Calculation of Rates for Radioactive Isotope Beam Production at TRIUMF

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Calculation of rates for radioactive isotope beam production at TRIUMF

by Fatima H. Garcia

B.Sc., University of Calgary, 2013

Thesis Submitted in Partial Fulﬁllment of the Requirements for the Degree of Master of Science

in the Department of Chemistry Faculty of Science

c Fatima H. Garcia 2016 SIMON FRASER UNIVERSITY Fall 2016

All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced without authorization under the conditions for “Fair Dealing.” Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately. Approval

Name: Fatima H. Garcia Degree: Master of Science (Chemistry) Title: Calculation of rates for radioactive isotope beam production at TRIUMF Examining Committee: Chair: Dr. Charles J. Walsby Associate Professor

Dr. Corina Andreoiu Senior Supervisor Associate Professor

Dr. Peter Kunz Co-Supervisor Adjunct Professor

Dr. Daniel B. Leznoﬀ Supervisor Professor

Dr. Anne J. Trudel Supervisor Chief Safety Oﬃcer TRIUMF

Dr. John D’Auria Internal Examiner Professor Emeritus

Date Defended/Approved: November 15th, 2016

ii Abstract

Access to new and rare radioactive isotopes is imperative for establishing fundamental knowledge and for its application in nuclear science. Rare Isotope Beam (RIB) facilities around the world, such as TRIUMF, work towards development of new target materials to generate increasingly exotic species, which are used in nuclear medicine, astrophysics and fundamental physics studies. At Simon Fraser University and TRIUMF, a computer simulation of the RIB targets used at the Isotope Separation and ACceleration (ISAC) facility of TRIUMF was built, to compliment existing knowledge and to support new target material development. The simulation was built using the GEANT4 nuclear transport toolkit, and can simulate the production rate of isotopes from user-deﬁned beam and target characteristics. The simulation models the bombardment of a production target by an incident high-energy particle beam and calculates isotope production rates via ﬁssion, fragmentation and spallation. In-target production rates from the simulation were analysed and compared to production mechanisms within the simulation environment, other nuclear transport algorithms and to the experimentally measured yield rates from the ISAC yield station. Additionally, preliminary studies were conducted using these in-target production rates as illustrative examples, showing the capabilities and power of the simulation. Keywords: GEANT4; Radioactive Isotope Beams; Monte Carlo Simulations; TRIUMF- ISAC.

iii Acknowledgements

Completion of this work would have not been possible without the presence of some very important people. I would like to take this opportunity to acknowledge and thank them for their unwavering support. Without it, the document before you would not exist.

To my mentor, Dr. Jason Donev, thank you for being a constant source of encouragement. Your stories and advice have helped me navigate through the academic maze.

To my supervisors, Dr. Corina Andreoiu and Dr. Peter Kunz, thank you for the opportunity to work on this project and for the constant support during its completion. Your expertise has been invaluable and your lessons essential to my success.

To the members of my committee, Dr. Daniel Leznoﬀ and Dr. Anne Trudel, thank for your insight and presence. The support I’ve received from you has been integral to this work.

To current and former members of the SFU Nuclear Science groups, specially Jennifer Pore, thank you. For answering all my silly questions, helping me through the struggles and teaching me about how to survive and thrive in graduate school. I am very fortunate to have shared an oﬃce with such a brilliant group of people.

To my friends at SFU and at home, particularly Yumeela and Allison, thank you. For all the early mornings, late nights, agonizing times, and hilarious moments. They, and you, have been an essential and amazing and fundamental part of my graduate school experience.

Finally, to my parents, Vilma and Roberto Garcia and my sister Lourdes Garcia, thank you. For everything. For your unyielding support, tireless encouragement, and uncon- ditional love. Your sacriﬁces, wisdom and example have been an unfaltering source of inspiration and motivation throughout my entire life. I know that, without you, I could not have made it this far. Los quiero mucho.

iv Table of Contents

Approval ii

Abstract iii

Acknowledgements iv

Table of Contents v

List of Tables viii

List of Figures ix

List of Abbreviations xi

1 Introduction 1 1.1 Nuclear Theory ...... 2 1.1.1 The Nucleus ...... 2 1.1.2 The Nuclide Chart ...... 3 1.1.3 Nuclear Reactions ...... 5 1.1.4 Nuclear Decay ...... 8 1.2 TRIUMF and ISAC ...... 13 1.2.1 ISOL Technique ...... 13 1.2.2 ISAC at TRIUMF ...... 14 1.2.3 ISAC Yield Station ...... 18 1.3 GEANT4 ...... 22 1.3.1 Framework ...... 22 1.3.2 Geometry ...... 24 1.3.3 Physics Lists ...... 25

2 Simulation 30 2.1 Simple Geometry ...... 30 2.1.1 Primary particle description ...... 30 2.1.2 Standard Target Description ...... 31

v 2.1.3 Detection Mechanisms ...... 32 2.2 Data Output & Processing ...... 33 2.3 Error Analysis ...... 34 2.4 Benchmark Products ...... 34 2.5 Competing Models ...... 35 2.5.1 Silberberg-Tsao ...... 35 2.5.2 FLUKA ...... 35 2.5.3 IAEA Benchmark of Spallation Models ...... 36 2.6 Extended Geometry ...... 36 2.7 Current Simulation Status ...... 38

3 Results 39 3.1 Physics Lists Results ...... 39 3.2 In-target Production Values ...... 42 3.2.1 Results Summary ...... 51

4 Further Studies 52 4.1 Cross Section Veriﬁcation ...... 52 4.1.1 Bertini & Binary Cascades ...... 53 4.1.2 Liège Cascade ...... 53 4.1.3 Validation conclusion ...... 53 4.2 Energy Deposition ...... 56 4.3 Release Times and Release Eﬃciencies ...... 57 4.4 Thorium Oxide Target Runs ...... 61 4.4.1 Medical Isotope Production ...... 63 4.5 Alternative targets ...... 64 4.6 Incident Particle Energy Dependence ...... 65 4.7 Kinetic Energy Distributions ...... 68 4.8 In-target decay ...... 71 4.9 Summary ...... 74

5 Future Work & Conclusions 75 5.1 Outlook ...... 75 5.2 Summary ...... 76

Bibliography 78

Appendix A Errors 84

Appendix B Alkaline Earth Metals 88

vi Appendix C Liège cascade cross section validation 92

Appendix D Target Materials 96

vii List of Tables

Table 2.1 The materials implemented in the GEANT4 simulation include all materials previously used at TRIUMF, along with a selection of other materials...... 32

Table 4.1 Energy deposition values returned from GEANT4 simulation and SRIM software [70], using the standard target disk conﬁguration...... 57 Table 4.2 Half lives of francium isotopes between 211Fr and 214Fr, given in seconds. Values taken from NNDC [12]...... 58 Table 4.3 Release parameters for the three intra-nuclear cascades used in the simulation...... 60 Table 4.4 In-target production rates of medical isotopes produced by a thorium target and compared to a uranium target, given in particles per second and normalized to reﬂect 1 μA of proton current through the target. . 64 Table 4.5 Materials implemented into the simulation. Target materials are either analogous to target materials used at ISAC or they have been implemented as hypothetical targets. Lead and aluminium targets have thickness of 87.3 and 12.1 mg/cm2 respectively, while all other targets have a thickness of 0.05 mol/cm2, and a target length of 5 cm. . . . . 65 Table 4.6 Stopping ranges for rubidium isotopes with kinetic energy between 30 MeV and 150 MeV, as calculated by the SRIM Software [70], with a target composed of 238U with a density of 2.38 g/cm3...... 70 Table 4.7 Parameters used to study in-target decay. The production rates are calculated using the Liège cascade with 109 incident protons at 480 MeV and scaled to 1 μA...... 73

viii List of Figures

Figure 1.1 The Chart of Nuclides, a nuclear analogue to the periodic table, depicts all known nuclei in terms of their proton and neutron number. The colour contours represent the half-lives of each isotope. Data from [12]. The valley of stability, in the center of chart, clearly deviates from the Z = N trend...... 4 Figure 1.2 Decays along the chart of nuclides. Adapted from Nucleonica [13]. . 4 Figure 1.3 Illustration of proton induced fission process...... 6 Figure 1.4 Illustration of a proton induced spallation reaction...... 7 Figure 1.5 Illustration of the fragmentation process. The nucleus absorbs the projectile which leads to an unstable configuration. The nucleus splits into two fragments, which may then undergo nucleon evaporation...... 7 Figure 1.6 Approximate mass distribution of products due to the proton induced reactions from protons of differing energies impinging on 209Bi. Image from [15]...... 8 Figure 1.7 The TRIUMF Isotope Separation and ACceleration (ISAC) facility [3]. 15 Figure 1.8 Cyclotron motion and production of high energy protons at TRIUMF. 16 Figure 1.9 Uranium carbide before and after processing for a uranium ISAC target. Reprinted from [27], with permission from Elsevier...... 17 Figure 1.10 ISAC target containers. Reprinted from [29], with permission from Elsevier ...... 18 Figure 1.11 The ISAC Yield Station. Reprinted from [31], with permission of AIP publishing...... 19 Figure 1.12 Yield measurement of 30Mg. Reprinted from [31], with permission from AIP Publishing...... 20 Figure 1.13 Yields measured at the ISAC Yield Station, for all targets used during ISAC’s operating history. Data from [30]...... 21 Figure 1.14 The GEANT4 framework [32]...... 23

ix Figure 1.15 A simple intranuclear cascade scheme shows that a proton enters the nucleus and interacts with a nucleon. This interaction may cause scattering or ejection of the participants or generation of other particles, such as a π-meson. Adapted from [40]...... 26

Figure 2.1 Simple target geometry visualization...... 31 Figure 2.2 The virtual sensitive detector...... 33 Figure 2.3 Visualiztion of extended target geometry...... 37

Figure 3.1 The in-target production rates generated by the Bertini intra-nuclear cascade, using the depleted uranium standard target...... 40 Figure 3.2 The in-target production rates generated by the Binary intra-nuclear cascade, using the depleted uranium standard target...... 41 Figure 3.3 The in-target production rates generated by the Liège intra-nuclear cascade, coupled with the ABLA de-excitation code, using the depleted uranium standard target...... 41 Figure 3.4 Yields from uranium targets as measured by the ISAC yield station. 42 Figure 3.5 In-target production rate results for the alkali metals as a function of mass number, A...... 45 Figure 3.6 In-target production rates for the alkali metals, produced by GEANT4 and compared with FLUKA and Silberberg-Tsao models...... 47 Figure 3.7 In-target production rate results for the alkali metals, produced by GEANT4 and compared to yield measurements from the ISAC Yield Database [30]...... 49

Figure 4.1 Production cross section values calculated from the bombardment of a 12.13mg/cm2 27Al by 800 MeV protons. Bertini and Binary cascade results compared to data from [67]...... 54 Figure 4.2 Production cross section values calculated from the bombardment of a 0.087 g/cm2 208Pb by 1 GeV protons. Liège cascade results compared with data from [68]...... 55 Figure 4.3 Energy deposited onto a 1 cm, 0.05 mol/cm2 depleted uranium standard target disk, plotted in MeV as a function of target distance in mm. The red points are values returned by the simulation. The line represents the linear fit to the data...... 57 Figure 4.4 Release efficiency fits for 211−214Fr...... 59 Figure 4.5 Nuclide chart produced using a thorium target with a 0.05 mol/cm2 thickness, 109 480 Mev protons and the Liège intra-nuclear cascade. 62 Figure 4.6 Comparison of production rates between a uranium and a thorium standard target...... 62

x Figure 4.7 In-target production rates from a thorium target, when compared to a uranium target, with thickness 0.05 mol/cm2...... 63 Figure 4.8 In-target isotope production rates from the Liège cascade, with respect to primary beam energy for isotopes of rubidium and cesium. 66 Figure 4.9 Production rates, from the Liège cascade, of isotopes of rubidium and cesiums as a function of energy...... 67 Figure 4.10 Kinetic energy distribution of rubidium isotopes, produced by the Bertini intra-nuclear cascade ...... 68 Figure 4.11 Kinetic energy distribution of rubidium isotopes, produced by the Binary intra-nuclear cascade ...... 68 Figure 4.12 Kinetic energy distribution of rubidium isotopes, produced by the Liège intra-nuclear cascade ...... 69 Figure 4.13 Sandwich target. The target material, in black, is used to produce isotopes, while the stopping material, in blue, is used to allow for faster diffusion and effusion out of the target...... 70 Figure 4.14 Examples of potential in-target decay chains. The production of the daughter isotope may be artificially enhanced by the production and subsequent decay of its parents...... 73

xi List of Abbreviations

ARIEL Advanced Rare IsotopE Laboratory. ASCII American Standard Code for Information Interchange. BERT Bertini. CERN European Organization for Nuclear Research. FLUKA FLUktuierende KAskade. FWHM Full Width Half Maximum. GEANT4 Geometry and Tracking 4. GUI Graphical User Interface. HPGe High Purity Germanium. ISAC Isotope Separator and ACcelerator. ISOL Isotope Separation On-Line. LN2 Liquid Nitrogen. NIST National Institute of Standards and Technology. NNDC National Nuclear Data Centre. RIB Radioactive Isotope Beam. SRIM Stopping and Range of Ions in Matter. ST Silberberg-Tsao.

The following are proper names:

ABLAv3 Model for nuclear ablation. GSI GSI Helmholtz Centre for Heavy Ion Research. MINUIT Numerical minimization computer program. PIN Type of diode, with characteristic p-type and n-type regions separated by an intrinsic semiconductor region. ROOT Particle physics data analysis software. TRIUMF Canada’s National Laboratory for Particle and Nuclear Physics.

xii Chapter 1

Introduction

Our collective understanding of the fundamentals of nuclear science has improved dra- matically over the last few decades, due in part to the advent of powerful and complex experimental infrastructure that has allowed for confirmation of existing theories, and dis- covery of new properties. The ability to isolate, generate and utilize isotopes for scientific investigations has also allowed us to test new theories and refine and expand current knowledge [1]. Further studies are needed to expand the nuclide chart, as the elements contained therein are important not only for fundamental science but for other areas such as stel- lar nucleo-synthesis and other astrophysical processes, and for applications such as nuclear medicine where isotopes are used to image, probe or treat diseases. These studies are carried out at facilities around the world, where stable and radioactive isotopes are generated, either to be examined for their fundamental properties or used as probes for other experiments [2]. The method of isotope generation varies with each facility and is dependent upon a wide set of parameters. These parameters must all be accounted for and optimized in order to carry out these scientific studies, and thus understanding the processes and generation mechanisms is crucial to the success of these experiments. In an effort to make predictions and plan for new experiments, scientists use computer models to aid in their preparations. These software models are an important tool as they can guide the experimenter towards optimization or reveal flaws. The purpose of this project was to employ computational simulation to support the need for radioactive isotope beam development. The goal was to build, characterize and implement the first stage of a comprehensive computer simulation that mimics the rare isotope beam production mechanisms in place at TRIUMF’s rare isotope beam facility Isotope Separator and ACcelerator (ISAC) [3]. This simulation will aid current experimental work, in providing estimates on the production of isotopes, as well as also becoming a tool for future experiments, leading to insights regarding potential materials and configurations.

1 This thesis discusses the work that was done to build, characterize and test the computer simulation of the isotope production mechanisms used at TRIUMF, built using the GEANT4 [4] nuclear transport framework.

1.1 Nuclear Theory

1.1.1 The Nucleus

A nucleus forms the core of an atom. It is composed of protons and neutrons, which together are known as nucleons. The nucleus is characterized by its proton number Z and its neutron number N, with its mass number or nucleon number being the addition of Z and N. This conﬁguration of protons and neutrons exhibits a set of interesting properties:

Nuclear Radius

The radius of the nucleus is smaller than that of the atom and can be approximated using the nucleon number A, 1/3 R = R0A , (1.1) where R0 = 1.2 fm is the charge radius, experimentally established using electron scattering experiments [5], which diﬀers from the mass radius, Rm = 1.4 fm.

Nuclear Strong Force

The nucleus is bound via the nuclear strong force. Unlike the Coulomb force, the nuclear strong force is not charge dependent, acting on protons and neutrons equally, and is only appreciable within the range of the nucleus, at ∼1 fm [6]. Given that the radius of a nucleus is dependent upon the number of nucleons in the system, the nuclear density is eﬀectively constant in all nuclei,

Mass A ρ = = , Volume (4π/3)R3 A = , 1/3 3 (4π/3)(R0A ) 1 = 3 . (1.2) (4π/3)R0

The stability of a nucleus X can be deﬁned by its binding energy, EB, which is the energy released when its individual particles come together to form a nucleus,

A A 2 EB(Z X) = [Zmp + Nmn − m(Z X)]c , (1.3)

2 where mp is the mass of a proton, mn is the mass of a neutron, and c is the speed of light [7]. This value determines how strongly bound the nucleus is and how likely it is to be unstable to certain types of decay.

