 2 ML 2ν S2! R ] 1ν I2! (M L 1) εm L ν S2! (ν I1! 1) S α = 1 [ [ R1 1+ ν ] S1!ν I2! (M L 1) R ] 2ν S1! T M εm L ν S2! (ν I1! 1) ε 2 ML 2ν S2! (2.47) (2.48) R 1 = T M τw(1) m 1! (2.49) R 2 = T M [m 2! 12 ] τw(2) m 1! 2 R 3 = T M [m 3! m 2! m 1! + 13 ] τw(3) m 1! 3 (2.50) (2.51) w(1) = 1 (2.52)

32 20 w(2) = 1 1 λτ (1 e λτ ) (2.53) w(3) = 1 1 2λτ (3 4e λτ +e 2λτ ) (2.54) For active interrogation measurements, the following equations are used: R 3 = ε 2 M 2 (M L 1) L R 1 (ν I1! 1) R 2 (M L 1) = εm L ν I2! R 1 (ν I1! 1) ( ) (M L 1) ν I3! +2ν I2! (ν I2! 1) (2.55) (2.56) In active interrogation an external source of neutrons is used instead of an internal spontaneous-fission source. Active interrogation must be used with uranium samples because 233 U, 235 U, and 238 U have very small spontaneous-fission rates [9]. This property allows us to simplify the equations, because we now assume that the source neutrons are not correlated and thus F s = 0 [7]. 2.6 The Rossi-α Distribution and the Neutron Coincidence Counting Method While the Hage-Cifarelli and Hansen-Dowdy methods both use the Feynman Distribution, there is another method, which based on an entirely different distribution, called the Rossi-α distribution. The Rossi-α distribution has been in use longer than the Feynman distribution and uses much simpler electronics for the initial data collection. The Rossi-α distribution and the Neutron Coincidence Counting method are more commonly used in the IAEA and international safeguard applications. This section will provide a brief overview of the method. This method is presented in depth in PANDA [9] and Application Guide to Neutron Multiplicity Counting [13].

33 21 Figure 2.6: An example of a Rossi-α distribution from Application Guide to Neutron Multiplicity Counting [13] Rossi-α Distribution The Rossi-α is a time-to-next event distribution. It is generated by taking the n th neutron event and calculating the time to the n+1, n+2, n+3 neutron events. This time-to-next calculation continues for all neutrons that are produced within a pre-determined time. The process then starts over for the next neutron. The actual Rossi-α distribution is formed by making a histogram of the time-to-next events. A figure of a Rossi-α distribution from Application Guide to Neutron Multiplicity Counting [13] is shown in Figure 2.6. A Rossi-α distribution is a decaying exponential, which is the correlated neutrons, on top of a constant plateau, which represent the uncorrelated neutrons. A pure Poisson process would produce a flat Rossi-α distribution The Neutron Coincidence Counting Method of Multiplicity Analysis Once the Rossi-α distribution is constructed, two gates are formed. The first gate, after a short pre-delay, is opened at the start of the Rossi-α distribution and represents the correlated and uncorrelated events. This gate is called the Reals + Accidentals gate. After a delay, another similar gate is opened called the Accidentals Gate. The Neutron Coincidence Counting Method (NCC) uses

34 22 the following equations [14]: f d = e Tg λ (2.57) m n (i) = 1 N g ( x i m b (i) = 1 N g ( x i ) xmax x=i ) xmax x=i N(x) (2.58) B(x) (2.59) S = S m n (0) (2.60) D = S (m n (1) m b (1)) (2.61) where ( ) x x! i =, N(i) is the Reals+Accidentals Gate (R+A) and B(i) is ((x i)!i!) the Accidentals Gate (R+A), N g is the total number of neutrons detected, and λ is the average neutron lifetime in the detector. The leakage multiplication can be found by using and D = ε 2 f d ML S = FεMν s1 (1+α) (2.62) ( ν s2 + ( ML 1 ) )ν S1 (1+α)ν I1 ν I1 1 (2.63) where Tg is the size of the (R+A) and A gate, and ν s1, ν s2, ν I1, and ν I2 are the factorial moments of the distributions and are defined as: ν 1! = n np n (2.64) ν 2 = n n(n 1)P n (2.65) ν 3 = n n(n 1)(n 2)P n (2.66)

35 23 I did not use the Neutron Coincidence Counting Method much in this dissertation due to an inability to sufficiently verify the reliability of the method as implemented. 2.7 Similar Dissertation by Steven Nolen In 2000, Steven Nolen from Texas A&M University in conjunction with Los Alamos National Laboratory published his PhD dissertation entitled Chain-Length Distributions in Subcritical Systems [15]. Nolen s dissertation was a Monte-Carlo style simulation of the same system that R. Feynman described. My simulation is similar to the work done by S. Nolen. I used the same assumptions about the physics of the system but did not consult his work prior to writing the program. In addition, Nolen wrote a 3-Dimensional simulation, while I only simulated point kinetics. Two plots from his dissertation are presented as Figures 1.1 and 1.2. Figure 1.2 shows the tail of the distribution as a smooth function, whereas my simulations, such as Figure 4.1, show a tail that is affected by statistical variation. From my time at Los Alamos National Laboratory, I distinctly remember seeing statistical variations in measured data. At this time, I do not have any explanation for this discrepancy between my simulation and that by Nolen. Since I do not directly use any of Nolan s work, this issue is only a curiosity. 2.8 Prasad and Snyderman s Closed Form Solution of the Feynman Distribution N. Snyderman and M. Prasad were at Lawrence Livermore National Laboratory (LLNL) in 2002, when they derived the closed-form solution to predicting the Feynman distribution. However, they did not openly publish their results until 2012 with the paper Statistical Theory of Fission Chains and Generalized Poisson Neutron Counting Distributions [5]. Their model uses the same physical assumptions as Hage-Cifarelli and Hansen-Dowdy. Instead of deriving expressions that can be solved for the multiplication knowing the moments of the distribution, they derived the form for the Feynman distribution itself. Their equations are complex

36 24 and difficult to code, but they do accurately predict the distribution. Prasad and Snyderman compare their theoretical calculation to measured data taken with a 239 Pu multiplying system. Prasad and Snyderman also presented all the nuclear values that they used for their calculation, which allowed me to simulate the same system. I present the results of that simulation in Section Prasad and Snyderman s paper is very dense and the equations are presented in a different order than one would use to do the calculation. Without re-writing all the equations, I outline the correct equations and order needed to reproduce Snyderman and Prasad s calculations. In this section, I use the equation numbering from Statistical Theory of Fission Chains and Generalized Poisson Neutron Counting Distributions [5]. The first step is to calculate the Ψ(n,m) matrix as introduced in Equation 43. However, Equation 43 is not the proper equation to use, for it has a tendency to be numerically unstable. The correct method to calculate Ψ(n,m) is to use the method as discussed in the last paragraph of Section III.D of Reference [5]. The next step is to use Equations 44 and 45 to calculate P ν (p), which is the probability of the multiplying system to create ν neutrons for a given value of p. This output is the equivalent of the probability distribution function of the output of Chains as shown on Page 32 of Reference [5]. If there is spontaneous fission in the system, one must use Equation 108 to factor those into a new Pν spontaniouse (p) distribution. The next step is to use Equation 8 with Equations 9 and 10 to calculate Λ(j), which is the probability of getting j neutrons from a chain after taking into account detector die-away time and a finite time gate. The final step is to use Equations 2 and 3 to calculate b n, which is the actual Feynman Distribution that one can measure. This final step takes into account the various combinatorial ways one can get n neutrons from overlapping fission chains. I have reproduced Snyderman and Prasad s calculations and I found that they match my simulations. (The derivation presented by Snyderman and Prasad is worthy of a dissertation in of itself and is not the focus of this dissertation.)

