Neutron Induced Fission Fragment Angular Distributions and Momentum Transfer Measured with the Niffte Fission Time Projection Chamber

NEUTRON INDUCED FISSION FRAGMENT ANGULAR DISTRIBUTIONS AND MOMENTUM TRANSFER MEASURED WITH THE NIFFTE FISSION TIME PROJECTION CHAMBER

by David Hensle c Copyright by David Hensle, 2019

All Rights Reserved A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy (Applied Physics).

Golden, Colorado Date

Signed: David Hensle

Signed: Dr. Uwe Greife Thesis Advisor

Golden, Colorado Date

Signed: Dr. Uwe Greife Professor and Head Department of Physics

ii ABSTRACT

Nuclear fission is the process by which a large nucleus splits into two heavy fragments and is often induced by an incident neutron. Understanding the probability by which an incident neutron will cause fission, i.e. the neutron induced fission cross section, is an important input into fission applications such as nuclear energy and stockpile stewardship. The Neutron Induced Fission Fragment Tracking Experiment (NIFFTE) Collaboration built a fission time projection chamber (fissionTPC) to study the fission process in a novel way in the hopes of achieving unprecedented precision on fission cross section measurements. With the fissionTPC’s ability to do three-dimensional tracking of fission fragments and other ionizing radiation, systematics of previous cross section measurements using fission chambers can be further explored. Because the fissionTPC records a wealth of data for every fission event, other physics can be measured concurrently with the cross section. In particular, fission fragment angular distributions and the linear momentum transferred from the incident neutron to the target nucleus as a function of incident neutron energy from 130 keV to 250 MeV will be the focus of this work. Angular anisotropy values for 235U and 238U and neutron linear momentum transfer for 235U, 238U, and 239Pu will be presented.

iii TABLE OF CONTENTS

ABSTRACT ...... iii

LISTOFFIGURES ...... vii

LISTOFTABLES ...... xiv

ACKNOWLEDGMENTS ...... xv

CHAPTER 1 NUCLEAR FISSION BACKGROUND ...... 1

1.1 FissionBasics ...... 1

1.2 NeutronInducedFission ...... 7

1.2.1 NIFFTECrossSectionFormulation ...... 10

1.3 FissionFragmentAngularAnisotropy ...... 12

1.4 NeutronLinearMomentumTransfer ...... 14

1.5 Motivation for Precise Fission Measurements ...... 16

CHAPTER 2 NIFFTE TIME PROJECTION CHAMBER ...... 18

2.1 Fission Time Projection Chamber Hardware and Operation ...... 18

2.1.1 ElectronicsandTriggering ...... 20

2.1.2 FissionTPCFillGas ...... 24

2.2 ActinideTargets ...... 28

2.3 NeutronSourceandTimeofFlightMeasurement ...... 32

2.3.1 CarbonFilterData...... 34

2.3.2 NeutronBeamWraparound ...... 35

CHAPTER 3 TPC DATA RECONSTRUCTION AND CORRECTIONS ...... 37

iv 3.1 DataReconstruction: SignalstoTracks ...... 37

3.2 NeutronTimeofFlightReconstruction ...... 42

3.3 CorrectionstoData...... 43

3.3.1 Californium-252Measurement ...... 44

3.3.2 PadSaturation ...... 45

3.3.3 ElectronDriftSpeed ...... 50

3.3.4 ElectronDiﬀusionAnisotropy ...... 52

3.4 CorrectionParametersAppliedtoTargets ...... 59

CHAPTER 4 NEUTRON LINEAR MOMENTUM TRANSFER ...... 61

4.1 Measuring Fragment Folding Angle Distributions ...... 62

4.2 SimulationofFragmentOpeningAngles ...... 64

4.3 LinearMomentumTransferError ...... 69

4.4 Linear Momentum Transfer Results and Discussion ...... 75

CHAPTER 5 FISSION FRAGMENT ANGULAR ANISOTROPY ...... 80

5.1 MeasuredAnisotropy...... 80

5.2 UncertaintyBudget...... 84

5.3 MultipleAnisotropyMeasurements ...... 92

5.4 MeasuredAnisotropyResults ...... 96

CHAPTER 6 SUMMARY AND OUTLOOK ...... 102

REFERENCESCITED ...... 106

APPENDIX A CONVERSION BETWEEN LAB FRAME AND CENTER OF MASSFRAME ...... 117

APPENDIX B DERIVATION OF THE ELECTRON DIFFUSION ANISOTROPY CORRECTION ...... 120

v APPENDIXCTABULATEDDATA ...... 124

vi LIST OF FIGURES

Figure 1.1 A cartoon depiction of neutron induced ﬁssion of 235U...... 2

Figure 1.2 A visual representation of each of the terms in 1.1 for the semi-empirical mass formula describing the binding energy associated withtheliquiddropmodel...... 3

Figure 1.3 (a) Potential energy curves as a function of L2 deformation (minimized with respect to the L4 component) for diﬀerent nuclei calculated by the liquid drop model (dashed curves) and the liquid drop model with shell corrections added (solid curves). Figure from . (b) Nuclear deformation described by the second and fourth order Legendre polynomial components,from...... 5

Figure 1.4 A geometrical interpretation of the partial cross sections for thermal neutrons incident on a 235Utarget...... 8

Figure 1.5 Fission fragment mass yields vary as a function of incident neutron energy where symmetric ﬁssion is more preferred at high incident neutronenergies.Figurefrom...... 9

Figure 1.6 Depiction of the total angular momentum J and projection M and K onto the beam axis and symmetry axis, respectively. From these quantum numbers, the angular distribution of the ﬁssion fragments can bepredicted...... 12

Figure 2.1 Schematic representation of the ﬁssion time projectionchamber...... 19

Figure 2.2 (a) A schematic representation of an alpha particle’s ionization charge cloud being drifted to the anode and ampliﬁed by the MICROMEGAS. (b) A projection of a charge cloud onto the hexagonal readout pads on theanodes...... 20

Figure 2.3 (a) A picture of one of the two highly segmented anodes. (b) The ﬁssionTPC fully assembled with all of the readout electronics attached. The drift chamber is 15 cm in diameter and the central cathode is 5.4 cm fromeachanode...... 21

vii Figure 2.4 (Left) Picture of an EtherDAQ card. (Center) A schematic layout of an EtherDAQ card containing the analog amplifiers on the bottom where the input signal enters, and the digital electronics and fiber readout on top. (Right) Work flow diagram of the FPGA firmware that controls thedataflowfortriggeringandprocessing...... 22

Figure 2.5 A representation of how low energy Argon recoils can produce more ionization density in the electron avalanche than a ﬁssion fragment and causesparking.Source...... 27

Figure 2.6 (a) A picture of the Plutonium side of the 239Pu/235U thick backed aluminum target before insertion into the ﬁssionTPC. (b)Start vertices for alpha particles emitted from a thick target with 235U on each side. Contamination of small quantities of 239Pu can be seen on the cathode. . 29

Figure 2.7 List of targets from which neutron induced ﬁssion data was taken. Actinides include 235U (blue), 238U (purple), and 239Pu(green). Aluminum target holders are 2cm in diameter and actinide deposits are drawnroughlytoscale...... 31

Figure 2.8 On overview of the Los Alamos Neutron Science Center. The location of the 90L WNR beamline where the TPC is operated is circled in red. . 32

Figure 2.9 Neutron time of flight spectrum for fission from a U235 target. The insert shows the photofission peak and fit to determine a time of flight resolution. Neutron wraparound is shown as the red line...... 33

Figure 2.10 (a) Inelastic neutron scattering cross section for Carbon-12 showing strong resonance feature at 2.078 MeV, from ENDF/B-VIII . (b) Neutron time of ﬂight spectrum as measured by the ﬁssionTPC showing location of the Carbon-12 resonance at around 400 ns...... 34

Figure 2.11 (a) The end of the last micropulse is fit with a sum of two decaying exponentials. This fit is then “folded” to get a prediction for the number of wraparound neutrons. (b) Percent of fission events caused by wraparound neutrons for a 235U target as a function of neutron energy calculated by taking the ratio of counts above and below the red wraparoundlineinFigure2.9...... 36

Figure 3.1 Signals from an alpha particle and the subsequent diﬀerentiation used in the digit forming process. Considering all of the pads have signals occurring at almost the exact same time, this set of signals corresponds to an alpha particle that was emitted very parallel to the anode plane, equating to a polar angle of cos(θ)veryclosetozero...... 38

viii Figure 3.2 A typical fission event from a thin carbon backed target. Along with the back to back fission fragments, small tracks from inelastic scattering of neutrons off protons in the gas are also visible. The red lines through the digits are the assigned tracks. Color scale denotes the ADC (charge)valueofeachdigit...... 39

Figure 3.3 Track length vs total track energy is a commonly used parameter space toperformparticleidentiﬁcation...... 41

Figure 3.4 Raw high speed cathode signal (blue) induced from a fission event and its trapezoidal filtered counterpart (red). Where a straight line fit to the rising edge of the filtered signal intersects with 0 is taken as the timeofflightfortheevent...... 43

Figure 3.5 (a) Fission fragment start vertices and (b) Length vs ADC from the 252Cf source taken at the 210V MICROGMEGAS setting ...... 44

Figure 3.6 (a) Saturated signal readouts from the pads for a ﬁssion fragment directed towards the anode. (b) A cut on the end portion of a charge cloud projected on the pads showing the hole created by the saturated signals...... 46

Figure 3.7 ADC vs cos θ for 252Cf fragment events at the four diﬀerent gain settings showing how fragments emitted towards the pads lose energy due to signal saturation and how gain increases with an increase in MICROMEGASvoltage...... 47

Figure 3.8 252Cf spontaneous alpha particle start positions as measured in the ﬁssionTPC at the four diﬀerent gain settings. A dead card is visible at roughly (-3,-3) and a small amount of contamination can be seen on the cathodeat210Vand230V...... 48

Figure 3.9 ADC vs cos(θ) (a) before and (b)after applying the attempted saturationﬁxtothe210Vdata...... 49

Figure 3.10 (a) Percent of initial ADC added when correcting for the saturation as a function of initial cos θ. (b) The change in cos(θ) after ﬁlling adding themissingcharge...... 50

Figure 3.11 Changing the electron drift speed in the reconstruction changes the measuredpolarangleofthetrack...... 51

ix Figure 3.12 (a) An example fit to the spontaneous alpha distribution from a 238U source after being flattened by changing the drift speed values. (b) The slope in from fits of (a) as a function of the electron drift speed as well as showing sensitivity to different fit ranges...... 52

Figure 3.13 Alpha and fragment polar angle distributions from a 252Cf source taken with the MICROMEGAS voltage set to (a) 200V and (b) 210V after setting the drift speed by ﬂattening the alpha distribution...... 53

Figure 3.14 An example Bragg curve showing the charge density along the a ﬁssion fragment track. The curve is approximated by a Gaussian start and a linear end using the start, end, and peak of the ionization proﬁle. . . . . 55

Figure 3.15 (a) A 2-D projection of the charge clouds of many ﬁssion fragments from the same angular bin overlaid and (b) projected along y axis. The average standard deviation for fragments in this angular bin is taken as the standard deviation of the (b) and, depending on how the charge 2 2 cloud was projected, used to extract σr0 or σz0 for the diﬀusion correction...... 56

Figure 3.16 An example electron diﬀusion anisotropy correction for fragments, as calculatedby3.5...... 58

Figure 3.17 Alpha and fragment polar angle distributions from a 252Cf source taken with the MICROMEGAS voltage set to (a) 200V and (b) 210V after applying the electron diﬀusion anisotropy and drift speed corrections. . . 58

Figure 4.1 A cartoon depiction of kinematic boost to the fission fragments in the lab frame. In the center of mass frame, the opening angle of the fission fragments are necessarily 180 degrees due to the fission process being essentiallyatwobodydecay...... 62

Figure 4.2 An example opening angle distribution ﬁt with a Gaussian to get the mean value. The dashed red Gaussian is the wraparound that gets subtractedfromthedistribution...... 63

Figure 4.3 Measured mean opening angle as a function of incident neutron energy withwraparoundcorrectionapplied...... 64

Figure 4.4 Overview of how one ﬁssion event is simulated. See text for more details...... 65

Figure 4.5 Simulated ﬁssion events and their folding angle as a function of momentum transferred for a particular incident neutron energybin. . . . 68

x Figure 4.6 (a) The diﬀerence between wraparound variations for the 239Pu target as well as what the linear momentum transfer would look like without the wraparound correction. The dashed line represents full momentum transfer...... 71

Figure 4.7 Effects of changing the electron diffusion by ±10% to the 235U from the 238U/235U target. Looking at the fractional momentum transfer better showcases the differences at low incident neutron energies. The dashed linedenotesfullmomentumtransfer...... 72

Figure 4.8 Error budgets for the linear momentum transfer measurements from the 238U/235U target (top) and the 239Pu/235U target. See the text for the explanationofthemethodology...... 73

Figure 4.9 Fraction of the total incident linear momentum transferred to the ﬁssion fragments as a function of incident neutron energy. The weighted average for each actinide set to one for points below 2 MeV by adjusting the drift speed diﬀerence between the upstream and downstreamvolumes...... 76

Figure 4.10 Momentum transfer plotted against other published data, including alpha induced fission , deuteron induced fission , and proton induced fission . To the best of our knowledge, this is the first measurement of linear momentum transfer from neutron induced fission...... 78

Figure 5.1 Track length vs the maximum value of the Bragg curve from a 235U downstream data set containing the recoil ions stripes on the left and the double humped ﬁssion distribution on the right. The dashed vertical line is the cut used to separate the ﬁssion fragments from the recoilions. Notethelogscaleinbinintensity...... 81

Figure 5.2 Example polar angle distribution for a particular incident neutron energy bin from the a 235U downstream target after all the corrections are made. The red line shows the ﬁt to the data and the two ranges highlighted by the vertical lines show the intervals for the min and max ﬁtrangesemployedintheerroranalysis...... 83

Figure 5.3 Variations in the 235U/235U downstream anisotropy due to changes in fragmentselection...... 86

Figure 5.4 Variations in the 235U/235U downstream anisotropy due to changes in themaximumﬁtrange...... 87

xi Figure 5.5 Variations in the 235U/235U downstream anisotropy due to error in the linear momentum transfer measurement as well as lab frame anisotropy and anisotropy calculated from the assumption of full momentum transfer...... 88

Figure 5.6 Variations in the 235U/235U downstream anisotropy due to error in the electron diﬀusion correction as well as anisotropy measured without applyingthecorrection...... 89

Figure 5.7 Variations in the 235U/235U downstream anisotropy due to ﬁtting with only a second order or with second and fourth order Legendre polynomials...... 90

Figure 5.8 Uncertainty budgets from anisotropy measurement from target: 235Pu/235U thick (1) upstream and (2) downstream, 235U/235U thick (3) upstream and (4) downstream, 235U/238U thin downstream (5) 238U and (6)235U, and 239Pu/238U thick (7) upstream and (8) downstream. See Figure 2.7 for target list. 238U uncertainties blow up below 1.2 MeV due beingbelowtheﬁssionthreshold...... 91

Figure 5.9 Anisotropy results from each target for (a)235U and (b)238U...... 93

Figure 5.10 Polar angle distributions and fourth order ﬁts to a variety of neutron energy bins. Top row shows a comparison of the upstream (blue) and downstream (green) 235U measurements of the 235U/235U thick target. Middle row compares the downstream 235U measurements from the thin backed 235U/238U target (blue) and the thick backed 239Pu/235U target (green). Bottom row compares the upstream (blue) and downstream (green) 238U measurements from the thick backed 235Pu/238Utarget. . . . 94

Figure 5.11 Second and fourth order Legendre components compared for (a) 235U anisotropy from the thick 235U/235U target and (b) 238U anisotropy from the thick 239Pu/238U target where the blue and green squares are the second order Legendre component for upstream and downstream measurements, and the cyan and black circles are the fourth order Legendre components for upstream and downstream measurements, respectively...... 95

Figure 5.12 235U ﬁssion fragment anisotropy plotted with other published data from EXFOR...... 97

Figure 5.13 238U ﬁssion fragment anisotropy plotted with other published data from EXFOR...... 98

xii Figure A.1 Diagram depicting the transformation between the lab frame and the center of mass frame where the incident neutron is traveling in the positive x-direction and up and down refer to the upstream and downstreamfragments,respectively...... 117

xiii LIST OF TABLES

Table 2.1 Conﬁguration and results of stability against sparking...... 26

Table 3.1 : Parameter values used in the corrections for each target, actinide, and volume...... 60

Table C.1 Neutron linear momentum transferred to 238U...... 124

Table C.2 Neutron linear momentum transferred to 235U in the thin-backed target containing 235U/238U...... 125

Table C.3 Neutron linear momentum transferred to 235U in the thin-backed target containing 235U/239Pu...... 126

Table C.4 Neutron linear momentum transferred to 239Pu...... 127

Table C.5 235U Anisotropyresults...... 128

Table C.6 238U Anisotropyresults...... 129

xiv ACKNOWLEDGMENTS

No one who achieves success does so without the help of others. A great many people have contributed to my success in this project, and if I forget to thank you here, I sincerely apologize and you should know that your contributions are greatly appreciated. I would like to start with thanking all of my primary education teachers that made science fun and interesting at a young age, from talking about electricity in third grade to the AP physics classes in high school. Without the accumulation of all your efforts, I would never have gained a passion for physics. A huge thank you goes to the fantastic professors at Gonzaga University, including Allan Greer, Jeff Bierman, Eric Kincanon, Erik Aver, and Christopher LaSota. I received a top- class physics education thanks to your insightful lectures and willingness to listen to my confused mumblings during office hours. Without the strong foundation formed studying at Gonzaga, I would not be where I am today. Go Zags! My excellent physics education continued into my graduate studies with Colorado School of Mines professors Lincoln Carr, Susanta Sarkar, Zhigang Wu, Kyle Leach, Alex Flournoy, Fred Sarazin, Alan Sellinger and especially David Wood who taught a solid third of my graduate classes while never failing to keep class interesting with his passion and personality. I learned much more physics than I thought possible in a year and a half of classes, and that is a testament to your teaching dedication. Another massive thanks goes to Barbara Shellenberger and Lee Anna Grauel – without you I am pretty sure the physics department at Mines would simply cease to exist. To all of my fellow graduate students, I would not be half the physicist that I am today without you. While our professors provided the physics curriculum, most of my physics and problem-solving skills are a direct result of our collaborative efforts. The list includes Alex Wilhelm, Dan Higgins, Marie McLain, Johannes Eser, Matt Jones, Eric Jones, Felix

xv Therrian, Alex Lidiak, and many more bright Mines students. A special thanks goes to Kirsten Blagg and Kevin Merenda for your engaging research discussions, physical insight, and general support, and David Rodriquez Perez for partnering in my physics journey for the last eight years(!) dating back to freshman physics at Gonzaga. Thanks for making physics fun! The NIFFTE collaboration has really made this research possible, and I would like to thank each and every member that has influenced this project, for without you, this thesis does not exist. In particular, I would like to give extra thanks to Verena Kleinrath who did a lot of work on the anisotropy before I even started graduate school, Michael Mendenhall for your brilliant work on tracking and electron diffusion, and Luke Snyder for holding my hand through the monster that is the fissionTPC hardware and always lending an ear when I needed an expert opinion. Jeremy Bundgaard deserves a special round of applause for taking me under his wing and teaching me an immeasurable amount about the fissionTPC and the coding nuances required to analyze the data. Your assistance literally saved me years on the learning curve and I am eternally grateful. To my adviser Uwe Greife, thank you for your physical insight, wise perspective, trust, and inerrant combination of direction and freedom. You were the perfect fit for me. And last, but certainty not least, thank you to my parents, Bob and Val Hensle, and partner in life, Emily Zupsic. This accomplishment is as much yours as it is mine. Without your unconditional love and support, I would not be the person I am today and this project would never have even begun. I owe this all to you.

xvi CHAPTER 1 NUCLEAR FISSION BACKGROUND

Nuclear fission is an extremely complex process that has been studied extensively since its discovery roughly 80 years ago. The purpose of this chapter is to provide sufficient background on the fission process to understand the experiments and analyses presented in the following chapters, not to provide an exhaustive description. Nuclear fission is a strongly coupled, many-body quantum mechanical process with many different observables, and as such, is a particularly difficult phenomenon to describe accurately from a first principles approach. Presented here are the general traits of fission and a few in-depth items that are important for understanding the topics of this thesis – neutron linear momentum transfer and fission fragment angular distributions.