Nuclear Saturation

The constant nuclear density, in Equation 1.2, and the constant nuclear binding energy between nucleons, given in Equation 1.3, requires that a nucleon interact only with those nucleons in its vicinity, rather than interacting with all nucleons in the system. This eﬀect causes a saturation of the nucleus, meaning no other nucleons can be bound in the system [6]. This eﬀect gives rise to boundaries in the nuclide chart, referred to as the proton and neutron drip lines. The placement of these boundaries is an important parameter in theoretical calculations, because they delineate the extent of nuclear binding [8]. Their location is also important as nuclear structure appears to change as nuclei approach the drip lines [9]. More details regarding the properties of the nucleus can be found in [7, 10, 11].

1.1.2 The Nuclide Chart

While the periodic table of elements is an excellent way to present important chemical and physical properties of each of the known elements, it does not describe the nuclear properties of each elemental isotope. Instead, nuclear scientists organize the isotopes into the so-called Chart of Nuclides. As seen in Figure 1.1, the chart arranges all known isotopes by neutron number N (on the x-axis) and by proton number Z (on the y-axis), their mass A given by the addition of both proton and neutron number. This arrangement means that all isotopes of one element will be found along a row, and all isotones, elements having the same neutron number, but diﬀerent proton number, are found along a column. The isobars, isotopes with the same mass number but diﬀerent Z and N combinations, increase in a diagonal trend along the chart.

3 Figure 1.1: The Chart of Nuclides, a nuclear analogue to the periodic table, depicts all known nuclei in terms of their proton and neutron number. The colour contours represent the half-lives of each isotope. Data from [12]. The valley of stability, in the center of chart, clearly deviates from the Z = N trend.

As isotopes undergo reactions, such as the decay modes discussed in Section 1.1.4, their products can be followed using the chart, as described in Figure 1.2.

Figure 1.2: Decays along the chart of nuclides. Adapted from Nucleonica [13].

The chart also has very distinct regions; the valley of stability, present in the middle of the chart, describes all known stable isotopes. Noting the path of the decay modes, it is evident that all radioactive species decay towards this valley, as the conﬁgurations in this region are the most stable.

4 The shape of the chart also contains features associated with nuclear structure. The valley of stability does not occur, as may be expected along the Z = N line, instead it deviates from this trend at around silicon, Z = 14, and appears to favour the more neutron rich species. This is due to the charged nature of the proton. Recall that a proton must overcome the repulsive Coulomb force barrier in order to become part of the nucleus. This Coulomb barrier increases as more protons are added, until no more can be bound into the nucleus. In contrast, since they are not charged, neutrons do not suﬀer from this eﬀect, resulting in a large number of neutron-rich isotopes.

1.1.3 Nuclear Reactions

Nuclear reactions are the mechanisms by which scientists tested their theories, formulated new ideas, and discovered new elements. They are still used today in order to push the boundary of nuclear science and to apply this knowledge to diverse ﬁelds that have all beneﬁted from this endeavour.

Cross Section Theory

Nuclear reactions are governed by the nuclei and particles involved, and their intrinsic properties, and may be spontaneous or induced by bombardment with particles. Scientists became interested in this last type of reaction, because with it, they could systematically study reaction mechanisms and they could produce new and exceedingly rare species. These reactions, it was found, were dependent upon the energy of the bombarding particle, the number of target nuclei present and some proportionality constant. The number of reactions during time interval dt, can be expressed as,

R = σreaction × Ntarget × nprojectile × dx × dt, (1.4) where Ntarget is the number of target nuclei, nprojectiledxdt the number of projectile particles moving through an area dx over some time dt (known as the particle flux φprojectile), and this proportionality constant, σreaction, now known as the total reaction cross section, usually expressed in millibarns (1 mb = 10−27 cm2) [14]. The cross section has units of area as it represents the area projected by a target, as seen by a projectile. It denotes the probability, that a reaction between these two participants will take place. Distinctions are drawn between different types of cross sections. There are cross sections associated with specific processes such as fission, and particles such as neutron [11]. The production cross sections encompass all cross sections that are involved in generating a particular product, and as a result they are key in examining isotope production mechanisms.

5 Fission

The process of fission was discovered in the late 1930’s, proposed by Meitner and observed by Hahn and Strassman [5], shortly after the finding of the neutron by Chadwick in 1932. Once this particle was isolated, scientists began to use it as a probe for their experiments, during which they discovered that it could induce break-up of the nucleus. Hahn and Strassman were able to chemically separate barium from uranium by bombarding a sample of uranium with the newly discovered neutron, thereby inducing fission. The break-up is caused by different factors, giving rise to the different fission classifications. Spontaneous fission occurs when an unstable nucleus breaks up in order to achieve a more stable state. This type of fission was first observed in naturally occurring uranium, by Flerov and Petrazhak in 1940 [14]. It typically occurs in the heavy isotopes, particularly in the mass region of the heaviest elements at ∼ Z = 92 and usually emits two fragments, with similar mass, and up to two or three neutrons. In contrast to spontaneous fission, induced fission occurs when a sample of material is bombarded by particles, such as protons, neutrons or photons. Seen in Figure 1.3, the fission reaction will cause the break of the nucleus, resulting in two unstable products, which will then emit nucleons to reach a more stable state. Further discussions on fission can be found in [7, 15].

Figure 1.3: Illustration of proton induced ﬁssion process. A nucleus is composed of protons, in red, and neutrons, in gray. It is bombarded by a proton which is absorbed, resulting an in unstable conﬁguration of the nucleus. The nucleus then splits into two unstable fragments, which may then eject neutrons to come to a more stable state. This process can also be achieved with other particles, such as neutrons.

Spallation

During a spallation reaction involving a high energy incident particle, the incoming particle will penetrate the surface of the nucleus, and may cause the ejection of a particle or a particle cluster. This reaction leaves a highly excited nucleus, that must then de-excite via nucleon emission, a process known as nucleon evaporation. Spallation products are typically found in the mass region surrounding the target nucleus accounting for the region where 2 Afragment ≥ 3 Atarget [5], as the process will only remove small amounts of nucleons. The reaction favours products occurring along the valley of stability [15]. The majority of the

6 products of this reaction are clustered below the target mass, such that most of the resulting nuclei will be a few nucleons lighter than the target material [16].

Figure 1.4: Illustration of a proton induced spallation reaction. The incident particle causes the nucleus to eject single nucleons or small groups of nuclei, leaving a highly unstable nucleus.

Fragmentation

Due to the fact that products from fragmentation reactions occur within the same mass region as spallation and fission products, it is difficult to isolate a fragmentation region. The process occurs when a high energy incident particle breaks up the nucleus. This results in an excited state which then relaxes via the splitting of the nucleus. This reaction typically leaves one high-mass fragment and one low-mass fragment. Thus production in the lower masses, 15 < A < 40, for a heavy target, may be attributed to this process [15], though the heavy mass fragmentation partner becomes difficult to disentangle from the fission or spallation regions. The fragmentation reaction is followed by a de-excitation of the remaining nucleus, which then relaxes by ejection, or evaporation, of neutrons, thus causing the reaction to favour neutron-deficient products.

Figure 1.5: Illustration of the fragmentation process. The nucleus absorbs the projectile which leads to an unstable conﬁguration. The nucleus splits into two fragments, which may then undergo nucleon evaporation.

Reaction Products

Each of the reactions previously described generate a range of product masses, and thus it may be possible to determine the reactions responsible for generating certain regions of a distribution of masses. Figure 1.6 shows the approximate mass distributions of products when protons in three diﬀerent energy ranges impinge upon a 209Bi target.

7 Figure 1.6: Approximate mass distribution of products due to the proton induced reactions from protons of diﬀering energies impinging on 209Bi. Image from [15].

From this image it is possible to see the reactions that are responsible for generating isotopes in certain regions of the nuclide chart. The 40 MeV protons produce isotopes in the region closest to that of the target mass. At these energy, it is the spallation reaction which dominates, as the proton forms a compound nucleus, which then decays to a lower energy state. Increasing the energy to 400 MeV will result in production of fission products, evident in the two peaks surrounding the mass region A ∼ 150. At 4 GeV, the spallation, fragmentation and fission reactions all contribute to the production of isotopes, and thus the mass distribution becomes a continuum blurring out the effects of each individual reaction.

1.1.4 Nuclear Decay

Nuclear decay occurs as a result of the instability of a nuclear conﬁguration, and produces a speciﬁc set of outgoing particles. Since these remnants have distinct signatures, they are then used to determine the reaction that took place, the original precursors and the fundamental processes that were required to carry out the reaction. Seen in Equation 1.3, the binding energy is a measure of the stability of the nucleus. The energetics of nuclear decay can be expressed in terms of the mass-energy relation E = mc2, where a mass m is converted into energy E, using the speed of light c as a proportionality constant. The likelihood that a nucleus will undergo a decay can be determined by Equation 1.5,

2 Q = (mreactants − mproducts)c , (1.5)

8 where m is the nuclear mass of the associated nucleus and Q represents the energy diﬀerence between the rest energies of the reactants and the products. This energy is imparted to the products in the form of kinetic energy. If Q < 0, the reaction is energetically unfavoured and will not occur spontaneously.

α Decay

This type of decay occurs when a nucleus ejects, either through spontaneous or induced 4 means, an α particle, otherwise known as a 2He nucleus.

A A−4 4 Z X →Z−2 Y +2 He. (1.6)

This decay may be attributed to Coulomb repulsion, an eﬀect which scales as the square of the proton number,growing much faster than the strength of the strong nuclear force, which scales as a function of nucleon number. As the number of protons increase, the Coulomb repulsing overcomes the strong nuclear force binding the nucleus together. Following the formulation in Equation 1.5, the energetics of the system will be,

A A−4 2 Qα = [m(Z X) − m(Z−2X) − m(α)]c . (1.7)

A spontaneous α decay occurs only if this Q value is greater than zero [7]. Emission of an α particle occurs due to the strength of the bound helium nucleus. It requires more energy to emit nucleons individually than it does to remove the group. A more detailed treatment of α decay is found in [17].

β Decay

In contrast to α decay, β decay is due to the weak nuclear force and manifests in one of two modes: β−, with emission of a electron and an electron anti-neutrino, and β+, the emission of a positron and an electron neutrino. This decay will result in the emission of a nucleus, a β particle and the β particle’s neutrino partner. The energetics of this system, characterized in the same way as Equation 1.5, account for the masses of the constituent particles. For the case of β− decay,

A A − Z X → Z+1YN−1 + e +ν ¯e, (1.8) A A 2 Qβ− = [m(Z X) − m(Z+1YN−1) − mν¯e ]c ., (1.9) where the mass of the neutrino is so small that it will make no appreciable contribution to the Q value, when compared to the other masses involved. The β− decay process transforms on neutron into a proton. In contrast, β+ decay produces a daughter that has one more

9 neutron rather than one more proton,

A A + Z X → Z−1YN+1 + e + νe, (1.10) A A 2 Qβ+ = [m(Z X) − m(Z−1YN+1) − 2me]c . (1.11)

Additionally to these two decay types, electron capture can also occur, and involved the nucleus capturing a nearby electron. The electron capture process competes with the β+ process, generating the same product nucleus [18]. This process occurs when an electron in the shell closest to the nucleus, the K shell, is captured, it then combines with a proton and generates a neutron and a neutrino, as seen in Equation 1.12.

A − A Z X + e → Z−1YN+1 + ν, (1.12) A A 2 Q = [m(Z X) − m(Z−1Y )]c − BK , (1.13) where BK refers to the binding energy, the energy required to remove an electron from the K shell. Since this case of electron capture does not require the production of a positron, it can occur in cases where β+ decay is energetically forbidden. Further discussion on the types of β can be found in [11].

γ Decay

Characterized by the emission of photons, γ decay occurs when an excited nucleus decays to a lower energy state. The energetics in this type of decay are characterized by the energy of the emitted γ-ray and the kinetic energy imparted to the remaining nucleus,

A ∗ A Z X →Z X + γ, (1.14)

A ∗ where Z X refers to the nucleus X in an excited state. The emission does not necessarily indicate the nucleus decaying into the ground state, as it can be emitted when the nucleus falls from any higher quantum state to a lower one. Additionally, a transition to a lower state need not be a single step, but may be a multi-step process, with each step emitting a γ-ray. The energy of a decay can also cause the ejection of an orbital electron. In this case of internal conversion, a γ-ray is not emitted, and instead an orbital electron absorbs the energy from the excited nucleus, and if this energy is higher than its binding energy, the electron will be liberated from the nucleus,

A ∗ A + − A Z X → Z X + e →Z X (1.15)

Qγ − EB = Edaughter + Ee− , (1.16) where this EB is the binding energy of the ejected electron [19].

10 Decay Kinetics

Nuclear decay is a stochastic process, meaning that there is no sure way to detect when a single, speciﬁc nucleus will decay. However, a system containing a number of decaying nuclei can be described using statistical methods. The decay of a species is governed by the half life, t1/2, which is deﬁned as the time required for half of an original sample of the species to decay. The decay constant is related to the of the half life,

ln(2) λ = , (1.17) t1/2 and represents the average probability that a nucleus will decay in a given period [5]. Thus the number of decayed nuclei during a given period of time is,

−dN = λN. (1.18) dt

In the diﬀerential calculus formulation, the solution to Equation 1.18 takes the form of an exponential function: −λt N(t) = N0e , (1.19) where the quantity N0 is the total number of nuclei in the system at t = 0. The activity of the system can then be deﬁned as the number of decays over some time,

−dN A(t) = dt = λN −λt = λN0e −λt = A0e . (1.20)

This set of general decay equations can be further extended to include multiple decay steps. Consider the sequential decay of a parent into two generations of daughters;

λ λ A −−→A B −−→B C (1.21)

The number of parent nuclei is described by the formulation of Equation 1.18,

dNA = −λANAdt, (1.22) describing the disintegration of the parent. The number of parent nuclei requires integration of Equation 1.22,

−λAt NA(t) = NA,0e (1.23)

11 The number of daughter nuclei decays in a similar fashion, but it will also be increased by the decay of the parent, such that the diﬀerential equation to solve includes this feeding,

dNB = λANAdt − λBNBdt, (1.24)

Assuming the parent was pure at the beginning of decay, there is no starting daughter nuclei,

NB,0 = 0, and thus integrating Equation 1.24 results in the total number of daughter nuclei after time t, λ A −λAt −λB t NB(t) = NA,0 (e − e ). (1.25) λB − λA The total number of nuclei of granddaughter C, is then calculated using the same formulation as for daughter B [10]. Given these characteristic equations, two diﬀerent cases arise when the ratio of the parent and daughter decay constant changes.

If the daughter is short-lived compared to its parent, tA > tB, or λA < λB, a state known as transient equilibrium is reached. In this case, the ratio of parent to daughter nuclei will reach a constant value. If the decay is allowed to continue for a long enough time, the decay of the daughter will become small compared to the decay of the parent, thus Equation 1.25 becomes: λA −λAt NB(t) = NA,0e , (1.26) λB − λA and since the number of parent nuclei is given by Equation 1.23, the ratio between the two species in the system becomes: N λ − λ A = B A , (1.27) NB λA And thus the decay of the total system is governed by the half-life of the parent. Secular equilibrium is a more extreme case of this, and occurs in the case where the daughter is much shorter lived than the parent, or λA λB. In this case, the ratio of the parent to daughter nuclei will reduce to; N λ A = B (1.28) NB λA The case of no equilibrium occurs when the daughter is long-lived compared its parent,

λ1 > λ2. In this type of decay, a pure parent will decay, adding to the number of daughter nuclei, and after a long enough period, the system will be dominated by the half life of the daughter [15].

Constant Production

The above formalism is only valid in the case where the number of parent nuclei is not increasing during subsequent decay. In some cases, production of the parent will take place while it decays, requiring a change of the formalism.

12 The constant production R of the parent alters the form of Equation 1.22, as this production increases the number of parent nuclei in the system;

dN = R − λN, (1.29) dt which has a solution, R N(t) = (1 − e−λt). (1.30) λ Since this constant production is time dependent, given enough time, the decay rate satu- rates at constant production value R. In the case of target irradiation, whereby an incident particle impinges upon a production target, a product can be made by direct reactions, by the decay of the parent during irradiation, and by decay of the parent after irradiation is stopped. If the rates of production of parent and daughter are RA and RB respectively, then the total number of daughters in the system is given by,

0 00 000 NB = NB + NB + NB , (1.31)

where,

R 0 B −λB tON λB tOFF NB = (1 − e )e , (1.32) λB R R 00 A −λB tON A −λAtON λB tON λB tOFF NB = (1 − e ) + (e − e ) e , (1.33) λB λA − λB R 000 A −λB tON −λAtOFF −λB tOFF NB = (1 − e )(e − e ). (1.34) λB − λA

0 00 In this formalism, NB is the number of nuclei produced via direct production, NB is the 000 number of daughter nuclei produced during irradiation, at tON , and NB is the number of daughter nuclei produced after irradiation has stopped, at tOFF [14]. The previously described equations form the theoretical framework used in nuclear science.