37 The Relationship Between Number of Starter Events and Number of Detector Events For my simulation, I use N (or n) to represent the number of starting events, which are either (α, n) neutrons or a spontaneous fission neutrons. These starter neutrons are multiplied and eventually become a number of neutrons recorded by the detector system. The relationship between the number of starter events (N S ) and the number of detected neutrons (N D ) is: N D = εm t ν S N S (2.67) where ε is the efficiency of the detector system, M t is the total multiplication and ν I is the neutron multiplicity for the starters. Note that ν I is equal to one for an (α, n) source The Ratio of Poisson Neutrons to Fission Neutrons (α) Both the Hage-Cifarelli and Hansen-Dowdy methods take into account Poisson distributed starting neutrons mixed with spontaneous fission neutrons, which are represented by the R value as shown in Table 2.2 and Equation 2.39 for Hansen- Dowdy. For the Hage-Cifarelli, its given by the S α value as shown in Table 2.3. For both methods, it is the ratio of the number of random starter neutrons to the number of correlated starter neutrons. R = S 0 ν 0 F 0 (2.39) For this dissertation, I define a variable α that represents the ratio of Poisson neutron decays to spontaneous fission decays. In relation to the multiplication factor, the β-fraction and ν i, α is given by α = Mβ ν i (2.68)

38 26 This definition does not completely mirror the one for Hage-Cifferali and Hansen- Dowdy because theirs is the ratio of neutrons, and mine is the ratio of decays. I have not found Equation 2.68 in any reference and it is published in this dissertation for the first time.

39 Chapter 3 Basic Simulation of Neutron Chains I wrote a Monte-Carlo simulation to calculate the Probability Density Function (PDF) of neutron chains in a sub-critical system. This simulation is zero dimensional (point kinetics) in space, which is a valid assumption for small systems. The simulation also distributes the neutrons in time in order to properly simulate neutron die-away in the detector and overlapping fission chains. The simulation data are output in list-mode data, which can then be analyzed by a variety of methods. List-mode data is where the time of each neutron detected is recorded in a sequential list. The goal is to simulate neutron chains and the associated Feynman distribution. Thesimulatingshouldbeasclosetowhatwouldbemeasuredbyadetectorsystem, given the practical constraint of maintaining computational speed. This simulation uses an explicit analog simulation method while reducing the physics to the bare essentials, as outlined in Section 3.1. The simulation also keeps the same physical assumptions that are used in the Hage-Cifarelli [4] and Hansen-Dowdy-Robba [3] derivations and the Prasad-Snyderman [5] formulation. The simulation replicates a variety of systems, such as uranium inside of a neutron well-counter. In this section, I outline the method and Mathematica algorithms that I used for the simulation before I added dead-time or delayed neutrons.

40 Physical Assumptions All geometry and cross sections are reduced to a single value, p, the probability that a neutron causes a fission. This simplification is valid, because, from a probability standpoint, itisirrelevantwhyaneutrondoesnotgoontocauseafission, itisonly relevant that it does or does not. This simplification also mirrors the definition in Nolen, who notes that p = k eff [6]. ν All absorption and leakage effects are reduced to another single value, ǫ, the probability that a neutron that does not cause a fission is detected in the detector. This simplification is valid, because, whether either neutron leaks, is absorbed without causing fission, or is below the energy corresponding to the lower-level discriminator (LLD) of the detector, the effect is the same; it is not detected. This definition also mirrors the definition of absolute efficiency as defined by Knoll [16]. Another assumption is that all neutrons in a chain are created at the same time. The time from one fission event to the next in a fast-neutron system is on the order of 10 ns and the detector response time is on the order of 20 µs. Thus, for even long fission chains, the fissions will happen so fast that the detector will detect them as if they are happening at the same time. The unit of measurement, the shake, is defined to be 10 ns time and is commonly used in weapons physics. Thefinalmajorassumptionisthatneutronemissionsarespreadoutintimeand thus they are represented by a single, decaying-exponential probability distribution function representing detector die-away time. In a thermalized system, the last two assumptions would not hold, because there would be a significant time delay, while the neutrons are being thermalized between fissions. Thus, one would have to take into account both the time distribution of thermalization and the time distribution for detection. 3.2 Method The program that I wrote, called Chains, is composed of subroutines, all of which are described in this section. Please note that I commonly use the Module[] command in Mathematica. This function allows me to create a subroutine that I can use later. Also note that

41 29 the Table[] command is an easy-to-use function that combines a For Loop and an output-array-creating function in a single command. The source of neutrons could either be from spontaneous fission or from an (α, n) source. The spontaneous-fission source distribution looks similar to an induced fission distribution, whereas, an (α, n) source always emits one neutron per reaction. I use the data either for a 252 Cf or for a 240 Pu spontaneous fission source, as given in The Neutron Specialist Handbook [7]. The program uses the Monte- Carlo table-lookup method to chose the number of source neutrons. For the induced-fission sources, the program uses the same method to determine the number of neutrons per fission. The program has the option to use the neutron multiplicity distribution for either 235 U or 239 Pu. The key algorithm of this simulation is a function that produces the number of neutrons created for a single decay of the source. The input is the probability (p) that a neutron induces fission. A Monte-Carlo history begins by determining how many starting neutrons the source produces and adds that value to a variable that represents the current number of live neutrons in the system. A uniformly distributed random number is then sampled and compared to p in order to determine if the neutron causes a fission or not. If it does, then fission multiplicity distribution is sampled to determine how many neutrons are created and that number is then added to the current number of live neutrons. If it does not cause fission, then the variable is reduced by one. This process continues for every live neutron until the counter reaches zero and returns the total number of neutrons leaked. If the value p is large enough for the system to be critical or supercritical, this function could run indefinitely. For each individual neutron leaked, the program samples a random number and compares it to ε, the detector efficiency, to determine if the neutron is detected. I assume that the timescale for fission is so fast that the approximation holds that all fission neutrons are created at the same time and enter the detector as a group. Once the neutrons enter the detector, the probability that they are detected at time t after t = 0 is represented by a decaying exponential-function with a time constant, λ. The units of time are commonly in µs. Once the total number of neutrons detected is calculated for a chain, the pro-

42 30 gram distributes them in time. The function uses the expression λlnξ, where ξ is a uniformly distributed random number between 0 and 1, to determine when an individual neutron is detected. This process is repeated for all neutrons in the chain and the results are then sorted by time. If the PDF is a decaying exponential function, then the Monte-Carlo method to sample that distribution is of the form lnξ [17]. In a physical system, the fissionable material is driven by a source that decays at a particular rate, Fs. The program again uses Fslnξ to determine when the next starting neutron appears. This method allows for overlapping fission chains, a very real problem, when the decay rate of the source is close to the die-away time of the detector. Overlapping fission chains make it difficult to properly determine how many neutrons were emitted from a single source-decay. At this point, the program produces a list of times when the neutrons are detected. These data can then be analyzed using any method desired Number of Neutrons per Fission The first part of Chains is a subroutine that randomly chooses how many neutrons are created from fission, based on the known PDF representing the number of neutrons emitted per fission. This process is done with a basic table look-up method. The first step is to input the initial data: fuel = "U235"; U235 = {{0, 0.033}, {1, 0.174}, {2, 0.334}, {3, 0.303}, {4, 0.124}, {5, 0.029}, {6, 0.003}}; fueldata = U235; The next step is to format the data for the table look-up method: data = Table[U235[[i, 2]], {i, 1, Length[U235]}]; i nfdata = Tablei-1, data[[j]], {i, 1, Length[data]} j=1 The next step is the actual implementation of the table look-up method:

43 31 Table 3.1: P ν values used. The plutonium data are from Statistical Theory of Fission Chains and Generalized Poisson Neutron Counting Distributions [5] and the uranium and californium data are from Neutron Specialist Handbook and Informational Text [7]. The number of significant figures in the table match those in the references. ν 240 Pu 252 Cf 235 U 239 Pu nneutrons[] := Module[{i}, i = 1; rand = Random[]; While[nFdata[[i, 2]] rand, i++]; i-1]; The product of this code is the function nneutrons[], which returns the number of neutrons per fission. A similar function, sneutrons[], is used to determine the number of source neutrons Source Selection The source of neutrons could either be from a spontaneous-fission source or from an (α, n) source. The spontaneous-fission-source distribution looks similar to an induced-fission distribution, but having a higher average number of neutrons, while an (α,n) source always emits one neutron per decay. I used P ν values for 240 Pu, 252 Cf, 235 U and 239 Pu, which can be found in Table 3.1. I wrote an algorithm using Mathematica s Which command that allows me to choose which starting source and multiplying material to use for nneutrons[] and sneutrons[]. The inputs are a-n (for (α, n)), Pu240, Cf252, U235 and

44 32 Pu Basic Chains The core of this simulation is the function Chain[p], that calculates the number of neutrons created for a single decay of the source. The input is p, which is the probability that a neutron induces fission. To start, it uses sneutrons[] to determine how many starting neutrons to have, and adds that quantity to the variable CurrentNeutrons, which is the current number of live neutrons. Then a random number is selected and compared to p to determine whether the neutron causes a fission or not. If it does, then nneutrons[] is used to determine how many neutrons are created, and that number is added to the current number of live neutrons. If it does not cause a fission, CurrentNeutrons is reduced by one. This process continues for every live neutron until none are left. At the end, it returns the total number of neutrons leaked. If the value p is large enough for the system to be critical or supercritical, this function could run indefinitely. Note that even with a supercritical system, not all neutron chains will be infinite length. Chain[p_] := Module[{n}, leaked = 0; n = sneutrons[]; CurrentNeutrons = n; lenght = n; While[CurrentNeutrons > 0, If[Random[] p, n = nneutrons[]; CurrentNeutrons = CurrentNeutrons- 1 + n; lenght = lenght+n, CurrentNeutrons = CurrentNeutrons- 1; leaked = leaked+1;]]; leaked] Detector Efficiency For each chain, the program takes each individual neutron that is leaked and selects a random number and compares it to ǫ, the detector efficiency. Detected[] returns

45 33 the total number of neutron detected. The inputs are n (from Chains[]), and ǫ. Detected[n_, _] := Module[{}, det = 0; For[i = 0, i < n, i++, If[Random[] <, det++] ]; det] Distribution in Time Caused by Detector Die-away Since fission chains in a fast neutron-system happen on the time scale of tens of nanoseconds and the detector response is on the order of tens of microseconds, we can assume, that from the perspective of the neutron detector, all neutrons are created at the same time in one burst [12]. Once the neutrons enter the detector, the probability that they are detected at time t after t=0 is represented a decayingexponential PDF with time constant dieaway. The units of time are commonly µs. This function incorporates Chain[] and Detected[]. Once the total number of neutrons detected is calculated for a chain, this function uses the well known Dieaway ln ξ expression to determine when an individual neutron is detected. This process is repeated for all neutrons in the chain, and the results are sorted and rounded to the nearest integer. DistinTime[p_, _, dieaway_] := Module[{}, chainlength = Chain[p]; chainlength = Detected[chainlength, ]; Sort[Round[Table[- Log[ Random[]] dieaway, {chainlength}]]]] For a neutron chain of 9 neutrons and a die-away of 200 µs, an example of the list-mode output of DistinTime[] is: {65, 70, 101, 111, 207, 216, 290, 421, 502}. The above array represents the time, in µs, when each of the nine neutrons is detected.

46 Distribution in Time Caused by Source Decay In a real system, the fissionable material is driven by a source that decays at a particular rate (decayrate). Chains starts at time t=0 and runs DistinTime[] to generate the times for which the neutrons from the first decay are detected. It then uses decayratelnξ to determine the time until the next decay of the source and adds that time to a running time variable (decaytime). It then runs DistinTime[] again and adds those times to the decay time variable in order to determine the detection times from the second source decay. This process is repeated for as many decays as desired. This method allows for overlapping fission chains, a very real problem when the decay-rate of the source is close to the die-away time of the detector and is difficult to determine accurately how many neutrons were emitted from a single source-decay. FullChains[a_, c_, b_, p_, _, dieaway_, decayrate_, n_] := Module[{}, Source[a, c, b]; decaytime = 0; temp2 = N[Table[ temp = DistinTime[p,, dieaway]+decaytime; decaytime = decaytime + Round[- Log[ Random[]] decayrate]; temp, {n}]]; times = Sort[Flatten[temp2]]] For four decays, with a die-away of one µs, and a decay rate of 1000 µs, an example of the list-mode output of FullChains[] is: {1, 1429, 1429, 1429, 1429, 1429, 1429, 1429, 1429, 1429, 1429, 1429, 1429, 1429, 1430, 1430, 1430, 1430, 1430, 1430, 1430, 1430, 1430, 1430, 1430, 1430, 1431, 1431, 1431, 1431, 1431, 1433, 2544, 2544, 2544, 2544, 2545, 2545, 2545, 3017, 3017} The four decays and their detected neutrons can clearly be seen. This example used a 1 µs die-away in order to resolve the chains clearly. While a 1 µs die-away time is not physically realistic, it is useful for testing and illustrative purposes.

47 35 FullChains[] outputs the event time data in list-mode as seen above and the event time data can then be passed on to Feynman[], RossiAlpha[], or another algorithm for further analysis. 3.3 Creating the Feynman Histogram The final step is to take the long list of times of detected neutrons and divide them into time intervals (as is done in the experimental side) and then to create a histogram of the intervals in order to produce the Feynman distribution, which is the end goal. Thislineconvertsthetimesofarrivaltoanarrayoftimeintervals. Forexample, with a time interval of 20 µs, a neutron that arrives at time 102 µs will be in time interval 6. temp = Round[times/binsize]; The above algorithm counts how many neutrons arrived in each time interval. tally = Tally[temp]; These two lines above create the histogram of the number of neutrons in each time interval, takes into account intervals with zero counts, and normalize the results. numberzeros = Max[temp4] - Length[tally]; temp5 = Sort[Table[tally[[i, 2]], {i, 1, Length[tally]}]]; temp6 = Tally[temp5]; temp7 = Insert[temp6, {0, numberzeros}, 1]; sum = Sum[temp7[[i, 2]], {i, 1, Length[temp7]}]; The full code is presented below:

48 36 Feynman[times_, binsize_] := Module[{}, temp4 = Round[times/ binsize]; tally = Tally[temp4]; numberzeros = Max[temp4]- Length[tally]; temp5 = Sort[Table[tally[[i, 2]], {i, 1, Length[tally]}]]; temp6 = Tally[temp5]; temp7 = Insert[temp6, {0, numberzeros}, 1]; sum = Sum[temp7[[i, 2]], {i, 1, Length[temp7]}]; Table[{i-1, 1.temp7[[i, 2]]/sum}, {i, 1, Length[temp7]}]] The output of the Feynman Histogram algorithm is: {{1, }, {2, }, {3, }, {4, }, {5, }, {6, }, {7, }, {8, }, {9, }, {10, }, {11, }, {12, }, {13, }, {14, }, {15, }, {16, }, {17, }, {18, }, {19, }, {20, }} The above array indicates that the simulation detected one neutron in the time interval 14% of the time, two neutrons 17.9% of the time, and so on. This array of data can then be plotted or passed on to analytical algorithms. The final command to run Chains is FullChains[s, p, ǫ, dieaway, decayrate, binsize, n], where s is the source to be used, p is the probability of fission, ǫ is the detection probability, dieaway is the detector die-away time, decayrate is the decay rate of the source, binsize is the size of the histogram interval and n is the number of source decays to run. 3.4 Implementation of the Hansen- Dowdy Method The Mathematica algorithms that I used to implement the Hansen-Dowdy Method as introduced in Section 2.4 is shown below.

49 37 Hansen[pp_, test_] := Module{}, Clear[Mt, MValue]; 1 MTotal = 1-fpp ; MLeak = 1-pp 1-fpp ; If[IntegerQ[],, = 0]; = 0; = Sim -1 ; D0 = n=0 Length[sourcedata]-1 n(n-1)sourcedata[[n+1, 2]] ; s R = 0; D0Bar = D0 1-R ; D1 = n=0 Length[fueldata]-1 n(n-1)fueldata[[n+1, 2]] ; f T = T0 - ; bt = T; g[bt] := 1-1--bt ; bt A = T T0 g[bt] -bt ; Cbar = n=1 Length[test] (n-1)test[[n, 2]] 1.; Length[test] test[[n, 2]] n=1 Cbar2 = n=1 Length[test] (n-1) 2 test[[n, 2]] 1.; Length[test] n=1 test[[n, 2]] Cbar3 = n=1 Length[test] (n-1) 3 test[[n, 2]] 1.; Length[test] n=1 test[[n, 2]] Cbar2 Cbar Ym = -Cbar-1 ; 1-4h Yc = Ym + 2 Cbar T0 ; Off[Solve::ratnz]; MValue = SolveYc == Mt (D0 +D1 (Mt -1)) T T0 On[Solve::ratnz]; Clear[]; 1-1--bt bt Print["MT = ", MTotal, " and ML = ", MLeak, " and Hansen-Dowdy gives Mt = ", MValue] && Mt > 0, Mt[[1, 1, 2]];

50 Creating the Rossi-α Distribution The following algorithm is used to replicate the data processing used to create Rossi-α distributions as introduced in Section RossiAlpha[times_, RossiBinSize_] := Module[{}, TempTimes = times; BinnedTimes = Table[ EndOfInterval = LengthWhile[TempTimes, # < (RossiBinSize +First[TempTimes]) &]; TempTimes2 = TempTimes; TempTimes = Drop[TempTimes, EndOfInterval]; Take[TempTimes2, EndOfInterval], {Floor[Last[times]/ RossiBinSize]}]; NCCData = Table[ Table[BinnedTimes[[j, i]]- BinnedTimes[[j, 1]], {i, 2, Length[BinnedTimes[[j]]]}], {j, 1, Length[BinnedTimes]}]; NCCData = DeleteCases[NCCData, {}]; Flatten[NCCData]] The output of this is a Reals+Accidentals and a Accidentals Histogram. 3.6 Implementation of the NCC Method The Mathematica algorithms that I used to implement the NCC Method as introduced in Section is shown below.

51 39 NCCMethodNew[times_, PreDelay_, Gate_, AGateLoc_, RossiBinSize_] := Module{}, NCCResults = NCC[times, PreDelay, Gate, AGateLoc, RossiBinSize]; RealsPlusAccidentals = NCCResults[[1]]; Accidentals = NCCResults[[2]]; Ng = Total[RealsPlusAccidentals[[2]]]; Nbins = Table[{RealsPlusAccidentals[[i, 1]], RealsPlusAccidentals[[i, 2]]/Total[RealsPlusAccidentals[[2]]]1.}, {i, 1, Length[RealsPlusAccidentals]}]; Bbins = Table[{Accidentals[[i, 1]], Accidentals[[i, 2]]/ s1 = s2 = s3 = I1 = I2 = I3 = = 0; Singles = Total[Accidentals[[2]]]1.}, {i, 1, Length[Accidentals]}]; Length[sourcedata]-1 (i)sourcedata[[i+1, 2]]; i=0 Length[sourcedata]-1 (i)(i-1)sourcedata[[i+1, 2]]; i=0 Length[sourcedata]-1 (i)(i-1)(i-2)sourcedata[[i+1, 2]]; i=0 Length[fueldata]-1 (i)fueldata[[i+1, 2]]; i=0 Length[fueldata]-1 (i)(i-1)fueldata[[i+1, 2]]; i=0 Length[fueldata]-1 (i)(i-1)(i-2)fueldata[[i+1, 2]]; i=0 FsSim 1 Length[RealsPlusAccidentals] Binomial[Nbins[[x, 1]], 0] Nbins[[x, 2]]; Ng x=1 Doubles = FsSim Length[RealsPlusAccidentals] 1 Binomial[Nbins[[x, 1]], 2] Nbins[[x, 2]] - Ng x=2 Length[Accidentals] 1 Binomial[Bbins[[x, 1]], 2] Bbins[[x, 2]] ; Ng x=2 results = SolveDoubles == 1 2 fdml s2+ Ml -1 I1-1 Print["NCC Method gives ", results] s1(1+) I2, Ml, Reals;

52 Parallel Processing Chains In a physical experiment, the first neutron would be generated at time t=0, and each neutron would be tagged sequentially until the end of the experiment. In my stimulations, each run of FullChains[] starts at time t=0 also and thus five runs of Chains would look like this: Run 1: {15, 23, 32, 46, 68, 72, 74, 76, 83, 93, 116, 127, 137, 139, 154}, Run 2: {28, 61, 75, 75, 80, 82, 101, 110, 128, 131, 184, 250, 265, 299, 315}, Run 3: {37, 78, 94, 135, 173}, Run 4: {31, 83, 154, 158, 189, 196}, Run 5: {25, 56, 122, 152, 357} However, for the neutron analysis methods to work, the neutrons times in the listmode needs to be monotonically increasing. It was necessary to write an algorithm that would make the first list-mode time for one run of Chains come after the last time in the previous run of Chains. An example of this is shown below: Run 1: {70, 104, 114, 130, 131, 140, 150, 168, 233, 240}, Run 2: {382, 391, 467, 467, 485, 485, 490, 499, 529}, Run 3: {645, 651, 653, 682, 706, 706, 738, 763, 888}, Run 4: {922, 924, 964, 985, 993, 1045, 1047, 1056, 1075, 1091, 1105, 1111, 1144}, Run 5: {1348, 1372, 1381, 1418, 1425, 1473, 1508, 1516, 1543, 1553, 1557, 1568, 1581, 1588, 1595} My solution was, for the running of FullChains[] on the first processor, to keep track of the time of what would have been the next source decay after the last source decay that was used and add that time to the t=0 time of the output of FullChains[] from the second processor. This process continued for the outputs of all the processors used in the parallel processing calculation. The code that I used is presented below.