1.1 Fission Basics

Nuclear fission is the process by which a nucleus splits into two daughter nuclei of comparable size called fragments. The amount of energy release in a typical fission event is on the order of 200 MeV, with roughly 80% going into the kinetic energy of the two fission fragments and the remaining 20% spent on the emission of gamma rays and neutrons from the excited states of the fission fragments after fission. For example, thermal neutron induced fission of 235U produces anywhere from zero to six prompt gamma rays from each fragment per fission event [1] and an average of about 2.4 prompt neutrons per fission event [2]. A cartoon depiction of a neutron induced fission event can be seen in Figure 1.1. In 1934, Enrico Fermi published a paper [4] in which he and his co-workers bombarded natural uranium with neutrons to create transuranic elements through neutron capture and subsequent β emission. These transuranic elements were of great interest to chemists, and Hahn and Strassman [5] conducted thorough experiments to show that isotopes of barium and lanthanum were made in the process of neutron bombardment on uranium. Meitner and

1 Figure 1.1: A cartoon depiction of neutron induced ﬁssion of 235U [3].

Frisch (1939) then realized that if a nucleus were to split into two comparable fragments, which they termed nuclear fission in analogy to the process of division of biological cells, the Coulombic repulsion between the two fragments would result in roughly 200 MeV of total kinetic energy [6]. This energy estimate was experimentally confirmed by Frisch [7]. Then, also in 1939, Bohr and Wheeler published a comprehensive paper treating the nucleus as a liquid drop [8] which is still taught today in introductory nuclear physics classes. (For some quick, interesting historical context of the development of the liquid drop model, see [9].) From the theoretical framework of the liquid drop model, the Bethe-Weizsäcker semi- empirical mass formula was developed which describes the binding energy of the drop as [10]

−3/4 apA Z even, N even Z(Z − 1) (A − 2Z)2 B(Z,A)= a A−a A2/3−a −a + 0, even, odd or odd, even v s c A1/3 sym A  −3/4 −apA Z odd, N odd (1.1)  where each one of these terms are easily interpreted in the context of Figure 1.2. The ﬁrst term is the volume term which is simply proportional to the volume of the nucleus. Due to the short range of the strong force extending only to each individual nucleon’s neighbors, this volume term is proportional to A and not A(A−1), which would be true if each nucleon

2 interacted with every other nucleon. However, the volume term does not take into account that the nucleons on the edge of the nucleus do not have as many neighbors. As such, a surface term must be subtracted that is proportional to the surface area, or A2/3. The binding energy must also include the repulsive force of the positively charged protons pushing against each other. Since the Coulomb range is large enough for each proton to interact with every other proton in the nucleus and the charge is expected to be distributed uniformly throughout the nucleus, the term is then proportional to Z(Z − 1)/A1/3.

Figure 1.2: A visual representation of each of the terms in 1.1 for the semi-empirical mass formula describing the binding energy associated with the liquid drop model.

At this point with the volume, surface, and Coulomb terms included, the semi-empirical mass formula would favor nuclei with no Coulomb repulsion, meaning nuclei made exclusively of neutrons. However, most nuclei are observed with Z ≈ A/2, so a term to take this into account is needed. This asymmetry term must favor Z = A/2 for smaller nuclei, but decrease in importance for larger nuclei which need more neutrons to overcome the increased Coulomb repulsion. Therefore, the asymmetry term is chosen to be proportional to (A − 2Z)2/A. The final term is to account for the observation that nucleons tend to couple pairwise for stable configurations. When there is an odd number of nucleons, this term is not applicable, but there are only four nuclei with odd N and Z (2H, 6Li, 10B, and 14N) whereas there are 167 with even N and Z. This pairing energy is typically expressed seen in Figure 1.2. Combining all of these terms together with empirically determined coefficients provides an estimate for the total amount of binding energy contained within a nucleus of given A and Z.

3 Nuclear ﬁssion is often thought of as progressive shape distortions of the original spherical nucleus into a prolate spheroid that has an eventual “neck pinch” to progress to the complete split between the two diﬀerent halves, as seen in Figure 1.3. In the context of the liquid drop model, potential energy changes based on the deformation of the shape of the nucleus can be discussed [11]. Consider the small deformation of a sphere where the radius can be described as

R(θ)= R0(1 + α2L2(cosθ)) (1.2)

where L2 is the second Legendre polynomial. The surface and Coulomb energy terms are then changed by 2 1 ∆E = α2E0 and ∆E = − α2E0 (1.3) s 5 2 s c 5 2 c meaning that the nucleus is stable against small distortions when the decrease in Coulomb energy is less than the increase in the surface energy. In other words, the drop will become unstable when

0 2 Ec Z 2as 0 or when ≥ ≈ 49 (1.4) 2Es A ac which says that nuclei with approximately Z ≥ 114 and A ≥ 265 should be above the spontaneous fission barrier and decay within the timescale of a single nuclear vibrational period. This simple model has some experimental validation in that most unstable nuclei beyond thorium decay by spontaneous fission and their mean life times decrease exponentially with increasing Z2/A [12]. In order to better describe the nuclear deformation, a fourth order Legendre polynomial is added to describe the ”pinching” of the neck of the liquid drop. Scission is defined as the act of the neck snapping and disconnecting the two fission fragment. By plotting the potential energy change as a function of shape deformation, the fission barrier can be calculated which corresponds to the maximum amount of potential energy to overcome to fission, as seen in Figure 1.3.

4 (a) Fission Barrier (b) Length vs ADC

Figure 1.3: (a) Potential energy curves as a function of L2 deformation (minimized with respect to the L4 component) for diﬀerent nuclei calculated by the liquid drop model (dashed curves) and the liquid drop model with shell corrections added (solid curves). Figure from [13]. (b) Nuclear deformation described by the second and fourth order Legendre polynomial components, from [14].

5 While the liquid drop model provides an intuitive, qualitative description and reasonable accuracy of the nuclear binding energy over the entire nuclear chart, it fails in accounting for regions of stability associated with specific numbers of N and Z. To describe these “magic numbers,” a single particle or shell model was developed which treats each individual nuclei as being in a spherically symmetric potential created by the collective interaction of all of the nucleons in the nucleus [15]. Included in this model is a component dependent on the spin and orbital angular momentum of the nucleon [16] and is very analogous to the spin-orbit coupling seen in atomic physics. From this inclusion of the nuclear spin-orbit coupling, it was found that for certain numbers of protons and neutrons, orbital “shells” would close and produce a very stable nucleus, thus explaining the “magic numbers.” This same phenomenon describes how electron orbital shells close in atomic nuclei and explain why noble gases have tightly bound electrons, for example. (For her description of the nuclear shell model, Maria Goeppert Mayer is one of three women to receive a Nobel prize in physics [17].) Even though the shell model successfully describes many nuclear phenomena, attempts to describe the fission barrier from single particle effects have been unsuccessful since values of the fission barrier are approximately 10−3 times smaller than the total amount of nuclear energy for fissionable nuclei [18]. Thus, Strutinsky proposed a model where single particle effects are treated as small corrections to the liquid drop model deformation [19], providing better estimates for fission barrier heights and predicting multiple humped barriers, important for isomeric fission [11]. The differences in fission barrier height and shape from the liquid drop model compared to the liquid drop model with shell corrections can be seen in Figure 1.3. Modern nuclear fission theory relies heavily on computation to produce predictions for fission observables. In principle, any nuclear fission theory will start with a set of collective variables to describe the nucleus, compute the potential energy landscape as a function of those variables, and then compute the dynamical evolution through that potential energy space. There are two primary models that modern theorists are using to do to this:

6 macroscopic-microscopic models, and nuclear energy density functional theory [20]. The macroscopic-microscopic model is the modern variant of the liquid drop picture with shell model effects where the nucleus is described by single particle orbitals and collective defor- mations, and is typically evolved in time by Langevin equations. This is in contrast to time dependent nuclear energy density functional theory which attempts to describe the entire quantum mechanical system by using a simplified Hamiltonian, and evolving the quantum mechanical system through time in a way that conserves the energy of the system. Time dependent nuclear density functional theory requires very high computational cost and is dependent on the many simplifying assumptions needed to produce a tractable solution from strongly-coupled, many body quantum mechanics. Another approach to modern theoretical fission observables is to use semi-empirical models fit to experimental data like fission product yields and neutron emission to make predictions for fission from different actinides and energy states. See, for example the General Description of Fission (GEF) [21] and the Fission Reaction Event Yield Algorithm (FREYA) [22] codes. In short, modern theoretical advances in fission are progressing despite dealing with an extremely complicated problem, albeit at a slow pace.

1.2 Neutron Induced Fission

With fission barrier heights of several MeV for the heavy actinides, spontaneous fission is often less probable than spontaneous emission of an alpha particle. Therefore, fission is typically induced by supplying the nucleus with enough energy that it can get over the fission barrier. The most common mechanism to induce fission is to supply a neutron with some amount of kinetic energy to the target nucleus such that it overcomes the fission barrier and splits, a process called neutron induced fission. However, upon absorption of a neutron, the target nucleus is also energetically able to de-excite through other processes like neutron and gamma emission and not undergo fission. The probability that a target actinide will absorb a neutron and then undergo fission is called the neutron induced fission cross section, expressed in units of area, and is usually reported in barns = 10−28 m2 = 100fm2.

7 Figure 1.4: A geometrical interpretation of the partial cross sections for thermal neutrons incident on a 235U target [23].

8 Figure 1.4 shows a geometrical interpretation of a cross section for thermal neutrons on a 235U target. Notice that the “actual size” of the target nucleus is very small compared to the total cross section for interaction. In this context, the capture portion of the cross section refers to the process by which a neutron is absorbed by the target nuclei, but the compound nucleus decays via a different process than fission. The values for both the total and partial cross sections are dependent on both the incident neutron energy and the target nuclei. Other observables besides the cross section can also vary quite drastically as a function of incident neutron energy due to the fact that the target nucleus will have different excitation energies from which to fission. A few examples are the fission fragment product yield (shown in Figure 1.5), neutron and gamma ray emission, fragment angular distributions (measured in 5) and the neutron’s linear momentum transfer to the target nucleus (measured in 4).

Figure 1.5: Fission fragment mass yields vary as a function of incident neutron energy where symmetric ﬁssion is more preferred at high incident neutron energies. Figure from [24].

Some actinides, like 238U, need additional kinetic energy carried by the incident neutron to get over the ﬁssion barrier. After the 238U absorbs an incident thermal neutron, a 239U nucleus is formed, which has a ﬁssion barrier of 6.2 MeV [11], but only receives about 5 MeV worth of excitation energy from the addition of a slow neutron, meaning 238U needs

9 the incident neutron to transfer roughly an additional 1.2 MeV worth of kinetic energy in order to fission [25]. 235U and 239Pu, the other two actinides important for this work, have no neutron energy fission threshold, so even upon the absorption of a thermal neutron, the resultant 236U and 240Pu have excitation energy above their respective fission barriers.

1.2.1 NIFFTE Cross Section Formulation

Consider the reaction A(a, b)B, which is stating that an a particle interacts with A to form reaction products b and B. The cross section for this reaction, in its simplest form, can be thought of as [26]

# of particles b emitted σ = (# of incident a particles/unit area)(# of target nuclei A within the beam area) (1.5) The probability that particle b is emitted in a particular polar θ and azimuthal angle φ is called the differential cross section dσ/dΩ. While spontaneous emission of radiation is emitted isotropically in the rest frame of the decaying nucleus due to the lack of a predefined axis, induced reactions are rarely fully isotropic. Relating the differential cross section to the total reaction cross section is simply done via

4π σ = (dσ/dΩ)dΩ (1.6) Z0 Looking at this simplistic picture of the cross section formulation, it becomes clear that changes in the angular distribution of the reaction products directly impact the cross section. In the context of this thesis and the NIFFTE collaboration, whose primary goal is to measure fission cross sections, this means that the angular anisotropy of the fission fragments due to the quantum mechanical state of the fissioning nucleus and the kinematic focusing of the fragments from the transfer of momentum from the incident neutron to the target nucleus must both be taken into account for a proper cross section measurement as these both change the measured angular distribution.

10 For neutron induced fission, measuring cross sections is challenging because the detection of neutrons is very difficult, so any measurement of the incident number of neutrons to put into the cross section formulation has very large errors. In order to remove this problem, neutron induced reaction cross sections are often done in ratio to a standard, or known, cross section by placing two different target actinides back-to-back such that the measurement of the cross section ratio cancels out the number of incident particles. The NIFFTE collaboration has developed a more experimentally practical formulation of the cross section ratio

σ (E) N (x,y) · Φ (E; x,y) w (E) ǫ (E) C (E) a = κ · XY b b · b · b · a (1.7) σ (E) att N (x,y) · Φ (E; x,y) w (E ǫ (E) C (E) b PXY a a a a b where C is the number ofP detected fission events, ǫ is the detection efficiency for fission events, N is the number of target atoms, Φ is the neutron beam flux, w is the detector

live-time, and κatt is the attenuation of the beam through the thick target backing. Fission chambers have been the preferred method of making these cross section measurements in the past and consist of, in their simplest form, two separated gas volumes with a central cathode for neutron time of flight measurements and an anode on each side to collect the ionized electrons from the gas and detect fission events [27]. The fissionTPC has an advantage over conventional fission chambers to explore more of the systematic errors associated with cross section measurements when compared to fission chambers because of its ability to do three-dimension particle tracking [28]. With the unique ability to do vertexing of ionizing radiation, i.e. find the start positions, the correction for the spatial non-uniformities between the actinide target and the neutron flux can be explored. Measuring the efficiency with the fissionTPC is also much more thorough due to the wealth of information from which to explore and fit [29]. Even counting the number of fission events is more advanced due to the fissionTPC’s ability to perform more robust particle identification. With all of these advantages, the NIFFTE collaboration is hoping to measure the 239Pu/235U cross section ratio to unprecedented precision while also providing valuable insight into the systematics

11 of previous cross section measurements.

1.3 Fission Fragment Angular Anisotropy

When an incident neutron is absorbed, the compound nucleus is put into a particular angular momentum state. This angular momentum state can be described by three quantum numbers: the total angular momentum J, and the projections M and K onto the neutron beam axis and the ﬁssion symmetry axis, respectively, as depicted in Figure 1.6.

Figure 1.6: Depiction of the total angular momentum J and projection M and K onto the beam axis and symmetry axis, respectively. From these quantum numbers, the angular distribution of the ﬁssion fragments can be predicted [30].

Treating the nucleus as a spinning top, the angular part of the Schr¨odinger equation reads

h¯2 1 ∂ ∂Ψ (cosθ ∂ − ∂ )2Ψ h¯2 ∂2Ψ sinθ + ∂χ ∂φ + + EΨ=0 (1.8) 2J sinθ ∂θ ∂θ sin2θ 2J ∂χ2 ⊥ " # k where θ is the angle between the nuclear symmetry axis and the space-fixed (beam) axis, φ is the azimuthal angle around the space-fixed axis, χ is the angle around the nuclear symmetry axis, and Jk and J⊥ are moments of inertia with respect to the symmetry axis. This equation was actually solved before the discovery of fission [31] as,

2J +1 iMχ iMφ J Ψ= 2 e e dM,K (θ) (1.9) r 8π

12 J where the dM,K (θ) functions are deﬁned by [32]

J dM,K (θ)= (J + M)!(J − M)!(J + K)!(J − K)! (−1)n[sin(θ/2)]K−M+2n[cos(θ/2)]2J−K+M−2n p × (1.10) (J − K − n)!(J + M − n)!(n + K − M)!n! n X and n = 0,1,2,3... and contains all terms for which there are no negative values in the denominator of the sum for any of the terms in parenthesis. Bohr first suggested that the nucleus is actually thermodynamically cold due to most of the excitation energy being stored in the deformation of the fissioning nucleus [33], opening the door to describing the fissioning nucleus through only one or a few channels. This idea developed into the transition state model used to describe fission fragment angular distributions from nuclei very close to their fission barrier. The transition state model assumes that the fission fragments are emitted along the symmetry axis and that the angular momentum numbers J, M, and K are conserved through the fission process (implying they are “good” quantum numbers). With these assumptions, the probability of emitting a fragment at polar angle θ from an angular momentum state described by J, M, and K is given by [11] 2J +1 P J (θ)= |dJ (θ)|22πR2sinθdθ (1.11) M,K 4πR2 M,K

J J where the angular distribution PM,K (θ) is then obtained by dividing PM,K by sinθ to get 2J +1 W J (θ)= |dJ (θ)|2 (1.12) M,K 2 M,K Note that only the polar angle fragment distributions are anisotropic as there is no dependence on the azimuthal angle. In comparison to this transition state model which assumes the nucleus is fissioning through only one angular momentum state, the standard saddle- point statistical model, outlined in [30], treats the angular momentum states as continuous functions, but the general idea is the same – angular distributions of fission fragments are described by the angular momentum state of the fissioning nucleus.

13 For a given neutron energy, the target nucleus will have a certain distribution of angular momentum states. Evaluating 1.12 and 1.10 with a given distribution of angular momentum states will directly produce the angular distribution of fission fragments; however, the distribution of angular momentum states of a fissioning nucleus is not known. Experimentally, fission fragment angular distributions can be measured for a given incident neutron energy bin and contain the fragments decaying from many angular momentum states. A sum of even order (to preserve the forward-backward symmetry) Legendre polynomials are used as an empirical fit in order to extract the anisotropy parameter

W [cos θ = 1] Counts parallel to neutron beam A = = , (1.13) W [cos θ = 0] Counts perpendicular to neutron beam From this anisotropy parameter, the distribution of angular momentum states of the ﬁssion- ing nucleus, as it evolves as a function of incident neutron energy, can be inferred. See, for example, [34, 35].

1.4 Neutron Linear Momentum Transfer

At high neutron energies (above ∼ 10 MeV), other reaction mechanisms arise that complicate the picture of complete neutron capture followed by ﬁssion, namely direct reactions and pre-equilibrium reactions, which produce less than full momentum transfer from the incident neutron to the ﬁssioning nucleus. A lengthy discussion of direct and pre-equilibrium reactions can be found in [18], an outline of which will follow here. In simple compound nuclear reactions, an incident particle is completely absorbed by the target nucleus and shares its energy throughout the entire nucleus in the process of achieving statistical equilibrium before decaying. The time scale for achieving such statistical equilibrium is typically ∼ 10−18 – 10−16s [36]. After a statistical equilibrium has been achieved, the resultant nucleus was hypothesized by Bohr [37] to have no “memory” of the process by which it got to its current state, meaning that transfer reactions are good ways to study nuclei that are out of reach via other means. But this lack of memory also means that emission from a nucleus that has reached equilibrium is expected to be isotropic in the

14 center of mass frame (when ignoring the effects of angular momentum discussed in 1.3). As a concrete example, the additional neutron from second chance fission (which sees an additional neutron emitted from the compound nucleus before fission, i.e. U(n,nf)) is expected to be emitted isotropically in the rest frame of the compound nucleus. In the context of the momentum transfer measurement of this thesis, this means that post-equilibrium emission does will not affect the average momentum transfer because the isotropic emission will cancel out over many events. Direct reactions are characterized by the timescale (∼ 10−22) in which it takes an incident projectile to roughly traverse the target nucleus. The incident particle can interact with a single nucleon in the nucleus, a group of nucleons, or even the whole nucleus, but emission of incident energy happens immediately. Several types of direct reactions can take place including elastic scattering where the target nucleus is left in its ground state, inelastic scattering in which the incident particle transfers some amount of energy to the target but is not captured, transfer reactions where the incident particle either gives or receives nucleons from the target, and knockout reactions where the incident particle removes a piece of the target nucleon (also called spallation reactions). All of these direct reactions necessarily do not see all of the momentum of the incident particle transferred to the target due to a compound nucleus never being formed. However, at incident neutron energies high enough for a direct reaction, fission of the target nucleus can still occur after the direct reaction, which means measuring a net deficiency from full momentum transfer. In the case of pre-equilibrium reactions, the incident neutron interacts with only a few nucleons in the target nucleus, but doesn’t immediately separate them from the target nucleus like in a direct reaction. However, the energy deposited to those few nuclei is so large that before the energy can be dissipated to the rest of the nucleus, particle emission occurs [38, 39]. Both spallation and pre-equilibrium reactions have similar signatures in that they both produce the emission of light particles of alphas and lighter (especially for incident neutrons in the energy range up to 250 MeV), and the emission of these particles is for-

15 ward focused, but their timescale and energy distribution are slightly different [40]. Again, since the emission of these pre-equilibrium particles is forward focused, they produce a net decrease in the amount of linear momentum carried by the fissioning nucleus. Momentum transfer from the incident neutron to the target nucleus shows up in the opening angle of the fragments, and by measuring the fragment opening angles, the momentum of the fissioning nucleus can be measured – a method first outlined in [41] and expanded upon in this thesis. While the fissionTPC cannot measure the high energy outgoing particles from pre-equilibrium and direct reactions explaining the lack of complete momentum transfer, it can make precise measurements of the fission fragment opening angle to infer just how much linear momentum was actually transferred.

1.5 Motivation for Precise Fission Measurements

Neutron induced fission has many relevant applications including astrophysics, weapons, and nuclear reactors [42]. One of the most important physical parameters that these applications are dependent on is the neutron induced fission cross section [43]. Evaluators compile many data sets and measurements together to get a best estimate of the cross section, but most of the underlying data sets share systematic errors and have individual uncertainties of 3-5 % [44]. In particular, fission chambers [27] have been the detector of choice to perform cross section measurements, but have limitations in their achievable precision [44]. Thus, the NIFFTE fissionTPC was constructed to produce an independent measurement of neutron induced fission cross sections and provide insight into the systematic errors of previous measurements [28]. The fissionTPC is an upgrade over older fission chambers in that it has the ability to reconstruct the ionizing radiation’s path through the detector to get full three- dimensional tracks. Because the fissionTPC has a wealth of data for every fission event, other physics can be extracted from the data beyond just a cross section measurement. Fission fragment angular distributions and neutron linear momentum transfer, for example, are the focus of this thesis.