1.2 TRIUMF and ISAC

As Canada’s National Laboratory for Particle and Nuclear Physics, TRIUMF is operated by a consortium of Canadian Universities and employs the ISOL technique to produce rare radioactive species, which are used by a plethora of experimental groups around the world.

1.2.1 ISOL Technique

The Isotope Separation On-Line (ISOL) [20] technique is one of the methods used around the world to generate rare isotopic species.

13 This technique involves the use of a primary particle beam impinging onto a production target, both of which can be optimized to produce a specific isotope. The species are produced and ionized to facilitate extraction at the production target. From the target, the isotope is ionized and sent as an ion beam via electro-static transfer lines to a mass separator [21], that serves to remove a portion of beam contaminants. From there, it is delivered to a given experiment for use as an incident particle or for study of its properties. In order to generate the maximal number of needed isotopes, the ISOL technique must consider the production rate of the isotopes, the speed of production, purification and delivery, the selectivity and the efficiency of the entire process. The production rates in the ISOL technique, the rates at which an isotope is generated, are governed by the incident particle and target combination. Once the incident particle collides with the target, it will induce the reactions mentioned in Section 1.1.3. These reactions should, in theory, generate all isotopes up to the mass of the target, and are subject to the cross sections, with their dependence given by,

Iproduction = σNtargetΦp, (1.35)

where Iproduction is the intensity, or the production rate of a particular isotope, σ is the production cross section, Ntarget is the number of target atoms per surface area, and

Φp is the number of incident particles moving through a given area, known as the particle flux [20]. The targets within the ISOL technique are often chosen for production of very specific isotopes, for their durability or for their performance. Distinctions are made between so- called thin and thick targets, where a thin target is defined as a target with a "thickness" on the order of a few mg/cm2, while a thick target often has a thickness of many g/cm2. Thin targets can be used for fusion evaporation reactions as well fragmentation reactions, and are ideal for study of short lived species, as the species are able to exit the target quickly and be measured. Thick targets, while not optimal for very short lived species, are able to produce isotopes at a much higher rate than the thin targets due to their thickness and ability to withstand high primary beam intensities.

1.2.2 ISAC at TRIUMF

Located in Vancouver, British Columbia, TRIUMF was proposed by researchers at Simon Fraser University, the University of British Columbia and the University of Victoria as a means to study fundamental nuclear science and its potential applications. The facility was conceived in 1965, and came to house the world’s largest cyclotron in 1974. Since then it has become a world-class facility for the investigation of fundamental nuclear properties and a site for diverse applications of this knowledge [22].

14 Figure 1.7: The TRIUMF Isotope Separation and ACceleration (ISAC) facility [3].

The Isotope Separation and ACceleration (ISAC) Facility at TRIUMF specializes in producing the radioactive species used for scientiﬁc study at TRIUMF. Seen in Figure 1.7, the facility contains a multitude of equipment and experiments that are used for probing nuclei for their properties via diﬀerent mechanisms. ISAC can take the proton current, up to the maximal 300 μA [23], generated by the TRIUMF cyclotron, and produce radioactive species using the ISOL technique, discussed in Section 1.2.1.

Incident Proton Beam

The main TRIUMF cyclotron is the driver of the ISOL technique at the facility. It is responsible for delivering the high energy proton beam used to generate isotopes. To produce a high energy proton, negatively charged hydrogen ions, H− are injected into a cyclotron. The cyclotron generates a magnetic field and provides an alternating electric field, subjecting the injected particle to the Lorentz force, seen in Equation 1.36. The magnetic field causes the hydrogen ion to move in circular orbits, while the alternating electric field causes the ions to be accelerated, which in turn increases the radius of their orbit. F~ = q[E~ + (~v × B~ )], (1.36) where F~ is the Lorentz force, E~ and B~ are the electric and magnetic fields respectively, ~v is the speed of the particle and q is its charge. Once the hydrogen ion has reached the optimal speed, the H− beam collides with a thin carbon foil, which strips the electrons,

15 thus generating a high energy proton p+. Using this technique, the TRIUMF cyclotron can produce protons with energy of up to 520 MeV, and deliver a proton current of up to 300 μA [24]. The TRIUMF cyclotron, in Figure 1.8, is composed of six arms, or sectors, which serve as the magnets used to induce the circular orbits. Their shape provides the necessary vertical beam focusing required to produce the proton beam. This conﬁguration is also suitable for multiple extraction points. The TRIUMF cyclotron has four diﬀerent extractions points, three of which are currently in use, with the fourth coming online with the completion of the Advanced Rare IsotopE Laboratory (ARIEL) facility [25].

Figure 1.8: Cyclotron motion and production of high energy protons at TRIUMF. The H− ions are injected into the cyclotron (denoted by the x), and due to the Lorentz force, Equation 1.36, they travel in a circular path. The alternating magnetic ﬁeld accelerates the particles to the required speed. Once they have reached the appropriate speed, they are stripped of their electrons by a stripping foil, which removes the electrons from the ion, and they exit the cyclotron as high energy protons.

ISAC Production Targets

The target material is a key component of the ISOL technique. The induced reactions can, in theory, produce every species up to the mass of the target material. This allows for two diﬀerent approaches to be used; the target material can be optimized to maximize generation of a speciﬁc isotope, or a heavy target can be used to generate a larger assortment of isotopes, that can then be separated and used. The ISOL technique at TRIUMF requires the use of thick targets. These targets must be able to withstand the demanding conditions imposed by irradiation of the high energy proton beam. Non-metal target materials used at TRIUMF are manufactured in-house, using the slip casting technique, in which a suspension of target material powder is deposited

16 on to a backing surface and allowed to dry. This results in a sheet of target material that can then be cut into the appropriate shapes [26]. The uranium carbide targets, which are often used at TRIUMF, are fabricated by ﬁrst milling ﬂakes of uranium carbide, UC2, Figure 1.9(a), into powder. Additives are added to this powder to help with binding to the target backing and elasticity. A suspension of this mixture is then made and deposited on a graphite backing, producing a sheet of target material which is then cut into D-shaped target disks, Figure 1.9(b).

(a) Raw uranium carbide ﬂakes. (b) D-shaped target foils of uranium carbide.

Figure 1.9: Uranium carbide before and after processing for a uranium ISAC target. (a) Raw uranium carbide ﬂakes are milled into a powder, alongside other components to help deposition and elasticity. (b) Once a suspension of uranium carbide has dried and adhered to the graphite backing, the resulting sheets are cut into D-shapes, which are then stacked into the ISAC target containers. Reprinted from [27], with permission from Elsevier.

These target disks are then stacked one after the other inside a target container. The disks are stacked such that the total thickness of the composite target is on the order of 0.05 - 0.1 mol/cm2. For the uranium carbide target this means stacking 200 disks for a total target thickness of 0.063 mol/cm2 [27]. The target container must also be able to withstand constant bombardment and energy deposition from the primary proton beam. The container material must be chemically inert to ensure it does not interfere with isotope production from the intended target material. The choice metal for an ISAC target container is tantalum, as it is a highly refractory metal, meaning that it can withstand the high temperatures required for RIB production. Since the targets operate at temperatures up to 2400 oC, it is important that the container material remain structurally intact during the course of its lifetime [16, 28]. The two types of ISAC target containers, seen in Figure 1.10(a), are of the same construction, but for the ﬁnned geometry of the high power targets. They consist of a solid cylinder of tantalum, with two supports found at either end of the tube, used to place the

17 target along the beam line. The upper part of the containers houses the ion source, used to ionize the products to more easily extract them from the target.

(a) ISAC Production Target Containers. (b) Schematic of a low energy ISAC target con- Top: Low power production target container [29]. tainer. Bottom: High power production target container.

Figure 1.10: ISAC target containers. Reprinted from [29], with permission from Elsevier. (a) The two types of containers share the same features, with the only diﬀerence being the metallic ﬁns in the high energy target, which are used to dissipate the power deposited by an intense primary beam. (b) This scheme shows the features of an ISAC target, including the supports and dimensions of the target container.

During its operating history, TRIUMF has used a set of diﬀerent production targets; oxides of nickel, thorium, and carbides of silicon, titanium, zirconium, uranium as well as niobium and tantalum metal foil targets. The isotope yield rates of these targets are contained within the ISAC Yield data base [30], and are available for experimental design and proposals.

1.2.3 ISAC Yield Station

The ISAC Yield Station is the ﬁrst experiment that the isotope beams can encounter along the experimental hall. It is responsible for diagnostics and characterization of beam components. The measurements done at the yield station provide a value for the expected yield of a particular species, given in units of particles per second. These measurements are known as the yields of a given radioactive isotope, and refer to the intensity of the radioactive isotope beam. The intensity, or the number of atoms in the beam is characterized using the decay modes of the species in question.

18 (a) ISAC Yield Station experimental set up. (b) Schematic of the Yield Station.

Figure 1.11: The ISAC Yield Station. Reprinted from [31], with permission of AIP publishing. (a) The Yield Station consists of a steel vacuum chamber which houses the α- and β-decay detectors used to measure and characterize the beam. The beam is brought up from the mass separator hall below and implanted on a mobile aluminized Mylar tape, housed inside this box. (b) The chamber contains a cycling aluminized Mylar tape, upon which the beam is implanted. The decay of the species is then detected using the α and β detectors housed inside the chamber. The γ-rays are measured using a mobile High Purity Germanium (HPGe) detector attached to a Liquid Nitrogen (LN2) dewar.

Mechanisms & Detection

The Yield Station chamber houses several different detection devices, responsible for measuring different types of decay. Three PIN diodes measure the α decay process; two are placed at 45o to the implantation spot, while the third detector is placed on a movable track, allowing for implantation right at the face of the detector. A set of four plastic scin- tillators detect the electrons and positrons produced in β− and β+ decay respectively, with one placed directly behind the implantation spot. The High Purity Germanium (HPGe), used for γ decay measurement, is housed outside of the yield station vacuum chamber. It is attached to a liquid nitrogen (LN2) dewar that is placed on a linear track, and can be moved to account for the solid angle, allowing for adjustment of detection efficiency [16]. As seen in Figure 1.12, during a yield measurement, the radioactive beam is implanted onto a piece of aluminized Mylar tape for a period of time, denoted as RIB ON and chosen by considering the decay properties of the species. The beam is then blocked, denoted as RIB OFF, allowing the species to decay with its characteristic half-life and products. The decay is measured using the detectors inside and outside the box. In this figure, a yield measurement is done for the 30Mg isotope, with a characteristic half-life of 335 ms [12].

19 After the beam is turned oﬀ, the decay is observed, shown as the solid red line. Seen here, the 30Al daughter is present, arising from the β− decay of the magnesium parent. The yield Y , which is the beam intensity is calculated using Equation 1.37,

P Y = , (1.37) nb where P is the production rate, n is the number of measurement cycles, is the total detection eﬃciency and b is the branching ratio of the decay channel used to measure the decay. The production rate P used in the yield station calculation depends upon the half-life of the implanted isotope and the duration of the implantation and decay, denoted as RIB ON and RIB OFF respectively, with their relation given in Equation 1.38.

λ N P = , (1.38) (1 − e−λtcollect ) (1 − e−λtdecay ) where λ is the decay constant, dependant upon half-life, as per Equation 1.17, tcollect is the duration of the implantation, tdecay is the decay time, and N is the total number of counts measured during the decay time tdecay [31].

Figure 1.12: Yield measurement of 30Mg. Reprinted from [31], with permission from AIP Publishing.

Yield measurements are conducted during dedicated beam time, prior to experiments or as needed during the cyclotron operation schedule.

20 Yield Database

The yield rates measured at the Yield Station refer to the intensity of the produced isotope beam implanted on the tape system inside the experiment. The values for each measured isotopes are recorded in the TRIUMF Yield Database [30], and are publicly available to facilitate experiment planning. The yields are categorized by either the target material used or by the isotope produced. The database states the isotope, the isomer, the incident proton current and the yield in particles per second, among other important quantities. Figure 1.13 shows the yields present in the ISAC Yield Database, with yields from all targets that have been used at the ISAC facility during its operating history.

Figure 1.13: Yields measured at the ISAC Yield Station, for all targets used during ISAC’s operating history. The colour contours represent the yields, in particles per second. The light yellow shape represnts all known isotopes. Data from [30].

Production Rates vs. Yield Rates

The production rate is the rate at which an isotope is produced in the target via the mechanisms discussed in Section 1.1. The yield rate refers to the intensity of the beam implanted at the yield station. Though these two values are related, there is an important diﬀerence between them. The production rate depends on the cross section of the process, the number of target nuclei and the incident proton beam, as given by Equation 1.35.

21 The yield rate is related to the production rate, but is reduced due to losses associated with the eﬃciencies inherent in a physical experiment. The yield rates can be calculated from the production rates, as seen in Equation 1.39.

Iyield = Iproduction = σNtargetΦp (1.39)

The efficiency, , can be further expanded into specific effects, Equation 1.40, all which reduce the value of the beam intensity.

= releaseionizationtransport (1.40)

The efficiencies are associated with different processes that occur after in-target production [20]. The transport efficiency, transport refers to the likelihood that the species will survive transport from the production target, through the mass separator and finally to the Yield Station; this efficiency is dependent upon the isotope transport system. The ionization efficiency, ionization, relates to how easily ionized the species of interest is, a value which is element or molecule specific. The final efficiency is the release efficiency, release, an efficiency which quantifies the likelihood that the species will survive long enough to be ionized, extracted, transported and measured; this efficiency is isotope specific. These efficiencies reduce the value of the in-target production rates, such that the measured intensity is less than the produced intensity. As a result, gaining an understanding on these efficiencies is key. The ionization efficiency can, in some cases, be identified based on the ionization potentials of the species of interest. The transport efficiency can be accounted for by measuring the RIB intensity at the target and then again at the measurement location. The release efficiency can be established by characterizing the time between production and measurement using a time tag. Adjusting an experimental set-up to account for these efficiencies may be realistically unachievable, but understanding the efficiencies and their effects on different species leads to better understanding of the isotope production mechanisms in place. Though experiments to quantify these losses are, in theory, possible, their design and testing are outside the scope of this work. During the course of this work, the effects of some of these efficiencies have been considered and analysed, and though done in a rather limited capacity, these preliminary studies serve to provide the first steps of the quantification of the effects.

1.3 GEANT4

1.3.1 Framework

GEANT4 is a Monte Carlo based nuclear transport toolkit written in the C++ computer language. It was developed at CERN to simulate the interactions of particles as they are transported through matter. The toolkit is very large and robust, comprised of millions

22 of lines of computer code, and supported by a worldwide collaboration, working to ensure that it is properly validated, benchmarked and constantly updated to provide the predictive power necessary in the nuclear applications in which it is used. The package consists of several categories, each responsible for a particular aspect of the simulation of particles being transported through matter. Seen in Figure 1.14, the categories may depend on other categories in order to carry out their tasks. Each of these categories is part of an overarching domain, which serves to diﬀerentiate the simulation steps [32]. The domains are categorized as events, tracking, user actions, geometry and physics processes.

Figure 1.14: The GEANT4 framework. The arrows denote that the category at the end requires inputs from the previous cagetory. The highlighed categories are those important to this work. Adapted from [32].

The "event" category is responsible for the generation of primary particles, which defines the beginning of the simulation. It is also used to handle single events in the simulation, an event being the primary unit of the simulation. In the interest of computational power, some event information is not permanently stored, but it can still be extracted by conversion of the information that persists. The propagation and transport of particles during simulation are handled by the "tracking" category within GEANT4. The particles are moved in a step-wise manner, the length and duration of which are determined based on their properties, and the properties of the materials through which they pass during that time. This is done by defining first a pre-step point, and checking where in the geometry the particle is before it is propagated; interactions as particles move through the material(s) are then applied in the along-step point; the

23 final state, the post-step point, of the particle and its production of secondaries are invoked once the particle has come to rest. The "user" action category serves to supplement the toolkit. GEANT4 contains a large set of particles, materials and physical processes, but should all of the above be insufficient for the simulation needs of the user, they are free to implement their own processes, materials and tracking. Each of the relevant categories contains a function call, where the user can define the specifics of their experiment. A physical process, a new particle or a different tracking mode can be adapted into the framework. The physical processes within GEANT4 are contained inside the "physics" category, which houses the processes, algorithms and phenomenological information required to properly determine what occurs during particle propagation and transport. This category is subdivided into the types of models, whether they use real parametrizations, depend upon a theoretical formulation or a combination, and these are further divided by the energy range in which they operate. Discussion of pertinent physics models appears in Section 1.3.3. The "geometry" category defines the experimental set up. Definition of detector volumes and materials is done within this category. Within the toolkit, the user is able to completely describe their specific geometry, using the materials they require. As discussed in Section 1.3.2, there are several methods with which the user can define their experimental set up.