53 41 testdata = 0; MulipleChains = Parallelize[Table[FullChains["Pu240", 0, "Pu239", pp,, Sim, FsSim, n], {Repeates}]]; DividedChains = Table[ TimeBetween = Round[- Log[ Random[]] FsSim] + Max[testdata]; testdata = MulipleChains[[i]] + TimeBetween, {i, 1, Repeates}]; CombinedChains = Flatten[DividedChains]; 3.8 Combined Input The full set of commands used to run Chains on a single processor is given below. Source["Pu240", 0, "Pu239"] Mult = 50; n = ; T = 289; = ; = ; FsSim = ; pp = Mult-1 Mult f ; testdata = FullChains["Pu240", 0, "Pu239", pp,,, FsSim, n]; test = Feynman[testdata, T] Hansen[pp, test] NCCMethodNew[CombinedChains, 1, Tg, 300, 500] below. The full set of commands used to run Chains on multiple processors is given

54 42 Source["Pu240", 0, "Pu239"] Mult = 10; Tg = 40; Repeates = 4; n = ; T = T0 = 289; = Sim = ; = ; FsSim = Mult-1 ; pp = Mult f -1 ; testdata = 0; MulipleChains = Parallelize[Table[FullChains["Pu240", 0, "Pu239", pp,, Sim, FsSim, n], {Repeates}]]; DividedChains = Table[ TimeBetween = Round[- Log[ Random[]] FsSim] + Max[testdata]; testdata = MulipleChains[[i]] + TimeBetween, {i, 1, Repeates}]; CombinedChains = Flatten[DividedChains]; test = Feynman[CombinedChains, T0]; Hansen[pp, test] NCCMethodNew[CombinedChains, 1, Tg, 300, 500] 3.9 Summary In this section, I outline the method and Mathematica algorithms that I used for the simulation before I added the simulation of dead-time or delayed neutrons. I present the implementation of the Hansen-Dowdy and NCC methods of neutron analysis. I discuss my method of modifying Chains for parallel processing. Chapter 4 covers the simulations and benchmarks of the algorithms presented in this chapter.

55 Chapter 4 Running Chains 4.1 Results Figure 4.1 shows the simulated dependence of k eff on the Feynman histogram. In this plot, the difference in multiplicity between systems of varying fission probability can clearly be seen. Their input values are; efficiency = 1, neutron source rate = 0.05 µs 1, detector dieaway = 0.1 µs 1 and time interval = 10 µs. Probability of Chain Length Chain Length keff =0.3 keff =0.5 keff =0.7 keff =0.9 keff =0.99 Figure 4.1: Feynman histogram for k eff = 0.3,.05, 0.7, 0.9 and This plot was created using an (α,n) starting source and 235 U as the multiplying material.

56 Benchmarking MeasuredData TheoryData ChainsMonte-CarloSimulation Probability of Chain Length Chain Length Figure 4.2: Comparison of the LLNL experimental and theoretical Feynman histogram, and Chains. This is the same data as in Table 4.1. Prasad and Snyderman, in Statistical Theory of Fission Chains and Generalized Poisson Neutron Counting Distributions [5], present measurements on a 239 Pu system with 240 Pu starting neutrons, which they compare to the results from their calculated model. Their input values are; efficiency = , neutron source rate = µs 1, detector dieaway = µs 1, time interval = 289 µs, and p = The significant figures are the same as in the original. I ran 400 million starting events using the same input values and formed the Feynman histogram from the results. The results are presented in Table 4.1 and shown in Figure 4.2. My simulation results closely match the experimental Feynman histogram data and theory, demonstrating that even this simplified simulation can accurately reproduce the Feynman histogram from a physical experiment. The source of the theoretical Feynman histogram data is described in Section 2.8. The Feynman histogram data from a physical experimental are from the same paper,

57 45 but no further details are given regarding the specifics of the experimental setup or nuclear material used. Table 4.1: Comparison of LLNL experimental Feynman histogram and theory from Statistical Theory of Fission Chains and Generalized Poisson Neutron Counting Distributions [5], and Chains simulation. These are the same data points as in Figure 4.2. The number of significant figures matches those found in the reference. The percent difference column compares the measured data to Chains. Multiplicity Measured Theory Simulated % Data with Chains Difference A Note About Simulation Parameters Unless otherwise specified, for this entire dissertation, I use the same input parameters as used for the benchmarking in Section Using the same values

58 46 M T M L Hansen-Dowdy Multiplication From Simulation Input Leakage Multiplication Figure 4.3: A plot of input values of M vs the output as given by the Hansen- Dowdy method. The parameters are as specified in Section with 240 Pu starting neutrons and 239 Pu as the multiplying material. M T is the total multiplication and M L is the leakage multiplication. was done in order to keep the simulations for this dissertation as similar to the measured physical system as possible. The input values are; efficiency = , neutron source rate = µs 1 (84,200 decay s ), detector dieaway = µs 1 (54.2 µs). I also use the same Feynman time interval of 289 µs. The number of significant figures shown above are exactly as presented in the reference Extracting M from Chains An important test for my simulation and analysis is how well I can input a specific subcritical multiplication for Chains, run Chains, and then use one or more of the analysis methods outlined in Chapter 2 to extract the subcritical multiplication. For this experiment, I used the parameters as specified in Section with 240 Pu emitting the starting neutrons and 239 Pu as the multiplying material. For most of

59 47 this dissertation, I only used the Hansen-Dowdy neutron analysis method. I did not use the Hage-Cifarelli method because, as it is shown, Hansen-Dowdy works very well and I only needed one method. I used the Neutron Coincidence Counting (NCC) method only for comparison with the Hansen-Dowdy method. The results for my experiment using the Hansen-Dowdy method are shown in Figure 4.3. The dotted line represents M L and the dashed line represents M T. The solid line shows the Hansen-Dowdy analysis. Hansen-Dowdy consistently gave an M T value that is a greater then the M L value but not as high as the M T value. Hansen-Dowdy is expected to give the M T value, so the results were lower than expected. However, Hansen-Dowdy gave consistent results that were within M T and M L using the same time interval value for the entire range. From this result, and the benchmarking in Section 4.1.1, I conclude that the Hansen-Dowdy method is reliable and accurate. Source["Pu240", 0, "Pu239"] time = AbsoluteTime[]; Mult = 3; Tg = 100; = 0; Repeates = 4; n = 50000; T = T0 = 289; = Sim = ; = 0.015; FsSim = Mult-1 ; pp = Mult f -1 ; time = AbsoluteTime[]; Print["Without Delayed Neutrons, n = ", n, " x ", Repeates] testdata = 0; progress = 0; SetSharedVariable[progress]; ProgressIndicator[Dynamic[progress]] MulipleChains = Parallelize[Table[{FullChainsPrint["Pu240", 0, "Pu239", pp,, Sim, FsSim, n], decaytime}, {Repeates}]]; i ListofLastDecays = Table MulipleChainsj, 2, {i, 1, Repeates}; j=1 ListofLastDecays = Insert[ListofLastDecays, 0, 1];