16 Fission fragment angular distributions are needed for detection efficiency in cross section measurements and for fundamental understanding of the fission process. For example, previous measurements of cross sections at the JRC-IRMM in Geel [45], the nToF collaboration at CERN [46], and the recent NIFFTE cross section publication [29], all include corrections for fission fragment anisotropy. Fission fragment angular distributions also provide insight into the state of the transition nucleus at the saddle point considering the quantum angular momentum numbers are the source of the anisotropy, as explained in 1.3. The level parameters are then used in reaction codes such as EMPIRE [47] and TALYS [48] that provide input to fission cross sections evaluations [49]. Since fission fragment anisotropy is defined in the center of mass frame, a conversion from the lab frame angles to center of mass angles is required. This conversion cannot be done correctly without knowing the amount of linear momentum transferred from the neutron to the target nucleus. Thus, a measurement of linear momentum transfer is necessary for an accurate measurement of fission fragment angular distributions. Other measurements such as total kinetic energy and fission fragment product yields also require a conversion from lab frame to center of mass frames and therefore need linear momentum transfer as an input parameter [24]. Also included in understanding the fission detection efficiency for cross section measurements is a term for the kinematic boost of the fission fragments due to the linear momentum transfer from the incident neutron [29]. If this kinematic boost is not taken into account correctly because of the incorrect assumption of full momentum transfer, an incorrect efficiency will be propagated into the cross section, thereby skewing the results, particularly at high incident neutron energies.

17 CHAPTER 2 NIFFTE TIME PROJECTION CHAMBER

There have been many different instruments used to study nuclear fission since its discovery in the 1930s. The Neutron Induced Fission Fragment Tracking Experiment (NIFFTE) collaboration has, for the first time, used a time projection chamber [50] built specifically to study the fission process: the fissionTPC. Presented here is an outline of the fissionTPC’s hardware, targets, and operation with a focus on aspects important for fission fragment angular anisotropy and neutron linear momentum transfer measurements presented in the following chapters.

2.1 Fission Time Projection Chamber Hardware and Operation

The power of time projection chambers (TPCs), first proposed in 1975 by David Nygren [51], is their ability to reconstruct full three-dimensional tracks of ionizing radiation. TPCs consist of a chamber, typically filled with gas, in which incident charged particle radiation creates an ionization track in the chamber to produce a three-dimensional electron charge cloud. This electron cloud is then drifted by a uniform electric field down to a segmented anode, and the resultant signal is read out after amplification. The position where the electrons impact the anode provides the x- and y-coordinates whereas the z-coordinate can be reconstructed by knowing the speed at which the electron cloud moves through the drift medium and projecting back in time from when the incident particle created the electron cloud (thus giving the TPC its name). Most TPCs have an additional magnetic field orthogonal to the drifting electric field that will curve the charged particle as it creates the track in the drift medium to measure the particle’s charge to mass ratio and also helps reduce the diffusion of the electron cloud as it drifts to the anode. All of this together allows TPCs to be high rate detectors capable of a large amount of track multiplicity that achieve strong particle tracking and identification. [52]

18 Figure 2.1: Schematic representation of the ﬁssion time projection chamber.

A schematic drawing of the fissionTPC can be seen in Figure 2.1, which depicts the neutron beam entering from the left side, through the upstream anode, and interacting with an actinide target sitting in the middle of the central cathode. Fission fragments and other ionizing radiation, e.g. spontaneous alpha emission and (n,n’) reactions, strip electrons from the fill gas in both upstream and downstream volumes. The drift chamber is 15 cm in diameter and the central cathode is 5.4 cm from each anode. An applied electric field of roughly 500 V/cm drifts the electron charge clouds to the highly segmented anodes, each with 2,976 hexagonal readout pads, 2 mm in pitch. To amplify the electron charge cloud signal, both fissionTPC anodes are also covered by a MI- CROMEGAS (MICRO MEsh Gaseous Structure) [53] where the electric field rises to about 5 kV/cm, creating a Townsend avalanche to increase the number of electrons striking the readout pads to produce a measurable signal. Figure 2.2 provides another schematic representation of how the electron charge clouds are drifted to the anodes. The 1000 wire/inch

19 (a) Electron Cloud Drift (b) Electron Cloud Projection

Figure 2.2: (a) A schematic representation of an alpha particle’s ionization charge cloud being drifted to the anode and ampliﬁed by the MICROMEGAS. (b) A projection of a charge cloud onto the hexagonal readout pads on the anodes.

MICROMEGAS mesh sits on pedestals, approximately 75 µm tall, placed at every pad cor- ner and has a voltage typically set at ∼ 200V . The signals from the hexagonal anode pads are then fed through the pad plane, shown in Figure 2.3, which is a 16 layer, 1.57 mm thick printed circuit board. Each one of the pads is connected through the pad plane to custom readout electronics [54]. Due to the large number of readout channels and a large sampling rate, a three or four month run in the neutron beam can produce 50-60 Tb of data that need to be processed. A picture of the ﬁssionTPC with all electronics connected is also shown in Figure 2.3.

2.1.1 Electronics and Triggering

Custom readout electronics, called the EtherDAQ (detailed in [54]), were required to meet the unique constraints of the ﬁssionTPC including small space requirements, high channel density, and short readout times. Each EtherDAQ card, capable of servicing up to 32 pads for a total of 192 cards, is responsible for taking the raw signal from a pad, seeing if that signal is above threshold, converting to a digital readout, and sending an Ethernet packet

20 (a) FissionTPC Anode (b) Fully Assembled FissionTPC

Figure 2.3: (a) A picture of one of the two highly segmented anodes. (b) The fissionTPC fully assembled with all of the readout electronics attached. The drift chamber is 15 cm in diameter and the central cathode is 5.4 cm from each anode. to the data storage. This processing flow as well as the hardware of an EtherDAQ card can be seen in Figure 2.4. Each EtherDAQ card consists of two boards, one for the digital electronics, and one for the analog amplifiers, called preamps. This was done to separate the more expensive digital electronics from the cheaper analog electronics for more efficient repairs. Once the signal is passed through the amplifiers, they are passed to the analog to digital converters (ADCs) to produce a 12-bit signal that goes into the field programmable gate array (FPGA) which splits the bit string into two. One string is stored while waiting for the other string to be differentiated and checked against a trigger threshold. If the trigger threshold is passed, the stored bit string is then passed to the Small form Factor Pluggable (SFP) module connector that sends a 770 to 860 nm, near-infrared wavelength over the optical fiber to the event builder at a rate up to 1.25 GB/s. The fissionTPC electronics consums about 3.1 kW of power, and due to this large power consumption, cooling is required by a pack of 24 fans that blow refrigerated air for high altitude operation at LANL.

21 Figure 2.4: (Left) Picture of an EtherDAQ card. (Center) A schematic layout of an Ether- DAQ card containing the analog amplifiers on the bottom where the input signal enters, and the digital electronics and fiber readout on top. (Right) Work flow diagram of the FPGA firmware that controls the data flow for triggering and processing.

22 There are four connectors on each EtherDAQ card: one at the bottom to read in the signals from the anode pads to the preamps, one connecting the preamp board to the digital electronics board, one at the top of the digital electronics board that supplies a 24V power supply, clock, and triggering, and one optical fiber connection at the top to connect the card to the data storage hardware. The clock input at the top connector is very important as this ensures that all ADCs are synchronized in time, allowing the correct timestamp to be sent from each card, thus making the data reconstruction possible. The EtherDAQ cards operate at a sampling rate of 65MHz (or 20ns) and the FPGA can store up to 120 samples before and after the trigger. Typically, the fissionTPC operation includes writing out 20 samples before the trigger to decrease the storage of unnecessary data. A 65 MHz sampling rate works well for the anode readout due to the spacial extent of the ionizing radiation being on the order of 1cm in width and an electron drift speed of about 3 cm/µs, but the neutron time of flight requires a timing resolution of a couple nanoseconds. To achieve a GHz sampling rate, an EtherDAQ card with a special preamp card sends sequentially delayed signals to 20 channels of the card. Using a Data Delay Devices 1520SA- 100-500 chip, a 1ns signal delay is sent to each one of the channels, effectively providing a GHz sampler with the same timebase as the main fissionTPC clock [55]. The waveforms produced from these electronics can be seen in Figure 3.1. Triggering on the fissionTPC can be done in a number of ways. One method is to allow each pad to trigger individually over its preset trigger threshold. While this removes any additional complexity, it can lead to an unsustainable and unnecessary amount of data output for a high activity source, like 239Pu. To deal with the extremely large data rate produced by a 0.5 MBq source, a global trigger hold-off can be sent that only allows the pads to readout over a certain time period. Typical uses of the global trigger hold-off include only opening the pad triggers for a certain number of beam macropulses or when trying to take autoradiograph data with a high activity source. A data rate of 20MB/s for a high activity source is reasonable for sources with MBq activities, which means typically supressing 97%

23 of the data volume. However, using the global trigger hold-off when operating in beam essentially cuts the fission rate and statistics needed for fission studies. To ensure no fission events are missed while also keeping the data rate reasonable, a cathode fission trigger was developed [56] which uses the cathode signal to spot a fission event. If the cathode sees a fission signal, the anode pads have the trigger hold-off release and can thus capture the subsequent fission track. Since the fission rate is roughly 5 orders of magnitude lower than a MBq alpha source, operating the cathode fission trigger has no significant impact on the data rate.

2.1.2 FissionTPC Fill Gas

For the data taken in this thesis, the fill gas in the drift chamber of the fissionTPC consisted of 95% Argon and 5% Isobutane at a pressure of 550 Torr. This composition was chosen because it was stable in the operation of the fissionTPC and had a high enough pressure to ensure that spontaneous α tracks would be fully contained within the active area of the detector. NIFFTE did a stability study [55] on different fill gases to assess the best gas that is suitable for both the MICROMEGAS amplification stage and the three-dimensional charged particle tracking application of the fissionTPC. An acceptable gas for the fissionTPC is one that can be run at a pressure high enough to stop a roughly 5 MeV alpha particle in the chamber, have a high enough drift speed to clear the chamber with a roughly 0.5 MBq 239Pu source, and be stable in the high flux, high energy neutron beam. Here, an unstable gas means that a high electron density avalanche going through the gain stage of the MICROMEGAS will cause a spark to travel from the readout anode pads to the MICROMEGAS mesh, resulting in a high voltage trip (the trip limit was set to 500nA). In the stability study, if no spark was observed within an hour of beam time, the gas is said to be stable. If a spark was observed in less than 60 min, the test was repeated for confirmation and an additional 60 min test with no beam was conducted to ensure the sparking was beam induced.

24 The physical process explaining the sparking process was described by Raether [57] where a fundamental limit of about 107 - 108 electron-ion pairs can form before sparking begins. UV photons emitted from the electron avalanche cause discharges at a lower gain than expected from the Raether limit through a process called photon feedback. If these emitted photons strike the MICROMEGAS, they produce more electrons, leading to sparking. Gases that have high photoabsorption will ”shield” the MICROMEGAS from the emitted photons, thus decreasing the photon feedback and be more stable against sparking. By quenching these photons, the electron avalanche will be more localized and the detector will be more robust against sparking.

Gases tested for the ﬁssionTPC were Argon paired with C4H10, CH4, CF4, and CO2 in diﬀering concentrations. Results of the study can be seen in Table 2.1. Gases with lower photoabsorption cross sections (CH4 and CF4) will reach the Raether limit with lower pressure and gain levels than the gases that have higher photoabsorption cross sections (CO2 and C4H10).

The most commonly used gas in ionization chambers is P10 (90% Argon and 10% CH4) due to its relatively inexpensive cost and fast drift speeds. However, P10 was only found to be stable at gas pressures below 530 Torr – too low for the α-particle tracks to stay in the active area of the detector. CF4 is another gas with a fast electron drift speed, useful for performance in high rate applications, but considering its low photoabsorption cross section

[58], it was not as stable when compared to C4H10 and CO2.

Even though CO2 has a low diffusion coefficient and is effective at cooling electrons [59], the gas used in the cross section measurements taken by the fissionTPC was a gas mixture of 95% Ar and 5% C4H10 because it contains hydrogen and a possible future measurement is the cross section ratio between 239Pu(n,f) and p(n,n′)p (a paper detailing the methodology

for such a measurement is currently in review). Since CO2 obviously lacks hydrogen, that gas would not work in a measurement of the neutron-proton inelastic scattering cross section.

25 Table 2.1: Conﬁguration and results of stability against sparking ρ(Torr) Conc. (%) E/ρ(V/cm/Torr) Stable Ar + CH4 530 10 65.4 Yes 530 10 67.9 No 760 10 50.7 No Ar + C4H10 600 1 43.8 Yes 760 1 33.3 Yes 550 5 48.5 Yes 550 5 54.5 Yes 800 5 39.8 Yes 800 5 40.6 No Ar + CO2 600 1 62.0 Yes 600 1 71.8 Yes 760 1 49.7 Yes 760 1 50.6 Yes 900 1 45.6 Yes Ar + CF4 600 30 75.2 Yes 760 25 61.2 Yes 760 25 62.6 No 760 30 66.2 No

26 MICROMEGAS sparking can be induced due to mesh defects, rate-induced effects, and high electron densities passing through the mesh. Since the fissionTPC was also run without beam, and the beam-induced rates account for less than 2% of the total charge generated in the gain stage (the rest coming primarily from the high rate 239Pu spontaneous α activity), the sparking in the fissionTPC must be coming from high electron density tracks. The fissionTPC was also tested in the presence of spontaneous fission from a 252Cf source and remained stable. This leaves the likely culprit of some of the 14000 Ar recoils per second in the fissionTPC as the cause of the sparking.

Figure 2.5: A representation of how low energy Argon recoils can produce more ionization density in the electron avalanche than a ﬁssion fragment and cause sparking. Source [55].

Even though a 100 MeV 99Mo ﬁssion fragment in 1 atm of argon gas has an ionization density more than 10 times that of a 1 MeV Ar ion in the same gas, the ﬁssion fragment must drift the 5.4cm from the central cathode that holds the target actinide to the MICROMESH above the anode. An Ar recoil can occur anywhere throughout the drift chamber, and a 1

27 MeV Ar ion drifting 1mm has an ionization density about 5 times that of the drifted ﬁssion fragment. Increasing the Ar ion to 10 MeV increases the ionization density by a factor of 20 over its 1 MeV counterpart. The fact that decreasing the pressure and thus reducing the ionization density of these ion recoils produces more stability also adds conﬁdence to this hypothesis [55]. A schematic representation of Ar recoils causing sparking can be seen in Figure 2.5.

2.2 Actinide Targets

Targets used in the ﬁssionTPC have speciﬁc requirements for the purposes of cross section measurements[60, 61]. Desired target qualities include:

• A very uniform actinide density such that neutron beam spacial variations can be mitigated. Increased target density non-uniformity, combined with the spacial non- uniformity of the neutron beam, necessarily introduces more error into the beam-target overlap correction seen in 1.2.1.

• A small amount of actinide deposited such that the fission fragments that come from within the target and/or fragments that are emitted at shallow angles have as little energy straggling within the target as possible; however, targets that are too thin do not provide a sufficient number of fission events for proper statistics in the cross section measurements. Target thicknesses used in the fissionTPC are typically between 50 - 200 µg/cm2.

• A very isotopically pure target to reduce uncertainty related to reactions on impurities. The NIFFTE collaboration has been fortunate to work with >99.9% isotopic purity actinide materials from ”private stocks” of U.S. national laboratories [61].

FissionTPC targets are deposited in one of two ways: vapor deposition or molecular plating [62]. The preferred process of vapor deposition creates very uniform targets in a process where the sample material is placed in a vacuum and vaporized. The vaporized

28 material then travels in a line of sight to the deposition site, creating very uniform (≤ 1.5% spacial variation) targets, but also wasting more than 90% of the sample material. Molecular plating must be used for rare actinide samples where the deposition process has a much higher yield, but spacial variation across the target can exceed 10% [61]. For the ﬁssionTPC, this means that the 235U and 238U targets were made via vapor deposition, and the 239Pu and 252Cf deposits were made via molecular plating.

(a) 239Pu target (b) Autoradiograph of 235U/238U target

Figure 2.6: (a) A picture of the Plutonium side of the 239Pu/235U thick backed aluminum target before insertion into the ﬁssionTPC. (b)Start vertices for alpha particles emitted from a thick target with 235U on each side. Contamination of small quantities of 239Pu can be seen on the cathode.

The two different target backings used in the fissionTPC are a thin carbon backing and a thick aluminum backing. Using a thin Carbon backing (∼ 100µg/cm2) allows for fission fragments to travel through the backing and be detected in the other volume. This allows for both fission fragments to be detected in the fissionTPC, making the measurements of the fission folding angle possible (see Chapter 4). However, thin carbon backings are not as useful for cross section ratio studies in that actinides cannot be placed back-to-back and the neutron beam flux therefore does not cancel. Thin carbon backings are also much more prone to breaking or tearing in the installation process of the target in the fissionTPC. Using a thick aluminum backing (∼ 0.5mm), as pictured in Figure 2.6, allows for two actinides to be placed back-to-back in order to cancel the neutron beam flux in cross section measurements,

29 but the aluminum backing is too thick for the fission fragments to pass through, meaning only one fission fragment from each fission event can be detected. All of the targets depicted in Figure 2.7 were loaded into the fissionTPC and put in beam at the LANSCE 90L beam path. The first target, a thin Carbon backing with a top half- moon of 238U and a bottom half-moon of 235U, was analyzed extensively for a cross section shape measurement [29]. The other thin Carbon backed target consists of a left half-moon of 239Pu and a right half-moon of 235U. While these targets were originally placed with the half-moons rotated 90 degrees without much thought, this rotation actually turned out to be important due to the neutron beam profile. Higher energy neutrons are more likely to come from the start of the Tungsten target before the proton beam loses energy through the Tungsten, leading to a beam spot on the fissionTPC that varies as a function of neutron energy. Since the spacial variation of the beam happened to be in the horizontal direction (parallel with the proton direction), the vertical deposits of the thin 239Pu/235U saw different beam fluxes and a cross section ratio measurement was not possible. However, this target data is still useful for the purposes of this thesis and can also be used for other physics that are not dependent on knowing the neutron flux, e.g. fission product yields and total kinetic energy analysis. These two thin target data sets also differ from each other in that the actinide deposits in the 238U/235U target were placed in the downstream volume of the fissionTPC compared to the 239Pu/235U deposits which were placed in the upstream volume. Thick backed targets used include 239Pu/235U, 239Pu/238U, and 235U/235U. The deposits on the two thick backed targets containing 239Pu are both 1 cm in diameter, but the 235U/235U target had deposits of 1.8cm in diameter. Having a 1 cm diameter deposit allows for the entire actinide to be covered by the beam spot such that the strong neutron flux gradient at the edge of the neutron beam does not have to be taken into account. However, in order to validate the entire beam shape, a 1.8cm radius 235U/235U was used which will also produce fissionTPC validation studies.

30 Figure 2.7: List of targets from which neutron induced ﬁssion data was taken. Actinides include 235U (blue), 238U (purple), and 239Pu(green). Aluminum target holders are 2cm in diameter and actinide deposits are drawn roughly to scale.

All of the thick backed targets were also rotated with respect to the neutron beam such that both sides of each target took a turn in the upstream and downstream volumes. This is an important check for the fissionTPC in that it reverses the kinematic boost, changes the neutron beam overlap with a non-spatially uniform deposit, and ensures that the fissionTPC can reproduce its own cross section and anisotropy measurements. The procedure for rotating the targets with respect to the neutron beam consisted of simply picking up the fissionTPC and rotating it 180 degrees. There was no actual fissionTPC pressure chamber break and re-loading of the targets during rotation, but the readout cards were removed and inserted back in locations that were not required to be the same. When actual target replacement takes place, the fissionTPC must be completely disassembled, removing all of the electronics and at least one anode from the pressure vessel in order to gain access to the target. In addition to the targets shown in Figure 2.7, an additional thick-backed target of 252Cf was used to investigate fission in a no neutron beam environment. With a large spontaneous fission chance of approximately 3% [63] compared to spontaneous α emission, the 252Cf target allows for the simultaneous study of isotropic emission of alpha particles and fission

31 fragments to validate the ﬁssionTPC’s reconstruction abilities, as discussed in 3.

2.3 Neutron Source and Time of Flight Measurement

The fissionTPC is operated at the Los Alamos Neutron Science Center (LANSCE) Weapons Neutron Reasearch (WNR) 90L beam line shown in Figure 2.8. An 800 MeV linear accelerator impinging protons upon a tungsten target produces an unmoderated spallation neutron source that is collimated into the flight path where the fissionTPC is placed. The beam pulse spacing consists of macropulses at a rate of 100 Hz, with each macropulse consisting of about 375 micropulses per macropulse. Each micropulse consists of a 250 ps width and they are separated by 1.8 µs [64].

Figure 2.8: On overview of the Los Alamos Neutron Science Center. The location of the 90L WNR beamline where the TPC is operated is circled in red. [65]

Neutron energies are deduced at the fissionTPC via time of flight measurements where the start time is generated when the accelerator’s proton beam impinges on the tungsten target and the stop time is taken as the signal induced on the cathode by a fission event. The central cathode of the fissionTPC is connected to a readout channel that has a sampling rate of 1 ns, as discussed in 2.1.1. The stop signal on the cathode is induced due to the fast image charge from high electron density clouds produced by fission fragments [28], and the algorithm to extract the time of flight from the cathode signal is discussed in 3.