1.3.2 Geometry

The geometry within GEANT4 describes the physical dimensions and the materials in the environment. The simulation is built in several different stages, and the toolkit makes distinctions between different types of simulation volumes. The so-called mother volume is defined as the box in which the entirety of the simulation takes place. Every component, incident particle, produced nucleus and physical process is defined, begins and ends its path within this mother volume. The volume itself can be of any shape and be filled with any material, but it must be big enough to contain all simulation components. Each of the simulation volumes is defined first through its shape. Simple and curved solids are available within the toolkit, but definition of more complicated components can be done through the use of Boolean solids or imported from external software [33]. This solid shape is then placed inside a logical volume, which defines the container where the solid shape sits, and is responsible for handling the material that comprises the shape. The component volume is then placed inside the mother volume. To save memory and computation time, a volume can be defined once and then placed inside the mother volume as many times as needed to generate segmented geometries. The materials database used inside GEANT4 may be populated from the information available through the National Institute of Standards and Technology (NIST) database [34]. The database contains information about isotope mass, natural isotope composition,

24 material densities and other data. The GEANT4 database also contains some common materials such as Kevlar and environments such as vacuum [33]. Material implementation is very straight-forward, allowing the user to deﬁne a wide range of materials that represent those available in nature.

1.3.3 Physics Lists

Within the GEANT4 framework, a physics list contains the necessary processes and models that must be called in order to simulate a physical reaction. A process contains information on initial and final states, defined by a precise cross section and a mean lifetime for the process, while a model is responsible for generated secondary particles; several different models may be contained within a single process [32]. The toolkit contains a vast number of physics lists, describing possible models and processes for a wide range of particles. It is left to the user to determine which physics list is appropriate for their particular application. The physics lists range between being empirically driven processes to theoretically based models. Empirical data-driven models are useful where experimental and measured data are available, and due to their use of precise data, they are the best models to use when predicting reaction outcomes. Theoretical models, in contrast, rely on the theoretical formulation of physics processes, and are thus most often useful in the regions where no data are available. The key consideration to make in choosing a model is the energy range of the simulation, as each of the physics lists is valid within a specific energy range, outside of which they begin to have questionable accuracy [4]. Since the simulation must be in line with the capabilities of the facilities at TRIUMF, it is the 500 MeV protons produced by the cyclotron that will dictate the energy range for this simulation. There are three intranuclear cascade physics lists that can accurately describe a interactions of proton travelling with an energy 500 MeV and the matter inside a target: the BERT-inspired cascade [35], the Binary cascade [36] and the Liège Cascade [37] models. These physics lists operate as intra-nuclear cascades in order to accurately predict the outcome of nuclear reactions.

Intranuclear Cascade Models

Formulated by Serber in the late 1940s [38], the intra-nuclear cascade was an attempt to numerically approximate the interactions occurring in the nuclear reactions between heavy ions. This methodology proposed that nuclear reactions could be approximated as interactions between two single participants. Whether it was an incident particle colliding with a nucleon inside the nucleus, or a nucleon scattering oﬀ another nucleon in the system, the overall eﬀects could be accounted for by treating the system as a series of binary reactions inside the nucleus. A simple picture of this intranuclear cascade is seen in Figure 1.15.

25 In this formulation, the nucleus is defined by randomly placing its constituent nucleons inside the nuclear volume at random, but within the confines of a defined density and distribution. The nucleons are then propagated along straight lines until two come within interacting distance. At this stage the total cross section for their interaction is applied, and if they interact, they are scattered. This scattering leaves a highly excited nucleus, a configuration which must be passed on to other algorithms to handle de-excitation [39].

Figure 1.15: A simple intranuclear cascade scheme shows that a proton enters the nucleus and interacts with a nucleon. This interaction may cause scattering or ejection of the participants or generation of other particles, such as a π-meson. Adapted from [40].

BERT-inspired Cascade

The Bertini-inspired intra-nuclear cascade within GEANT4 is a theoretical approach based on a reworking of the solution to the Boltzmann equation formulated by Bertini in 1968. The algorithm within GEANT4 includes the intra-nuclear cascade description, and a series of models describing pre-equilibrium, explosion, ﬁssion and evaporation. This cascade treats the nucleus as a series of up to three concentric shells, each with changing density. It employs relativistic kinematics in its particle transport, and the cascade is stopped once all particles that can be ejected have been removed from the nucleus. The products and reactions are then checked to ensure that they satisfy the conservation of energy. The cascade can use protons, neutrons and πmesons as the projectile particle, while any possible material can be used as a target, and is valid in the incident particle energy range between 0 MeV and 10 GeV [41]. A typical run of the Bertini cascade consists of a set of sequential steps. First the cascade model evaluates the point on the nuclear sphere where the incident particle will enter the nucleus. The particle is propagated using free particle-particle cross sections and the nucleon

26 density in the region. Once the particle path and its collisions have been established, its four-momentum and that of its collision partners are calculated. At this point, if the Pauli exclusion principles allows for it, the paths of the products are calculated and the cascade continues. This stage comes to an end when all particles have been propagated through the cascade. The four-momentum information for all particles is then passed onto the pre-equilibrium and de-excitation models [35]. First the pre-equilibrium model uses an exciton model, which determines the number of particle-hole pairs inside the nucleus. Using this information, it tests whether the nuclear configuration meets the particle-hole selection rules, ∆p = 0, ±1, ∆h = 0, ±1, ∆n = 0, ±2, where p is the number of particles, h is the number of holes and n is the number of excitons, given by n = p + h. If the nuclear configuration is allowed, it takes the target excitation data and proton and neutron exciton configurations and ejects nucleons from the system as needed. The nuclear configuration is then passed onto the evaporation models which are responsible for further de-exciting the nucleus. Fission is used if necessary, using a phenomenological formulation, and particle emission is computed until the excitation energy of the nucleus falls below 0.1 MeV [4]. The accuracy of the Bertini cascade in the production of radionuclides has been tested and verified by the GEANT4 collaboration [35,42] to ensure that it has been properly implemented into the framework, and to ensure that the results are in line with the experimental data.

Binary Cascade

Operating in the energy region between 0 MeV and 3 GeV [41], the Binary intra-nuclear cascade computes the outcome of interactions between incident particles and target materials. It propagates primary and secondary particles in a binary fashion, such that interactions take place between two single particles. The products of these reactions then scatter with the other components of the system, forming the basis of the nuclear cascade. Within the Binary cascade, the nucleus is approximated to be a sphere, with isotropic density, and nucleons are placed inside the nucleus using theoretical approximations for nuclear density distributions [36]. The movement and interactions are carried in a binary process, by numerically integrating the equations of motion, establishing a time step. Each time step occurs between the start of the particle trajectory and its interaction with another particle; between its entrance and exit from the nucleus, or its decay. The Binary cascade algorithm deals only with the interactions occurring within the nucleus. Once all particles have left the nucleus, and all other interactions have taken place, the nuclear conﬁguration is passed onto the GEANT4 pre-equilibrium and de-excitation codes.

27 The pre-equilibrium model packaged into the GEANT4 framework takes information about the nucleus, such as mass A, proton number Z, its four-momentum, its excitation energy and the number of particle-hole pairs n, and checks whether the selection rule ∆n = 0, ±2 allows this conﬁguration of the nucleus. This model allows for the emission of single nucleons, deuterium, tritium and helium nuclei [4]. The GEANT4 evaporation model then takes the output of the pre-equilibrium algorithm. This last model is responsible for further relaxing the nucleus. If the nucleus has atomic mass A > 65, ﬁssion is evaluated, and carried out if allowed. Then the model will further de-excite the nucleus by emitting fragments such as α and γ-ray particles and other light nuclei [4]. The Binary cascade, coupled with the GEANT4 pre-equilibrium and evaporation models, has been benchmarked by several groups inside the GEANT4 collaboration [42], showing that the model can be used to accurately predict nuclear reactions and their products.

Liège Intranuclear Cascade

Developed primarily as a parametrized model, the Liège cascade operates in the energy range between 1 MeV and 20 GeV. This physics list aims to accurately describe systems in which spallation reactions dominate [43]. Within the GEANT4 framework, the Liège cascade models the nucleus as a free Fermi gas in a static potential well, using a realistic profile for the nuclear density. The model attempts to use the minimal number of free parameters, in order to more accurately predict collision reactions and products. This is done by either taking needed parameters from phenomenological values, such as the radius of a nucleus and reaction cross sections, or by fixing known parameters, as in the case for stopping time [4]. The point at which an incident particle enters the nucleus is chosen at random, by calculating the impact parameter, which is the distance between the incident particle and the center of the nucleus. The model propagates particles along straight paths of varying lengths, where the length is dependent upon the time step required between the previous interaction and the next one. During this process, the cascade is able to eject nucleons and nucleon clusters, with up to A = 8, where the formation of a cluster is determined by whether its neighbours, if close enough to the surface of the nucleus, are sufficiently close enough to be ejected. This cluster-coalescence algorithm may mimic the process that causes spallation reactions to emit small clusters of nucleons. As with the other models, the Liège model has been extensively benchmarked [44, 45], as well as constantly updated, ensuring its predictive power [37, 46] Though the standard de-excitation and evaporation models within GEANT4 are the standards coupled with the Liège model, another set of these models can be used to simulate nuclear reactions. ABLAV3 is an algorithm that models the de-excitation of a highly excited nucleus to a more stable state. At its core, it takes the principle of abrasion and ablation into

28 consideration [47], where both refer to the removal of nucleons from the excited nucleus, effectively handling the fragmentation processes. This model can be separated into three parts, according the reactions taking place, namely abrasion, evaporation and fission. The abrasion routine is of particular interest when the collision partners are heavy nuclei, and it computes the outcome of the collision by distributing the nucleons involved as either participants or spectators. The evaporation model is then called, and checks for the likelihood of fragment emission. Finally the phenomenological fission model is applied [48], and the three processes return their products. The coupling of the Liège model and ABLA has been extensively tested [37, 49], and can accurately reproduce the experimental data available.

29 Chapter 2

Simulation

The GEANT4-based simulation built was to mimic the TRIUMF production mechanisms as closely as possible. This required the proper deﬁnition of the initial particle beam and the target materials and geometry. There were three areas of the simulation to consider: the incident particle and target combination, the quantities to extract and the mechanism to use, and the physics models to implement. The simulation is built using GEANT4 version 10.01.p02 [4]. The choice of GEANT4 was made to compliment existing FLUKA software to predict isotope production at the ISAC facility, as no previous GEANT4 version was available. It thus is meant to serve as a compliment to the FLUKA simulation and to provide an alternate prediction method. The analysis is done using ROOT version 5 [50].

2.1 Simple Geometry

To achieve a suitable comparison model between GEANT4 and other nuclear transport toolkits, as well as to more precisely compare the diﬀerent physics lists within GEANT4 itself, a simple geometric model of the target was built, and a realistic incident particle beam deﬁned.

2.1.1 Primary particle description

The primary particles are defined as protons incident along the z-axis on the mother volume. The user is free to determine the kinetic energy of the incident beam at the beginning of a simulation run, which is passed to the simulation as part of a terminal command line option used to begin a simulation run. The proton beam is generated using the General Particle Source class available in GEANT4, which allows for the definition of the incident particle, event origin, default energy and particle distribution. The proton events are initiated 80 mm from the target, and travel along the length of the target. To more accurately simulate the realistic case, the protons are given a Gaussian distribution profile with a Full Width Half Maximum

30 (FWHM) of 7 mm in both x- and y-axes, matching the beam distribution delivered to ISAC by the TRIUMF cyclotron. The simulation also supports the use of other primary particles, such as neutrons and electrons.

2.1.2 Standard Target Description

The production target is built as a solid cylinder, composed entirely of the target material. The cylinder has radius 9.5 mm, and is 5 cm long. To more accurately depict a segmented target, this cylinder is composed of 5 disks, each 1 cm thick, to complete the 5 cm length of the target. The stacked disks are deﬁned by setting the dimensions of one disk and placing this volume ﬁve times along the mother volume, one after the other, while ensuring no overlap. Figure 2.1 shows a visual representation of the target material geometry.

(a) Single target disk. (b) Full target material geometry.

Figure 2.1: Simple target geometry visualization. (a) A single target disk is 1 cm thick with a radius of 9.5 mm. (b) The target is composed of five targets, stacked beside one another to simulate a compound target. (c) The "wireframe" visualization of the target shows the composition of the target, with the five disks. (d) A visualization of a ten particle run shows different events occuring in the target. The blue tracks are protons, green tracks are electrons, red tracks are photons and the yellow tracks are produced nuclei.

31 This simpler version of the ISAC thick target geometry was used to ensure a standard to compare to other nuclear models, such as FLUKA and Silberberg-Tsao models, described in Section 2.5. The GEANT4 simulation takes into account the different components within each of the defined target materials, details of which are found in Table 2.1. The common factor between all the described target materials is the so-called thickness, set at 0.05 mol/cm2. This thickness describes the number of atoms along the path of the incident particle. Each material was defined within GEANT4 using the G4Material class, which allows the user to use materials available in the NIST database [34], or specify materials based on their elemental components and density.

Table 2.1: The materials implemented in the GEANT4 Simulation include all materials previously used at TRIUMF, along with a selection of other materials. The target densities are all calculated to account for a 0.05 mol/cm2 thickness. The materials are all deﬁned in terms of the proton and neutron content, such that they are pure isotopes. The material densities are those found in the CRC Handbook of Chemistry and Physics [51].

Material Components Target density (g/cm3) Material density (g/cm3) 99.77% 238U Depleted Uranium 2.38 19.1 00.23% 235U Thorium 232Th 2.32 11.7 Tantalum 181Ta 1.80 16.4 Niobium 93Nb 0.93 8.57 Zirconium 90Zr 0.91 6.52 Nickel 58Ni 0.59 8.90 Titanium 48Ti 0.48 4.51 Silicon 28Si 0.28 2.33

Table 2.1 outlines all target materials that have been implemented into the simulation, which encompass the materials that have been used at ISAC. The results presented in Chapter 3 are those extracted from simulation runs using the depleted uranium target material.

2.1.3 Detection Mechanisms

Information regarding the isotopes produced with this incident particle-target combination is extracted using a GEANT4 Sensitive Detector. This class is responsible for recording, or scoring, important quantities during simulation. The user is responsible for activating the sensitive detector and for deﬁning the types of values it is to record [52]. Available information includes proton number, baryon number and kinetic energy.

32 Within this simulation, the sensitive detector is built as a virtual "check" volume. This check volume is a so-called virtual volume because there is no material associated with it, and it does not interact with the materials or the products, it is simply used as a recoding mechanism. The dimensions of this check volume are 1 mm larger than those of the target, such that the length is 5.1 cm, and the radius is 10.5 mm. The check volume is deﬁned as a solid 5 cm long cylinder, surrounding the target material, instead of the segmented method of the target material.

(a) Virtual sensitive detector visualization. (b) Run with sensitive detector visualization.

Figure 2.2: The virtual sensitive detector. It is used to count and identiﬁy the produced isotopes. (a) The virtal sensitive detector encases the target disks, and is deﬁned as a continual cylinder, with a radius of 10.5 mm and a thickness of 5.1 mm, such that it is larger than the target volume. (b) The virtual sensitive detector is used only to detect produced nuclei, and its material properies are those for empty space. As such all outgoing particles do not interact with this volume.

2.2 Data Output & Processing

The simulation returns data in the form of a comma-separated ASCII file, containing a single product nucleus per line, given as proton number, number of nucleons and the kinetic energy. The raw files of a 108 primary particle run are on the order of 1 gigabyte, and contain information about all isotopes generated during the run and hit the virtual detector, as well as all the neutrons that have hit the virtual detector. In order to efficiently count the instances of each isotope, this ASCII file is passed to the data analysis software package ROOT [53]. A ROOT script reads the ASCII file, takes the proton number, Z, and number of nucleons, and from these calculates the neutron number, N. The script also populates the chart of nuclides, built as a 3D histogram, with the number of neutrons, N, on the x-axis, the number of protons, Z, on the y-axis, and the quantity of each isotope, captured in a colour map.

33 The script can also be used to count speciﬁc elements. This must be implemented into the script, and will return the exact number of a particular isotope produced during a simulation. These values, for a given isotope, are manually entered into a spreadsheet for further analysis.

2.3 Error Analysis

To ensure the precision as well as the accuracy of the simulated results, an analysis of the errors was conducted. To calculate the errors in this Monte Carlo simulation, ten simultaneous and independent runs were conducted, using the same number of primaries at the same energy. The isotope production rates of these identical runs are recorded. The standard deviation of the runs is then divided by the average between the runs, returning a percent error associated with the simulation, as described by Equation 2.1,

σ % Error = × 100, (2.1) 1 Pn=10 N 1 Pn where P is the production rate of a speciﬁc isotope, n is the run number, from 1 to 10, σ is the standard deviation between the runs. This method of standard error provided error approximations in the simulation results. A more detailed description and example of this analysis is found in Appendix A.