60 48 DividedChains = Table[ TimeBetween = Round[- Log[ Random[]] FsSim] + ListofLastDecays[[i]]; testdata = MulipleChains[[i, 1]] + TimeBetween, {i, 1, Repeates}]; CombinedChains = Flatten[DividedChains]; test = Feynman[CombinedChains, T0]; Hansen[pp, test] NCCMethod[CombinedChains, 1, Tg, 300, 500] Speak["Completed."] time = AbsoluteTime[]-time /M Approach to Criticality I simulated a measurement of a (1/M) approach-to-criticality experiment using both an (α, n) and a spontaneous-fission starting neutron source. The multiplying material is 235 U. The independent variable is p. The results are shown in Figure 4.4. Asexpected, thepredictedvalueofpatcriticalityis1/ν = 1/2.414 = 0.424for both cases. These results also demonstrate the invariance of the neutron population on criticality. The criticality of a system is determined completely by the balance between neutron creation and neutron loss though leakage or non-fission absorption. This concept is introduced in all introduction nuclear engineering textbooks such as Lamarsh and Baratta s Introduction to Nuclear Engineering [18] and in advanced textbooks such as Duderstadt and Hamilton s Nuclear Reactor Analysis [19]. An interesting result of criticality analysis is that flux and neutron population are not variables and are thus irrelevant. The criticality of a system does not in anyway depend on the neutron population. Thus, changing the starting source from an (α, n) source to a spontaneous-fission source will not change the criticality of the system, and my simulation shows just that result Feynman-β Curve I simulated a Feynman-β curve, which is used to measure the neutron die-away time in the detector as previously mentioned in Section The results are shown in Figure 4.5. The fit gives a neutron die-away time of 10.4 µs, which matches the input value of 10 µs within 4%.

61 /M Predicted Criticality (p) /M Predicted Criticality (p) Figure 4.4: Dependence of 1/M plot on p, the probability of fission, with an (α,n) (top) and spontaneous fission (bottom) starting neutron sources Time to Next Event Conducting a time-to-next-event calculation of the list-mode data is an instructive exercise, because in a multiplying medium there are two visible bands. A comparison between theoretical Feynman histogram created by Prasad [20] and simulated Feynman histogram from Chains is shown in Figure 4.6. The top band corresponds

62 Y IntervalSize (s) Figure 4.5: Feynman-β plot from Chains. The points are from the simulation and line is the β fit. to the decay rate of the source neutrons and the bottom band corresponds to the multiplication and numerically corresponds to the detector die-away time. This plot is sometimes called a waterfall plot. 4.2 Minimum Number of Events Needed for the Hansen-Dowdy Method In order to determine the minimum number of events needed for the Hansen-Dowdy method to produce acceptable results, I ran Chains with a different number of starting neutrons (N S ), and ran Chains 10 times for each value of N. From those runs. I made a table of the number of events recorded, the average M-value from the Hansen-Dowdy method, the standard deviation of M from those 10 runs, and the percent standard deviation. The true value of M is the value once M reaches equilibrium with a larger number of starting neutrons. I also made a column which gives the average M value for that value of N divided by the true value of M. I

63 51 Time to Next Event Event Number Figure 4.6: A time to next event plot from Chains (top) compared to one pub- lished by Prasad (bottom) [20]. The upper bands is from the source decay and its location is determined by the decay constant. The lower band represents the correlated neutrons and its location is determined by the detector die-away time. The two plots do not use the same input parameters. defined acceptable results when the standard deviation was less than 3% and the average M value as within 3% of the M value from the true value. I simulated values of M = 1.5, 2, 3, 5, 7, 10, 15, and 20. I also ran each simulation for efficiency values of 1.5%, 4.4% and 25%. The 1.5% value is representative of portable neutron systems such as the Ortec Fission Meter, the 4.4% value is the standard value used for the rest of this dissertation as discussed in Section 4.1.2

64 52 and the 25% value is representative of a neutron well-counter. Note that for this section, I refer to the number of recorded events, which is the number of neutrons that the detector actually recorded, after taking into account efficiency Results Figure 4.7 shows the dependence of the minimum number of events needed to for the Hansen-Dowdy method to produce the true M value for a 239 Pu system with a 240 Pu spontaneous fission starting source and a 235 U system with an (α,n) starting source. They are plotted on a logarithmic scale to highlight the effects. As expected, more events are needed as the sub-critical multiplication increases. However, there is an unexpected result at low values of M where more events are needed as M decreases. It is notable that from M=7 and on, the log plots appear linear, which implies a exponential rise in the minimum number of events needed. There are two effects that contribute to the shape of this plot. The first is that, at very low values of M, there is significant variation from one measurement to the next, resulting in a high standard deviation. This effect can be seen in Figure 4.8. The very slow decrease of standard deviation is also clearly demonstrated by a table of the same simulation results which are given in Table 4.2. Both effects are minimal around subcritical multiplication of 5 and a plot of M=5 is shown in Figure 4.9. At M=5 the standard deviation of the results shrinks and the mean reaches its true value rapidly as the number of events increases. For high values of M, there is a different effect. In those cases, the standard deviation is low even for a very few number of neutron events, however it takes a large number of recorded events for the calculated M-value to reach the correct value. This effect can be seen in Figure This result is significant, because, if not enough events are recorded in a physical measurement, then the resulting calculated value of M would be low, and repeated measurements would only confirm the erroneous value.

65 53 Eff =1.5% Eff =4.4% Eff =25% Minimum Number of Events M Eff =1.5% Eff =4.4% Eff =25% Minimum Number of Events M Figure 4.7: Thedependenceoftheminimumnumberofeventstogetthetruevalue of M on the input M-value. The results for a 239 Pu system with a 239 Pu starting source is on the top and the results for an 235 U with (α,n) starting neutron system is on the bottom.

66 M Number of Events Recorded M Number of Events Recorded Figure 4.8: The dependence of the number of neutron events recorded on the M value and standard deviation for M=1.5 and efficiency of 1.5% for a 239 Pu system with 240 Pu starting neutrons. The top plot illustrates the large standard deviation of M values at low multiplicity and the bottom plot illustrates the slow decrease of the standard deviation of M values as the number of events increases.

67 M Number of Events Recorded Figure 4.9: The dependence of the number of neutron events recorded on the M value and standard deviation for M=5 and efficiency of 4.4% for a 239 Pu system. For this case, as the number of recorded events increases, the standard deviation rapidly decreases and the average M value quickly reaches its true value M Number of Events Recorded Figure 4.10: The dependence of the number of neutron events recorded on the M value and standard deviation for M=20 and efficiency of 25% for a 239 Pu system with 240 Pustartingneutrons. Thestandarddeviationbarsarepresentinthefigure, however they are too small to be seen.

68 56 Table 4.2: The data table used to determine the minimum number of events for M=1.5 and efficiency of 1.5% for a 239 Pu system with 240 Pu starting neutrons. The column with the magnitude of the standard deviation is not relevant and is not shown. Input N Average Number of Starters Average M % Standard Deviation M Ratio

69 Summary I developed a point neutron-chain simulation and successfully benchmarked it to both measured Feynman histogram data and derived theoretical values. I also used the program to simulate an 1/M experiment and a Feynman beta curve. These results demonstrate the simulation can be to be used to replicate a variety of experiments. Our simulation runs much faster than a 2D or 3D simulation, while still providing accurate results. I also ran simulations in order to determine the minimum number of recorded events needed for the Hansen-Dowdy method to produce accurate results.