32 In order to calibrate the time of flight spectrum, photon induced fission is used. When the 800MeV protons hit the tungsten target, a gamma flash is also produced along with the spallation products. These gammas also travel down the flight path and induce fission, providing a time of flight stop signal. Measuring the flight path length (see 2.3.1) and knowing the speed of light, the photofission peak can be placed at the correct time of flight, namely 26.88 ns. A time of flight resolution estimate of ∼ 3ns is provided by the FWHM of the photofission peak seen in the insert of Figure 2.9. Relativistic neutron kinetic energies can then be calculated by

2 1 Ekin = mnc − 1 (2.1) 1 − β2 ! v L where β = c = ct with neutron velocity v, flightp path length L, and time of flight t. The calibrated neutron time of flight spectrum for fission events from a 235U target can be seen in Figure 2.9

Figure 2.9: Neutron time of flight spectrum for fission from a U235 target. The insert shows the photofission peak and fit to determine a time of flight resolution. Neutron wraparound is shown as the red line.

33 2.3.1 Carbon Filter Data

The path length from spallation source to actinide target used to calculate the incident neutron energy is measured by placing a block of carbon just after the collimator, before interaction with the fissionTPC, to take advantage of a very distinct neutron scattering resonance in the carbon inelastic scattering cross section. As depicted in Figure 2.10, the neutron flux at the actinide target is decreased due to the carbon filter knocking neutrons from the flight path at the strong resonant energy of the neutron cross section, thus causing a dip in the time of flight spectrum at that neutron energy, namely 2.078 MeV.

(a) 12C inelastic neutron scattering cross section (b) Neutron ToF with Carbon Filter

Figure 2.10: (a) Inelastic neutron scattering cross section for Carbon-12 showing strong resonance feature at 2.078 MeV, from ENDF/B-VIII [49]. (b) Neutron time of ﬂight spectrum as measured by the ﬁssionTPC showing location of the Carbon-12 resonance at around 400 ns.

Solving 2.1 for the flight path length L, and inserting the neutron time of flight and neutron energy of the carbon resonance at 2.078 MeV, the flight path length is measured to be 8.060 ± 11mm. Thus, the photo-fission peak can now be set at the proper time of flight according to the formula L t = . (2.2) γ c

34 The flight path measurement has been repeated over the many years that the fissionTPC has been operating at the 90L flight path, and every subsequent measurement of the flight path has agreed within errors thanks to the fissionTPC being mounted to a table which has not been moved since installation into the flight path. It is also interesting to note that while the Carbon filter is inserted in the flight path, the fissionTPC’s operating stability decreases dramatically due to all of the electronics being sprayed with neutrons.

2.3.2 Neutron Beam Wraparound

As mentioned in 2.3, a proton beam packet hits the tungsten target at a high rate of 1.8 µs (setting the time of flight window as seen in Figure 2.9), producing a large neutron flux; however, this high rate also produces a larger amount of “wraparound” neutrons – those that have a longer flight time than 1.8 µs are instead counted in the subsequent beam packet and will be assigned an incorrect neutron time of flight. For example, an incident neutron with a time of flight of 2 µs will cause a fission event in the fissionTPC tagged as coming from an incident neutron with a time of flight of 200 ns. The beam pulse spacing therefore introduces a lower limit on neutron energy detectable by the fissionTPC running at the 90L beam path. For the fissionTPC flight path of 8.06 m, the wraparound neutrons are less than 130 keV , and the highest energy wraparound neutrons can be seen in the “pedestal” before the photofission peak in Figure 2.9. Since no neutrons can travel faster than the gamma rays produced from the spallation target, these events before the photofission peak must be generated by wraparound neutrons. While wraparound events cannot be determined on an individual basis, the overall magnitude can be estimated by looking at the decay tail of the last micropulse. Fitting the tail of the last micropulse with a sum of two decaying exponentials, as shown in Figure 2.11, and then “folding” this function to account for all micropulses gives an estimate of the wraparound neutron contribution. Matching the wraparound fit function to the pedestal in Figure 2.9 ensures the fit function is folded correctly. The functional form to describe the

35 wraparound is

−C2x −C4x F itwrap = C0 C1e + C3e (2.3) " # micropulsesX where the parameters C1 through C4 are set by fitting the last micropulse tail and C0 is a scaling parameter to ensure the function matches the wraparound pedestal in the time of flight spectrum. By taking the ratio of the counts above and below the wraparound line in Figure 2.9, a percentage of wraparound events for each neutron energy bin can be calculated. The percent wraparound shown in Figure 2.11 is then used to correct the anisotropy and neutron linear momentum transfer measurements presented in the following chapters. It should be noted that 238U requires no wraparound correction because it has a fission threshold of roughly 1.4 MeV and all wraparound neutrons have an energy less than 130 keV .

(a) FissionTPC Anode (b) Fully Assembled FissionTPC

Figure 2.11: (a) The end of the last micropulse is fit with a sum of two decaying exponentials. This fit is then “folded” to get a prediction for the number of wraparound neutrons. (b) Percent of fission events caused by wraparound neutrons for a 235U target as a function of neutron energy calculated by taking the ratio of counts above and below the red wraparound line in Figure 2.9.

36 CHAPTER 3 TPC DATA RECONSTRUCTION AND CORRECTIONS

Considering the fissionTPC has almost 6000 readout channels and a data rate of 20MB/s, data processing and analysis plays an integral part in producing physics results from the fissionTPC. This chapter will focus on the process of going from the individual pad signals through ionizing track creation and fitting, as well as some post-processing corrections that need to be applied to the data before performing the linear momentum transfer and anisotropy analysis in the following chapters. Data from a calibration source of 252Cf is also presented to demonstrate and benchmark these corrections.

3.1 Data Reconstruction: Signals to Tracks

After all of the signals are read out from the anode pads and the cathode, an extensive reconstruction process is needed to extract information about the ionizing radiation that created the signals. The first step is to turn the signals from each pad into “digits” – a voxel of charge containing x, y, z, and charge information. In order to form a digit, the signal from each pad is first passed through a five point derivative filter to find the slope of the signal at each time step. This forms a peak that corresponds with the rise and plateau of the signal from the pad, as seen in Figure 3.1. The value of the differentiated signal at each time step is then taken as the amount of charge, in Analog to Digital Converter (ADC) units, that was deposited on that pad at that particular clock time. Knowing the pad location on the anode plane provides absolute valus of the x and y position for each digit, and the z (drift) position is calculated according to

z = trel × vdrift (3.1) where trel is the time relative to the start of the event and vdrift is the electron drift speed through the gas. Because the ﬁssionTPC can only measure when an electron interacts with

37 Figure 3.1: Signals from an alpha particle and the subsequent diﬀerentiation used in the digit forming process. Considering all of the pads have signals occurring at almost the exact same time, this set of signals corresponds to an alpha particle that was emitted very parallel to the anode plane, equating to a polar angle of cos(θ) very close to zero.

38 the anode and not when those electrons are freed in the gas, the fissionTPC does not have an absolute time measurement for each digit. This means that there is no absolute z position in the fissionTPC – z is only relative to the other digits in each event that pass the threshold trigger. While knowing absolute z is not important for the analysis presented in this thesis, it can be estimated for events with spontaneous alphas or fission fragments in that the start of these tracks must come from the target at the center of the drift chamber. The digits plot Figure 3.2 shows absolute z being set such that the upstream and downstream fragments both originate from the target, where they are expected.

Figure 3.2: A typical fission event from a thin carbon backed target. Along with the back to back fission fragments, small tracks from inelastic scattering of neutrons off protons in the gas are also visible. The red lines through the digits are the assigned tracks. Color scale denotes the ADC (charge) value of each digit.

After forming all of the digits for each event, adjacent digits are grouped together to form individual tracks. When ionization events are separated in time or are far away from each other in the chamber, the digit grouping stage is easy considering digits from diﬀerent tracks are not close by. However, a MBq source like 239Pu has a much higher chance of alpha particle charge clouds overlapping in the chamber, meaning there is no clear separation of

39 digits between different tracks. In order to separate pile-up events, a topology graph is constructed that selects points where the different line segments intersect. Vertices of the topology graph are used as separation points between the two different charge clouds and line segments that are co-linear are merged together. Fission fragments have extra protection against their digits being used in other tracks by giving grouping preference to higher energy segments. [66] Once the digits are grouped into individual tracks, they are fit to extract track parameters such as start and end positions. Since the fissionTPC does not operate with a magnetic field, all of the charge profiles are straight eliminating the need to use complicated tracking algorithms like the Kalman filter [67]. Therefore, principal component analysis (PCA) is used to fit each of the grouped charge clouds to produce the eigenvalues and eigenvectors of each charge cloud. The eigenvector with the largest eigenvalue corresponds to the principal axes of charge cloud and sets the charge cloud direction. Start and end positions of each track are set along the principal axes where the ionization profile hits a particular threshold. This threshold is dependent on electron diffusion, track orientation, and the gain levels set on the fissionTPC. By tuning this threshold, track vertices are “focused” until dots of 239Pu contamination have their smallest radial extent (see Figure 2.6, for example). Finely tuned track start positions are not important for the work presented in this thesis because it does not change the polar angle, but exact track vertices can be important in the context of the actinide deposit spatial variation that is used for the beam-target overlap correction in the cross section measurement discussed in 1.2.1. An example fission event from a thin carbon backed actinide deposit can be seen in Fig- ure 3.2, showing two back-to-back fission fragments as well as other small ionizing radiation from neutron beam interactions with the fill gas. The red lines are the fits from the principal component analysis. From these fits, parameters such as track start and end, track length, and track direction can be extracted. Track polar and azimuthal angles are computed directly from the start and end positions of the tracks. Fission fragment polar angles are the

40 focus of the anisotropy and linear momentum transfer analyses in the following chapters. Integrating the total amount of charge associated with each track also provides the total amount of energy the track deposited in the fissionTPC, quoted here in uncalibrated analog to digital converter (ADC) units. Projecting the charge onto the fit produces the Bragg curve – energy deposited per unit length(dE/dx) by the track as a function of distance – and values for the Bragg peak and Bragg position can also be extracted. The Bragg parameters are important for correcting the electron diffusion anisotropy, as discussed in 3.3.4. If the ionizing radiation produced a signal large enough for the cathode to read, a neutron time of flight is also assigned to the track, as discussed in 3.2.

Figure 3.3: Track length vs total track energy is a commonly used parameter space to perform particle identiﬁcation.

Using all of these different track parameters provides a large number of parameter spaces, each representing different aspects of the data. One of the primary parameter spaces used for particle identification can be seen in Figure 3.3 where track length and total track energy in ADC units are plotted as a heat map using the thin carbon backed target with half moons of 235U and 238U placed in the neutron beam. The different types of ionizing radiation are

41 labeled on the plot including p(n,n’)p, spontaneous alphas emitted from the target material, and recoil ions (X(n,n’)X) from inelastic scattering on the higher mass gas atoms and other materials throughout the fissionTPC. Note the log scale on the x-axis showing significant separation between fission fragments and all other ionizing radiation.

3.2 Neutron Time of Flight Reconstruction

Neutron energy resolution is an important part of any neutron induced fission measurement, and with a neutron flight path length of only 8m, it is vital that the fissionTPC has neutron time of flight precision on the order of a couple nanoseconds to produce accurate energy resolution for high energy neutrons. For example, the time of flight for a 250 MeV neutron on an 8m flight path is 43.8ns, which differs only slightly from the 41.2ns for a 300 MeV neutron. As mentioned in Chapter 2, the central cathode used for the neutron time of flight measurement has a 1 GHz (or 1 ns) sampling rate. However, a fission event on the cathode has a rise time of several hundred nanoseconds, so a procedure to process the rise time in a consistent way to produce a time of flight reconstruction with a couple ns timing resolution is needed. If the cathode signal passes a given threshold, it will be processed by a standard imple- mentation of the trapezoidal filter (see, for example, [68]). The raw cathode signal (blue) and its trapezoidal filtered counterpart (red) can be seen in Figure 3.4. A straight line is then fit on the rising edge of the filtered waveform between 40% and 60% of the maximum value. Where this line intersects with an amplitude of 0 is set as the neutron time of flight. Once a time of flight signal is acquired in the fissionTPC, it is compared to the signal from the accelerator and shifted to be relative to the latest micro- and macro-pulses from the neutron beam. This procedure is sufficient for a timing resolution of 2-4 ns at FWHM, as determined by fitting the photofission peak, seen in Figure 2.9.

42 Figure 3.4: Raw high speed cathode signal (blue) induced from a fission event and its trapezoidal filtered counterpart (red). Where a straight line fit to the rising edge of the filtered signal intersects with 0 is taken as the time of flight for the event.

3.3 Corrections to Data

The reconstruction process of the events through initial processing is not perfect and some corrections are required to better align events with their physical track parameters coming out of the target. Corrections applied for the linear momentum transfer and anisotropy analyses presented in this thesis include a slight change in the electron drift speed and a slight angle correction due to the electron diffusion. Detector saturation effects were also studied and a proposed solution is presented here, but this correction was deemed unnecessary and is just presented here for future detector improvement. Additionally, a source of 252Cf was used to ensure these corrections can accurately reproduce the angular distribution for isotropic emission of fission fragment and alphas, and explicitly show the effects of pad saturation on the fragment angles.

43 3.3.1 Californium-252 Measurement

Understanding the detector response and efficiency is a vital part of any physics measurement. The fissionTPC is unique in the study of fission in that it is capable of measuring ionizing radiation ranging in energy from less than a 1 MeV proton all the way to 100 MeV fission fragments. Being able to cover two orders of magnitude in energy from very different types of ionizing radiation poses certain problems. Some examples of different detector response covering recoil protons to fission fragments include vertexing abilities, electron diffusion differences, track length differences, and differences in signal amplification. While this is certainly not the first study of 252Cf in the fissionTPC (see [56] for example), this is the first measurement with 252Cf under operating conditions as close as possible to the operating conditions used when running in the neutron beam. In particular, this measurement was focused on understanding the polar angle distributions of alpha particles and fission fragments to benchmark the corrections that are discussed next. Previous measurements have focused more on acquiring precise counts for fission fragments and alphas in direct relation to neutron induced fission cross section measurements. Additionally, this measurement was taken with four different gain settings on the MICROMEGAS mesh: 190V, 200V, 210V, and 230V. The effects of these different gain settings are discussed 3.3.2.

(a) Fragment Start Positions (b) Length vs ADC

Figure 3.5: (a) Fission fragment start vertices and (b) Length vs ADC from the 252Cf source taken at the 210V MICROGMEGAS setting

44 Californium-252 has one of the highest spontaneous fission branching ratios at 3.09%, with the remaining 96.91% of decays through emission of an alpha particle [63], thus providing a good source for studying the different detector response between alphas and fragments with comparable statistics. The particular source used in this measurement is somewhat unique in that the majority of the actinide was deposited near the edge of the aluminum disk, as seen in the reconstruction of the start positions of the fission fragments shown in Figure 3.5. While the target deposit is unusual, the target is still centered in the fissionTPC and the fragment and alpha tracks still ionize the gas just the same as if the actinide was deposited solely in the center of the target. Also shown in Figure 3.5 is the length vs ADC values for this target showing the large separation between the fission fragments and the alpha particles. There is a significant amount of noise at very low ADC values that comes from the pad triggers being slightly too low, but these can easily be removed in the following analysis with an energy cut. The alpha band shown in this plot has more straggling than seen in autoradiograph data taken for the other targets listed in Figure 2.7 due to the alpha particles not being completely contained within the gas volume of the fissionTPC. Gas pressure, the main determining factor in how far alpha particles will travel, is set such that the roughly 5 MeV alphas from 235U, 238U, and 239Pu will cover almost the entire active volume. Since the primary alpha decay of 252Cf comes from 6.1 MeV alphas [63], they travel far enough in the gas to often times hit the field cage or anode. While this results in not all of the energy of these alphas being detected, this is not a problem for the determination of the alpha polar angle – the primary focus of this study. The gas pressure was not to changed to fully contain 252Cf alpha decays because of the desire for the operating conditions to be as close to in-beam measurements as possible for the purpose of reproducing diffusion parameters and tracking uncertainties.

3.3.2 Pad Saturation

Having enough sensitivity to detect the start of alpha particles all the way to 100+ MeV fragments is a diﬃcult requirement for the range of the electronics used in the ﬁssionTPC.

45 Because alpha start positions are important for the beam-target overlap term and the measure of the number of target atoms in the cross section calculation in 1.2.1, gain levels must be set high enough to provide proper alpha vertexing.

(a) Fragment Signals (b) Saturation Hole

Figure 3.6: (a) Saturated signal readouts from the pads for a ﬁssion fragment directed towards the anode. (b) A cut on the end portion of a charge cloud projected on the pads showing the hole created by the saturated signals.

However, fission fragments that come directly from the target towards the anode, i.e. cos θ = 1, deposit their entire charge cloud on a much smaller number of pads compared to fragments that come out parallel to the anode. This causes the analog electronics on the readout cards to saturate and the signal readout to reach some maximum value. Because the reconstruction process takes a derivative of the signal to find the digits, no digits are extracted following the saturation and a hole forms in the center of the part of the charge cloud that reaches the anode last, equating to the start of the fragment track. Since the saturation occurs on the analog section of the cards, the actual cutoff for saturation is pad dependent, meaning there is no set level from which to determine when a signal saturates. This signal saturation and the hole it forms is demonstrated in Figure 3.6. To demonstrate and understand the effects of the pad saturation and gain levels on fragment polar tracks, four different gain levels were set on the MICROMEGAS over the

46 Figure 3.7: ADC vs cos θ for 252Cf fragment events at the four diﬀerent gain settings showing how fragments emitted towards the pads lose energy due to signal saturation and how gain increases with an increase in MICROMEGAS voltage.

47 course of the 252Cf measurement. For the first gain level of 190 V, alpha particle start vertices are almost completely missing, but the fission fragments show no signs of saturation. The second gain level of 200 V still has trouble with the alpha start vertices from the 252Cf source and the first signs of saturation in the fission fragments begin to show. Moving up to 210 V allows for alpha vertexing that begins to look much more like the fission fragment start vertices shown in Figure 3.5, but the saturation of the fission fragments extends to tracks with cos θ ≈ 0.9. Further increasing the gain to 230 V focuses the alpha vertices into a clear image of the actinide deposit, but the majority of fission fragments with cos θ > 0.5 have significant saturation. These observations are demonstrated in Figure 3.8 and Figure 3.7. Changing between all of these different voltages demonstrates that a single gain setting cannot provide proper alpha particle tracking without producing saturation in the fission fragments – an in-between voltage minimizing the downside of both must be used.

Figure 3.8: 252Cf spontaneous alpha particle start positions as measured in the ﬁssionTPC at the four diﬀerent gain settings. A dead card is visible at roughly (-3,-3) and a small amount of contamination can be seen on the cathode at 210V and 230V.

48 Due to the hole in the center of their charge clouds, the saturated fission fragments have their polar angle skewed in the fitting process, causing a ”bump” to form near cos θ = 1, as seen in Figure 3.13 and Figure 3.17. Because the anisotropy analysis in Chapter 5 relies on fitting the angular distribution, an attempt to correct the saturation was made to quantify the impact of the saturation. This attempted fix relies on the ability to recover the information lost by the saturated signal to fill in the hole that forms due to the lost digits. The fix consists of first identifying saturated signals as those with a value higher than 3000 ADC (an estimated threshold above which saturation is possible). Because the fission fragment charge clouds should always have more charge in the center of the cloud, saturated digits are identified as coming from a saturated signal candidate and having at least four neighboring digits (in x and y, not z) with a higher ADC value. If a saturated digit was found, that digit was set to have the value of its largest neighboring digit, thus filling in the saturation hole.

(a) Before Saturation Fix (b) After Saturation Fix

Figure 3.9: ADC vs cos(θ) (a) before and (b)after applying the attempted saturation ﬁx to the 210V data.

Shown in Figure 3.9 is this attempted saturation ﬁx applied to the 210V data demonstrat- ing that putting charge back into the saturation hole changes both the total ADC and the reconstructed polar angle. This is shown explicitly in Figure 3.10. While an improvement is deﬁnitely made when comparing the two plots in Figure 3.9, this procedure is certainly not

49 perfect. However, this attempted fix shows how restricted the angular change is in cos θ. Be- cause the saturated events for a moderate amount of saturation are restricted to high cos θ, and the fix was not perfect in its ability to fully patch the saturated events, the anisotropy results and linear momentum transfer results do not apply this saturation fix to the data used in the analysis, but rather they simply cut out the events at high cos θ. Most data sets from the targets seen in Figure 2.7 have a level of saturation somewhere between the 200V and 210V 252Cf data.

(a) Energy Change From Fix (b) Angle Change From Fix

Figure 3.10: (a) Percent of initial ADC added when correcting for the saturation as a function of initial cos θ. (b) The change in cos(θ) after ﬁlling adding the missing charge.