2.4 Benchmark Products

TRIUMF’s Yield Database contains a large set of measurements, for a wide range of isotope beams. In order to more efficiently characterize the simulation, and to mitigate some of the considerations discussed in Section 1.2.1, a smaller subset of the measured data was examined. As seen in Equations 1.39 and 1.40, the measured intensity of the beam, or the yield rate is governed by the ionization efficiency. The alkali metals, comprising the first column of the periodic table, have very high ionization efficiencies [54], due to the configuration of their orbital electron shell, which only contains one electron. They thus do not suffer greatly from ionization efficiency losses. Furthermore, they are fairly volatile, and with the exception of lithium, do not form stable compounds at typical target operating temperatures [55], allowing for relatively good release efficiencies. Their varying masses ensure that this set of elements is produced via all three primary reactions discussed in Section 1.1.3, ensuring that analysis of this subset will include all three production mechanisms. These factors thus allow for more meaningful comparison between the in- target production rates and the yield measurements.

34 2.5 Competing Models

GEANT4 is one of many nuclear transport codes, each with their advantages and limita- tions. To ensure accuracy in modelling and implementation two other nuclear codes were consulted.

2.5.1 Silberberg-Tsao

The Silberberg-Tsao (ST) model was developed by R. Silberberg and C.H. Tsao in the early 1970s, with the goal of describing the nuclear interactions that occur in space, in an attempt to characterize and understand cosmic radiation. It consists of semi-empirical formulae for the cross sections of diﬀerent reaction mechanisms that occur in heavy targets [56]. The main formulation contains several parameters that change depending on the type of reaction described, the mass of the target used, and the mass of the isotope produced. The type of reaction is taken into account by using its associated cross section, and the distribution of cross sections as a function of atomic number is also addressed, among other important parameters. Each of the parameters in the formulation are varied and depend on the region of the nuclide chart of the target used and the products [56, 57]. For this work, a Fortran script containing the ST algorithm was used to predict in-target production rates.

2.5.2 FLUKA

Developed by the Istituto Nazionale di Fisica Nucleare (INFN) and CERN, FLUKA [58] is a FORTRAN-based, full integrated particle physics simulation package. It is a multi- purpose software, which uses Monte-Carlo statistics to simulate interactions and transport of different particles with and through matter [59]. Similar to GEANT4, FLUKA takes into consideration all physical processes that are the cause of or the products of different nuclear reactions with different incident particle and target combinations. It is also able to define and work with complex geometries and handle different target materials. Emphasis is placed upon using the least number of free parameters, thus ensuring the completely accurate simulation of real events. A simulation using FLUKA requires a series of so-called "cards", lists of commands that guide the simulation. These cards are used to establish the geometry, the incident particle and the quantities to evaluate among other useful properties. The physical processes are all included into the framework; the user need only select and activate the cards of interest in order for the physical processes to take effect. FLUKA has been in use at TRIUMF for a number of years to estimate radionuclide production for radiation safety and yield rates. Since the expertise in using this framework already exists, comparison results were generated by requesting simulation runs be completed by A. Laxdal, Particle Source Development Engineer, working with the ISAC

35 Facility Operations and Development department of TRIUMF. Given speciﬁcations as to the incident particle energy and target geometry, she was able to complete simulation runs using FLUKA and its Flair Graphical User Interface (GUI).

2.5.3 IAEA Benchmark of Spallation Models

The International Atomic Energy Agency sought to compare and benchmark the available nuclear transport codes, with a particular focus on examining their ability to reproduce spallation products [60]. The results from this systematic evaluation of the currently available models encompass information such as the double differential cross sections for neutrons, protons, among other particles; the mass, charge and isotope distributions and various excitation functions [61]. This is done for a wide range of energies and lists the authors responsible for the data that the spallation codes are compared to. This extensive database contains the results from the GEANT4 Bertini and Binary cascades as well as results from the Liège cascade, coupled with ABLA07, a more recent version of the ABLAV3 de-excitation code. The purpose of compiling the database was not to determine the best computation model for spallation reactions, but rather to compare the results from different formulations and to determine whether there are large discrepancies between models, or gaps in isotope production that required modifications. The database of the spallation result is available to the public and may be used to examine the validity of the presented models. The author of the accompanying paper [60] does, however, mention that the main goal of the benchmarking process was to establish strengths and weaknesses of the overall understanding and modelling of spallation reactions, rather than to determine which model is the absolute best at describing this complex reaction. The database was briefly consulted during the course of this work for consistency purposes.

2.6 Extended Geometry

To more closely mimic the complex target geometry of the TRIUMF targets, an extended geometry was built. This extended geometry included the addition of the tantalum target tube, with a 10.5 mm radius, and the finned geometry used in the high energy target containers. As opposed to the segmented target material geometry, the tube was built as a single solid tube with cylindrical cavity in the middle, by using Boolean geometry; two tubes were built, one with a 10.5 mm radius, and another with a 9.5 mm radius, then the inner tube was subtracted from the definition of the larger tube, thus creating a cylindrical casing. The tantalum fins were built in much the same way, employing Boolean geometry. The foils were defined as squares with the appropriate dimensions, 55 mm by 55 mm with a

36 thickness of 380 μm [62]. The circular cut-out was defined by taking the same cylinder defined for the target container and subtracting it some from foiled geometry. A total of 90 fins were added to the target using a looped routine. The twisted supports were also added to the geometry. Though the elements aforementioned were all coded into the geometry, for comparison only the simple geometry described in Section 1.3.2 was used for the purposes of benchmarking, comparison and simulation. The elements can be activated in the source code if needed. A visual representation of the extended target geometry is seen in Figure 2.3.

(a) Extended geometry. (b) Extended geometry with target container.

Figure 2.3: Visualization of extended target geometry. (a) The extended geometry consists of the target disks, in purple, the fins and supports in white and the virtual container, also in white, surrounding the whole geometry. (b) In addition to the fins, the extended geometry also contains a cylindrical target container, seen in red. (c) In this cross section, the construction of the target container is shown, where the target material is in purple, the target container in red, and the fins in grey. (d) A target run using the extended geometry shows similar features to the simple geometry run. Blue tracks represent protons, green tracks are photons, red tracks are electrons and yellow tracks represent produced nuclei.

37 2.7 Current Simulation Status

The current simulation requires GEANT4.10 to run the particle propagation and production and ROOT version 5.34 (or later) to generate the production rate nuclide charts. The simulation must be run in a Linux terminal via a line command, with which the user can set the primary particle energy and the number of primaries. A complete version of the code is housed in the TRIUMF document server [63]. As is stands, the geometry provides the user with the five target disks defined in Section 1.3.2, but this can be changed as required inside the source code. The current materials defined are those in Tables 2.1 and 4.5, but may be extended as necessary. The GEANT4 toolkit provides access to many other simulation parameters and results, which can be implemented in future versions of this simulation.

38 Chapter 3

Results

The results from the simulation of the primary particle, described in Section 2.1.1, and the target geometry, described in Section 2.1.2, were extracted in several different ways. First, the overall results of the Bertini, Binary, and the Liège (coupled with the ABLAV3 de-excitation code) cascades were examined, accounting for all isotopes produced by each list. These isotopes were plotted in 2-Dimensional plots with respect to their proton number Z, and neutron number N, effectively creating nuclide charts, in Section 3.1. For a more specific comparison, the in-target production rate values of each of the alkali metals were extracted and plotted as a function of mass number, A. These in-target production rates were compared between the three physics lists, in Section 3.2, to competing nuclear transport codes, in Section 3.2, and to the measured yield rates, in the ISAC Yield Database [30], in Section 3.2. To further examine the capabilities of the simulation, further studies were conducted, including cross section verification, characterization of release times and efficiencies and implementation of alternative targets, results of which are summarized in Chapter 4.

3.1 Physics Lists Results

Due to their different algorithms, each of the physics lists discussed in Section 1.3.3 produced different isotopes, in varying quantities from runs with identical sets of starting conditions. The in-target production rate of each isotope, measured in units of isotopes per second, is plotted as a function of proton number Z and neutron number N. This generates a nuclide chart, similar the one seen in Figure 1.1, where the production rate values are presented by the colour contour in these 2D plots. The charts, shown in Figures 3.1, 3.2 and 3.3, respectively, serve as an overall comparison of the production of each of the lists. Each contains different features. The results depicted in the charts were all produced by bombarding the depleted uranium standard target with

39 109 protons at an energy of 480 MeV, a process which would take on average 480 CPU hours. In Figure 3.1, the results from the Bertini cascade are plotted, with the in-target production rates plotted as a function of mass number A. This nuclide chart contains two very interesting features, namely the gaps in the mass regions 29 < A < 48 and 167 < A < 185. This intra-nuclear cascade also depicts very prominent ﬁssion regions, seen as two regions of higher production around mass A = 114. In comparison to the Bertini cascade, both the Binary and the Liège cascades, in Figures 3.2 and 3.3, respectively, produce isotopes across the entire N,Z space. In this region, the Binary cascade produces isotopes at lower rates than the Liège cascade. The same trend is noted in the light mass region, up to A = 30, where the Liège cascade appears to generate isotopes at a higher rate.

Figure 3.1: The in-target production rates generated by the Bertini Cascade, using the depleted uranium standard target. The Bertini cascade has two distinct gaps, in the mass regions between 29 < A < 48 and 167 < A < 185.

40 Figure 3.2: The in-target production rates generated by the Binary intra-nuclear cascade, using the depleted uranium standard target. The Binary cascade appears to produce a large number of isotopes in the mass region 95 < A < 125, compared to the other two lists.

Figure 3.3: The in-target production rates generated by the Liège intra-nuclear cascade results, coupled with the ABLA de-excitation code, using the depleted uranium standard target. The Liège cascade produces a higher number of isotopes in the mass region 165 < A < 180, compared to the Binary cascade.

41 Figure 3.4: Yields from uranium targets as measured by the ISAC yield station.

Figure 3.4 shows the yields meausred from the uranium targets that have been used at TRIUMF during the course of its operating history. This ﬁgure provides a visual comparison between the simulated in-target production data and the experimentally measured yield data. Though this visual comparison may provide insights as to the capability of each list to predict the measured data, a more robust analysis requires the detailed comparison of a subset of the simulated data.

3.2 In-target Production Values

In-target production rates were extracted for the six alkali metals: lithium, sodium, potassium, rubidium, cesium and francium. This subset of elements was chosen due to ionization efficiency considerations, as discussed in Section 2.4, and due to the wide range of half- lives present in their isotopes. The in-target production rate was calculated by counting the number of times an isotope was produced by the simulation. This production rate is plotted as a function of mass number A, for each specific element. The results from each of the physics lists, the Bertini, Binary and Liège cascades, are plotted in Figures 3.5. Figure 3.6 compares the rates produced by the GEANT4 physics lists to those of FLUKA and the Silberberg-Tsao model. The in-target production rates are also compared to the yield measurements from the ISAC yield database [30], plotted in Figure 3.7. Note that, as discussed in Section 1.2.3, the production rates are related to the yield rates but are modified by efficiencies associated with conducting a physical experiment.

42 Comparison Scaling

In order to attain a useful comparison, the in-target production rates and the measured yield rates were scaled to reﬂect 1 μA of proton current impinging on the target. This was done by obtaining the production rate from a full simulation, dividing it by the number of primary protons, thus converting the in-target production rate to isotopes per primary. This value was then scaled using the number of particles in 1 μA of current, as seen in Equation 3.1

A P (isotopes/1μA) = P, (3.1) Nproton where P (isotopes/1μA) is the scaled production rate, P is the production rate calculated from the simulation, Nproton is the number of primary protons moving through the target material, and the constant A = 6.24×1012, corresponding to the number of protons in 1 μA of current. This scaling returns the scaled production rate in terms of isotopes per second per 1 μA of proton current.

GEANT4 Results

In order to establish which of the three physics lists is most appropriate, a comparison between the three lists was done using the alkali metals. The in-target production rates for the light alkalis, shown in Figure 3.5(a), depict some important features. For the lithium isotopes, the Bertini cascade appears to produce four orders of magnitude less isotopes than both the Binary and the Liège cascades. The sodium production rate results show that the Bertini and the Binary cascades are unable to produce the neutron-rich isotopes of sodium, and the physics lists vary in the quantity of isotope produced. The Bertini produces only 104 of 24,25Na, and the Binary cascade produces 109 of the same isotopes. The most striking feature of the potassium in-target production rate results is that the Bertini cascade does not produce isotopes of this element, when using 480 MeV incident protons and 109 incident primaries. The Binary and the Liège cascades, in contrast, are able to produce isotopes both on the neutron-deﬁcient and the neutron-rich sides. There is, however, still a visible diﬀerence in the produced quantities, with the Liège cascade producing an order of magnitude more than the Binary cascade.

The features of the heavy alkalis span a wider range of masses, when compared to the light alkali metals. The rubidium isotopes, in Figure 3.5(b), are produced in very similar quantities by all three physics lists, though their peaks do not occur at the same isotope mass. While the Binary and the Liège cascades have their maxima at 91Rb and 92Rb,

43 respectively, the Bertini cascade predicts a maximum at 87Rb, favouring neutron-deficient rubidium isotopes. For the cesium isotopes, the Bertini cascade also results in a production curve peak at a lighter mass of 130Cs, when compared to the other two physics lists. The in-target production rates of cesium isotopes, produced by the Binary and the Liège cascades, peak at 138Cs and 144Cs, respectively. This behaviour deviates from the agreement of the two lists, seen in the rubidium isotopes. The Liège cascade also appears to produce constant values of the cesium isotopes 130−143Cs, a trend which is not seen in either of the two other physics lists. The francium isotopes also show a shift in the in-target production rate peak, but in this instance, the Bertini cascade predicts a peak at 220Fr, the Binary cascade predicts a peak at 213Fr and the Liège cascade predicts a peak at 214Fr. For this element, the Bertini cascade also predicts two orders of magnitude less francium than both the Binary and the Liège cascades, a trend similar to that in the rubidium and cesium cases. There is also an interesting feature in the neutron-rich francium isotopes predicted by the Binary cascade; the production rate results decrease with respect to higher mass numbers, as expected, but then increase again at 231Fr, a trend which is not seen in the other two lists. From this graphical analysis, it is evident that the Binary and the Liège cascades favour the neutron neutron-rich isotopes of cesium and the neutron-deficient isotopes of francium. The Bertini cascade favours the opposite case, producing more neutron-deficient isotopes of rubidium and more neutron-rich isotopes of francium. As mentioned in Section 2.3, the production rates depicted in Figure 3.5 have calculated uncertainties. Details of error calculations, and figures with errors are found in Appendix A.

44 (a) Production rates for the light alkali metals. (b) Production rates for the heavy alkali metals

Figure 3.5: In-target production rate results for the alkali metals as a function of mass number, A. (a) Production rates for the light alkali metals; lithium, sodium and potassium. (b) Production rates for the heavy alkali metals; rubidium, cesium and francium.

45 GEANT4, FLUKA and Silberberg-Tsao models

Described in Sections 2.5.2 and 2.5.1, respectively, the FLUKA [58] and Silberberg-Tsao (ST) [56] computer codes were used to simulate the same GEANT4 standard target. They were to provide a baseline ensuring the proper implementation of the geometry in GEANT4 and providing a comparison for the GEANT4 simulations. Figure 3.6 shows the comparison between the GEANT4 in-target production rates, those seen in Figure 3.5, and the production rates predicted by FLUKA and Silberberg-Tsao. The light alkali metals, in Figure 3.6(a), show that in the lithium isotopes, both FLUKA and the Silberberg-Tsao model (ST) agree with the in-target production rates produced the Binary cascade. In the sodium isotopes, FLUKA appears much closer to the Liège cascade, while the ST produces rates that are an order of magnitude higher than FLUKA, closer to the results from the Binary cascade. The potassium production rate results generated by FLUKA are very similar in magnitude to the Liège cascade, while the ST model generates these isotopes with two orders of magnitude higher production than both the GEANT4 lists and the FLUKA code. For rubidium, in Figure 3.6(b), FLUKA produces a curve very similar in shape and in value to the Binary cascade production curve, while the ST code is more in line with the results from the Bertini cascade. The cesium isotopes are very similar to the rubidium isotopes; FLUKA follows the shape of the Binary cascade. However in this case the ST model does not follow the Bertini cascade, instead generating more neutron-rich cesium isotopes and following the Liège cascade on the neutron-deficient side. In the francium isotopes, it appears that FLUKA does not produce the appropriate quantities of neutron-rich isotopes, while again following the trend of the Binary cascade for the neutron-deficient francium isotopes. The ST code appears to follow the shape of the Liège cascade, with values that are an order of magnitude lower. With this comparison between GEANT4, FLUKA and Silberberg-Tsao, it is evident that while the Binary cascade appears to employ similar production mechanism algorithms as FLUKA, while the Silberberg-Tsao deviates from the other codes, likely due to its optimization for fission rather that full consideration of the other production mechanisms. However, the in-target production rates predicted by GEANT4 are in line with those predicted by FLUKA and the Silberberg-Tsao codes. This indicates that the geometry discussed in Section 1.3.2 has properly been implemented into the GEANT4 framework. This also verifies that the production mechanisms used within GEANT4 are on par with those used by other similar available tools. It is important to note that this comparison can only be done in this qualitative manner, as a full quantitative comparison would require the complete analysis of the production mechanisms and algorithms within each of the three codes.