70 Chapter 5 Delayed Neutrons 5.1 Calculating α As discussed in Section 2.10, α is the ratio of Poisson starting neutrons to fission starting neutrons and is given by α = Mβ ν i (5.1) In order to test this equation I modified the core Chain algorithm given in Section to count and output the number of induced fissions. ChainFission[p_] := Module[{n}, FissonCounter = 0; leaked = 0; n = sneutrons[]; CurrentNeutrons = n; lenght = n; While[CurrentNeutrons > 0, ]; If[Random[] < p, FissonCounter = FissonCounter + 1; n = nneutrons[]; CurrentNeutrons = CurrentNeutrons- 1 + n; lenght = lenght+n, CurrentNeutrons = CurrentNeutrons- 1; leaked = leaked+1;] {leaked, FissonCounter}]

71 59 I wrote a simple algorithm to calculate the number of delayed neutrons produced per source neturon. For each source neutron, the algorithm generates random number and compares that to the value of β to determine if a delayed neutron is created. DelayedNeutronsProduced[n_, _] := Module[{}, delayed = 0; For[i = 0, i < n, i++, If[Random[] <, delayed++] ]; delayed] I then wrote CalculateAlpha[] that uses ChainFission[] and DelayedNeutronsProduced[] and calculates the α value. It works by running ChainFission[] a large number of times, and it calculates the total number of fissions from all the runs. It then runs DelayedNeutronsProduced[] for all those fissions and then takes the ratio of the number of delayed neutrons produced to the number of runs. In order to match the definition of α given in Section 2.10, CalculateAlpha[] always uses an (α, n) starting source. CalculateAlpha[multiplier_, p_, _, runs_] := Module[{}, Source["a-n", 0.0, multiplier]; totalfissions = Total[Table[ChainFission[p][[2]], {runs}]]; totaldelayedneutrons = DelayedNeutronsProduced[totalfissions, ]; totaldelayedneutrons/ runs 1.] For 235 U, I used a β = and a ν i = 2.41, which gives a β ν i of This calculated value compares to a simulated value of as shown in Figure 5.1. For 239 Pu, I used a β = and a ν i = which gives n β ν i of This calculated value compares to a simulated value of , as shown in Figure 5.2. Now that Equation 5.1 has been verified, I use Equation 5.1 every time I need to calculate α. Using Equation 5.1 instead of running a Monte Carlo process saves significant computational time.

72 Alpha M Figure 5.1: The dependence of M on α for a 235 U system. The linear fit to the line is α = M. 5.2 Simulating Delayed Neutrons As previously discussed in Chapter 1, delayed neutrons have two effects. The first is that they change the source ν distributions, as shown in Figure 5.3. The second is that they increase the source-neutron rate. In order to simulate the first effect, I modified the algorithm that selects the number of starting neutrons to randomly produce a single neutron, with the probability weighted by α. The αα = α 1+α expression in the code is used to properly normalize the probability to fall between 0 and 1. The algorithm is shown below. = 1+ ; sneutrons[] := Module[{i}, If[Random[] <, 1, i = 1; rand = Random[]; While[sFdata[[i, 2]] rand, i++]; i-1]];

73 Alpha M Figure 5.2: The dependence of M on α for a 239 Pu system. The linear fit to the line is α = M. Taking into account the increased source rate was done by multiplying the original source rate by (1+α). Note that this modification appears in the code as (1/1 + α) because the source-rate input is inverted. The results of the delayed neutron simulations are shown in Table 5.1 and Table 5.2. No significant effect of delayed neutrons was not observed up to a multiplication of 300. I was unable to simulate the cases using 235 U with (α,n) starting neutrons beyond M = 300 and the cases using 239 Pu with 240 Pu spontaneous fission starting neutrons beyond M = 200 because of insignificant memory on a Lawrence Livermore National Laboratory node, which has 250 GB of RAM. As discussed in Section 4.2, as one simulates higher multiplication values, the number of simulated neutrons needed also increases. In any meaningful manner, a multiplication of 200 or 300 is absurdly high, and simulating an multiplication even higher would have no realistic value. These simulations demonstrate that delayed neutrons do not have any effect on the measurement of subcritical systems for any reasonable value of M. The complete algorithm used to generate the results of these simulations is given at the end of this chapter.

74 62 a) b) c) d) Figure 5.3: Source ν distributions for 240 Pu with delayed neutron-contributions. The figures show neutrons fractions of a) 0, b) 0.1, c) 0.5 and d) 1 with effective ν values of a) 2.154, b) 1.95, c) 1.55 and d) Using Elevated β Values In order to determine what conditions would be necessary for delayed neutrons to have an effect, I re-ran the delayed neutron simulation with elevated β-values. I found that there is a noticeable effect with β-value of about 50% and a multiplication value of 50. The results of an experiment with a β-value of 50% is shown Table 5.1: The effect of delayed neutrons on a 235 U system with (α,n) starting neutrons. M L M T α M HD without DN M HD with DN

75 63 Table 5.2: The effect of delayed neutrons on a 239 Pu system with 240 Pu starting neutrons. M L M T α M HD without DN M HD with DN in Table 5.3. It is worth noting that multiplying systems using 235 U may be more sensitive to the effects of delayed neutrons than systems using 239 Pu. This effect could be, because the decrease of the effective source ν is partially offset by the increased effective source-rate. I modeled the multiplying system using 235 U with an (α,n) starting source and thus the source ν remained a constant equal to 1. The addition of delayed neutrons for the normal β value case resulted in lower multiplications, while the addition of delayed neutrons for the elevated β value case resulted in higher multiplications. This discrepancy may be caused by running an insufficient number of neutrons at these high multiplications and thus I do not consider this discrepancy significant. Table 5.3: The effect of a non-physical β-value of 50% on delayed neutrons. M L of 50 M L of U M HD without DN U M HD with DN Pu M HD without DN Pu M HD with DN

76 Summary In this chapter I present the modifications to Chains that enabled the simulation of the effects of delayed neutrons on a subcritical system. I verified the expression for the ratio of delayed neutrons to starting neutrons as presented in Equation I simulated subcritical systems with delayed neutrons up to a multiplication of 300 and found that delayed neutrons have no effect. Delayed-neutrons do have an effect with non-physical β-values. Systems with a spontaneous-fission starting source may be less sensitive to the effects of delayed neutrons.

77 65 Source["a-n", 0, "U235"] Mult = 10; Tg = 50; = 0; frac = 0.7; Repeates = 4; n = ; T = T0 = 289; = Sim = ; = ; FsSim = Mult-1 ; pp = Mult f -1 ; Print["Without Delayed Neutrons and n = ", n, " x ", Repeates, " = ", nrepeates] testdata = 0; MulipleChains = Parallelize[ Table[{FullChains["a-n", 0, "U235", pp,, Sim, FsSim, n], decaytime}, {Repeates}]]; i ListofLastDecays = Table MulipleChains[[j, 2]], {i, 1, Repeates}; j=1 ListofLastDecays = Insert[ListofLastDecays, 0, 1]; DividedChains = Table[ testdata = MulipleChains[[i, 1]] + ListofLastDecays[[i]], {i, 1, Repeates}]; CombinedChains = Flatten[DividedChains]; test = Feynman[CombinedChains, T0]; Hansen[pp, test] NotebookSave[EvaluationNotebook[]]; Print["With Delayed Neutrons and n = ", n, " x ", Repeates, " = ", nrepeates] testdata = 0; alpha = Mult frac/f; Print["Alpha = ", alpha] MulipleChains = Parallelize TableFullChains"a-n", alpha, "U235", pp,, Sim, decaytime, {Repeates}; FsSim (1+alpha), n, i ListofLastDecays = Table MulipleChains[[j, 2]], {i, 1, Repeates}; j=1 ListofLastDecays = Insert[ListofLastDecays, 0, 1]; DividedChains = Table[ testdata = MulipleChains[[i, 1]] + ListofLastDecays[[i]], {i, 1, Repeates}]; CombinedChains = Flatten[DividedChains]; test = Feynman[CombinedChains, T0]; Hansen[pp, test]