3.3.3 Electron Drift Speed

The electron charge cloud drifts through the fissionTPC fill gas at a constant velocity, and knowing this electron drift speed is important in determining the polar angle. Since electron drift speed is not measured directly in the fissionTPC, an estimated guess must be used when first reconstructing the data. Because reconstructing with a larger or smaller drift speed changes the polar angle, as depicted in Figure 3.11, the correct drift speed must be used. To find the drift speed for each target and volume, spontaneous alphas emitted from the actinide deposit are used as a known isotropic source which must have a flat distribution in

50 Figure 3.11: Changing the electron drift speed in the reconstruction changes the measured polar angle of the track.

cos θ. (The cosine function takes care of the solid angle in the detector geometry to yield a ﬂat distribution for isotropic sources, and because of this, cos θ is often used instead of just θ). The drift speed changes the polar angle of the fragment tracks by changing the length of the track in the z direction:

vmeas ∆Znew = (zend − zstart) (3.2) vreco

2 2 2 Lnew = ∆X +∆Y +∆Znew (3.3) p ∆Znew cos θnew = (3.4) Lnew where vmeas is the drift speed measured using the prescription shown in Figure 3.12, vreco is the drift speed used in the initial reconstruction of signals into tracks, and Lnew is the recalculated track length based on ∆Znew. Thus, due to this geometric relation between drift speed and the track polar angle, increasing or decreasing the electron drift speed will raise or lower the slope of the cos θ distribution, respectively. The drift speed can be changed until the slope of the line fit to the distribution is flat, thus giving an estimate for the electron drift speed, as seen in Figure 3.12. For the 95% Argon + 5% Isobutane at 550 Torr and 500 V/cm used in this data taking, the electron drift speed is approximately 3 cm/µs, but can vary by a few percent depending on the gas temperature and the voltage setting on the MICROMEGAS affecting the overall drift field. An error of ±0.01cm/µs is attributed to the measurement of the drift speed in each data set

51 (a) Spontaneous Alphas (b) Electron Drift Velocity Estimation

Figure 3.12: (a) An example fit to the spontaneous alpha distribution from a 238U source after being flattened by changing the drift speed values. (b) The slope in from fits of (a) as a function of the electron drift speed as well as showing sensitivity to different fit ranges. which takes into account variation based on the choice of fit range. It is assumed that fission fragment charge clouds and alpha charge clouds exhibit the same drift velocity. Lowering the drift field to roughly 300 V/cm actually increases the electron drift speed to roughly 4 cm/µs for this gas and pressure, but this lower field strength is not used because the assumption of equal drift speeds for alphas and fragments breaks down [56]. A significant portion of the fragment electron charge cloud is stretched towards the large amount of positive ions and the electron cloud is substantially elongated in the drift direction. See the discussion of the ”ghost tails” in [56] for more information.

3.3.4 Electron Diﬀusion Anisotropy

As the electron charge cloud drifts from the origination point, the electrons will diffuse throughout the gas, slightly changing the shape of the charge cloud by the time it is measured at the anodes. The larger the ionization density of the track and the longer the diffusion distance, the more the charge clouds will change, thus meaning that fission fragment charge clouds are more distorted from the electron diffusion compared to alpha particles which have ionization densities less than an order of magnitude smaller compared to the fission

52 fragments.

(a) Before Correction: 200V (b) Before Correction: 210V

Figure 3.13: Alpha and fragment polar angle distributions from a 252Cf source taken with the MICROMEGAS voltage set to (a) 200V and (b) 210V after setting the drift speed by ﬂattening the alpha distribution.

If the electron diffusion was the same in every direction, then the best fit of that charge cloud through the PCA analysis described in 3.1 would be the same, and looking at the fission fragment tracks visually, e.g. Figure 3.2, shows no obvious sign of the electron diffusion creating asymmetric tracks, indicating that the correction for the electron diffusion must be relatively small. However, comparing the spontaneous emission of alpha particles and fission fragments from a 252Cf source shows that even after setting the drift speed by flattening the alpha distribution, the fission fragment polar distribution is still not flat, as shown in Figure 3.13. There are two explanations for this: fission fragment charge clouds have a different drift speed compared to the alpha particles, or the electron diffusion is not symmetric about the track axis, or some combination of both. As discussed in 3.3.3, by increasing the drift field from 300 to 500 V/cm, the very clear stretching of the fragment charge cloud towards the positive ions was eliminated, indicating that the drift field is large enough to overpower the smaller local field created by the positive ions. This leaves the likely culprit to be the electron diffusion being anisotropic and creating a slight angle change which, when aggregated over many fission tracks, skews the fission fragment angular distribution.

53 The phrase “electron diffusion anisotropy” refers to the phenomenon that the electron diffusion coefficients are not the same in the drift and radial directions of the fissionTPC. Because the drift field is only being applied in the z-direction, the electron drift coefficients perpendicular (r) and parallel (z) to the field are not the same. This effect causes a shape distortion of the electron cloud as it drifts through the chamber to the anode pads. A correction can be made based on a Gaussian perturbation (which has different spreading, or Gaussian sigma, in the radial and drift directions) to the principal component tracking analysis that fits the charge clouds in the reconstruction. The correction to the polar angle, as derived in Appendix B, is given by

sin2 θ cos θ δ cos θ = (σ2 − σ2 ) (3.5) a2q(a)da/Q r0 z0

where σz0 and σr0 are the diffusionR parameters in the radial and drift directions respectively, q(a) is the charge deposited per unit length along the track, a is the distance along the track, and θ is the usual polar angle. It is important to note that in the derivation, a coordinate shift was imposed such that the a appearing in the denominator is in relation to the center of the charge cloud and not the start of the track. Looking at 3.5, many different variables need to be assigned for each fragment track that are not directly obvious, so stepping through each of the terms individually is a worthwhile exercise. Starting with the easiest, sin θ and cos θ can be computed directly from the reconstruction processes already described in 3.1, and the total amount of charge Q is just the total ADC value assigned to each fragment in the digit clustering stage. However, due to the large number of fission fragments taken and the extreme data requirements needed to save each of the digits, it would be impractical to save the complete Bragg curve q(a) for each fragment, thus requiring an approximation of the Bragg curve that can be used post-processing for this correction. The quantities that are saved relating to the Bragg curve that are accessible post-processing include the start and end positions of the track where q(a) drops roughly to zero, and the location and value of the Bragg peak. This leaves only

54 three points from which to approximate the Bragg curve for ﬁssion fragments.

Figure 3.14: An example Bragg curve showing the charge density along the a ﬁssion fragment track. The curve is approximated by a Gaussian start and a linear end using the start, end, and peak of the ionization proﬁle.

In order to approximate the Bragg curve with only these three points, a piece-wise function is constructed which includes a Gaussian before the Bragg peak and a linear tail, as seen in Figure 3.14. The center of the Gaussian used to approximate the start of the curve is set to the Bragg peak location and the amplitude is set as the value of the Bragg peak, while the width of the Gaussian is set such that the FWHM is 1.25 times the distance from the Bragg peak to the start of the track. For the linear tail, the y-intercept is set such that the function must match the first Gaussian half at the Bragg peak and the slope is set such that q(a)da = Q. This is done by first integrating the Gaussian start, comparing how muchR of the total ADC is left, and setting the slope such that the integration of the linear tail exactly matches the difference between the integral of the Gaussian and the total ADC.

55 Approximating q(a) with these functions allows for the straight forward calculation of the integral in the denominator of Equation 3.5 and also allows for the calculation of the average drift distance, which is given by 1 = l(a)q(a)da (3.6) Q Z where l(a)=5.4−a·cos θ and 5.4 cm is the maximum drift distance from the cathode to the anode. While the approximation of the Bragg curve using these two functions is certainly not perfect, these functions perform well in these integrals and the overall correction is insensitive to changes in the parameters of these functions. For example, changing the Gaussian FWHM from 1.25 to 2 times the distance from the Bragg peak and setting the slope of the linear tail such that it intersects with at the end point of the track rather than being set by the total ADC have no noticeable eﬀect on the ﬁnal correction shown in Figure 3.16.

(a) Fragment Charge Clouds (b) Fragment Cloud Projection

Figure 3.15: (a) A 2-D projection of the charge clouds of many ﬁssion fragments from the same angular bin overlaid and (b) projected along y axis. The average standard deviation for fragments in this angular bin is taken as the standard deviation of the (b) and, depending 2 2 on how the charge cloud was projected, used to extract σr0 or σz0 for the diﬀusion correction.

Values for the electron diﬀusion parameters can be taken by measuring the widths of the detected charge clouds, as demonstrated in Figure 3.15. The process is as follows. First, a run must be re-processed and saved with the digits. Using the fragment track data that is

56 extracted from the digits, the digits are shifted in (x,y,z) such that the track would appear to start from the origin and then each of the tracks are rotated in their polar and azimuthal angles such that they lie on top of each other. Projecting these charge clouds into a two- dimension histogram gives (a) in Figure 3.15. The charge cloud width, or σ, can then be calculated by a further one-dimensional projection, as demonstrated in (b). There are two different widths for asymmetric charge clouds, and both of these can be calculated depending on which way the charge clouds are projected. In particular, the charge clouds of the fission fragments that are emitted very parallel to the anode with 0 < cos θ<.05 have their principal axis pointing purely in the radial direction. That leaves then the other two principal axes to be pointing in either the drift direction or also in the radial direction. By measuring the widths of these particular fission fragment charge clouds, values for σr and σz are obtained. These widths are then directly related to the diffusion parameters needed in the diffusion correction by the drift length as

2 2 2 2 σr = σr0l(a) and σz = σz0l(a) (3.7) where the drift length l(a) for these fragments is 5.4 cm since they come out parallel to the anode.

252 The values for σr and σz measured from the Cf sample are 0.1545 cm and 0.134 cm for the 200 V setting on the MICROMEGAS, and 0.156 cm and 0.133 cm for the 210V setting, respectively. Because these charge clouds are measured directly from the digits of the fission fragments, they vary slightly with run conditions including the gain setting, individual trigger levels on the pads, and the total amount of energy deposited by fission fragments in the gas which varies between different actinides.

Thus, the measurements of σr0 and σz0 demonstrate a larger diffusion coefficient in the radial direction than in the drift direction, meaning that the electron diffusion correction computed from Equation 3.5 is non-zero. The magnitude of the correction is shown in Figure 3.16 where the double band structure comes from the light and heavy fragments having different ionization densities.

57 Figure 3.16: An example electron diﬀusion anisotropy correction for fragments, as calculated by 3.5.

(a) After Correction: 200V (b) After Correction: 210V

Figure 3.17: Alpha and fragment polar angle distributions froma 252Cf source taken with the MICROMEGAS voltage set to (a) 200V and (b) 210V after applying the electron diﬀusion anisotropy and drift speed corrections.

58 After this correction has been applied, the fission fragments and alpha polar distributions are flat within error, demonstrated by Figure 3.17. This is significant in that it demonstrates that the procedure outlined above successfully reproduces a known isotropic source for both fission fragments and alphas and shows that the fissionTPC has close enough to 100% efficiency for cos θ> 0.2 or 0.3 to properly reconstruct polar angle distributions. By doing this correction and verifying it with a 252Cf source, the anisotropy measurements in Chapter 5 do not need to be normalized by any additional efficiency curve as long as the fit to the angular distribution is above the cos θ turnoff due to losing fragments in the target material.

3.4 Correction Parameters Applied to Targets

Due to slight variations in operating conditions such as gain settings, gas temperature and pressure, and pad thresholds, each target and data set requires measurement of correction parameters. Table 3.1 shows the values for each data set used in the anisotropy and linear momentum analysis. The“BR” and “AR” designations refer to before and after the TPC rotation. The thin targets where not rotated. For the drift speed measurement, a reconstruction drift speed of 3.05 cm/µs was assumed. However, in the NIFFTE reconstruction code, there is a “ﬂoat” to “int” conversion to ensure an integer number of z buckets to ﬁll with charge in the digit formation processes, and this

conversion means that the actual vreco used in Equation 3.2 is 3.0682 cm/µs.

59 Table 3.1: : Parameter values used in the corrections for each target, actinide, and volume.

Target Actinide & Orientation Drift Speed [cm/µs] σr [cm] σz [cm] u8-up 2.752 0.159 0.137 u8-down 2.756 0.161 0.137 u8u5-thin u5-up 2.752 0.159 0.135 u5-down 2.756 0.161 0.137 p9-up 2.945 0.1585 0.1405 p9-down 2.883 0.1565 0.138 p9u5-thin u5-up 2.945 0.1595 0.138 u5-down 2.883 0.158 0.137 u5-up(AR) 3.085 0.1515 0.131 p9u5-thick u5-down(BR) 3.055 0.1525 0.1315 u8-up(BR) 3.072 0.1505 0.14 p9u8-thick u8-down(AR) 3.085 0.151 0.1405 u5-up(BR) 3.074 0.155 0.126 u5u5-thick u5-down(BR) 3.010 0.1545 0.135

60 CHAPTER 4 NEUTRON LINEAR MOMENTUM TRANSFER

In neutron induced fission, the incident neutron carries a defined amount of linear momentum, and upon interacting with the target nucleus, that momentum must be conserved. At low enough incident energies, the incident neutron is completely absorbed by the target nucleus which then forms an unstable compound nucleus that undergoes fission. This complete absorption of the incident neutron followed by fission implies that the totality of the linear momentum of the incident neutron must have been transferred to the compound nucleus, and subsequently to the fission fragments. As described in Chapter 1 however, pre-equilibrium emission and direct reactions complicate this picture such that the incident neutron does not transfer all of its momentum to the fission fragments. The effect of the linear momentum on the fragments can be seen when looking at the fission fragment’s folding angle. Because fission is essentially a two body decay (ternary fission is discussed Section 4.4), the two fission fragments must be emitted at an angle of 180 degrees with respect to each other in the rest frame of the fissioning nucleus. However, moving into the lab frame where the target nucleus was originally at rest, the compound nucleus is now moving as it acquired linear momentum from the incident neutron. This causes the fragments to receive a kinematic boost in the lab frame and decreases the folding angle to less than 180 degrees, as depicted in Figure 4.1. Therefore, an experimental measurement of the linear momentum transfer is feasible if a detector is able to measure the folding angle between fission fragments as a function of neutron energy. This chapter focuses on the fissionTPC’s ability to measure the fission fragment folding angle and also the analysis required to extract the average linear momentum transfer from the incident neutron to the fissioning nucleus from that measured folding angle.

61 Figure 4.1: A cartoon depiction of kinematic boost to the fission fragments in the lab frame. In the center of mass frame, the opening angle of the fission fragments are necessarily 180 degrees due to the fission process being essentially a two body decay.

4.1 Measuring Fragment Folding Angle Distributions

The measurement of fission fragment folding angles is made possible by the fissionTPC’s dual chambered design coupled with a deposit of actinide targets on a thin carbon backing. Two different targets featuring three different actinides were used for the linear momentum transfer analysis – a target with half-moons of 238U and 235U on the downstream side of the carbon backing and a target with half-moons of 239Pu and 235U on the upstream side of the carbon backing, as shown in Figure 2.7. Coincident fragments from these targets are considered to be those within the same digitized event, have start vertices that differ by less than 2 mm, and both have more than 3,000 ADC of energy. (The 3,000 ADC cut is used here for practical considerations to eliminate the large amounts of data associated with alpha tracks and proton recoils as well as to remove recoil ions that start in the upstream volume and traverse through the target into the downstream volume.) A good example of how coincident fragments look in the fissionTPC can be seen in Figure 3.2. 235U fragments that originate from the left half of the 238U/235U target have a starting vertex on top of two dead pads, and are therefore removed [29]. Also excluded from this analysis are all fragments that have a measured lab angle of

62 cos θ>.95 due to the saturation issue discussed in Section 3.3.2. Once coincidence fragments are selected, they are separated into incident neutron energy bins and their angles are corrected for drift speed and electron diﬀusion using the same procedures outlined in Sections 3.3.3 and 3.3.4 respectively. Then, an opening angle can be calculated by simply taking the angle between the upstream and downstream fragments. An example distribution is shown in Figure 4.2 with the Gaussian ﬁt to get the mean opening angle for this incident neutron energy bin.

Figure 4.2: An example opening angle distribution ﬁt with a Gaussian to get the mean value. The dashed red Gaussian is the wraparound that gets subtracted from the distribution.

The amount of neutron beam wraparound subtracted from the distribution is also shown in Figure 4.2 as the dashed Gaussian. The center of this wraparound Gaussian is placed at 180 degrees and the amplitude is the associated percentage of wraparound as determined using the method in 2.3.2. The wraparound corrected mean folding angle as a function of incident neutron energy is shown in Figure 4.3 where the vertical error bars represent the statistical error on the mean of the Gaussian ﬁt. (238U does not need a wraparound correction because it has a neutron induced ﬁssion threshold of ∼ 1.2MeV .)

63 Figure 4.3: Measured mean opening angle as a function of incident neutron energy with wraparound correction applied.

4.2 Simulation of Fragment Opening Angles

Now that the mean folding angle has been measured as a function of incident neutron energy, the question becomes: what momentum transfer of the incident neutron to the target nucleus is needed to reproduce the measured folding angle? If the pre-neutron emission fission fragment masses, lab energies, and lab angles are known on an event by event basis, the linear momentum needed to reproduce the measured folding angle can easily be solved using the equations and derivations shown in Appendix A. However, the fissionTPC does not measure the masses of the fragments, and the fissionTPC analysis framework is not yet developed enough for an accurate determination of the fragment energies considering the roughly 6000 channels that need to be gain matched and calibrated. Since the fissionTPC cannot measure all of the necessary components needed to determine the linear momentum transfer on an event by event basis, a new approach must be undertaken. That new approach consists of a Monte Carlo simulation, explained below and displayed in Figure 4.4. The simulation compiles a representative set of fission events for

64 each incident neutron energy bin and then adds linear momentum to that distribution until the mean folding angle of the simulated set of events matches the measured mean folding angle. The following steps are taken to simulate the folding angle of a single ﬁssion event.

Figure 4.4: Overview of how one ﬁssion event is simulated. See text for more details.

1. Since the folding angle is dependent on the cos θ distribution, the simulated fission polar angle distribution must be representative of the measured polar angle distribution. The simplest way to demonstrate that the folding angle is polar angle dependent is to consider two different fission events, one in which the fission fragments are emitted in the center of mass frame completely perpendicular to the incident neutron beam direction and the other in which they are emitted completely parallel to the incident neutron beam direction. Applying momentum in the direction of the neutron beam will cause the folding angle of the perpendicularly emitted fission fragments to change, but the parallel fragments will always be at 180 degrees.

65 To account for this polar angle dependence, a sample is selected randomly from the measured downstream cos θ distribution constructed from the fragments used to mea-

down sure the folding angle to give a starting θLab . This step will match polar angle de- pendencies between the simulation and the measured data to account for factors like absolute electron drift speed, detection eﬃciency for coincidence fragments, and angular anisotropy.

2. The initial downstream fragment angle is converted from the lab frame to the center of mass frame using the derivation in Appendix A and Equation A.11, restated here for convenience.

EL CM d,u 2 L cos θd,u = 1 − CM (1 − cos θd,u). (4.1) s Ed,u L Solving A.8 for Ed provides the input for 4.1

m p2 1 m p cos θL 2ECM m2 EL = ECM + d n cos2 θL − + n n d CN + p2 (cos2 θL − 1). d d m2 d 2 m2 m n d CN CN s d (4.2) The input parameters are then the total kinetic energy in the center of mass frame

CM CM (Ed + Eu ), the mass of each fragment (md,u), and a selected momentum transfer

of the neutron (pn) which will be varied in order to deduce the mean of the measured distribution. Mass distributions are taken from the GEF code [21], but it should be noted that the GEF code does not provide distributions past neutron energies of 100 MeV , so the energy bins past that threshold use the last available energy bin distribution. This is not an issue due to the mass distribution at these energies being primarily symmetric ﬁssion, and a broader or sharper symmetric ﬁssion hump does not change the mean of the folding angle calculated. Total kinetic energy distributions are also taken from the GEF code for consistency. Note that since the opening angle is determined before prompt neutron emission, the pre-neutron emission TKE is used.

66 The TKE is split between the fragments in such a way that conserves the momentum of the two body decay in the center of mass frame:

CM CM Etot md,u Ed,u = (4.3) md + mu

CM where Etot is the total kinetic energy and md,u refers to the downstream and upstream fragment masses.

3. Since the ﬁssion process is essentially a two body decay, a 180 degree folding angle is assumed in the center of mass frame for the upstream fragment. This then automatically

up down gives the upstream center of mass angle: θCM = θCM .

4. Using the same input parameters as used in Step 2, the upstream fragment is converted back from the center of mass frame to the lab frame using A.8 and A.9.

5. The opening angle for the ﬁssion event is then calculated via

down up θopen = 180+ θLab − θLab (4.4)

and Gaussian smearing is added to θopen with a sigma of 3.8 degrees to match the detector resolution. No smearing will produce a double humped distribution due to the diﬀerence in opening angle depending on whether the large or small fragment is emitted downstream. By smearing, these eﬀects are averaged out, as seen in the measured data.

All of these above steps are used to simulate a single fission event, and are repeated to get a distribution of simulated folding angles for a particular incident neutron energy bin. The particular choice of momentum transfer that is inserted into these simulated events is also changed, allowing for the average simulated opening angle to be plotted as a function of chosen momentum transferred for each measured incident neutron energy bin. These simulated folding angle distributions are then fit and the mean value plotted as a function of the chosen momentum transfer, as shown in Figure 4.5. A line is fit to the average

67 Figure 4.5: Simulated ﬁssion events and their folding angle as a function of momentum transferred for a particular incident neutron energy bin.