46 (a) Production rates for the light alkali metals. (b) Production rates for the heavy alkali metals

Figure 3.6: In-target production rates for the alkali metals, produced by GEANT4 and compared with FLUKA and Silberberg-Tsao models. (a) The light alkali metals; lithium, sodium and potassium and (b) The heavy alkali metal; rubidium, cesium and francium.

47 GEANT4 and Yield Measurements

The comparison between the three physics lists within GEANT4 was necessary to examine the way the different algorithms affected the production of the alkali metals. The comparison between FLUKA and the Silberberg-Tsao models was needed in order to ensure that the geometry was properly implemented into the simulation, and to verify that the algorithms within GEANT4 qualitatively agreed with the other tools available for the simulation of particles travelling through matter. The final comparison is drawn between the in-target production rates predicted by the simulation and the yield rates measured at the ISAC yield station [31]. The inherent differences between the in-target production and the measured yield rates, mentioned in Section 1.2.3, must be stressed. Though related to one another, the yield rates suffer from the efficiencies associated with performing a physical experiment, and thus the in-target production rates are modified by these efficiencies, and once measured are recorded as the yield rates, stated in Equation 1.39. It is important to note, that as the measured yield rates suffer from efficiency losses, the production rates of decay products may also be enhanced by short-lived decays of their parents. If the isotope of interest is a daughter nucleus of a produced parent, the subsequent decay will increase the production, an effect which will be reflected in the measured yield of the daughter. A more detailed description of this effect is found in Section 4.8. The production rate results presented in Figure 3.7 must be carefully considered. In this figure, the in-target production rates are plotted alongside the measured yield rates, taken from the ISAC Yield Database [30], are plotted as points, with their associated errors. The comparison between the in-target production rates and the measured yield rates is important, because it illustrates the inherent difference between the two quantities. The in- target production rate of an isotope is effectively the highest yield rate that can be measured, but as discussed, the efficiencies reduce this production rate to the final measured yield rate. Thus, if the in-target production rate values predicted by the simulation are lower than the yield rates measured, it is an indication that the particular physics list is not adequate for the purpose of simulating the isotopes in question. In Figure 3.7(a), the in-target production rates for the lithium isotopes produced using the Binary and the Liège cascades are above those of the measured yield rates. However, the Bertini cascade predicts production rate values lower than the actual measured yield rates. The production rate results for the sodium isotopes exhibit the same trend, where the Bertini cascade predicts production rates lower than those measured. The Binary and the Liège cascades both predict higher production rates, with the Binary cascade predicting rates three orders of magnitude higher than the measured rates.

48 (a) Production rates for the light alkali metals. (b) Production rates for the heavy alkali metals.

Figure 3.7: In-target production rate results for the alkali metals, produced by GEANT4 compared to the yield measurements from the ISAC Yield Database [30]. (a) The light alkali metals, lithium, sodium and potassium and (b) The heavy alkali metals, rubidium, cesium and francium. 49 As discussed before, the Bertini cascade is unable to predict isotopes of potassium, at least within the scope of this simulation. The Binary and the Liège model are able to produce these isotopes, though the Binary cascade predicts production rates lower than the measured yields. This indicates that the Binary cascade is not the adequate physics list to use for production of potassium. For the heavy alkalis, shown in Figure 3.7(b), the rubidium isotope production rates are predicted to be in line with the measured yield rates, with only the Binary cascade predicting lower values for the neutron-deficient isotopes than those of the yield measurements. The neutron-deficient isotopes of cesium have production rates higher than the yield measurements, except the 129,130Cs isotopes, where the Binary cascade predicts lower values than those that have been measured. On the neutron-rich side, the Bertini predicts lower values, while both the Binary and the Liège cascades predict production rates which are higher than the yield measurements gathered at the yield station. Finally, the francium isotope in-target production rates are all higher that the measured yields, but the Bertini cascade cannot reproduce the trend of the yield measurements. The Binary cascade is able to produce the trend up until the increase in 231Fr, a feature which is not seen in the yield measurements. The Liège cascade is able to produce the general trend in the yield measurement. The sharp decrease in the region 214−219Fr is due to the half-lives of these isotopes. Discussed in Section 1.2.3, the measured yield rates are subject to the half-life of each isotopes. The francium isotopes in this region are very short lived, t1/2 < 1 ms [12], and thus are measured with lower rates than neighbouring isotopes. Since the simulation does not take this effect into account, the production curves do not mimic this behaviour. However, as the isotopes should be produced with similar rates as their neighbours, this half-life effect is not relevant for this comparison, though it is expected in the measured yields. The implications of this effect are discussed in Section 4.3. It must be noted, that the yield measurements, as in the case of the GEANT4 in-target production rates curves, have associated uncertainties. Information on the uncertainty calculation is found in Appendix A. Having compared the values and trends of the production rate results generated by the GEANT4 physics lists, and the yield measurements obtained at the ISAC yield station [31], it is evident that the Bertini intra-nuclear cascade is not adequate for the simulation of isotope production using a thick target, such as those used at TRIUMF-ISAC. However, as the Bertini cascade is adequate for fission reactions, it should still be considered when new target material implementations are required, particularly in the case where the target material used is known to have high fission cross sections, when compared to its spallation and fragmentation cross sections. The Binary cascade predicts lower production rates than the yields measured in the potassium and rubidium isotopes, while depicting a sudden increase in the neutron-rich

50 francium isotopes. From this behaviour, it is apparent that the Binary cascade may not be the most suitable for this particular application. In contrast to the other two lists, the Liège cascade predicts production rate values higher than the measured yields, and it is able to reproduce the trends seen in the yield rates. This indicated that, of the three physics lists, the Liège cascade is the most appropriate for simulating the proton bombardment of an ISAC-like target at proton energies similar to those produce by the TRIUMF cyclotron. However, since each of the cascade results depend on the incident particle energy as well as target materials, the Bertini and the Binary cascade results should be considered alongside results from the Liège cascade when varying the energy of the incident proton beam. The alkali earth metals show similar trends and can be used to come to the same conclusion as their alkali neighbours. The production rate curves for the alkaline earth metals were also extracted can be found in Appendix B.

3.2.1 Results Summary

The Bertini and the Binary intra-nuclear cascades each have their own features and uses. However, for the purpose of describing all isotope production mechanisms discussed in Sec- tion 1.1.3, using the TRIUMF-ISAC target speciﬁcations, the combination of the Liège intra-nuclear cascade with the ABLAV3 de-excitation code is the most adequate GEANT4 physics list to use. Based on the analysis above, it appears that this combination of the Liège cascade and ABLAV3 de-excitation model produces in-target production rates consistent with those of the other nuclear transport codes used, namely FLUKA and the Silberberg- Tsao, as well as being able to predict appropriate production rates given the yield measurements from the ISAC Yield Database. Nevertheless, the Bertini and the Binary cascades are still included in the simulation package and should be considered as viable options, to ensure that a more robust and complete simulation is available for use. Futhermore, due to its portability, the simulation may also be used to model the production of isotopes due to protons with diﬀerent energies, such as the cyclotrons used for nuclear medicine, or even with other projectile particles, as will be the case at the new ARIEL facility. The simulation is available to the TRIUMF community [63].

51 Chapter 4

Further Studies

In addition to the in-target production rate analysis, several more studies were conducted in order to evaluate the capacities of the transport code, to examine the versatility of the simulation, and to gauge its adaptability to other applications. It must be mentioned that the analyses that follow were done on a selection of the full data sets produced. The interesting results from this small subset of isotopes, however, is able to demonstrate the power of the simulation and the potential applications that can beneﬁt from the model.

4.1 Cross Section Veriﬁcation

The cross section of a reaction, seen in Equation 1.35, plays an important role in the quantity of a species that is produced during nuclear reactions. This cross section is an observable quantity, that can be measured using diﬀerent experimental techniques [64–66]. Each of the physics lists within GEANT4 are validated by the GEANT4 Collaboration, using experimental cross sections, ensuring that the models are properly implemented. The cross sections involved in proton bombardment for the Bertini and Binary cascades were veriﬁed by using a thin 27Al target [42], while the Liege cascade was validated using a thin 208Pb target [44]. These validations indicate that the theoretical formulations contained in these lists were adequate at predicting the measured cross sections involved in the respective experiments. To ensure that the cascades were properly implemented into the current simulation environment, the simulations carried out by Koi et al. [42] and Boudard et al. [44] were replicated. Using the same target material, target thickness and incident proton energy, the in-target production rates for the pertinent isotopes were extracted. The reaction cross sections were then calculated using the in-target production rates,

P σreaction = , (4.1) ΦpNtarget

52 were P is the production rate, Φp is the proton ﬂux, and the Ntarget is the number of nuclei in the target.

4.1.1 Bertini & Binary Cascades

The validation of the Bertini and the Binary cascades was carried out by the Hadronic Working group of the GEANT4 Collaboration. It was done by reproducing the experimental set up used by Vonach et al. [67], which used an 27Al target with thickness 0.012 g/cm2 and an 800 MeV incident proton beam, obtaining cross sections measurements for isotopes of sodium, magnesium, oxygen, ﬂuorine and neon. Figure 4.1 shows the results from this simulation, plotted alongside the validation data in [67]. Within errors, the Bertini and the Binary cascades are able to accurately reproduce the production cross sections of the speciﬁed elements given the target thickness and the incident proton energy used in the experiment. This illustrates that the Bertini and Binary cascades use algorithms which are suitable for the modelling of reactions caused by incident proton beams impinging on a thin target.

4.1.2 Liège Cascade

Boudard et al. [44] validated the Liège cascade using experimentally determined lead cross sections. Using a thin 208Pb target, 0.087 g/cm2 thick and a 1 GeV incident proton beam, Enqvist et al. were able to measure the production cross sections of the elements between dysprosium (Z = 66), and lead (Z = 82) [68]. The speciﬁcations of the experiment were input into the simulation, and the cross sections calculated, using Equation 4.1. Figure 4.2 shows a subset of the results from the simulation, compared to the results from the experiment by Enqvist et al. It shows that, within error, the Liège cascade reproduces these isotopes using the appropriate cross sections, val- idating that it can be used to model the production of isotopes by proton bombardment onto a target. The full set of results is found in Appendix C.

4.1.3 Validation conclusion

This analysis replicated the studies by Koi et al. [42] and Boudard et al. [44], in which they compare the results of the Bertini and Binary cascades, and the Liège cascade, respectively, to the data published by Vonach et al. [67] and Enqvist et al. [68]. This work served to verify that the Bertini, Binary and Liège cascades are suitable for the modelling of the bombardment of a target by a high energy proton beam. It also served to verify that the physics lists were properly implemented into the current simulation, accomplished by successfully replicating the published cross section results.

53 (a) Isotopes of sodium and magnesium.

(b) Isotopes of oxygen, ﬂuorine and neon.

Figure 4.1: Production cross section values calculated from the bombardment of 27Al by 800 MeV protons. Bertini and Binary cascade results compared to data from [67]. The cross sections are plotted in millibarns as a function of mass number A. The errors in the simulated results are calculated as per Appendix A, while the data errors are those quoted in [67].

54 (a) Isotopes of dysprosium, holmium and erbium. (b) Isotopes of platnium, gold and mercury.

Figure 4.2: Production cross section values calculated from the bombardment of 208Pb by 1 GeV protons. Liège cascade results compared with data and errors from [68]. Errors for the simulated data are calculated based on the technique illustrated in Appendix A.

55 4.2 Energy Deposition

Another test to ensure that the materials were properly implemented involved the study of the energy deposited by the incident particles. In this particular case, as the protons move through the standard target, defined in Section 2.1.2, they impart energy onto the target nuclei and induce electromagnetic and nuclear interactions. Figure 4.3 depicts the energy deposited by 480 MeV protons moving through a 1 cm disk of the 0.05 mol/cm2 standard target. The distribution is due to the method used to calculate the energy deposited by the protons within GEANT4. The toolkit evaluates the energy difference of the protons between the previous interaction and the next one. This can result in a distribution of the deposited energy as the protons undergo various interactions while they move between the different atoms in the target material. In this bombardment the largest losses are caused by electromagnetic interactions. Despite the spread in the distribution, there is a clear trend, depicted by the line of the best fit shown. Given that this figure plots the energy deposited as a function of target length, the slope of this fit is the energy deposition value, dE/dx, a value which is an intrinsic property of each material. The apparent increase in deposited energy at the 10 mm position is due to the boundary encountered by the particles as they travel. The energy deposition values were extracted from the target material volume rather than the virtual detector volume described in Section 2.1.3. The position was extracted as the difference between the position of the particle before interaction and after interaction, relative to the volume the particle is in. If the particle encounters a boundary, the final position, relative to the volume, will be the end of the boundary. This results in an artificial increase of deposited energy at the 1 cm position, as it is the end of the target volume. The target material properties were input into the Stopping and Range of Ions in Mat- ter (SRIM) software [69, 70] in order to calculate the energy deposition value. Developed by J. Ziegler and collaborators, this software is able to calculate the stopping range and energy deposition values for a wide range of target materials and configurations. The standard target composition was input into the SRIM software, and the energy deposition was calculated.

56 Figure 4.3: Energy deposited onto a 1 cm, 0.05 mol/cm2 depleted uranium standard target disk, plotted in MeV as a function of target distance in mm. The red points are values returned by the simulation. The line represents the linear ﬁt to the data.

The energy deposition values, with errors, from Figure 4.3 and SRIM calculations are found in Table 4.1.

Table 4.1: Energy deposition values returned from GEANT4 simulation and SRIM software [70], using the standard target disk conﬁguration.

Energy deposited dE/dx (MeV/mm) GEANT4 0.308398 ± 2.107×10−5 SRIM 0.337200 ± 6.3×10−5

Though the values have a 8% diﬀerence, they are still in reasonable agreement, verify- ing that the target material and target density have been properly implemented into the geometry. This lends conﬁdence to the production rate results presented in Section 3.2.

4.3 Release Times and Release Eﬃciencies

The diﬀerence between the in-target production and measured yield rates is due to the eﬃ- ciencies associated with the ion extraction and transport mechanisms inherent in a physical experiment, as discussed in Section 1.2.3.

57 In order to better understand the simulation as well as the system, the release efficiency and release times of a selection of francium isotopes were studied. Recalling that the release efficiency is the likelihood that an isotope is released from the target, the release time is then the time required for this process to occur. This release time plays an important role in whether an isotope will be measured at the ISAC yield station. This is due to the fact that an isotope may decay with its characteristic half-life before or during extraction. If the release time of the element in question is longer than the isotope’s half-life, the isotope will decay before it can be measured at the yield station. This means that of the total number of isotopes produced, only a fraction will survive long enough to be measured. The production rates of a series of isotopes were used to examine release efficiencies and release times. To examine the effect of release on in-target isotope production rates, a comparison between the simulated production rates and the measured yield rates was done. A simple exponential function, Equation 4.2, was fit to the ratio between the yield and production rates,

−trelease ln(2) IX t = releasee 1/2 , (4.2) PX where X is a given isotope, IX is the measured yield rate for that isotope, PX is the in-target production rate, and t1/2 is the half-life of the isotope. The release efficiency is expressed as the fraction between the measured yield rate and the in-target production rate. The release is a constant efficiency factor and, along with the release time trelease, was left as a fit parameter. This function was fit to the production-yield rate ratio using the MINUIT [71] algorithm available in ROOT, which is used to fit complex multi-parameter functions. For this analysis, the isotopes 211−214Fr were chosen, as they have a broad range of half- lives, with a particularly abrupt change between 213Fr and 214Fr. The half-lives of these isotopes are given in Table 4.2.

Table 4.2: Half lives of francium isotopes between 211Fr and 214Fr, given in seconds. Values taken from NNDC [12].

Isotope Half-life (s) 211Fr 186.0 ± 1.2 212Fr 1200.0 ± 36 213Fr 34.82 ± 0.14 214Fr 0.005 ± 0.0002

58 The in-target production rates from each of the Bertini, Binary and Liège cascades were compared to the measured yield rates present in the ISAC Yield Database [30]. The resulting ﬁt was plotted, and can be found in Figure 4.4.

(a) Bertini cascade results (b) Binary cascade result.

Figure 4.4: Release efficiency fits for 211−214Fr. The points are calculated using Equation 4.2, and the fit is done using the MINUIT minimization algorithm available in ROOT [71].

The abrupt change in the half-lives between 213Fr and 214Fr is seen in the 4.4 as a drop, expected due to the short half-life of 214Fr. The production rate of 214Fr is comparable to the production of its neighbouring 213Fr. However, the half-life of 214Fr means that only a small fraction of the isotope will survive long enough to exit the target, travel to the yield station and be measured. This implies that the release time of francium is larger than the half-life of the 214Fr isotope. Release time and eﬃciency parameters returned by the ﬁts are found in Table 4.3.