78 Chapter 6 Simulation of Dead-Time Dead-time in neutron measurements involves some added complications that are not present in other detector systems. One complication is that most neutron systemshavemorethanonedetector; and, insomesystems, everydetectorhasitsown electronics; and, in other systems, multiple detectors share a pre-amplifier/ amplifier/discriminator chain. Another complication is that, while these systems are not normally used in high count-rate environments, the nature of the fission chains leads to sudden bursts of neutrons that could cause dead-time in detectors at the precise moment when one would want all detectors to be active. Mathematically this dead-time can be modeled by a variable detector efficiency. In most neutron-detection systems, if two neutrons create ionizations in the same detector at the same time, it would result in a pulse that is twice as large as if one neutron was detected. In some systems, the two neutrons would be counted as one pulse, and in others the pulse could be higher than the upperlevel discriminator and thus not counted at all. For the dead-time simulations presented in this dissertation, two neutrons that enter one detector at the same time are counted as one. 6.1 Paralyzable and Non-paralyzable Detectors may be either paralyzable or non-paralyzable. In a non-paralyzable system, the detector is dead for a constant period of time after an event and then it is active again. Any additional events occurring in the detector during the dead-

79 67 time do not extend the dead-time. In a paralyzable system, the length of the dead-time is extended, when an additional event occurs in the detector during the dead-time [16]. Both cases of dead-time are simulated. 6.2 Simulation of Count-Rate Only Non-paralyzable The Mathematica code that I used for the non-paralyzable model is shown below. It starts by creating a list of n counts with a specified decay rate. This program repeats the same method used in Section After the list is created, the program takes each event and checks if the next event is greater than the deadtime. If it is, then it adds one to a counter, and if it is not, nothing happens. This algorithm works by keeping track of when the last event occurred and subtracting the i th event from the i th+1 event and checking if that value is less than the deadtime. The program then divides the total number of counts recorded by the time of the last event to obtain a count-rate, which becomes the output. NonParDet[decayrate_, deadtime_, n_] := Module{}, detected = 1; decaytime = 0; decaytime = Tabledecaytime = decaytime-log[random[]]decayrate -1, {n}; oldtime = decaytime[[1]]; For[i = 0, i < n-1, i++, newtime = decaytime[[i+2]]; If[newtime - oldtime > deadtime, oldtime = newtime; detected ++;] ]1.detected/decaytime[[n]] Paralyzable The Mathematica code that I used for the paralyzable model is shown below. It is the same for the non-paralyzable model with the exception that, if two counts occur within the dead-time, then the dead-time is extended. This extension is

80 68 done by re-setting the time of last event, if the i th+1 event happens during the dead-time. ParDet[decayrate_, deadtime_, n_] := Module{}, detected = 1; decaytime = 0; decaytime = Tabledecaytime = decaytime-log[random[]]decayrate -1, {n}; oldtime = decaytime[[1]]; For[i = 0, i < n-1, i++, newtime = decaytime[[i+2]]; If[newtime - oldtime > deadtime, oldtime = newtime; detected ++;, oldtime = newtime;] ]1.detected/decaytime[[n]] Results The results of the simulations are shown in Figure 6.2. For the case of no deadtime, the dependence of the counting rate was calculated using the paralyzable simulation with dead-time set to zero. The goal of this simulation is to recreate the dead-time effects of paralyzable and non-paralyzable detectors as shown by Knoll in Figure 6.1 [16]. Figure 6.2 shows those same effects and thus, I conclude that my method of simulating dead-time is accurate. 6.3 Simulation of Dead Time with Chains To use the dead-time simulation with Chains, it is necessary to make two modifications. The first is to take into account multiple detectors or groups of detectors, and the second is to change the output to list mode List Mode The algorithm for simulating list mode is very similar to the basic dead-time method shown in Section 6.2. The only difference is that instead of keeping track

81 69 Figure 6.1: Plot of paralyzable and non-paralyzable dead-time. n is the true interaction rate, m is the recorded count rate and τ is the system dead-time. This plot is Figure 4.8 in Knoll [16]. of the total counts, each event is saved to an array that becomes the output. The algorithm for the non-paralyzable case is shown below. ListDetNonPar[decaytimes_, deadtime_] := Module[{}, n = Length[decaytimes]; detectedtimes = ConstantArray[0, n]; j = 2; oldtime = decaytimes[[1]]; detectedtimes[[1]] = decaytimes[[1]]; For[i = 0, i < n-0, i++, newtime = decaytimes[[i+1]]; If[newtime - oldtime > deadtime, oldtime = newtime; detectedtimes[[j]] = newtime; j++;] ]; DeleteCases[detectedtimes, 0]]

82 70 Measured Count-Rate NoDeadTime Non-Paralyzable Paralyzable Real Count-Rate Figure 6.2: A count-rate only dead-time simulation without multiplication. This plot matches Figure 4.8 in Knoll [16] Multiple Detectors Most neutron-measurement systems use multiple detectors. In some systems every detector has its own electronics and in other systems, multiple detectors share a pre-amplifier/amplifier/discriminator chain. These options are illustrated in Figure 6.3. Multiple detectors are modeled by taking the list-mode output of Chains, as showninsection3.2.6,andtakingeachcountandrandomlyassigningitanumber1 to n, corresponding to the number of detectors in the system. If multiple detectors are grouped to one electronics chain, they are treated as one detector. For example, if a system has 20 detectors in groups of 4, then the number of detectors modeled would be 5. The efficiency is taken for the system as a whole. The counts are then divided into separate arrays for each detector and then each array is passed through the dead-time algorithm to remove counts. The algorithm that implements multiple detectors for the non-paralyzable case is shown below.

83 71 Figure 6.3: The left image shows 16 detectors connected to individual preamplifiers and the right image shows a possible method of grouping multiple detectors to one pre-amplifier. Illustration by Alissa Pendorf. DeadTimeNonPar[times_, deadtime_, NumberofDetectors_] := Module[{}, dividedcounts = Table[{RandomInteger[{1, NumberofDetectors}], times[[i]]}, {i, 1, Length[times]}]; dividedcounts2 = Table[Cases[dividedcounts, {i, _}][[2]], {i, 1, NumberofDetectors}]; dividedcounts3 = Table[ListDetNonPar[dividedcounts2[[i]], deadtime], {i, 1, NumberofDetectors}]; Sort[Flatten[dividedcounts3]]] 6.4 Dead-Time Runs and Results After the algorithm was written and tested, I ran Chains with the dead-time algorithms as shown below. For each case, I also calculated the percentage of events lost due to dead-time. Chains was run once for each multiplication value and for that M-value the same list-mode data were used for all subsequent analysis. For all cases, 240 Pu was used as the starting source and 239 Pu as the multiplying material. For all runs, I used a dead-time of 1 µs.