68 opening angle as a function of neutron momentum transfer and the ﬁt is then used to extract what momentum transfer would produce the measured folding angle. This process can also be seen in Figure 4.5 where the dashed horizontal lines represent the wraparound corrected folding angle measurement for the particular neutron energy bin with associated error taken from Figure 4.2. Where these lines intersect with the ﬁt is the amount of neutron momentum that is required to produce the measured folding angle, as designated by the dashed vertical lines. The vertical red line denotes the mean incident neutron momentum for the particular neutron energy bin, calculated via

E2 +2E m c2 p = n n n . (4.5) p c This process is performed for all incident neutron energy bins to extract the momentum transfer as a function of incident neutron energy. Thus, from this simulation procedure, the amount of linear momentum for a given folding angle measurement can be extracted; but before showing the results, errors must be discussed and quantiﬁed.

4.3 Linear Momentum Transfer Error

In order to estimate the error on the momentum transfer measurement and simulation, a variational approach was taken with the majority of input parameters that went into both the simulation and the measurement of the folding angle. This approach consists of remeasuring the opening angle and rerunning the simulation using the error range of each input parameter. The error is then taken as the diﬀerence between that set and the primary set of parameters. The uncertainties considered are as follows:

• Statistics: Statistical errors from the folding angle measurement are taken directly from the error on the mean of the Gaussian ﬁt, and the magnitude of the statistical error can be seen directly in Figure 4.3. This statistical error is propagated through to the ﬁnal linear momentum transfer result by matching the simulated folding angle to the low and high estimates of the measured opening angles. This can be seen in Figure 4.5, where the two horizontal dashed lines represent the statistical error range

69 of the measured folding angle distribution and the two vertical dashed lines are the resultant momentum transfer range. The mean result for the linear momentum transfer is computed as the average of the high and low statistical errors, i.e. the middle of the two vertical lines in Figure 4.5.

• Wraparound: The error on the magnitude of events coming from the neutron beam wraparound comes directly from uncertainty on the wraparound ﬁt parameters. Di- rectly propagating those uncertainties through the integration of the ﬁt produces the percent wraparound error seen in Figure 2.11. The measured folding angle distributions then have their folding angle distributions subtracted more or less in accordance of the error in the amount of wraparound, i.e. the dashed Gaussian in Figure 4.2 has its amplitude changed according to the wraparound uncertainty. Since this is not an input into the simulation, the shifted measured folding angles are evaluated just like the primary values, as shown in Figure 4.5. Figure 4.6 shows an example for the linear momentum transfer with and without the wraparound correction as well as the small variation due to the error in the wraparound magnitude.

• Total Kinetic Energy: Data sets of measured total kinetic energy of the fission fragments do not have particularly good agreement [69]. Additionally, measurements of total kinetic energy even for these widely studied actinides is scarce at very high incident neutron energies. Thus, as was mentioned earlier, the GEF TKE input was used for all three actinides in order to be consistent. To account for the likely possibility that these values are systematically off, the simulation was re-run for TKE values ±2MeV from the GEF result. This range should be large enough to encompass the difference between GEF and the measured experiments that GEF benchmarks itself on, as well as experimental systematic errors missed in the measurements.

• Mass of the Compound Nucleus: The mass of the compound nucleus used is A+1, but considering incomplete momentum transfer stems from one or more particles being

70 Figure 4.6: (a) The diﬀerence between wraparound variations for the 239Pu target as well as what the linear momentum transfer would look like without the wraparound correction. The dashed line represents full momentum transfer.

emitted before fission, including the effect of a smaller fissioning nucleus is important. By reducing the mass of the compound nucleus by 5 amu, and taking that 5 amu from either the upstream or the downstream fragment, the effect of losing mass can be estimated. This term also encompasses the mass distributions from GEF. The variation from a 5 amu difference between the upstream and downstream fragment will have a larger overall effect on the opening angle than if the large or small mass peak of the GEF code were to have moved by 5 amu. Due to the assumption that the heavy and light masses have equal probability of being emitted downstream or upstream, just moving the center of the large and small mass peaks from GEF would average out and no change on the linear momentum transfer result would be observed. Taking the 5 amu only from the upstream or downstream fragment instead ensures maximum impact of the mass distributions changing.

71 • Electron Drift Speed As shown in Section 3.3.3, the electron drift speed will change the polar angle distribution of the fragments. Considering the downstream ﬁssion fragment angular distribution is an input to the simulation, varying the downstream drift speed will demonstrate the sensitivity of the linear momentum result to the input angular distribution. Input values for the drift speed are varied by ±.03cm/µs, or about one percent of the absolute value.

Figure 4.7: Effects of changing the electron diffusion by ±10% to the 235U from the 238U/235U target. Looking at the fractional momentum transfer better showcases the differences at low incident neutron energies. The dashed line denotes full momentum transfer.

• Electron Diffusion Anisotropy As discussed in 3.3.4, electron diffusion also affects the polar angle distribution of the fragments. In order to account for the error in this correction, a 10% variation in the correction strength was applied. In other words, δ cos θ from 3.5 was multiplied by factors of 1.1 or 0.9 and the folding angle and linear momentum transfer results are re-computed to determine the sensitivity to the electron diffusion anisotropy correction. An example of the magnitude of this correction is demonstrated in Figure 4.7

72 The error associated with each of the above parameters is taken as the diﬀerence of the momentum calculated with the expected parameter set and the momentum calculated when changing the speciﬁc parameter, or σparameter = pmain − pparameter. Not all parameters have a linear relationship to the linear momentum transfer result, so σparameter could be asymmetric about the main result. For example, the mass of the compound nucleus can only decrease from A+1 since there is no way for the target actinide to gain additional nucleons beyond the incident neutron. All of the σparameter’s are treated as independent and are summed in quadrature to get a σtotal. Because of the asymmetry introduced by some of the parameters,

σtotal is also asymmetric about the mean.

Figure 4.8: Error budgets for the linear momentum transfer measurements from the 238U/235U target (top) and the 239Pu/235U target. See the text for the explanation of the methodology.

73 Figure 4.8 shows the uncertainty budgets for all of the linear momentum measurements

+ − taken. The percent error is calculated by dividing the largest of σtotal or σtotal by the momentum transfer result. At low incident neutron energies, statistical errors are very large due to the very precise measurement of the folding angle required to be able to determine the difference in linear momentum transfer between low energy incident neutrons. A few other sources of systematic errors were considered, but not included. First, this momentum transfer analysis has assumed that fission is a two body decay, but this assumption ignores ternary fission [11]. Ternary fission, the process by which three particles are emitted simultaneously during the scission process – typically two fragments and an additional high energy (10+ MeV) alpha particle – necessarily changes the folding angle of the two primary fission fragments as the ternary particle carries away a significant amount of momentum. However, ternary fission consists of less than 1% of all fission events [11], and on this fact alone, can be ignored in the momentum transfer analysis. Even if ternary fission had a high enough cross section to merit significant statistical impact, there is no evidence for an angular relation of the ternary particle with respect to an incident neutron beam. While it is well documented that ternary particles have the highest probability to be emitted perpendicular to the primary fission axis, there is no evidence to suggest a preference of this particle to be emitted upstream or downstream with respect to the beam. This means that the mean folding angle of the fission distribution is not affected; and since the measurement presented here focuses on the average linear momentum transfer, this measurement is also unaffected. Another source of possible systematic error is the emission of neutrons from the fission fragments changing the measured folding angle. Considering how fast fragments produce the electron charge cloud, only prompt neutron emission can affect the measured folding angle. But, just like for ternary fission, neutron emission can only impact the folding angle measurement if there is an emission asymmetry with respect to the incident neutron beam. It is generally assumed that neutron emission is emitted isotropically in the rest frame of the

74 fission fragment it is emitted from, but there are two competing phenomenon that challenge this assumption. Angular momentum of the fission fragment can change the isotropic emission of the neutrons, but the manner in which this occurs is very similar to the kinematic boost the neutron receives from the moving fission fragment and is thus difficult to disen- tangle, especially since the effect is small. The other possibility is the existence of scission neutrons which actually act in a manner which looks like a decrease in the kinematic focusing effect. Both of these effects are very small, and their existence and scale are debated. An extensive study and discussion can be seen in [70]. Whatever the outcome of the debate, it is safe to say that neutron emission is a negligible effect for this work. Having the two different targets placed on opposite sides of the carbon backing with respect to the beam also eliminates another possible systematic error relating to fragments passing through the carbon backing. It is possible that a portion of fragments that pass through the carbon are Coulomb scattered off the backing material in a way that systematically changes the measurement of the folding angle. The 238U/235U actinide was in the downstream volume and the 239Pu/235U target was in the upstream volume; but for both targets, the Monte Carlo simulation samples from the downstream cos θ distribution. Since there is no statistically significant difference between the two targets in the measurement of the linear momentum transfer result shown in Figure 4.10, this means that any possible bias introduced by the carbon backing is not a problem.

4.4 Linear Momentum Transfer Results and Discussion

The ﬁnal step to the linear momentum transfer result consists of forcing the folding angle measurement to give full momentum transfer for incident neutrons with less than two MeV. All of the linear momentum from incident neutrons below two MeV must be absorbed by the target nucleus because pre-equilibrium emission and direct reactions have energy thresholds higher than two MeV. However, such small linear momentum transfer requires a very precise measurement of the fragment folding angle. All of the errors discussed in 4.3 are applied to both the upstream

75 Figure 4.9: Fraction of the total incident linear momentum transferred to the ﬁssion fragments as a function of incident neutron energy. The weighted average for each actinide set to one for points below 2 MeV by adjusting the drift speed diﬀerence between the upstream and downstream volumes.

76 and downstream volume simultaneously, but the folding angle measurement depends more on the differences between the upstream and downstream volumes. Due to the electron drift speed and diffusion corrections not being accurate enough, an additional electron drift speed difference was applied to the actinides such that the linear momentum transfer for low energy incident neutrons is complete. The required drift speed difference was implemented as a subtraction of the upstream drift speed by 0.070 cm/µs for the 235U/238U target, and 0.107 cm/µs and 0.075 cm/µs for the 235U and 239Pu target, respectively. These corrections are roughly 2% of the drift speed measured in Section 3.3.3. While this drift speed difference applied to the two sides is greater than the 0.03 cm/µs error that is associated with the drift speed measurement, it also compensates for the uncertainties in the electron diffusion between the two volumes. This “normalization” procedure does not apply to the 238U measurement because of its neutron induced fission threshold at roughly 1.2MeV , so the drift speed difference between the upstream and downstream volumes is set to the same as for the 235U half that is sharing the drift chambers. However, this threshold also means the 238U has no wraparound neutron events contaminating the photo-fission peak. Since the 238U nucleus has such a high mass compared to the amount of momentum a several MeV photon can impart, there should be very minimal effect. Measuring the folding angle for these photon induced fission events after the normalization confirms this hypothesis with a measurement of 179.98 ± 0.14 degrees – consistent with no momentum transfer. Implementing this normalization changes the measured folding angle by only very small fractions of a degree, but since that precision is required to distinguish small momentum transfer differences, it makes a big impact in the fractional momentum transfer at low energies shown in Figure 4.9. On the other hand, very minute differences in the folding angle does not actually correspond to large differences in the absolute momentum transferred, meaning that this normalization plays an almost insignificant role for incident neutron energies greater than 10 MeV. So, the very large deviation from full momentum shown in Figure 4.10 at

77 high incident neutron energies, arguably the most important and impactful region of this measurement, is immune to any issues associated with the normalization.

Figure 4.10: Momentum transfer plotted against other published data, including alpha induced fission [71, 72], deuteron induced fission [72], and proton induced fission [71–74]. To the best of our knowledge, this is the first measurement of linear momentum transfer from neutron induced fission.

The result of the simulation, error calculations, and drift speed normalization can be seen in Figure 4.10, along with other published data of momentum transferred to fissioning nuclei for proton, deuteron, and alpha induced fission. (Tabulated data of this measurement can be found in Table C.1, Table C.2, Table C.3, and Table C.4 in Appendix C.) Based on our literature search, this may be the first measurement of linear momentum transfer by a neutron to a fissioning nucleus and appears to show a different trend than the previous measurements of proton induced fission.

78 An in-depth theoretical study of why neutrons linear momentum transfer appears to differ greatly from proton linear momentum transfer is outside the scope of this thesis, but this difference may have something to do with the fact that protons are charged and neutrons are not, so their interactions with the target are necessarily different. An additional study on whether the small deviation from full momentum transfer in Figure 4.9 in the energy range from a few MeV to about 20MeV is related to incident linear momentum being converted to angular momentum of the fissioning nucleus is also outside the scope of this thesis, but is hypothesized here for completeness.

79 CHAPTER 5 FISSION FRAGMENT ANGULAR ANISOTROPY

As described in Section 1.3, fission fragment emission is not isotropic due to the angular momentum of the fissioning nucleus. Measuring just how anisotropic the fragments are is an important input into cross section measurements and provides valuable insight into the state of the fissioning nucleus at scission. The amount of anisotropy is described by the anisotropy parameter, expressed as

W [cos θ = 1] Counts parallel to neutron beam A = = (5.1) W [cos θ = 0] Counts perpendicular to neutron beam where W is a sum of even order Legendre polynomials ﬁt to the center of mass polar angle distribution of the fragments. This chapter will describe 235U and 238U anisotropy as measured by the ﬁssionTPC and compare them to previous measurement techniques.

5.1 Measured Anisotropy

Considering the fissionTPC measures the polar angles directly from the charge cloud fitting described in 3.1, the actual procedure of measuring the anisotropy after that point is straight forward – separate the fragments into incident neutron energy bins, convert from the lab frame to the center of mass frame, and then fit the polar angle distributions and calculate the anisotropy. However, some nuance goes into each step. First of all, fission fragment events must be differentiated from spontaneous alpha decays from the target and recoil ions from neutron collisions with various gas and detector atoms. The method applied to this anisotropy analysis consists of simply making a cut on the peak of the Bragg curve to select fission fragments. Recoil ions cannot produce as high of dE/dx as the fission fragments since they are not as positively charged and less energetic, separating them in the parameter space shown in Figure 5.1. Fragments that come out at very low cos θ begin to overlap with the Argon recoils, limiting the low cos θ fit range.

80 Figure 5.1: Track length vs the maximum value of the Bragg curve from a 235U downstream data set containing the recoil ions stripes on the left and the double humped ﬁssion distribution on the right. The dashed vertical line is the cut used to separate the ﬁssion fragments from the recoil ions. Note the log scale in bin intensity.

Next, the drift speed and electron diffusion corrections from Chapter 3 must be applied to all of the fission fragments. The values used in these corrections for each target and actinide are listed in Table 3.1. Successfully performing these corrections on the 252Cf data has significant implications for the anisotropy measurement presented here in that the efficiency of detecting a fragment as a function of polar angle must be understood before the fitting procedure can take place. Flattening the spontaneous alpha and fission fragment cos θ distributions simultaneously for the 252Cf measurement demonstrates that the fissionTPC has full detection efficiency over the fit range. This means that no additional efficiency normalization is required in the anisotropy measurement presented here. (235U angular anisotropy was measured before in the fissionTPC, but that measurement did not have a correction for the electron diffusion anisotropy and thus had to do an efficiency normalization by dividing every energy bin by the angular distribution of the lowest energy bin [75]. However, as will be shown in 5.4, the assumption of isotropic emission at 0.13 MeV is not supported by our data.)

81 After applying the corrections from Chapter 3, a conversion from the lab angle into the center of mass frame is done on an event by event basis. The equations used to convert from the lab frame to the center of mass frame are derived in Appendix A, but the necessary equations are stated here for convenience:

2 CM L pnmd,u pn L L Ed,u = Ed,u + 2 ∓ 2md,uEd,ucosθd,u (5.2) 2mCN mCN q EL CM d,u 2 L cosθd,u = 1 − CM (1 − cos θd,u). (5.3) s Ed,u where the unknown parameters not directly measured by the fissionTPC for each event are the mass of the compound nucleus and fragments, and the momentum transferred by the neutron. Fragment masses are taken to be the mean masses for 14 MeV neutron induced fission of 235U [11], namely 95.5 amu for the high energy peak and 137 amu for the high energy peak, while the compound nucleus mass is taken to be A+1. The amount of momentum transferred by the neutron to the fragment is taken from the results of Chapter 4. Once all corrections are made and the angles converted into the center of mass frame, the fragments are sorted into incident neutron energy bins and their polar angle distributions are plotted, as seen in Figure 5.2. There are several observed features to point out in this plot. First, the drop-off at low cos θ is due to fragments that do not make it out of the sample material at shallow angles. The surface roughness of the target primary determines the shape of this drop-off. Increasing surface roughness will cause more “mountains and valleys” and therefore more fragments will be born in a valley and hit mountains, causing no ionization in the gas and thus will not be detected [29]. The second feature to notice in Figure 5.2 is the “bump” near cos θ = 1. This bump is due to fragments that deposit all of their energy on only a few number of pads and those pads saturate, as discussed in Section 3.3.2. Since the signals are differentiated in the reconstruction process, this saturation causes a hollow structure in the center of the digit cloud. Because the tracking algorithms are weighted by the ADC count of each digit, having a hole in the center of the charge cloud skews the angle of the track and creates the

82 Figure 5.2: Example polar angle distribution for a particular incident neutron energy bin from the a 235U downstream target after all the corrections are made. The red line shows the ﬁt to the data and the two ranges highlighted by the vertical lines show the intervals for the min and max ﬁt ranges employed in the error analysis.

83 bump. Note that the saturation is only a problem for the fragments that are emitted directly towards the anode as those are the only events that are able to deposit enough charge onto a single pad in order to saturate it. As mentioned, a sum of even order Legendre polynomials (to preserve the upstream and downstream symmetry inherent in the two body ﬁssion decay) is used to ﬁt the angular distributions, namely

W (cos θCM )= a0 + a2L2(cos θCM )+ a4L4(cos θCM ) (5.4) where 1 L (cos θ )= (3cos2 θ − 1) (5.5) 2 CM 2 CM 1 L (cos θ )= (35 cos4 θ − 30cos2 θ +3) (5.6) 4 CM 8 CM CM

Solving for W (cos θCM = 1)/W (cos θCM = 0) gives the anisotropy parameter

a0 + a2 + a4 Ameas = (5.7) a0 − a2/2+3a4/8 However, neutron wraparound is included in that measurement and needs to be corrected for. The eﬀect of the anisotropy of the wraparound events is

Ameas = (1 − pwrap)Areal + pwrapAwrap (5.8) where pwrap is the percentage of wraparound events in each neutron energy bin, as determined in Figure 2.11, Awrap is the anisotropy of the wraparound events, and Areal is the actual anisotropy. Working under the preliminary assumption that the low energy wraparound neutrons produce isotropic ﬁssion, then Awrap = 1 and solving for Areal gives

Ameas − pwrap Areal = . (5.9) 1 − pwrap

5.2 Uncertainty Budget

Much like the linear momentum transfer analysis, a variational approach to the systematic errors is used to gauge the sensitivity of most of the parameter choices that go into

84 measuring the anisotropy. This means that the angular distributions are remade using different correction parameters or cuts and then re-ﬁt. How the anisotropy result changes based on the changing of each parameter provides the error estimate related to each term. The errors that are considered are as follows:

• Statistics: The statistical error is calculated by standard error propagation

∂A ∂A ∂A σ = ( )2δa2 +( )2δa2 +( )2δa2 (5.10) stat ∂a 0 ∂a 2 ∂a 4 r 0 2 4 where the errors of each parameter errors comes from the ﬁt and the partial derivatives are

∂A −8(12a2 +5a4) = 2 (5.11) ∂a0 (8a0 − 4a2 +3a4)

∂A 8(12a0 +7a4) = 2 (5.12) ∂a2 (8a0 − 4a2 +3a4)

∂A 8(5a0 +7a2) = 2 (5.13) ∂a4 (8a0 − 4a2 +3a4) In general, the neutron energy binning is such that the statistical error is roughly constant at around 2% for all data sets at all energies, as seen in Figure 5.8.

• Wraparound: The error term associated with the wraparound can be calculated directly through standard error propagation as

∂A 1 − A σ = real δp = meas δp (5.14) wrap ∂p wrap p2 − 2p +1 wrap wrap wrap wrap where δpwrap are the vertical error bars shown in Figure 2.11.

• Fragment Selection: As seen in Figure 5.1, a simple cut on the value of the track’s Bragg peak was used to select fragments. This energy threshold was varied by ± 50 ADC/mm to get an approximation of the error associated with this cut. It should be mentioned that the fragment events that appear at lower Bragg peak values are those that are emitted very perpendicular to the beam at low cos θ and lose a lot of their

85 Figure 5.3: Variations in the 235U/235U downstream anisotropy due to changes in fragment selection.

energy through the target material. Thus, the fragment selection cut can dictate how low the ﬁt can go in cos θ. On the other hand, the cut needs to be high enough to exclude the argon recoils that overlap with the energy straggled ﬁssion fragments in Figure 5.1. Argon recoils that appear in this overlap region are very forward peaked and come from high energy neutrons as those are the only way to get such energetic Argon recoils. Because of this, the last few energy bins see an asymmetric increase in error towards higher anisotropy values due to the inclusion of some high energy argon recoils as seen in Figure 5.3.