59 Table 4.3: Release parameters for the three intra-nuclear cascades used in the simulation. The release eﬃciency factors are constants while the release times are given in seconds.

Cascade Release time (s) Release eﬃciency factor Bertini 0.05203 ± 0.00145 32.4051± 3.30970 Binary 0.04575 ± 0.00146 0.26187 ± 0.02719 Liège 0.05294 ± 0.00145 0.90593 ± 0.09160

The release parameters returned by the fit can be used to infer information about the efficacy of the physics lists, and to quantify the time scales involved in isotope release. While all three of the physics lists depict the expected drop in the yield/production rate ratio, the scales of the ratio are drastically different in the Bertini cascade when compared to the values in the Binary and Liège cascade fits. The Bertini cascade effectively predicts that more isotopes are measured than are actually produced in the target. Though this may occur due to in-target decay, as discussed in Section 4.8, this decay will not increase the production of isotopes by a factor of five, as shown in Figure 4.4(a). Possible contributions of parent in-target decay in this case are not significant. The release times are all on the order of ∼50 ms, corresponding to about ten half-lives of the 214Fr isotope. Since this isotope has been previously measured at the yield station, it is expected that its release time is comparable to its half-life. If the true release time was much longer, this isotope would not be measured at the yield station. In light of this, the release times returned by the fit from all three of the physics lists are reasonable. Though Equation 4.2 is very simple, it is able to return reasonable values for the release time and release efficiency factor, demonstrating that it can be used for other elements. This fitting routine may be applied to other elemental isotopes, so as to establish the release times and release efficiency factors for other elements that have been measured at the ISAC yield station. It is important to note that this analysis was done using simulated in-target production rates. Though it is possible to obtain experimentally measured release times and efficiencies, as done at GSI [72,73], this cannot be done at TRIUMF-ISAC at this time. This is due to the different types of targets used at ISAC and GSI. The thick target, described in Section 1.2.2, does not allow for the selective production and tagging methods used at GSI. Thus, for the moment, this simulated analysis is the only method that can be used to study release times and efficiencies from the thick targets used at the ISAC facility.

60 4.4 Thorium Oxide Target Runs

During 2014, a new thorium oxide target was fabricated and used, for the first time, at the ISAC facility. Since this was the first target of its kind at ISAC, there was a need to compare the isotope yield rates from this target to the existing uranium carbide targets. As previously discussed in Section 1.2.3, the simulation cannot yet predict the measured yield rate values. However, it still had, at the time of the target run, the capacity to provide an upper value on the isotope yields that could be expected during this target run. Since there had been no previous record of yield rates from this target, this tool provided the only quantitative value for expected rates. For this purpose, a thorium target was implemented into the simulation, with the same 5 cm total target length, and a target thickness of 0.05 mol/cm2, as per the target specifications. The same incident protons at 480 MeV were used, with runs totalling 109 incident protons. The Liège intra-nuclear cascade was used as the physics list for this simulation, as it was shown to be the optimal physics list for isotope production at the ISAC facility, seen in Chapter 3. Figure 4.5 summarizes the isotopes produced using the described thorium target and Liège intra-nuclear cascade combination. In the interest of comparing the results from the uranium and thorium target materials, nuclide charts depicting the ratios between uranium and thorium are found in Figure 4.6. Figure 4.6(a) shows the ratio between thorium and uranium, while Figure 4.6(b) shows the ratio between uranium and thorium. In both cases the rates from the latter are divided by those of the former. It is evident, from this graphical comparison, that each target is better able to produce certain isotopes than its counterpart. The results from the Liège cascade shows that the thorium target is better able to produce isotopes in the spallation region, closer to the target mass (A=228) and centred around mass A=210, while the uranium target is better suited to generating neutron-rich fission products, found in the middle of the chart. For further study, this graphical analysis must be complimented with statistical comparisons, by systematically observing pertinent isotopes. As this target run was completed during the course of this work, this analysis was not included as part of simulation effort, but rather was implemented to test the on-line capabilities of the simulation.

61 Figure 4.5: Nuclide chart produced using a thorium target with a 0.05 mol/cm2 thickness, 109 480 Mev protons and the Liège intra-nuclear cascade.

(a) Production rates from a thorium target, (b) Production rates from a uranium target, when compared to a uranium target. when compared to a thorium target.

Figure 4.6: Comparison of production rates between a uranium and a thorium standard target. The targets both have thickness 0.05 mol/cm2, and both simulations bombarded the target with 109 protons with initial energy 480 MeV. (a) The ratio of nuclei when the production rates from thorium are divided by those of uranium. (b) The ratio of nuclei when the production rates from uranium are divided by those of thorium.

62 4.4.1 Medical Isotope Production

One of the potential applications of thorium targets is their ability to generate isotopic species used in nuclear medicine. Of particular interest was the production of actinium, radium and thorium isotopes, specially 225Ac, 223Ra and 224Ra. Both 225Ac and 223Ra can be used as precursors to other nuclear medicine isotopes, or can themselves be used in targeted alpha therapy [74, 75]. These medical isotopes can be generated by bombardment of thorium targets [76, 77]. It is thus important to consider the production rates of these isotopes when a thorium target is used, and to compare these values to their production using a uranium target. A result of this comparison can be seen in Figure 4.7. Here the isotope production rates of the thorium target have been normalized to the production rates of the uranium target, both with the same target length and thickness.

Figure 4.7: In-target production rates from a thorium target, when compared to a uranium target, with thickness 0.05 mol/cm2. From this comparison, it is clear that the thorium target is better suited to produce the isotopes of radium, at Z = 88, actinium, at Z = 89, and thorium, at Z = 90. Note negative values which might result from the subtraction are not present on the logarithmic scale used for this plot.

63 Table 4.4: In-target production rates of medical isotopes produced by a thorium target and compared to a uranium target, given in particles per second and normalized to reﬂect 1 μA of proton current through the target.

Production rates (isotopes/sec) Isotope Ratio (Th/U) Thorium Uranium 227Th 3.46E9 3.99E8 8.66 228Th 5.60E9 5.25E8 10.7 223Ra 1.51E9 1.93E8 7.80 224Ra 1.34E9 1.54E8 8.71 225Ra 1.14E9 1.22E8 9.36 225Ac 2.99E9 3.56E8 8.39 227Ac 3.22E9 3.72E8 8.66 228Ac 3.44E9 3.63E8 9.48

Based on the data presented in Figure 4.7 and in Table 4.4 it is clear that the thorium target generates higher quantities of radium (Z = 88), actinium (Z = 89) and thorium (Z = 90) isotopes than the uranium target, when using the same target thickness, incident particle type and particle ﬂux. Thus, according to this brief analysis and for the purposes of nuclear medicine, a thorium target may be a more desirable material to generate these important precursor and usable isotopes.

4.5 Alternative targets

To test the extensions of the model, new target materials were implemented into the geometry. These targets include existing materials that have been used as part of the target runs at ISAC, and hypothetical targets used to examine their isotope production viability.

64 Table 4.5: Materials implemented into the simulation. Target materials are either analogous to target materials used at ISAC or they have been implemented as hypothetical targets. Lead and aluminium targets have thickness of 87.3 and 12.1 mg/cm2 respectively, while all other targets have a thickness of 0.05 mol/cm2, and a target length of 5 cm.

Material Formula Density (g/cm3) Type Depleted Uranium 238U + 235U 2.38 ISAC Thorium natTh 2.32 ISAC Tantalum natTa 1.81 ISAC Niobium natNb 0.93 ISAC Zirconium natZr 0.91 ISAC Nickel natNi 0.58 ISAC Titanium natTi 0.47 ISAC Silicon natSi 0.28 ISAC Boron Nitride BN 0.2482 Hypothetical Lead 208Pb 0.0873 Hypothetical Aluminium 27Al 0.0243 Hypothetical

4.6 Incident Particle Energy Dependence

The cross sections which govern the interactions between nuclei and thus the production of isotopes are known to be energy dependent [12,78]. Some of these cross sections have been experimentally measured, while some are extrapolated using theoretical models. To establish how the energy dependence affects in-target isotope production, the incident proton energy was varied and the effects studied. As was seen in Section 4.4, changing the target material leads to a change in the production of isotopes. Varying the energy of the incident particle has a similar effect. To study this effect on isotope production, different incident proton energies were used to impinge upon the standard uranium target. Six different energies were chosen, ranging between 100 MeV and 2 GeV, in order to study the production trends in the medium to high energy range. To illustrate this dependence, the in-target production rates of rubidium and cesium isotopes were extracted, plotted first against the isotope mass number, seen in Figure 4.8, and then against energy, for a particular isotope, in Figure 4.9.

65 (a) Rubidium isotope in-target production rates (b) Cesium isotope in-target production rates as a function of mass number A. as a function of mass number A.

Figure 4.8: In-target isotope production rates with respect to primary beam energy for isotopes of rubidium and cesium. The incident energy change appears to shift the distribution of the production towards the neutron deﬁcient isotopes for all three of the GEANT4 physics lists.

66 It is clear from Figure 4.8 that increasing the energy of the incident proton beam produces more neutron-deficient isotopes, causing the production rate peaks to become broader, a trend which is evident in all three of the physics lists. Figure 4.9 shows the production of each isotope as a function of energy, rather than mass number. This figure illustrates that there is in fact an optimal energy that can be used to maximize production of rubidium and cesium isotopes. After the peak is reached, production of isotopes is effectively constant, meaning that increasing the incident proton energy past this point does not have an significant effect. Particularly in the case of the cesium isotopes, it appears that increasing the incident proton energy causes a decrease in production, at a incident proton energy of ∼1500 MeV. It is important to note that the isotopes presented in Figures 4.9(a) and 4.9(b) are only a small subset of the isotopes of rubidium and cesium, which themselves are only two of the elements produced in the proton bombardment of a uranium target. This cursory analysis nonetheless serves to illustrate the power of this simulation. Even with only the first iteration of this simulation, it is possible to study the energy dependence of isotope production, which can lead to optimizations in energy in order to increase production of certain isotopes.

(a) Production rates of rubidium isotopes as (b) Production rates of cesium isotopes as a a function of energy. function of energy.

Figure 4.9: Production rates, from the Liège cascade, of isotopes of rubidium and cesium as a function of energy. It is clear that production reaches a saturation point at given energies for both elements.

67 4.7 Kinetic Energy Distributions

The kinetic energy of each produced isotope be examined within the simulation. This information can be plotted to reﬂect the number of isotopes with a particular kinetic energy, which gives an energy distribution as a function of mass. As an example, the rubidium isotopes are produced by all three physics lists within the mass range 78 < A < 103, and may be plotted with their kinetic energy as a function of mass number, A. The Bertini cascade results, shown in Figure 4.10, produce two distinct regions, though most isotopes appear to have kinetic energy values below 150 MeV.

Figure 4.10: Kinetic energy distribution of rubidium isotopes, produced by the Bertini intra-nuclear cascade

Figure 4.11: Kinetic energy distribution of rubidium isotopes, produced by the Binary intra-nuclear cascade

68 Figure 4.12: Kinetic energy distribution of rubidium isotopes, produced by the Liège intra- nuclear cascade

The Binary cascade, in Figure 4.11, shows a large, mostly homogeneous distribution with a wide range of energies. The Liège cascade, Figure 4.12, produces most of the rubidium isotopes with kinetic energies between 100 MeV and 110 MeV. This graphical representation not only verifies that the physics lists are indeed using different algorithms to generate isotopes, given the same incident particle type, incident particle energy and target material, but may contain interesting information regarding how far into the target different isotopes are able to travel. To illustrate this last application, a Stopping Range In Matter (SRIM) [70] was run for different isotopes of rubidium. The standard uranium target was applied into the SRIM software, and the stopping ranges of the rubidium isotopes 81−87Rb were calculated. The kinetic energy of the isotopes were varied between 30 MeV and 150 MeV, as per the most diffuse range seen in Figure 4.11 and produced by the Binary cascade. Table 4.6 shows the results for the given isotopes. By using the kinetic energy distributions produced by GEANT4 and the stopping range tables in the SRIM software, the distances travelled by these isotopes within the targets can be established. For example in the rubidium case, from Table 4.6, the maximum distance travelled by isotopes with 30 MeV kinetic energy is on the order of ∼25 μm, while those with 150 MeV travel for ∼60 μm before being stopped.

69 Table 4.6: Stopping ranges for rubidium isotopes with kinetic energy between 30 MeV and 150 MeV, as calculated by the SRIM Software [70], with a target composed of 238U with a density of 2.38 g/cm3.

Stopping range (μm) Isotope 30 MeV 150 MeV 81Rb 22.50 60.33 82Rb 22.64 60.62 83Rb 22.77 60.92 84Rb 22.93 61.28 85Rb 23.03 61.50 86Rb 23.12 61.70 87Rb 23.29 62.08

The information provided by this analysis can be used to develop not only new target materials, but new target types. For example, consider a sandwich type target, as seen in Figure 4.13. In this type of target, two different materials would be used to maximize isotope extraction. The target material would be used to generate the isotope of interest; and the stopping material would be used to promote the diffusion and effusion out of the target, resulting in higher extractions. The stopping materials could also be used to suppress effusion and diffusion of unwanted products from the induced nuclear reactions, potentially purifying in the beam before the separation stage.

Figure 4.13: Sandwich target. The target material, in black, is used to produce isotopes, while the stopping material, in blue, is used to allow for faster diﬀusion and eﬀusion out of the target.

70 This analysis is by no means complete. The case of the rubidium isotopes is used merely as a proof-of-principle study. Similar plots can be generated for other produced isotopes, providing a large dataset for examination.

4.8 In-target decay

An important aspect that was examined is the potential for in-target decay of produced isotopes. This refers to the possibility that a species, once created, will decay while travelling through the length of the target, before it exits the material. This effect has ramifications when considering the population of daughter isotopes. The magnitude of the effect on the production of daughter isotopes can be studied using the decay kinematics equations, seen in Sections 1.1.4 and 1.1.4. In the case of constant production, the production of an isotope can come from several different factors, summarized in Equation 1.34. The isotope will be produced via direct production while the beam impinges on the target, via decay of its parent while the beam impinges on the target, and via subsequent decay after the beam is turned off. This effect may be referred to as in-target decay. These effects will then increase the number of daughter isotopes produced, and if not accounted for, may be mislabelled as direct production. Furthermore, the release of isotopes complicates the process; if an isotope is too short-lived, it will decay and not exit, causing an increase in the daughter isotope; if the isotope it too long-lived, it will exit the target and be measured; if an isotope has a half-life comparable to that of its element release time, some fraction will exit the target, while another will contribute to production of the daughter. Thus, understanding the magnitude of this effect is crucial to establish realistic in-target production rates. In order to study this effect, an ISAC yield station fitting program was re-purposed to accept values for the production rates, half-lives and release times of parent and daughter isotopes. This fitting algorithm modelled the in-target decay by modifying the Bateman equations to reflect in-target production and release times:

dN = λ N − λ N + P − µ N . (4.3) dt i−1 i−1 i i i i i where i is the species counter (1,2,3 ...), λi and Ni are the half-life and the number of atoms of the ith species, and Pi is the in-target production rate of the ith species. The factor µi represents release, 1 µi = (4.4) trelease,i Three different isotope chains were considered in order to illustrate different effects. Table 4.7 contains the inputs used for three decay chains. The production rates used were those of the Liège cascade with 109 incident protons at 480 MeV and scaled to 1 μA. The release times for the francium isotopes were those estimated in Section 4.3.

71 The α decay chain 214Fr → 210At, seen in Equation 4.5 and Figure 4.14(a), demonstrates the case where the half-life of the parent isotope is much smaller than the release time of the element. In this instance, the francium isotope reaches saturation after about 0.01 seconds, while the astatine isotope continues to build up due to francium decay.

214Fr −−−−−−−→α 210At. (4.5) t1/2=0.005 s

This is an instance where the in-target decay of the parent significantly contributes to production of the daughter isotope. The α decay of 213Fr into 209At, Equation 4.6, reflects the case where the release time is shorter than the decay time. Figure 4.14(b) shows this effect, where contribution from the 213Fr parent decay does not contribute significantly to the total number of the 209At daughter. This occurs because most 213Fr atoms will be released before they can decay into their daughter isotope.

213Fr −−−−−−−→α 209At. (4.6) t1/2=34.82 s

Finally, the β− decay chain of 239U, Equation 4.7, demonstrates the case where the the production of a parent dominates the system. In this case, it is the production and decay of 239U which accounts for the largest production of the atoms in the system. The direct production of 239Np and 239Pu have comparatively small contributions. In Figure 4.14(c), the uranium and neptunium contributions have been added together. The green curve in this ﬁgure, indicates the activity of 239Np reached saturation after ∼7 days of irradiation, assuming the 239U and 239Np are not released. The blue curve denotes the build-up of the 239Pu daughter, once again assuming no release. However, yield measurements done on an implanted 238Pu sample [30] indicate that this isotope is released at a higher rate than predicted from direct production.

β− β− 239U −−−−−−−−→ 239Np −−−−−−−→ 239Pu. (4.7) t1/2=23.45 m t1/2=2.36 d These three examples showcase different cases of the effects of release time effects and in-target decay on in-target production rates. They are, however, only a small subset of the possible combinations of effects that can occur inside the target. Further study and consideration is thus necessary to fully understand the magnitude of the effects. Implementation of the GEANT4 decay algorithms would streamline this analysis, and would provide a more realistic in-target production model.

72 Table 4.7: Parameters used to study in-target decay. The production rates are calculated using the Liège cascade with 109 incident protons at 480 MeV and scaled to 1 μA.

Isotope Production rate (isotope/sec) t1/2 (s) 214Fr 2.42E08 0.005 210At 9.27E07 2.88E04 213Fr 2.23E08 34.82 209At 1.19E8 1.95E04 239U 1.22E10 1.41E03 239Np 1.54E07 2.03E05 239Pu 1.24E04 7.60E11

(a) In-target decay of 214Fr into 210At. (b) In-target decay of 213Fr → 209At.

Figure 4.14: Examples of potential in-target decay chains. The solid lines represent the total number of atoms in the system given some time. (a) 214Fr decays into 210At. The half-life of the francium parent, 5 ms, is much shorter than the ∼50 ms release time estimated in Section 4.3. (b) 213Fr decays into 209At. The half-life of the parent isotope, 34.82 s, is much longer than the release time of the 50 ms release time. (c) 239U decays into 239Np which then decays to 239Pu. The production of the parent isotopes dominates, and is ultimately responsible for the total activity in the system.

73 4.9 Summary

The analyses presented in this chapter have covered studies that were done using selected production rates extracted from the simulation. They represent the first stage of the types of analyses that can be done with the simulation, demonstrating its power and its capabilities. With this simple model, the user can examine how production rates fluctuate with energy, release times and efficiencies of elements, production rates from different targets, kinetic energy distributions of products and the effects of in-target isotope decay. Each of the studies must be further explored. This first iteration of the simulation is able to predict in-target production rates which can be used for the previously described applications.

74 Chapter 5

Future Work & Conclusions

5.1 Outlook

The simulation may be further expanded to include more realistic processes as well as to take consideration of other factors that aﬀect the production rates and processes. These factors include user-imposed eﬀects such as cross section biasing and variable dependent production rates.

Cross section biasing

Due to the number of primaries used in the simulation, it is likely that some production cross sections, for more exotic reactions, have not been accessed, diminishing the predictive power of the simulation. In order to access these lower cross sections, the user may either set up a simulation run with more primaries, at the cost of computing time, or they may artiﬁcially increase the cross sections via biasing mechanisms within GEANT4. This cross section biasing can be done using physics-based biasing described in the Event Biasing Techniques section of the GEANT4 User’s Guide for Application Developers [52].

Variable dependent production rates

The production rates presented in this work were generated using a very specific primary particle and target combination. Changing any of these standard parameters could give rise to interesting trends in the production of isotopes, and perhaps shed light on the advantage of using different conditions within the ISAC facility. The dependence of the production rates with respect to the target material has been illustrated in Section 4.4. However, further study on how the rates change with respect to different target materials may prove to be incredibly useful, particularly for production of medical isotopes and undeveloped isotope beams at TRIUMF-ISAC. Though the exact form of the relation is not known, the energy dependent nature of reaction cross sections is well established. Studies have been done to determine how the

75 energy of the projectile affects the production of given isotopes [79–81]. Since the projectile energy is a simulation input, this type of study can be performed using the ISAC target geometry. The first stage of this type of analysis was presented in Section 4.6. Further study of this dependence can be used to determine the optimal energy required to generate needed isotopes, and may lead to the development of new radioactive isotope beams. The simulation is currently built to bombard a target material with a given number of protons, but the particle type can easily be changed within the simulation source code. As a consequence, this tool may be expanded to perform studies on the incident particle dependence of the production rates. By changing the incident particle, the simulation may be used to guide further studies in other sections of the TRIUMF facility, particularly where heavy ions and other particles are used as primary beams. It may also be useful in evaluating the strength of the methods used at the TRIUMF-ISAC facility with respect to other ISOL facilities operating with different particle driver beams. Inclusion of the radioactive decay in the production of isotopes would further improve the simulation by producing more realistic in-target production rates. As it stands, the GEANT4 toolkit does not treat time as an accessible variable; rather it is left to the user to implement time into their simulation using different mechanisms. Radioactive decay must be considered in combination with release properties. As discussed in Section 4.8, only isotopes with release times with similar to larger time-scales to their half-lives will contribute to build-up of daughter isotopes via decay. Coupling the GEANT4 radioactive decay package to the current simulation would produce more realistic results. There are also different processes that can be added or improved within the GEANT4 toolkit that will further increase the accuracy of the simulation. New functionalities such as production of isomers would improve the simulation reliability. Finally, improvements of the packaged physics lists, the Bertini, Binary, and Liège cascades in particular, and updates to supplementary codes like the ABLA de-excitation model, will improve the overall accuracy of the simulation.

5.2 Summary

A simulation of the isotope production mechanisms used at TRIUMF-ISAC has been built using the GEANT4 nuclear transport toolkit. Production rates produced using three can- didate physics lists, the Bertini, Binary, and Liège intra-nuclear cascades, have been examined, with an emphasis on the isotopes of the alkali metals. These production rates have been compared to other nuclear transport algorithms, namely FLUKA and Silberberg-Tsao, demonstrating that the target geometry has been properly implemented into the GEANT4 framework. Furthermore, production rates from the simulation have been compared to pertinent yield measurements from the ISAC Yield Station [30]. From these comparisons, a recommendation to use the Liège intra-nuclear cascade is proposed. However, the Bertini

76 and the Binary cascades must also be included in future analyses as their results may prove to be more useful in certain regions of the nuclide chart. The tool can currently be used to provide production rate values, which may be used as upper limits of the yields to be measured at the ISAC yield station, allowing for its use during current experiments. It may also be used as a testing ground for the implementation of new target materials, allowing it be used for future studies involving medical isotopes, astrophysical investigations and fundamental studies.

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83 Appendix A

Errors

To ensure the consistency of the results, errors were calculated for each of the isotopes presented. The production curves, plotted as the number of a particular isotope as a function of mass number A, are plotted in ﬁgures A.1 and A.2 with appropriate error bars. Each of these points, representing a particular isotope of an element has an associated error, computed by calculating the standard deviation and dividing this number by the average of ten identical simulation runs. Table A.1: Data on lithium isotopes from a depleted uranium standard target produced by the Liège intra-nuclear cascade. Production rates given in isotopes/sec/1μm.

Run number 6Li 7Li 8Li 9Li 1 4490 5940 11571 0 2 4365 5768 11470 0 3 4575 5802 11690 0 4 4570 5812 11951 0 5 4520 5763 11736 0 6 4453 5885 11622 0 7 4437 5825 11722 1 8 4405 5897 11638 0 9 4489 5920 11648 0 10 4606 5832 11575 0

For example, consider the isotopes of lithium produced by the Liège intra-nuclear cascade, data which can be found in table A.1. The above-mentioned formulation, expressed in equation A.1, returns a percent error, which is then applied to the production value of each isotope. σ % Error = × 100, (A.1) 1 Pn=10 N 1 Pn where σ is the standard deviation between the ten simulation runs, Pn is the production rate of the nth run, where 1 ≤ n ≤ 10, for ten runs, and N is the total number of each isotope in all ten runs.

84 The percent errors, found in table A.2, are then applied to the total number of each isotope, providing error bars for the production curve values. This is done for each of the physics list, providing unique errors, associated with the speciﬁc mechanisms used by each list.

Table A.2: Errors on the lithium isotope production rates produced by the Liège intra- nuclear cascade.

Isotope 6Li 7Li 8Li 9Li Average 4491 5844 11662 0.1 Standard deviation 78 62 128 0.31 % Error 1.74 1.07 1.10 316

The errors on the yield measurements, taken from [30], are calculated according to equation A.2 √ Yield Error = N, (A.2) where N is the measured yield rate. It must be noted that this is a simple assumption on the yield measurement errors. It does not account for systematic deviations between individual yield data points, errors which may arise due to low transmission or low statistics. Each of the values in figures A.1 and A.2 have associated errors, though they may be far too small to resolve even with the point size used in these plots. Also, it must be noted that the errors on the most neutron-deficient and neutron rich isotopes, particularly in the case of the sodium and francium isotopes appear to be large. This effect is due to the low statistics with which these isotopes were produced.

85 (a) Production curves for the light (b) Production curves for the heavy alkali metals with errors. alkali metals with errors

Figure A.1: Production curves for the alkali metals, compared with FLUKA and Silberberg- Tsao models with associated errors. (a) Production curves for the light alkali metals, lithium, sodium and potassium. (b) Production curves for the heavy alkali metals, rubidium, cesium and francium. The errors on some data points are contained within the space of the squares themselves.

86 (a) Production curves and errors for the light (b) Production curves and errors for the heavy alkali metals, compared to yield measurements. alkali metals, compared to yield measurements.

Figure A.2: Production curves for the alkali metals, compared to the yield measurements from the ISAC Yield Database [30] with associated errors. (a) Production curves for the light alkali metals. (b) Production curves for the heavy alkali metals. The errors on some data points are contained within the space of the squares themselves.

87 Appendix B

Alkaline Earth Metals

The second group of the periodic table, the alkaline earth metals, share similar ionization properties with the alkali metals. They can thus also be used as benchmark products within the simulation. The in-target production rates for the alkali earth metals were generated using the same initial conditions as the alkali metals: 109 480 MeV incident protons, with the productions scaled to 1 μA of current, as per Equation 3.1. The comparison of the three nuclear codes, GEANT4, FLUKA and Silberberg-Tsao (ST) for the three lightest alkaline metals, Figure B.1(a) contains much the same information on the in-target production rate trends, as their alkali counterparts. While the Binary and the Liège intra-nuclear cascades appear to be using similar algorithms for production of beryllium isotopes, with a diﬀerence of two orders of magnitude, the Bertini cascade is only able to produce three isotopes and is in line with results from the FLUKA transport code. The magnesium in-target production rates depict a sharp decrease in value between 28Mg and 29Mg in the Binary cascade, whereas the other physics lists and the other codes do not exhibit this decrease. In this element, the Liège cascade is in line with results with FLUKA, while the Bertini cascade produces these isotopes two orders of magnitude less frequently than the Liège cascade and four orders of magnitude than the Binary cascade, at least in the isotopes it is able to produce. Like the potassium isotopes, the Bertini cascade is unable to produce isotopes of calcium. Once again the Liège cascade is in line with FLUKA results, while the Binary cascade appears to have a similar trend, but an order of magnitude lower than the Liège cascade. Once again, within the light alkaline earth metals, the Bertini cascade appears to be inad- equate when simulating this set of isotopes.

88 (a) Production curves for the light alkali (b) Production curves for the heavy alkali earth metals. earth metals

Figure B.1: In-target production rates for the alkaline earth metals, compared to FLUKA and Silberberg-Tsao nuclear codes.

89 The heavy alkaline earth metals, Figure B.1(b), are also very similar to the alkali metals case. The in-target production rates for the strontium isotopes appear to be consistent with one another, except for a maximum on the neutron-rich side of the isotopes predicted by the Liège cascade. The neutron-deficient barium isotopes produced by the Bertini cascade are more favoured than their neutron-rich counterparts, a trend which is not seen in the other physics lists within GEANT4 nor FLUKA or the Silberberg-Tsao. The opposite is evident in the radium isotopes, where the Bertini cascade appears to favour the more neutron-rich isotopes. The Binary cascade predicts a decrease and then an increase in the production of the neutron-rich radium isotopes, in the mass region 230 < A < 235. The comparison between the GEANT4 in-target production rates and the ISAC measured yield rates, gathered from the ISAC Yield Database [30] is shown in Figure B.2. Though the beryllium isotopes have been measured at ISAC, they have not yet been measured using a uranium target, and thus are not included in the in-target production rate and measured yield rate comparison. The yield measurements of the magnesium isotopes appear to be in line with the Liège cascade, except for the 25Mg isotope. These isotopes are also predicted to have lower in- target production rates than the yield rates measured by the Bertini cascade and the Binary cascades. With only two measured yield values, it is difficult to make a distinct comparison for the calcium isotopes. The Binary and the Liège cascades both predict in-target production rates higher than the measured yields, while the Bertini is unable to produce these isotopes, as noted previously. The comparison for heavy alkali earth metals are summarized in Figure B.2(b). The strontium isotopes are all produced at higher rates than their measured yields, which is as expected. As with the calcium isotopes, a comparison cannot be made for the barium isotopes, due to the availability of only one yield measurement from an uranium target. The radium isotopes appear to be best described by the Liège cascade, as the Bertini appears to favour the more neutron-rich isotopes while unable to produce the neutron-deficient side; the Binary is unable to predict the proper trend in the most neutron-rich isotopes, instead predicting an increase in the production. As their light counterparts, the heavy alkali earth metals appear to show that of the three physics lists, the Liège intra-nuclear cascade is the most suitable for producing this set of isotopes using a uranium target with a incident proton beam, as per the specifications of TRIUMF-ISAC. However, it is important to also consider the in-target production values for both the Bertini and the Binary cascades in future target material studies, as their capabilities and algorithms may be superior to those of the Liège model in other target systems.

90 (a) Production curves for the light alkali (b) Production curves for the heavy alkali earth metals. earth metals

Figure B.2: Production curves for the alkaline earth metals, compared to the yield measurements from the ISAC Yield Database [30].

91 Appendix C

Liège cascade cross section validation

The cross sections measured by Enqvist et al. [68], were the production cross sections for the elements between dysprosium, Z = 66, and lead Z = 82. This was done using a 208Pb target and a 1 GeV incident proton beam.This set up was simulated, and the cross sections calculated. Figures C.1, C.2 and C.3 show the entire set of results. The simulation results are plotted alongside the data gathered by Enqvist et al.

92 (a) Isotopes of thullium, ytterbium and lutetium.(b) Isotopes of dysprosium, holmium and erbium.

Figure C.1: Production cross section values calculated from the bombardment of 208Pb by 1 GeV protons, for elements 66 ≤ Z ≤ 71. Liège cascade results compared with data from [68]. The errors on some data points are contained within the space of the squares themselves.

93 (a) Isotopes of rhenium, osmium and iridium. (b) Isotopes of hafnium, tantalum and tungsten.

Figure C.2: Production cross section values calculated from the bombardment of 208Pb by 1 GeV protons, for elements 72 ≤ Z ≤ 77. Liège cascade results compared with data from [68]. The errors on some data points are contained within the space of the squares themselves.

94 (a) Isotopes of thallium and lead.

(b) Isotopes of platinum, gold and mercury.

Figure C.3: Production cross section values calculated from the bombardment of 208Pb by 1 GeV protons, for elements 78 ≤ Z ≤ 82. Liège cascade results compared with data from [68]. The errors on some data points are contained within the space of the squares themselves.

95 Appendix D

Target Materials

Though the uranium carbide targets have been the work-horse targets at ISAC in recent years, the facility has used other target materials in the past. In an effort to reflect the results from these target materials alongside the results from the depleted uranium targets, these other targets have been implemented into the simulation geometry. Specified in Table 4.5 of Section 4.5, the targets have been implemented to reflect the 0.05 mol/cm2 target thickness typically used at ISAC. Using the Liège cascade, each of the targets was bombarded with 107 incident protons at 480 MeV. The results, plotted as nuclide charts can be seen in figure D.1 Alongside these existing target materials, there were other materials that were implemented into the geometry for proof-of-principle purposes: in order to determine whether a hypothetical target could be implemented into the geometry with ease. These targets had the same target thickness of 0.05 mol/cm2, but were composed of carbon, fluorine and boron nitride. The carbon and the fluorine targets were used to test the ease of implementation and the boron nitride target was used to test whether a complex target could be defined. The resulting nuclide charts are found in Figure D.2. These results were tabulated in the nuclide charts, but were not analysed in-depth due to the quantity of data. They serve as evidence that the simulation of the incident particle and target material is useful for other studies, aside from production of isotopes from the main uranium carbide target.

96 (a) Isotopes produced from a silicon target. (b) Isotopes produced from a titanium target.

(e) Isotopes produced from a tantalum target. (f) Isotopes produced from a nickel target.

Figure D.1: Nuclide charts produced using the naturally abundant compositions of the materials that have previously been used at the ISAC facility.

97 (a) Isotopes produced from a carbon target. (b) Isotopes produced from a ﬂuorine target.

Figure D.2: Nuclide charts produced using carbon (D.2(a)), ﬂuorine (D.2(b)) and boron nitride (D.2(c)).