• Fit Range: The minimum and maximum fit location is demonstrated in Figure 5.2 where the solid vertical lines denote the chosen fit range and the error associated with the minimum and maximum fit range is ±0.03cos θ, denoted by the vertical dashed lines. Each one of the four dashed lines are varied independently to provide separate estimates for both the minimum and maximum fit locations. For all of the taken anisotropy data, the maximum fit location has a larger associated error than the minimum fit location because of two factors: first, the bump at large cos θ from the

86 Figure 5.4: Variations in the 235U/235U downstream anisotropy due to changes in the maximum ﬁt range.

pad saturation discussed in Section 3.3.2, and second, the fact that the fourth order Legendre polynomial plays a more significant role at high cos θ, also compounding the saturation issue. An example of how much the anisotropy varies with changes with the maximum fit range is seen in Figure 5.4. Very little change in the anisotropy is seen for varying the low cos θ fit range even though the minimum value is very close to the efficiency falloff from fragment straggling in the target.

• Lab to Center of Mass Frame: The value of the linear momentum transfer applied to the ﬁssioning nucleus input into 5.2 has an uncertainty on it from the error analysis

done in Chapter 4. New polar angle distributions are formed for the values of pn ± δpn and re-ﬁt to determine the error based on the conversion from the lab to center of mass frame. Anisotropy values assuming full momentum transfer and in the lab frame are also computed and compared to the anisotropy using the results from the linear momentum transfer in Figure 5.5. The example shown is in the downstream volume, meaning that the lab frame anisotropy increased due to the kinematic focusing. If an

87 upstream example were also shown, the opposite would be true and the black lab frame line would be on the bottom instead of the maroon full momentum transfer line.

Figure 5.5: Variations in the 235U/235U downstream anisotropy due to error in the linear momentum transfer measurement as well as lab frame anisotropy and anisotropy calculated from the assumption of full momentum transfer.

• Electron Drift Speed: The method to determine the electron drift speed is laid out in Section 3.3.3, and, based on that measurement technique, an error of ±.01cm/µs is associated with that measurement for the anisotropy analysis. Again, new angular distributions are prepared based on the different drift speed corrections and the error due to the drift speed is taken as the difference in anisotropy between those distributions and the ones made from the central drift speed value. Because the effect of drift speed on the angular distributions is roughly degenerate with the second order Legen- dre polynomial, the uncertainty from the drift speed measurement is roughly constant across all neutron energy bins at 1% of the anisotropy.

• Electron Diﬀusion: Based on the assumptions of the electron diﬀusion correction derived in Appendix B and the procedure described in 3.3.4, a 10% error was assigned to the correction, meaning new anisotropy values are computed by multiplying δ cos θ

88 Figure 5.6: Variations in the 235U/235U downstream anisotropy due to error in the electron diﬀusion correction as well as anisotropy measured without applying the correction.

in Equation 3.5 by 1.1 and 0.9. Like the electron drift speed, this electron diffusion correction is also roughly degenerate with the second order Legendre component leading to a roughly constant error contribution across all incident neutron energies. Shown in Figure 5.6 is an example showing the difference between anisotropy values with and without the diffusion corrections applied.

The difference between fitting the angular distributions with second and fourth order Legendre polynomials to compute the anisotropy is not considered in this error analysis for a few reasons. First, evidence of a fourth order component in the anisotropy has been shown, particularly in anisotropy values deviating drastically from one [76]. Such strong evidence for a fourth order component can also be seen for this work in Figure 5.11 where the 238U anisotropy values peak. Second, if there is no fourth order component, the fit should produce a minimal contribution from the fourth order term. This can also be seen in Figure 5.11 where the fourth order component of 235U is small or consistent with zero through the entire incident energy range. The difference between fourth order contributions in 238U and 235U suggests that when a fourth order term is important for large anisotropy

89 Figure 5.7: Variations in the 235U/235U downstream anisotropy due to fitting with only a second order or with second and fourth order Legendre polynomials. values, it will turn on and be accounted for, otherwise the anisotropy is unaffected. This is demonstrated in Figure 5.7 where the difference between the anisotropy calculated with fits of second or fourth order components is smaller than the statistical errors of the fit. In summary, fitting with a fourth order component is necessary to describe large anisotropy values and in cases where the fourth order component is small, the anisotropy values are statistically not different. Fitting with Legendre components larger than fourth order is not applicable due to the limited fit range the current fissionTPC measurements are restricted to considering the extrapolation of these higher order terms would produce large fluctuations in the resultant anisotropy values. Just like in the uncertainty analysis of the linear momentum transfer, the error of each of the stated parameters is taken as the difference between the “main” anisotropy result and the anisotropy resultant from the parameter change. This leads to asymmetric errors for some components, especially for fragment selection at high incident neutron energies and the maximum fit location interacting with the saturation bump. All of the estimated errors from each of the different parameters are treated as independent and added in quadrature

90 Figure 5.8: Uncertainty budgets from anisotropy measurement from target: 235Pu/235U thick (1) upstream and (2) downstream, 235U/235U thick (3) upstream and (4) downstream, 235U/238U thin downstream (5) 238U and (6)235U, and 239Pu/238U thick (7) upstream and (8) downstream. See Figure 2.7 for target list. 238U uncertainties blow up below 1.2 MeV due being below the fission threshold. 91 to get an estimate of the total error, which can be seen in Figure 5.8. Counting statistics for bins below 1.2 MeV for the 238U measurements drop off by about three orders of magnitude due to being below the fission threshold. At energies below this, the fraction of a percent of other actinide contaminates in the target to provide a few events. Any points below 1.2 MeV are not reported for 238U. Also, due to the poor counting statistics of the 235U anisotropy from the thin backed 235U/238U source, the last two highest energy points are removed, as seen in Figure 5.9. Statistics is essentially the limiting factor on the error of the anisotropy measurements, which constitutes about a two percent error on the anisotropy result. After statistics, the next error that contributes the most is the maximum fit location. This is due to the saturation bump explained in Section 3.3.2 at high cos θ and the fourth order Legendre polynomial having more impact at larger cos θ. The maximum fit range is also closely related to the statistics – data sets with less statistics generally have a larger maximum fit location error. Since the angular distributions are refit with a different fit range, it makes sense that the fluctuations related to changing the maximum fit location are roughly the same magnitude as the statistical error of the fit. Considering these fluctuations are not seen when changing the minimum fit range, the majority of the fit uncertainty must be related to the high cos θ region. The drift speed measurement detailed in Section 3.3.3 contributes a roughly one percent error across all energy ranges. At high energies, fragment selection can surpass all other errors considering the high energy recoil ions can pass the fragment cut and add more counts towards high cos θ (since the recoil ions are focused in the beam direction). All of the other errors considered here have magnitudes less than one percent of the anisotropy result and are not substantial contributors to the total error.

5.3 Multiple Anisotropy Measurements

All of the error analysis is done for each anisotropy measurement from each target in Figure 2.7. The thin-backed target containing 239Pu/235U was not used in the anisotropy analysis due to a different cathode amplifier not being able to assign neutron time of flight

92 to all of the fission events at lower cos θ, decreasing the efficiency and making an anisotropy measurement impossible. (This target could still be used for the linear momentum transfer analysis considering the simulation’s first step is sampling from the cos θ distribution, thereby removing any dependence on polar angle efficiency.) Additionally, the volume of the thin targets opposite of the actinide deposit are also not great for anisotropy measurements as fragments at low cos θ are more likely to not make it through the carbon backing and be detected, thus breaking the polar efficiency demonstrated by the 252Cf measurement.

(a) 235U Anisotropy Measurements (b) 238U Anisotropy Measurements

Figure 5.9: Anisotropy results from each target for (a)235U and (b)238U.

For all of the thick-backed targets, the fissionTPC was picked up and rotated 180 degrees with respect to the neutron beam, as described in Section 2.2. This means that for all thick targets, the kinematic boost was applied in the opposite direction, providing a good test for the application of the linear momentum transfer measurement in Chapter 4, as well as providing more quality data to measure the anisotropy. Since there is no statistically significant difference between the upstream and downstream anisotropy measurements for both 235U and 238U, this confirms that the linear momentum transfer measurement from Chapter 4 is appropriate. In general, there is good agreement between all of the anisotropy measurements through all energy bins. Looking at the 235U anisotropy results, the black dots corresponding to the

93 Figure 5.10: Polar angle distributions and fourth order ﬁts to a variety of neutron energy bins. Top row shows a comparison of the upstream (blue) and downstream (green) 235U measurements of the 235U/235U thick target. Middle row compares the downstream 235U measurements from the thin backed 235U/238U target (blue) and the thick backed 239Pu/235U target (green). Bottom row compares the upstream (blue) and downstream (green) 238U measurements from the thick backed 235Pu/238U target.

94 one thin target measurement seem to be systematically higher than the other targets. There are a few differences between this target and other measurements, namely that this target was on a thin-backing (although the 238U half does not appear systematically high compared to the thick backed measurements), and a vertex cut on 235U was applied to remove the left half of the actinide deposit due to two dead pads on that side (see [29]). However, these differences are not expected to cause any systematic differences and looking at the angular distributions, as in Figure 5.10, shows no red flags. A possible explanation for the systematic offset, and the one assumed here, is that a few parameters that systematically shift the anisotropy happen to be at the edge of their suspected errors, thus producing a result that is systematically different, in accordance with the general error bar agreement. The electron drift speed and diffusion corrections, for example, both produce roughly constant systematic shifts across all energy bins.

(a) 235U Anisotropy Measurements (b) 238U Anisotropy Measurements

Figure 5.11: Second and fourth order Legendre components compared for (a) 235U anisotropy from the thick 235U/235U target and (b) 238U anisotropy from the thick 239Pu/238U target where the blue and green squares are the second order Legendre component for upstream and downstream measurements, and the cyan and black circles are the fourth order Legendre components for upstream and downstream measurements, respectively.

Comparisons for individual polar angle distributions between measurements can be seen in Figure 5.10. These angular distributions are normalized such that the ﬁts will intersect on the y-axis at one, meaning that the anisotropy (pre-wraparound correction) is just the

95 value of the fit at cos θ = 1. Another comparison between data sets that highlights the earlier discussion in 5.2 on the fit order is to plot the second and fourth order Legendre fit components, as in Figure 5.11.

5.4 Measured Anisotropy Results

The advantage of having multiple data sets to measure the anisotropy, beyond just showing reproducibility, is that additional statistics can be performed to get a more accurate measure of the measured anisotropy results. Namely, a weighted average is used between all the data sets to provide the best estimate

w A A¯ = i i (5.15) w P i where the weights are given by P

2 2 w = (5.16) i σ+ + σ− i i which is replacing typical σ used as the weight with the average between the two asymmetric error bars. The error on the best estimate is the greater of the standard weighted average error 1 σA¯ = (5.17) wi or the weighted standard deviation pP

2 wi(Ai − A¯) σ ¯ = (5.18) A (N−1) P wi sP N where N is the number of data points with non-zero weight. The need to compute two diﬀerent errors arises from the typical weighted average error not taking into account the spread of the data points and the weighted standard deviation giving an error of zero if all points lie on top of each other, regardless of the size of the error bars. Taking the largest error between these two methods ensures that both the spread of the data and size of the error bars are taken into account.

96 Figure 5.12: 235U ﬁssion fragment anisotropy plotted with other published data from EXFOR [77].

97 Shown in Figure 5.12 and Figure 5.13 are the best estimates and error for the anisotropy measurements of 235U and 238U for incident neutron energies from 0.145 to 230 MeV. (Tab- ulated anisotropy data can be found in Appendix C, Table C.5 and Table C.6.) There are a few interesting features in the anisotropy structure to point out. First is that the anisotropy peaks at the transitions from first to second chance fission and again from second to third chance fission near seven and 15 MeV, respectively, where the reaction goes from U(n,f) → U(n,nf) → U(n, 2nf).

Figure 5.13: 238U ﬁssion fragment anisotropy plotted with other published data from EXFOR [77].

The anisotropy measured here shows sideways peaked angular distributions at low and high incident neutron energies. It is often assumed in many of these measurements that low energy incident neutrons produce isotropic ﬁssion distributions, but that assumption needs to be challenged with new measurement campaigns. Smirenkin [78] also presents anisotropy

98 data below one at energies lower than 0.1 MeV, which do not fall in the plot range of Figure 5.12. Angular distributions that are peaked at right angles with respect to the beam implies that the orbital angular momentum is oriented in the beam direction despite the angular momentum of the incident neutron with respect to the nuclear center being perpendicular. An explanation for this at high incident projectile energies is given by Halpern [79] as a combination of two factors. First is that at high incident energies, there are relatively many collisions between the incident particle and the individual nucleons in the target nucleus that result in the angular momentum being oriented along the beam axis. The second is that these collisions do not produce large amounts of excitation energy in the nucleus as small excitation energies more favorably produce anisotropy values farther from one. Besides the 235U anisotropy data shown here, which shows a strong trend towards anisotropy values below one at high incident neutron energies, there has also been measurements of 185 MeV proton induced fission of natural uranium producing an anisotropy of .91 ± .05 [80]. Also, proton induce fission of 238U at energies up to 1 GeV shown in [81] shows a very strong trend towards very sideways peaked angular distributions as incident proton energies increase. Also of note is that the anisotropy measurements tend to be systematically lower than previously published data in particular energy ranges, particularly in the 235U measurement, so comparing previous measurements to this work is necessary. Crucial for anisotropy measurements is ensuring that the angular efficiency is understood and incorporated. Events that are being lost through the target material, coupled with less than full solid angle coverage results in fragments at lower cos θ being lost. Fitting a distribution with missing events at lower cos θ will artificially inflate anisotropy values. For the fissionTPC, artificially low anisotropy values would mean more efficiency at low cos θ than high cos θ, and this is not likely. Besides, with the measurement of the 252Cf source, the efficiency of the fissionTPC was shown to be one for cos θ> ∼ 0.3.

99 Looking at the other experiments more closely, the Vorobyev data [82] uses two position sensitive multiwire proportional counters to detect their fission fragments. Generally good agreement can be seen between the 238U Vorobyev data and this work in Figure 5.13, but the Vorobyev 235U data is systematically higher than the results presented here. The 235U Vorobyev data retrieved from EXFOR [77] has a timestamp of 2015 and refers to a paper that assumes full efficiency for cos θ>.4 [82]; however, the 238U data retrieved has a timestamp of 2017, after further work showed their detection efficiency barely exceeds 0.8 [83]. Clarity in their final 235U anisotropy result is needed, and the disagreements between that work and this work should not be taken at face value. Next, the Meadows data [84] used a double sided Frisch-gridded ionization chamber. This detector is very similar to the fissionTPC in that the target is placed in the middle of the upstream and downstream gas filled volume producing essentially 4π coverage. Meadows

252 235 shows flat angular distributions for a Cf and U(nth,f) down to cos θ ≈ 0. This is in direct contrast to the efficiency drop-off at cosθ ≈ 0.25 shown in 3 and the angular distributions in Figure 5.10. The fissionTPC demonstrates that 4π gas coverage does not ensure 100% detection efficiency, even with target thicknesses of only 50−200µg/cm2. Frisch- gridded ionization chambers do not measure the track angle directly like the fissionTPC, but instead rely on the ratio of the signals on the Cathode compared to the Frisch-grid. It is possible that an uncontrolled systematic error is introduced in this process of finding the fission fragment angles. With this being said, good agreement with this work and the Meadows data occurs below 2 MeV (notice the Meadows point hiding under the second energy bin), but at higher neutron energies, the two data sets diverge. However, in the fast neutron regime, Meadows does an extra step in dividing the measured cos θ distribution by

235 the U(nth,f) cos θ distribution “in order to correct for distortions induced by the uranium deposit and by the angular resolution” [84]. If the low neutron energy data that Meadows normalized to was actually sideways peaked, as indicated in this measurement, that would explain the sudden systematic shift.

100 The other measurements for angular distributions of 235U were all similar in that they used fission counters to detect fission fragments at five or six different angular locations surrounding the target. Proportional ionization chambers were used by Simmons [85] and Nesterov [86] where the determination of the detection efficiency for both measurements was done by moderating fast neutrons from their neutron source through a paraffin block to decrease the neutron energies to the point where they are expected to produce an isotropic distribution of fission fragments, also using the assumption of isotropic fission for low neutron energies. Musgrove [87] used six surface barrier detectors at angles ranging from 9 to 90 degrees with efficiency also determined by thermal neutrons and Smirenkin [78] used glass detectors, but the normalization procedure is not stated. To summarize, the systematic shift between the data shown here and almost all of the other data sets is that low energy neutron induced fission was used to understand the detection efficiency. If low energy neutrons produce anisotropies below one, this will systematically shift all measured values high, thus explaining the discrepancy between this work and previously published results. Another small difference between this measurement and other measurements, especially at high energies, is that for previous measurements, upstream and downstream anisotropies were found in the lab frame and then averaged to take into account the linear momentum transfer correction whereas this work applied the momentum transfer measured in Chapter 4 on an event by event basis.

101 CHAPTER 6 SUMMARY AND OUTLOOK

The Neutron Induced Fission Fragment Tracking Experiment (NIFFTE) collaboration has built a fission Time Projection Chamber (fissionTPC) to study neutron induced fission in a novel way by implementing three-dimensional tracking capability of ionizing radiation with the goal of measuring neutron induced fission cross sections to unprecedented precision. Constituting of two highly segmented anodes and a total of about 6000 readout channels, the fissionTPC has the ability to measure the electron charge clouds that fission fragments and other ionizing radiation make as they travel through the fill gas. Converting the raw signals from the individual pads to digits, and then grouping and fitting those digits provides the reconstruction of ionization events in the chamber. From this reconstruction, several track parameters can be directly extracted, including track angles, energies, lengths, and even ionization profiles. Because of the wealth of information from every fission event, the fissionTPC has the ability to explore systematic errors from previous cross section measurements while simultaneously collecting data that can be analyzed for other physical observables including ternary fission, total kinetic energy release, and, presented in this thesis, fission fragment angular anisotropy and linear momentum transferred from the incident neutron to the target nucleus. Linear momentum transfer and angular anisotropy are both needed to fully understand detection efficiency for cross section measurements, and angular anisotropy can be used to infer the angular momentum state of the fissioning nucleus. A few examples of detector systematics relating to reconstructed fission fragment polar angles were explored in this thesis with the help of performing a dedicated 252Cf measurement which provided an isotropic source of spontaneous alpha particles and fission fragments. By changing the voltage setting on the MICROMEGAS (MICRO MEsh Gaseous Structure) which provides the signal amplification on the anodes, different detection thresholds were

102 explored. The lowest voltage setting presented caused the majority of alpha particle tracks to be either missed entirely or only a partial segment of the track to pass the pad thresholds, but would be able to capture all of the energy of the fission fragments. Increasing the gain to higher levels saw the increase in fidelity of the alpha tracks, but the analog section of the readout cards would saturate for fission fragments that are directed towards the anodes such that most of their energy is deposited on a small number of pads. Due to the process of digit creation which takes a derivative of the raw signal, a hole is formed in the center of these fission fragment charge clouds, skewing the fitting algorithm away from the hole and creating an artificial “bump” at high cos θ in the angular distributions. A proposed solution was presented, but was not implemented as the reconstruction was not perfect and the saturation problem was shown to be contained to a very specific polar angle range. Future work on a better algorithm for fixing the saturation is needed in order to extend the polar angle range used in the anisotropy and momentum transfer analysis presented here. Alternatively, new readout electronics can be implemented that have a larger dynamic range such that the saturation does not occur but are still sensitive enough to pick up the low energy events. Another necessary systematic to understand polar angle measurements in the fissionTPC is the electron drift speed through the fill gas which directly influences relative distance between the start and end z-positions of the reconstructed tracks. By changing the length of the track in the drift direction, the polar angle of that track will also change. Because the fissionTPC does not measure the electron drift speed directly, the drift speed must be inferred through the measured angular distributions of spontaneously emitted alphas. Taking advantage of the fact that spontaneous alphas are emitted isotropically, a method for finding the drift speed through flattening the cos θ distribution was devised and applied to fission fragments on an event by event basis. However, even after correcting for the electron drift speed to flatten the alpha particle polar angle distribution, the spontaneous fission fragment angular distribution from the

103 252Cf source was shown not to be flat. An additional correction rising from the electron diffusion coefficients not being the same in the drift and radial direction of the fissionTPC chamber was needed. After applying this correction, it was shown that the fissionTPC can successfully reproduce an isotropic source and any further efficiency normalization for anisotropy measurements was shown to not be needed. By placing the target actinide on a thin carbon backing, both fission fragments from a fission event can be detected simultaneously by the fissionTPC, and a procedure for measuring the average opening angle was presented. Since fission is a two body decay, the opening angle between the two fission fragments must be 180 degrees in the rest frame of the fissioning nucleus. However, the fissioning nucleus is moving in the lab frame after acquiring some amount of linear momentum from the incident neutron, causing a kinematic focusing of the fission fragments that decreases the opening angle away from 180 degrees in the lab frame. Using a Monte Carlo simulation, a method for extracting the linear momentum transfer required to reproduce the measured opening was presented. To our knowledge, this is the first measurement of linear momentum transfer by an incident neutron to a fissioning nucleus and shows a considerable difference in shape compared to proton, deuteron, and alpha induced fission. Theoretical exploration to explain this shape is needed. Also presented in this thesis was a measurement of the fission fragment angular anisotropy for 235U and 238U using the direct measurement of the fragment angles through the fis- sionTPC reconstruction process and applying the necessary corrections. Considering the fission fragment anisotropy parameter is defined to be in the center of mass frame, the linear momentum transfer measurement was applied to do the conversion from the lab frame. A best estimate for the anisotropy measurements was a combination of many different targets in both upstream and downstream orientations with respect to the beam, and the agreement between upstream and downstream data sets was a good confirmation of the linear momentum transfer measurement.

104 While the 235U anisotropy measurement (and also possibly 238U) is low compared to other previously measured data sets, this systematic difference can be explained by the normalization of detector efficiency of previous measurements to low neutron energy fission. This measurement relied on the development of the electron drift speed measurement and the application of the electron diffusion correction to show the successful reconstruction of an isotropic fission source of 252Cf, meaning that an additional normalization to low energy neutrons was not needed. A dedicated measurement of fission fragment angular distributions at low incident neutron energies is needed. For example, a measurement campaign of the fissionTPC at the Lujan center [64] would address this. Unfortunately, a measurement of 239Pu fragment anisotropy was outside the scope of this thesis. Because of the high rate of alpha decay from 239Pu, additional positive space charge accumulates in the fissionTPC, which causes an angular distortion of the fission fragments similar to the effects of the fourth order Legendre polynomial term. The original idea was to use the thin-backed target containing half-moons of 235U and 239Pu, which has half the amount of 239Pu compared to the other targets, to show that space charge in that measurement was not a compounding factor in that data set. By measuring the 235U anisotropy on that target and showing there is not significant space charge effects, the anisotropy of the 239Pu half could also be trusted. Unfortunately, a cathode amplifier problem on this data set made it such that a significant portion of fission fragments at low cos θ did not get assigned a neutron time of flight, making an anisotropy measurement impossible with the methods presented in this thesis. Further investigation of the space charge problem is needed before an anisotropy measurement of 239Pu can be made in the fissionTPC.

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116 APPENDIX A CONVERSION BETWEEN LAB FRAME AND CENTER OF MASS FRAME

This is a semi-relativistic derivation of fragment energies and angles from the lab frame to the center of mass frame. Namely, the incident momentum from the neutron is treated relativistically, but the compound system after interaction is treated classically. A 250 MeV neutron has a relativistic gamma of about 1.27, so that translates to almost a 30% diﬀerence between the classical and relativistic momentum. However, even if the compound nucleus takes in 1 GeV/c of momentum, that still only translates to a relativistic gamma factor of 1.0001 – much less than a fraction of a percent diﬀerence, meaning the approximation of a classical treatment of the velocity of the center of mass is a good one. The diagram shown in Figure A.1 will be used to guide the derivation and an analogous derivation can be found in [88].

Figure A.1: Diagram depicting the transformation between the lab frame and the center of mass frame where the incident neutron is traveling in the positive x-direction and up and down refer to the upstream and downstream fragments, respectively.

117 The compound, ﬁssioning nucleus has acquired some amount of linear momentum from the incident neutron along the beam direction

L pCN = mCN VCN = pn (A.1) where pn is the amount of momentum transferred to the compound nucleus from the neutron. At this point, only one particle is considered – the compound nucleus – and therefore the velocity of the center of mass frame is equal to the velocity of the compound nucleus in the lab frame

~ CM ~ L ~pn V = VCN = (A.2) mCN The velocity of the upstream and downstream fragments can now be written as

~ L ~ CM ~ CM Vd,u = Vd,u + V (A.3)

~ L ~ CM ~pn Vd,u = Vd,u + (A.4) mCN Projection of A.4 onto the x and y axes gives (with the incident neutron direction chosen to be along the positive x direction)

L L CM CM pn Vd,u cos θd,u = Vd,u cos θd,u ± (A.5) mCN

L L CM CM Vd,u sin θd,u = Vd,u sin θd,u (A.6)

where the (+) and (-) refer to the downstream and upstream fragments, respectively. Squar- ing A.5 and A.6, adding them together, and using the identity sin2 θ + cos2 θ = 1 yields

2 L2 CM 2 pn CM CM pn Vd,u = Vd,u + 2 ± 2Vd,u cos θd,u . (A.7) mCN mCN 1 2 To convert from velocity to energy, multiply by md,u/2 and substitute E = 2 mV to get

2 L CM md,upn pn CM CM Ed,u = Ed,u + 2 ± 2md,uEd,u cos θd,u (A.8) 2mCN mCN q which gives the conversion of fragment kinetic energies from the center of mass frame to the lab frame. In order to get the conversion from center of mass angle to lab angle, square A.6,

118 1 2 2 2 multiply by md,u/2, substitute E = 2 mV , and use the trig identity sin θ =1 − cos θ to get

ECM L d,u 2 CM cos θd,u = 1 − L (1 − cos θd,u ). (A.9) s Ed,u Thus, conversion fragment of energy and angle from the center of mass frame to the lab frame can be accomplished by using A.8 and A.9. Note that the conversion from the lab frame to the center of mass frame is just a simple re-arrangement of A.8 and A.9:

2 CM L md,upn pn L L Ed,u = Ed,u + 2 ∓ 2md,uEd,u cos θd,u (A.10) 2mCN mCN q EL CM d,u 2 L cos θd,u = 1 − CM (1 − cos θd,u). (A.11) s Ed,u

119 APPENDIX B DERIVATION OF THE ELECTRON DIFFUSION ANISOTROPY CORRECTION

Since principal component analysis can essentially be deﬁned as the eigenvectors of the covariance matrix of a distribution, the strategy for deriving a correction for the electron diﬀusion anisotropy is to treat the correction as a perturbation of the convolution matrix used to do the PCA where the perturbation takes the form of Gaussian smearing of the charge cloud. [66] The center of a charge cloud q(a) with total charge Q ≡ q(a)d3a is computed by 1 R < a >≡ aq(a)d3a. (B.1) Q Z Then, the convolution matrix is given by

x2 xy xy M ≡ aaT q(a+ < a >)d3a = yx y2 yz q(a+ < a >)d3a (B.2)   Z Z zx zy z2 For notational convenience, a coordinate shift to the center of the charge cloud is performed such that < a >= 0 and that term can thus be suppressed. After calculation of the covariance matrix M, the orthogonal eigenvectors can be calculated with the eigenvectors pointing along the ”principle axes” of the charge cloud distribution. The eigenvector xˆ0 with the largest eigenvector λ0 is taken to be ”ﬁt” to the charge cloud in the reconstruction process described in 3.1. At this point, it is useful to show the simple case where a 1-D charge density lies purely along unit vector direction xˆ0 such that the PCA covariance matrix becomes

T T 2 M = (axˆ0)(axˆ0) q(a)da = xˆ0xˆ0 a q(a)da (B.3) Z Z 2 By inspection, the largest principle component has direction xˆ0 and eigenvalue λ = a q(a)da.

For this particularly simple case, the other eigenvalues are degenerate (λ1 = λ2 R= 0), and their eigenvectors may be chosen in any mutually orthogonal direction to xˆ0.

120 Now consider some convolution kernel C(u) (normalized to C(u)d3u = 1) that smears the charge distribution out as a function of displacement u fromR each initial point. The smeared charge distribution from resulting from this convolution is then

(C ∗ q)(a) ≡ C(u)q(a − u)d3u (B.4) Z Plugging this convolution kernel into B.2 and employing a shift of integration variable a → a + u, the convolution-perturbed PCA matrix becomes

M′ = (a + u)(a + u)T C(u)q(a)d3ud3a (B.5) Z Z By multiplying out the factors, remembering the shift in coordinates such that < a >= 0 and that the convolution kernel is normalized to C(u)d3u = 1), the convolution matrix can be simpliﬁed to R

M′ = M + Q uuT C(u)d3u ≡ M + δM (B.6) Z which explicitly shows that by perturbing the charge cloud, the PCA convolution matrix is perturbed by the covariance of the convolution used to smear the digits. One could directly calculate a new PCA result from B.6, but in the context of understanding the change the electron diffusion has on a fission fragment charge cloud, more insight and practical use is gained by analytically approximating the change from the original PCA by assuming a small perturbation. While the initial PCA provides the unperturbed eigenvalues and eigenvectors such that Mxi = λixî, under a small perturbation such that M′ → M + δM, the eigenvectors change to first order by

T xˆj δMxˆi δxˆi = xˆj (B.7) λi − λj Xj6=i Looking at just the principal axes, since that is where the track polar angle is determined, and plugging in the expression for δM gives

xˆj T T 3 δxˆ0 = xˆj uu C(u)d u xˆ0 (B.8) (λ0 − λj)/Q Xj6=0 Z

121 For the context of the ﬁssionTPC, which has cylindrical symmetry in the radial r direction and the drift ﬁeld in the z, a form for the normalized convolution kernel was assumed to be

2 2 u 2 −3/2 ux y uz (2π) − 2 − 2 − 2 u 2σr 2σr 2σz C( )= 2 e (B.9) σzσr which causes Gaussian smearing in the radial and drift directions of the charge cloud. Be- cause C(u) is even under negation of any one coordinate, the calculation of the oﬀ diagonal terms of the perturbation of the covariance matrix are all 0, leaving only

2 σr T 3 2 δM = Q uu C(u)d u = Q σr (B.10)  2 Z σz Since the fission fragment angular anisotropy and neutron linear momentum transfer analysis depend only on the polar angle, and not the azimuthal angle, it is advantageous to gather an expression for how much to correct the polar angle due to the electron diffusion anisotropy. Since the convolution kernel and the fissionTPC have cylindrical symmetry about

the z axis, a coordinate rotation such that the principal component eigenvector xˆ0 lies in

T the x-z plane is performed, giving xˆ0 = (sin θ, 0, cos θ) and allowing the other orthogonomal

T T eigenvectors to be chosen as xˆ1 = yˆ and xˆ2 =(− cos θ, 0, sin θ). using this choice provides

2 σr sinθ T 2 2 2 xˆ2 δM = Q −cosθ 0 sinθ σr 0 = Q(σr − σz ) (B.11)  σ2 cosθ z     Plugging this term back into B.7, pulling out the z direction, and using the eigenvalue derivation for a simple line charge produces a correction for the polar angle in the form of

sin2 θ cos θ δ cos θ = zˆ · δxˆ = (σ2 − σ2) (B.12) 0 a2q(a)da/Q r z This is an acceptable form for the correction,R but measuring the amount of Gaussian smearing for every single fission fragment is not practical for the fissionTPC reconstruction process. A better way is to measure an average electron diffusion coefficient related to the Gaussian smearing by the drift distance, namely

2 2 2 2 σz = σz0l(a) and σr = σr0l(a) (B.13)

122 where the smearing σ2 expands proportionally to the drift distance l(a). Using the same convolution kernel B.9, but replacing σr → σr(a) and σz → σz(a) changes the correction to

(σ2(a) − σ2(a))q(a)da δ cos θ = r z sin2θ cos θ (B.14) a2q(a)da/Q R Pulling out the diffusion coefficients fromR the integral gives the final form of the correction used in the analysis

sin2θ cos θ δ cos θ = (σ2 − σ2 ) (B.15) a2q(a)da/Q r0 z0 where the average drift distance isR given by 1 = l(a)q(a)da. (B.16) Q Z

123 APPENDIX C TABULATED DATA

Tabulated data is presented below for neutron linear momentum transfer and anisotropy results seen in Figure 4.10, Figure 5.12, and Figure 5.13.

Table C.1: Neutron linear momentum transferred to 238U.

− + Neutron Energy [MeV] Bin Width [MeV]

[MeV/c] σ

[MeV/c] 1.28 0.11 49.13 5.63 5.79 1.51 0.12 54.52 3.77 2.78 1.78 0.15 53.63 2.50 2.38 2.10 0.17 56.31 2.54 2.39 2.48 0.20 63.32 2.91 2.22 2.92 0.24 66.84 2.42 2.25 3.44 0.28 74.42 2.57 2.48 4.06 0.33 81.66 4.02 2.14 4.79 0.39 86.87 3.40 2.27 5.64 0.46 97.10 2.81 2.79 6.65 0.55 102.70 2.50 3.37 7.84 0.64 112.11 4.77 2.03 9.24 0.76 120.04 2.31 2.84 10.89 0.89 132.13 5.93 2.41 12.84 1.05 139.00 2.51 3.09 15.14 1.24 156.92 3.31 2.91 17.84 1.46 160.98 4.45 2.70 23.07 3.76 186.57 3.36 2.37 32.05 5.22 211.52 2.72 3.58 44.54 7.26 237.44 3.37 2.78 61.88 10.09 258.80 4.58 3.79 85.98 14.02 271.92 5.26 3.75 119.47 19.47 260.83 5.87 4.16 166.01 27.06 248.58 6.80 7.12 230.67 37.60 234.07 12.24 10.94

124 Table C.2: Neutron linear momentum transferred to 235U in the thin-backed target containing 235U/238U.

− + Neutron Energy [MeV] Bin Width [MeV]

[MeV/c] σ

[MeV/c] 0.14 0.01 20.14 5.57 6.00 0.18 0.02 21.73 4.60 4.56 0.25 0.05 21.10 3.01 2.94 0.35 0.05 26.31 2.61 2.68 0.45 0.05 30.18 2.62 2.82 0.55 0.05 33.72 2.65 2.63 0.65 0.05 38.47 3.51 2.63 0.75 0.05 38.51 2.67 2.72 0.85 0.05 36.70 2.71 2.90 0.95 0.05 42.32 2.88 3.06 1.09 0.09 47.30 4.88 2.20 1.28 0.11 51.60 2.59 2.91 1.51 0.12 52.93 2.40 2.30 1.78 0.15 56.04 2.68 2.17 2.10 0.17 60.41 2.62 1.96 2.48 0.20 65.25 2.07 2.48 2.92 0.24 71.66 2.44 2.26 3.44 0.28 73.33 2.68 2.16 4.06 0.33 83.92 3.13 2.45 4.79 0.39 86.33 2.71 2.63 5.64 0.46 94.88 3.27 2.91 6.65 0.55 105.62 2.70 2.94 7.84 0.64 115.87 2.50 3.48 9.24 0.76 122.43 3.21 2.61 10.89 0.89 136.63 3.04 2.99 12.84 1.05 144.51 3.41 3.20 15.14 1.24 157.84 3.99 3.05 17.84 1.46 166.51 3.37 4.74 23.07 3.76 186.03 3.47 2.45 32.05 5.22 210.44 4.54 2.65 44.54 7.26 236.28 4.63 3.46 61.88 10.09 255.00 5.68 4.44 85.98 14.02 249.64 5.35 4.31 119.47 19.47 251.74 6.78 5.94 166.01 27.06 231.17 8.39 7.50 230.67 37.60 221.52 13.09 13.84

125 Table C.3: Neutron linear momentum transferred to 235U in the thin-backed target containing 235U/239Pu.

− + Neutron Energy [MeV] Bin Width [MeV]

[MeV/c] σ

[MeV/c] 0.14 0.01 16.94 5.45 5.44 0.18 0.02 20.39 4.46 4.52 0.25 0.05 19.55 2.90 2.83 0.35 0.05 28.15 2.54 2.58 0.45 0.05 29.98 2.52 2.63 0.55 0.05 32.00 2.77 2.73 0.65 0.05 37.17 2.74 2.55 0.75 0.05 39.31 3.29 2.57 0.85 0.05 41.96 2.55 2.91 0.95 0.05 36.40 2.47 4.09 1.09 0.09 45.31 2.19 2.44 1.28 0.11 47.60 1.98 3.64 1.51 0.12 50.47 2.86 1.99 1.78 0.15 57.21 2.54 2.03 2.10 0.17 61.83 1.85 2.69 2.48 0.20 67.27 2.21 2.15 2.92 0.24 74.63 4.51 2.02 3.44 0.28 78.00 3.05 2.10 4.06 0.33 78.53 2.13 3.65 4.79 0.39 92.56 2.46 2.42 5.64 0.46 101.68 2.40 2.94 6.65 0.55 107.29 4.40 2.15 7.84 0.64 117.40 6.01 2.09 9.24 0.76 120.96 2.33 2.79 10.89 0.89 136.56 3.36 3.12 12.84 1.05 149.25 3.49 3.12 15.14 1.24 159.71 3.42 2.78 17.84 1.46 176.23 4.08 3.03 23.07 3.76 194.50 3.44 2.81 32.05 5.22 214.80 3.21 4.21 44.54 7.26 242.78 5.23 2.86 61.88 10.09 256.92 5.20 3.46 85.98 14.02 256.03 5.55 4.27 119.47 19.47 259.43 7.13 5.75 166.01 27.06 257.78 9.51 8.99 230.67 37.60 216.36 12.77 13.17

126 Table C.4: Neutron linear momentum transferred to 239Pu.

− + Neutron Energy [MeV] Bin Width [MeV]

[MeV/c] σ

[MeV/c] 0.14 0.01 28.46 6.54 6.06 0.18 0.02 32.70 5.15 5.18 0.25 0.05 20.35 3.35 3.31 0.35 0.05 25.15 3.34 2.99 0.45 0.05 28.75 3.43 2.87 0.55 0.05 32.15 2.61 3.24 0.65 0.05 35.68 2.58 2.77 0.75 0.05 39.10 2.54 2.97 0.85 0.05 45.01 3.10 3.04 0.95 0.05 40.12 4.05 2.91 1.09 0.09 48.91 2.44 2.99 1.28 0.11 45.11 2.05 3.11 1.51 0.12 52.32 2.04 3.41 1.78 0.15 56.25 2.57 2.16 2.10 0.17 59.47 2.87 2.03 2.48 0.20 64.16 2.05 2.05 2.92 0.24 73.50 2.43 2.82 3.44 0.28 78.17 3.31 2.20 4.06 0.33 80.51 2.52 2.25 4.79 0.39 86.73 2.31 2.86 5.64 0.46 98.81 3.71 2.80 6.65 0.55 107.34 2.49 2.97 7.84 0.64 115.54 3.58 2.78 9.24 0.76 123.83 2.72 2.72 10.89 0.89 135.07 5.08 2.96 12.84 1.05 146.40 3.95 3.30 15.14 1.24 154.59 5.03 3.51 17.84 1.46 177.26 5.48 3.68 23.07 3.76 187.54 3.30 4.16 32.05 5.22 215.04 5.21 3.24 44.54 7.26 231.31 3.81 5.04 61.88 10.09 247.18 5.41 4.63 85.98 14.02 244.04 6.31 5.16 119.47 19.47 249.82 7.38 7.19 166.01 27.06 239.82 10.60 9.15 230.67 37.60 240.48 19.87 19.08

127 Table C.5: 235U Anisotropy results.

Neutron Energy [MeV] Neutron Bin Width [MeV] Anisotropy Anisotropy Error 0.14 0.01 0.923 0.016 0.18 0.02 0.971 0.015 0.25 0.05 0.979 0.009 0.35 0.05 1.016 0.012 0.45 0.05 1.028 0.010 0.55 0.05 1.068 0.010 0.65 0.05 1.069 0.017 0.75 0.05 1.073 0.015 0.85 0.05 1.097 0.010 0.95 0.05 1.101 0.010 1.09 0.09 1.101 0.014 1.28 0.11 1.117 0.015 1.51 0.12 1.122 0.014 1.78 0.15 1.133 0.009 2.10 0.17 1.148 0.012 2.48 0.20 1.145 0.013 2.92 0.24 1.143 0.012 3.44 0.28 1.135 0.010 4.06 0.33 1.115 0.010 4.79 0.39 1.112 0.011 5.64 0.46 1.131 0.011 6.65 0.55 1.220 0.011 7.84 0.64 1.290 0.014 9.24 0.76 1.256 0.015 10.89 0.89 1.213 0.012 12.84 1.05 1.233 0.013 15.14 1.24 1.286 0.014 17.84 1.46 1.259 0.016 23.07 3.76 1.234 0.012 32.05 5.22 1.198 0.017 44.54 7.26 1.135 0.014 61.88 10.09 1.082 0.012 85.98 14.02 1.041 0.012 119.47 19.47 0.987 0.021 166.01 27.06 0.949 0.018 230.67 37.60 0.932 0.033

128 Table C.6: 238U Anisotropy results.

Neutron Energy [MeV] Neutron Bin Width [MeV] Anisotropy Anisotropy Error 1.28 0.11 1.428 0.065 1.51 0.12 1.537 0.027 1.78 0.15 1.288 0.025 2.10 0.17 1.237 0.021 2.48 0.20 1.274 0.037 2.92 0.24 1.214 0.034 3.44 0.28 1.222 0.021 4.06 0.33 1.206 0.022 4.79 0.39 1.202 0.021 5.64 0.46 1.241 0.023 6.65 0.55 1.551 0.043 7.84 0.64 1.611 0.024 9.24 0.76 1.448 0.034 10.89 0.89 1.343 0.050 12.84 1.05 1.252 0.026 15.14 1.24 1.359 0.028 17.84 1.46 1.350 0.028 23.07 3.76 1.378 0.020 32.05 5.22 1.328 0.043 44.54 7.26 1.251 0.020 61.88 10.09 1.184 0.024 85.98 14.02 1.127 0.040 119.47 19.47 1.074 0.025 166.01 27.06 1.087 0.033 230.67 37.60 1.030 0.088

129